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--- abstract: 'Mean-reverting assets are one of the holy grails of financial markets: if such assets existed, they would provide trivially profitable investment strategies for any investor able to trade them, thanks to the knowledge that such assets oscillate predictably around their long term mean. The modus operandi of cointegration-based trading strategies [@tsay2005analysis §8] is to create first a portfolio of assets whose aggregate value mean-reverts, to exploit that knowledge by selling short or buying that portfolio when its value deviates from its long-term mean. Such portfolios are typically selected using tools from cointegration theory [@granger; @johansen], whose aim is to detect combinations of assets that are stationary, and therefore mean-reverting. We argue in this work that focusing on stationarity only may not suffice to ensure profitability of cointegration-based strategies. While it might be possible to create synthetically, using a large array of financial assets, a portfolio whose aggregate value is stationary and therefore mean-reverting, trading such a large portfolio incurs in practice important trade or borrow costs. Looking for stationary portfolios formed by many assets may also result in portfolios that have a very small volatility and which require significant leverage to be profitable. We study in this work algorithmic approaches that can take mitigate these effects by searching for maximally mean-reverting portfolios which are sufficiently sparse and/or volatile.' author: - | Marco Cuturi\ Graduate School of Informatics\ Kyoto University\ `[email protected]`\ \ Alexandre d’Aspremont\ D.I., UMR CNRS 8548\ Ecole Normale Supérieure, `[email protected]` title: | Mean-Reverting Portfolios:\ Tradeoffs Between Sparsity and Volatility --- Introduction ============ Mean-reverting assets, namely assets whose price oscillates predictably around a long term mean, provide investors with an ideal investment opportunity. Because of their tendency to pull back to a given price level, a naive contrarian strategy of buying the asset when its price lies below that mean, or selling short the asset when it lies above that mean can be profitable. Unsurprisingly, assets that exhibit significant mean-reversion are very hard to find in efficient markets. Whenever mean-reversion is observed in a single asset, it is almost always impossible to profit from it: the asset may typically have very low volatility, be illiquid, hard to short-sell, or its mean-reversion may occur at a time-scale (months, years) for which the borrow-cost of holding or shorting the asset may well exceed any profit expected from such a contrarian strategy. ### Synthetic Mean-Reverting Baskets Since mean-reverting assets rarely appear in liquid markets, investors have focused instead on creating synthetic assets that can mimic the properties of a single mean-reverting asset, and trading such synthetic assets as if they were a single asset. Such a synthetic asset is typically designed by combining long and short positions in various liquid assets to form a *mean-reverting portfolio*, whose aggregate value exhibits significant mean-reversion. Constructing such synthetic portfolios is, however, challenging. Whereas simple descriptive statistics and unit-root test procedures can be used to test whether a single asset is mean-reverting, building mean-reverting portfolios requires finding a proper vector of algebraic weights (long and short positions) that describes a portfolio which has a mean-reverting aggregate value. In that sense, mean-reverting portfolios are made by the investor, and cannot be simply chosen among tradable assets. A mean-reverting portfolio is characterized both by the pool of assets the investor has selected (starting with the dimension of the vector), and by the fixed nominal quantities (or weights) of each of these assets in the portfolio, which the investor also needs to set. When only two assets are considered, such baskets are usually known as long-short trading pairs. We consider in this paper baskets that are constituted by more than two assets. ### Mean-Reverting Baskets with Sufficient Volatility and Sparsity A mean-reverting portfolio must exhibit sufficient mean-reversion to ensure that a contrarian strategy is profitable. To meet this requirement, investors have relied on cointegration theory [@granger; @maddala1998urc; @johansen2005cointegration] to estimate linear combinations of assets which exhibit stationarity (and therefore mean-reversion) using historical data. We argue in this work, as we did in earlier references [@alex; @cuturi2013mean], that mean-reverting strategies cannot, however, only rely on this approach to be profitable. Arbitrage opportunities can only exist if they are large enough to be traded without using too much leverage or incurring too many transaction costs. For mean-reverting baskets, this condition translates naturally into a first requirement that the gap between the basket valuation and its long term mean is large enough on average, namely that the basket price has sufficient variance or volatility. A second desirable property is that mean-reverting portfolios require trading as few assets as possible to minimize costs, namely that the weights vector of that portfolio is sparse. We propose in this work methods that maximize a proxy for mean reversion, and which can take into account at the same time constraints on variance and sparsity.\ \ We propose first in Section \[s:crit\] three proxies for mean reversion. Section \[s:opt\] defines the basket optimization problems corresponding to these quantities. We show in Section \[s:sdp\] that each of these problems translate naturally into semidefinite relaxations which produce either exact or approximate solutions using sparse PCA techniques. Finally, we present numerical evidence in Section \[s:numres\] that taking into account sparsity and volatility can significantly boost the performance of mean-reverting trading strategies in trading environments where trading costs are not negligible. Proxies for Mean-Reversion {#s:crit} ========================== Isolating stable linear combinations of variables of multivariate time series is a fundamental problem in econometrics. A classical formulation of the problem reads as follows: given a vector valued process $x=(x_t)_t$ taking values in $\RR^n$ and indexed by time $t\in\NN$, and making no assumptions on the stationarity of each individual component of $x$, can we estimate one or many directions $y\in\RR^n$ such that the univariate process $(y^Tx_t)$ is stationary? When such a vector $y$ exists, the process $x$ is said to be cointegrated. The goal of cointegration techniques is to detect and estimate such directions $y$. Taken for granted that such techniques can efficiently isolate sparse mean reverting baskets, their financial application can be either straightforward using simple event triggers to buy, sell or simply hold the basket [@tsay2005analysis §8.6], or more elaborate optimal trading strategies if one assumes that the mean-reverting basket value is a Ohrstein-Ullenbeck process, as discussed in [@jurek; @liu2010optimal; @elie:hal-00573429]. Related Work and Problem Setting -------------------------------- @granger provided in their seminal work a first approach to compare two non-stationary univariate time series $(x_t,y_t)$, and test for the existence of a term $\alpha$ such that $y_t-\alpha x_t$ becomes stationary. Following this seminal work, several techniques have been proposed to generalize that idea to multivariate time series. As detailed in the survey by @maddala1998urc [§5], cointegration techniques differ in the modeling assumptions they require on the time series themselves. Some are designed to identify only one cointegrated relationship, whereas others are designed to detect many or all of them. Among these references, @johansen proposed a popular approach that builds upon a VAR model, as surveyed in [@johansen2005cointegration; @johansen2009cointegration]. These approaches all discuss issues that are relevant to econometrics, such as de-trending and seasonal adjustments. Some of them focus more specifically on testing procedures designed to check whether such cointegrated relationships exist or not, rather than on the robustness of the estimation of that relationship itself. We follow in this work a simpler approach proposed by @alex, which is to trade-off interpretability, testing and modeling assumptions for a simpler optimization framework which can be tailored to include other aspects than only stationarity. @alex did so by adding regularizers to the predictability criterion proposed by @box1977cam. We follow in this paper the approach we proposed in [@cuturi2013mean] to design mean-reversion proxies that do not rely on any modeling assumption. Throughout this paper, we write $\symm_n$ for the $n\times n$ cone of positive definite matrices. We consider in the following a multivariate stochastic process $x=(x_t)_{t\in\NN}$ taking values in $\RR^n$. We write $\Acal_k= \Expect[x_t x_{t+k}^T], k\geq 0$ for the lag-$k$ autocovariance matrix of $x_t$ if it is finite. Using a sample path $\bx$ of $(x_t)$, where $\bx=(\bx_1,\ldots,\bx_T)$ and each $\bx_t\in\RR^n$, we write $A_k$ for the *empirical* counterpart of $\Acal_k$ computed from $\bx$, $$\label{eq:autos} A_k\defeq \frac{1}{T-k-1}\sum_{t=1}^{T-k} \tilde{\bx}_t \tilde{\bx}_{t+k}^T,\; \tilde{\bx}_t\defeq \bx_t-\frac{1}{T}\sum_{t=1}^T \bx_t.$$ Given $y\in\RR^n$, we now define three measures which can all be interpreted as proxies for the mean reversion of $y^Tx_t$. **Predictability** – defined for stationary processes by @box1977cam and generalized for non-stationary processes by @Bewl94 – measures how close to noise the series is. The **portmanteau** statistic [@Ljun78] is used to test whether a process is white noise. Finally, the **crossing statistic** [@ylvisaker1965expected] measures the probability that a process crosses its mean per unit of time. In all three cases, low values for these criteria imply a fast mean-reversion. Predictability {#subsec:pred} -------------- We briefly recall the canonical decomposition derived in [@box1977cam]. Suppose that $x_t$ follows the recursion: \[eq:ar1\] x\_t= \_[t-1]{} + \_t, where $\hat{x}_{t-1}$ is a predictor of $x_t$ built upon past values of the process recorded up to $t-1$, and $\varepsilon_t$ is a vector of i.i.d. Gaussian noise with zero mean and covariance $\Sigma \in \symm_n$ independent of all variables $(x_{r})_{r<t}$. The canonical analysis in [@box1977cam] starts as follows. ### Univariate case Suppose $n=1$ and thus $\Sigma\in\RR_+$, Equation (\[eq:ar1\]) leads thus to $$\Expect[x_t^2]=\Expect[\hat{x}_{t-1}^2]+\Expect{[\varepsilon_t^2]}, \text{ thus } 1=\frac{\hat{\sigma}^2}{\sigma^2}+\frac{\Sigma}{\sigma^2},$$ by introducing the variances $\sigma^2$ and $\hat{\sigma}^2$ of $x_t$ and $\hat{x}_t$ respectively. @box1977cam measure the *predictability* of $x_t$ by the ratio $$\lambda\defeq\frac{\hat{\sigma}^2}{\sigma^2}.$$ The intuition behind this variance ratio is simple: when it is small the variance of the noise dominates that of $\hat{x}_{t-1}$ and $x_t$ is dominated by the noise term; when it is large, $\hat{x}_{t-1}$ dominates the noise and $x_t$ can be accurately predicted on average. ### Multivariate case Suppose $n>1$ and consider now the univariate process $(y^Tx_t)_{t}$ with weights $y\in\RR^{n}$. Using (\[eq:ar1\]) we know that $y^Tx_t =y^T\hat{x}_{t-1}+y^T\varepsilon_t$, and we can measure its predicability as \[eq:pred\] (y), where $\hat{\Acal}_0$ and $\Acal_0$ are the covariance matrices of $x_t$ and $\hat{x}_{t-1}$ respectively. Minimizing predictability $\lambda(y)$ is then equivalent to finding the minimum generalized eigenvalue $\lambda$ solving \[eq:pred2\] (\_0 - \_0) =0. Assuming that $\Acal_0$ is positive definite, the basket with minimum predictability will be given by $y=\Acal_0^{-1/2}y_0$, where $y_0$ is the eigenvector corresponding to the smallest eigenvalue of the matrix $\Acal_0^{-1/2} \hat{\Acal}_0 \Acal_0^{-1/2}$. ### Estimation of $\lambda(y)$ All of the quantities used to define $\lambda$ above need to be estimated from sample paths. $\Acal_0$ can be estimated by $A_0$ following Equation . All other quantities depend on the predictor $\hat{x}_{t-1}$. @box1977cam assume that $x_t$ follows a vector autoregressive model of order $p$ – VAR(p) in short – and therefore $\hat{x}_{t-1}$ takes the form, $$\hat{x}_{t-1}=\sum_{k=1}^p \Hca_k x_{t-k},$$ where the $p$ matrices $(\Hca_k)$ contain each $n\times n$ autoregressive coefficients. Estimating $\Hca_k$ from the sample path $\bx$, @box1977cam solve for the optimal basket by inserting these estimates in the generalized eigenvalue problem displayed in Equation . If one assumes that $p=1$ (the case $p>1$ can be trivially reformulated as a VAR(1) model with adequate reparameterization), then $$\hat{\Acal}_0=\Hca_1 \Acal_0 \Hca_1^T \text{ and }\Acal_1=\Acal_0 \Hca_1,$$ and thus the Yule-Walker estimator [@lutkepohl2005nim §3.3] of $\Hca_1$ would be $H_1=A_0^{-1} A_1$. Minimizing predictability boils down to solving in that case $$\min_{y} \hat{\lambda}(y), \; \hat{\lambda}(y)\defeq \frac{y^T \left( H_1 A_0 H_1^T\right) y}{y^T A_0 y}=\frac{y^T \left( A_1 A_0^{-1} A_1^T\right) y}{y^T A_0 y},$$ which is equivalent to computing the smallest eigenvector of the matrix $A_0^{-1/2}A_1 A_0^{-1} A_1^T A_0^{-1/2}$ if the covariance matrix $A_0$ is invertible. The machinery of @box1977cam to quantify mean-reversion requires defining a model to form $\hat{x}_{t-1}$, the conditional expectation of $x_t$ given previous observations. We consider in the following two criteria that do without such modeling assumptions. Portmanteau Criterion {#ss:portm} --------------------- Recall that the [*portmanteau*]{} statistic of order $p$ [@Ljun78] of a centered univariate stationary process $x$ (with $n=1$) is given by $$\por_p(x)=\frac{1}{p}\sum_{i=1}^p \left(\frac{\Expect[x_t x_{t+i}]}{\Expect[x_t^2]}\right)^2$$ where ${\Expect[x_t x_{t+i}]}/{\Expect[x_t^2]}$ is the $i$th order autocorrelation of $x_t$. The portmanteau statistic of a white noise process is by definition $0$ for any $p$. Given a multivariate $(n>1)$ process $x$ we write $$\phi_p(y)=\por_p(y^T x)=\frac{1}{p}\sum_{i=1}^p\left(\frac{y^T \Acal_i y}{y^T \Acal_0 y}\right)^2,$$ for a coefficient vector $y\in\RR^n$. By construction, $\phi_p(y)=\phi_p(ty)$ for any $t\ne 0$ and in what follows, we will impose $\|y\|_2=1$. The quantities $\phi_p(y)$ are computed using the following estimates [@Hami94 p.110]: \[eq:portm\] \_p(y)=\_[i=1]{}\^p()\^2. Crossing Statistics {#ss:cross} ------------------- @Kede94 [§4.1] define the [*zero crossing rate*]{} of a univariate $(n=1)$ process $x$ (its expected number of crosses around $0$ per unit of time) as \[eq:cross-rate\] (x)=, A result known as the cosine formula states that if $x_t$ is an autoregressive process of order one AR(1), namely if $|a|<1$, $\varepsilon_t$ is i.i.d. standard Gaussian noise and $x_t=a x_{t-1} + \varepsilon_t$, then [@Kede94 §4.2.2]: $$\gamma(x)=\frac{\arccos(a)}{\pi}.$$ Hence, for AR(1) processes, minimizing the first order autocorrelation $a$ also directly maximizes the crossing rate of the process $x$. For $n>1$, since the first order autocorrelation of $y^Tx_t$ is equal to $y^T\Acal_1y$, we propose to minimize $y^T\Acal_1y$ and ensure that all other absolute autocorrelations $\abs{y^T\Acal_ky}$, $k>1$ are small. Optimal Baskets {#s:opt} =============== Given a centered multivariate process $\bx$, we form its covariance matrix $A_0$ and its $p$ autocovariances $(A_1,\ldots,A_p)$. Because $y^TAy=y^T(A+A^T)y/2$, we symmetrize all autocovariance matrices $A_i$. We investigate in this section the problem of estimating baskets that have maximal mean reversion (as measured by the proxies proposed in Section\[s:crit\]), while being at the same time sufficiently volatile and supported by as few assets as possible. The latter will be achieved by selecting portfolios $y$ that have a small “0-norm”, namely that the number of non-zero components in $y$, $$\|y\|_0\defeq \#\{1\leq i\leq d | y_i\ne 0\},$$ is small. The former will be achieved by selecting portfolios whose aggregated value exhibits a variance over time that exceeds a given threshold $\nu>0$. Note that for the variance of $(y^Tx_t)$ to exceed a level $\nu$, the largest eigenvalue of $A_0$ must necessarily be larger than $\nu$, which we always assume in what follows. Combining these two constraints, we propose three different mathematical programs that reflect these trade-offs. Minimizing Predictability {#ss:opt-pred} ------------------------- Minimizing Box-Tiao’s predictability $\hat{\lambda}$ defined in §\[subsec:pred\] while ensuring that both the variance of the resulting process exceeds $\nu$ and that the vector of loadings is sparse with a 0-norm equal to $k$, means solving the following program: \[eq:P1\] & y\^T M y\ & y\^T A\_0y,\ & y\_2=1,\ & y\_0=k, in the variable $y\in\RR^n$ with $M\defeq A_1 A_0^{-1} A_1^T$, where $M,A_0\in\symm_n$. Without the normalization constraint $\|y\|_2=1$ and the sparsity constraint $\|y\|_0=k$, problem  is equivalent to a generalized eigenvalue problem in the pair $(M,A_0)$. That problem quickly becomes unstable when $A_0$ is ill-conditioned or $M$ is singular. Adding the normalization constraint $\|y\|_2=1$ solves these numerical problems. Minimizing the Portmanteau Statistic {#ss:opt-portm} ------------------------------------ Using a similar formulation, we can also minimize the order $p$ portmanteau statistic defined in §\[ss:portm\] while ensuring a minimal variance level $\nu$ by solving: \[eq:P2\] &\_[i=1]{}\^[p]{}(y\^T A\_i y)\^2\ & y\^T A\_0y ,\ & y\_2=1,\ & y\_0=k, in the variable $y\in\RR^n$, for some parameter $\nu>0$. Problem  has a natural interpretation: the objective function directly minimizes the portmanteau statistic, while the constraints normalize the norm of the basket weights to one, impose a variance larger than $\nu$ and impose a sparsity constraint on $y$. Minimizing the Crossing Statistic {#ss:opt-portm2} --------------------------------- Following the results in §\[ss:cross\], maximizing the crossing rate while keeping the rest of the autocorrelogram low, \[eq:P3\] & y\^TA\_1y + \_[k=2]{}\^[p]{}(y\^T A\_k y)\^2\ & y\^T A\_0y ,\ & y\_2=1,\ & y\_0=k, in the variable $y\in\RR^n$, for some parameters $\mu,\nu>0$, will produce processes that are close to being AR(1), while having a high crossing rate. Semidefinite Relaxations and Sparse Components {#s:sdp} ============================================== Problems ,  and   are not convex, and can be in practice extremely difficult to solve, since they involve a sparse selection of variables. We detail in this section convex relaxations to these problems which can be used to derive relevant sub-optimal solutions. A Semidefinite Programming Approach to Basket Estimation {#subsec:asemidefinite} -------------------------------------------------------- We propose to relax problems ,  and   into Semidefinite Programs (SDP) [@vandenberghe1996semidefinite]. We show that these semidefinite programs can handle naturally sparsity and volatility constraints while still aiming at mean-reversion. In some restricted cases, one can show that these relaxations are tight, in the sense that they solve exactly the programs described above. In such cases, the true solution $y^\star$ of some of the programs above can be recovered using their corresponding SDP solution $Y^\star$. However, in most of the cases we will be interested in, such a correspondence is not guaranteed and these SDP relaxations can only serve as a guide to propose solutions to these hard non-convex problems when considered with respect to vector $y$. To do so, the optimal solution $Y^\star$ needs to be *deflated* from a large rank $d\times d$ matrix to a rank one matrix $yy^T$, where $y$ can be considered a good candidate for basket weights. A typical approach to deflate a positive definite matrix into a vector is to consider its eigenvector with the leading eigenvalue. Having sparsity constraints in mind, we propose to apply a heuristic grounded on sparse-PCA [@zou2006sparse; @d2007direct]. Instead of considering the lead eigenvector, we recover the leading *sparse* eigenvector of $Y^\star$ (with a $0$-norm constrained to be equal to $k$). Several efficient algorithmic approaches have been proposed to solve approximately that problem; we use the SPASM toolbox [@sjostrand2012spasm] in our experiments. Predictability {#predictability} -------------- We can form a convex relaxation of the predictability optimization problem  over the variable $y\in\RR^n$, $$\BA{ll} \mbox{minimize} & y^T M y\\ \mbox{subject to} & y^T A_0y\geq \nu\\ & \|y\|_2=1,\\ & \|y\|_0=k, \EA$$ by using the lifting argument of @Lova91, writing $Y=yy^T$, to solve now the problem using a semidefinite variable $Y$, and by introducing a sparsity-inducing regularizer on $Y$ which considers the $L_1$ norm of $Y$, $$\norm{Y}_1\defeq \sum_{ij}\abs{Y_{ij}},$$ so that Problem  becomes (here $\rho>0$), $$\BA{ll} \mbox{minimize} & \Tr(MY) + \rho \norm{Y}_1\\ \mbox{subject to} & \Tr(A_0Y)\geq\nu\\ & \Tr(Y)=1,~\Rank(Y)=1,~Y\succeq 0. \EA$$ We relax this last problem further by dropping the rank constraint, to get \[eq:SDP1\] & (MY) + \_1\ & (A\_0Y)\ & (Y)=1, Y0 which is a convex semidefinite program in $Y\in\symm_n$. Portmanteau ----------- Using the same lifting argument and writing $Y=yy^T$, we can relax problem  by solving \[eq:SDP2\] & \_[i=1]{}\^p (A\_iY)\^2 + \_1\ & (BY)\ & (Y)=1, Y0, a semidefinite program in $Y\in\symm_n$. Crossing Stats -------------- As above, we can write a semidefinite relaxation for problem : \[eq:SDP3\] & (A\_1Y)+ \_[i=2]{}\^p (A\_iY)\^2 + \_1\ & (BY)\ & (Y)=1, Y0 ### Tightness of the SDP Relaxation in the Absence of Sparsity Constraints Note that for the crossing stats criterion (with $p=1$ and no quadratic term in $Y$) criteria, the original problem \[eq:P3\] and its relaxation \[eq:SDP3\] are equivalent, taken for granted that no sparsity constraint is considered in the original problems and $\mu$ set to $0$ in the relaxations. This relaxations boil down to an SDP’s that only has a linear objective, a linear constraint and a constraint on the trace of $Y$. In that case, @Bric61 showed that the range of two quadratic forms over the unit sphere is a convex set when the ambient dimension $n\geq 3$, which means in particular that for any two square matrices $A,B$ of dimension $n$ &{(y\^TAy,y\^TBy): y\^n, y\_2=1}=&\ &{((AY),(BY)): Y\_n, Y=1, Y0}& We refer the reader to [@Barv02 §II.13] for a more complete discussion of this result. As remarked in [@cuturi2013mean], the same equivalence holds for \[eq:P1\] and \[eq:SDP1\]. This means that, in the case where $\rho,\mu=0$ and the 0-norm of $y$ is *not* constrained, for any solution $Y^\star$ of the relaxation  there exists a vector $y^\star$ which satisfies $\norm{y}_2^2=\Tr(Y^\star)=1$, $y^{\star T} A_0 y^\star=\Tr(BY^\star)$ and $y^{\star T}My^\star=\Tr(MY^\star)$ which means that $y^\star$ is an optimal solution of the original problem . @Boyd:1072 [App.B] show how to explicitly extract such a solution $y^\star$ from a matrix $Y^\star$ solving . This result is however mostly anecdotical in the context of this paper, in which we look for sparse and volatile baskets: using these two regularizers breaks the tightness result between the original problems in $\RR^d$ and their SDP counterparts. Numerical Experiments {#s:numres} ===================== ![**Option implied volatility** for Apple between January 4 2004 and December 30 2010.[]{data-label="fig:vol"}](aapl.pdf){width=".7\textwidth"} In this section, we evaluate the ability of our techniques to extract mean-reverting baskets with sufficient variance and small 0-norm from a universe of tradable assets. We measure performance by applying to these baskets a trading strategy designed specifically for mean-reverting processes. We show that, under realistic trading costs assumptions, selecting sparse and volatile mean-reverting baskets translates into lower incurred costs and thus improves the performance of trading strategies. Historical Data --------------- We consider daily time series of option implied volatilities for 210 stocks from January 4 2004 to December 30 2010. A key advantage of using option implied volatility data is that these numbers vary in a somewhat limited range. Volatility also tends to exhibit regime switching, hence can be considered piecewise stationary, which helps in extracting structural relationships. We plot a sample time series from this dataset in Figure \[fig:vol\] that corresponds to the implicit volatility of Apple’s stock. In what follows, we mean by asset the implied volatility of any of these stocks, whose value can be efficiently replicated using option portfolios. Mean-reverting Basket Estimators -------------------------------- We compare the three basket selection techniques detailed here, **predictability**, **portmanteau** and **crossing statistic**, implemented with varying targets for both sparsity and volatility, with two cointegration estimators that build upon principal component analysis [@maddala1998urc §5.5.4]. By the label ‘PCA’ we mean in what follows the eigenvector with smallest eigenvalue of the covariance matrix $A_0$ of the process [@stock1988tct]. By ‘sPCA’ we mean the sparse eigenvector of $A_0$ with 0-norm $k$ that has the smallest eigenvalue, which can be simply estimated by computing the leading sparse eigenvector of $\lambda I-A_0$ where $\lambda$ is bigger than the leading eigenvalue of $A_0$. This sparse principal component of the covariance matrix $A_0$ should not be confused with our utilization of sparse PCA in Section \[subsec:asemidefinite\] as a way to recover a vector solution from the solution of a positive semidefinite problem. Note also that techniques based on principal components do not take explicitly variance levels into account when estimating the weights of a co-integrated relationship. @jurek Trading Strategy ----------------------- While option implied volatility is not directly tradable, it can be synthesized using baskets of call options, and we assimilate it to a tradable asset with (significant) transaction costs in what follows. For baskets of volatilities isolated by the techniques listed above, we apply the [@jurek] strategy for log utilities to the basket process recording out of sample performance. @jurek propose to trade a stationary autoregressive process $(x_t)_{t}$ of order $1$ and mean $\mu$ governed by the equation $x_{t+1} = \rho x_t +\sigma \varepsilon_t$, where $\abs{\rho}<1$, by taking a position $N_t$ in the asset $x_t$ which is proportional to $$\label{eq:jurek} N_t = \frac{\rho (\mu-x_t)}{\sigma^2}W_t$$ In effect, the strategy advocates taking a long (resp. short) position in the asset whenever it is below (resp. above) its long-term mean, and adjust the position size to account for the volatility of $x_t$ and its mean reversion speed $\rho$. Given basket weights $y$, we apply standard AR estimation procedures on the in-sample portion of $y^T\bx$ to recover estimates for $\hat{\rho}$ and $\hat{\sigma}$ and plug them directly in Equation . This approach is illustrated for two baskets in Figure \[fig:syn\]. -2.5cm![**Three sample trading experiments, using the PCA, sparse PCA and the Crossing Statistics estimators**. \[a\] Pool of 9 volatility time-series selected using our fast PCA selection procedure. \[b\] Basket weights estimated with in-sample data using either the eigenvector of the covariance matrix with smallest eigenvalue, the smallest eigenvector with a sparsity constraint of $k=\lfloor 0.5 \times 9\rfloor=4$ and the Crossing Statistics estimator with a volatility threshold of $\nu=0.2$, a constraint on the basket’s variance to be larger than $0.2 \times$ the median variance of all $8$ assets. \[c\] Using these 3 procedures, the time series of the resulting basket price in the in-sample part \[c\] and out-sample parts \[d\] are displayed. \[e\] Using the [@jurek] trading strategy results in varying positions (expressed as units of baskets) during the out-sample testing phase. \[f\] Transaction costs that result from trading the assets to achieve such positions accumulate over time. \[g\] Taking both trading gains and transaction costs into account, the net wealth of the investor for each strategy can be computed (the Sharpe over the test period is displayed in the legend). Note how both sparsity and volatility constraints translate into portfolios composed of less assets, but with a higher variance.[]{data-label="fig:syn"}](example_3basks2.pdf "fig:"){width="140.00000%"} Transaction Costs ----------------- We assume that fixed transaction costs are negligible, but that transaction costs per contract unit are incurred at each trading date. We vary the size of these costs across experiments to show the robustness of the approaches tested here to trading costs fluctuations. We let the transaction cost per contract unit vary between 0.03 and 0.17 cents by increments of 0.02 cents. Since the average value of a contract over our dataset is about 40 cents, this is akin to considering trading costs ranging from about 7 to about 40 Base Points (BP), that is 0.07 to 0.4%. Experimental Setup ------------------ We consider 20 sliding windows of one year (255 trading days) taken in the history, and consider each of these windows independently. Each window is split between 85% of days to estimate and 15% of days to test-trade our models, resulting in 38 test-trading days. We do not recompute the weights of the baskets during the test phase. The 210 stock volatilities (assets) we consider are grouped into 13 subgroups, depending on the economic sector of their stock. This results in 13 sector pools whose size varies between 3 assets and 43 assets. We look for mean-reverting baskets in each of these 13 sector pools. Because all combinations of stocks in each of the 13 sector pools may not necessarily mean-reverting, we select smaller candidate pools of $n$ assets through a greedy backward-forward minimization scheme, where $8\leq n\leq 12$. To do so, we start with an exhaustive search of all pools of size 3 within the sector pool, and proceed by adding or removing an asset using the PCA estimator (the smallest eigenvalue of the covariance matrix of a set of assets). We use the PCA estimator in that backward-forward search because it is the fastest to compute. We score each pool using that PCA statistic, the smaller meaning the better. We generate up to 200 candidate pools per each of the 13 sector pools. Out of all these candidate pools, we keep the best 50 in each window, and use then our cointegration estimation approaches separately on these candidates. One such pool was, for instance, composed of the stocks `{BBY,COST,DIS,GCI,MCD,VOD,VZ,WAG,T}` observed during the year 2006. Figure \[fig:syn\] provides a closeup on that universe of stocks, and shows the results of three trading experiments using either PCA, sparse PCA or the Crossing Stats estimator to build trading strategies. Results ------- ### Robustness of Sharpe Ratios to Costs In Figure \[fig:sharpe\], we plot the average of the Sharpe ratio over the $922$ baskets estimated in our experimental set versus transaction costs. We consider different PCA settings as well as our three estimators using, in all 3 cases, the variance bound $\nu$ to be $0.3$ times the median of all variances of assets available in a given asset pool, and the 0-norm to be equal to 0.3 times the size of the universe (itself between 8 and 12). We observe that Sharpe ratios decrease the fastest for the naive PCA based method, this decrease being somewhat mitigated when adding a constraint on the 0-norm of the basket weights obtained with sparse PCA. Our methods require, in addition to sparsity, enough volatily to secure sufficient gains. These empirical observations agree with the intuition of this paper: simple cointegration techniques can produce synthetic baskets with high mean-reversion, large support, low variance. Trading a portfolio with low variance which is supported by multiple assets translates in practice into high trading costs which can damage the overall performance of the strategy. Both sparse PCA and our techniques manage instead to achieve a trade-off between desirable mean-reversion properties and, at the same time, control for sufficient variance and small basket size to allow for lower overall transaction costs. ### Tradeoffs Between Mean Reversion, Sparsity, and Volatility In the plots of Figure \[fig:sharpeCrossing\] and \[fig:sharpeCrossing2\], this analysis is further detailed by considering various settings for $\nu$ (volatility threshold) and $k$. To improve the legibility of these results we summarize, following the observation in Figure \[fig:sharpe\] that the relationship between Sharpes and transactions costs seems almost linear, each of these curves by two numbers: an intercept level (Sharpe ratio when costs are low) and a slope (degradtion of Sharpe as costs increase). Using these two numbers, we locate all considered strategies in the intercept/slope plane. We first show the spectral techniques, PCA and sPCA with different levels of sparsity, meaning that $k$ is set to $\lfloor u \times d\rfloor$ where $u\in\{0.3,0.5,0.7\}$ and $d$ is the size of the original basket. Each of the three estimators we propose is studied in a separate plot. For each we present various results characterized by two numbers: a volatility threshold $\nu\in\{0,0.1,0.2,0.3,0.4,0.5\}$ and a sparsity level $u\in\{0.3,0.5,0.7\}$. To avoid cumbersome labels, we attach an arrow to each point: the arrow’s length in the vertical direction is equal to $u$ and characterizes the size of the basket, the horizontal length is equal to $\nu$ and characterizes the volatility level. As can be seen in these 3 plots, an interesting interplay between these two factors allows for a continuum of strategies that trade mean-reversion (and thus Sharpe levels) for robustness to cost level. ![Average Sharpe ratio for the @jurek trading strategy captured over about 922 trading episodes, using different basket estimation approaches. These 922 trading episodes were obtained by considering 7 disjoint time-windows in our market sample, each of a length of about one year. Each time-window was divided into 85% in-sample data to estimate baskets, and 15% outsample to test strategies. On each time-window , the set of 210 tradable assets during that period was clustered using sectorial information, and each cluster screened (in the in-sample part of the time-window) to look for the most promising baskets of size between 8 and 12 in terms of mean reversion, by choosing greedily subsets of stocks that exhibited the smallest minimal eigenvalues in their covariance matrices. For each trading episode, the same universe of stocks was fed to different mean-reversion algorithms. Because volatility time-series are bounded and quite stationary, we consider the PCA approach, which uses the eigenvector with the smallest eigenvalue of the covariance matrix of the time-series to define a cointegrated relationship. Besides standard PCA, we have also consider sparse PCA eigenvectors with minimal eigenvalue, with the size $k$ of the support of the eigenvector (the size of the resulting basket) constrained to be 30%, 50% or 70% of the total number of considered assets. We consider also the portmanteau, predictability and crossing stats estimation techniques with variance thresholds of $\nu=0.2$ and a support whose size $k$ (the number of assets effectively traded) is targeted to be about $30\%$ of the size of the considered universe (itself between 8 and 12). As can be seen in the figure, the sharpe ratios of all trading approaches decrease with an increase in transaction costs. One expects sparse baskets to perform better under the assumption that costs are high, and this is indeed observed here. Because the relationship between sharpe ratios and transaction costs can be efficiently summarized as being a linear one, we propose in the plots displayed in Figures \[fig:sharpeCrossing\] and  \[fig:sharpeCrossing2\] a way to summarize the lines above with two numbers each: their intercept (Sharpe level in the quasi-absence of costs) and slope (degradation of Sharpe as costs increase). This visualization is useful to observe how sparsity (basket size) and volatility thresholds influence the robustness to costs of the strategies we propose. This visualization allows us to observe how performance is influenced by these parameter settings.\[fig:sharpe\]](ex2-eps-converted-to.pdf){width=".8\textwidth"} ![Relationships between Sharpe in a low cost setting (intercept) in the $x$-axis and robustness of Sharpe to costs (slope of Sharpe/costs curve) of a different estimators implemented with varying volatility levels $\nu$ and sparsity levels $k$ parameterized as a multiple of the universe size. Each colored square in the figures above corresponds to the performance of a given estimator (Portmanteau in subfigure $(a)$, Predictability in subfigure $(b)$) using different parameters for $\nu\in\{0,0.1,0.2,0.3,0.4,0.5\}$ and $u\in\{0.3,0.5,0.7\}$. The parameters used for each experiment are displayed using an arrow whose vertical length is proportional to $\nu$ and horizontal length is proportional to $u$.\[fig:sharpeCrossing\]](Portmanteau___-eps-converted-to.pdf "fig:"){width="\textwidth"}\ (a) ![Relationships between Sharpe in a low cost setting (intercept) in the $x$-axis and robustness of Sharpe to costs (slope of Sharpe/costs curve) of a different estimators implemented with varying volatility levels $\nu$ and sparsity levels $k$ parameterized as a multiple of the universe size. Each colored square in the figures above corresponds to the performance of a given estimator (Portmanteau in subfigure $(a)$, Predictability in subfigure $(b)$) using different parameters for $\nu\in\{0,0.1,0.2,0.3,0.4,0.5\}$ and $u\in\{0.3,0.5,0.7\}$. The parameters used for each experiment are displayed using an arrow whose vertical length is proportional to $\nu$ and horizontal length is proportional to $u$.\[fig:sharpeCrossing\]](Predictability___-eps-converted-to.pdf "fig:"){width="\textwidth"}\ (b) ![Same setting as Figure \[fig:sharpeCrossing\], using the crossing statistics (c).\[fig:sharpeCrossing2\]](Crossing___-eps-converted-to.pdf "fig:"){width="\textwidth"}\ (c) Conclusion ========== We have described three different criteria to quantify the amount of mean reversion in a time series. For each of these criteria, we have detailed a tractable algorithm to isolate a vector of weights that has optimal mean reversion, while constraining both the variance (or signal strength) of the resulting univariate series to be above a certain level and its 0-norm to be at a certain level. We show that these bounds on variance and support size, together with our new criteria for mean reversion, can significantly improve the performance of mean reversion statistical arbitrage strategies and provide useful controls to adjust mean-reverting strategies to varying trading conditions, notably liquidity risk and cost environment.

ArXiv

--- abstract: 'We investigate the occurrence of anomalous diffusive transport associated with acoustic wave fields propagating through highly-scattering periodic media. Previous studies had correlated the occurrence of anomalous diffusion to either the random properties of the scattering medium or to the presence of localized disorder. In this study, we show that anomalous diffusive transport can occur also in perfectly periodic media and in the absence of disorder. The analysis of the fundamental physical mechanism leading to this unexpected behavior is performed via a combination of deterministic, stochastic, and fractional-order models in order to capture the different elements contributing to this phenomenon. Results indicate that this anomalous transport can indeed occur in perfectly periodic media when the dispersion behavior is characterized by anisotropic (partial) bandgaps. In selected frequency ranges, the propagation of acoustic waves not only becomes diffusive but its intensity distribution acquires a distinctive L[é]{}vy $\alpha$-stable profile having pronounced heavy-tails. In these ranges, the acoustic transport in the medium occurs according to a hybrid transport mechanism which is simultaneously propagating and anomalously diffusive. We show that such behavior is well captured by a fractional diffusive transport model whose order can be obtained by the analysis of the heavy tails.' author: - Salvatore Buonocore - Mihir Sen - Fabio Semperlotti bibliography: - 'ref.bib' title: 'Occurrence of anomalous diffusion and non-local response in highly-scattering acoustic periodic media' --- Introduction {#Introduction} ============ In recent years, several theoretical and experimental studies have shown that field transport processes in non-homogeneous and complex media can occur according to either hybrid or anomalous mechanisms. Some examples of these physical mechanisms include anomalous diffusive transport (such as non-Fourier [@povstenko2013fractional; @borino2011non; @ezzat2010thermoelectric], or non-Fickian diffusion [@benson2000application; @benson2001fractional; @cushman2000fractional; @fomin2005effect] with heavy-tailed distribution) or hybrid wave transport (characterized by simultaneous propagation and diffusion [@mainardi1996fractional; @mainardi1996fundamental; @mainardi1994special; @mainardi2010fractional; @chen2003modified; @chen2004fractional]). Simultaneous hybrid and anomalous transport has also been observed, particularly in wave propagation problems involving random scattering media. Electromagnetic waves traveling through a scattering material[@yamilov2014position] such as fog [@belin2008display] or murky water [@zevallos2005time] are relevant examples of practical problems where such transport process can arise. A distinctive feature of anomalous transport is the occurrence of heavy-tailed distributions of the representative field quantities [@benson2001fractional]. In this case, the diffusion process does not follow a classical Gaussian distribution but instead is characterized by a high-probability of occurrence of the events associated with large variance (i.e. those described by the “heavy” tails). This behavior is typically not accounted for in traditional field transport theories based on integer order differential or integral models. Purely numerical methods, such as Monte Carlo or finite element simulations[@huang1991optical; @ishimaru2012imaging; @mosk2012controlling; @sebbah2012waves; @gibson2005recent], can capture this response but are very computationally intensive and do not provide any additional insight in the physical mechanisms generating the macroscopic dynamic behavior. The ability to accurately predict the anomalous response and to retrieve information hidden in diffused fields remains a challenging and extremely important topic in many applications. Acoustical and optical imaging, non-intrusive monitoring of engineering and biomedical materials are just a few examples of practical problems in which the ability to carefully predict the field distribution is of paramount importance to achieve accurate and physically meaningful solutions. Nevertheless, in most classical approaches, information contained in the heavy tails is typically discarded because it cannot be properly captured and interpreted by integer-order transport models. Hybrid and anomalous diffusive transport mechanisms are pervasive also in acoustics. This type of transport can arise when acoustic fields propagate in a highly scattering medium such as a urban environment [@albert2010effect; @remillieux2012experimental], a forest [@aylor1972noise; @tarrero2008sound], a stratified fluid (e.g. the ocean) [@baggeroer1993overview; @dowling2015acoustic; @casasanta2012fractional], or a porous medium [@benson2001fractional; @schumer2001eulerian; @fellah2003measuring; @fellah2000transient]. From a general perspective, classical diffusion of wave fields occurs within a range where the wavelength is comparable to the size of the scatterers, the so-called Mie scattering regime. Any deviation from classical diffusion, being either sub-diffusion [@metzler2000random; @goychuk2012fractional] (typically linked to Anderson localization) or super-diffusion (typically linked to L[é]{}vy-flights) [@barthelemy2008levy; @bertolotti2010multiple], still arises within the same regime. The two dominant factors are either the relation between the transport mean free path and the wavelength, or the statistical distributions of the scattering paths in presence of disorder. When a wave field interacts with scattering elements, it undergoes a variety of physical phenomena including reflection, refraction, diffraction, and absorption that significantly alter its initial characteristics. Depending on the quantity, distribution, and properties of the scatterers the momentum vector of an initially coherent wave can become quickly randomized. For most processes, the Central Limit Theorem (CLT) guarantees that the distribution of macroscopic observable quantities (e.g. the field intensity) converges to a Gaussian profile in full agreement with the predictions from classical Fourier diffusion. At the same time, the transition to a macroscopic diffusion behavior leads to an inevitable coexistence of diffusive and wave-like processes at the meso- and macro-scales. There are numerous physical processes in nature whose *basin of attraction* is given by the normal (Gaussian) distribution. On the other hand, when the distribution of characteristic step-length has infinite variance, the diffusion process no longer follows the standard diffusion theory, but rather acquires an anomalous behavior with a basin of attraction given by the so-called $\alpha$-stable L[é]{}vy distribution. In the latter case, the unbounded value of the variance of the step-length distribution is due to the non-negligible probability of existence of steps whose lengths greatly differ from the mean value; these are usually denoted as L[é]{}vy flights. The distinctive feature of the $\alpha$-stable L[é]{}vy distributions is the occurrence of heavy tails having a power-law decay of the form $p(l) \sim l^{-(\alpha+1)}$. This characteristic suggests that transport phenomena evolving according to L[é]{}vy statistics are dominated by infrequent but very long steps, and therefore their dynamics are profoundly different from those predicted by the random (Brownian) motion. Many of the complex hybrid transport mechanisms mentioned above fall in this category, and therefore cannot be successfully described in the framework of classical diffusion theory. In addition, these complex transport mechanisms are typically not amenable to closed-form analytical solutions therefore requiring either fully numerical or statistical approaches to predict the field quantities under various input conditions. Typical modeling approaches rely on random walk statistical models [@metzler2000random; @bouchaud1990anomalous] or on semi-empirical corrections to the fundamental diffusive transport equation via renormalization theory [@asatryan2003diffusion; @cobus2016anderson]. These approaches imply a considerable computational cost and do not provide physical insight in the operating mechanisms of the anomalous response. A few studies have also indicated that, for this type of processes, the macroscopic governing equation describing the evolution of the wave field intensity could be described by a generalization of the classical diffusion equation using fractional derivatives [@bertolotti2010multiple; @metzler2000random; @bertolotti2007light]. To-date, the occurrence of anomalous diffusion of wave fields has been connected and observed only in random and disordered media [@barthelemy2008levy; @burresi2012weak; @bouchaud1990anomalous; @asatryan2003diffusion; @cobus2016anderson]. In this study, we show theoretical and numerical evidence that anomalous behavior can occur even in presence of perfectly periodic media and in absence of disorder. We present this analysis in the context of diffusive transport of acoustic waves although the results could be generalized to other wave fields. In particular, we investigate the specific case of propagation of acoustic waves in a medium with identical and periodically distributed hard scatterers. We develop a theoretical framework for multiple scattering in super-diffusive periodic media. We first show, by full field numerical simulations, that under certain conditions, acoustic waves propagating through a periodic medium are subject to anomalous diffusion. Then, we propose an approach based on a combination of deterministic and stochastic methodologies to explore the physical origin of this unexpected behavior. Ultimately, we show that fractional order models can predict, more accurately and effectively, the resulting anomalous field quantities. More important, we will show that the analysis of the heavy tails provide a reliable means to extract the equivalent fractional order of the medium. Anomalous diffusion in acoustic periodic media: overview of the method ====================================================================== We consider the generic problem of an acoustic bulk medium made of periodically-distributed cylindrical hard scatterers in air (Fig. \[Fig\_1\]). We assume a monopole-like acoustic source, located in the center of the lattice, which emits at a selected harmonic frequency chosen within the scattering regime. The main objective is to characterize the propagation of acoustic waves in such medium based on different regimes of dispersion. As previously anticipated, in selected regimes the propagation of acoustic waves will exhibit anomalous diffusive transport properties. The reminder of this study will be dedicated to investigating the causes leading to the occurrence of such phenomenon. In order to identify the fundamental mechanisms at the origin of this behavior, we have designed a multi-folded approach capable of characterizing the different processes contributing to the macrosopic anomalous response. The approach consists of the following components. First, we investigate via full-field numerical simulations the propagation of acoustic waves in either a 1D or a 2D periodic scattering medium. The numerical results will allow making important observations on the different propagation mechanisms occurring in the two systems and on the corresponding diffusive processes. Then, the radiative transfer theory will be applied to interpret the evolution of the wave intensity distribution and analyze the nature of the diffusive phenomena in the context of a renormalization approach. In order to identify the physical mechanism at the origin of the anomalous diffusion, a multiple scattering analysis based on the multipole expansion method will be applied in order to characterize the interaction between different scatterers. In particular, this approach was intended to identify and quantify possible long-range interactions between pairs of scatterers. Based on the results of the multiple scattering analysis, a Monte Carlo model is used to confirm that the anomalous transport is in fact originated by the long-range interactions between different directions of propagation in the lattice. Finally, we show that the behavior of the lattice can be effectively described in a homogenized sense, by a fractional continuum diffusion model whose fractional order can be identified by fitting an $\alpha$-stable distribution to the heavy tails of the wave intensity. This approach can be seen as an equivalent *fractional homogenization* of the medium. Of particular interest is the fact that the fractional (homogenized) model allows a closed-form analytical solution the agrees very well with the numerical predictions. Scattering and diffusive transport {#Overview} ================================== From a general perspective, it is possible to identify four different wave propagation regimes in scattering media which are classified based on the relative ratio of quantities such as the transport mean free path $l_t$, the wavelength of the propagating field $\lambda$, and the characteristic size of the scattering domain $L$. The four regimes are: 1. The *homogenized regime*: it occurs when the wavelength of the incident wave field is much larger than the typical characteristic size $d$ of the scatterer, that is $\lambda \gg d$. 2. The *diffusive regime*: it occurs when the wavelength $\lambda$ satisfies the relation $\lambda/2\pi \ll l_t \ll L $. In this regime the wave intensity can be approximated by the diffusion equation. 3. The *anomalously diffusive regime*: it occurs when the interference of waves causes the reduction of the transport mean free path $l_{t}$ and consequently a renormalization of the macroscopic diffusion constant $D$. In this regime, the transport mean free path varies according to the size of the cluster and to the degree of disorder. 4. The *localization regime*: it occurs in the range $\lambda/2\pi \geq l_{t}$ and corresponds to a diffusion constant $D$ tending to 0. As mentioned in the classification above, there are regimes in which the intensity of the wave field can be properly described by the *diffusion approximation*, that is it varies in space as prescribed by the field evolution in a diffusion equation. In particular, when the incident wave has a wavelength smaller than the length-scale characterizing the material and/or of the geometric variations of the physical medium, the wave field undergoes multiple scattering with a consequent randomization of its phase and direction of propagation. In order to characterize this phenomenon a statistical description based on random walk models is typically employed. These models rely on phenomenological quantities such as the scattering $l_s$ and the transport $l_t$ mean free paths. From a physical perspective, $l_s$ represents the average distance between two successive scattering events, while $l_t$ is the mean distance after which the wave field loses memory of its initial direction and becomes randomized [@van1999multiple]. When the filling factor $f$ (which describes the density of scatterers) is low, $l_{s}$ and $l_{t}$ are defined as [@ishimaru1978wave]: $$\begin{aligned} \label{ls_lt} l_{s} &=& \dfrac{1}{\rho\sigma_{t}}\\ l_{t} &=& \dfrac{l_{s}}{1-\langle\cos{\theta}\rangle} \nonumber \label{eq:lt}\end{aligned}$$ where $\sigma_{t}$ is the total scattering cross section, $\langle\cos{\theta}\rangle$ is the *anisotropy factor* and $\rho$ is the scatterers concentration. Note that the relations in Eq. (\[ls\_lt\]) are valid only for low filling factors, approximately in the range $f \leq 0.1$. For increased values of the filling factor, the scattering cross section $\sigma_t$ needs to be rescaled. The rescaling factor in the range $0.1 \leq f \leq 0.6$ is given by $\sigma_t \rightarrow \sigma_t (1-f)$, while higher filling factors require a more elaborated rescaling procedure [@ishimaru1978wave]. The scattering cross section plays a crucial role in the characterization of multiple scattering phenomena and in two dimensions takes the form: $$\begin{aligned} \label{sigma_t} \begin{split} \sigma_t = \int_{2\pi}\sigma_d(\theta) d\theta . \end{split}\end{aligned}$$ The integrand $\sigma_d$ is the differential scattering cross section defined as: $$\begin{aligned} \label{diff_scatt_cross_sect} \begin{split} \sigma_d(\theta)= \lim_{R \rightarrow \infty} R\left [ (S_s(\theta))/S_i\right ]. \end{split} \end{aligned}$$ . In Eq. (\[diff\_scatt\_cross\_sect\]), the term $S_s$ is the scattered power flux density at a distance $R$ from the scatterer in the direction $\mathbf{\hat{o}}$ caused by an incident power flux density $S_i$. The azimuthal angle $\theta$ is the angle between the incident ($\mathbf{\hat{i}}$) and the scattered wave fields ($\mathbf{\hat{o}}$). The *scattering phase function* is obtained by normalizing the differential scattering cross section with respect to $\sigma_t$: $$\begin{aligned} \label{} p\mathbf{( \hat{o},\hat{i})}=p(\cos\theta) = \frac{\sigma_d(\theta)}{\sigma_{t}} \end{aligned}$$ and represents the probability that a wave field impinging on the scatterer from a given direction will be scattered by an angle $\theta$. The mean value of the previous probability distribution defines the *anisotropy factor*: $$\begin{aligned} \label{} \langle \cos \theta \rangle = \int_{2\pi} p(\cos \theta) \cos(\theta) d \theta. \end{aligned}$$ This factor varies between 0 and 1, and it accounts for the existence of preferential scattering directions. For $\langle\cos{\theta}\rangle=0$ all the scattering directions have the same probability and the scattering is isotropic. As $\langle\cos{\theta}\rangle$ approaches 1, the forward scattering becomes the most probable event. These quantities will be used in the following analyses in order to identify the different scattering regimes. One-dimensional medium ====================== Consider a one-dimensional bulk scattering medium composed of $N$ hard cylindrical scatterers equally distributed in an air background (Fig. \[Fig\_3\]). This system can be interpreted by all means as a classical 1D acoustic metamaterial. The radius of the individual scatterer is $a = 0.2 d$, where $d$ indicates the distance between two neighboring cylinders. The filling fraction for this particular cluster is $f = \pi a^2/d^2 \approx 0.1257$. The waveguide is excited by a monochromatic acoustic monopole $S$ that replaces the center cylinder. The response of the system is obtained numerically by means of a commercial finite element software (Comsol Multiphysics) and using symmetric boundary conditions on the top and bottom edges and perfectly matched layers (PML) on the left and right edges. The frequency of excitation is selected in the first bandgap (see Fig. \[Fig\_10\] for the general dispersion properties of this waveguide) and has a non-dimensional value $\Omega = 0.0831$. In this excitation regime the diffusive behavior is expected. Remember that, in the absence of disorder or trapping mechanisms and in the range of excitation frequencies where the diffusion approximation holds, the variance of the step-length distribution characterizing the multiple scattering process of the acoustic field is expected to be finite and, if the steps are independent (by virtue of the Central Limit Theorem) the limit distribution should be the Normal distribution as predicted by the standard diffusion model. The resulting normalized magnitude of the acoustic pressure field generated in the waveguide is shown in Fig. \[Fig\_4\](a) in terms of a contour plot and in Fig. \[Fig\_4\](b),(c) in terms of the intensity profile along the mid-line of the waveguide as defined later in §\[Modell\]. From Fig. \[Fig\_4\](b),(c) a characteristic exponential decay of the type $e^{-x/l_s}$ (consistent with the Beer-Lambert law) is very well identifiable. This trend represents the decay of the coherent part of the intensity and corresponds to the squared absolute value of the Green’s function solution. In more general terms, this result shows a solution which is perfectly consistent with the classical diffusion behavior. This is a well expected result and it is reported here only for comparison with the results that will be presented below. Two-dimensional medium ====================== Radial lattice -------------- The immediate extension of the previous scenario to a two-dimensional system corresponds to a radial distribution of equally distributed hard scatterers. As in the 1D case, the system is excited by a monochromatic monopole source located in the center of the 2D lattice at point $S$. The source is monochromatic and it is actuated at the non-dimensional frequency $\Omega = 0.0831$, that belongs to the first bandgap. The normalized magnitude of the acoustic pressure field for this system is numerically calculated and shown in Fig. \[Fig\_6\](a). Fig. \[Fig\_6\](b) provides a closeup view of the field around the source (the area within the black dashed line). The acoustic intensity profile along the $x$-axis direction shows an exponential decay as illustrated by Fig. \[Fig\_7\]. All radial directions (not shown) exhibit an identical response as expected due the azimuthal symmetry of the system. As in the 1D case, this linear decay of the intensity distribution was expected and confirms that, in systems with a high degree of symmetry, a classical diffusion behavior should be recovered. From a practical perspective, this radial lattice could be seen as a radial arrangement of 1D waveguides. Rectangular lattice {#rect_lattice} ------------------- The dynamic behavior of the lattice changes quite drastically when the axial-symmetry is removed. Consider the square lattice of scatterers schematically illustrated in Fig. \[Fig\_1\]. Assume each scatterer having an individual radius of $a = 0.2 d$, where $d$ is the distance between two neighboring cylinders. The filling fraction for this periodic cluster is $f = \pi a^2/d^2 \approx 0.1257$. ### Dispersion analysis {#DispersionRelation} In order to understand the dynamic behavior of this lattice and interpret the results that will follow, we start analyzing the fundamental dispersion structure of the square lattice. The dispersion was calculated using finite element analysis and the band structure is plotted along the irreducible part of the first Brillouin zone, as shown in Fig. \[Fig\_10\]. The results highlight the existence of anisotropy in terms of directions of propagation. These directions are connected to the existence of a partial bandgap in the $\Gamma-X$ direction between the non-dimensional frequencies $\Omega = 0.0824$ and $\Omega = 0.1103$. When the system is excited at a frequency within the bandgap, the propagation acquires an anisotropic distribution (see § \[Forced\] and Fig. \[Fig\_8\]), because propagation can only occur in the $\Gamma-M$ direction. This is not an unexpected result and, in fact, it is fully consistent with the propagation behavior expected in square periodic lattice. However, we will show that these dispersion characteristics play a key role in the occurrence of anomalous behavior. ### Forced response {#Forced} The forced response of the lattice was also numerically evaluated. In this case, the lattice is excited by a monochromatic acoustic monopole $S$ that replaces the center cylinder. As for the previous two lattices, the total acoustic pressure field is calculated numerically using the finite element method and reported in Fig. \[Fig\_8\]. More specifically, Fig. \[Fig\_8\](a) presents the response to an excitation outside the first bandgap, while Fig. \[Fig\_8\](b) reports the case just inside the first bandgap. Note that due to symmetry considerations, only a quarter of the domain was solved. As the acoustic wave fronts propagate through the medium in the radial directions and interact with the scattering particles, the rays are scattered in multiple directions. In both cases it is evident that the propagation is strongly anisotropic and occurs mostly along the diagonal directions of the lattice. The response of the medium is shown in Fig. \[Fig\_8\] in terms of the normalized magnitude of the acoustic pressure distribution. Contrarily to what observed for the radial lattice, in this case the intensity distribution does not decay linearly. This behavior is very evident by performing a numerical fit of the simulation data, as shown in Fig. \[Fig\_9\]. These results suggest the occurrence of an unexpected mechanism of diffusion despite the lattice periodicity. This is a remarkable departure compared with available results in the literature that, to-date, have highlighted the occurrence of anomalous diffusion only in connection with random distributions of geometric or material properties. Radiative transport approach {#Modell} ============================ The results presented above illustrated that in case of anisotropic propagation a departure from the classical diffusive behavior is observed. In this section, we use a traditional radiative transport approach with renormalization to show that this observed behavior can be mapped to anomalous diffusion. We investigate the presence of anomalous diffusion for wavelength ranges in the passband and in the bandgap. As already pointed out, within the regime $\lambda/2 \pi<l_{t}<L$ the diffusion approximation applies and the spatial evolution of the wave amplitude can be predicted by a diffusion equation for the wave intensity. Starting from a cluster of particles, as schematically illustrated in Fig. \[Fig\_11\], and applying the diffusion approximation the 2D diffusion equation for harmonic excitation and lossless scatterers is given by: $$\begin{aligned} \label{Diffusion equation} \begin{split} \nabla^2 I =-\frac{P_0}{\pi l_t}\delta(\vec{r}-\vec{r}_s) \end{split}\end{aligned}$$ where $I$ is the intensity of the acoustic wavefield, $P_0$ is the total emitted acoustic power, $\vec{r}$ and $\vec{r}_s$ are the position vectors of the source $S$ and of a generic point $P$, respectively. The average acoustic intensity of a monochromatic monopole source can be obtained as $\langle I \rangle = ||0.5*Re(p \cdot v')||$, where $p$ is the pressure field, and $v'$ is the complex conjugate of the velocity field. The diffusion equation Eq. (\[Diffusion equation\]) requires the following boundary conditions at the edge of the domain to be solved: $$\begin{aligned} \begin{split} I\mathbf{(r_s)}-\frac{\pi l_t}{4}\frac{\partial }{\partial n} I\mathbf{(r_s)} = 0 \end{split}\end{aligned}$$ where $\mathbf{\hat{n}}$ is the unit inward normal. These boundary conditions are obtained by the requirement of zero inward flux at the boundaries[@ishimaru1978wave]. The numerical value of this boundary condition on the intensity was obtained by the previous finite element model. By enforcing this boundary condition, Eq. (\[Diffusion equation\]) can be solved analytically: $$\begin{aligned} \begin{split} I = -\frac{P_0}{2\pi^2 l_t}ln\frac{|\vec{r}-\vec{r}_s|}{L}+ I_{0} \end{split} \label{Solution}\end{aligned}$$ where $I_{0}$ is the value of the intensity at the boundary of the cluster of scatterers and $L$ is the size of the computational domain. In order to be able to solve Eq. (\[Solution\]), we need to estimate the parameters $l_t$ and $l_s$ and characterize the specific regime of propagation. To achieve this result, we first plot $\langle l_{s}\rangle$ and $\langle l_{t}\rangle$ versus the wavelength $\lambda$ as shown in Fig. \[Fig\_12\]. These curves were numerically determined using the model presented in §\[rect\_lattice\] and the Eqs. (\[ls\_lt\]). The transport mean free path $\langle l_{t}\rangle$ is always expected to be greater than $\langle l_{s}\rangle$ and to converge asymptotically to $\langle l_{s}\rangle$ for large wavelengths. In fact, for long wavelengths the wavefield is marginally affected by the presence of the scatterers. In the short wavelength limit, $l_s/d$ tends to 1 because the wave fronts are highly directional (this is the range of validity of ray acoustics approximation) and **$l_s$** is approximately given by the average distance between two neighboring scatterers. Fig. \[Fig\_13\] shows a detailed view of the previous curves in the frequency range corresponding to the first bandgap and within the diffusive regime. The labels $A$ and $B$ indicate the non-dimensional wavelengths corresponding to the excitation conditions analyzed in the following sections. Note that these curves provide the foundation to investigate the different regimes of propagation and to implement the renormalization approach. ### Renormalization and anomalous diffusion {#Numerical_1} Fig. \[Fig\_14\] shows the acoustic intensity distribution $I$ along the $x$ axis for the two excitation conditions identified by the labels A and B. The red circles show the numerical solution obtained by the FE model and provide a one-dimensional section of the data in Fig. \[Fig\_8\] along the $x$-axis. The continuous blue line is the analytical solution of the diffusion equation Eq. (\[Solution\]) after having rescaled the transport mean free path. In particular, for excitation wavelengths in the first passband the value $\langle l_{t}\rangle/d\approx 1.211$ was rescaled to $\langle l_{t}\rangle/d\approx 0.32 \pm 0.02$ (label $A^*$ in Fig. \[Fig\_13\]), while for the first bandgap the value $\langle l_{t}\rangle/d\approx 1.42$ was rescaled to $\langle l_{t}\rangle/d\approx 0.48 \pm 0.02$ (label $B^*$ in Fig.  \[Fig\_13\]). These results show that, in order to be able to predict the numerical data by using the diffusion approximation, a renormalization of the transport mean free path (and consequently of the diffusion coefficient) must take place. The renormalization requires smaller values of the transport parameters which is a clear indication of superdiffusive anomalous behavior. Causes of anomalous diffusion {#NumericalViewPer} ============================= In the previous sections we showed the occurrence of anomalous diffusion of acoustic waves in perfectly periodic square lattices and suggested that the possible origin of this mechanism is linked to the anisotropy of the dispersion properties (i.e. to the anisotropy of the bandgaps). In this section we will present theoretical and numerical models with the intent of uncovering the physical mechanism leading to this unexpected propagation modality. It is anticipated that the occurrence of anomalous diffusion will be connected to the existence of long range interactions between different directions of propagation governed by either bandpass or stopband behavior. We will use a combination of both deterministic and stochastic methods in order to quantify the long-range interactions and to demostrate that they are at the origin of the macroscopic anomalous diffusion mechanism. More specifically, we will use a scattering matrix approach to quantify the interaction between different scatterers in different regimes. Then, we will use a discrete random walk diffusion model (which uses probability density functions obtained from the scattering model) to show that, under these assumptions, the anomalous diffusion process matches well with the numerically predicted behavior. The scattering matrix --------------------- In order to evaluate and quantify the strength of the interaction between different scatterers in the lattice, we use a multiple scattering approach based on the multipole expansion method. According to this method, after applying the Jacobi’s expansion and the Graf’s addition theorem, the general solution of the wave field can be expressed as: $$\begin{aligned} \begin{split} \label{Eq.206} p(\vec{r}_m)=\sum_{j=-\infty}^{\infty} (e^{i\vec{k}\cdot \vec{P}_m} e^{ij(\pi /2-\psi_0)}J_j(\vec{r}_m)+A_j^m H_j(\vec{r}_m))+\\ \sum_{n=1,n\neq m}^{N}\sum_{q=-\infty}^{\infty}A_n^q\sum_{j=-\infty}^{\infty}H_{q-j}(kR_{nm})e^{i(q-j)\Phi_{nm}}J_j(\vec{r}_m) \end{split}\end{aligned}$$ where $\vec{k}$ represents the wave vector, $\psi_0$ is the angle of the impinging wave field with respect to the $x$-axis, $\vec{P}_m$ is the position vector of the scatterer’s center $O_m$ with respect to the origin $O$ of the system of reference, $\vec{r}_m$ is the position vector of a generic point $P$ with respect to the scatterer’s center $O_m$, $R_{nm} = \left |\vec{P}_m - \vec{P}_n \right |$, $J_q(\cdot)$ are the Bessel functions of the first kind, $H_q(\cdot)$ are the Hankel functions of the first kind. To determine the unknown amplitude coefficients $A_m^q$ the boundary conditions at the surface of the $m$th cylinder must be enforced. The result is a linear set of equations as follows[@linton2005multiple; @martin2006multiple; @kafesaki1999multiple]: $$\begin{aligned} \begin{split} \label{Eq.207} A_m^p+Z_p\sum_{n=1,n\neq m}^{N}\sum_{q=-\infty}^{\infty}A_n^qH_{q-p}(kR_{nm})e^{i(q-p)\Phi_{nm}}= \\ -Z_p e^{i\vec{k}\cdot\vec{P}_m} e^{ip(\pi/2-\psi_0)},\\ \quad m=1,...,N ,\quad p=0,+1,-1,... \end{split}\end{aligned}$$ where $Z_p =J'_p(\cdot)/ H'_p(\cdot)$ specifies the Neumann boundary conditions on the surface of the cylinders. The unknown amplitude coefficients $A_m^q$ can be determined by solving the infinite system of algebraic equations with inner sum truncated at some positive integer $|q| =Q$. The information about the relative energy exchange between the scatterers can be obtained by rearranging Eq. (\[Eq.207\]) in matrix form as: $$\begin{aligned} \begin{split} \label{Eq.209} (I-TS)f=Ta \end{split}\end{aligned}$$ where $I$ is the unit matrix, $T$ is the block diagonal impedance matrix, the vector $f$ represents the unknown expansions of scattered waves, and the vector $a$ stands for the expansion vector of incident waves on all the scattering cylinders. Finally the matrix $S$ is the so called combined translation matrix that can be expressed as follows: $$\begin{aligned} \begin{split} \label{Eq.210} S=\begin{bmatrix} 0 & L_{12} &...& L_{1N}\\ L_{21} & 0 &...&L_{2N}\\ \vdots & \vdots & \ddots & \vdots \\ L_{N1}& L_{N2} & ... & 0 \end{bmatrix} \end{split}\end{aligned}$$ where the matrix $L_{nm}$ is defined as follows: $$\begin{aligned} \begin{split} \label{Eq.211} L_{nm}(q,p)=H_{q-p}(kR_{nm})e^{i(q-p)\Phi_{nm}}. \end{split}\end{aligned}$$ The matrix $L_{nm}$ represents the translation matrix between the $n$th and the $m$th cylinder, representing therefore the incident wave on the $n$th cylinder caused by the scattered wave off the $m$th cylinder. The elements of the translation matrix can be obtained from the addition theorem of cylindrical harmonics also known as Graf’s theorem. The generic term $S_{mn}$ quantifies the portion of the acoustic intensity scattered by the cylinder $m$ capable of reaching the cylinder $n$. Equivalently, it represents the fraction of the acoustic intensity reaching the cylinder $n$ due to the wave scattered by the cylinder $m$. This approach was applied to model both the 1D and the 2D waveguides. The normalized scattering coefficients for the 1D waveguide are shown in matrix form in Fig. \[Fig\_16\]. Each block of this matrix has a size $\bar{Q} = 2*Q+1$, where $Q$ is the total number of spherical harmonics used in the multipole series expansion. The total size of the matrix is $N*\bar{Q}$, where $N$ is the total number of cylinders in the cluster. The main diagonal represents the coefficient $S_{mm}$, that is the scattering of a given cylinder $m$ towards itself, and therefore these terms are all zero. The analysis of Fig. \[Fig\_16\] shows, as expected, that in a 1D waveguide the scatterers only interact with their closest neighbors. In other terms, there is no evidence of long-range interaction in 1D periodic waveguides. This is not surprising because we had already found from the full-field simulations in Fig. \[Fig\_4\](a) that the diffusive transport was following a purely Gaussian distribution (hence dominated by nearest neighbor interactions). In a similar way, the analysis can be repeated for the 2D waveguide with square lattice structure. The resulting scattering coefficients are shown in Fig. \[Fig\_17\]. Contrarily to the 1D example, this scattering matrix has the appearance of a tridiagonal matrix that highlights the substantial interactions between distant neighbors. In other terms, the rectangular lattice show strong evidence of long-range interactions. These results provide a first important observation concerning the cause of anomalous diffusion in periodic rectangular lattice, that is the anisotropy of the dispersion bands gives rise to long-range interactions that ultimately alter the diffusion process. Discrete random walk models: approximate acoustic intensity ----------------------------------------------------------- The previous analysis is not yet sufficient to provide conclusive evidence that the long-range interactions due to the bandgap anisotropy are the main cause of the anomalous wave diffusion. In order to identify this further logical link, we developed a discrete random walk (DRW) model capable of simulating the diffusion process resulting from the multiple scattering of the acoustic waves. The interaction between the different elements of a DRW model is typically represented by probability density functions (*pdf*). In the following, the *pdf*s are synthesized based on the coefficients of the scattering matrix. The model can then be numerically solved in order to predict the approximate acoustic intensity resulting from the scattered field. ### 1D discrete random walk model The DRW model for a 1D waveguide is composed of a series of boxes (see Fig. \[Fig\_18\]), each one representing a scatterer. This model can be seen as the direct discrete equivalent of the 1D waveguide in Fig. \[Fig\_3\]. The dots in each box represent the different acoustic rays impinging on a given scatterer and being refracted towards different (scattering) elements. This model follows a ray acoustic approximation which is a reasonable assumption in the range of wavelength we have been considering. To simulate the monopole acoustic source located at the center of the waveguide, the center box (labeled $i$) contains a source term that serves as an omni-directional source of rays. In the 1D model, the rays emitted from the center box can be scattered both to the left and to the right according to the associated *pdf* synthesized based on the elements of the scattering matrix. At every time increment, the rays “jump” into another box following a Markovian process and a *pdf* proportional to the coefficients extracted from the scattering matrix. The equilibrium condition needed to solve the DRW model and simulate the evolution of the acoustic intensity upon scattering is given by imposing the conservation of rays: $$\begin{aligned} \label{eqn:Conservation number of particles} \begin{split} n_{i,j+1}=n_{i,j}+\sum_{k=1}^{N_L} n_{k,j}P(i-k)+\sum_{k=1}^{N_R}n_{k,j}P(k-i)- \\ \sum_{k=1}^{N_L}n_{i,j}P(i-k)-\sum_{k=1}^{N_R}n_{i,j}P(k-i) + B_i \end{split}\end{aligned}$$ where $i$ is the box index, $j$ is the time index, $n(i,j)$ is the number of rays at time $i$ entering the box $j$ (i.e. impinging on the scatterer $j$), $B_i$ is the source term, and $N_L$ and $N_R$ represent the number of boxes on the left and right side, respectively. The previous equation can be rearranged as follows: $$\begin{aligned} \label{eqn:Conservation number of particles 2} \begin{split} n_{i,j+1}=n_{i,j}+\sum_{k=1}^{N_L}(n_{k,j}-n_{i,j})P(i-k)- \\ \sum_{k=1}^{N_R}(n_{k,j}-n_{i,j})P(k-i)+ B_i. \end{split}\end{aligned}$$ The comparison between the intensity distributions obtained with the FE model and by the equivalent 1D DRW model is given in Fig. \[Fig\_19\]. The direct comparison of the results shows a very good agreement between the two models. Note that the DRW is a diffusive model therefore the comparison between the intensity distributions is meaningful only in the tail region. As expected the tails evolve according to a Gaussian distribution. The comparison with the 1D waveguide was provided to illustrate the validity of the proposed approach and to confirm that, under the given assumptions, the results from the DRW converge to the full-field simulations. ### 2D discrete random walk model The same approach illustrated above for the 1D waveguide can be applied to the analysis of the 2D square lattice. In this case, the DRW model is composed of a 2D distribution of boxes simulating the scatterers. The interactions between different boxes are again expressed in terms of *pdf*s that are synthesized based on the scattering coefficients obtained from the 2D multipole expansion model (Fig. \[Fig\_17\]). The equilibrium condition for the 2D DRW model is given by: $$\begin{aligned} \label{eqn:Conservation number of particles 2D} \begin{split} n_{i,h,j+1}=n_{i,h,j}+ \\ \sum_{k_h=1}^{N_D}\sum_{k_i=1}^{N_L}(n_{k_i,k_h,j}-n_{i,h,j})P(i-k_i,h-k_h)-\\ \sum_{k_h=1}^{N_U}\sum_{k_i=1}^{N_R}(n_{k_i,k_h,j}-n_{i,h,j})P(k_i-i,k_h-h)+ B_{ih} \end{split}\end{aligned}$$ where $i$ and $h$ are the box indices, $j$ is the time index, $n(i,h,j)$ is the number of particles at time $j$ in the box $(i,h)$, $B_{ih}$ is the source term and $N_L $, $N_R$, $N_U$ and $N_D$ represent the number of boxes on the left, right, up and down sides, respectively. The comparison between the intensity distributions obtained by numerical FE simulations and by equivalent 2D DRW model is reported in Fig. \[Fig\_20\]. Also in this case, the DRW model is in very good agreement with the FE simulations and, most important, is perfectly capable of capturing the anomalous (power-law) decay of the tails of the distribution. This result provides the conclusive proof that the anomalous behavior observed in the square lattice is in fact the result of long-range (L[é]{}vy flights) interactions due to scattering events occurring along different directions of propagation that are characterized by anisotropic dispersion. $\alpha$-stable distributions and fractional diffusion equation {#fractional diffusion} =============================================================== The renormalization criterion used in section §\[Numerical\_1\] to determine the existence of the anomalous diffusion regime is theoretically well-grounded but it does not allow a convenient approach to classify the anomalous regime. This classification typically requires the analysis of the time scales involved in the evolution of the moments of the distribution [@metzler2000random]. Here we suggest a different approach that, not only provides a more direct classification based on the available data, but opens new routes for an analytical treatment of the resulting diffusion problem. The intensity distributions reported in Fig. \[Fig\_14\] suggest a power-law behavior of the tails. Recent studies [@mainardi1996fractional; @mainardi1995fractional; @benson2000application; @benson2001fractional] have shown that, for physical phenomena exhibiting this characteristic distribution of the field variables, the governing equations are generalizations to the fractional order of the classical diffusion equation. Power-law distributions, associated with infinite variance random variables (the so called L[é]{}vy flights), are in the domain of attraction of $\alpha$-stable random variables also called L[é]{}vy stable densities (their properties are summarized in Appendix \[Appendix\]). On the other hand finite-variance random variables are in the Normal domain of attraction that is a subset of L[é]{}vy stable densities. This suggests that the trend of the tails carries information about the $\alpha$-stable order of the underlying distribution. In order to show that this situation occurs also in the present case, we performed numerical fits of the acoustic intensity profiles (Fig. \[Fig\_14\]) using $\alpha$-stable distributions. The four parameters defining the $\alpha$-stable distributions were obtained by numerically solving a nonlinear optimization problem. The most important parameter is the characteristic exponent $\alpha$ (also called the index of stability) that is also connected to the slope of the tails. For the square lattice distribution, the values of $\alpha$ determined with the optimization procedure are $\alpha = 0.89$ and $\alpha = 0.57$ for the passband and bandgap excitation wavelengths, respectively. In order to show that the order of the $\alpha$-stable distribution effectively describes the anomalous diffusive dynamics of the system, we use a generalized fractional diffusion equation [@mainardi2007fundamental]: $$\begin{aligned} \label{Fractional diffusion} \begin{split} _{x}\textrm{D}_{\theta}^{\alpha}u(x,t)=_{t}\textrm{D}_{*}^{\beta}u(x,t) \quad x \in \mathbb{R} , \quad t \in \mathbb{R}^+ \end{split}\end{aligned}$$ where $\alpha$, $\theta$, $\beta$ are real parameters always restricted as follows: $$\begin{aligned} \label{Restrictions} \begin{split} 0< \alpha \leq2, \quad |\theta|\leq min\left \{ \alpha,2-\alpha \right \}, 0 <\beta \leq 2. \end{split}\end{aligned}$$ In Eq. (\[Fractional diffusion\]), $u = u(x,t)$ is the field variable, $_{x}\textrm{D}_{\theta}^{\alpha}$ is the *Riesz-Feller* space fractional derivative of order $\alpha$, and $_{t}\textrm{D}_{*}^{\beta}$ is the Caputo time-fractional derivative of order $\beta$. The fractional operator in this equation exhibits a non-local behavior which makes it ideally suited to model dynamical systems dominated by long-range interactions. Mainardi [@mainardi2007fundamental] reported the Green’s function for a Cauchy problem based on the space-time fractional diffusion equation. The self-similar nature of the solution allows the application of a similarity method that separates the solution into a space dependent (the reduced Green’s function $K$) and a time dependent term. In our system, we use a harmonic (constant amplitude) source and we analyze the steady state response, that is we consider a self-similar problem. In other terms, the reduced Green’s function $K$ proposed by Mainardi coincides with the normalized solution of the forced fractional diffusion equation governing our problem: $$\begin{aligned} \label{Reduced Green function} \begin{split} \textrm{K}_{\alpha,\beta}^{\alpha}(x) = \frac{1}{\pi x}\sum_{n=1}^{\infty}\frac{\Gamma (1+\alpha n)}{\Gamma(1+\beta n)}\sin\left [ \frac{n\pi}{2}(\theta-\alpha) \right ](-x^{-\alpha})^n. \end{split}\end{aligned}$$ Note that this solution is valid in the case $\alpha<\beta$. In our case, $\beta=1$ to model a space fractional diffusion equation. Fig. \[Fig\_22\] shows the comparison between the normalized acoustic intensity from the FE numerical data and the result from the reduced Green’s function $K$ (Eq. (\[Reduced Green function\])) calculated for the order $\alpha$ obtained by the previous $\alpha$-stable fits. The above results clearly show that the fractional diffusion equation is able to capture the heavy-tailed behavior of the intensity distribution with good accuracy. They also confirm that the use of $\alpha$-stable fits provides a reliable approach to classify the anomalous behavior and to extract the corresponding fractional order of the operator. The above numerical results provide also further confirmation that the observed dynamic behavior from the full field numerical simulations is in fact dominated by anomalous diffusion. These results are particularly relevant if seen in a perspective of developing predictive capabilities for transport processes in highly inhomogeneous systems. As an example, fractional models would provide an excellent framework for the solution of inverse problems in imaging and remote sensing through highly scattering media. The ability to properly capture a mixed transport behavior, such as partially propagating and diffusive, would allow extracting more information from the measured response therefore improving the sensitivity and resolution of these approaches. From a broader perspective, this methodology has general applicability and could be extended to a variety of applications involving wave-like field transport such as those mentioned in the introduction. Conclusions {#Conclusion} =========== In this paper, we investigated the scattering behavior of sound waves in a perfectly periodic acoustic medium composed of a square lattice of hard cylinders in air. From a general perspective, the most remarkable result consists in the observation of the occurrence of anomalous hybrid transport in perfectly periodic lattice structures without disorder or random properties. This result is particularly relevant because the anomalous response of a scattering system was previously observed only in systems with either stochastic material or geometric properties. By using a combination of theoretical and numerical models, both deterministic and stochastic, it was determined that the existence of long-range interactions associated with the anisotropy of the dispersion bands was the driving factor leading to the occurrence of the anomalous transport behavior. The resulting diffused intensity fields were characterized by heavy-tails with marked asymptotic power-law decay, that were well described by $\alpha$-stable distributions. It was also shown that the $\alpha$-stable nature of the dynamic response provided a reliable approach for the classification and characterization of the non-local effect via the intrinsic parameters of $\alpha$-stable distributions. Observing that $\alpha$-stable distributions represent the fundamental kernel for the solutions of fractional continuum models, we showed that a space fractional diffusion equation having the order predicted by the $\alpha$-stable fit of the acoustic intensity was capable of capturing very accurately the characteristic features of the anomalous transport process. From a general perspective, this approach can be interpreted as a fractional order homogenization of the periodic medium which is capable of mapping the complex inhomogeneous system to a (fractional) governing equation that still accepts an analytical solution. This latter characteristic is particularly remarkable if seen from a practical application perspective because it could open the way to accurate and non-iterative inverse problems that play a critical role in remote sensing, imaging, and material design, just to name a few. Another key observation concerns the strong deviation of the tails of the acoustic intensity from the Gaussian distribution which highlights that much information is still contained in the tails. This aspect is particularly relevant for imaging and sensing in scattering media because traditional analytical methodologies typically assume a Gaussian distribution of the measured intensity field hence leading to two main drawbacks: 1) the loss of important information about the internal structure of the medium which is contained in the tails, and 2) the lack of a proper model capable of extracting and interpreting this information from measured data. Acknowledgments {#acknowledgments .unnumbered} =============== The authors gratefully acknowledge the financial support of the National Science Foundation under the grants DCSD CAREER $\#1621909$ and of the Air Force Office of Scientific Research under Grant No. YIP FA9550-15-1-0133. Appendix A {#Appendix} ========== This appendix summarizes some basic properties of the $\alpha$-stable distributions that have been used to analyze and interpret the simulation data in the paper. The family of $\alpha$-stable distributions are defined by the Fourier transform of their characteristics functions $\psi(w)$ that can be written in the explicit form as[@herranz2004alpha; @benson2002fractional]: $$\begin{aligned} \label{alpha-stab0} \psi(w)=\exp\left \{ i\mu w-\gamma\left | w \right |^\alpha B_{w,\alpha} \right \} \\ B_{w,\alpha}=\left\{\begin{matrix} \left [1+i\beta \operatorname{sgn}(w) \tan\frac{\alpha \pi}{2} \right ] \quad \alpha\neq1 \nonumber \\ \left [1+i\beta \operatorname{sgn}(w) \frac{2}{\pi} \log\left | w \right | \right ] \quad \alpha=1 \end{matrix}\right.\end{aligned}$$ where $0<\alpha\leq 2$, $-1\leq\beta\leq 1$, $\gamma>0$, and $-\infty<\mu<\infty$. The parameters $\alpha$, $\beta$, $\gamma$ and $\mu$ uniquely and completely identify the stable distribution. 1. The parameter $\alpha$ is the *characteristic exponent*, or the *stability parameter*, and it defines the degree of impulsiveness of the distribution. As $\alpha$ decreases the level of impulsiveness of the distribution increases. For $\alpha=2$ we recover the Gaussian distribution. A particular case is obtained for $\alpha=1$ and $\beta=0$ that corresponds to the Cauchy distribution. For $\alpha \notin (0,2]$ the inverse Fourier transform $\psi(w)$ is not positive-definite and hence is not a proper probability density function. 2. The parameter $\beta$ is the *symmetry*, or *skewness parameter*, and determines the skewness of the distribution. Symmetric distributions have $\beta=0$, whereas $\beta=1$ and $\beta=-1$ correspond to completely skewed distributions. 3. The parameter $\gamma$ is the *scale parameter*. It is a measure of the spread of the samples from a distribution around the mean. 4. The parameter $\mu$ is the *location parameter* and corresponds to a shift in the $x$-axis of the pdf. For a symmetric $(\beta=0)$ distribution, $\mu$ is the mean when $1<\alpha\leq 2$ and the median when $0<\alpha\leq 1$. The characteristic functions described in Eq. (\[alpha-stab0\]) are equivalent to a probability density function and do not have analytical solutions except for few special cases. The main feature of these characteristic functions is the presence of heavy-tails when compared to a Gaussian distribution. The probability density functions with tails heavier than Gaussian are also denoted as *impulsive*. An impulsive process is characterized by the presence of large values that significantly deviates from the mean value of the distribution with non-negligible probability. In this sense the $\alpha$-stable distribution represents a generalization of the Gaussian distribution that allows to model impulsive processes by using only four parameters instead of an infinite number of moments. The possibility of describing the distribution of particles in anomalous diffusion phenomena by using $\alpha$-stable distributions has numerous advantages: 1) many methods exist to perform statistical inference on $\alpha$-stable environments [@nikias1995signal; @janicki1993simulation], 2) these distributions are simple because they are completely characterized by only four parameters, 3) the use of $\alpha$-stable distributions finds a theoretical justification in the fact that they satisfy the generalized central limit theorem which states that the limit distribution on infinitely many [i.i.d.]{} random variables, is a stable distribution, 4) they include the Gaussian distribution as a particular case for a specific set of parameters. These distributions are stable since the output of a linear system in response to $\alpha$-stable inputs is again $\alpha$-stable.

ArXiv

--- abstract: 'We propose a new type of hidden layer for a multilayer perceptron, and demonstrate that it obtains the best reported performance for an MLP on the MNIST dataset.' bibliography: - 'strings.bib' - 'strings-shorter.bib' - 'ml.bib' - 'aigaion-shorter.bib' --- The piecewise linear activation function ======================================== We propose to use a specific kind of piecewise linear function as the activation function for a multilayer perceptron. Specifically, suppose that the layer receives as input a vector $x \in \mathbb{R}^D$. The layer then computes presynaptic output $z = x^T W + b$ where $W \in \mathbb{R}^{D \times N}$ and $b \in \mathbb{R}^N$ are learnable parameters of the layer. We propose to have each layer produce output via the activation function $h(z)_i = \text{max}_{j \in S_i} z_j$ where $S_i$ is a different non-empty set of indices into $z$ for each $i$. This function provides several benefits: - It is similar to the rectified linear units [@Glorot+al-AI-2011] which have already proven useful for many classification tasks. - Unlike rectifier units, every unit is guaranteed to have some of its parameters receive some training signal at each update step. This is because the inputs $z_j$ are only compared to each other, and not to 0., so one is always guaranteed to be the maximal element through which the gradient flows. In the case of rectified linear units, there is only a single element $z_j$ and it is compared against 0. In the case when $0 > z_j$, $z_j$ receives no update signal. - Max pooling over groups of units allows the features of the network to easily become invariant to some aspects of their input. For example, if a unit $h_i$ pools (takes the max) over $z_1$, $z_2$, and $z_3$, and $z_1$, $z_2$ and $z_3$ respond to the same object in three different positions, then $h_i$ is invariant to these changes in the objects position. A layer consisting only of rectifier units can’t take the max over features like this; it can only take their average. - Max pooling can reduce the total number of parameters in the network. If we pool with non-overlapping receptive fields of size $k$, then $h$ has size $N / k$, and the next layer has its number of weight parameters reduced by a factor of $k$ relative to if we did not use max pooling. This makes the network cheaper to train and evaluate but also more statistically efficient. - This kind of piecewise linear function can be seen as letting each unit $h_i$ learn its own activation function. Given large enough sets $S_i$, $h_i$ can implement increasing complex convex functions of its input. This includes functions that are already used in other MLPS, such as the rectified linear function and absolute value rectification. Experiments =========== We used $S_i = \{ 5 i, 5 i + 1, ... 5 i + 4 \}$ in our experiments. In other words, the activation function consists of max pooling over non-overlapping groups of five consecutive pre-synaptic inputs. We apply this activation function to the multilayer perceptron trained on MNIST by @Hinton-et-al-arxiv2012. This MLP uses two hidden layers of 1200 units each. In our setup, the presynaptic activation $z$ has size 1200 so the pooled output of each layer has size 240. The rest of our training setup remains unchanged apart from adjustment to hyperparameters. @Hinton-et-al-arxiv2012 report 110 errors on the test set. To our knowledge, this is the best published result on the MNIST dataset for a method that uses neither pretraining nor knowledge of the input geometry. It is not clear how @Hinton-et-al-arxiv2012 obtained a single test set number. We train on the first 50,000 training examples, using the last 10,000 as a validation set. We use the misclassification rate on the validation set to determine at what point to stop training. We then record the log likelihood on the first 50,000 examples, and continue training but using the full 60,000 example training set. When the log likelihood of the validation set first exceeds the recorded value of the training set log likelihood, we stop training the model, and evaluate its test set error. Using this approach, our trained model made 94 mistakes on the test set. We believe this is the best-ever result that does not use pretraining or knowledge of the input geometry.

ArXiv

--- abstract: 'In this paper the stability of a closed-loop cascade control system in the trajectory tracking task is addressed. The considered plant consists of underlying second-order fully actuated perturbed dynamics and the first order system which describes dynamics of the input. The main theoretical result presented in the paper concerns stability conditions formulated based on the Lyapunov analysis for the cascade control structure taking advantage of the active rejection disturbance approach. In particular, limitations imposed on a feasible set of an observer bandwidth are discussed. In order to illustrate characteristics of the closed-loop control system simulation results are presented. Furthermore, the controller is verified experimentally using a two-axis telescope mount. The obtained results confirm that the considered control strategy can be efficiently applied for mechanical systems when a high tracking precision is required.' author: - | Rados[ł]{}aw Patelski, Dariusz Pazderski\ Poznań University of Technology\ Institute of Automation and Robotics\ ul. Piotrowo 3a  60-965 Poznań, Poland bibliography: - 'bibDP.bib' title: 'Tracking control for a cascade perturbed control system using active disturbance rejection paradigm[^1]' --- Introduction ============ Set-point regulation and trajectory tracking constitute elementary tasks in control theory. It is well known that a fundamental method of stabilisation by means of a smooth static state feedback has significant limitations which come, among others, from the inability to measure the state as well as the occurrence of parametric and structural model uncertainties. Thus, for these reasons, various adaptive and robust control techniques are required to improve the performance of the closed-loop system. In particular, algorithms used for the state and disturbance estimation are of great importance here. The use of high gain observers (HGOs) is well motivated in the theory of linear dynamic systems, where it is commonly assumed that state estimation dynamics are negligible with respect to the dominant dynamics of the closed-loop system. A similar approach can be employed successfully for a certain class of nonlinear systems where establishing a fast convergence of estimation errors may be sufficient to ensure the stability, [@KhP:2014]. In a natural way, the HGO observer is a basic tool to support a control feedback when a plant model is roughly known. Here one can mention the free-model control paradigm introduced by Fliess and others, [@Fliess:2009; @FlJ:2013] as well as the active disturbance rejection control (ADRC) proposed by Han and Gao, [@Han:1998; @Gao:2002; @Gao:2006; @Han:2009]. It turns out that the above-mentioned control methodology can be highly competitive with respect to the classic PID technique in many industrial applications, [@SiGao:2005; @WCW:2007; @MiGao:2005; @CZG:2007; @MiH:2015; @NSKCFL:2018]. Furthermore, it can be regarded as an alternative control approach in comparison to the sliding control technique proposed by Utkin and others, [@Utk:77; @Bartol:2008], where bounded matched disturbances are rejected due to fast switching discontinuous controls. Thus, it is possible to stabilise the closed-loop control system, in the sense of Filippov, on a prescribed, possibly time-varying, sliding surface, [@Bart:96; @NVMPB:2012]. Currently, also second and higher-order sliding techniques for control and state estimations are being explored, [@Levant:1993; @Levant:1998; @Bartol:1998; @Cast:2016]. It is noteworthy to recall a recent control algorithm based on higher-order sliding modes to solve the tracking problem in a finite time for a class of uncertain mechanical systems in robotics, [@Gal:2015; @Gal:2016]. From a theoretical point of view, some questions arise regarding conditions of application of control techniques based on a disturbance observer, with particular emphasis on maintaining the stability of the closed-loop system. Recently, new results concerning this issue have been reported for ADRC controllers, [@SiGao:2017; @ACSA:2017]. In this paper we further study the ADRC methodology taking into account a particular structure of perturbed plant. Basically, we deal with a cascade control system which is composed of two parts. The first component is represented by second-order dynamics which constitute an essential part of the plant. It is assumed that the system is fully actuated and subject to matched-type disturbances with bounded partial derivatives. The second component is defined by an elementary first-order linear system which describes input dynamics of the entire plant. Simultaneously, it is supposed that the state and control input of the second order dynamics are not fully available. It can be seen that the considered plant well corresponds to a class of mechanical systems equipped with a local feedback applied at the level of actuators. As a result of additional dynamics, real control forces are not accessible directly which may deteriorate the stability of the closed-loop system. In order to analyse the closed-loop system we take advantage of Lyapunov tools. Basically, we investigate how an extended state observer (ESO) affects the stability when additional input dynamics are considered. Further we formulate stability conditions and estimate bounds of errors. In particular, we show that the observer gains cannot be made arbitrarily large as it is commonly recommended in the ADRC paradigm. Such an obstruction is a result of the occurrence of input dynamics which is not explicitly taken into account in the feedback design procedure. According to the best authors’ knowledge, the Lyapunov stability analysis for the considered control structure taking advantage of the ADRC approach has not been addressed in the literature so far. Theoretical results are illustrated by numerical simulations and experiments. The experimental validation are conducted on a real two-axis telescope mount driven by synchronous gearless motors, [@KPKJPKBJN:2019]. Here we show that the considered methods provide high tracking accuracy which is required in such an application. Additionally, we compare the efficiency of compensation terms, computed based on the reference trajectory and on-line estimates in order to improve the tracking performance. The paper is organised as follows. In Section 2 the model of a cascade control process is introduced. Then a preliminary feedback is designed and a corresponding extended state observer is proposed. The stability of the closed-loop system is studied using Lyapunov tools and stability conditions with respect to the considered control structure are formulated. Simulation results are presented in Section 3 in order to illustrate the performance of the controller. In Section 4 extensive experimental results are discussed. Section 5 concludes the paper. Controller and observer design ============================== Dynamics of a perturbed cascaded system --------------------------------------- Consider a second order fully actuated control system defined as follows $$\left\{ \begin{array}{cl} \dot{x}_{1} & =x_{2},\\ \dot{x}_{2} & =Bu+h(x_{1},x_{2})+q(x_{1},x_{2},u,t), \end{array}\right.\label{eq:general:nominal system}$$ where $x_{1},\,x_{2}\in\mathbb{R}^{n}$ are state variables, $B\in\mathbb{R}^{n\times n}$ is a non-singular input matrix while $u\in\mathbb{R}^{n}$ stands for an input. Functions $h:\mathbb{R}^{2n}\rightarrow\mathbb{R}^{n}$ and $q:\mathbb{R}^{2n}\times\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}^{n}$ denote known and unknown components of the dynamics, respectively. Next, it is assumed that input $u$ in is not directly accessible for a control purpose, however, it is governed by the following first order dynamics $$\dot{u}=T^{-1}\left(-u+v\right),\label{eq:general:input dynamics}$$ where $v\in\mathbb{R}^{n}$ is regarded as a real input and $T\in\mathbb{R}^{n\times n}$ is a diagonal matrix of positive time constants. In fact, both dynamics constitute a cascaded third order plant, for which the underlying component is represented by , while corresponds to stable input dynamics. Control system design --------------------- The control task investigated in this paper deals with tracking of a reference trajectory specified for an output of system - which is determined by $y:=x_1$. Simultaneously, it is assumed that variables $x_2$ and $u$ are unavailable for measurement and the only information is provided by the output. To be more precise, we define at least $C^3$ continuous reference trajectory $x_{d}(t):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and consider output tracking error $\tilde{y}:=x_d-x_1$. Additionally, to quantify a difference between $u$ and $v$, we introduce error $\tilde{u}:=v-u$. Since $v$ is viewed as an alternative input of , one can rewrite as $$\left\{ \begin{array}{cl} \dot{x}_{1} & =x_{2},\\ \dot{x}_{2} & =Bv-B\tilde{u}+h+q. \end{array}\right.\label{eq:general:nominal_system_input_v}$$ For control design purposes, the tracking error will be considered with respect to the state of system . Consequently, one defines $$e = \begin{bmatrix}e_1\\ e_2\end{bmatrix}:=\begin{bmatrix}\tilde{y}\\ e_2\end{bmatrix}=\begin{bmatrix}x_d-x_1\\ \dot{x}_d-x_2\end{bmatrix}\in\mathbb{R}^{2n}.$$ Accordingly, taking time derivative of $e$, one can obtain the following open-loop error dynamics $$\left\{ \begin{array}{cl} \dot{e}_{1} & =e_{2},\\ \dot{e}_{2} & =\ddot{x}_{d}-Bv+B\tilde{u}-h-q. \end{array}\right.\label{eq:general:tracking error dynamics}$$ In order to stabilise system in a vicinity of zero, the following preliminary control law is proposed $$v:=B^{-1}\left(K_{p}\left(x_{d}-\hat{x}_{1}\right)+K_{d}\left(\dot{x}_{d}-\hat{x}_{2}\right)-h_{u}+\ddot{x}_{d}-w_c\right),\label{eq:general:control law}$$ where $K_{p},K_{d}\in\mathbb{R}^{n}$ are diagonal matrices of constant positive gains, $\hat{x}_1\in\mathbb{R}^n$, $\hat{x}_2\in\mathbb{R}^n$ and $w_c\in\mathbb{R}^n$ denote estimates of states and a disturbance, respectively. These estimates are computed by an observer that is not yet defined. Term $h_{u}:\mathbb{R}^{4n}\rightarrow\mathbb{R}^{n}$ is a compensation function, designed in attempt to attenuate influence of $h$ on the closed system dynamics, and is defined using available signals as follows $$h_{u}:=h_{1}(\hat{x}_{1},\hat{x}_{2})+h_{2}(x_{d},\dot{x}_{d}),\label{eq:general:known dynamics compensation}$$ while $h_1$ and $h_2$ satisfy $$h_{1}(x_{1},x_{2})+h_{2}(x_{1},x_{2}):=h.\label{eq:general:known dynamics}$$ Next, in order to simplify design of an observer we rewrite dynamics . Firstly, we consider a new form which does not introduce any change to the system dynamics and is as follows $$\left\{ \begin{array}{cl} \dot{x}_{1} & =x_{2},\\ \dot{x}_{2} & =Bu+h_u+h-h_{u}+q. \end{array}\right.\label{eq:general:nominal system rewritten}$$ Secondly, according to active disturbance rejection methodology, it is assumed that $$z_{3}:=q+h-h_{u}$$ describes an augmented state which can be regarded as a total disturbance. Correspondingly, one can introduce extended state $z=\begin{bmatrix}z_{1}^{T} & z_{2}^{T} & z_{3}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{3n}$, where $z_1:=x_1$ and $z_2:=x_2$. As a result, the following extended form of dynamics can be established $$\left\{ \begin{array}{cl} \dot{z}_{1} & =z_{2},\\ \dot{z}_{2} & =Bu+h_{u}+z_{3},\\ \dot{z}_{3} & =\dot{q}+\dot{h}-\dot{h}_{u}. \end{array}\right.\label{eq:general:extended system-1}$$ Now, in order to estimate state $z$ we define the following Luenberger-like observer $$\left\{ \begin{array}{cl} \dot{\hat{z}}_{1} & =K_{1}\left(z_{1}-\hat{z}_{1}\right)+\hat{z}_{2},\\ \dot{\hat{z}}_{2} & =K_{2}\left(z_{1}-\hat{z}_{1}\right)+\hat{z}_{3}+h_{u}+Bv,\\ \dot{\hat{z}}_{3} & =K_{3}\left(z_{1}-\hat{z}_{1}\right), \end{array}\right.\label{eq:general:observer}$$ where $\hat{z}=\begin{bmatrix}\hat{z}_{1}^{T} & \hat{z}_{2}^{T} & \hat{z}_{3}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{3n}$ denotes estimate of $z$ and $K_{1},K_{2},K_{3}\in\mathbb{R}^{n\times n}$ are diagonal matrices of positive gains of the observer which are chosen based on linear stability criteria. Since estimates $\hat{z}$ are expected to converge to real values of $z$, let observation errors be expressed as $\tilde{z}:=z-\hat{z}$. Taking time derivative of $\tilde{z}$, using , and recalling one obtains the following dynamics $$\dot{\tilde{z}}=H_{o}\tilde{z}+C_{0}B\tilde{u}+C_{1}\dot{z}_{3}\label{eq:general:observator error dynamics}$$ where $$\label{eq:general:Ho_def} H_{o}=\begin{bmatrix}-K_{1} & I & 0\\ -K_{2} & 0 & I\\ -K_{3} & 0 & 0 \end{bmatrix}\in\mathbb{R}^{3n\times 3n},$$ $$C_{0}=\begin{bmatrix}0& -I& 0\end{bmatrix}^T,\ C_{1}=\begin{bmatrix}0& 0& I \end{bmatrix}^T\in\mathbb{R}^{3n},$$ while $I$ stands for the identity matrix of size $n\times n$. Here, it is required that $H_{o}$ is Hurwitz, what can be guaranteed by a proper choice of observer gains. Next, we recall tracking dynamics and feedback . It is proposed that compensating term in , which partially rejects unknown disturbances, is defined by an estimate provided by observer , namely $w_c:=\hat{z}_3$. Consequently, by substituting into the following is obtained $$\dot{e}=H_{c}e+W_{1}\tilde{z}+C_{2}B\tilde{u},\label{eq:general:regulator error dynamics}$$ where $$\label{eq:general:Hc_def} H_{c}=\begin{bmatrix}0 & I\\ -K_{p} & -K_{d} \end{bmatrix},W_{1}=\begin{bmatrix}0 & 0 & 0\\ -K_{p} & -K_{d} & -I \end{bmatrix},\ C_{2}=\begin{bmatrix}0\\ I \end{bmatrix}\in\mathbb{R}^{2n\times n}$$ and $H_{c}$ is Hurwitz for $K_{p}\succ 0$ and $K_{d}\succ 0$. Further, in order to facilitate the design and analysis of the closed-loop system, we take advantage of a scaling operator defined by $$\Delta_m\left(\alpha\right):=\mathrm{diag}\left\{\alpha^{m-1}I,\, \alpha^{m-2}I,\, \ldots,\, I\right\}\in\mathbb{R}^{mn\times mn},$$ where $\alpha>0$ is a positive scalar. Then we define the following scaled tracking and observation errors $$\begin{aligned} \bar{e}:=&\left(\kappa\omega\right)^{-1}\Delta_2\left(\kappa\omega\right)e,\label{eq:general:regulator auxiliary errors}\\ \bar{z}:=&\omega^{-2}\Delta_3\left(\omega\right)\tilde{z},\label{eq:general:observer auxiliary errors} \end{aligned}$$ where $\omega\in\mathbb{R}_{+}$ is scaling parameter which modifies the bandwidth of the the observer, while $\kappa\in\mathbb{R}_{+}$ denotes a relative bandwidth of the feedback determined with respect to $\omega$. Embracing this notation one can introduce the following scaled gains $$\label{eq:design:scaled_gains} \bar{K}_c:=\left(\kappa\omega\right)^{-1}K_c \Delta_2^{-1}\left(\kappa\omega\right),\, \bar{K}_o:=\omega^{-3}\Delta_3\left(\omega\right) \left[K_1^T\ K_2^T\ K_3^T\right]^T,$$ while $K_c:=\left[K_p\ K_d\right]\in\mathbb{R}^{n\times 2n}$. Additionally, exploring relationships outlined in the Appendix, one can rewrite dynamics and as follows $$\begin{aligned} \dot{\bar{e}}=&\kappa\omega\bar{H}_{c}\bar{e}+\kappa^{-1}\omega\bar{W}_{1}\Delta_3\left(\kappa\right)\bar{z}+\left(\kappa\omega\right)^{-1}C_{2}B\tilde{u},\label{eq:general:regulator auxiliary error dynamics}\\ \dot{\bar{z}}=&\omega\bar{H}_{o}\bar{z}+\omega^{-1} C_{0}B\tilde{u}+\omega^{-2} C_{1}\dot{z}_{3},\label{eq:general:observator auxiliary error dynamics}\end{aligned}$$ with $\bar{H}_c$ and $\bar{H}_o$ being Hurwitz matrices of forms , defined in terms of scaled gains $\bar{K}_c$ and $\bar{K}_o$, respectively. Similarly, $\bar{W}_1$ corresponds to $W_1$ parameterised by new gains. Since $\bar{H}_c$ and $\bar{H}_o$ are Hurwitz, one can state that the following Lyapunov equations are satisfied $$\bar{P}_c\bar{H}_c^{T}+\bar{H}_c\bar{P}_c=-\bar{Q}_c,\ \bar{P}_o\bar{H}_o^{T}+\bar{H}_o\bar{P}_o=-\bar{Q}_o \label{eq:general:Lyapunov equation}$$ for some symmetric, positive defined matrices $\bar{Q}_c,\, \bar{P}_c\in\mathbb{R}^{2n\times 2n}$ and $\bar{Q}_o,\, \bar{P}_o\in\mathbb{R}^{3n\times 3n}$. Stability analysis of the closed-loop cascaded control system ------------------------------------------------------------- Lyapunov stability of the closed-loop is to be considered now. For this purpose, a state which consists of tracking, observation and input errors is defined as $$\bar\zeta=\begin{bmatrix}\bar{e}^T&\bar{z}^T&\tilde{u}^T\end{bmatrix}^T\in\mathbb{R}^{6n}.\label{eq:general:stability:errors}$$ A positive definite function is proposed as follows $$V(\bar{\zeta})=\frac{1}{2}\bar{e}^{T}\bar{P}_c\bar{e}+\frac{1}{2}\bar{z}^{T}\bar{P}_o\bar{z}+\frac{1}{2}\tilde{u}^{T}\tilde{u}.\label{eq:general:stability:lyapunov proposition}$$ Its derivative takes form of $$\begin{aligned} \dot{V}(\bar{\zeta})=&-\frac{1}{2}\kappa\omega\bar{e}^{T}\bar{Q}_c\bar{e}-\frac{1}{2}\omega\bar{z}^{T}\bar{Q}_o\bar{z}+\kappa^{-1}\omega \bar{e}^T \bar{P}_c\bar{W}_1\Delta_3\left(\kappa\right)\bar{z}+\left(\kappa\omega\right)^{-1}\bar{e}^{T}\bar{P}_c C_2B\tilde{u}+\omega^{-1}\bar{z}^T\bar{P}_o C_o B\tilde{u}\\&+\omega^{-2}\bar{z}^{T}\bar{P}_o C_1\dot{z}_{3}-\tilde{u}^{T}T^{-1}\tilde{u}+\tilde{u}^{T}\dot{v}.\label{eq:general:stability:lyapunov derivative} \end{aligned}$$ Derivative of control law $v$ defined by can be expressed in terms of $\bar{\zeta}$ as (the details are outlined in the Appendix) $$\dot{v}=B^{-1}\left(\omega^{3}\left(\kappa^3\bar{K}_c\bar{H}_c\bar{e}+\left(\kappa\bar{K}_c\bar{W}_1 \Delta_3\left(\kappa\right)+\bar{W}_2\Delta_3\left(\kappa\right)\bar{H}_o\right)\bar{z}\right)-\dot{h}_u+\dddot{x}_{d}\right),\label{eq:general:stability:control law derivative}$$ where $\bar{K}_c:=\left[\bar{K}_p\ \bar{K}_d\right]\in\mathbb{R}^{n\times 2n}$ and $\bar{W}_2:=\left[\bar{K}_c\ I \right]\in\mathbb{R}^{n\times{3n}}$. Substituting (\[eq:general:stability:control law derivative\]) and $\dot{z}_{3}$ into (\[eq:general:stability:lyapunov derivative\]) leads to $$\begin{aligned} \dot{V}(\bar{\zeta})=&-\frac{1}{2}\kappa\omega\bar{e}^{T}\bar{Q}_c\bar{e}-\frac{1}{2}\omega\bar{z}^{T}\bar{Q}_o\bar{z}+\kappa^{-1}\omega\bar{e}^T\bar{P}_c\bar{W}_{1}\Delta_3\left(\kappa\right)\bar{z}+\left(\kappa\omega\right)^{-1}\bar{e}^{T}\bar{P}_c C_2B\tilde{u}+\omega^{-1}\bar{z}^T\bar{P}_o C_o B\tilde{u}\\&+\left(\kappa\omega\right)^3 \tilde{u}^{T}B^{-1}\bar{K}_c\bar{H}_c\bar{e}+\omega^3\tilde{u}^{T}B^{-1}\left(\kappa\bar{K}_c\bar{W}_1 \Delta_3\left(\kappa\right)+\bar{W}_2\Delta_3\left(\kappa\right)\bar{H}_o\right)\bar{z}-\tilde{u}^{T}T^{-1}\tilde{u}\\ &+\tilde{u}^{T}B^{-1}\dddot{x}_d+\tilde{u}^{T}B^{-1}\dot{h}_u+\omega^{-2}\bar{z}^{T}\bar{P}_o C_1\left(\dot h-\dot{h}_u\right)+\omega^{-2}\bar{z}^{T}\bar{P}_o C_1\dot{q}(z_{1},z_{2},u,t).\label{eq:general:stability:lyapunov derivative split} \end{aligned}$$ In order to simplify the stability analysis, derivative $\dot{V}$ will be decomposed into four terms defined as follows $$\begin{aligned} Y_1:=&-\frac{1}{2}\kappa\omega\bar{e}^{T}\bar{Q}_c\bar{e}-\frac{1}{2}\omega\bar{z}^{T}\bar{Q}_o\bar{z}+\kappa^{-1}\omega\bar{e}^T\bar{P}_c\bar{W}_{1}\Delta_3\left(\kappa\right)\bar{z}+\left(\kappa\omega\right)^{-1}\bar{e}^{T}\bar{P}_c C_2B\tilde{u}+\omega^{-1}\bar{z}^T\bar{P}_o C_o B\tilde{u}\\&+\left(\kappa\omega\right)^3 \tilde{u}^{T}B^{-1}\bar{K}_c\bar{H}_c\bar{e}+\omega^3\tilde{u}^{T}B^{-1}\left(\kappa\bar{K}_c\bar{W}_1 \Delta_3\left(\kappa\right)+\bar{W}_2\Delta_3\left(\kappa\right)\bar{H}_o\right)\bar{z}-\tilde{u}^{T}T^{-1}\tilde{u},\\ Y_2:=& \tilde{u}^{T}B^{-1}\dddot{x}_d,\,Y_3:=\tilde{u}^{T}B^{-1}\dot{h}_u+\omega^{-2}\bar{z}^{T}\bar{P}_o C_1\left(\dot h-\dot{h}_u\right),\,Y_4:= \omega^{-2}\bar{z}^{T}\bar{P}_o C_1\dot{q}(z_{1},z_{2},u,t). \end{aligned}$$ Each term of $\dot{V}$ will be now considered separately. Firstly, $Y_{1}$ which represents mainly influence of input dynamics on the nominal system will be looked upon. Negative definiteness of this term will be a starting point for further analysis of the closed-loop stability. Let it be rewritten using the matrix notation as $$Y_{1} =-\frac{1}{2}\omega\bar{\zeta}^{T}Q_{Y1}\bar{\zeta},\label{eq:general:stability:Y1}$$ where $$\begin{split} Q_{Y1}=\left[\begin{matrix}\kappa\bar{Q}_c &-\kappa^{-1}\bar{P}_c\bar{W}_1\Delta_3\left(\kappa\right)&Q_{Y1_{13}}\\-\kappa^{-1}\left(\bar{P}_c\bar{W}_1\Delta_3\left(\kappa\right)\right)^T&\bar{Q}_o&Q_{Y1_{23}}\\ Q_{Y1_{13}}^T&Q_{Y1_{23}}^T&2\omega^{-1} T^{-1} \end{matrix}\right]\in\mathbb{R}^{6n\times 6n} \end{split}$$ while $$\begin{aligned} Q_{Y1_{13}} =& -\kappa^{-1}\omega^{-2}\bar{P}_c C_2B-\kappa^3\omega^2\left(B^{-1}\bar{K}_c\bar{H}_c\right)^T,\\ Q_{Y1_{23}}=&-\omega^{-2}\bar{P}_o C_o B-\omega^2\left(B^{-1}\left(\kappa\bar{K}_c\bar{W}_1 \Delta_3\left(\kappa\right)+\bar{W}_2\Delta_3\left(\kappa\right)\bar{H}_o\right)\right)^T. \end{aligned}$$ It can be showed, that there may exist sets $\Omega_v, \mathrm{K}_v \subset \mathbb{R}_{+}$, such, that for every $\omega\in\Omega_v$ and $\kappa\in\mathrm{K}_v$ matrix $Q_{Y1}$ remains positive definite. Domains of both $\Omega_v$ and $\mathrm{K}_v$ strongly depend on inertia matrix $T$ and input matrix $B$ of nominal system. In the absence of other disturbances system would remain asymptotically stable for such a choice of both $\omega$ and $\kappa$ parameters. Influence of other elements of $\dot{V}(\bar{\zeta})$ will be considered in terms of upper bounds which can be imposed on them. \[assu:desired trajectory\]Let desired trajectory $x_{d}$ be chosen such, that norms of $x_{d},\dot{x}_{d},\ddot{x}_{d},\dddot{x}_{d}$ are bounded by, respectively, constant positive scalar values $x_{b0},x_{b1},x_{b2},x_{b3}\in\mathbb{R}_{+}$. Establishing upper bound for norm of $Y_{2}$ is straightforward by using Cauchy-Schwartz inequality. $$\begin{aligned} Y_{2} & =-\tilde{u}^{T}B^{-1}\dddot{x}_{d},\nonumber \\ \left\Vert Y_{2}\right\Vert & \leq\left\Vert \tilde{u}\right\Vert \cdot\left\Vert B^{-1}\dddot{x}_{d}\right\Vert \nonumber \\ & \leq\left\Vert \bar{\zeta}\right\Vert \left\Vert B^{-1}\right\Vert x_{b3}.\label{eq:general:stability:Y2}\end{aligned}$$ Now, $Y_{3}$ is to be considered. This term comes from imperfect compensation of known dynamics in nominal system and it can be further split into the following $$Y_{31}:=\omega^{-2}\bar{z}^{T}P_{o}C_{1}\left(\dot{h}-\dot{h}_{u}\right), Y_{32}:=\tilde{u}^{T}B^{-1}\dot{h}_{u}.\label{eq:general:stability:Y3}$$ \[assu:bounded dynamics\]Let functions $h_{1}(a,b)$ and $h_{2}(a,b)$ be defined such, that norms of partial derivatives\ $\frac{\partial}{\partial a}h_{1}(a,b)$,$\frac{\partial}{\partial b}h_{1}(a,b)$,$\frac{\partial}{\partial a}h_{2}(a,b)$,$\frac{\partial}{\partial b}h_{2}(a,b)$ are bounded for every $a,b\in\mathbb{R}^{n}$ by $h_{1a},h_{1b,}h_{2a,}h_{2b}\in\mathbb{R}_{+}$ respectively. By applying chain rule to calculate derivatives of each function and substituting difference of error and desired trajectory for state variables, term $Y_{31}$ can be expressed as $$Y_{31}=\omega^{-2}\bar{z}^{T}P_{o}C_{1}\left(W_{h1}\begin{bmatrix}\dot{x}_{d}\\\ddot{x}_{d}\end{bmatrix} - W_{h2}\left(\kappa\omega\bar{H}_{c}\bar{e}+\kappa^{-1}\omega\bar{W}_{1}\Delta_3(\kappa)\bar{z}+\left(\kappa\omega\right)^{-1}C_{2}B\tilde{u}\right)+W_{h3}\left(\omega\bar{H}_{o}\bar{z}+\omega^{-1}C_{0}B\tilde{u}\right)\right),\label{eq:general:stability:Y31 equation}$$ where $$\begin{aligned} W_{h1} & =\begin{bmatrix}\left(\frac{\partial h_{1}}{\partial z_{1}}+\frac{\partial h_{2}}{\partial z_{1}}-\frac{\partial h_{2}}{\partial x_{d}}-\frac{\partial h_{1}}{\partial\hat{z}_{1}}\right) & \left(\frac{\partial h_{1}}{\partial z_{2}}+\frac{\partial h_{2}}{\partial z_{2}}-\frac{\partial h_{2}}{\partial\dot{x}_{d}}-\frac{\partial h_{1}}{\partial\hat{z}_{2}}\right)\end{bmatrix}, \nonumber \\ W_{h2} & =\begin{bmatrix}\left(\frac{\partial h_{1}}{\partial z_{1}}+\frac{\partial h_{2}}{\partial z_{1}}-\frac{\partial h_{1}}{\partial\hat{z}_{1}}\right) & \kappa\omega\left(\frac{\partial h_{1}}{\partial z_{2}}+\frac{\partial h_{2}}{\partial z_{2}}-\frac{\partial h_{1}}{\partial\hat{z}_{2}}\right)\end{bmatrix}, \nonumber \\ W_{h3} & =\begin{bmatrix}\frac{\partial h_{1}}{\partial\hat{z}_{1}} & \omega\frac{\partial h_{1}}{\partial\hat{z}_{2}} & 0\end{bmatrix}. \nonumber\end{aligned}$$ This term can be said to be bounded by $$\begin{aligned} \left\Vert Y_{31}\right\Vert \leq & \omega^{-2}\left\Vert \bar{\zeta}\right\Vert \left\Vert P_{o}C_{1}\right\Vert \left(\left(2h_{1a}+2h_{2a}\right)x_{b1}+\left(2h_{1b}+2h_{2b}\right)x_{b2}\right) \nonumber \\ & +\left\Vert \bar{\zeta}\right\Vert ^{2}\left\Vert P_{o}C_{1}\right\Vert \left(\omega^{-1}\kappa\left\Vert W_{h2b}\right\Vert \left(\left\Vert \bar{H}_{c}\right\Vert +\left\Vert \bar{W}_{1}\right\Vert \right)+\omega^{-3}\kappa^{-1}\left\Vert W_{h2b}\right\Vert \left\Vert C_{2}B\right\Vert\right) \nonumber \\ & +\left\Vert \bar{\zeta}\right\Vert ^{2}\left\Vert P_{o}C_{1}\right\Vert \left(\omega^{-1}\left\Vert W_{h3b}\right\Vert \left\Vert \bar{H}_{o}\right\Vert +\omega^{-2}\left\Vert B\right\Vert h_{1b} \right). \label{eq:general:stability:Y31 bound}\end{aligned}$$ where $W_{h2b}=\begin{bmatrix}2h_{1a} + h_{2a} & \kappa\omega\left(2h_{1b} + h_{2b}\right)\end{bmatrix}$ and $W_{h3b} = \begin{bmatrix}h_{1a} & \omega h_{1b} & 0 \end{bmatrix}$. Having established upper bound of $Y_{31}$, we can perform similar analysis with respect to $Y_{32}$. Let $Y_{32}$ be rewritten as $$Y_{32} = \tilde{u}^T B^{-1} \left(W_{h4}\begin{bmatrix}\dot{x}_{d}\\\ddot{x}_{d}\end{bmatrix} - W_{h5}\left(\kappa\omega\bar{H}_{c}\bar{e}+\kappa^{-1}\omega\bar{W}_{1}\Delta_3(\kappa)\bar{z}+\left(\kappa\omega\right)^{-1}C_{2}B\tilde{u}\right)-W_{h6}\left(\omega\bar{H}_{o}\bar{z}+\omega^{-1}C_{0}B\tilde{u}\right)\right), \label{eq:general:stability:Y32 equation}$$ where $$\begin{aligned} W_{h4} & =\begin{bmatrix}\left(\frac{\partial h_{2}}{\partial x_{d}}+\frac{\partial h_{1}}{\partial \hat{z}_{1}}\right) & \left(\frac{\partial h_{2}}{\partial \dot{x}_{d}}+\frac{\partial h_{1}}{\partial \hat{z}_{2}}\right)\end{bmatrix}, \nonumber \\ W_{h5} & =\begin{bmatrix}\frac{\partial h_{1}}{\partial \hat{z}_{1}} & \kappa\omega\frac{\partial h_{1}}{\partial \hat{z}_{2}}\end{bmatrix}, \nonumber \\ W_{h6} & = W_{h3}. \nonumber \end{aligned}$$ An upper bound of norm of $Y_{32}$ can be expressed by the following inequality $$\begin{aligned} \left\Vert Y_{32} \right\Vert \leq & \omega^{-2}\left\Vert \bar{\zeta}\right\Vert \left\Vert \bar{P}_{o}C_{1}\right\Vert \left(q_{z1}x_{b1}+q_{z2}x_{b2}+\left\Vert B\right\Vert q_{z2}+\left\Vert T^{-1}\right\Vert q_{u}+\left\Vert \bar{P}_{o}C_{1}\right\Vert q_{t}\right) \nonumber \\ & +\kappa\omega^{-1}\left\Vert \bar{\zeta}\right\Vert ^{2}\left\Vert \bar{P}_{o}C_{1}\right\Vert \left\Vert W_{q2}\right\Vert \left(\left\Vert \bar{H}_{c}\right\Vert +\left\Vert \bar{W}_{1}\right\Vert \right)\label{eq:general:stability:Y_32 bound}\end{aligned}$$ where $W_{h5b} = \begin{bmatrix}h_{1a} & \kappa\omega h_{1b}\end{bmatrix}$ and naturally $W_{h6b}=W_{h3b}$. A remark can be made now about the structure of $W_{h2}$, $W_{h3}$, $W_{h4}$ and $W_{h5}$. It may be recognized, that elements of these matrices can be divided into group of derivatives calculated with respect to the first and the second argument. Former of these are not scaled by either observer or regulator bandwidth, while the latter is scaled by either $\kappa\omega$ or $\omega$ factor. As will be showed later in the analysis, this difference will have significant influence on the system stability and ability of the controller to reduce tracking errors. Lastly, some upper bound need to be defined for $Y_{4}$ to complete the stability analysis. This final term comes from nominal disturbance $q(z_1, z_2, u, y)$ alone. By chain rule it can be shown that $$Y_{4} = \omega^{-2}\bar{z}^{T}\bar{P}_{o}C_{1}\left(W_{q1}\begin{bmatrix}\dot{x}_{d}\\\ddot{x}_{d}\end{bmatrix}+\kappa\omega W_{q2}\bar{H}_{c}\bar{e}+\kappa^{-1}\omega W_{q2}\bar{W}_{1}\Delta_3\left(\kappa\right)\bar{z}+\left(\kappa\omega\right)^{-1}W_{q2}C_{2}B\tilde{u}-\frac{\partial q}{\partial u}T^{-1}\tilde{u}+\frac{\partial q}{\partial t}\right),\label{eq:general:stability:Y4}$$ where $W_{q1}=\begin{bmatrix}\frac{\partial q}{\partial z_1} & \frac{\partial q}{\partial z_2}\end{bmatrix}$ and $W_{q2} = \begin{bmatrix}\frac{\partial q}{\partial z_1} & \kappa\omega\frac{\partial q}{\partial z_2}\end{bmatrix}$. \[assu:disturbance derivatives\]Let partial derivatives $\frac{\partial}{\partial z_{1}}q(z_{1},z_{2},u,t),\frac{\partial}{\partial z_{2}}q(z_{1},z_{2},u,t),\frac{\partial}{\partial u}q(z_{1},z_{2},u,t),\frac{\partial}{\partial t}q(z_{1},z_{2},u,t)$ be defined in the whole domain and let their norms be bounded by constants $q_{z1},q_{z2},q_{u}$ and $q_{t}\in\mathbb{R}_{+}$, respectively. Under Assumption \[assu:disturbance derivatives\] the norm of $Y_{4}$ is bounded by $$\begin{aligned} \left\Vert Y_{4}\right\Vert & \leq\omega^{-2}\left\Vert \bar{\zeta}\right\Vert \left\Vert \bar{P}C\right\Vert \left(q_{z1}x_{b1}+q_{z2}x_{b2}\right)+\omega^{-1}\left\Vert \bar{\zeta}\right\Vert ^{2}\left\Vert \bar{P}CW_{5b}\right\Vert \left(\left\Vert \bar{H}\right\Vert +\omega^{-2}\left\Vert \bar{C}B\right\Vert \right)\label{eq:general:stability:Y4 bound}\\ & +\omega^{-2}\left\Vert \bar{\zeta}\right\Vert ^{2}\left\Vert \bar{P}C\right\Vert \left(\left\Vert T^{-1}\right\Vert q_{u}+q_{t}\right).\nonumber \end{aligned}$$ With some general bounds for each of $\dot{V}(\bar{\zeta})$ terms established, conclusions concerning system stability can be finally drawn. For the sake of convenience, let some auxiliary measure of Lyapunov function derivative negative definiteness $\Lambda_V$ and Lyapunov function derivative perturbation $\Gamma_V$ be defined as $$\begin{aligned} \Lambda_V := & \frac{1}{2}\omega\lambda_{\min}(Q_{Y1}) -\kappa\omega\left\Vert B^{-1}\right\Vert \left\Vert W_{h5b}\right\Vert \left(\left\Vert \bar{H}_{c}\right\Vert +\left\Vert \bar{W}_{1}\right\Vert \right)-\omega\left\Vert B^{-1}\right\Vert \left\Vert W_{h6b}\right\Vert \left\Vert \bar{H}_{o}\right\Vert \nonumber \\ & -2h_{1b}-\omega^{-1}\left\Vert W_{h3b}\right\Vert \left\Vert \bar{H}_{o}\right\Vert -\kappa\omega^{-1}\left\Vert P_{o}C_{1}\right\Vert \left(\left\Vert W_{h2b}\right\Vert +\left\Vert W_{q2}\right\Vert \right)\left(\left\Vert \bar{H}_{c}\right\Vert +\left\Vert \bar{W}_{1}\right\Vert \right) \nonumber \\ & -\omega^{-2}\left\Vert B\right\Vert h_{1b}-\omega^{-3}\kappa^{-1}\left\Vert W_{h2b}\right\Vert \left\Vert C_{2}B\right\Vert, \label{eq:general:stability:Lyapunov negative definiteness} \\ \Gamma_V := & \left\Vert B^{-1}\right\Vert \left\Vert \left(h_{2a}+h_{1a}\right)x_{b1}+\left(h_{2b}+h_{1b}\right)x_{b2}\right\Vert \nonumber \\ & \omega^{-2}\left\Vert \bar{P}_{o}C_{1}\right\Vert \left(q_{z1}x_{b1}+q_{z2}x_{b2}+\left\Vert B\right\Vert q_{z2}+\left\Vert T^{-1}\right\Vert q_{u}+\left\Vert \bar{P}_{o}C_{1}\right\Vert q_{t}\right), \label{eq:general:stability:Lyapunov perturbation}\end{aligned}$$ where $\lambda_{\min}(Q)$ stands for the smallest eigenvalue of matrix $Q$, then upper bound of $\dot{V}_{\bar\zeta}(\bar\zeta)$ can be expressed as $$\dot{V}_{\bar\zeta} \leq -\Lambda_V\left\Vert \bar\zeta \right\Vert^2 + \Gamma_V\left\Vert \bar\zeta \right\Vert. \label{eq:general:stability:Lyapunov derivative bound}$$ Now, following conditions can be declared 1. \[enum:general:stability:condition 1\] $\omega\in\Omega_v, \kappa\in\mathrm{K}_v$, 2. \[enum:general:stability:condition 2\] $\Gamma_V \geq 0$, and succeeding theorem concludes presented analysis. Perturbed cascade system (\[eq:general:nominal system\])-(\[eq:general:input dynamics\]) satisfying Assumptions \[assu:desired trajectory\]-\[assu:disturbance derivatives\], controlled by feedback (\[eq:general:control law\]) which is supported by extended state observer (\[eq:general:observer\]), remains practically stable if there exist symmetric, positive defined matrices $Q_o$ and $Q_c$ such, that conditions \[enum:general:stability:condition 1\] and \[enum:general:stability:condition 2\] can be simultaneously satisfied. Scaled tracking errors $\bar\zeta$ are then bounded as follows $$\label{eq:control:conclusion} \lim_{t\rightarrow\infty}\left\Vert \bar{\zeta}(t)\right\Vert\leq \frac{\Gamma_{V}}{\Lambda_{V}}.$$ Foregoing proposition remains valid only if Assumptions \[assu:desired trajectory\]-\[assu:disturbance derivatives\] are satisfied. While Assumption \[assu:desired trajectory\] considers desired trajectory only and can be easily fulfilled for any system with state $x_1$ defined on $\mathbb{R}^n$, a closer look at the remaining assumptions ought to be taken now. Similar in their nature, both concern imperfectly known parts of the system dynamics, with the difference being whether an attempt to implicitly compensate these dynamics is taken or not. As a known dynamic term satisfying Assumption \[assu:bounded dynamics\] can also be treated as an unknown disturbance, without a loss of generality, only Assumption \[assu:disturbance derivatives\] has to be commented here. It can be noted, that for many commonly considered systems this assumption cannot be satisfied. A mechanical system equipped with revolute kinematic pairs can be an example of such system, which dynamics, due to Coriolis and centrifugal forces, have neither bounded time derivative nor bounded partial derivative calculated with respect to second state variable. Engineering practice shows nonetheless that for systems, in which cross-coupling is insignificant enough due to a proper mass distribution, this assumption can be approximately satisfied, at least in a bounded set of the state-space, and the stability analysis holds. The requirement that partial derivatives of any disturbance in the system should be bounded is restrictive one, yet less conservative than commonly used in the ADRC analysis expectation of time derivative boundedness. In this sense, the presented analysis is more liberal than ones considered in the literature and it can be expected that enforced assumptions can be better justified. Numerical simulations ===================== In attempt to further research behaviour of the system in the presence of unmodelled dynamics governing the input signal numerical simulations have been conducted. Model of the system has been implemented in Matlab-Simulink environment. The second order, single degree of freedom system and the first order dynamics of the input have been modelled according to the following equations $$\left\{ \begin{array}{cl} \dot{x}_{1} & =x_{2},\\ \dot{x}_{2} & =u, \end{array}\right.\label{eq:simulation:system}$$ where $$\dot{u}=\frac{1}{T}\left(-u+v\right)\label{eq:simulation:input}$$ and $v$ is a controllable input of the system. Parameters $T$ and $\omega$ of the controller were modified in simulations to investigate how they affect the closed-loop system stability and the tracking accuracy. Chosen parameters of the system are presented in the table \[tab:simulation:gains\]. Desired trajectory $x_d$ was selected as a sine wave with unitary amplitude and frequency of $\unit[\frac{10}{2\pi}]{Hz}$. $\bar{K}_{1}$ $\bar{K}_{2}$ $\bar{K}_{3}$ $\bar{K}_{p}$ $\bar{K}_{d}$ $\kappa$ --------------- --------------- --------------- --------------- --------------- ---------- $3$ $3$ $1$ $1$ $2$ $0.01$ : Auxiliary gains of the observer and controller\[tab:simulation:gains\] Selected results of simulations are presented in Figs. \[fig:simulation:T01 adrc\]-\[fig:simulation:T1 pd\]. Tracking errors of two state variables are presented on the plots. Error of $x_{1}$ is presented with solid line, while $e_{2}$ has been plotted with dashed lines on each figure. Integrals of squared errors $e_1$ (ISE criterion) and integral of squared control signals $v$ (ISC criterion) have been calculated for each simulation and are presented above the plots to quantify obtained tracking results. Tests were performed for different values of $T$ and $\omega$, as well as for compensation term $w_c=\hat{z}_{3}$ enabled or disabled, cf. . It can be clearly seen that the existence of some upper bound of $\Omega$ is confirmed by simulation results as proposed by Eq. . As expected, value of this bound decreases with increase of time constant $T$. In the conducted simulations it was not possible to observe and confirm existence of any lower bound imposed on $\Omega$ and for an arbitrarily small $\omega$ stability of the system was being maintained. Secondly, an influence of disturbance rejection term $\hat{z}_{3}$ is clearly visible and is twofold. For $\omega$ chosen to satisfy stability condition \[enum:general:stability:condition 1\], it can be observed, that the presence of the disturbance estimate allows significant decreasing of tracking errors $e_2$ caused by the input dynamics which were not modelled during the controller synthesis. Basically, a residual value of error $e_2$ becomes smaller for a higher value of bandwidth $\omega$. Error trajectory $e_1$ is also slightly modified, however, this effect is irrelevant according to ISE criterion. Nonetheless, usage of the disturbance estimate leads to a significant shrink of $\Omega$ subset. It is plainly visible, that removal of $\hat{z}_{3}$ estimate may lead to recovering of stability of the system in comparison with simulation scenarios obtained using the corresponding ADRC controller. Experimental results ==================== Practical experiments have been undertaken in order to further investigate the considered control problem. All experiment were carried out using robotic telescope mount developed at Institute of Automatic Control and Robotic of Poznan University of Technology, [@KPKJPKBJN:2019]. The plant consists of a robotic mount and an astronomic telescope with a mirror of diameter 0.5 m. The robotic mount alone includes two axes driven independently by $\unit[24]{V}$ permanent magnet synchronous motors (PMSM) with high-precision ring encoders producing absolute position measurement with 32-bit resolution. Control algorithms has been implemented in C++ using Texas Instruments AM4379 Sitara processor with ARM Cortex-A9 core clocked at $\unit[600]{MHz}$. Beside control structure, prepared firmware contains several additional blocks necessary for conducting of proper astronomical research. Controller itself is implemented in a cascade form which consists of independent current and position loops. Both loops work simultaneously with frequency of $\unit[10]{kHz}$. The current loop designed to precisely track desired torque of the motor employs Park-Clark transformation of measured phase currents to express motor dynamics in *q-d* coordinated. Both *q* and *d* axes are then controlled by independent PI regulators with feedforward term and anti-windup correction which satisfy the following equation $$\begin{aligned}\dot{v} & =k_{i}\left(\tilde{i}-k_{s}\left(k_{p}\tilde{i}+v+u_{r}-\mathrm{sat}\left(k_{p}\tilde{i}+v+u_{r}\right)\right)\right),\\ u & =\mathrm{sat}\left(k_{p}\tilde{i}+v+u_{r},U_{m}\right), \end{aligned} \label{eq:experiments:current loop}$$ where $\tilde{i}$ stands for current tracking error, $v$ is integrator input signal, $u$ is regulator input, $u_{r}$ expresses feedforward term, $k_{p}$, $k_{i}$ and $k_{s}$ are positive regulator gains, and finally $\mathrm{sat}(u^{*},U_{m})$ is saturation function of signal $u^{*}$ up to value of $U_{m}$. Output voltage $u$ is generated using PWM output. Current in $d$ axis is stabilised at zero, while current of *q* axis tracks the desired current of the axis. Relation between desired torque and desired current is modelled as a constant gain equals $\unit[2.45]{\frac{Nm}{A}}$. Desired torque is computed in the position loop by the active disturbance rejection based controller designed for the second order mechanical system modelled as follows $$\left\{ \begin{array}{cl} \dot{x}_{1} & =x_{2}\\ \dot{x}_{2} & =B\tau+\smash{\underbrace{f_{c}\cdot\mathrm{tanh}(f_{t}\cdot x_{2})}_{h(x_{2})}}, \end{array}\right.\label{eq:experiment:model}\\*[0.625\normalbaselineskip]$$ where $x_{1}\in\mathbb{R}^2$ and $x_{2}\in\mathbb{R}^2$ are position and velocities of axes, $B$ is input matrix with diagonal coefficients equal $B_{1,1}=\frac{1}{5},B_{2,2}=\frac{1}{30}$, $f_c$ is the constant positive Coulomb friction coefficient while $f_{t}=10^{3}$ expresses scaling term which defines steepness of friction model. Velocity of the axis is approximated in the experiments using either observer estimate $\hat{z}_{2}$ or desired trajectory derivative $\dot{x}_{d}$. The assumed model of the friction force is strongly local, in the sense that different values of $f_{c}$ are required for different accelerations in a time instant when the sign of velocity changes. This locality was overcame during the experiments by manual changes of $f_{c}$ coefficient. While torque generated by the motor is treated as an input signal of the mechanical system, there exists residual dynamics defined by the current loop which is not modelled in the position loop. Here, we assume that this dynamics can be approximated by and thus we can infer about the stability according to mathematical analysis considered in Section 2. Other disturbances come chiefly from flexibility of the mount, ignored cross-coupling reactions between joints and torque ripples generated by synchronous motors. Though some of these disturbances globally do not satisfy assumptions accepted for theoretical analysis of the system stability, in the considered scenario an influence of these dynamics is insignificant. Due to small desired velocities chosen in the experiment, these assumptions can be approximately satisfied here. All gains of the controllers chosen for experiments are collected in Table \[tab:experiment:gains\]. Horizontal axis Vertical axis --------- ------------------ ------------------- $K_{1}$ $1.2\cdot10^{3}$ $2.4\cdot10^{2}$ $K_{2}$ $5.7\cdot10^{5}$ $2.28\cdot10^{4}$ $K_{3}$ $10^{8}$ $0.8\cdot10^{6}$ $K_{p}$ $225$ $225$ $K_{d}$ $24$ $24$ : \[tab:experiment:gains\]Gains of the controllers and observers Here we present selected results of the experiments. In the investigated experimental scenarios both axes were at move simultaneously and the desired trajectory was designed as a sine wave with period of $\unit[30]{s}$ and maximum velocity of $50v_{s}$, in the first experiment, and $500v_{s}$, in the second, where $v_{s}=\unit[7.268\cdot10^{-5}]{\frac{rad}{s}}$ stands for the nominal velocity of stars on the night sky. During the system operation significant changes of friction forces are clearly visible and the influence of compensation term can be easily noticed. Since friction terms vary significantly around zero velocity the tracking accuracy is decreased. In such a case the process of the disturbance estimation is not performed fast enough. Furthermore, in the considered application one cannot select larger gains of the observer due to additional dynamics imposed by an actuator and delays in the control loop. Here, one can recall relationship which clearly states that the tracking precision is dependent on the bound of $\Gamma_V$, cf. . Thus, one can expect that the tracking accuracy increases in operating conditions when disturbances become slow-time varying. This is well illustrated in experiments where friction terms change in a wide range. Each experiment presents results obtained with different approaches to $h_{u}$ term design. Once again integral squared error was calculated for each of the presented plots to ease evaluation of the obtained results. Series of conclusions can be drawn from the presented results. Due to inherently more disturbed dynamics of horizontal axis, any improvement using friction compensation for slow trajectories is hardly achieved. Meanwhile, the compensation term based on the desired trajectory, effectively decreases tracking error bound for all other experiments. As may be expected, compensation function based on estimates of state variables is unable to provide any acceptable tracking quality due to inherent noise in the signal and the existence of input dynamics. It can be noted, that in the first experiment the friction compensation term allows one to decrease the bound of tracking error while overall quality expressed by ISE criterion is worse in comparison to this obtained in experiment without the corresponding term in the feedback. This behaviour is not seen in the second experiment, in which significant improvement was obtained for both axes in terms of error boundary as well as ISE criterion. Conclusions =========== This paper is focused on the application of ADRC controller to a class of second order systems subject to differentiable disturbances. In particular, the system is analysed taking into account the presence of the first order input dynamics and unmodelled terms which may include cross-coupling effects between the state variables. By the means of Lyapunov analysis, general conditions of practical stability are discussed. It is proved that, even in the presence of additional input dynamics, boundedness of partial derivatives of total disturbance can be a sufficient requirement to guarantee stability of the closed-loop system. Using numerical simulations the considered controller is compared against a simple PD-based regulator. The obtained results confirm that in the case of input dynamics, the bandwidth of an extended observer is limited which restricts the effectiveness of the ADRC approach. Lastly, practical results of employing ADRC regulator in the task of trajectory tracking for a robotised astronomical telescope mount are presented. In this application, it is assumed that friction effects are modelled inaccurately and a local drive control-loop is treated as unknown input dynamics. The obtained results illustrate that the considered control algorithm can provide a high tracking accuracy. Further research in this topic may include attempts to explore in more details conditions for the feasible selection of the observer parameters in order to guarantee the stability of the closed-loop system. Other forms of input dynamics and observer models can also be considered in the future works. Appendix ======== Selected properties of scaled dynamics -------------------------------------- Assuming that errors and gains are scaled according to , and the following relationships are satisfied: $$\begin{aligned} \Delta_2\left(\kappa\omega\right)H_c \Delta_2^{-1}\left(\kappa\omega\right) =\kappa\omega\bar{H}_c,\ \Delta_3\left(\omega\right)H_o \Delta_3^{-1}\left(\omega\right) =\omega\bar{H}_o,\\ \Delta_2\left(\kappa\omega\right)W_1=W_1,\, W_1\Delta_3^{-1}\left(\omega\right)=\bar{W}_1\Delta_3\left(\kappa\right)\\ W_2=\Delta_3\left(\kappa\omega\right)\bar{W}_2.\label{eq:app:scalled_terms} \end{aligned}$$ Computation of $\dot{v}$ ------------------------ Taking advantage of estimate $\bar{z}$ and assuming that $w_c:=\hat{z}_3$ one can rewrite as follows $$v=B^{-1}\left(K_c e + K_c \begin{bmatrix}\tilde{z}_1^T&\tilde{z}_2^T \end{bmatrix}^T-h_u+\ddot{x}_d-\hat{z}_3\right),$$ where $K_c := \left[K_p\ K_d\right]$. Equivalently, one has $$v=B^{-1}\left(K_c e - W_2\tilde{z}-h_u+\ddot{x}_d-z_3\right).$$ Consequently, time derivative of $v$ satisfies $$\begin{aligned} \dot{v}&=B^{-1}\left(K_c \dot{e}+W_2\dot{\tilde{z}}-\dot{h}_u+\dddot{x}_d-\dot{z}_3\right){\stackrel{(\ref{eq:general:regulator error dynamics}),(\ref{eq:general:observator error dynamics})}{=}}B^{-1}\left(K_c H_ce+K_cW_1\tilde{z}+K_c C_2B\tilde{u}+W_2H_o\tilde{z}\right.\\ &\quad\left.+W_2C_oB\tilde{u}+W_2C_1\dot{z}_3-\dot{z}_3-\dot{h}_u+\dddot{x}_d\right)=B^{-1}\left(K_c H_ce+K_cW_1\tilde{z}+W_2H_o\tilde{z}-\dot{h}_u+\dddot{x}_d\right). \end{aligned}$$ Computations of $Y_3$ and $Y_4$ ------------------------------- By chain rule it can be shown that $$\dot{h}_1(z_1, z_2) = \begin{bmatrix}\frac{\partial h_{1}}{\partial z_{1}} & \frac{\partial h_{1}}{\partial z_{2}}\end{bmatrix}\begin{bmatrix}\dot{x}_{d}\\ \ddot{x}_{d} \end{bmatrix}-\begin{bmatrix}\frac{\partial h_{1}}{\partial z_{1}} & \kappa\omega\frac{\partial h_{1}}{\partial z_{2}}\end{bmatrix}\dot{\bar{e}},\ \dot{h}_2(z_1,z_2) = \begin{bmatrix}\frac{\partial h_{2}}{\partial z_{1}} & \frac{\partial h_{2}}{\partial z_{2}}\end{bmatrix}\begin{bmatrix}\dot{x}_{d}\\ \ddot{x}_{d} \end{bmatrix}-\begin{bmatrix}\frac{\partial h_{2}}{\partial z_{1}} & \kappa\omega\frac{\partial h_{2}}{\partial z_{2}}\end{bmatrix}\dot{\bar{e}},$$ $$\dot{h}_1(\hat{z}_1,\hat{z}_2)=\begin{bmatrix}\frac{\partial h_{1}}{\partial\hat{z}_{1}} & \frac{\partial h_{1}}{\partial\hat{z}_{2}}\end{bmatrix}\begin{bmatrix}\dot{x}_{d}\\ \ddot{x}_{d} \end{bmatrix}-\begin{bmatrix}\frac{\partial h_{1}}{\partial\hat{z}_{1}} & \kappa\omega\frac{\partial h_{1}}{\partial\hat{z}_{2}}\end{bmatrix}\dot{\bar{e}}-\begin{bmatrix}\frac{\partial h_{1}}{\partial\hat{z}_{1}} & \omega\frac{\partial h_{1}}{\partial\hat{z}_{2}} & 0\end{bmatrix}\dot{\bar{z}},\ \dot{h}_2(x_d,\dot{x}_d)=\begin{bmatrix}\frac{\partial h_{2}}{\partial x_{d}} & \frac{\partial h_{2}}{\partial\dot{x}_{d}}\end{bmatrix}\begin{bmatrix}\dot{x}_{d}\\ \ddot{x}_{d} \end{bmatrix}.$$ From here, following are true $$\begin{aligned} \dot{h} - \dot{h}_u &= W_{h1}\begin{bmatrix}\dot{x}_{d}\\ \ddot{x}_{d} \end{bmatrix}-W_{h2}\dot{\bar{e}}+W_{h3}\dot{\bar{z}},\ \dot{h}_u = W_{h4}\begin{bmatrix}\dot{x}_{d}\\ \ddot{x}_{d} \end{bmatrix}-W_{h5}\dot{\bar{e}}-W_{h6}\dot{\bar{z}}, \end{aligned}$$ what leads to solution of $Y_3$ by means of basic substitution. Now, the computation of term $Y_4$ will be taken into account. Disturbance term $q(z_{1},z_{2},u,t)$ can be expressed in form of $$\begin{aligned} \dot{q}(z_{1},z_{2},u,t)&=\frac{\partial q}{\partial z_{1}}\left(\dot{x}_{d}-\dot{e}_{1}\right)+\frac{\partial q}{\partial z_{2}}\left(\ddot{x}_{d}-\dot{e}_{2}\right)+\frac{\partial q}{\partial u}T^{-1}\left(-u+v\right)+\frac{\partial q}{\partial t}\\ &=W_{q1}\begin{bmatrix}\dot{x}_{d}\\ \ddot{x}_{d} \end{bmatrix}+W_{q2}\dot{\bar{e}}-\frac{\partial q}{\partial u}T^{-1}\tilde{u}+\frac{\partial q}{\partial t}\\ &=W_{q1}\begin{bmatrix}\dot{x}_{d}\\ \ddot{x}_{d} \end{bmatrix}+W_{q2}\left(\kappa\omega\bar{H}_{c}\bar{e}+\kappa^{-1}\omega\bar{W}_{1}D(\kappa)\bar{z}+\left(\kappa\omega\right)^{-1}C_{2}B\tilde{u}\right)-\frac{\partial q}{\partial u}T^{-1}\tilde{u}+\frac{\partial q}{\partial t}. \end{aligned}$$ [^1]: This work was supported by the National Science Centre (NCN) under the grant No. 2014/15/B/ST7/00429, contract No. UMO-2014/15/B/ST7/00429.

ArXiv

--- abstract: 'Star formation in galaxies is triggered by a combination of processes, including gravitational instabilities, spiral wave shocks, stellar compression, and turbulence compression. Some of these persist in the far outer regions where the column density is far below the threshold for instabilities, making the outer disk cutoff somewhat gradual. We show that in a galaxy with a single exponential gas profile the star formation rate can have a double exponential with a shallow one in the inner part and a steep one in the outer part. Such double exponentials have been observed recently in the broad-band intensity profiles of spiral and dwarf Irregular galaxies. The break radius in our model occurs slightly outside the threshold for instabilities provided the Mach number for compressive motions remains of order unity to large radii. The ratio of the break radius to the inner exponential scale length increases for higher surface brightness disks because the unstable part extends further out. This is also in agreement with observations. Galaxies with extended outer gas disks that fall more slowly than a single exponential, such as $1/R$, can have their star formation rate scale approximately as a single exponential with radius, even out to 10 disk scale lengths. H$\alpha$ profiles should drop much faster than the star formation rate as a result of the rapidly decreasing ambient density.' author: - 'Bruce G. Elmegreen' - 'Deidre A. Hunter' title: Radial Profiles of Star Formation in the Far Outer Regions of Galaxy Disks --- Introduction ============ The outer disks of spiral galaxies have a low level of star formation (Ferguson et al. 1998; LeLièvre & Roy 2000; Cuillandre, et al. 2001; de Blok & Walter 2003; Thilker et al. 2005; Gil de Paz et a. 2005), even though the gas is gravitationally stable by the Kennicutt (1989) condition. Triggering by other mechanisms, such as turbulence compression (Mac Low & Klessen 2004), supernovae, and extragalactic cloud impacts (Tenorio-Tagle 1981), might be the reason. As a result, radial light profiles should not drop suddenly at the stability threshold, but should taper slowly as various star formation processes get more and more unlikely and the gas supply diminishes. The purpose of this paper is to investigate a simple model of star formation with generalized triggering in a smoothly varying gas disk. We seek to determine what the overall radial light profile might be. The radial light profiles of spiral and irregular galaxies are typically exponential over 3 to 5 scale lengths (van der Kruit 2001) with rare examples, particularly among low-inclination spirals, going further (Courteau 1996; Barton & Thompson 1997; Weiner et al. 2001; Erwin, Pohlen, & Beckman 2005; Bland-Hawthorn et al. 2005). Some galaxies have another, steeper exponential in the inner disk bulge region (Courteau, de Jong & Broeils 1996), which does not concern us here as it may be the result of gas inflow or bar formation (Kormendy & Kennicutt 2004). Many galaxies also have a steep exponential in the far outer disk (Näslund & Jörsäter 1997; de Grijs, Kregel, & Wesson 2001; Pohlen et al. 2002). This outer exponential is the focus of our discussion. As the outer disk is significantly below the sky brightness and generally difficult to observe, its properties are not well known; it may not even be exponential. van der Kruit (1988) suggested that disk asymmetries can make what is really a sharp outer truncation appear much smoother when the light profiles are azimuthally averaged; he noted that very deep exposures of edge-on disks tend to show sharp edges instead of smooth outer exponentials. Florido et al. (2001) showed how a sharp function could be fitted to outer disk cutoffs. The outer disk profile also depends critically on the level and uniformity of the sky brightness subtracted from the image. The transition from the main disk exponential to the outer disk profile has several observed characteristics. The outer disk scale length is about half that of the inner disk for both spiral and dwarf irregular galaxies (Hunter & Elmegreen 2006, hereafter Paper I). The ratio of the transition, or “break,” radius, $R_{br}$, to the main disk scale length, $R_D$, is 3 to 4 for spiral galaxies (van der Kruit & Searle 1981; Barteldrees & Dettmar 1994; Pohlen, Dettmar, & Lütticke 2000; Schwarzkopf & Dettmar 2000; Kregel, van der Kruit & de Grijs 2002) and $\sim2$ for dwarf and spiral Irregulars (Paper I). There is a slight increase in this ratio for decreasing $R_D$ among spirals (Pohlen, Dettmar, & Lütticke 2000; Kregel, van der Kruit & de Grijs 2002; Kregel & van der Kruit 2004), and another slight increase for increasing central surface brightness among spirals (Kregel & van der Kruit 2004). The first of these two correlations does not hold for dwarf Irregulars, which have both small disk scale lengths and small ratios $R_{br}/R_D$. The second correlation does hold for dwarf Irregulars. If there is a universal reason for outer disk transitions (as in the present model), then correlations which apply to both spirals and irregulars would seem to be most important. Thus the second correlation, in which $R_{br}/R_D$ increases with central surface brightness, should be viewed as fundamental, and the first simply a result of the second along with the independent correlation between scale length and central surface brightness found by de Jong (1996) and Beijersbergen, de Blok, & van der Hulst (1999). The apparent ratio $R_{br}/R_D$ should also depend slightly on galaxy inclination as a result of a tendency to overestimate $R_D$ for edge-on spirals where central extinction flattens the radial profile. Exponential light profiles in galaxies have been attributed to several things. Cosmological collapse during galaxy formation, starting with a nearly uniform spheroid, can produce profiles that resemble exponentials out to $\sim 2-6$ scale-lengths (Freeman 1970; Fall & Efstathiou 1980). Exponential disks also arise through radial flows in viscously evolving disks if the star formation rate is proportional to the viscosity (e.g., Lin & Pringle 1987; Yoshii & Sommer-Larsen 1989; Zhang & Wyse 2000; Ferguson & Clarke 2001). Double exponential profiles have no previous explanation (see review in Pohlen et al. 2004). van der Kruit (1987) proposed that outer disk truncations arise during galaxy formation and the break radius is determined by the maximum angular momentum of the proto-galactic cloud. Kennicutt (1989) suggested that truncation arises where the gas disk drops below the threshold for gravitational instabilities. Elmegreen & Parravano (1994) and Schaye (2004) proposed it arises when the ISM converts to a mostly warm phase, as observed in the outer regions of spirals (Dickey, Hanson & Helou 1990; Braun 1997) and dwarfs (Young & Lo 1996, 1997). Dalcanton et al. (1997), Firmani & Avila-Reese (2000), Van den Bosch (2001), Abadi, et al. (2003), Governato et al. (2004) and Robertson et al. (2004, 2005) simulated galaxy formation with threshold star formation and obtained exponential profiles with an outer disk cutoff. None of these models actually obtained double exponentials, only sharp outer disk truncations. The theory of disk truncation is highly uncertain, however. The angular momentum in the outer parts of a galaxy can change over time during interactions. The gravitational stability threshold may not be sharp if the ISM cools (Elmegreen 1991) or magnetic forces remove angular momentum (Kim, Ostriker & Stone 2002) during compression. The phase transition may not occur if the outer gas disk tapers slowly, like $1/R$ (Wolfire et al. 2003). All of these uncertainties suggest that refined models may eventually obtain more gradual outer disk truncations. The presence of double exponentials in dwarf galaxies (Paper I) places immediate constraints on the models. Most dwarfs have nearly solid body rotation curves throughout a large fraction of their optical disks. This means there is little shear, so viscous evolution should not play a significant role in structuring disk profiles. There is also no correlation in our Paper I sample between the break radius and the radius where the rotation curve changes from near solid body in the inner regions to near flat in the outer regions. Thus even the outer exponential is not likely to result from radial migration and evolution related to shear. Collapse models could in principle be arranged to give the desired radial profiles, but the collapse models in cosmological simulations so far have just given inner exponential disks with relatively sharp outer cutoffs. There have been no suggestions yet about how conditions during galaxy formation could be tuned to give outer double exponentials. One possibility is that galaxy collapse gives a single exponential disk and then subsequent accretion of gas makes the far-outer disk with a different profile (Bottema 1996). This may explain a sudden decrement in the rotation speed at the optical disk edge of NGC 4013 (Bottema 1995; see also van der Kruit 2001), but the decrement could also come from the prominent warp in that galaxy. If the outer disk is accreted, then there is no obvious reason why the ratio of outer to inner scale lengths should be about the same from galaxy to galaxy, including the dwarfs (Paper I). Here we consider a star formation model that includes turbulence and other compressions as cloud formation mechanisms, in addition to spontaneous gravitational instabilities (see also Kravtsov 2003). Observations of dwarf galaxies have shown that star formation is not simply regulated by a threshold column density (see review in Paper I). Star formation clearly occurs in clouds that stand above the threshold even if the average column density is below the threshold, and it persists far out in the outer disks of dwarfs as it does in spirals. It has also become clear that the ISM in both dwarfs and spiral galaxies is highly structured into clouds of all sizes, presumably as a result of turbulence and other processes. For the dwarf galaxies, this conclusion follows from the log-normal shape of the probability density function of H$\alpha$ emission (Hunter & Elmegreen 2004), and from the power-law power spectra of H$\alpha$ and stellar emissions (Willett, Elmegreen & Hunter 2005). The same power laws for star formation are seen in spiral galaxies (Elmegreen, Elmegreen, & Leitner 2003; Elmegreen et al. 2003). Dwarfs also show power law or fractal structure in the HI gas, as in the M81 dwarfs (Westpfahl et al. 1999) and in the Small and Large Magellanic Clouds (Stanimirovic et al. 1999; Elmegreen, Kim & Staveley-Smith 2001). The same is observed in local HI (e.g., Dickey et al. 2001). All of these distributions resemble the characteristics of compressible turbulence as illustrated in simulations (see review in Elmegreen & Scalo 2004). These considerations lead to a model for star formation in a turbulent, self-gravitating medium. This model is more general than the instability model alone as it allows for more processes, including pressurized triggering of star formation, turbulence triggering, spiral density wave triggering, and swing-amplified instabilities. It should be useful for predictions of outer disk star formation rates and for semi-analytical models of star formation in cosmological studies. Multi-component Model of Star Formation ======================================= Many of the general properties of galaxy disks and star formation can be combined into a relatively simple model that gives the star formation rate as a function of radius. These properties lead to the basic assumptions of the model, as listed here: - Galaxies form with a smoothly distributed gas disk having an outer cutoff, as usually seen in cosmology simulations. This cutoff will enter the present discussion as the outermost point of the disk, significantly beyond any break radius that may appear. We assume in some models that the smooth gas disk is a single exponential, although other forms will have the same basic properties. It will be significant that the star formation rate takes an approximately double exponential profile even in a gas disk that is a single exponential. Other models assume an exponential gas disk with an outer $1/R$ extension. In these cases, the star formation profile can be a continuous exponential or a double exponential with the outer part flatter then the inner part. In all cases, the star formation profile will drop faster than the gas profile, but it will rarely truncate suddenly. - The ISM is turbulent and partly stirred by pressures related to existing stars. This means the velocity dispersion tends to decrease slightly with radius as the stellar disk decreases exponentially. Such a velocity dispersion decrease is observed for some spiral galaxies (Boulanger & Viallefond 1992). Theoretical discussions of the radial profile of gaseous velocity dispersion are in Jog & Narayan (2005). Equating the energy densities of turbulence and stellar energy input, this gives approximately $M^2\propto e^{-R/R_D}$ for Mach number $M$, radius $R$, and stellar exponential disk scale-length $R_D$. The precise form of this relation is not important to the model; other cases considered below use a constant Mach number and get about the same result. The exponential form assumes the gas density for stirring by supernovae and other stellar pressures is about constant with radius, as appropriate for the HI medium in the main disks of galaxies. Then the Mach number alone responds to the stellar energy density. This is in rough agreement with observations showing greater HI velocity dispersions for cool HI clouds near stellar associations (Braun 2005). This equation also emphasizes that the important Mach number for our model is the one that regulates the first step of cloud formation, i.e., the conversion of ambient gas into dense cloud complexes where stars form. Such emphasis places turbulence on an equal footing with large-scale gravitational instabilities. There may be other processes governing the radial profile of the Mach number inside individual dense clouds and the Mach number for the mass-weighted ISM as a whole. - The Mach number reaches a minimum value near unity in the outer disk as a result of either a transition to a warm-dominant thermal HI phase or a sustained low level of turbulence (Sellwood & Balbus 1999). In either case, cool clouds are still possible in the compressed regions, but turbulence compression is weak. Combined with the previous point, this means $$M^2\approx1+Ae^{-R/R_d}\label{eq:mach}$$ where $A$ is the square of the effective Mach number in the inner disk. We assume in some models below that $A=100$; the results do not depend on this value as long as the main part of the inner disk is Toomre unstable. The most important point for the model is that some level of turbulence remains in the outer disk so that turbulence-induced compression makes clouds even where the average disk is Toomre-stable. Thus $A=0$ gives acceptable results too. For the models shown below, Equation \[eq:mach\] will be used with either $A=100$ or $A=0$; a more detailed treatment might have the coefficient $A$ depend on the SFR per unit gas mass, or on other processes related to interstellar turbulence. - Isothermal turbulence produces clouds with a log-normal distribution of column density, as observed in simulations by Padoan et al. (2000), Ostriker et al. (2001), and Vázquez-Semadeni & García (2001). Then the probability of a region having a local column density $\Sigma_g$ is $$P(\Sigma_g)d\ln\Sigma_g=P_0 \exp\left(-0.5\left[\ln \Sigma_g/\Sigma_p\right]^2 /\sigma^2\right)d\ln \Sigma_g.$$ The column density at the peak of this distribution, $\Sigma_p$, will be determined at each radius to give the appropriate average column density (see below). The dispersion of the log-normal may scale with the Mach number of the turbulence, $$\sigma=\left(\ln\left[1+0.5M^2\right]\right)^{1/2} \label{eq:disp}$$ as in simulations by Nordlund & Padoan (1999). The log-normal is consistent with the pixel-to-pixel distribution of H$\alpha$ intensity in Im galaxies (Hunter & Elmegreen 2004). These last two points (with $A>1$) make the ISM more clumpy in the inner regions than in the outer regions. For the unstable inner part of the disk, this clumpiness does not matter much for the star formation rate because it is relatively easy for $\Sigma_g$ to exceed $\Sigma_c$ and also because the instabilities themselves drive turbulence and cloudy structure. In the stable outer parts, however, the turbulence-formed clumps and any outward propagating spirals from the inner disk are the primary regions where $\Sigma_g>\Sigma_c$ and star formation occurs only in them. This makes star formation very patchy in outer disks, and it proceeds at a low average rate. The rate is not zero even though the average gas column density, $<\Sigma_g>$, is significantly less than $\Sigma_c$ because star formation persists in the tail of the $P(\Sigma_g)$ function. The log-normal form for $P\left(\Sigma_g\right)$ is not critical for the double exponential radial profile. It is used here primarily for convenience and because of its presumed connection with turbulence. The important point is that $P\left(\Sigma_g\right)$ has a tail at high $\Sigma_g$ that gets wider with increasing Mach number, and that some low level of turbulence compression remains in the gravitationally stable outer disk. - The critical column density for gravity to overcome Coriolis and pressure forces is the Toomre value appropriate for gas. The general concept that galaxy edges result from below-threshold $\Sigma_g/\Sigma_c$ dates back to Fall & Efstathiou (1980), Quirk (1972), Zasov & Simakov (1988) and Kennicutt (1989). We write $\Sigma_c$ here in terms of the epicyclic frequency $\kappa$ and the Mach number $M$ instead of the velocity dispersion, $$\Sigma_c=C M\kappa/\left(\pi G\right). \label{eq:thres}$$ The constant of proportionality, C (units of velocity dispersion), absorbs the fixed rms speed and effective adiabatic index that is in the usual expression because we replaced the dispersion with the radial-varying Mach number. In our model, $C$ determines where the break radius might occur in the original exponential; whether it breaks or not depends also on the run of Mach number with radius. The instability condition does not indicate only the onset of swing-amplified or shear instabilities in thin disks, or the onset of ring instabilities, as originally devised by Safronov (1960) and Toomre (1964). It is also the condition for the stability of giant expanding shells (Elmegreen, Palous, & Ehlerova 2002) and most likely relevant to the collapse of turbulence-compressed regions too (Elmegreen 2002). This is because all of these processes involve gravity, rotation, and pressure. When $\Sigma_g>\Sigma_c$, gravity overcomes the Coriolis force during the contraction of the largest cloud that is initially in pressure-gravity equilibrium. The origin of the cloud does not matter. If $\Sigma_g<\Sigma_c$, then Coriolis forces disrupt collapsing spiral arms, expanding shells, turbulence-compressed clouds, and ISM structures before much star formation begins in them. Thus, the Toomre condition should be a general condition for star formation, independent of the detailed triggering processes, which may be quite varied (Elmegreen 2002). The situation is the same if the ambient medium cools during the compression, but then $\Sigma_c$ should be set equal to $C \gamma_{eff}^{1/2} M\kappa/\left(\pi G\right)$ for effective adiabatic index $\gamma_{eff}=c^{-2}dP/d\rho$ (Elmegreen 1991), considering pressure $P$, density $\rho$, and velocity dispersion $c$. Most likely this occurs in the outer disks where compression can convert warm HI into cool diffuse clouds. We do not consider this additional factor here. - The star formation rate is proportional to some power of the [*local*]{} gas column density $\Sigma_g$ when the threshold is exceeded. A 1.4 power was observed by Kennicutt (1998) for a wide range of conditions. A lower limit to the power is $\sim1$, which also fits the data in some models (Boissier et al. 2003; Gao & Solomon 2004). Note that these power law observations differ significantly from what one would get for the Toomre instability alone, where the maximum growth rate, $\kappa\left(\Sigma_g^2/\Sigma_c^2-1\right)^{1/2},$ increases from zero rapidly as $\Sigma_g$ begins to exceed $\Sigma_c$, and then asymptotically levels off to a dependence on $\Sigma_g^1$. This makes the star formation rate per unit area, which is $\Sigma_g$ times the growth rate, proportional to $\Sigma_g$ raised to a power greater than or equal to 2. If $\Sigma_g/\Sigma_c\sim1.5,$ for example, then the star formation rate should scale with $\Sigma_g^{2.8}$. The Toomre condition alone is not appropriate for star formation because all of the other dynamical processes that are involved (such as turbulence driven by young stars and thermal cooling inside compressed regions) change both $\Sigma_g$ and $\Sigma_c$ locally. The Toomre condition assumes an isothermal uniform gas. If this isothermal assumption is relaxed, then galaxy disks can become unstable for a wider range of conditions (e.g., Elmegreen 1991). The origin of the observed power law is not fully understood, but it is probably related to star formation processes that operate at the local dynamical rate in a medium that is structured by turbulence (Elmegreen 2002). It should follow naturally from a full hydrodynamical model that includes these effects (Kravtsov 2003; Li, Mac Low & Klessen 2005). These points incorporate the main processes that are believed to be involved with galactic-scale star formation: ISM turbulence, pressurized shell formation and other pressurized triggering, thermal equilibrium, and general gravitational instabilities. Turbulence and other compressions make the disk cloudy and this cloudy structure persists in the outer disk even where $<\Sigma_g>$ is less than $\Sigma_c$, allowing star formation to continue at large radii. The decrease in the Mach number means the cloudiness decreases with radius, so the ISM becomes less turbulent and more smooth in the outer regions, as observed for HI (Braun 1997). This combination of cloud-forming turbulence with a Mach number that converges asymptotically to near-unity, along with cloud-forming instabilities that become decreasingly important with radius, produces the transition from an inner near-exponential to an outer near-exponential in our model. The outer exponential is where the disk is Toomre-stable and the Mach number is of order unity. Both of these conditions are satisfied at about the same place when a clear double exponential appears (see below). If there are spiral arms in the outer disk, even if they are generated in the inner disk and radiate dissipatively to the outer disk, then this model should not change much because these spirals provide only one more possible source of cloudy structure and triggered star formation. As long as the gas becomes gradually less compressive in the outer regions, the star formation rate tapers off smoothly until the physical edge of the disk (or ionized edge of the gas) is reached. Thus the Mach number in our model should be interpreted as the ratio of rms bulk motion to sound speed, regardless of whether the bulk motions occur in spiral shocks, turbulence, or pressurized shells. Figure \[fig-sfr\] shows radial profiles of various quantities from models based on these principles. The models calculate the results in radial steps of $dR=0.1$ (arbitrary units) for a disk exponential scale length of $R_D=2.5$ and an outer disk cutoff of 20. At each radius, $R$, the average gas column density is determined from the initial exponential, $<\Sigma_g>=\Sigma_{g0}e^{-R/R_D}$, the Mach number is determined from Equation \[eq:mach\], and the dispersion of the probability distribution function for local column density is determined from the Mach number using Equation \[eq:disp\]. Then the peak column density in this distribution, $\Sigma_p$, is determined from the integral over $\Sigma_g P\left(\Sigma_g\right)$ by setting the average column density that results from this integral equal to $<\Sigma_g>$. The threshold column density, $\Sigma_c$, is also determined at this radius, from Equation \[eq:thres\]. After this setup for the average quantities, the model makes clouds and determines star formation rates. The local column density is determined by randomly sampling from the distribution function $P\left(\Sigma_g\right)$, and then the star formation rate is set equal to this local column density raised to a power of 1 or 1.5 in the two cases shown, provided the local column density exceeds $\Sigma_c$. If the local column density is less than $\Sigma_c$, then the star formation rate at this position is set to zero. To adequately sample the random assignments of column densities, we consider a number of azimuthal points at each radius equal to $R/dR$. That is, the size of a cloud is assumed to be constant with radius. When $R/dR$ random column densities and resulting star formation rates are determined at each $R$, we average together these column densities and star formation rates to give the plotted quantities. Figure \[fig-sfr\] shows results for a rotation curve appropriate for most galaxies (the rotation curve affects only $\kappa$). This rotation curve is rising in the inner part and flat in the outer part: $V = V_0 (r/R_D)/ [ 1+ r/R_D]$. The star formation rate is shown on the left with dashed lines tracing two exponential profiles to guide the eye. Other quantities for the same models are shown on the right: critical column density, $\Sigma_c$ (magenta), Mach number (black dashed), average gas surface density, $<\Sigma_g>$ (red), and local gas column density, $\Sigma_g$ (green). Five cases are considered. In the top two and bottom two panels, the star formation rate scales with the local column density to the 1.5 power, while in the middle panel, the rate scales with $\Sigma_g$ to the first power. The difference is that when SFR$\propto\Sigma_g^{1.5}$, the inner exponential in star formation is steeper than the inner exponential in gas; otherwise the SFR and the gas have the same profiles. The break radius does not depend noticeably on whether the SFR scales with $\Sigma$ or $\Sigma^{1.5}$. The bottom two panels show the difference between models with high and low critical column densities. When $\Sigma_g/\Sigma_c$ is lower (bottom panel), less of the disk is unstable and the break radius is smaller. This is consistent with our observation that the relative break radius is smaller in dwarfs than in spirals (Paper I). It occurs because the average surface density is lower compared to the critical value in dwarfs than in spirals. The top two panels in Figure \[fig-sfr\] show cases where the Mach number has constant values with radius: 1 (top) and 10 (second from top). When the Mach number is 10 throughout, cloud formation continues at a high rate in the outer part of the disk (i.e., $\sigma$ in the dispersion of $P(\Sigma_g)$ stays large), and there is no significant drop in SFR there. Consequently, there is no clear double exponential. When the Mach number is 1 throughout, the average SFR profile is almost exactly the same as for the exponential Mach number (compare to the second panel up from the bottom which has the same parameters except for the Mach number), but the rms scatter is much lower when $M=1$ than when $M\sim10$ in the inner disk. This illustrates how the Mach number is unimportant for the average star formation rate in the unstable inner part of the disk. That region is “saturated” with star formation from spiral shocks and instabilities and unable to produce more star formation even with more compression. However the Mach number in the inner disk is important for the detailed structure of star formation, i.e., for the variability of it and for the geometry of cloud structure. These models illustrate how a combination of increasing disk stability and moderate Mach number can create an approximate double exponential in the star formation rate when the overall gas distribution is more uniform. The break radius occurs slightly beyond the point where $<\Sigma_g>\sim\Sigma_c$ if the Mach number is of order unity there. It can occur further out if the Mach number is still high. The detailed profile of the star formation rate in regions where the average column density exceeds the threshold does not depend much on the rotation curve or Mach number. This is because once the threshold is exceeded, the threshold no longer enters into the star formation rate for the simple power law model. Two other types of radial profiles are found in spiral and irregular galaxies: those which continue in an exponential fashion out to the largest measured radius and those which have a shallower exponential in the outer part (Erwin, Beckman, & Pohlen 2005; Paper I). Galaxies of the first type, with a single exponential extending out to $\sim10$ scale lengths (Weiner et al. 2001; Bland-Hawthorn, et al. 2005), are difficult to understand with single-component star formation models because the outer disk should be far below the Kennicutt (1989) threshold. Our multi-component model can reproduce the observation, but the outer gas disk has to fall more slowly than an extrapolation of the inner exponential. The bottom panels in Figure \[fig-sfr2\] show an example. The average gas disk is exponential out to $6R_D$ and then it tapers beyond that as $1/R$ out to $10R_D$. This is consistent with the shallow outer HI profile in NGC 300 (Puche et al. 1990), which has its stellar disk extend continuously to $10R_D$ (Bland-Hawthorn et al. 2005). The profiles of Mach number (using $A=100$) and rotation speed are the same as in the previous examples, and the local star formation rate is $\propto\Sigma_g^{1.5}$. In the left-hand panel, the star formation rate becomes very patchy in the outer part, but the average rate (solid blue line) follows an overall exponential profile. In the right panel, the ratio of $\Sigma_c$ to $<\Sigma_g>$, which is the Toomre stability parameter, $Q$, is $5.7$ in the outer regions, indicating great stability on average. Still, there is a lot of cloud and star formation from local compressions. The top part of Figure \[fig-sfr2\] shows the case if the $1/R$ part of the gas disk begins at a smaller radius, $5.2R_D$, with all else being the same. Then the profile in the outer disk can be shallower than in the inner disk. This is the second type of profile mentioned above. This explanation differs from that in Paper I, where we suggested that dwarf galaxies of this second type, with relatively flat outer parts, could have their steepening in the central regions because of enhanced star formation there. This was because the central regions tended to be bluer than the outer regions; most BCD galaxies were examples of this. The origin of the flat-outer exponential profiles in barred S0 galaxies (Erwin, et al. 2005) cannot be due to intense inner-disk star formation because their Hubble types are too early. In these S0 galaxies the isophotal contours tend to become more round with distance beyond the break radius. Radial profiles determined from deprojected circular averages at all radii could then introduce false inflections. Other flat-outer profiles in the Erwin et al. sample have break radii associated with outer rings or outer Lindblad resonances; these would not be connected with star formation changes either. More observations of the various types of radial profiles and their associated gas and star formation properties are necessary before the relative importance of these models can be understood. Our discussion so far has concerned the radial profile of the star formation rate but not the radial profile of the resulting stars that form over a Hubble time. Our comparisons between the predicted star formation profiles and the observed surface brightness profiles are therefore premature. The next step in this analysis should be an integration of the star formation rate over time, but this requires some knowledge of the gas accretion rate, both as a function of radius and time. Two limiting cases may be discussed at this point. At the end of the star formation process, after all of the gas has been converted into stars, the stellar mass profile should reflect the total accreted gas profile, altered, if need be, by radial gas motions, stellar migration, minor mergers and tidal stripping. There is no reason to believe that this final stellar profile will resemble the star formation profile. The second limiting case is at the beginning of the star formation process, when the outer disk is still dominated by gas. Then the stellar component is only a small perturbation to the outer disk and the star formation profile should be about the same as the accumulated stellar mass profile, altered again by any accreted stars, radial migrations, and tidal effects. Fortunately the outer parts of late-type galaxy disks, beyond $R_{br}$, are usually in this second limit, i.e., gas-dominated. For dwarf galaxies, this was shown by Hunter, Elmegreen & Baker (1998). Thus we believe the star formation profiles derived here for the outer disk are a suitable explanation for the stellar surface density profiles. To check this possibility, we ran two of the SFR models shown in Figure \[fig-sfr\] (the top panel and the second one up from the bottom) over a sequence of timesteps. At each timestep and at each radius, we deducted 1% of the instantaneous SFR from the average gas mass, and then added this mass to the integrated star mass. The initial conditions were the same as in Figure \[fig-sfr2\], with no accumulated stars at first. Figure \[fig-evol\] shows the resultant SFR, average gas surface density, and average stellar surface density at three times: the beginning of the run (blue curves), after 20 timesteps (green curves), and after 50 timesteps (red curves). We label these timesteps as 20% and 50% of the gas consumption time, respectively, where this consumption time is considered to be the inverse of the percentage (1%) deducted each step. There is no gas accretion from outside the galaxy. The panels on the left assume the same exponential Mach number profile as the second panel up from the bottom in Figure \[fig-sfr2\] (i.e., A=100 in Eq. \[eq:mach\]) and the panels on the right have the same constant Mach number profile as the top panel in Figure \[fig-sfr2\] (i.e., $A=0$ and $M=1$). Evidently, the gas gets depleted most quickly in the inner regions, as expected, and the stars build up an exponential profile over time. The double exponential is still present in the accumulated stars. The Mach number profile does not matter much, as discussed previously. At 50 timesteps, the star mass in the steep exponential part of the outer disk is still much less than the gas mass, in agreement with the observations cited in the previous paragraph. We have not commented yet on the relation between star formation rate and H$\alpha$ surface brightness. The H$\alpha$ surface brightness should become an inadequate tracer of star formation in the very low-density regions of outer galaxy disks because the emission measure drops below detectability. The emission measure through the diameter of a classical Stromgren sphere with uniform density $n$ is $\left(6S_0n^4/\left[\pi\alpha\right]\right)^{1/3}$ for Lyman continuum photon luminosity $S_0$ and recombination rate $\alpha$. The scaling with $n^{4/3}$ implies that even if the $H\alpha$ flux does not escape the galaxy, the intensity of an HII region is at least $e^{16/3}=207$ times fainter at a position four scale lengths further out compared to the inner disk. This factor assumes the density is smaller by only the factor $e^{-4}$ without a corresponding increase in scale height. An increase in scale height with decreasing disk self-gravity, as $H=c^2/\left[\pi G\Sigma_T\right]$ for velocity dispersion $c$ and total disk column density $\Sigma_T$, makes the midplane density roughly proportional to $\Sigma^2$, in which case the H$\alpha$ intensity can drop by a factor $e^{32/3}\sim4\times10^4$ in four scale lengths of an exponential disk. Thus there should be a significant drop in the radial profiles of H$\alpha$ even with outer disk star formation of the type discussed here. This type of $H\alpha$ drop is in agreement with observations (e.g., Thilker et al. 2005; Paper I). The multi-component star formation proposed here also explains some of the irregularities with the Kennicutt (1989) prediction that were found in our previous papers on dwarf Irregulars. For example, star formation in dwarfs often occurs where the average column density is less than $\Sigma_c$, in contradiction to the Kennicutt model, as long as there are cool cloudy regions that locally have $\Sigma_g>\Sigma_c$. This peculiarity was noted before in many studies of dwarf galaxies (van der Hulst et al. 1993; Taylor et al. 1994; van Zee et al. 1997; Meurer et al. 1998; Hunter, Elmegreen, & van Woerden 2001). In the current star formation model, these cloudy regions form by turbulence and other processes (pressurized shells, external cloud impacts, end-of-bar flows, gaseous spirals, etc.) even in sub-threshold regions if the Mach number for the associated flows is still relatively high. The revised model also extends the Kennicutt result by allowing for a typical decrease in velocity dispersion with radius and by associating the threshold $\Sigma_c$ with any of a variety of cloud formation processes, and not just isothermal gravitational instabilities in initially smooth disks. Conclusions {#sec-disc} =========== Many processes of star formation combine to give the radial profiles of galaxies. In the inner main-disk regions where the gas is usually gravitationally unstable in spite of the large Coriolis and pressure forces, star formation should saturate to its maximum possible rate. This is the gravitational collapse rate for the conversion of low density to high density gas, multiplied by the fraction of the high density gas that is suitable for star formation, i.e., the fraction in the form of stellar-mass globules with masses exceeding the local thermal Jeans mass (e.g., Elmegreen 2002; Kravtsov 2003). The actual dynamics involved with the first step, dense cloud formation, will be varied, involving swing-amplified spiral instabilities, spiral density wave shocks, compression or shell formation around existing star formation sites, and turbulence compression. In galaxies with strong stellar spirals, the spiral shocks may dominate dense cloud formation, making most clouds spiral-like, as in M51 (e.g., Block et al. 1997), while in galaxies without such strong spirals, another mechanism should dominate, making most clouds shell-like (as in the LMC), globular, or hierarchically fractal. In all of these cases, the same star formation rate per unit area arises, all from the saturation condition. Thus they all give the Kennicutt-Schmidt law or something like it in a regular fashion, regardless of the detailed processes involved. In the outer parts of disks, some of these processes shut down completely. There should be no strong stellar spirals beyond the outer Lindblad resonance for the main (self-amplified) modal pattern speed, and there should be few swing-amplified stellar spirals if the Toomre Q parameter is high. Cold cloud formation should also be more difficult at low ambient pressure. However, a low level of star formation may sustain itself at large radii by driving shells and turbulence and by compressing the existing clouds. Also, gaseous spiral arms can propagate there from the inner disk, as they are able to penetrate the outer Lindblad resonance unlike the stellar spirals. Gaseous arms can also form by instabilities there if there is significant cooling during compression (because that lowers $\Sigma_c$ through the $\gamma_{eff}$ parameter). These processes maintain star formation at levels much lower than the saturation rate given above, and therefore lower than the Kennicutt-Schmidt law rate, primarily because an ever-decreasing fraction of the gas can make the transition from low density to high density in the first step. In this paper, we modelled all of these processes in a general way using the few simple rules just mentioned. The transition from saturated star formation in the inner disk to unsaturated in the outer disk was followed, and radial profiles were obtained that look moderately close to real profiles. In the first case considered, the profile was exponential in the main disk and it tapered off beyond that with a form that also resembled an exponential, but steeper for several scale lengths. The ratio of the break radius to the inner scale length varied with the surface density (higher surface densities have higher ratios) because more unstable inner disks have their inner exponentials extend further out before the transition occurs. This correlation can explain the observation among both spirals and dwarfs that $R_{br}/R_D$ increases with main disk surface brightness. In two other cases with shallower outer gas profiles, the star formation profile varied between a nearly pure exponential out to $\sim10$ scale lengths and a shallow outer exponential, depending on where the transition between the inner and outer gas profiles occurred relative to the stability threshold radius. In all cases, the H$\alpha$ profile should drop much more suddenly with radius than the star formation profile as the emission measure of individual HII regions drops rapidly below the detectability limits. The main ingredients of our star formation model are: a generally smooth decline of gas column density in the disk with a cutoff in the far outer part (usually beyond the observations), a turbulent Mach number that decreases with radius and then levels off to near unity, or remains near unity throughout, a distribution function for local column density with a high column-density tail and a dispersion that increases with Mach number, a column density threshold for self-gravity to overcome Coriolis and pressure forces, and a local star formation rate that increases with the local cloud density when the threshold is exceeded. For such a model, the inner exponential occurs where the average column density exceeds the threshold, almost regardless of Mach number or Mach number gradient. The outer profile occurs where the gas column density is sub-critical and the Mach number is relatively small but non-zero, e.g., near unity. The small but non-zero Mach number gives turbulence and other dynamical processes the ability to form clouds that locally exceed the stability threshold, but these processes are not likely to do this very often. As a result, star-forming clouds become very patchy in the outer disk, making the star formation gradient significantly steeper than the gas gradient. Only the peaks of the clouds stand above the stability threshold. Gas cooling during cloud formation is also an essential ingredient. Cooling to diffuse cloud temperatures and colder is assumed to follow any significant compression, as predicted elsewhere based on studies of interstellar thermal equilibrium. We are grateful to the referee for useful comments. Funding for this work was provided by the National Science Foundation through grants AST-0204922 to DAH and AST-0205097 to BGE. Abadi, M. G., Navarro, J. 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--- abstract: 'The equation of state (EOS) of the osmotic pressure for linear-polymer solutions in good solvents is universally described by a scaling function. We experimentally measured the osmotic pressure of the gelation process via osmotic deswelling. It is found that the same scaling function for linear-polymer solutions also describes the EOS of the osmotic pressure throughout the gelation process, involving both sol and gel states. Furthermore, we reveal that the osmotic pressure of polymer gels is universally governed by the semidilute scaling law of linear-polymer solutions.' author: - Takashi Yasuda - Naoyuki Sakumichi - 'Ung-il Chung' - Takamasa Sakai title: Universal Equation of State of Osmotic Pressure in Gelation Process --- [^1] The statistical mechanics of groups of chains is the basis of polymer physics [@flory1953principles; @de1979scaling; @lifshitz1978some; @panyukov1996statistical]. A remarkable example of this basis is the universality of linear-polymer solutions in good solvents [@de1979scaling; @oono1985statistical]. Their macroscopic collective properties are independent of the microscopic details of the system, because of the great length of polymer chains. This example of the universality of critical phenomena in the $O(n)$-symmetric universality classes ($n=1,2,3$ corresponding to the Ising, XY, and Heisenberg classes, respectively) is found in many systems, ranging from the fields of soft and hard condensed-matter physics to high-energy physics [@pelissetto2002critical]. The above polymer solutions correspond to the limit of $n\to 0$ (self-avoiding walks) in three dimensions [@de1979scaling; @pelissetto2002critical], for which the critical exponent (the excluded volume parameter) $\nu \simeq 0.588$ can be computed by Monte Carlo simulations [@clisby2010accurate; @clisby2016high], the $\epsilon$ expansion method [@kompaniets2017minimally], and the conformal bootstrap method [@shimada2016fractal; @hikami2018conformal]. Furthermore, not only the critical exponents but also the asymptotic scaling functions themselves can be experimentally measured, such as the osmotic pressure [@noda1981thermodynamic; @higo1983osmotic] and the correlation lengths of the density fluctuations [@wiltzius1983universality]. Here, we focus on the equation of state (EOS) of osmotic pressure for linear-polymer solutions in good solvents, which is universally described by the scaling function [@noda1981thermodynamic; @higo1983osmotic; @des1975lagrangian; @des1982osmotic; @ohta1982conformation; @ohta1983theory]: $$\hat{\Pi} = f\left({\hat{c}}\right), \label{eq:EOS}$$ where $\hat{\Pi} \equiv \Pi M/(cRT)$ is the reduced osmotic pressure, and $\hat{c}\equiv c/c^*$ is the reduced polymer concentration normalized by the overlap concentration $c^{*}\equiv1/(A_2 M)$. Here, $M$, $R$, $T$, and $A_2$ are the molar mass, gas constant, absolute temperature, and the second virial coefficient, respectively. The above definition of $c^{*}$ is proportional [@burchard1999solution] to the conventional definition of the overlap concentration $c^*_g \equiv 3M/(4\pi N_A R_g^3)$, at which the polymer chains begin to overlap each other to fill the space. Here, $N_A$ and $R_g$ are the Avogadro constant and the gyration radius of the polymer chain, respectively. ![ Universal EOS of polymer solutions and gelation process in a good solvent. Main image shows the $\hat{c}$-dependence of $\hat{\Pi}$ in a log–log plot, and the inset shows the $\hat{c}^{-1}$-dependence of $\hat{\Pi}/\hat{c}^{1.31}$. The triangles represent two kinds of linear polymer (poly(styrene) of $M=51$–$1900$ kg$/$mol [@higo1983osmotic] and poly($\alpha$-methylstyrene) of $M= 70.8$–$1820$ kg$/$mol [@noda1981thermodynamic]) in toluene solutions. These converge to the universal EOS (\[eq:EOS\]) (black solid curve), which is asymptotic to the van ’t Hoff law ($\hat{\Pi}=1$) as $\hat{c}\to 0$ and to the scaling law in Eq. (\[eq:scaling\]) as $\hat{c}\to\infty$ (black dotted lines). The black circles represent four-branched polymer (poly(ethylene glycol)) solutions of $M=10$ and $40$ kg$/$mol. The orange filled circles represent the gelation process in sol states with various degrees of connectivity ($p=0, 0.1, \dots, 0.5$) at a constant concentration ($c=20$ g$/$L). The red star in the inset corresponds to the universal EOS for polymer gels. []{data-label="fig:EOS"}](Fig1.pdf){width="\linewidth"} In the case of branched polymer solutions, it was reported that each EOS of regular star polymers with up to 18 arms exhibits only minor differences from the universal EOS (\[eq:EOS\]) of linear polymers [@higo1983osmotic; @adam1991concentration; @merkle1993osmotic; @burchard1999solution]. Here, $\hat{c}\equiv c/c^*$ is the only universal scaling parameter (up to multiplication by a constant) [@burchard1999solution]. In other words, $c/c^*_{g}$ is not a universal scaling parameter because $c_{g}^{*}/c^*=3\sqrt{\pi}\Psi^{*}$ includes the interpenetration factor $\Psi^{*}$, which is nonuniversal for a number of arms (e.g., $\Psi^{*}\simeq 0.24$ and $0.44$ for linear and four-branched polymer solutions, respectively [@rubio1996monte; @okumoto1998excluded]). Figure \[fig:EOS\] demonstrates that the two kinds of linear polymer solution and four-branched polymer solutions converge to the single universal EOS (\[eq:EOS\]). In the dilute regime ($c<c^{*}$), each molecular chain is sufficiently isolated such that the universal EOS (\[eq:EOS\]) is well described by the virial expansion [@flory1953principles]: $$\hat{\Pi} = f\left({\hat{c}}\right) = 1 + \hat{c} + \gamma\,\hat{c}^{2}+ \dots \quad(\mathrm{for}\,\,\, 0<\hat{c}<1), \label{eq:virial}$$ where $\gamma \simeq 0.25$ [@flory1953principles; @noda1981thermodynamic] is the dimensionless virial ratio. In the semidilute regime ($c^{*}<c$), molecular chains become interpenetrated and the universal EOS (\[eq:EOS\]) is asymptotic to the scaling law [@des1975lagrangian; @de1979scaling]: $$\hat{\Pi} = f\left({\hat{c}}\right) \simeq K\hat{c}^{\frac{1}{3\nu -1}} \qquad(\mathrm{for}\,\,\, \hat{c}\gg 1), \label{eq:scaling}$$ where $K\simeq 1.1$ is the numerical constant and $1/(3\nu -1)\simeq 1.31$ if $\nu=0.588$. In the present study, we experimentally investigate the EOS of the osmotic pressure of polymer gels, including the whole gelation process. We measured the osmotic pressure in both the sol and gel states via osmotic deswelling in external polymer solutions [@bastide1981osmotic; @horkay1986studies; @horkay2000osmotic]. Our findings are summarized in Fig. \[fig:EOS\]; the universality of EOS (\[eq:EOS\]) holds for both the sol (orange filled circles) and gel (red star) states with only minor variations, although these systems are comprised of highly branched three-dimensional polymer networks. When gelation proceeds at a constant concentration $c$, the average molar mass $M$ increases, and $c^{*}$ decreases. Thus, both $\hat{\Pi}$ and $\hat{c}$ continuously increase along the universal EOS (\[eq:EOS\]) in the sol states. After the gelation (i.e., sol–gel transition), because polymer gels correspond to $M\to\infty$ and $c^{*}\to 0$, both $\hat{\Pi}$ and $\hat{c}$ diverge to infinity in the gel states. According to the semidilute scaling law given by Eq. (\[eq:scaling\]), $\hat{\Pi}/\hat{c}^{1.31}$ is always constant in gel states (red star in the inset of Fig. \[fig:EOS\]). To statically reproduce the gelation process, we non-stoichiometrically tuned the mixing fractions $s$ ($0\leq s\leq 1/2$) of two kinds of precursor solution in an AB-type polymerization system (schematics in Fig. \[fig:gelation\]). Here, $s$ is the molar fraction of the minor precursors to all precursors. We define the connectivity $p$ ($0\leq p\leq1$) as the fraction of the reacted terminal functional groups, assuming reaction completion. By tuning $s$ in accordance with $p = 2s$ [@sakai2016sol; @yoshikawa2019connectivity], we can obtain a desired $p$. Before gelation ($0\leq p <p_\mathrm{gel}$), polymer chains crosslink to form a polydisperse mixture of highly branched polymers with increases in the average molar mass $M$. After gelation ($p_\mathrm{gel}\leq p\leq 1$), polymer networks crosslink to complete the reaction as the elasticity increases. Based on our findings shown in Fig. \[fig:EOS\], the “non-reduced” osmotic pressure $\Pi$ during the gelation process is illustrated in Fig. \[fig:gelation\]. Unlike the sol states, the gel states have elastic contributions to the swelling pressure. According to Flory and Rehner [@flory1943jr], the total swelling pressure in the gel states ($\Pi_{\mathrm{tot}}$) consists of two separate contributions as $\Pi_{\mathrm{tot}}=\Pi_{\mathrm{mix}}+\Pi_{\mathrm{el}}$, where $\Pi_{\mathrm{mix}}$ and $\Pi_{\mathrm{el}}$ are the mixing and elastic contributions, respectively. We regard $\Pi_{\mathrm{mix}}$ as being the osmotic pressure in the gel states because $\Pi_{\mathrm{mix}}$ corresponds to the osmotic pressure in the sol state $\Pi$. As the connectivity $p$ increases at a constant concentration $c$, the osmotic pressure $\Pi$ in the sol states decreases, because the chemical reaction decreases the number density of the molecules. When the samples enter the gel states, the osmotic pressure $\Pi_{\mathrm{mix}}$ reaches a constant; polymer gels are always in a semidilute regime with an infinite molar mass.\ ![ Osmotic pressure during the gelation process at a constant polymer concentration ($c=60$ g$/$L) and a constant molar mass of precursors ($M=10$ kg$/$mol). By measuring $\Pi_\mathrm{tot}$ and $G$, we obtain $\Pi_\mathrm{mix}=\Pi_\mathrm{tot}+G$ in gel states. As the connectivity $p$ increases, $\Pi$ and $\Pi_\mathrm{tot}$ decrease, but $\Pi_{\mathrm{mix}}$ remains constant (blue curves). After gelation ($p_{\mathrm{gel}}\leq p\leq1$), the elasticity (red curve) increases. Here, $p$ ($0\leq p\leq1$) is controlled by mixing two kinds of precursors non-stoichiometrically in an AB-type polymerization system. Gel samples with a low connectivity ($p_{\mathrm{gel}} \leq p < 0.7$) were difficult to characterize, because of the outflow of small polymer clusters. []{data-label="fig:gelation"}](Fig2.pdf){width="\linewidth"} *Materials and Methods*. — For a model system of AB-type polymerization in gelation, we used a tetra-PEG gel, synthesized by the AB-type cross-end coupling of two tetra-arm poly(ethylene glycol) (tetra-PEG) units of the same size [@sakai2008design]. Each end of the tetra-PEG is modified with mutually reactive maleimine (tetra-PEG MA) and thiol (tetra-PEG SH). We dissolved tetra-PEG MA and tetra-PEG SH (Nippon Oil & Fat Corporation) in a phosphate-citrate buffer with an ionic strength and pH of $200$ mM and $3.8$, respectively. For gelation, we mixed these solutions with equal molar masses $M$ and equal concentrations $c$ in various mixing fractions $s$. We held each sample in an enclosed space to maintain humid conditions at room temperature ($T\simeq 298\ K$) to allow the reaction to complete. ![ Osmotic deswelling in external polymer solutions to measure $\Pi$ and $\Pi_{\mathrm{mix}}$ in (a) sol and (b) gel samples, respectively. For each plot, precursors are $M=10$ kg$/$mol. Each line is the least squares fit to the data for each $p$. (a) Square root plots of $\Pi$ of sol samples on $c_{0}=20$ g$/$L for the degrees of connectivity $p=0, 0.1, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5$. We immersed samples with a micro-dialyzer in external polymer (PVP) solutions. We can determine $\Pi$, because $\Pi = \Pi_{\mathrm{ext}}$ at equilibrium. (b) Swelling ratio $Q$ of gel samples on $c_{0}=60$ g$/$L for $p=0.7,0.8,0.9,1$ in equilibrium state in external polymer (PVP) solutions. We directly immersed samples in the external solutions of various concentrations $c_\mathrm{ext}$. We can determine $\Pi_{\mathrm{mix}}$, because $\Pi_{\mathrm{mix}}=\Pi_{\mathrm{ext}}+G$ at equilibrium. []{data-label="fig:measurement"}](Fig3.pdf){width="\linewidth"} We prepared the four-branched polymer (precursor) solutions ($p=0$) by dissolving tetra-PEG MA with molar masses of $M=10$ and $40$ kg$/$mol and initial concentrations $c_{0} = 20$–$120$ g$/$L. Herein, we define the polymer concentration ($c_{0}$ and $c$) as the precursor weight divided by the solvent volume, rather than the solution volume, to extend the universality of the EOS (\[eq:EOS\]) to higher concentrations (see Supplemental Material, Sec. S1). We prepared the sol and gel samples in the gelation process, dissolving precursors with $M=10$ kg$/$mol. For $c_{0} = 20$ g$/$L, we set $p=2s=0.1, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5$ (sol samples). For $c_{0}=40, 60, 80, 120$ g$/$L, we set $p=2s=0.1, 0.2, 0.3$ (sol samples) and $0.7, 0.8, 0.9, 1$ (gel samples). Sec. S2 of Supplemental Material describes the determination of these measurement ranges. We measured the osmotic pressures of the sol states $\Pi$, using controlled aqueous poly(vinylpyrrolidone) (PVP, K90, Sigma Aldrich) solutions whose concentration dependence of osmotic pressure $\Pi_{\mathrm{ext}}$ was measured by Vink [@vink1971precision] (Supplemental Material, Sec. S3). As shown in the schematic in Fig. \[fig:measurement\](a), each solution sample was placed in a micro-dialyzer (MD300, Scienova), which had a semipermeable membrane with a mesh size of $3.5$ kDa. We immersed each dialyzer in aqueous polymer (PVP) solutions at a certain concentration $c_\mathrm{ext}$ with stirring. This system achieved equilibrium at $\Pi = \Pi_{\mathrm{ext}}$. (Achievement of swelling equilibrium was assured as described in Sec. S4 of Supplemental Material.) At that time, each solution sample was changed in weight and concentration from its initial to equilibrium states as $w_0 \to w$ and $c_0 \to c$, respectively. Assuming a constant weight density and small deformation for the sample, we obtained the concentration at equilibrium as $c=c_{0}/Q$, where $Q=w/w_{0}$ is the swelling ratio. In examining the gelation process (orange filled circles in Fig. \[fig:EOS\] and Fig. \[fig:gelation\]), we compared $\Pi$ of the “as-prepared” sol samples at equal concentrations $c=c_0$ with various values of $p$. We determined $\Pi$ of each as-prepared sol sample by measuring the swelling ratio $Q$ as a linear function of $c_\mathrm{ext}$. (The method used to determine $\Pi$ of the as-prepared samples is the same as that used for gels, as detailed below.) To evaluate the parameters $M$ and $c^{*}$ from $\Pi = \Pi (c)$ measured at each $p$, we used the square root plots [@flory1953principles], as shown in Fig. \[fig:measurement\](a). From the virial expansion (\[eq:virial\]), we have $\hat{\Pi} = \left[1+\hat{c}/2 + \left(\gamma-1/4 \right)\hat{c}^2/2 \right]^2 + O\left(\hat{c}^3\right) $. Together with $\gamma \simeq 1/4$ (Supplemental Material, Sec. S5) for certain solutions of few-branched polymers, we have $\sqrt{\Pi/c} \simeq \sqrt{RT/M} \left[1+c/(2c^{*})\right]$ for small $c/c^{*}$. Thus, the intercepts and slopes of each fitting line in Fig. \[fig:measurement\](a) give $M$ and $c^{*}$, respectively, for each $p$. The obtained $M$ and $c^{*}$ values are consistent with the scaling prediction of $c^{*} \sim M^{1/(3\nu-1)}$ with $\nu=0.588$ (Supplemental Material, Sec. S6). We measured the osmotic pressure in the as-prepared gel states $\Pi_\mathrm{mix}$ via the osmotic deswelling. As shown in the schematic in Fig. \[fig:measurement\](b), we immersed each gel sample directly in the external aqueous polymer (PVP) solutions with various concentrations $c_\mathrm{ext}$, because the surfaces of the gels function as semipermeable membranes. Then, each gel sample swells or deswells from the as-prepared state to the equilibrium state as $\Pi_{\mathrm{mix}}+\Pi_{\mathrm{el}}=\Pi_{\mathrm{ext}}$, changing its weight and concentration from the initial to equilibrium states as $w_0 \to w$ and $c_0 \to c$, respectively. The swelling ratio $Q=w/w_{0}$ was measured and interpolated as a linear function of $c_\mathrm{ext}$ for each gel sample (e.g., the samples of various connectivity $p$ at $c_0=60$ g$/$L, as given in Fig. \[fig:measurement\](b)). By using the $c_\mathrm{ext}$-dependence of $Q$, we evaluated $c_\mathrm{ext}$ and $\Pi_\mathrm{ext}$ such that each gel sample maintained its weight ($Q = 1$) and concentration ($c = c_{0}$). We assumed that $\Pi_{\mathrm{el}} = -G$ [@james1949simple] and evaluated $\Pi_{\mathrm{mix}} = \Pi_{\mathrm{ext}} + G$ of each as-prepared gel sample, where $G$ is the shear modulus as measured by rheometry (Supplemental Material, Sec. S7).\ ![ Osmotic pressure during the gelation process. The molar mass of precursors is $M=10$ kg$/$mol, corresponding to the overlap concentration $c^*\simeq58$ g$/$L at $p=0$. (a) Osmotic pressure in the unreacted four-branched polymer solutions (black circles) and in the reaction-completed polymer gels (red filled circles). The former and latter are in very good agreement with the universal EOS (\[eq:EOS\]) (black curve) and with the semidilute scaling law $\Pi\propto c^{2.31}$ (red line), respectively. Here, $3\nu/(3\nu -1)\simeq 2.31$ for $\nu \simeq 0.588$. The black dotted curve is the virial expansion (\[eq:virial\]) up to the third-order terms. As $p$ increases (green triangles), $\Pi$ decreases in sol states ($0\leq p<p_\mathrm{gel}$) and becomes constant in gel states ($p_\mathrm{gel}<p\leq1$). The inset shows the osmotic pressure during the gelation process at a constant polymer concentration $c=c_{0}=40, 60, 80, 120$ g$/$L. The green triangles ($c=40$ g$/$L) are the same as those in the main panel. The blue circles ($c=60$ g$/$L) are used in Fig. \[fig:gelation\]. (b) Connectivity ($p$) dependence of $\hat{\Pi}/{\hat{c}}^{1.31}$. The symbols and data are the same as those in the inset of (a). In gel states, $\hat{\Pi}/{\hat{c}}^{1.31}$ converge to the universal value $1.1$, which is independent of $p$ and $c$. []{data-label="fig:result"}](Fig4.pdf){width="\linewidth"} *Results and Analysis*. — The main panel in Fig. \[fig:result\](a) shows the $c$-dependence of the osmotic pressure in the unreacted four-branched polymer solutions ($p = 0$) and in the reaction-completed polymer gels ($p = 1$). In the wide concentration range $c$, the experimental results of the former and latter are in remarkably good agreement with the universal EOS (\[eq:EOS\]) for *linear* polymer solutions and with the semidilute scaling law $\Pi\propto c^{3\nu/(3\nu -1)}$, respectively. With the increase in $c$, $\Pi$ in polymer solutions (black curve) is asymptotic to $\Pi_{\mathrm{mix}}$ in polymer gels (red line). This asymptotic relationship suggests that the $\Pi_{\mathrm{mix}}$ of polymer gels is governed by the semidilute scaling law in Eq. (\[eq:scaling\]) with $K\simeq 1.1$ for polymer solutions. The inset in Fig. \[fig:result\](a) shows the $p$-dependence of $\Pi$ and $\Pi_\mathrm{mix}$ throughout the gelation process ($0\leq p \leq1$). In the sol states ($0\leq p<p_\mathrm{gel}$), $\Pi$ decreases as $p$ increases, because the average molar mass $M$ increases. As $c$ increases, the extent of the decrease in the osmotic pressure itself decreases. In particular, for $c=120$ g$/$L, $\Pi$ and $\Pi_\mathrm{mix}$ are constant throughout the gelation process ($0\leq p \leq1$), because the precursor solution is in the semidilute regime even at $p=0$. In the gel states ($p_\mathrm{gel}<p\leq1$), $\Pi_\mathrm{mix}$ is constant even if $p$ increases. In general, the osmotic pressure is dependent and independent of the average molar mass in the dilute and semidilute regimes, respectively [@de1979scaling]. Thus, the constant $\Pi_\mathrm{mix}$ in the gel states ($p_\mathrm{gel}<p\leq1$) indicates that polymer gels are always in the semidilute regime, because of the infinite molar mass of the polymer networks. We can interpret $\Pi$ during the gelation process in the sol states ($0\leq p<p_\mathrm{gel}$) in terms of the universal EOS (\[eq:EOS\]). By using $M$ and $c^{*}$ evaluated in Fig. \[fig:measurement\](a) at each $p$, we changed the state variables (from $c$ and $\Pi$ to $\hat{c}$ and $\hat{\Pi}$), yielding the orange filled circles in Fig. \[fig:EOS\]. Remarkably, the gelation process in sol states ($p=0, 0.1, \dots, 0.5$) obeys the universal EOS (\[eq:EOS\]), although these systems continue to form multi-branched polymer clusters. Considering this in tandem with the semidilute scaling law observed in the gel states ($\Pi_{\mathrm{mix}}\propto c^{2.31}$), it is expected that $\hat{\Pi}_\mathrm{mix}$ in the gel states ($p_\mathrm{gel}<p\leq1$) will conform to the semidilute scaling law given by Eq. (\[eq:scaling\]) of *linear*-polymer solutions (red line in Fig. \[fig:result\](a)) with $K\simeq 1.1$. Based on the above expectation, we propose a universal EOS of osmotic pressure $\Pi_\mathrm{mix}$ for polymer gels as $$K=\frac{\hat{\Pi}_\mathrm{mix}}{\hat{c}^{1/(3\nu-1)}} \equiv \frac{M{c^{*}}^{1/(3\nu-1)} \Pi_\mathrm{mix}}{RTc^{\,3\nu/(3\nu-1)}}, \label{eq:scaling-gel}$$ where $K\simeq 1.1$. We note that $\hat{\Pi}_\mathrm{mix}/\hat{c}^{1/(3\nu-1)}$ is finite, although both $\hat{c}\equiv c/c^{*}$ and $\hat{\Pi}_\mathrm{mix}\equiv \Pi_\mathrm{mix} M/(cRT)$ diverge to infinity, because gels correspond to infinite molar mass $M\to\infty$ and $c^{*}\to 0$. In Fig. \[fig:result\](b), we demonstrate that $\hat{\Pi}_\mathrm{mix}/\hat{c}^{1/(3\nu-1)}$ converge to the universal value $K\simeq1.1$, which is independent of $p$ and $c$, after the gelation ($p_\mathrm{gel}\leq p\leq 1$). Thus, in the inset of Fig. \[fig:EOS\], the gel states are positioned at $(1/\hat{c},\hat{\Pi}_\mathrm{mix}/{\hat{c}}^{1.31}) \simeq (0, 1.1)$ (red star). We obtained Fig. \[fig:result\](b) by setting a constant value for $M{c^{*}}^{1/(3\nu-1)}$ and substituting $\Pi$ and $\Pi_\mathrm{mix}$ (shown in the inset of Fig. \[fig:result\](a)) into Eqs. (\[eq:scaling\]) and (\[eq:scaling-gel\]), respectively. (Further details are given in Sec. S6 of Supplemental Material.) This procedure demonstrates that we can determine $\Pi_\mathrm{mix}$ for any polymer gel by measuring a non-universal parameter $M{c^{*}}^{1/3\nu-1}$.\ *Concluding remarks*. — We have experimentally measured the osmotic pressure of polymer gels throughout the gelation process. We find that the universal EOS (\[eq:EOS\]) of the osmotic pressure for *linear*-polymer solutions describes the osmotic pressure throughout the gelation process involving both the sol and gel states (Fig. \[fig:EOS\]). In the sol states, both $\hat{\Pi}$ and $\hat{c}$ continuously increase according to the universal EOS (\[eq:EOS\]) with an increase in the average molar mass (orange filled circles in Fig. \[fig:EOS\]). In the gel states, the osmotic pressure of polymer gels is universally governed by the semidilute scaling law (\[eq:scaling-gel\]) (red star in Fig. \[fig:EOS\] and Fig. 4(b)). Here, both $\hat{\Pi}$ and $\hat{c}$ diverge to infinity, because the gel states correspond to the average molar mass $M\to\infty$ and the overlap concentration $c^{*}\to 0$. In addition, we have demonstrated that Eq. (\[eq:scaling-gel\]) enables the determination of $\Pi_\mathrm{mix}$ for any polymer gel by measuring a non-universal parameter $M{c^{*}}^{1/3\nu-1}$. Because polymer gels are open systems that can swell and deswell by exchanging solvents with the human body, an understanding of the osmotic pressure is essential for controlling the swelling of polymer gels. Our findings are not only conceptually important to polymer physics, but also practically useful, encouraging biomedical applications of polymer gels such as soft contact lenses, adhesion barriers, sealants, and artificial vitreous humor. We thank Masao Doi, Yuichi Masubuchi, Takashi Uneyama, and Xiang Li for their useful comments. This work was supported by the Japan Society for the Promotion of Science (JSPS) through the Grants-in-Aid for Early Career Scientists grant number 19K14672 to N.S., Scientific Research (B) grant number 18H02027 to T.S., and Scientific Research (S) grant number 16H06312 to U.C. This work was also supported by the Japan Science and Technology Agency (JST) CREST grant number JPMJCR1992 to T.S. and COI grant number JPMJCE1304 to U.C. [32]{} url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ** (, , ). , ** (, , ). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). [^1]: These authors contributed equally: T. Yasuda, N. Sakumichi

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--- abstract: 'The relevant integration of wind power into the grid has involved a remarkable impact on power system operation, mainly in terms of security and reliability due to the inherent loss of the rotational inertia as a consequence of such new generation units decoupled from the grid.' address: - 'Dept. of Electrical Engineering, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain' - 'Dept. of Hydraulic, Energy and Environmental Engineering, Universidad Politécnica de Madrid, 28040, Spain' - 'Dept. of Civil Engineering, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain' author: - 'Ana Fernández-Guillamón' - José Ignacio Sarasúa - Manuel Chazarra - 'Antonio Vigueras-Rodríguez' - 'Daniel Fernández-Muñoz' - 'Ángel Molina-García' bibliography: - 'biblio.bib' title: 'Frequency Control Analysis based on Unit Commitment Schemes with High Wind Power Integration: a Spanish Isolated Power System Case Study' --- Power system stability, Wind energy integration, Wind frequency control, Unit commitment Nomenclature {#nomenclature .unnumbered} ============ Introduction {#sec.introduction} ============ Conventional power plants with synchronous generators have traditionally determined the inertia of power systems [@tielens16]. However, during the last decades, most countries have promoted large-scale integration of Renewable Energy Sources (RES) [@zhang17; @fernandez19power]. RES are usually not connected to the grid through synchronous machines, but through power electronic converters electrically decoupled from the grid [@junyent15; @tian16]. As a consequence, by increasing the amount of RES and replacing synchronous conventional units, the effective rotational inertia of power systems can be significantly reduced [@akhtar15; @yang18; @fernandez19analysis]. Actually, Albadi *et al.* consider that the impact of RES on power systems mainly depends on the RES integration and the system inertia [@albadi10], being the RES negative effects more severe in isolated power systems [@martinez18]. Among the different RES, wind power is the most developed and relatively mature technology [[@BREEZE2014223]]{}, especially variable speed wind turbines (VSWT) [@edrah15; @syahputra16; @artigao18; @cardozo18; @li18]. Indeed, Toulabi *et al.* affirm that the participation of wind power into frequency control services becomes inevitable due to the relevant integration of such resource [@toulabi17]. Power imbalances between generation and demand can occur, among others, due to the loss of power generators [@sokoler16]. Actually, this loss of power generators can be the most severe contingency in case it is the largest one [@zhang18]. These imbalances cause frequency fluctuations, and subsequently the grid becomes unstable, even leading to black-outs [@khalghani16; @marzband16]. Hence, the frequency control services are playing an essential role for secure and reliable power systems [@ozer15]. Moreover, frequency stability is the most critical issue in isolated power systems due to their low rotational inertia [@aghamohammadi14; @jiang15; @jiang16]. Frequency control has a hierarchical structure, and in Europe is usually organized up to five layers: $(i)$ frequency containment, $(ii)$ imbalance netting, $(iii,iv)$ frequency restoration (automatic and/or manual) and $(v)$ replacement, from fast to slow timescales [@entsoe_europe]. According to the specific literature, several studies have proposed wind frequency control approaches. However, authors notice that in those contributions: $(i)$ energy schedule scenarios considered are usually arbitrary and unrealistic, without considering a unit commitment (UC) scheme and individual generation units [@keung09; @el11; @mahto17; @abazari18; @alsharafi18]; $(ii)$ the power imbalance is usually taken as a fixed random value (between 3 and 20%), excluding the $N-1$ criterion [@ma10; @wang13; @kang16; @ochoa18]; $(iii)$ load shedding is not taken into account in the frequency response analysis [ [@bevrani11; @ma18; @8667397]]{}; and $(iv)$ only a few wind power integration scenarios are commonly analyzed to evaluate the wind frequency controller —usually one or two different scenarios— [@zhang12; @mi16; @bao18; @chen18]. As a consequence, simulations can address unrealistic and inaccurate results. For instance, recent studies considering two wind integration share rates provide different —and even opposite— conclusions regarding frequency nadir and RoCoF (Rate of Change of Frequency): some authors conclude that these parameters can improve [@wang13; @wilches15], others that they could get worse [@alsharafi18; @aziz18] or even be similar [@kang16] as wind penetration increases. By considering previous contributions, the aim of this paper is to analyze the frequency response of an isolated power system with high integration of wind power generation including wind frequency control and load shedding. These energy schedule scenarios are determined by a UC model, taking into account some technical and economical constraints [@farrokhabadi18] and guaranteeing the frequency system recovery after the largest power plant outage ($N-1$ criterion) [@teng16]. A realistic load shedding program is also included, as well as wind frequency control. With this aim, our study is carried out in Gran Canaria Island power system, in the Canary island archipelago (Spain), where the wind power integration has increased from 90 to 180 MW in the last two years. Moreover, in the Canary island archipelago, more than 200 loss of generation events per year were registered between 2005 and 2010. In fact, the number of this kind of incidents even surpassed 300 per year, subsequently suffering from the activation of the load shedding programs [@padron2015reducing]. This analysis can be extended to other isolated power systems with relevant wind energy potential, such as Madagascar [@praene17] or Japan [@meti15]. The main contributions of this paper can be thus summarized as follows: - Evaluation of wind frequency control responses, including load shedding and rotational inertia changes from realistic operation conditions under generation unit tripping. - Analysis of frequency deviations (nadir, RoCoF) in isolated power systems with high penetration of wind power, using energy schedules and unit comments obtained from an optimization model. The rest of the paper is organized as follows: Section \[sec.power\_system\_description\] describes the Gran Canaria power system and the generation scheduling process. The power system model, including both optimization and dynamic models, are described in Section \[sec.power\_system\_model\]. Simulation results are analyzed and discussed in Section \[sec.results\]. Finally, Section \[sec.conclusions\] outlines the main conclusions of the paper. Power System and Generation Scheduling Process {#sec.power_system_description} ============================================== Preliminaries ------------- [\[sec.power\_system\_general\_overview\]]{} Different frequency analysis studies have been carried out by authors based on specific power systems. For instance, Zerket *et al.*. considered a modified Nordic 32-bus test system [@zertek12]; in [@nazari2014distributed], the power system of Flores Island, and the electric power system of Sao Miguel Island were used; Moghadam *et al.*. focused on the power system of Ireland [@moghadam2014distributed]; in [@wang2017system], the Singapore power system was used; Pradhan *et al.* tested the three-area New England system [@pradhan2019online]; and [@sarasua2019analysis] considered the Spanish isolated power system located in El Hierro Island. In this paper, authors have focused on Gran Canaria Island (Spain), where the wind power integration has increased from 90 to 180 MW in the last two years. Gran Canaria Island belongs to Canarian archipelago, one of the outermost regions of the European Union. Canarian archipelago is located in the north-west of the African Continent. From the energy point of view, Gran Canaria Island is an isolated power system. Traditionally, Gran Canaria Island’s generation has been exclusively associated with fossil fuels: diesel, steam, gas and combined cycle units from two different power plants: *Jinámar power plant* and *Barranco de Tirijana power plant*. However, this fossil fuel dependence has involved an important economic and environmental drawback. To overcome theses problems, the Canary Government promoted the installation of wind power plants in the 90’s, accounting for 70 MW in 2002. In the following decade, the installation of wind power plants stopped around 95 MW and, since 2015, wind power capacity has been doubled, nearly reaching 180 MW. Regarding the wind power generation and system demand in Gran Canaria Island along 2018, both are shown in Fig. \[fig.demand\]. The system demand is discretized for six different intervals, considering the lowest and highest demand of Gran Canaria Island. Wind power generation is discretized for five intervals. According to the ranges in the system demand and the wind power generation shown in Fig. \[fig.demand\], thirty energy scenarios are proposed to analyse the frequency response of the system including wind frequency control. Each energy scenario is based on a pair demand-wind power generation as it is further described in Section \[sec.results\]. {width="0.75\linewidth"} Generation Scheduling Process ----------------------------- [\[sec.power\_system\_generation\_scheduling\]]{} The generation scheduling of the Gran Canaria Island power system is ruled in [@miet12; @miet15]. It is carried out by the Spanish Transmission System Operator (TSO) according to the economic criterion of variable costs of each power plant. The schedules are obtained according to different time horizons: weekly or daily. Each energy schedule depends on the previous time horizon and, subsequently, weekly and daily schedules are required to determine the hourly generation scheduling, which is used in the present paper. An overview of these schedules is summarized in Fig. \[fig.schedule\]. \ 1. *Weekly scheduling:* Estimation of the hourly start-up and shut-down decisions from each Saturday (00:00 h) to the following Friday (23:59 h). This initial generation schedule is determined following two steps: $(i)$ an economic dispatch is carried out to minimize the total variable costs to meet the net power system demand (i.e., the power system demand minus the renewable generation). The result of such economic dispatch includes both the hourly energy and the reserve schedules (labeled as [`Schedule-A`]{}). $(ii)$ an economic and security dispatch is determined taking into account the transmission lines and minimizing the total variable costs to support the net power system demand and a certain level of power quality. The result of this economic and security dispatch is also a hourly energy and reserve schedule (labeled as [`Schedule-B`]{}). 2. *Daily scheduling:* Updates the [`Schedule-B`]{} of a certain day $D$ from the updated available information of the power system: generation from suppliers, demand from consumers and the state of the transmission lines. The result of the daily scheduling in the day $D$ is a new hourly energy and reserve schedule (labeled as [`Schedule-C`]{}). It is obtained before 14:00 h of the day $D-1$. This [`Schedule-C`]{} is determined following a similar process as in the weekly scheduling: $(i)$ an economic dispatch is firstly carried out and $(ii)$ an economic and security dispatch is then calculated. The daily scheduling processes aims to minimize the total variable costs to meet the net power system demand with a minimum certain level of power quality. Methodology: Unit Commitment and Frequency Model {#sec.power_system_model} ================================================ Frequency deviations are analyzed according to possible generation tripping and power system reserves by considering explicitly individual generation units and technologies. With this aim, a series of energy scenarios for each scenario of system demand and wind power generation based on a real isolated power system (the Gran Canaria Island) are estimated and evaluated accordingly, considering current wind power integration percentages and load shedding programs. Fig. \[fig.flow\_chart\] shows the proposed methodology, highlighting the novelties and differences presented in this paper compared to other approaches focused on frequency analysis (i.e., carry out a UC to determine the energy scenarios, consider the loss of the largest power plant as imbalance, and include load shedding with and without wind frequency control). The following subsections describe respectively the unit commitment model and frequency models used in this work. Fig. \[fig.power\_system\] shows a simplified scheme of the modelled Gran Canaria Island power system, where conventional and wind power plants are depicted. ![[]{data-label="fig.flow_chart"}](flowchart.pdf){width="0.9\linewidth"} {width="0.595\linewidth"} Unit commitment model: creation of scenarios {#sec.optimization_model} -------------------------------------------- In order to analyze frequency deviations in the Gran Canaria power system, a UC model is required to estimate the number of thermal units connected to the grid for each generation mix scenario. These thermal units remain unchanged during the subsequent frequency control analysis. The UC model used in this paper has been recently proposed by the authors in [[@FernandezMunoz2019]]{}, based on [[@fernandez19]]{}, which is a deterministic thermal model based on mixed integer linear programming (MILP). Other contributions focused on probabilistic unit commitment can be also found in the specific literature. In this way, [[@AZIZIPANAHABARGHOOEE2016634]]{} proposes an optimal allocation of up/down spinning reserves under high integration of wind power. The planning horizon of our model is adapted to 24 hours with a time resolution of one hour consistent with the approach used by the TSO for the next-day generation scheduling [[@pezic13]]{}; and the hydropower technology is excluded from the model formulation in order to be consistent with the generation mix of the Gran Canaria power system. The model formulation is partially based on [[@morales12]]{} which is, to the author’s knowledge, the most computationally efficient formulation available in the literature when considering different types of start-up costs of thermal units. The Gran Canaria power system is operated in a centralized manner by the TSO in order to minimize the total system costs, according to [@miet15]. Therefore, the objective function of the UC model here used consists in minimizing the start-up cost, the fuel cost, the operation and maintenance cost and the wear and tear cost of all thermal units. The optimal solution of the UC model is formed by the hourly energy schedule of each thermal unit of the system, taking into account their minimum on-line and off-line times. Among others, the energy schedule is restricted by the following constraints. The production-cost curve of each thermal unit is modeled as a piece-wise linear function discretized by ten pieces. The number of pieces is determined as a trade-off between the accuracy of the solution and the computation time cost limits. In addition, the system demand and the spinning reserve requirements must be fulfilled in each hour. According to the P.O. SEIE 1 in [@miet12], the hourly spinning reserve must be higher or equal to the maximum of the following three values: $(i)$ the expected inter-hourly increase in the system demand between two consecutive hours; $(ii)$ the most likely wind power loss calculated by the TSO from historical data; $(iii)$ the loss of the largest spinning generating unit in each hour. It is important to bear in mind that there are two conceptual differences between the formulation presented in [@morales12] and the one used here: - The meaning of each start-up type of each thermal unit is different. In [@morales12], each start-up type corresponds to a different power trajectory of the thermal unit whereas the approach of the model here used is the following: each start-up type refers to the start-up cost as a function of the time that the unit has remained off-line since the previous shut-down. The start-up cost calculation and the involved parameters of the thermal units are defined in [@miet15]. - For those thermal units that have a start-up process longer than one hour, a single output power trajectory ranging from zero to the unit’s minimum output power is considered. Further details of the UC model formulation can be found in [@FernandezMunoz2019]. {#sec.dynamic_model} ### General overview {#sec.dynamic_model_general_overview} Frequency deviations in power systems are usually modeled by means of an aggregated inertial model. This assumption has been successfully applied to isolated power systems, as the Irish power system [@mansoor00]. In this paper, frequency system variations are the result of an imbalance between the supply-side (*Barranco de Tirijana* power plant $P_{T}$, *Jinámar* power plant $P_{J}$ and wind power plants $P_{w}$, which are explicitly considered for simulations) and the demand-side $P_{d}$. A load frequency sensitivity parameter $D$ is also included to model the load sensitivity under frequency excursions [@o14], $$\label{eq.swing} f\,\dfrac{df}{dt}=\dfrac{1}{T_{m}(t)}\left(P_{T}+P_{J}+P_{w}-P_{d}-D\cdot\Delta f \right) ,$$ being $T_{m}(t)$ the total inertia of the power system, which corresponds to the equivalent addition of the rotational inertia of all synchronous generators under operation conditions, in terms of the system base, $$\label{eq.Tm} T_{m} (t)= \sum_{m=1}^{4}2\;H_{s,m}+\sum_{q=1}^{5}2\;H_{g,q}+\sum_{k=1}^{5}2\;H_{ds,k}+\sum_{l=1}^{2}2\;H_{cc,l}\,.$$ Frequency and power variables also depend on time, but it is not explicitly included to simplify the expressions. Fig. \[fig.dynamic\_model\] shows the general block diagram of the proposed simulation model. It has been developed in Matlab/Simulink. The block [`Power system`]{} in Fig. \[fig.dynamic\_model\] contains eq. ([\[eq.swing\]]{}), modeled with the corresponding block diagram. The power provided by the power plants, $P_{T}$ and $P_{J}$ respectively, are the addition of the power supplied by each thermal generation unit (steam $s$, gas $g$, combined cycle $cc$ and diesel $ds$) under operating conditions, expressed as follows: $$\label{eq.pt} P_{T}=\sum_{m=1}^{2}P_{s,m}+\sum_{q=4}^{5}P_{g,q}+\sum_{l=1}^{2}P_{cc,l}\;,$$ $$\label{eq.pj} P_{J}=\sum_{m=3}^{4}P_{s,m}+\sum_{q=1}^{3}P_{g,i}+\sum_{k=1}^{5}P_{ds,k}\;.$$ The $k$, $l$, $m$ and $q$ indexes refer to the number of diesel, combined cycle, steam and gas units of each power plant, respectively. The proposed dynamic model depicted in Fig. \[fig.dynamic\_model\] is thus composed by the different thermal units belonging to *Barranco de Tirijana* and *Jinámar* power plants —explicitly considered in the model—, as well as the wind power plants, the power system and the power load of consumers. This load consumer block includes the load shedding program, activated when the grid frequency exceed certain thresholds. The dynamic response of each generation unit is simulated according to the transfer functions discussed in Section [\[sec.thermal\_units\]]{}. {width=".6\linewidth"} ### Thermal generation units {#sec.thermal_units} The frequency response of the thermal generation units has been modeled through the transfer functions proposed in [@kundur94; @neplan16]. Parameters have been selected from typical values, see Table \[tab.thermal\]. The combined cycle generation unit frequency behavior is supposed similar to the gas generation units, see Fig. \[fig.thermal\_generation\_models\]. The two inputs for these three frequency models are $(i)$ frequency deviations —including constraints provided by the frequency containment—, and $(ii)$ AGC conditions for the frequency restoration in isolated power systems. Both inputs are linked by the corresponding droop. {width="0.755\linewidth"} According to the Spanish insular power system requirements, the AGC system is in charge of removing the steady-state frequency error after the frequency containment control. This is usually known as ‘frequency restoration’, and modeled in a similar way to [@perez14]. The equivalent regulation effort $\Delta RR$ is then estimated as: $$\label{eq.rr} \Delta RR = - K_{f} \cdot \Delta f.$$ This expression is included in the block diagram shown Fig. \[fig.dynamic\_model\], in block [`AGC`]{}. $K_{f}$ is determined following the ENTSO-E recommendations [@entsoe]. This regulation effort is conducted by all thermal generation units and distributed depending on the participation factors $K_{u,i}$, assuming that: $(i)$ all thermal generation units connected to the power system equally participate in secondary regulation control and $(ii)$ the participation factors are obtained as a function of the speed droop of each unit. The participation factors $K_{u,i}$ of thermal generation units disconnected from the grid are considered as zero [@wood12]: $$\label{eq.pref} \begin{split} \Delta P_{ref,th}=\dfrac{1}{T_{u,th}}\int\Delta RR\cdot K_{u,th}\;dt = \\ = \dfrac{-1}{T_{u,th}} \cdot K_{u,th} \cdot K_f \int \Delta f \;dt. \end{split}$$ $$\label{eq.Kuth} \begin{split} \sum K_{u,th}=\sum_{m=1}^{4}K_{u,s,m}+\sum_{q=1}^{5}K_{u,g,q} +\\ +\sum_{k=1}^{5}K_{u,d,k}+\sum_{l=1}^{2}K_{u,cc,l}=1 \end{split}$$ ### Load shedding {#sec.load_shedding} A realistic load shedding scheme is considered in the proposed model by means of sudden load disconnections when frequency excursions are higher than certain thresholds. Table \[tab.load\] summarizes these frequency thresholds, time delay and load shedding values for different scenarios. This load shedding scheme depends on the islanding power system operation conditions required by the Spanish TSO and thus, the responses are in line with certain frequency excursion thresholds. When the scenario corresponds to an intermediate load case, the load shedding value is interpolated from the corresponding steps. ### Wind power plants {#sec.wind_turbines} One equivalent VSWT with $n_{WT}$-times the size of each one model the wind power penetration —being $n_{WT}$ the number of wind turbines [@pyller03; @mokhtari14]—, is proposed as aggregated model for wind power plants. An equivalent averaged wind speed ($v_{w}=10.25$ m/s) is assumed for the simulations. This wind speed is considered as constant, which has been previously used in the specific literature for short-time period frequency analysis including wind power plants [@chang10; @erlich10; @margaris12; @zertek12; @vzertek12]. With this wind speed, the wind generation accounts for 80% of the installed wind power capacity. Wind turbines are modeled according to the turbine control model, mechanical two-mass model and wind power model described in [@ullah08; @clark10]. The two-mass model assumes the rotor and blades as a single mass, and the generator as another mass [@jafari17; @liu17]. Huerta *et al.* consider that the two-mass model is the most suitable to evaluate the grid stability [@huerta17]. Wind turbines also include a frequency control response. The strategy for VSWTs implemented in this paper is based on the technique described in [@fernandez18; @fernandez18fast], see Fig. \[fig.control\]. It was evaluated in [@fernandez18] for isolated power systems with up to a 45% of wind power integration and compared to the approach of [@tarnowski09], providing a more appropriate frequency response under power imbalance conditions. In [@fernandez18fast], the proposed frequency control strategy was studied for multi-area power systems. Three operation modes are considered: $(i)$ normal operation mode, $(ii)$ overproduction mode and $(iii)$ recovery mode. Different set-point active power $P_{sp}$ values are then determined aiming to restore the grid frequency under power imbalance conditions. Fig. \[fig.power\] depicts the VSWTs active power variations $\Delta P _{WF}$ submitted to an under-frequency excursion, being $\Delta P_{w}=P_{sp}-P_{MPPT}(\Omega_{MPPT})$. With the aim of reducing load shedding contributions under high wind power integration scenarios, two modifications are carried out to the preliminary frequency controller, both in overproduction and recovery periods. According to [@fernandez18], the overproduction power $\Delta P_{OP}$ is estimated proportionally to the frequency excursion evolution, with a maximum value of 10%. In this paper, the maximum $\Delta P_{OP}$ is increased to 15%, in order to provide more power after the imbalance and minimizing load shedding situations. In the recovery mode, the power of point $P_{2}$ is defined as $P_{MPPT} (\Omega_{V})+x\cdot \left( P_{mt}(\Omega_{V})-P_{MPPT} (\Omega_{V})\right) $ —see Fig. \[fig.control\]—, being $x$ an scale factor considered as 0.75 in the original approach [@fernandez18]. However, in this case, the recovery time period of the wind turbines is faster than the AGC action of the frequency restoration control, and subsequently obtaining an undesirable frequency evolution, see Fig. \[fig.potencia\]. As a consequence, $x$ has been increased to 0.95, smoothing and slowing down the recovery period of the wind frequency controller. This alternative frequency controller is included in the VSWT model as seen in Fig. \[fig.aero\_control\]. {width="0.75\linewidth"} Results {#sec.results} ======= Scenarios under consideration ----------------------------- According to the demand distribution in Gran Canaria Island along 2018 previously discussed in Section [\[sec.power\_system\_general\_overview\]]{}, six different power demand conditions are considered for the study. Each system demand is analyzed under different wind power generation percentages following Fig. \[fig.demand\]. Thus, thirty different energy scenarios are under study, which is significantly higher than other contributions focused on frequency control under contingencies including wind power plants [@lalor2005frequency; @sigrist2009representative]. To determine the energy schedule of each supply-demand scenario, the Unit Commitment model described in Section \[sec.optimization\_model\] is run with GAMS software and Cplex 12.2 solver, which uses a branch and cut algorithm to solve MILP problems. Fig. \[fig.scenarios2\] depicts the energy schedule of each scenario aggregated by generation technology. From the $N-1$ criterion, the largest generation unit is suddenly disconnected under a contingency. As a consequence, a different generation group is disconnected in each scenario, depending on the energy schedule obtained by the UC model and subsequently addressed a variety of power imbalance situations. Fig. \[fig.after\_imbalance\] summarizes the energy schedule of each scenario after these disconnections, pointed out the technology and generation unit tripping under such circumstances. Due to these sudden disconnections, the equivalent rotational inertia of the power system is reduced according to eq. ([\[eq.Tm\]]{}). ![Scenarios under study[]{data-label="fig.scenarios2"}](scenarios.pdf){width="0.995\linewidth"} ![Generation mix after disconnections[]{data-label="fig.after_imbalance"}](scenarios_mod.pdf){width="0.995\linewidth"} Frequency response analysis {#sec.simulation_results} --------------------------- \ \ With the aim of evaluating frequency deviation and power system performance under the sudden generation disconnection established with the $N-1$ criterion, grid frequency response is analyzed $(i)$ excluding wind frequency control and only considering conventional units; and $(ii)$ including conventional units and wind frequency control strategy. Firstly, nadir and RoCoF results for the 30 simulated scenarios according to the generation unit tripping obtained for the UC model and depicted in Fig. \[fig.after\_imbalance\] are compared to results obtained following methodologies of previous contributions [@keung09; @ma10; @el11; @alsharafi18]. They usually assume a constant 10% imbalance, neglect any inertia power system modification and do not include load shedding scheme in their frequency analysis models. With this aim, Fig.s \[fig.nadir\] and \[fig.rocof\] summarize nadir and RoCoF respectively, including (or not) wind frequency response. RoCoF is calculated between 0.3 and 0.5 s after the sudden disconnection of the largest conventional generation unit for each energy scenario. As can be seen, clear differences are identified between both approaches. In fact, most obvious results are determined with a constant power imbalance, as was to be expected, see Fig. \[fig.nadir\_sin\] and \[fig.nadir\_con\] for nadir comparison values. If a simplified power system modeling is considered for frequency control analysis, with typical 10% power imbalance conditions —usually assumed in previous contributions as was previously discussed— 49.4 Hz nadir and 0.5 Hz/s RoCoF values are obtained for all cases, which provides significant discrepancies with our proposal, see Fig. \[fig.nadir\] and \[fig.rocof\] respectively. Indeed, nadir lies in between 48.54 and 49.15 Hz when wind power plants are excluded from frequency control, depending on each scenario —see Fig. [\[fig.nadir\_sin\_2\]]{}—. In fact, these values were even worse if the load shedding program was not considered, as it is activated in 21 of the 30 scenarios analyzed. However, a larger wind power integration without frequency control —see Fig. [\[fig.nadir\_sin\_2\]]{}— doesn’t imply a worse nadir response, which could be deduced a priori, due to the loss of the larger power plant (which is different, depending on the scenario). When wind frequency control is considered for simulations, the minimum frequency is increased 110 mHz in average for all cases. Moreover, the more wind power integration providing frequency control, the lower nadir is obtained. For instance, for wind power integration over 50%, the minimum frequency is reduced around 200 mHz. It can’t be then deduced an homogeneous response of the considered power system submitted to realistic generation unit tripping. In this way, and based on the proposed methodology and modeling, it is important to point out that higher wind power integration excluding frequency control does not always imply a worse frequency response, see Fig. \[fig.nadir\_sin\_2\]. With regard to RoCoF, it varies between 0.97 and 1.93 Hz/s initially, see Fig. [\[fig.rocof\_sin\_2\]]{}, but slighted 185 mHz/s in average by including wind frequency control, Fig. [\[fig.rocof\_con\_2\]]{}. These results are substantially different from those obtained in the simplified power system analysis (where RoCoF was around 0.5 Hz/s); this is mainly due to the inertia change considered in this study and neglected in the previous one, as low inertia is related to a faster ROCOF [@daly15]. Therefore, including wind power frequency control can lead to lower frequency deviations under imbalances, as was expected. \ The proposed wind frequency response analysis allows us to evaluate the wind frequency control impact on load shedding actions in islanding power systems under different imbalances. Fig. \[fig.deslastre\_res\] summarizes the load shedding for the 30 simulated scenarios and considering the generation unit tripping obtained for the UC model, see Fig. \[fig.after\_imbalance\]. In this way, Fig. [\[fig.deslastre\_sin\_2\]]{} and Fig. [\[fig.deslastre\_con\_2\]]{} shows the corresponding load shedding responses by including or not wind frequency control for the considered energy scenarios. Both nadir and RoCoF improvements lead to a load shedding reduction in 11 scenarios. Moreover, in these 11 scenarios, the average load shedding reduction is 80%, getting up to a 100% reduction in 5 scenarios —for example, compare 30.80 MW load shedding under the range \[500, 550\] MW power demand and \[105, 120\] MW wind power generation, Fig. [\[fig.deslastre\_sin\_2\]]{}, to 0 MW load shedding under the same demand and wind power values when wind frequency control is included, Fig. [\[fig.deslastre\_con\_2\]]{}—. Table \[tab.overview\] shows a comparison of results between the proposed analysis described in this paper and conventional methodologies previously considered where a constant imbalance is assumed, inertia of the power system is kept constant during the imbalance and load shedding is not included for simulations. In the table, the average $\mu$ and variance $\sigma^{2}$ of nadir, RoCoF inertia change and load shedding values for the 30 different generation mix and imbalance scenarios are shown with and without wind frequency control. Conclusion {#sec.conclusions} ========== The case study is focused on the real isolated power system located in the Gran Canaria Island (Spain), which has doubled its wind power capacity in the last two years. With regard to the frequency analysis, by including wind power generation into frequency control, nadir and RoCoF are reduced in most of energy scenarios considered (110 mHz and 185 mHz/s in average, respectively). Regarding load shedding, it is reduced in 11 out of the 30 power imbalance analyzed. This improvement is more significant in high wind power integration scenarios (regardless of the power demand), and for high power demands (regardless of the wind power integration). Therefore, wind frequency control can be considered a remarkable solution to reduce load shedding in islanding power systems with high wind power integration. Acknowledgment ============== Authors thank Ignacio Ares for the preliminary analyses that he did as part of his final master project. Funding ======= This work has been partially supported the Spanish Ministry of Economy and Competitiveness under the project ‘Value of pumped-hydro energy storage in isolated power systems with high wind power penetration’ of the National Plan for Scientific and Technical Research and Innovation 2013-2016 (Ref. ENE2016-77951-R) and by the Spanish Education, Culture and Sports Ministry (Ref. FPU16/04282).

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--- author: - Kaifeng Huang - Bihuan Chen - Bowen Shi - Ying Wang - Congying Xu - Xin Peng bibliography: - 'src/reference.bib' title: 'Interactive, Effort-Aware Library Version Harmonization' ---

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--- author: - - - bibliography: - '../bibliography.bib' title: 'Generating Optimal Privacy-Protection Mechanisms via Machine Learning\' ---

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--- abstract: 'At present, Babcock-Leighton flux transport solar dynamo models appear as the most promising model for explaining diverse observational aspects of the sunspot cycle. The success of these flux transport dynamo models is largely dependent upon a single-cell meridional circulation with a deep equatorward component at the base of the Sun’s convection zone. However, recent observations suggest that the meridional flow may in fact be very shallow (confined to the top 10% of the Sun) and more complex than previously thought. Taken together these observations raise serious concerns on the validity of the flux transport paradigm. By accounting for the turbulent pumping of magnetic flux as evidenced in magnetohydrodynamic simulations of solar convection, we demonstrate that flux transport dynamo models can generate solar-like magnetic cycles even if the meridional flow is shallow. Solar-like periodic reversals is recovered even when meridional circulation is altogether absent, however, in this case the solar surface magnetic field dynamics does not extend all the way to the polar regions. Very importantly, our results demonstrate that the Parker-Yoshimura sign rule for dynamo wave propagation can be circumvented in Babcock-Leighton dynamo models by the latitudinal component of turbulent pumping – which can generate equatorward propagating sunspot belts in the absence of a deep, equatorward meridional flow. We also show that variations in turbulent pumping coefficients can modulate the solar cycle amplitude and periodicity. Our results suggest the viability of an alternate magnetic flux transport paradigm – mediated via turbulent pumping – for sustaining solar-stellar dynamo action.' author: - Soumitra Hazra and Dibyendu Nandy title: 'A Proposed Paradigm for Solar Cycle Dynamics Mediated via Turbulent Pumping of Magnetic Flux in Babcock-Leighton type Solar Dynamos' --- Introduction ============ The cycle of sunspots involves the generation and recycling of the Sun’s toroidal and poloidal magnetic field components. The magnetohydrodynamic (MHD) dynamo mechanism that achieves this is sustained by the energy of solar internal plasma motions such as differential rotation, turbulent convection and meridional circulation. The toroidal field is generated through stretching of the poloidal component by differential rotation [@park55] and is believed to be stored and amplified at the overshoot layer [@moren92] beneath the base of the solar convection zone (SCZ). Strong toroidal flux tubes are unstable to magnetic buoyancy and erupt through the surface producing sunspots, which are strongly magnetized and have a systematic tilt [@hale08; @hale19]. The poloidal field is believed to be regenerated through a combination of helical turbulent convection (traditionally known as the mean-field $\alpha$-effect; [@park55]) in the main body of the SCZ and the redistribution of the magnetic flux of tilted bipolar sunspot pairs (the Babcock-Leighton process; [@bab61; @leigh69]). Despite early, pioneering attempts to self-consistently model the interactions of turbulent plasma flows and magnetic fields in the context of the solar cycle [@gilm83; @glat85] such full MHD simulations are still not successful in yielding solutions that can match solar cycle observations. This task is indeed difficult, for the range of density and pressure scale heights, scale of turbulence and high Reynolds number that characterize the SCZ is difficult to capture even in the most powerful supercomputers. An alternative approach to modelling the solar cycle is based on solving the magnetic induction equation in the SCZ with observed plasma flows as inputs and with additional physics gleaned from simulations of convection and flux tube dynamics. These so called flux transport dynamo models have shown great promise in recent years in addressing a wide variety of solar cycle problems [@char10; @ossen03]. In particular, solar dynamo models based on the Babcock-Leighton mechanism for poloidal field generation have been more successful in explaining diverse observational features of the solar cycle [@dikp99; @nandy02; @chat04; @chou04; @guer07; @nandy11; @chou12; @haz14; @pass14]. Recent observations also strongly favor the Babcock-Leighton mechanism as a major source for poloidal field generation [@dasi10; @munoz13]. In this scenario, the poloidal field generation is essentially predominantly confined to near-surface layers. For the dynamo to function efficiently, the toroidal field that presumably resides deep in the interior has to reach the near-surface layers for the Babcock-Leighton poloidal source to be effective. This is achieved by the buoyant transport of magnetic flux from the Sun’s interior to its surface (through sunspot eruptions). Subsequent to this the poloidal field so generated at near-surface layers must be transported back to the solar interior, where differential rotation can generate the toroidal field. The deep meridional flow assumed in such models (See Fig. 1, left-hemisphere) plays a significant role in this flux transport process and is thought to govern the period of the sunspot cycle [@char20; @hatha03; @yeat08; @ghaz14]. Moreover, a fundamentally crucial role attributed to the deep equatorward meridional flow is that it allows the Parker-Yoshimura sign rule [@park55; @yosh75] to be overcome, which would otherwise result in poleward propagating dynamo waves in contradiction to observations that the sunspot belt migrates equatorwards with the progress of the cycle [@chou95; @ghaz14; @pass15; @belus15]. While the poleward meridional flow at the solar surface is well observed (Hathaway & Rightmire 2010; 2011) the internal meridional flow profile has remained largely unconstrained. A recent study utilizing solar supergranules [@hatha12] suggests that the meridional flow is confined to within the top 10% of the Sun (Fig. 1, right-hemisphere) – much shallower than previously thought. Independent studies utilizing helioseismic inversions are also indicative that the equatorward meridional counterflow may be located at shallow depths [@mitra07; @zhao13]. The latter also infer the flow to be multi-cellular and more complex. These studies motivate exploring alternative paradigms for flux transport dynamics in Babcock-Leighton type models of the solar cycle which are crucially dependent on meridional circulation linking the two segregated dynamo source regions in the SCZ. This leads us to consider the role of turbulent pumping. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Magnetoconvection simulations supported by theoretical considerations have established that turbulent pumping preferentially transports magnetic fields vertically downwards [@brand96; @tobias01; @ossen02; @dorch01; @kapla06; @pip09; @racine11; @rogach11; @aug15; @warn16; @sim16] – likely mediated via strong downward convective plumes which are particularly effective on weak magnetic fields (such as the poloidal component). In strong rotation regimes, there is also a significant latitudinal component of turbulent pumping. In particular, two studies, one utilizing mean-field dynamo simulations [@brand92] and the other utilizing turbulent three dimensional magnetoconvection simulations [@ossen02] recognized the possibility that turbulent pumping may contribute to the equatorward propagation of the toroidal field belt. We note that most Babcock-Leighton kinematic flux transport solar dynamo models do not include the process of turbulent pumping of magnetic flux. The few studies that exist on the impact of turbulent pumping in the context of flux transport dynamo models show it to be dynamically important in flux transport dynamics, the maintenance of solar-like parity and solar-cycle memory [@guer08; @kar12; @jiang13]. In their model with turbulent pumping, Guerrero & de Gouveia Dal Pino (2008) used a spatially distributed $\alpha$-coefficient in the near-surface layers to model the Babcock-Leighton poloidal source and a meridional circulation whose equatorward component penetrated up to $0.8R_\odot$, i.e., more than half the depth of the SCZ; therefore, from this modelling it is not possible to segregate the contributions of turbulent pumping and meridional flow (the peak latitudinal component of the former coincides with the equatorward component of the latter) to the toroidal field migration. Here, utilizing a newly developed state-of-the-art flux transport dynamo model where a double-ring algorithm is utilized to model the Babcock-Leighton process, we explore the impact of turbulent pumping in flux transport dynamo models with nonexistent, or shallow meridional circulation. Our results indicate the possibility of an alternative flux transport paradigm for the solar cycle in which turbulent pumping of magnetic flux resolves the problems posed by a shallow (or inconsequential) meridional flow. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  (a)                                          (b)        (c) -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Model ===== Our flux transport solar dynamo model solves for the coupled, evolution equation for the axisymmetric toroidal and poloidal components of the solar magnetic fields: $$\label{Eq_2.5DynA} \frac{\partial A}{\partial t} + \frac{1}{s}\left[ \textbf{v}_p \cdot \nabla (sA) \right] = \eta\left( \nabla^2 - \frac{1}{s^2} \right)A + {S_{BL}},$$ $$\begin{aligned} \label{Eq_2.5DynB} \frac{\partial B}{\partial t} + s\left[ \textbf{v}_p \cdot \nabla\left(\frac{B}{s} \right) \right] + (\nabla \cdot \textbf{v}_p)B = \eta\left( \nabla^2 - \frac{1}{s^2} \right)B \nonumber \\ + s\left(\left[ \nabla \times (A\bf \hat{e}_\phi) \right]\cdot \nabla \Omega\right) + \frac{1}{s}\frac{\partial (sB)}{\partial r}\frac{\partial \eta}{\partial r},~~~~~\end{aligned}$$ where, $B$ is the toroidal component of magnetic field and $A$ is the vector potential for the poloidal component of magnetic field. ${\textbf v}_p$ is the meridional flow, $\Omega$ is the differential rotation, $\eta$ is the turbulent magnetic diffusivity and $s = r\sin(\theta)$. For the differential rotation and diffusivity profile, we use an analytic fit to the observed solar differential rotation (the near-surface shear layer is not included) and a two-step turbulent diffusivity profile (which ensures a smooth transition to low levels of diffusivity beneath the base of the convection zone) (For detailed profile, see Hazra & Nandy 2013). We use the same meridional flow profile as defined in Hazra & Nandy (2013). Our flow profile has penetration depth of $0.65R_\odot$ to represent deep meridional flow situation, and $0.90~R_\odot$ to represent shallow meridional flow situation. We set the peak speed of the meridional flow to be 15 ms$^{-1}$ (near mid-latitudes). The second term on the RHS of the toroidal field evolution equation acts as the source term for the toroidal field (rotational shear), while in the poloidal field evolution equation, the source term, ${S_{BL}}$, is due to the Babcock-Leighton mechanism. Here we use a double-ring algorithm for buoyant sunspot eruptions that best captures the Babcock-Leighton mechanism for poloidal field generation [@durney97; @nandy01; @munoz10; @haz13] and which has been tested thoroughly in other contexts. Specifics about our double ring algorithm can be found in Hazra & Nandy (2013) and Hazra (2016; PhD Thesis). --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Results ======= To bring out the significance of the recent observations, we first consider a single cell, shallow meridional flow, confined only to the top 10% of the convection zone (Fig. 1, right-hemisphere). In the first scenario we seek to answer the following question: Can solar-like cycles be sustained through magnetic field dynamics completely confined to the top 10% of the Sun? In these simulations initialized with antisymmetric toroidal field condition (with initial B $\sim$ 100 kG), we first allow magnetic flux tubes to buoyantly erupt from 0.90 $R_{\odot}$ (i.e., the depth to which the shallow flow is confined) when they exceed a buoyancy threshold of $10^4$ Gauss (G). In this case we find that the simulated fields fall and remain below this threshold (at all latitudes at 0.90 $R_{\odot}$) with no buoyant eruptions, implying that a Babcock-Leighton type solar dynamo cannot operate in this case. Dikpati [*et al.*]{} (2002) considered the contribution of the near-surface shear layer in their simulations (which we have not) and concluded that this near-surface layer contributes only about 1 kG to the total toroidal field production and hence insufficient to drive a large-scale dynamo. Guerrero & de Gouveia Dal Pino (2008) also utilized a near-surface shear layer with radial pumping and found solar-like solutions only under special circumstances; however, given that for this particular case they utilized a local $\alpha$-effect for the latter simulations (with a spatially distributed $\alpha$-effect in the near-surface layer) it is not evident that these simulations are relatable to the Babcock-Leighton solar dynamo concept. The upper layers of the SCZ is highly turbulent and storage and amplification of strong magnetic flux tubes may not be possible in these layers [@park75; @moren83] and therefore this result is not unexpected. While Brandenburg (2005) has conjectured that the near-surface shear layer may be able to power a large-scale dynamo, this remains to be convincingly demonstrated in the context of a Babcock-Leighton dynamo. In the second scenario with a shallow meridional flow, we allow magnetic flux tubes to buoyantly erupt from 0.71 $R_{\odot}$, i.e. from base of the convection zone. In this case we get periodic solutions but analysis of the butterfly diagrams (taken both at the base of SCZ and near solar surface) shows that the toroidal field belts have almost symmetrical poleward and equatorward branches with no significant equatorward migration (see Fig. 2). Moreover, as already noted by Guerrero, G. & de Gouveia Dal Pino (2008), the solutions with shallow meridional flow always display quadrupolar parity in contradiction with solar cycle observations. Clearly, a shallow flow poses a serious problem for solar cycle models. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  (a)                                               (b)      (c) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- We now introduce both radial and latitudinal turbulent pumping in our dynamo model to explore whether a Babcock Leighton flux transport dynamo can operate with meridional flow which is much shallower than previously assumed; we also extend this study to the scenario where meridional flow is altogether absent. The turbulent pumping profile is determined from independent MHD simulations of solar magnetoconvection [@ossen02; @kapla06]. Profiles for radial and latitudinal turbulent pumping ($\gamma_r$ and $\gamma_\theta$) are: $$\begin{aligned} \gamma_r = - \gamma_{0r} \left[ 1 + \rm{erf}\left( \frac{r - 0.715R_\odot}{0.015R_\odot}\right) \right] \left[ 1 - \rm{erf} \left( \frac{r-0.97R_\odot}{0.1R_\odot}\right) \right] \nonumber \\ \times \left[ \rm{exp}\left( \frac{r-0.715R_\odot}{0.25R_\odot}\right) ^2 \rm{cos}\theta +1\right] ~~~~\end{aligned}$$ $$\begin{aligned} \gamma_\theta = \gamma_{0\theta} \left[1+\mathrm{erf}\left(\frac{r-0.8R_\odot}{0.55R_\odot}\right)\right] \left[1-\mathrm{erf}\left(\frac{r-0.98R_\odot}{0.025R_\odot}\right)\right] \times \cos \theta \sin^4 \theta ~~~~\end{aligned}$$ The value of $\gamma_{0r}$ and $\gamma_{0\theta}$ determines the amplitude of $\gamma_r$ and $\gamma_\theta$ respectively. Fig. 3 (top and bottom plot) shows that radial pumping speed (dashed lines) is negative throughout the convection zone corresponding to downward advective transport and vanishes below $0.7R_\odot$. The radial pumping speed is maximum near the poles and decreases towards the equator. Fig. 3 (top and bottom plot) shows that the latitudinal pumping speed (solid lines) is positive (negative) in the convection zone in the northern (southern) hemisphere and vanishes below the overshoot layer. This corresponds to equatorward latitudinal pumping throughout the convection zone. Dynamo simulations with turbulent pumping generate solar-like magnetic cycles (Fig. 4 and Fig. 5). Now the toroidal field belt migrates equatorward, the solution exhibits solar-like parity and the correct phase relationship between the toroidal and poloidal components of the magnetic field (see Fig. 5). Evidently, the coupling between the poloidal source at the near-surface layers with the deeper layers of the convection zone where the toroidal field is stored and amplified, the equatorward migration of the sunspot-forming toroidal field belt and correct solar-like parity is due to the important role played by turbulent pumping. We note if the speed of the latitudinal pumping in on order of 1.0 ms$^{-1}$ the solutions are always of dipolar parity irrespective of whether one initializes the model with dipolar or quadrupolar parity. Interestingly, the latitudinal migration rate of the sunspot belt as observed is of the same order. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  (a)                                              (b)      (c) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- The above result begs the question whether flux transport solar dynamo models based on the Babcock-Leighton mechanism that include turbulent pumping can operate without any meridional plasma flow. To test this, we remove meridional circulation completely from our model and perform simulations with turbulent pumping included. We find that this model generates solar-like sunspot cycles with periodic reversals (see Fig. 6) which are qualitatively similar to the earlier solution with both pumping and shallow meridional flow. However, we find that the surface magnetic field dynamics related to polar field reversal is limited to within 60 degrees latitudes in both the hemispheres. At higher latitudes (near the poles) the field is very weak and almost non-varying over solar cycle timescales. This is expected if the surface magnetic field dynamics is governed primarily by diffusion. Based on this result, we argue that this scenario of non-existent meridional circulation is not supported by current observations of surface dynamics which seem to suggest that the fields do migrate all the way to the poles. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Two important characteristics associated with the solar magnetic cycle are its amplitude and periodicity. While the periodicity of the cycle predominantly depends on the recycling time between toroidal and poloidal field, its amplitude depends on a variety of factors including dynamo source strengths and relative efficacy of transport timescales with respect to the turbulent diffusion timescale. We explore the dependency of the solar cycle period and amplitude to variations in the transport coefficients to explore the subtleties of the interplay between diverse flux transport processes. Figure 7 shows the dependency of cycle amplitude and periodicity on different velocity components like turbulent pumping and (shallow) meridional flow. A parametric analysis of this dependency yields the following relationships for cycle period ($T$) and cycle amplitude (Amp): $$T \simeq 9.7 ~ \gamma_r^{-0.25} \gamma_\theta^{-0.26} v^{-0.068},$$ $$Amp \simeq 11.76 ~ \gamma_r^{1.07} \gamma_\theta^{-0.27} v^{-0.16},$$ which is gleaned from simulations within the following ranges: $0.25 ~ms^{-1} \leq \gamma_r \leq 1.25 ~ms^{-1}$, $0.25 ~ms^{-1} \leq \gamma_\theta \leq 1.25 ~ms^{-1}$ and $2 ~ms^{-1} \leq v \leq 15 ~ms^{-1}$; $\gamma_r$ and $\gamma_\theta$ are radial and latitudinal turbulent pumping speeds, and $v$ is the shallow meridional flow speed. This analysis shows that cycle period and amplitude are both governed by diverse transport coefficients such as meridional flow speed, and radial and latitudinal components of turbulent pumping. As radial turbulent pumping carries the flux directly to the base of the convection zone where toroidal field is amplified, increase in the radial turbulent pumping speed leads to a decrease in cycle period. Increasing latitudinal pumping also has a similar effect on period which is similar to what is achieved by increasing meridional flow speed, namely a faster transport through the shear layer and thus shorter cycle periods. The cycle amplitude decreases on increasing the latitudinal pumping or meridional flow speed and this is due to the fact that less time is available for toroidal field induction when it is swept at a faster rate through the rotational shear layers. In surface flux transport models, a similar effect is found but due to a different reason – wherein a faster meridional flow reduces the polar field strength because it takes flux of both polarity and deposits this at the poles (in effect carrying less net flux to the poles); in these simulations with a shallow meridional flow and the double-ring algorithm a similar mechanism could also be contributing to an overall reduction of the field strength. What is interesting to note though is the positive dependence of cycle amplitude on the radial pumping speed. We believe that a faster radial pumping moves the poloidal field down to the generating layers of the toroidal field in the deeper parts of the convection zone faster, thus allowing for less turbulent decay in the poloidal field strength; this eventually results in a stronger poloidal field in the SCZ which generates a stronger toroidal component. We note that the derived exponents for the cycle period above differ from that determined by Guerrero & de Gouveia Dal Pino (2008). The cycle period in our simulations is more strongly dependent on the latitudinal speed of turbulent pumping and less so on meridional circulation, whereas in Guerrero & de Gouveia Dal Pino (2008) it is the exact reverse. In our model the meridional flow is very shallow and limited to only the top $10 \%$ of the SCZ, whereas in the model setup of Guerrero & de Gouveia Dal Pino (2008), the meridional flow penetrates down to about 0.8 $R_\odot$; this we believe makes their dynamo cycle periods more sensitive to meridional flow as compared to latitudinal pumping. Generally, we find solar-like solutions in a modest turbulent pumping speed range on the order of 1 ms$^{-1}$. This parameter study shows that our result are robust to reasonable variations in turbulent pumping coefficients and also points to how the latter may determine solar cycle strength and periodicity. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --   ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- As there is some uncertainty regarding the exact details of turbulent pumping profiles, we have tested an alternative turbulent pumping profile based on Warnecke *et al.* (2016). Recent magnetoconvection simulations performed by Warnecke et al. (2016) suggest that radial pumping is downward throughout the convection zone below $45^{\circ}$ and upward above $45^{\circ}$, while latitudinal pumping is poleward at the surface and equatorward at the base of the convection zone. Our generated turbulent pumping profiles in the northern hemisphere (defined within $0 \leq \theta \leq \pi/2$) based on the suggestions of Warnecke *et al.* (2006) are: $$\begin{aligned} \gamma_r = - \gamma_{0r} \left[ 1 + \rm{erf}\left( \frac{r - 0.715R_\odot}{0.05R_\odot}\right) \right] \left[ 1 - \rm{erf} \left( \frac{r-0.98R_\odot}{0.08R_\odot}\right) \right] \nonumber \\ \times \sin(4 \theta), ~~~~\end{aligned}$$ $$\begin{aligned} \gamma_\theta = \left\{\begin{array}{cc} \gamma_{0\theta} \sin \left[\frac{2 \pi (r- R_p)}{R_0-R_p}\right] \times \cos \theta \sin^4 \theta ~~~~ & r \geq R_p\\ 0 & r<R_p \end{array}\right.,\end{aligned}$$ where $R_p= 0.76R_\odot$ i.e. the penetration depth of the latitudinal pumping. The amplitudes of $\gamma_r$ and $\gamma_\theta$ are determined by the value of $\gamma_{0r}$ and $\gamma_{0\theta}$ respectively. Turbulent pumping profiles in the southern hemisphere are generated by replacing colatitude $\theta$ by $(\pi- \theta)$. Fig. 8 (the top and middle panels) show that our generated turbulent pumping profiles capture the basic essence of the suggestions made by Warnecke *et al.* (2016). Our simulations (with shallow meridional flow) and the more complex turbulent pumping profile gleaned from Warnecke *et al.* (2016) reproduce broad features of the solar cycle and are qualitatively similar to those detailed earlier. Discussions =========== In summary, we have demonstrated that flux transport dynamo models of the solar cycle based on the Babcock-Leighton mechanism for poloidal field generation does not require a deep equatorward meridional plasma flow to function effectively. In fact, our results indicate that when turbulent pumping of magnetic flux is taken in to consideration, dynamo models can generate solar-like magnetic cycles even without any meridional circulation although the surface magnetic field dynamics does not reach all the way to the polar regions in this case. Our conclusions are robust across a modest range of plausible parameter space for turbulent pumping coefficients and also indicate some tolerance for diverse pumping profiles. These findings have significant implications for our understanding of the solar cycle. First of all, the serious challenges that were apparently posed by observations of a shallow (and perhaps complex, multi-cellular) meridional flow on the very premise of flux transport dynamo models stands resolved. Turbulent pumping essentially takes over the role of meridional circulation by transporting magnetic fields from the near-surface solar layers to the deep interior, ensuring that efficient recycling of toroidal and poloidal field components across the SCZ is not compromised. While these findings augur well for dynamo models of the solar cycle, they also imply that we need to revisit many aspects of our current understanding if indeed meridional circulation is not as effective as previously thought. For example, our simulations indicate that variations in turbulent pumping speeds can be an effective means for the modulation of solar cycle periodicity and amplitude. It has been argued earlier that the interplay between competing flux transport processes determine the dynamical memory of the solar cycle governing solar cycle predictability [@yeat08]. If turbulent pumping is the dominant flux transport process as seems plausible based on the simulations presented herein, the cycle memory would be short and this is indeed supported by independent studies [@kar12] and solar cycle observations [@munoz13]. It is noteworthy that on the other hand, if meridional circulation were to be the dominant flux transport process, the solar cycle memory would be relatively longer and last over several cycles. This is not borne out by observations. Previous results in the context of the maintenance of solar-like dipolar parity have relied on a strong turbulent diffusion to couple the Northern and Southern hemispheres of the Sun [@chat04], or a dynamo $\alpha$-effect which is co-spatial with the deep equatorward counterflow in the meridional circulation assumed in most flux transport dynamo models [@dikp01]. However, our results indicate that turbulent pumping is equally capable of coupling the Northern and Southern solar hemispheres and aid in the maintenance of solar-like dipolar parity. This is in keeping with earlier, independent simulations based on a somewhat different dynamo model [@guer08]. Most importantly, our results point out an alternative to circumventing the Parker-Yoshimura sign rule constraint [@park55; @yosh75] in Babcock-Leighton type solar dynamos that would otherwise imply poleward propagating sunspot belts in conflict with observations. Brandenburg [*et al.*]{} (1992) and Ossendrijver [*et al.*]{} (2002) had already pointed towards this possibility in the context of mean-field dynamo models. While a deep meridional counterflow is currently thought to circumvent this constraint and force the toroidal field belt equatorward, our results convincingly demonstrate that the latitudinal component of turbulent pumping provides a viable alternative to overcoming the Parker-Yoshimura sign rule in Babcock-Leighton models of the solar cycle (even in the absence of meridional circulation). We note however that our theoretical results should not be taken as support for the existence of a shallow meridional flow, rather we point out that flux transport dynamo models of the solar cycle are equally capable for working with a shallow or non-existent meridional flow, as long as the turbulent pumping of magnetic flux is accounted for; this is particularly viable when turbulent pumping has a dynamically important latitudinal component. Taken together, these insights suggest a plausible new paradigm for dynamo models of the solar cycle, wherein, turbulent pumping of magnetic flux effectively replaces the important roles that are currently thought to be mediated via a deep meridional circulation within the Sun’s interior. Since the dynamical memory and thus predictability of the solar cycle depends on the dominant mode of magnetic flux transport in the Sun’s interior, this would also imply that physics-based prediction models of long-term space weather need to adequately include the physics of turbulent pumping of magnetic fields. We acknowledge the referee of this manuscript for useful suggestions. We thank Jörn Warnecke for helpful discussions related to the adaptation of the turbulent pumping profile from Warnecke [*et al.*]{} 2016. 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--- abstract: 'We discuss properties of $L^2$-eigenfunctions of Schrödinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce new ones.' address: - | Institute of Mathematics USC RAS, Chernyshevskii str., 112,\ Ufa, 450000, Russia\ & Bashkir State Pedagogical University, October rev. st., 3a,\ Ufa, 450008, Russia,\ E-mail: [email protected]\ matem.anrb.ru & www.bspu.ru - | Fakultät für Mathematik, Reichenhainer Str. 41,\ Chemnitz, D-09126, Germany,\ www.tu-chemnitz.de/\~mtau - | Fakultät für Mathematik, Reichenhainer Str. 41,\ Chemnitz, D-09126, Germany,\ www.tu-chemnitz.de/stochastik/ author: - 'D. Borisov' - 'M. Tautenhahn' - 'I. Veselić' title: Equidistribution estimates for eigenfunctions and eigenvalue bounds for random operators --- Introduction ============ In this note we present recent results in Harmonic Analysis for solutions of (time-independent) Schr[ö]{}dinger equations and other partial differential equations. They are motivated by interest in techniques relevant for proving localization for random Schr[ö]{}dinger operators. The mentioned Harmonic Analysis results which we present are a quantitative unique continuation principle and an equidistribution property for eigenfunctions, which is scale-uniform. These results, and variants thereof, go under various names, depending on the particular field of mathematics: They are called observability estimate, uncertainty relation, scale-free unique continuation principle, or local positive definiteness. The latter term signifies that a self-adjoint operator is (strictly) positive definite when restricted to a relevant subspace, while it is not so on the whole Hilbert space. For the purpose of motivation we discuss this property in the next section. The term *localization* refers to the phenomenon, that quantum Hamiltonians describing the movement of electrons in certain disordered media exhibit pure point spectrum in appropriately specified energy regions. The corresponding eigenfunctions decay exponentially in space. The (time-dependent) wavepackets describing electrons stay localized essentially in a compact region of space for all times. Nota bene, all mentioned properties hold *almost surely*. This is natural in the context of random operators. An important partial result for deriving localization are Wegner estimates. These are bounds on the expected number of eigenvalues in a bounded energy interval of a random Schr[ö]{}dinger operator restricted to a box. The localization problem has been studied for other classes of random operators beyond those of Schr[ö]{}dinger type. An example are random divergence type operators, see e.g. Refs.  and . This are partial differential operators with randomness in coefficients of higher order terms. In paricular, the second order term is no longer the Laplacian, but a variable coefficient operator. In this context one is again lead to consider the above mentioned questions of Harmonic Analysis for eigenfunctions of differential operators. In this note we present an exposition of recently published results, and an announcement of a quantitative unique continuation principle and an equidistribution estimate for eigenfunctions for a class of elliptic operators with variable coefficients. Motivation: Moving and lifting of eigenvalues {#ss:motivation} --------------------------------------------- Here we discuss some aspects of eigenvalue perturbation theory. It will provide an accessible explanation why one is interested in the results presented in Sections \[sec:schroedinger\] and \[sec:elliptic\] below in the context of random Schr[ö]{}dinger operators and elliptic differential operators, respectively. In fact, to illustrate the main questions it will be for the moment completely sufficient to restrict our attention to the finite dimensional situation, i.e. to perturbation theory for finite symmetric matrices. The focus will be on how (local) positive definiteness of the perturbation relates to lifting of eigenvalues. Let $A$ and $B$ be symmetric $n \times n$ matrices, with $B\geq b>0$ positive definite. The variational min-max principle for eigenvalues shows that for any $k \in \{1,\dots, n\}$ and $t\geq 0$ $$\label{eq:positive_definite_perturbation} \lambda_k(A+tB) \geq \lambda_k(A) + b \, t$$ where $\lambda_k(M)$ denotes the $k$th lowest eigenvalue, counting multiplicities, of a symmetric matrix $M$. Note that the dimension $n$ does not enter in the bound . Without the positive definiteness assumption on $B$ this universal bound will fail, most blatantly if $$A =\begin{pmatrix} A_1 & \ 0 \\ 0 & A_2 \end{pmatrix} \quad \text{and} \quad B =\begin{pmatrix} \operatorname{Id} & \ 0 \\ 0 & -\operatorname{Id} \end{pmatrix} .$$ In this case, all eigenvalue $\lambda_k(A+tB)$ *will move*, even with constant speed w.r.t. the variable $t$, albeit in different directions. If $B$ is singular, some eigenvalues may not move at all. However, for appropriate classes of symmetric matrices $A$, and of positive semidefinite matrices $B$, one may still aim to prove $$\label{eq:positive_semidefinite_perturbation} \forall \, t\geq 0, k \in \{1,\dots, n\} \, \exists \, \kappa >0 \text{ such that } \lambda_k(A+tB) \geq \lambda_k(A) + \kappa t$$ Note however, that $\kappa $ is now not a uniform bound but depends on the class of symmetric matrices from which $A$ is chosen, the class of semidefinite matrices from which $B$ is chosen, the range from which the coupling $t$ is chosen, and the range from which the index $k \in \{1,\dots, n\}$ is chosen. In the case of random operators or matrices one in is interested in the situation where $$\label{eq:multiparameter_family} A(\omega) =A_0+\sum_{j \in Q} \omega_j B_j =\Big(A_0+\sum_{j \in Q, j\neq 0} \omega_j B_j \Big)+ \omega_0 B_0$$ is a multi-parameter pencil. Here $Q$ is some subset of ${\mathbb{Z}}^d$ containing $0$. The real variables $\omega_j$ model random coupling constants determining the strength of the perturbation $B_j$ in each configuration $\omega=(\omega_j)_{j \in Q}$. Now, already suggest to write $A(\omega)$ as $$A(\omega_0^\perp)+tB \quad\text{where} \quad t=\omega_0,\ B=B_0,\ \text{and} \ \omega_0^\perp=(\omega_j)_{j \in Q, j\neq0} .$$ This highlights that if we consider $A(\omega)$ as a function of the single variable $t=\omega_0$, it is clearly a one-parameter family of operators, albeit the “unperturbed part” $A(\omega_0^\perp)$ of $A(\omega)=A(\omega_0^\perp)+tB$ is not a single operator, but varying over the ensemble $(A(\omega_0^\perp))_{\omega_0^\perp}$. To have a useful version of in this situation, the constant $\kappa $ needs to have a uniform lower bound $\inf_{A} \kappa $ where $A=A(\omega_0^\perp)$ varies over all matrices in the ensemble. In what follows we present rigorous results of the type , but where $A$ and $B$ are not finite matrices, but differential and multiplication operators. The relevant operators have all compact resolvent, ensuring that the entire spectrum consists of eigenvalues. Equidistribution property of Schrödinger eigenfunctions {#sec:schroedinger} ======================================================= The following result is taken from Ref. . It is an equidistribution estimate for Schr[ö]{}dinger eigenfunctions, which is uniform w.r.t. the naturally arising length scales, and has strong implications for the spectral theory of random Schrödinger operators. We fix some notation. For $L>0$ we denote by $\Lambda_L = (-L/2 , L/2)^d$ a cube in ${\mathbb{R}}^d$. For $\delta>0$ the open ball centered at $x\in {\mathbb{R}}$ with radius $\delta$ is denoted by $B(x, \delta)$. For a sequence of points $(x_j)_j$ indexed by $j \in {\mathbb{Z}}^d$ we denote the collection of balls $\cup_{j \in {\mathbb{Z}}^d} B(x_j , \delta) $ by $S$ and its intersection with $\Lambda_L$ by $S_L$. We will be dealing with certain subspaces of the standard second order Sobolev space $W^{2,2}({\Lambda}_L)$ on the cube. Let $\Delta$ be the $d$-dimensional Laplacian. Its restriction to the cube ${\Lambda}={\Lambda}_L$ needs boundary conditions to be self-adjoint. The domain of the Dirichlet Laplacian will be denoted by ${\mathcal{D}}(\Delta_{{\Lambda},0})$ and the domain of the Laplacian with periodic boundary conditions by ${\mathcal{D}}(\Delta_{{\Lambda},\mathrm{per}})$. Let $V \colon {\mathbb{R}}^d\to {\mathbb{R}}$ be a bounded measurable function, and $H_L = (-\Delta + V)_{\Lambda_L} $ a Schrödinger operator on the cube $\Lambda_L$ with Dirichlet or periodic boundary conditions. The corresponding domains are still ${\mathcal{D}}(\Delta_{{\Lambda},0})$ and $ {\mathcal{D}}(\Delta_{{\Lambda},\mathrm{per}})$, respectively. Note that we denote a multiplication operator by the same symbol as the corresponding function. The following theorem was proven in Ref. . \[thm:RojasVeselic\] Let $\delta, K_{V} > 0$. Then there exists $C_{\rm sfUC} \in (0,\infty)$ such that for all $L \in 2{\mathbb{N}}+1 $, all measurable $V : {\mathbb{R}}^d \to [-K_{V} , K_{V}]$, all real-valued $\psi \in {\mathcal{D}}(\Delta_{{\Lambda},0}) \cup {\mathcal{D}}(\Delta_{{\Lambda},\mathrm{per}})$ with $(-\Delta + V)\psi = 0$ almost everywhere on $\Lambda_L$, and all sequences $(x_j)_{j \in {\mathbb{Z}}^d} \subset {\mathbb{R}}^d$, such that for all $j \in {\mathbb{Z}}^d$ the ball $B(x_j , \delta) \subset \Lambda_1 + j$, we have $$\label{eq:observability} \int_{S_L} \psi^2 \geq C_{\rm sfUC} \int_{\Lambda_L} \psi^2 .$$ The value of the result is not in the *existence* of the constant $C_{\rm sfUC}$, but in the *quantitative control* of the dependence of $C_{\rm sfUC}$ on parameters entering the model. The very formulation of the theorem states that $C_{\rm sfUC}$ is independent of the position of the balls $B(x_j,\delta)$ within $\Lambda_1 +j$, and independent of the scale $L\in2\mathbb{N} +1$. From the estimates given in Section 2 of Ref.  one infers that $C_{\rm sfUC}$ depends on the potential $V$ only through the norm $\lVert V \rVert_\infty$ (on an exponential scale), and it depends on the small radius $\delta>0$ polynomially, i.e. $C\gtrsim \delta^N$, for some $N\in\mathbb{N}$ which depends on the dimension on $d$ and $\lVert V \rVert_\infty$. The theorem states a property of functions in the kernel of the operator. It is easily applied to eigenfunctions corresponding to other eigenvalues since $$H_L\psi=E\psi \Leftrightarrow (H_L-E)\psi=0 .$$ As a consequence of the energy shift the constant $K_{V}$ has to be replaced with $K_{V-E}$, which may be larger than $K_{V}$. It may always be estimated by $K_{V-E}\leq K_V+|E|$. There is a very natural question supported by earlier results, which was spelled out in Ref. , namely does the following generalisation of Theorem \[thm:RojasVeselic\] hold: Given $\delta >0$, $K\geq0$ and $E\in\mathbb{R}$ there is a constant $C>0$ such that for all measurable $ V\colon \mathbb{R}^d \rightarrow [-K,K] $, all $L \in 2{\mathbb{N}}+1$, and all sequences $(x_j)_{j\in\mathbb{Z}^d} \subset \mathbb{R}^d$ with $B(x_j,\delta) \subset\Lambda_1 +j$ for all $j \in {\mathbb{Z}}^d$ we have $$\label{eq:uncertainty} \chi_{(-\infty,E]} (H_L) \, W_L \, \chi_{(-\infty,E]} (H_L) \geq C~ \chi_{(-\infty,E]} (H_L) ,$$ where $W_L=\chi_{S_L}$ is the indicator function of $S_L$ and $\chi_{I} (H_L)$ denotes the spectral projector of $H_L$ onto the interval $I$. Here $C=C_{\delta, K, E}$ is determined by $\delta, K, E$ alone. Klein obtained a positive answer to the question for sufficiently short subintervals of $(-\infty,E]$. \[thm:Klein-13\] Let $d \in {\mathbb{N}}$, $E\in {\mathbb{R}}$, $\delta\in (0,1/2]$ and $V:{\mathbb{R}}^d \to {\mathbb{R}}$ be measurable and bounded. There is a constant $M_d>0$ such that if we set $$\gamma = \frac{1}{2} \delta^{M_d \bigl(1 + (2\lVert V \rVert_\infty + E)^{2/3}\bigr)} ,$$ then for all energy intervals $I\subset (-\infty, E]$ with length bounded by $2\gamma$, all $L \in 2{\mathbb{N}}+1$, $L\geq 72 \sqrt{d}$ and all sequences $(x_j)_{j\in\mathbb{Z}^d} \subset \mathbb{R}^d$ with $B(x_j,\delta) \subset\Lambda_1 +j$ for all $j \in {\mathbb{Z}}^d$ $$\chi_{I} (H_L) \, W_L \, \chi_{I} (H_L) \geq \gamma^2\chi_{I} (H_L) .$$ This does not answer the above posed question question completely due to the restriction $|I| \leq 2\gamma$. However, the result is sufficient for many questions in spectral theory of random Schrödinger operators. For a history of the questions discussed here and earlier results we refer to Ref. . Random Schr[ö]{}dinger operators {#ss:rSo} -------------------------------- Let ${{\Lambda}_L}$ be a cube of side $L\in2{\mathbb{N}}+1$, $(\Omega, {\mathbb{P}})$ a probability space, $V_0 \colon {{\Lambda}_L}\to {\mathbb{R}}$ a bounded, measurable deterministic potential, $V_\omega \colon {{\Lambda}_L}\to {\mathbb{R}}$ a bounded random potential and $H_{\omega,L}= (-\Delta + V_0+V_\omega)_{{\Lambda}_L}$ a random Schrödinger operator on $L^2({{\Lambda}_L})$ with Dirichlet or periodic boundary conditions. We assume that the random potential is of Delone-Anderson form $$V_\omega(x):= \sum_{j \in{{\mathbb{Z}}^d}} \ \omega_j u_j(x) .$$ The random variables $\omega_j, j\in {{\mathbb{Z}}^d},$ are independent with probability distributions $\mu_j$, such that for some $m>0$ an all $j\in {{\mathbb{Z}}^d}$ we have $\operatorname{\operatorname{supp}}\mu_j \subset [-m, m]$. Fix $0 < \delta_- < \delta_+<\infty$ and $0 < C_- \leq C_+ <\infty$. The sequence of measurable functions $u_j \colon {\mathbb{R}}^d \to {\mathbb{R}}$, $j \in {{\mathbb{Z}}^d}$, is such that $$\begin{aligned} \forall j \in {{\mathbb{Z}}^d}: \quad C_- \chi_{B(z_j,\delta_-)} \leq u_j \leq C_+ \chi_{B(z_j,\delta_+)}, \ \text{and} \ B(z_j,\delta_-) \subset {\Lambda}_1 + j . \end{aligned}$$ Lifting of eigenvalues {#ss:lifting} ---------------------- Let $\lambda_k^L(\omega)$ denote the eigenvalues of $H_{\omega,L}$ enumerated in non-decreasing order and counting multiplicities and $\psi_k=\psi_k^L(\omega)$ the normalised eigenvectors corresponding to $\lambda_k^L(\omega)$. While we suppress the dependence of $\psi_k$ on $L$ and $\omega$ in the notation, it should be kept in mind. Then $$\lambda_k^L(\omega) = {\langle}\psi_k, H_{\omega,L} \psi_k{\rangle}= \int_{{\Lambda}_L} \overline{\psi_k} ( H_{\omega,L} \psi_k ) .$$ Define the vector $ e=(e_j)_{j\in{{\mathbb{Z}}^d}}$ by $e_j=1$ for $j\in{{\mathbb{Z}}^d}$. Consider the monotone shift of $V_\omega$ $$V_{\omega+ {t} \cdot e} = \sum_{j \in{{\mathbb{Z}}^d}} (\omega_j+ {t} ) u_j$$ and set ${Q}={Q}_L= \Lambda_L \cap {{\mathbb{Z}}^d}$. By first order perturbation theory we have $$\frac{\rm d}{{\rm d}{\tau}} \lambda_k^L(\omega+ {\tau} \cdot e) |_{\tau=t} = \langle \psi_k, \sum_{k \in {Q}} u_j \, \psi_k {\rangle}.$$ Note that the right hand side depends on $t$ implicitly through the eigenfunction $\psi_k$. Let us fix some $E_0\in{\mathbb{R}}$ and restrict our attention only to those eigenvalues satisfying $\lambda_n^L(\omega) \leq E_0$. By Theorem \[thm:RojasVeselic\] there exists a constant $C_{\rm sfUC}$ depending on the energy $E_0$, $\delta_-$ and the overall supremum $$\label{eq:Vsupremum} \sup_{|s|\leq m} \ \sup_{|\omega_j|\leq m} \ \sup_{x\in{\mathbb{R}}^d} \big|V_{0}(x) +V_\omega(x) +s \sum_{j\in Q} u_j \big|$$ of the potential, such that $$\sum_{k \in {Q}} {\langle}\psi_k, u_j \, \psi_k {\rangle}\geq C_- \sum_{k \in {Q}}{\langle}\psi_k, \chi_{B(z_k,\delta_-)}\psi_k {\rangle}\geq C_-\cdot C_{\rm sfUC} =: \kappa .$$ Here we used that $\|\psi\|_{L^2 (\Lambda)}=1$. (Note that the quantity $\kappa$ depends a-priori on the model parameters.) Integrating the derivative gives $$\begin{aligned} \nonumber \lambda_k^L(\omega+ {t} \cdot e) &= \lambda_k^L(\omega) + \int_0^{t} \frac{{\mathrm{d}}\lambda_k^L(\omega+ \tau \cdot e) }{{\mathrm{d}}\tau}|_{\tau=s} \, {\mathrm{d}}s \\ & \geq \lambda_k^L(\omega) + \int_0^{t} \kappa \, {\mathrm{d}}s = \lambda_k^L(\omega) + t \kappa . \label{eq:lifting}\end{aligned}$$ This is the lifting estimate for eigenvalues of random (Schrödinger) operators alluded to in §\[ss:motivation\]. It should be compared with there. Indeed, due to the uniform nature of the estimate in Theorem \[thm:RojasVeselic\] we have $$\label{eq:uniform_kappa} \inf_{ L \in 2{\mathbb{N}}+1} \ \inf_{\omega \text{ s.t. } \forall \, k : |\omega_j|\leq m} \ \inf_{ |{t}|\leq m} \ \inf_{n \text{ s.t. } \lambda_n^L(\omega)\leq E_0} \kappa >0 .$$ Thus eigenvalues lifting estimate is almost as uniform as . A parameter, with respect to which the lifting estimate is *not* uniform is the cut-off energy $E_0$. Indeed, if we add in an infimum over $E_0>0$ on the left hand side, it becomes zero, unless $\sum_k\chi_{B(z_k,\delta_-)}\geq 1$ almost everywhere on ${\mathbb{R}}^d$. Wegner estimates ---------------- Here we present a Wegner estimate. Such estimates play an important role in the proof of localization via the multiscale analysis. The latter is an induction argument over increasing length scales. The Wegner bound is used to prove the induction step. Let $ s\colon [0,\infty) \to[0,1]$ be the global modulus of continuity of the family $\{\mu_j\}_{j\in {{\mathbb{Z}}^d}}$, that is, $$\label{definition-s-mu-epsilon} s(\epsilon):= \sup_{j \in {{\mathbb{Z}}^d}} \sup_{a \in {\mathbb{R}}} \, \mu_j\Big(\Big[a-\frac{\epsilon}{2},a+\frac{\epsilon}{2}\Big]\Big)$$ The main result of Ref.  on the model described in the last paragraph is a Wegner estimate which is valid for all compact energy intervals. \[t:Wegner\] Let $H_{\omega,L}$ be a random Schrödinger operator as in §\[ss:rSo\]. Then for each $E_0\in {\mathbb{R}}$ there exists a constant $C_W$, such that for all $E\le E_0$, $\epsilon \le 1/3$, and all $L\in 2{\mathbb{N}}+1$ we have $$\label{eq:WE} {\mathbb{E}}\{{{\mathop{\mathrm{Tr} \,}}}[ \chi_{[E-\epsilon,E+\epsilon]}(H_{\omega, L}) ]\} \le C_W \ s(\epsilon) \, \lvert \ln \, \epsilon \rvert^d \ \lvert \Lambda_L \rvert .$$ The Wegner constant $C_W$ depends only on $E_0$, $\|V_0\|_\infty$, $m$, $C_-$, $C_+$, $\delta_-$, and $\delta_+$. Klein[@Klein-13] obtains an improvement over this result based on his above quoted Theorem \[thm:Klein-13\]. There are many earlier, related Wegner estimates. For an overview we refer to Ref. . Comparison of local $L^2$-norms ------------------------------- An important step in the proof of Theorem \[thm:RojasVeselic\] is the following result which compares $L^2$-norms of the restrictions of a PDE-solution to two distinct subsets. In our applications the solution will be an eigenfunction of the Schrödinger operator. Various estimates of this type have been given in Refs. , and . We quote here the version from the last mentioned paper. \[thm:quantitative-UCP\] Let $K, R, \beta\in [0, \infty), \delta \in (0,1]$. There exists a constant $C_{\rm qUC}=C_{\rm qUC}(d,K, R,\delta, \beta) >0$ such that, for any $G\subset {\mathbb{R}}^d$ open, any $\Theta\subset G$ measurable, satisfying the geometric conditions $$\operatorname{diam} \Theta + \operatorname{dist} (0 , \Theta) \leq 2R \leq 2 \operatorname{dist} (0 , \Theta), \quad \delta < 4R, \quad B(0, 14R ) \subset G,$$ and any measurable $V\colon G \to [-K,K]$ and real-valued $\psi\in W^{2,2}(G)$ satisfying the differential inequality $$\label{eq:subsolution} \lvert \Delta \psi \rvert \leq \lvert V\psi \rvert \quad \text{a.e.on } G \quad \text{ as well as } \quad \int_{G} \lvert \psi \rvert^2 \leq \beta \int_{\Theta} \lvert \psi \rvert^2 ,$$ we have $$\label{eq:aim} \int_{B(0,\delta)} \lvert \psi \rvert^2 \geq C_{\rm qUC} \int_{\Theta} \lvert \psi\rvert^2 .$$ plot\[smooth cycle\] coordinates[(1,1) (1,6) (4,6) (8,7.5) (10,7) (9,2) ]{}; plot\[smooth cycle\] coordinates[(1,1) (1,6) (4,6) (8,7.5) (10,7) (9,2) ]{}; (5.5,3.5) circle (0.5cm); (5.5,3.5) circle (0.5cm); (5.5,3.5) circle (1pt); (4.4,3.7) node [$B(0,\delta)$]{}; (7,4) rectangle (8,5); (7,4) rectangle (8,5); (8.3,4.5) node [$\Theta$]{}; (5.51,3.51)–(7,4.1); (5.5,3.48)–(5.8,1.095); (6.35,4.1) node [$R$]{}; (5.3,2) node [$14R$]{}; (4,6.3) node [$G$]{}; Equidistribution property eigenfunctions of second order elliptic operators {#sec:elliptic} =========================================================================== Notation -------- Let ${\mathcal{L}}$ be the second order partial differential operator $${\mathcal{L}}u = -\sum_{i,j=1}^d \partial_i \left( a^{ij} \partial_j u \right)$$ acting on functions $u$ on ${\mathbb{R}}^d$. Here $\partial_i$ denotes the $i$th weak derivative. Moreover, we introduce the following assumption on the coefficient functions $a^{ij}$. \[ass:elliptic+\] Let $r,{\vartheta}_1 , {\vartheta}_2 > 0$. The operator ${\mathcal{L}}$ satisfies $A(r,{\vartheta}_1 , {\vartheta}_2)$, if and only if $a^{ij} = a^{ji}$ for all $i,j \in \{1,\ldots , d\}$ and for almost all $x,y \in B(0,r)$ and all $\xi \in {\mathbb{R}}^d$ we have $$\label{eq:elliptic} {\vartheta}_1^{-1} \lvert \xi \rvert^2 \leq \sum_{i,j=1}^d a^{ij} (x) \xi_i \xi_j \leq {\vartheta}_1 \lvert \xi \rvert^2 \quad\text{and}\quad \sum_{i,j=1}^d \lvert a^{ij} (x) - a^{ij} (y) \rvert \leq {\vartheta}_2 \lvert x-y \rvert .$$ A quantitative unique continuation principle -------------------------------------------- We first present an extension of the quantitative continuation principle, formulated for Schrödinger operators in Theorem \[thm:quantitative-UCP\], to elliptic operators with variable coefficients. \[thm:qUC-elliptic\] Let $R\in (0,\infty)$, $K_V, \beta \in [0,\infty)$ and $\delta \in (0, 4 R]$. There is an $\epsilon> 0$, such that if $ A(14R, 1+\epsilon, \epsilon)$ holds then there is a constant $C_{\rm qUC} > 0$, such that for any open $G\subset {\mathbb{R}}^d$ containing the origin and $\Theta \subset G$ measurable satisfying $$\operatorname{diam} \Theta + \operatorname{\operatorname{dist}}(0 , \Theta) \leq 2R \leq 2 \operatorname{\operatorname{dist}}(0 , \Theta) \quad \text{and} \quad B(0,14R) \subset G,$$ any measurable $V : G \to [-K_V , K_V]$ and real-valued $\psi \in W^{2,2} (G)$ satisfying the differential inequality $$\label{eq:psi} \lvert {\mathcal{L}}\psi \rvert \leq \lvert V\psi \rvert \quad \text{a.e.\ on $G$} \quad \text{as well as} \quad \frac{\lVert \psi \rVert_G^2}{\lVert \psi \rVert_\Theta^2} \leq \beta ,$$ we have $$\lVert \psi \rVert_{B(x,\delta)}^2 \geq C_{\rm qUC} \lVert \psi \rVert_{\Theta}^2 .$$ Scale-free unique continuation principle ---------------------------------------- We move on to discuss the equidistribution property or scale-free unique continuation principle for eigenfunctions. The aim is to formulate an analog of Theorem \[thm:RojasVeselic\] for variable coefficient elliptic operators. As presented below, for the moment we have solved only the situation where the second order term is sufficiently close to the Laplacian. As before, we denote by ${\Lambda}_L$ a box of side $L\in {\mathbb{N}}$. By $V$ we indicate a bounded measurable potential on ${\mathbb{R}}^d$ taking values in $[-K_V,K_V]$, where $K_V$ is a positive constant. We restrict the operator ${\mathcal{L}}$ on ${\Lambda}_L(0)$ and add either periodic or Dirichlet boundary conditions. In the former case we denote such an operator by ${\mathcal{L}}_{L,0}$, and its domain ${\mathcal{D}}({\mathcal{L}}_{L,0})$ is the subspace of $W^{2,2}({\Lambda}_L)$ consisting of functions vanishing on $\partial {\Lambda}_L$. The notation for the operator with periodic boundary condition is ${\mathcal{L}}_{L,\mathrm{per}}$ and its domains ${\mathcal{D}}({\mathcal{L}}_{L,\mathrm{per}})$ consists of the functions in $W^{2,2}({\Lambda}_L)$ satisfying periodic boundary conditions. \[ass:periodicCoefficients\] For each pair $i,j$ the function $a^{ij}\colon {\mathbb{R}}^d \to {\mathbb{R}}$ is ${\mathbb{Z}}^d$-periodic. Assume that in the case of operator ${\mathcal{L}}_{L,0}$ its coefficients $a^{ij}$, $i\not= j$ vanish on the sides of box ${\Lambda}_L$, while the coefficients $a^{ii}$ satisfy periodic boundary conditions on the sides of box ${\Lambda}_L$. In the case of operator ${\mathcal{L}}_{L,\mathrm{per}}$ suppose that all its coefficients satisfy periodic boundary conditions on the sides of box ${\Lambda}_L$. \[thm:equidistribution-elliptic\] Fix $K_V\in [0,+\infty)$, $\delta\in(0,1]$. Assume $A(\sqrt{d},1+\epsilon,\epsilon)$ with $\epsilon>0$ as in Theorem \[thm:qUC-elliptic\] . Assume \[ass:periodicCoefficients\]. Then there exists a constant $C_{sfUC}>0$ such that for any $L\in 2{\mathbb{N}}+1$, any sequence $$\label{d1.1} Z:=\{z_k\}_{k\in{\mathbb{Z}}^d} \ \text{ in }\ {\mathbb{R}}^d \quad \text{such that} \ B(z_k,\delta)\subset {\Lambda}_1(k) \text{ for each } k\in{\mathbb{Z}}^d,$$ any measurable $V: {\Lambda}_L\mapsto [-K_V,K_V]$ and any real-valued $\psi\in{\mathcal{D}}({\mathcal{L}}_{L,0})$, respectively $\psi\in {\mathcal{D}}({\mathcal{L}}_{L,\mathrm{per}})$ satisfying $$\label{d1.2} |{\mathcal{L}}\psi|\leqslant |V\psi|\quad \text{a.e.}\quad {\Lambda}_L$$ we have $$\label{d1.3} \int\limits_{S_L} |\psi(x)|^2 dx=\sum\limits_{k\in Q_L} \|\psi\|_{L_2(B(z_k,\delta))}^2\geqslant C_{sfUC} \|\psi\|_{L_2({\Lambda}_L)}^2,$$ where $S_L:=S\cap{\Lambda}_L=\cup_{k\in Q_L} B(z_k,\delta)$, $Q_L={\Lambda}_L\cap {\mathbb{Z}}^d$, and $S:=\cup_{k\in {\mathbb{Z}}^d} B(z_k,\delta)$. As a *Corollary* we obtain immediately an eigenvalue lifting estimate analogous to , where $\kappa$ is again uniform w.r.t. many parameters, as spelled out in subsection \[ss:lifting\] explicitly. The proof of Theorem \[thm:equidistribution-elliptic\] is based on the strategy implemented in Ref. . First one uses the conditions on the coefficients $a^{ij}$ described in Assumption \[ass:periodicCoefficients\] to extend $\psi$ as well as the differential expression ${\mathcal{L}}$ to the whole of ${\mathbb{R}}^d$ while keeping the $W^{2,2}$-regularity and the differential inequality originally satisfied by $\psi$. Then one uses the comparison Theorem \[thm:qUC-elliptic\] for local $L^2$-norms. Note that now the condition concerning the minimal distance to the boundary of $G$ plays no role, since $\psi$ has been extended to the whole of ${\mathbb{R}}^d$. From this point the combinatorial and geometric arguments of Ref  take over. In fact, one can prove a abstract meta-theorem: Once the comparison of local $L^2$-norms of $\psi$ holds up to the boundary, an equidistribution property for $\psi$ follows. Interestingly, such an argument no longer uses the fact that $\psi$ is a solution of an differential equation or inequality. Acknowledgments {#acknowledgments .unnumbered} =============== D.B. was partially supported by RFBR, the grant of the President of Russia for young scientists - doctors of science (MD-183.2014.1), and the fellowship of Dynasty foundation for young mathematicians. M.T. and I.V. have been partially supported by the DAAD and the Croatian Ministry of Science, Education and Sports through the PPP-grant “Scale-uniform controllability of partial differential equations”. M.T. and I.V. have been partially supported by the DFG. [1]{} A. Figotin and A. Klein, [*Commun. Math. Phys.*]{} [**180**]{}, 265 (1996). P. Stollmann, [*Isr. J. Math.*]{} [**107**]{}, 125 (1998). C. Rojas-Molina and I. Veseli[ć]{}, [*Commun. Math. Phys.*]{} [**320**]{}, 245 (2013). A. Klein, [*Commun. Math. Phys.*]{} [**323**]{}, 1229 (2013). F. Germinet and A. Klein, [*J. Eur. Math. Soc.*]{} [**15**]{}, 53 (2013). J. Bourgain and A. Klein, [*Invent. Math.*]{} [**194**]{}, 41 (2013). D. I. Borisov, M. Tautenhahn and I. Veseli[ć]{}, Equidistribution properties of eigenfunctions of divergence form operators, in preparation.

ArXiv

--- abstract: 'We have explored a simple microscopic model to simulate a thermally activated rate process where the associated bath which comprises a set of relaxing modes is not in an equilibrium state. The model captures some of the essential features of non-Markovian Langevin dynamics with a fluctuating barrier. Making use of the Fokker-Planck description we calculate the barrier dynamics in the steady state and non-stationary regimes. The Kramers-Grote-Hynes reactive frequency has been computed in closed form in the steady state to illustrate the strong dependence of the dynamic coupling of the system with the relaxing modes. The influence of nonequilibrium excitation of the bath modes and its relaxation on the kinetics of activation of the system mode is demonstrated. We derive the dressed time-dependent Kramers rate in the nonstationary regime in closed analytical form which exhibits strong non-exponential relaxation kinetics of the reaction co-ordinate. The feature can be identified as a typical non-Markovian dynamical effect.' --- =0.0cm =-0.0cm =-1.0cm =21.0cm =15.5cm =0.2cm =0.5cm [**[Jyotipratim Ray Chaudhuri$^{\rm a}$, Gautam Gangopadhyay$^{\rm b}$,\ Deb Shankar Ray$^{\rm a}$]{}**]{} $^{\rm a}$[**[Indian Association for the Cultivation of Science]{}**]{}\ [**[Jadavpur, Calcutta 700 032, INDIA.]{}**]{} $^{\rm b}$[**[S. N. Bose National Centre for Basic Sciences]{}**]{}\ [**[JD Block, Sector III, Salt Lake City, Calcutta 700 091, INDIA.]{}**]{} **[I.Introduction]{}** More than half a century ago Kramers$^{1}$ considered the problem of activated rate processes by using a model Brownian particle trapped in a one dimensional well which is separated by a barrier of finite height from a deeper well. The particle was supposed to be immersed in a medium such that the medium exerts a frictional force on the particle but at the same time thermally activate it so that the particle may gain enough energy to cross the barrier. Over several decades the model has been the standard paradigm in many areas of physics and chemistry$^{2}$. The Kramers problem was to find the rate of escape from the well to the barrier. The motion of the particle is governed by the following phenomenological Langevin equation, $$\ddot{x}=-\frac{1}{m}\frac{\partial V(x)}{\partial x} - \gamma\dot{x} + \frac{1}{m} F(t) \hspace{0.2cm},$$ where $x$ is the coordinate of the particle of mass $m$ moving in a potential $V(x)$. $\gamma$ and $F(t)$ are the damping rate and the Gaussian stationary random force provided by the thermal bath respectively. The properties of noise can be summarized by the following two relations, $$\langle F(t)\rangle=0 \hspace{0.4cm}, \hspace{0.4cm} \langle F(0)F(t)\rangle=2 \gamma mKT \delta(t) \hspace{0.2cm}.$$ The Langevin equation (1) is equivalent to the Fokker-Planck equation for probability distribution $p=p(x,v,t)$ \[also known as Kramers equation\], $$\frac{\partial p}{\partial t}=\frac{1}{m}\frac{\partial V(x)}{\partial x} \frac{\partial p}{\partial v}-v\frac{\partial p}{\partial x} + \gamma \left[\frac{KT}{m} \frac{\partial^{2} p}{\partial v^{2}} +\frac{\partial} {\partial v}(vp) \right] \hspace{0.2cm}.$$ Kramers$^{1}$ obtained the steady state escape rate $k$ in the limiting cases of high and low damping rates in the following form, $$k=\left\{\begin{array}{lllll} \frac{\omega_{0}\omega_{b}}{2\pi\gamma}\exp[-\frac{E_{b}}{KT}] & & & \gamma\longrightarrow\infty \\ \gamma\frac{E_{b}}{KT}\exp[-\frac{E_{b}}{KT}] & & & \gamma\longrightarrow 0 \end{array}\right. \hspace{0.2cm},$$ where $\omega_{o}$ and $\omega_{b}$ are the frequencies associated with the curvature of the potential at the bottom of the well and at the barrier top, respectively. $E_{b}$ refers to the depth of the well. Kramers has also derived an expression for ‘intermediate’ value of $\gamma$ : $$\begin{aligned} k=\frac{\omega_{0}}{2\pi\omega_{b}}\left\{\left[ \left(\frac{\gamma}{2} \right)^{2}+\omega_{b}^{2}\right]^{\frac{1}{2}}-\frac{\gamma}{2}\right\} \exp(-E_{b}/KT)\hspace{0.2cm}.\end{aligned}$$ For non-Markovian random processes where one takes into account of the short internal time scales of the system compared to that of the thermal bath, the Langevin equation(1) gets replaced by its non-Markovian counterpart$^{3,4}$, sometimes called the generalized Langevin equation (GLE); $$\ddot{x}=-\frac{1}{m}\frac{\partial V(x)}{\partial x}-\int_{0}^{t}d\tau Z(t-\tau) \dot{x}(\tau) + \frac{1}{m}R(t) \hspace{0.2cm},$$ where $R(t)$ is Gaussian but non-Markovian such that $$\langle R(t) \rangle = 0 ,\hspace{1.0cm}\langle R(0)R(t) \rangle = Z(t)mKT \hspace{0.2cm}.$$ The memory function $Z(t)$ is expressed in terms of Fourier-Laplace components $$Z_{n}(\omega) = \int_{o}^{\infty} dt Z(t) e^{-in\omega t}$$ with $Z_{0}(\omega) = \gamma$ Based on equation (5) Adelman$^{5}$ obtained the generalized Fokker-Planck equation for a Brownian oscillator with a parabolic potential as given by ; $$\frac{\partial p}{\partial t} = -{\bar{\omega}}_{b}^{2} x \frac{\partial p} {\partial v} -v\frac{\partial p}{\partial x} + {\bar{\gamma}}\frac{\partial}{\partial v} (vp)+ {\bar{\gamma}} \frac{KT}{m}\frac{\partial^{2}p}{\partial v^{2}} + \frac{KT}{m}\left(\frac {{\bar{\omega}}_{b}^{2}}{\omega_{b}^{2}}-1\right)\frac{\partial^{2}p} {\partial v\partial x} \hspace{0.2cm},$$ where ${\bar{\gamma}}$ = ${\bar{\gamma}}(t)$ and ${{\bar{\omega}}_{b}^{2}} ={{\bar{\omega}}_{b}^{2}}(t)$ are now functions of time \[although bounded , they may not always provide long time limits\] which play a decisive role in the calculation of non-Markovian Kramers rate. Various workers have made use of generalized Langevin equation to treat the different aspects of the escape problem in the non-Markovian regime. For example, Grote and Hynes$^{4}$ considered the average motion of the particle in the vicinity of the barrier governed by GLE and found that on the average the particle is slowed down by friction and defining a reactive frequency $\lambda_{r}$ they showed that the average motion goes as $\exp(\pm\lambda_{r} t)$. The analysis of Hänggi and Mojtabai$^{6}$ on the other hand is based on the generalized Fokker-Planck equation of Adelman with a parabolic potential in the high friction limit. The generalized FP approach has also been adopted by Carmeli and Nitzan$^{7}$ to derive the expression for the steady-state escape rate in the high and low friction limit in the Markovian as well as non-Markovian regimes. A comprehensive overview has been given in Ref.(2). While the early post-Kramers development as summarized above is largely phenomenological, an interesting advancement in the theory of activated rate processes was made when the generalized Langevin equation was realized in terms of a microscopic model which comprises a system coupled linearly to a discrete set of harmonic oscillators. Using the properties of the bath and a normal mode analysis it was shown$^{8}$ that the reactive frequency $\lambda_{r}$ defined by Grote and Hynes$^{4}$ for the average motion across the barrier is actually a renormalised effective barrier frequency. The object of the present paper is twofold : First is to consider a simple variant of the system-heat bath model$^{9,10,11}$ to simulate the activated rate processes, where the associated bath is in a nonequilibrium state. The model incorporates some of the essential features of Langevin dynamics with a fluctuating barrier which had been heuristically and phenomenologically proposed earlier in several occasions.$^{10,13-17}$ While the majority of the treatments of the phenomenological fluctuating barrier rest on the reduction of the equations to overdamped limit$^{5,10,14}$, thus restricting the validity of the solutions in the large time limit, we take full account of the inertial terms in our calculation of barrier dynamics and probability distribution function both in the long time and in the short time nonstationary regimes. The Fokker-Planck description allows us to calculate Kramers-Grote-Hynes reactive frequency pertaining to these situations for non-Markovian dynamics in closed form. Second, since the theories of activated processes traditionally deal with stationary bath, the nonstationary activated processes has remained largely overlooked so far. We specifically address this issue and examine the influence of initial excitation and subsequent relaxation of a bath modes on the activation of the reaction co-ordinate. We show that relaxation of the nonequilibrium bath modes may result in strong non-exponential kinetics and a nonstationary Kramers rate. The physical situation that has been addressed is the following : We consider that at $t=0_-$, the time just before the system and the bath is subjected to an external excitation, the system is appropriately thermalized. At $t=0$, the excitation is switched on and the bath is thrown into a nonstationary state which behaves as a nonequilibrium reservoir. We follow the stochastic dynamics of the system mode after $t>0$. The important separation of the time scales of the fluctuations of the nonequilibrium bath and the thermal bath (to which it relaxes) is that the former effectively remains stationary on the fast correlation of the thermal noise. The outline of the paper is as follows; Following Ref. \[10\] we discuss in Sec.II a microscopic model to simulate an activated rate process where the system in question is not initially thermalized. Appropriate elimination of reservoir degrees of freedom leads to a nonlinear non-Markovian Langevin equation which governs the dynamics of a particle with a fluctuating barrier, stochasticity being contributed by both (additive) thermal noise and a slower (multiplicative) noisy relaxing nonequilibrium modes. The Fokker-Planck description is provided in Sec.III. The standard Markovian description and the generalized FP equation of Adelman’s form can be recovered in the appropriate limits. In Sec.IV we derive the expression for Kramers rate of barrier crossing in the non-Markovian but steady state regime and show that the Kramers-Grote-Hynes “reactive frequency” can be explicitly realized in this model in closed form. Sec.V is devoted to nonstationary aspect. We solve the time-dependent FP equation for nonstationary probability density and calculate the corresponding current. An expression for Kramers rate in the nonstationary regime in closed analytical form is derived. The paper is concluded in Sec.VI. **[II. The model and the Langevin equation]{}** We consider a model consisting of a system mode coupled to a set of relaxing modes considered as a semi-infinite dimensional system ({$q_k$}-subsystem) which effectively constitutes a nonequilibrium bath. This, in turn, is in contact with a thermally equilibrated reservoir. Both the reservoirs are composed of two sets of harmonic oscillators characterized by the frequency sets $\{\omega_{k}\}$ and $\{\Omega_{j}\}$ for the nonequilibrium and the equilibrium bath, respectively. The system-reservoir combination evolves under the total Hamiltonian $$\begin{aligned} H=\frac{p^{2}}{2m}+V(x)+\frac{1}{2}\sum_{j}(P_{j}^{2}+\Omega_{j}^{2}Q_{j}^{2}) +\frac{1}{2}\sum_{k}(p_{k}^{2}+\omega_{k}^{2}q_{k}^{2})\nonumber\\ -x\sum_{j}K_{j}Q_{j}-g(x)\sum_{k}q_{k}-\sum_{j,k}\alpha_{jk}q_{k}Q_{j} \hspace{0.2cm},\end{aligned}$$ the first two terms on the right hand side describe the system mode. The Hamiltonian for the thermal and nonequilibrium baths are described by the sets $\{Q_{j},P_{j}\}$ and $\{q_{j},p_{j}\}$ for coordinates and momenta, respectively. The coupling terms containing $K_{j}$ refers to the usual system-thermal bath linear coupling. The last two terms indicate the coupling of the nonequilibrium bath to the system and the thermal bath modes, respectively. Since in the present problem, H is considered to be classical and temperature, T high for the thermally activated problem we note that quantum effects do not play any significant role. Hamiltonian (9) is a simpler variant of that treated in Ref.\[10\]. For simplicity we take $m=1$ in (9) and for rest of the treatment. As shown in Ref. \[10\] the model (9) captures the essential features of fluctuating barrier dynamics. We recall the relevant aspect in the following discussions. Eliminating the equilibrium reservoir variables $\{Q_{j},P_{j}\}$ in an appropriate way $^{9,10}$ one may show that the nonequilibrium bath modes obey the following equations of motion, $${\ddot{q}}_{k}+\gamma{\dot{q}}_{k}+\omega_{k}^{2}q_{k}=g(x)+\eta_{k}(t) \hspace{0.2cm}.$$ This takes into account of the average dissipation ($\gamma$) of the nonequilibrium reservoir modes $q_{k}$ due to its coupling to thermal reservoir which induces fluctuations $\eta_{k}(t)$ characterized by $\langle \eta_{k}(t)\rangle=0$ and the usual fluctuation-dissipation theorem $\langle \eta_{k}(t)\eta_{k}(0)\rangle=2\gamma KT\delta(t)$. We mention here that moving from Eq.(9) to (10) generate cross terms of the form $\sum_j \gamma_{kj} q_j$, which are neglected for $j \neq k$. Proceeding similarly to eliminate the thermal reservoir variables from the equations of motion of the system mode one obtains $${\ddot{x}}+\gamma_{\rm eq}{\dot{x}}+V'(x)=\xi_{\rm eq}(t)+g'(x)\sum_{k}q_{k} \hspace{0.2cm},$$ where $\gamma_{\rm eq}$ refers to the dissipation coefficient of the system mode due to its direct coupling to the thermal bath providing fluctuations $\xi_{\rm eq}(t)$. Here we have $$\begin{aligned} \langle \xi_{\rm eq}(t)\rangle=0 \hspace{0.2cm}{\rm and} \hspace{0.2cm} \langle \xi_{\rm eq}(t)\xi_{\rm eq}(0)\rangle=2\gamma_{\rm eq}KT\delta(t) \hspace{0.2cm}.\end{aligned}$$ Now making use of the formal solutions of Eq.(10)$^{10}$ which takes into account of the relaxation of the nonequilibrium modes and integrating over the nonequilibrium modes with a Debye type frequency distribution of the form, $$\begin{aligned} \rho(\omega)=\left\{ \begin{array}{lllll} 3\omega^{2}/2\omega_{c}^{3} & & , & {\rm for} |\omega| \le \omega_{c}\\ 0 & & , & {\rm for}|\omega| > \omega_{c} \end{array} \right.\end{aligned}$$ where $\omega_{c}$ is the high frequency Debye cut-off, one finally arrives at the following Langevin equation of motion for the system mode, $${\ddot{x}}+\Gamma(x){\dot{x}}+\tilde{V}'(x)=\xi_{\rm eq}(t)+g'(x)\xi_{\rm neq} (t)\hspace{0.2cm}.$$ Here $\Gamma(x)$ is a system coordinate dependent dissipation constant composed of $\gamma_{\rm eq}$ and $\gamma_{\rm neq}$ as follows, $$\Gamma(x)=\gamma_{\rm eq}+\gamma_{\rm neq}[g'(x)]^{2}\hspace{0.2cm}.$$ $\xi_{\rm neq}$ refers to the fluctuations of the nonequilibrium bath modes which effectively cause a damping of the system mode by an amount $\gamma_{\rm neq}[g'(x)]^{2}$. Eq.(12) also includes the modification of the potential $V(x)$ in which the particle moves as $${\tilde{V}}(x)=V(x)-\frac{\omega_{c}}{\pi} \; \gamma_{\rm neq} \; g^{2}(x) \hspace{0.2cm}.$$ Eq.(12) thus describes the effective dynamics of a particle in a modified barrier, where the metastability of the well originates from the dynamic coupling $g(x)$ of the system mode with the nonequilibrium bath modes. It is necessary to stress here that $g(x)$, in general, is nonlinear. This nonlinearity has two immediate consequences. First, by virtue of the term ${\tilde{V}}'(x)$ in Eq.(12) it gives rise to a fluctuating barrier. Second, the term $g'(x) \xi_{\rm neq} (t)$ imparts a multiplicative noise term in Eq.(12) in addition to the usual additive noise term $\xi_{\rm eq} (t)$. We point out here that the problem of diffusion over a fluctuating barrier$^{13-17}$ of similar nature has been addressed earlier by a number of workers from the phenomenological point of view. For example, Stein et.al.$^{14}$ have calculated the decay of probability from the metastable state in the white noise limit and also for short finite correlation times for the fluctuating part of the potential. Riemann and Elston$^{15}$ have calculated an asymptotic rate formula when the particle is subjected to both dichotomous and thermal noise. The treatment followed in the aforesaid cases concerns overdamped situation and, in general, the validity is restricted to long time limit. In the present problem, however, we look at the stochastic process right from the moment the nonequilibrium excitation (followed by the relaxation) sets in. We are therefore forced to take into consideration of the inertial term in Eq.(12) on its usual footing. We now turn to the another aspect of the problem. In order to define the problem described by Eq.(12) completely, it is further necessary to state the properties of fluctuations of the nonequilibrium bath $\xi_{\rm neq}(t)$. We have first for Gaussian noise $$\begin{aligned} \langle \xi_{\rm neq} (t) \rangle = 0 \; \; .\end{aligned}$$ Also the essential properties of $\xi_{\rm neq} (t)$ explicitly depend on the nonequilibrium state of the intermediate oscillator modes $\{ q_k \}$ through ${\cal U}(\omega, t)$, the energy density distribution function at time $t$ in terms of the following fluctuation-dissipation relation$^{10}$ for the nonequilibrium bath, $$\begin{aligned} {\cal U}(\omega , t) & = & \frac{1}{4\gamma_{\rm neq}} \int_{-\infty}^{+\infty} d\tau \; \langle \xi_{\rm neq} (t) \xi_{\rm neq} (t+\tau) \rangle \; e^{i\omega \tau} \nonumber \\ & = & \frac{1}{2} KT \; + \; e^{-\gamma t/2} \; \left [ {\cal U}(\omega,0) - \frac{1}{2} KT \right ] \; \; ,\end{aligned}$$ $\left [ {\cal U}(\omega,0) - \frac{1}{2} KT \right ] $ is a measure of departure of energy density from thermal average at $t=0$. The exponential term implies that this deviation due to the initial excitation decays asymptotically to zero as $t\rightarrow \infty$, so that one recovers the usual fluctuation-dissipation relation for the thermal bath. With the above specification of correlation function of $\xi_{\rm neq}$ Eq.(15) thus attributes the nonstationary character of the {$q_k$}-subsystem. In passing, we stress that the above derivation$^{10}$ is based on the assumption that $\xi_{\rm neq}$ is effectively stationary on the fast correlation of the thermal modes. This is a necessary requirement for the systematic separation of time scales involved in the dynamics. We point out that the effective dynamics sets no choice on any special form of coupling $g(x)$ between the system mode and the relaxing mode and as such this may be of arbitrary nonsingular type for our problem we have considered here. **[III.The generalized Fokker-Planck description]{}** Eq.(12) is the required Langevin equation for the particle moving in a modified potential ${\tilde{V}}(x)$ \[Eq.(14)\] and damped by a coordinate-dependent friction $ \Gamma (x) $ \[Eq.(13)\] due to its linear coupling to a thermal bath and nonlinear coupling to the $\{q_k\}$-subsystem characterized by fluctuations $\xi_{\rm neq} (t)$. Before proceeding further a few pertinent points are to be noted to stress some distinct and important aspects of the model. First, depending on the system-{$q_k$}-subsystem coupling $g(x)$ both the modified potential ${\tilde{V}}(x)$ as well as $\Gamma(x)$ are, in general, nonlinear. So the stochastic differential equation (12) is nonlinear. Again, the stochasticity in Eq.(12) is composed of two parts : $\xi_{\rm eq} (t)$ is an additive noise due to thermal bath while $\xi_{\rm neq} g'(x)$ is a multiplicative contribution due to nonlinear coupling to {$q_k$}-subsystem. It is thus important to note that the presence of multiplicative noise and a fluctuating barrier are associated with nonlinearity in $g(x)$. Second, the Langevin equation (12) is non-Markovian. The origin of this non-Markovian nature lies in the decaying term in Eq.(15) where the decay explicitly expresses the initial nonequilibrium nature of the $\{ q_k\}$-subsystem following the sudden excitation at $t=0$. This non-Markovian feature is thus not to be confused with that arises due to the usual frequency dependence of the dissipation constant. Third, although the modification of $V(x)$ is due to the specific choice of the Debye model for the mode density which has so far been commonly used, the theory remains effectively unchanged as one goes over to more complicated spectrum. We now rewrite Eq.(12) in the form, $$\left.\begin{array}{l} \dot{u}_{1}=F_{1}(u_{1},u_{2},t ; \xi_{\rm neq},\xi_{\rm eq}) \dot{u}_{2}=F_{2}(u_{1},u_{2},t ; \xi_{\rm neq},\xi_{\rm eq}) \end{array}\right\}\hspace{0.2cm},$$ where we use the following abbreviations, $$\left.\begin{array}{l} u_{1}=x\\ u_{2}=v \end{array}\right\}$$ and $$\left.\begin{array}{l} F_{1}=v\\ F_{2}=-\Gamma(x)v-\tilde{V}^{'}(x)+ \xi_{\rm eq}(t)+g'(x) \xi_{\rm neq}(t)\end{array}\right\} \hspace{0.2cm}.$$ The vector $u$ with components $u_{1}$ and $u_{2}$ thus represents a point in a 2-dimensional ‘phase space’ and the Eq.(16) determines the velocity at each point in this phase space. The conservation of points now asserts the following linear equation of motion for density $\rho(u,t)$ in ‘phase space’, $$\begin{aligned} \frac{\partial}{\partial t}\rho(u,t)=-\sum_{n=1}^{2}\frac{\partial}{\partial u_{n}} F_{n}(u,t;\xi_{\rm neq},\xi_{\rm eq})\rho(u,t)\hspace{0.2cm},\end{aligned}$$ or more compactly $$\frac{\partial \rho}{\partial t}=-\nabla\cdot F\rho\hspace{0.2cm}.$$ Our next task is to find out a differential equation whose average solution is given by $\langle \rho \rangle$ where the stochastic averaging has to be performed over two noise processes $\xi_{\rm neq}$ and $\xi_{\rm eq}$. To this end we note that $\nabla \cdot F$ can be partitioned into two parts ; a constant part $\nabla \cdot F_0$ and a fluctuating part $\nabla \cdot F_1 (t)$, containing these noises. Thus we write $$\nabla \cdot F(u,t;\xi_{\rm neq},\xi_{\rm eq}) = \nabla\cdot F_0(u) + \epsilon \nabla \cdot F_1 (u,t;\xi_{\rm neq},\xi_{\rm eq}) \; \; ,$$ where $\epsilon$ is a parameter (we put it as an external parameter to keep track of the order of the perturbation expansion in $\epsilon \tau_c$, where $\tau_c$ is the correlation time of fluctuation of $\xi_{\rm neq} (t)$ ; we put $\epsilon=1$ at the end of calculation) and also note that $\langle F_1(t) \rangle =0$. Eq.(19) therefore takes the following form , $$\dot{\rho} (u,t) = (A_0 \; + \; \epsilon A_1) \; \rho (u,t) \; \; ,$$ where $A_0=-\nabla \cdot F_0$, $ A_1=-\nabla \cdot F_1$. The symbol $\nabla$ is used for the operator that differentiate everything that comes after it with respect to $u$. Making use of one of the main results for the theory of linear equation of the form (21) with multiplicative noise, we derive an average equation for $\rho$ \[$\langle \rho \rangle = p(u,t)$, the probability density of $u(t)$ ; for details refer to Van Kampen$^{12}$\], $$\dot{p} =\left \{ A_0 + \epsilon^2 \int_0^\infty \langle A_1(t) \; \exp (\tau A_0) \; A_1 (t-\tau) \rangle \; \exp (-\tau A_0) \right \} \; p \; \; .$$ The above result is based on second order cumulant expansion and is valid in the case that fluctuations are small but rapid and the correlation time $\tau_c$ is short but finite, i.e., $$\begin{aligned} \langle A_1(t) \; A_1(t') \rangle =0 \; \; {\rm for} \; |t-t'| > \tau_c \; \; .\end{aligned}$$ The Eq.(22) is exact in the limit correlation time $\tau_c$ tends to zero. Using the expressions for $A_0$ and $A_1$ we obtain $$\begin{aligned} \frac{\partial p(u,t)}{\partial t} = \{ -\nabla\cdot F_0 \; + \; \epsilon^2 \int_0^\infty d\tau \; \langle \nabla\cdot F_1(t) \; \exp(-\tau \nabla\cdot F_0) \; \nabla\cdot F_1(t-\tau) \rangle \nonumber \\ \exp(\tau\nabla\cdot F_0) \} \; p(u,t) \; \; .\end{aligned}$$ The operator $\exp(-\tau\nabla\cdot F_0)$ in the above equation provides the solution of the equation $$\frac{\partial f(u,t)}{\partial t} = - \nabla \cdot F_0 \; f(u,t) \; \; ,$$ ($f$ signifies the unperturbed part of $\rho$) which can be found explicitly in terms of characteristics curves. The equation $$\dot{u} = F_0(u)$$ for fixed $t$ determines a mapping from $u(\tau=0)$ to $u(\tau)$, i.e., $u\rightarrow u^\tau$ with inverse $(u^\tau)^{-\tau}=u$. The solution of Eq.(24) is $$f(u,t) = f(u^{-t},0) \left | \frac{d (u^{-t})}{d(u)} \right | = \exp(-t \nabla\cdot F_0) f(u,0) \; \; ,$$ $\left | \frac{d (u^{-t})}{d(u)} \right |$ being a Jacobian determinant. The effect of $\exp(-t\nabla\cdot F_0)$ on $f(u)$ is as follows ; $$\exp(-t\nabla\cdot F_0) \; f(u,0) = f(u^{-t},0) \left | \frac{d (u^{-t})}{d(u)} \right | \; \; .$$ The above simplification when put in Eq.(23) yields $$\begin{aligned} \frac{\partial}{\partial t}p(u,t)=\nabla\cdot \left\{-F_{0}+\epsilon^{2} \int_{0}^{\infty}\left|\frac{d(u^{-\tau})}{d(u)}\right| \langle F_{1}(u,t)\nabla_{-\tau}\cdot F_{1}(u^{-\tau},t-\tau) \rangle\right.\nonumber\\ \left.\left|\frac{d(u)}{d(u^{-\tau})}\right| d\tau\right\} p(u,t)\hspace{0.2cm}.\end{aligned}$$ $\nabla_{-\tau}$ denotes differentiation with respect to $u_{-\tau}$. We put $\epsilon = 1$ for the rest of the treatment. We now identify, $$\left.\begin{array}{l} u_{1}=x\\ u_{2}=v\\ F_{01}=v\hspace{0.2cm},\hspace{0.2cm}F_{11}=0\\ F_{02}=-\Gamma(x)v-\tilde{V}'(x)\hspace{0.2cm},\hspace{0.2cm} F_{12}=\xi_{\rm eq}(t)+g'(x)\xi_{\rm neq}(t) \end{array} \right \} \; \; .$$ In this notation Eq.(28) now reduces to $$\begin{aligned} \frac{\partial p}{\partial t}=-\frac{\partial}{\partial x}(vp)+\frac{\partial} {\partial v}\left\{\Gamma v+\tilde{V}'(x)\right\}p\hspace{7.5cm}\nonumber\\ \nonumber\\ +\frac{\partial}{\partial v}\int_{0}^{\infty}d\tau\langle\left [\xi_{\rm eq}(t) +g'(x)\xi_{\rm neq}(t)\right ]\left [\frac{\partial}{\partial v^{-\tau}}\{ \xi_{\rm eq}(t-\tau)+g'(x^{-\tau})\xi_{\rm neq}(t-\tau)\}\right]\rangle p \hspace{0.2cm},\end{aligned}$$ where we have used the fact that the Jacobian obey the equation $^{12}$ $$\frac{d}{dt}\log\left|\frac{d(x^{t},v^{t})}{d(x,v)}\right| =\frac{\partial}{\partial x}v+\frac{\partial}{\partial v}\{-\Gamma v+\tilde{V} '(x)\} =-\Gamma$$ so that Jacobian equals to $e^{-\Gamma t}$. As a next approximation we consider the ‘unpurterbed’ part of Eq.(16) and take the variation of $v$ during $\tau_{c}$ into account to first order in $\tau_{c}$. Thus we have $$x^{-\tau}=x-\tau v\hspace{0.2cm};\hspace{0.2cm}v^{-\tau}=v+\Gamma\tau v+\tau \tilde{V}'(x)\hspace{0.2cm}.$$ Neglecting terms ${\cal O}(\tau^{2})$ Eq.(32) yields, $$\frac{\partial}{\partial v^{-\tau}}=(1-\Gamma\tau)\frac{\partial}{\partial v} +\tau \frac{\partial}{\partial x}\hspace{0.2cm}.$$ Taking into consideration of Eq.(33), Eq.(30) can be simplified after some algebra to the following form, $$\begin{aligned} \frac{\partial}{\partial t}p(x,v,t)=-\frac{\partial}{\partial x}(vp)+ \frac{\partial}{\partial v}\left\{ \Gamma (x)v+\tilde{V}'(x)-2g'(x)g''(x) I_{nn}\right \}p\nonumber\\ \nonumber\\ +\left \{ I_{ee}+[g'(x)]^{2}I_{nn} \right \}\frac{\partial^{2} p}{\partial v\partial x}\hspace{5.0cm}\nonumber\\ \nonumber\\ +\left \{ J_{ee}-\Gamma(x)I_{ee}+[g'(x)]^{2}J_{nn}-\Gamma(x)[g'(x)]^{2}I_{nn} -vg'(x)g''(x)I_{nn}\right\}\frac{\partial^{2} p}{\partial v^{2}}\hspace{0.2cm},\end{aligned}$$ where, $$\left.\begin{array}{l} I_{ee}=\int_{0}^{\infty} d\tau \langle\xi_{\rm eq}(t)\xi_{\rm eq}(t-\tau)\rangle\tau \\ I_{nn}=\int_{0}^{\infty} d\tau \langle\xi_{\rm neq}(t)\xi_{\rm neq}(t-\tau)\rangle\tau \\ J_{ee}=\int_{0}^{\infty} d\tau \langle\xi_{\rm eq}(t)\xi_{\rm eq}(t-\tau)\rangle \\ J_{nn}=\int_{0}^{\infty} d\tau \langle\xi_{\rm neq}(t)\xi_{\rm neq}(t-\tau)\rangle \\ \end{array}\right\}\hspace{0.2cm}.$$ The subscripts $ee$ and $nn$ in the above expressions for the integrals over the correlation functions refer to equilibrium and nonequilibrium baths, respectively. In deriving the last Eq.(34) we have assumed that the two reservoirs are uncorrelated. Eq.(34) is the required generalized Fokker- Planck equation for our problem. In order to allow ourselves a fair comparison with Fokker-Planck equation of other forms$^{5,6,7}$, we first turn to the diffusion terms in Eq.(34). The coefficients are coordinate $(x)$ dependent. It is customary to get rid of this dependence by approximating the coefficients at the barrier top (say, $ x=0$) \[one may also use mean field or steady state solutions of Eq.(34) obtained by neglecting the fluctuation terms and putting appropriate stationary condition in the diffusion coefficients\]. The drift term in Eq.(34) refers to the presence of a dressed potential of the form, $$\begin{aligned} R(x)=\tilde{V}(x)-[g'(x)]^{2}\; I_{nn}\end{aligned}$$ or $$R(x)=V(x)-\frac{\omega_{c}}{\pi} \gamma_{\rm neq} g^{2}(x)-[g'(x)]^{2} \; I_{nn} \; \; .$$ The modification of the potential is essentially due to the nonlinear coupling of the system to the nonequilibrium modes. $I_{nn}$ is a non-Markovian small contribution and therefore the third term in (36) may be neglected without any loss of generality. For the rest of the treatment we use $R(x)\simeq \tilde{V}(x)$. At the vicinity of the barrier top $x=0, \tilde{V} '(x)$ may be approximated, as usual, by a parabolic potential, i.e., $$\tilde{V}(x)\simeq \bar{E}_{b}-\frac{1}{2}\bar{\omega}_{b}^{2}x^{2}$$ with $$V(x)\simeq E_{b}-\frac{1}{2}\omega_{b}^{2}x^{2}\hspace{0.2cm}.$$ For convenience, one may set $g(0) = 0$ in the Taylor series expansion for $g(x)$ (carried out at the barrier top $x=0$ ), without any loss of generality. And one obtains $$\bar{E_{b}} = E_{b}$$ and $$\bar{\omega_b}^{2} = {\omega_b}^{2} + \frac{2 \omega_{c} \gamma_{neq}}{\pi} [g'(0)]^{2}\; \; .$$ In the linearized description, the Fokker-Planck Eq.(34) is now reduced to the following form, $$\frac{\partial p}{\partial t}=-v\frac{\partial p}{\partial x}+\Gamma p+ [\Gamma v-\bar{\omega}_{b}^{2}x]\frac{\partial p}{\partial v}+ A\frac{\partial^{2} p}{\partial v^{2}}+B\frac{\partial^{2} p}{\partial v \partial x}\hspace{0.2cm},$$ where we have used the following abbreviations; $$A=J_{ee}-\Gamma(0)I_{ee}+[g'(0)]^{2}J_{nn}-\Gamma(0)[g'(0)]^{2}I_{nn}$$ and $$B=I_{ee}+[g'(0)]^{2}I_{nn}\hspace{0.2cm}.$$ From the last two relations we have $$A=\left[ J_{ee}+g'(0)^{2}J_{nn}\right]-\Gamma(0)B$$ Defining $A$ and $B$ as $$A=\bar{\gamma}KT \hspace{0.2cm}{\rm and}\hspace{0.2cm}B=\bar{\beta}KT$$ one obtains $$\begin{aligned} \frac{\partial p}{\partial t}=-v\frac{\partial p}{\partial x}-\bar{\omega}_{b} ^{2}x\frac{\partial p}{\partial v}+\Gamma\frac{\partial}{\partial v}(vp)+ \bar{\gamma}KT\frac{\partial^{2} p}{\partial v^{2}}\nonumber\\ \nonumber\\ +KT\left[ \frac{J_{ee}+g'(0)^{2}J_{nn}}{\Gamma(0) KT}-\frac{\bar{\gamma}} {\Gamma(0)}\right]\frac{\partial^{2} p}{\partial x\partial v}\hspace{0.2cm}.\end{aligned}$$ Identifying $$\bar{\Omega}^{2}=\Omega^{2}\left[ \frac{J_{ee}+g'(0)^{2}J_{nn}}{\Gamma(0) KT} \right]\hspace{0.2cm},$$ Eq.(46) may be rewritten as, $$\begin{aligned} \frac{\partial}{\partial t}p(x,v,t)=-v\frac{\partial p}{\partial x} -\bar{\omega}_{b}^{2}x\frac{\partial p}{\partial v} +\Gamma\frac{\partial}{\partial v}(vp)+ \bar{\gamma}KT\frac{\partial^{2} p}{\partial v^{2}}\nonumber\\ \nonumber\\ +KT\left[ \frac{\bar{\Omega}^{2}(t)}{\Omega^{2}}-\frac{\bar{\gamma}} {\Gamma(0)} \right ] \frac{\partial^{2} p}{\partial x\partial v}\hspace{0.2cm}.\end{aligned}$$ Here $\bar{\gamma}(t)$ and $\bar{\Omega}(t)$ are functions of time (due to the relaxation of the nonequilibrium modes) as defined by Eqs.(45) and (47). Or in other words nonstationary nature of the bath makes $\bar{\Omega}(t)$ time-dependent through $J_{nn}$ term which is essentially a non-Markovian modification. Now the fluctuation-dissipation relations for equilibrium and nonequilibrium baths stated in Sec.II may be invoked. For equilibrium baths as noted earlier we have the usual result; $$J_{ee}=\int_{0}^{\infty} d\tau \langle\xi_{\rm eq}(t)\xi_{\rm eq}(t-\tau) \rangle=\gamma_{\rm eq}KT$$ For the nonequilibrium version, Eq.(15) may be rearranged further to note that $$J_{nn}=\int_{0}^{\infty} d\tau \langle\xi_{\rm neq}(t)\xi_{\rm neq}(t-\tau) \rangle=\gamma_{\rm neq}KT(1+r e^{-\frac{\gamma}{2} t})$$ where $r$ is a measure of the deviation from equilibrium at the initial instant and is given by $r=\left \{ \frac{{\cal U}(\omega_{\rightarrow 0},0)}{2KT} -1 \right\}$. Here ${\cal U}(\omega,t)$ defines the energy density distribution at time $t$. Using (49) and (50) we obtain from Eq.(47) $$\frac{\bar{\Omega}^{2}(t)}{\Omega^{2}}=1+\frac{r \gamma_{\rm neq}e^{-\frac{\gamma}{2} t}} {\gamma_{\rm eq}+\gamma_{\rm neq}[g'(0)]^{2}}\hspace{0.2cm}.$$ In the long time limit the relation reduces to $$\left.{\cal L}t\right._{t \rightarrow \infty} \bar{\Omega}(t)=\Omega\hspace{0.2cm}.$$ It is interesting to note that with the replacement $\frac{\bar{\Omega}^{2} (t)}{\Omega^{2}} \sim \frac{\bar{\omega}_{b}^{2}}{\omega_{b}^{2}}$ (terms are of order $1+{\cal O}(\gamma)$) and $\Gamma(0) \sim \bar{\gamma}$ one recovers the Fokker-Planck equation in the Adelman’s form$^{5}$ (Eq.(8)). **[IV.Non-Markovian steady state Kramers rate]{}** We now proceed to analyze our generalized Fokker-Planck equation (48) and calculate the steady state current and the Kramers escape rate over the barrier. The procedure we follow in this section is similar to that of Kramers supplemented by Hänggi and Mojtabai’s earlier analysis$^{6}$. As usual we make the ansatz $$p(x,v,t)=F(x,v,t)\exp\left [ -\frac{\frac{v^{2}}{2}+\tilde{V}(x)}{KT}\right]$$ with $\tilde{V}(x)$ as approximated by a parabolic potential of the form \[ see Eqs. (37-40) \] $$\begin{aligned} \tilde{V}(x)\simeq \bar{E}_{b}-\frac{1}{2}\bar{\omega}_{b}^{2}x^{2}\end{aligned}$$ with$\hspace{5.0cm}\bar{E}_{b}=E_{b}$ and$\hspace{5.0cm}\bar{\omega}_{b}^{2}=\omega_{b}^{2}+ \frac{2 \; \omega_{c} \; \gamma_{\rm neq}}{\pi} [g'(0)]^{2}$ as stated earlier. We seek an equation for $F$ of the form $$F(x,v,t)=F(u,t)\hspace{0.2cm},\hspace{0.2cm}u=v+ax\hspace{0.2cm}.$$ Inserting (53) and (54) in Eq.(48) we obtain $$\begin{aligned} \frac{\partial F}{\partial t}=\left\{(\Gamma-\bar{\gamma})-\frac{1}{KT} (\Gamma-\bar{\gamma})v^{2}-\frac{\bar{\omega}_{b}^{2}}{KT}\Delta xv\right\}F \nonumber\\ \nonumber\\ +\left[\left\{(\Gamma-2\bar{\gamma})-a(1+\Delta)\right\}v-\bar{\omega}_{b}^{2} (1-\Delta)x\right]\frac{\partial F}{\partial u}\nonumber\\ \nonumber\\ +KT(\bar{\gamma}+\Delta a)\frac{\partial^{2}F}{\partial u^{2}}\hspace{0.2cm},\end{aligned}$$ where $$\Delta=\frac{\bar{\Omega}^{2}(t)}{\Omega^{2}}-\frac{\bar{\gamma}}{\Gamma(0 )}\hspace{0.2cm}.$$ Using (51), $\Delta$ may be rewritten as $$\Delta\simeq \frac{r \gamma_{\rm neq}e^{-\frac{\gamma}{2} t}}{\gamma_{\rm eq}+\gamma_{ \rm neq}[g'(0)]^{2}}$$ for $\Gamma\sim\bar{\gamma}$. Assuming $\frac{\Delta}{KT}$ and $(\Gamma-\bar{\gamma})$ to be very small we obtain $$\frac{\partial F}{\partial t} = KT\frac{\partial^{2} F}{\partial u^{2}} -\left[ \frac{\Gamma+a(1+\Delta)}{\Gamma+\Delta a}v+\frac{\bar{\omega}_{b} ^{2}(1-\Delta)}{\Gamma+\Delta a}x\right]\frac{\partial F}{\partial u} \hspace{0.2cm},$$ which may be written in the form $$\frac{\partial F}{\partial t}=KT\frac{\partial^{2} F}{\partial u^{2}}+ \bar{\alpha}u\frac{\partial F}{\partial u}\hspace{0.2cm},$$ with $$\bar{\alpha}=-\frac{\Gamma+a(1+\Delta)}{\Gamma+ \Delta a}\hspace{0.2cm},$$ and $a$ is a solution of the quadratic equation $$a^{2}(1+\Delta)+\Gamma a-\bar{\omega}_{b}^{2}(1-\Delta)=0\hspace{0.2cm}.$$ Since $\left.{\cal L}t\right._{t\rightarrow\infty}\Delta=0$, the long time or steady state solution of Eq.(58) is satisfied by $$KT\frac{\partial^{2} F}{\partial u^{2}}+ \bar{\alpha}u\frac{\partial F}{\partial u}=0$$ with $$\left.{\cal L}t\right._{t\rightarrow\infty}\bar{\alpha}(t)=-\frac{\Gamma+a} {\Gamma}=\alpha({\rm say})\hspace{0.2cm}.$$ Since the Eqs.(62) and (63) are identical in form to the expressions obtained in the usual Kramers theory one can have the usual expressions for the probability density $p(x,v)$ and the current $j_{s}$ as $$p(x,v,\infty)= N\left[\left(\frac{\pi KT}{2 \alpha}\right)^{\frac{1}{2}} +\int_{0}^{v-|a|x}dz \exp\left(-\frac{\alpha z^{2}}{2KT}\right)\right] \exp\left[ -\frac{\frac{v^{2}}{2}+\tilde{V}(x)}{KT}\right]$$ with $$\begin{aligned} F_{s} = N\left[\left(\frac{\pi KT}{2 \alpha}\right)^{\frac{1}{2}} +\int_{0}^{v-|a|x}dz \exp\left(-\frac{\alpha z^{2}}{2KT}\right)\right] \; \; ,\end{aligned}$$ (here the subscript $s$ in $F_s$ refers to steady state $F$) and $$j_{s}=\int_{-\infty}^{+\infty}dv \hspace{0.1cm}vp(x,v) =N(KT)^{\frac{3}{2}}\left(\frac{2\pi}{\alpha+1}\right)^{\frac{1}{2}} \exp\left(-\frac{E_b}{KT}\right)\hspace{0.2cm},$$ where we have used the linearized version of $\tilde{V}(x)$ near the top of the barrier at $x=0$, $$\begin{aligned} \tilde{V}(x)=\bar{E}_{b}-\frac{1}{2}\bar{\omega}_{b}^{2}x^{2}\hspace{0.2cm},\end{aligned}$$ with ${\bar{E}}_b = E_b$ and ${\bar{\omega}}_b$ is as given in Eq.(40) and $N$ is the normalization constant. Employing the asymptotic distribution (just before the system is subjected to the shock at $t=0$) of $P_{w}(x,v)$ for $x\rightarrow -\infty$ and at $t=0_-$ from $p(x,v,t)$, where $P_{w}(x,v)=p(x\rightarrow -\infty,v;t=0)$ \[see Sec. V for calculation of $p(x,v,t)$\], one obtains the total number of particles in the well, $$n_{a}=N \int_{-\infty}^{+\infty}dv \int_{-\infty}^{+\infty}dx P_{w}(x,v) =N \frac{2\pi KT}{\omega_{0}}\left(\frac{2\pi KT}{\alpha}\right)^{\frac{1}{2}} \hspace{0.2cm}.$$ Here $\omega_{0}$ is the frequency at the bottom of the left well. We have set the potential energy at the bottom of the left well equal to zero, for convenience. The final result for the rate of escape in the steady state is given by $$k=\frac{j_{s}}{n_{a}}=\frac{\omega_{0}\lambda}{2\pi\bar{\omega}_{b}} e^{-E_{b}/KT}\hspace{0.2cm},$$ where $$\lambda=\left[\left\{\left(\frac{\Gamma}{2}\right)^{2}+\bar{\omega}_{b} ^{2}\right\}^{\frac{1}{2}}-\frac{\Gamma}{2}\right]\hspace{0.2cm}.$$ It is evident that $\lambda$ is reminiscent of the ‘reactive frequency’ $\lambda_{r}$ of Grote and Hynes$^{4}$ . Microscopically the non-Markovian character of the dynamics in $\lambda$ enters through the explicit structure of $\Gamma$ and $\bar{\omega}_{b}$ which are given by $$\Gamma=\gamma_{\rm eq}+\gamma_{\rm neq}[g'(0)]^{2}$$ and $$\bar{\omega}_{b}^{2}=\omega_{b}^{2} \; + \; \frac{2\omega_c \gamma_{\rm neq}} {\pi} [g'(0)]^2 \; \; .$$ The appearance of the reactive frequency $\lambda$ is suggestive of the fact that the particle on the average is not moving on the bare barrier with frequency $\omega_{b}$ but on a dressed barrier frequency $\bar{\omega}_{b} $ corrected by $\lambda$. Pollak$^{8}$ has shown that the reactive frequency $\lambda$ is exactly an imaginary frequency of a barrier that has been modified by the bath modeled as a discrete set of harmonic oscillators linearly coupled to the system. The effect of $\lambda$ is to slow down the particle by friction near the barrier. In the present model where the generalized Langevin equation(12) describes the motion of the particle over a fluctuating barrier the essential modification of $\lambda$ and $\omega_{b}$ rests on the nonlinear coupling of the nonequilibrium relaxing modes with the system. Thus in addition to the properties of the bath, dynamic nature of the system-bath coupling is also significant in governing the barrier dynamics. We note in passing that the usual Markovian limit can be recovered if one puts $\gamma_{\rm neq}=0$ in Eq.(67) and associated quantities. Before closing this section one pertinent point need to be mentioned. A closer look into the derivation makes it clear that Eq.(67) results from an ansatz of the form (53) where we use $\tilde{V}(x)$ in the Boltzmann factor. This choice is basically guided by the fact that the potential $V(x)$ gets dressed at $t=0$ by initial excitation of nonequilibrium modes. This choice also makes the stationary current independent of position. However, if one uses the bare potential $V(x)$ and assume a weak dependence of $x$ on $j_{s}$, one obtains Eq.(67) with $\bar{\omega}_{b}$ in the denominator getting replaced by $\omega_{b}$ itself. The main lesson is that the modification of Kramers rate (67) is essentially due to $\lambda$, the reactive frequency of Grote-Hynes, which has been recognized as an important result in view of some experimental evidence$^{18}$ of relatively weak dependence of rate on damping in the large friction limit. **[V.Time-dependent solution of the generalized Fokker-Planck equation ; nonstationary Kramers rate ; nonexponential relaxation kinetics]{}** We now turn to Eq.(55). Rearranging the time-dependent $\Delta$-containing terms it may be rewritten as $$\frac{1}{\Gamma}\frac{\partial F}{\partial t}=-\left[\frac{ (\Gamma+a)v+\bar {\omega}_{b}^{2}x}{\Gamma}\right]\frac{\partial F}{\partial u}+KT \frac{\partial^{2} F}{\partial u^{2}}+\Delta\left[\frac{aKT}{\Gamma} \frac{\partial^{2} F}{\partial u^{2}}-\frac{(av-\bar{\omega}_{b}^{2}x)} {\Gamma}\frac{\partial F}{\partial u}\right]\hspace{0.2cm},$$ where $\Delta$ is defined in Eq.(57). Let us write $$\frac{ (\Gamma+a)v+\bar{\omega}_{b}^{2}x}{\Gamma}=-\alpha u$$ and $$\frac{(av-\bar{\omega}_{b}^{2}x)}{\Gamma}=-\lambda u$$ Here $\alpha$ is as defined in (63) and $\lambda$ is to be determined. In terms of the relations (72) and (73), Eq.(71) reduces to a more compact form. $$\frac{1}{\Gamma}\frac{\partial F}{\partial t}=\alpha u \frac{\partial F}{\partial u}+KT\frac{\partial^{2} F}{\partial u^{2}} +\Delta\left[\frac{aKT}{\Gamma} \frac{\partial^{2} F}{\partial u^{2}}+\lambda u \frac{\partial F}{\partial u}\right]\hspace{0.2cm}.$$ Eq.(72) may be used to calculate the value of $a$ as obtained from the solution of the algebraic equation $$a^{2}+\Gamma a-\bar\omega_{b}^{2}=0\hspace{0.2cm}.$$ Only the negative root of the above equation (say $a_{-}$) is the physically realizable solution corresponding to the steady state solution. This value of $a$ determines uniquely the value of $\lambda$ as defined in Eq.(73) to obtain $$\lambda=-\alpha\hspace{0.2cm}.$$ We now seek a solution $F(u,t)$ of Eq.(74) in the form $$F(u,t)=F_{s}(u)e^{-\phi(t)}\hspace{0.2cm},$$ where $F_{s}(u)$ is the steady state solution obtained in the earlier section, i.e., it satisfies $$\alpha u \frac{\partial F_{s}}{\partial u}+KT\frac{\partial^{2}F_{s}} {\partial u^{2}}=0\hspace{0.2cm}\hspace{0.2cm}.$$ We require further $$\left.{\cal L}t\right._{t\rightarrow\infty}\phi (t)=0\hspace{0.2cm}.$$ Substituting (77) in Eq.(74) it may be shown that the ‘space’ and the time part is separable. We obtain, $$-\frac{1}{\Gamma}\frac{\partial \phi}{\partial t} e^{\frac{\gamma}{2} t}=\frac{C} {F_{s}}\left[\lambda u\frac{\partial F_{s}}{\partial u}+\frac{aKT}{\Gamma} \frac{\partial^{2}F_{s}}{\partial u^{2}} \right]={\rm constant}=D({\rm say}) \hspace{0.2cm},$$ where we have made use of the Eq.(78) and also $$\begin{aligned} \Delta=Ce^{-\frac{\gamma}{2} t}\hspace{0.2cm} {\rm with} \hspace{0.2cm}C=\frac{r \gamma_{\rm neq}}{\gamma_{\rm eq}+\gamma_{ \rm neq}[g'(0)]^{2}}\hspace{0.2cm}.\end{aligned}$$ On integration over time we obtain from Eq.(80), the solution $$\phi (t)=2 D\frac{\Gamma}{\gamma}e^{-\frac{\gamma}{2} t}$$ where $D$ is determined by the initial condition. The time-dependent solution of Eq.(71) therefore reads as $$F(u,t)=F_{s}(u)\exp\left[-\frac{2D\Gamma}{\gamma} e^{-\frac{\gamma}{2} t}\right] \hspace{0.2cm}.$$ Thus the corresponding probability distribution is given by, $$\begin{aligned} p(x,v,t)=N\left[\left(\frac{\pi KT}{2\alpha}\right)^{\frac{1}{2}}+\int_{0}^ {v-|a|x}dz \exp\left(-\frac{\alpha z^{2}}{2KT}\right)\right]\nonumber\\ \nonumber\\ \exp\left[-\frac{\frac{v^{2}}{2}+\tilde{V}(x)}{KT}\right] e^{-\frac{2D\Gamma}{\gamma}\left[\exp(-\frac{\gamma}{2} t)\right]}\hspace{0.2cm}.\end{aligned}$$ To determine $D$ we now demand that just at the moment the system (and the nonthermal bath) is subjected to external excitation at $t=0$ and $x\rightarrow -\infty$ the distribution (75) must coincide with the usual Boltzmann distribution where the energy term in the Boltzmann factor in addition to usual kinetic and potential terms contains the initial fluctuation of energy density $\Delta {\cal U}$ \[$\Delta {\cal U}={\cal U}(\omega,0)-\frac{1}{2} KT$\] due to excitation of the system at $t=0$ \[see Eq.(15)\]. $$\begin{aligned} p(x,v,t)\stackrel{t\rightarrow 0} {\longrightarrow} N\left(\frac{2\pi KT}{\alpha}\right)^{\frac{1}{2}} e^{-2D\frac{\Gamma}{\gamma}} e^{-\frac{1}{KT}\left(\frac{v^{2}}{2}+\tilde{V}(x)\right)}\nonumber\\ \nonumber\\ =N\left(\frac{2\pi KT}{\alpha}\right)^{\frac{1}{2}} e^{-\frac{1}{KT}\left(\frac{v^{2}}{2}+{\tilde{V}}(x)+\Delta {\cal U}\right)}, \hspace{0.2cm}{\rm for} (x\rightarrow -\infty)\hspace{0.2cm}.\end{aligned}$$ The last equality demands that $$D=\frac{\gamma}{2\Gamma} \; \frac{\Delta {\cal U}}{KT}$$ \[for the current to be coordinate independent the parabolic approximation of $\tilde{V}(x)$ is to be used\]. $D$ is thus determined in terms of the relaxing mode parameters and fluctuations of the energy density distribution at $t=0$. The time-dependent probability density therefore allows us to construct nonstationary current, $$j(t)=\int_{-\infty}^{+\infty}dv\hspace{0.1cm}v\hspace{0.1cm}p(x,v,t)= j_{s}e^{-\frac{2D\Gamma}{\gamma}\exp(-\frac{\gamma}{2} t)}\hspace{0.2cm},$$ where $j_{s}$ is the stationary or steady state current as derived in the last section. By Eq.(74) we have, $$p_{w}(x,v)=p(x\rightarrow -\infty,v,t=0_{-})\hspace{0.2cm},$$ which was used to calculate the number of particles $n_{a}$ initially in the well just before the system was subjected to shock at $t=0$. Thus non-stationary Kramers rate of transition is given by $$k(t)=\frac{\omega_{0}}{2\pi\bar{\omega}_{b}}\left[\left\{\left(\frac{\Gamma} {2}\right)^{2}+\bar{\omega}_{b}^{2}\right\}^{\frac{1}{2}}-\frac{\Gamma}{2} \right]e^{-\frac{E_{b}}{KT}} e^{-\left[\frac{2D\Gamma}{\gamma}\exp(-\frac{\gamma}{2} t)\right]}\hspace{0.2cm},$$ or in terms of the steady state Kramers rate $k$ $$k(t)=k\exp\left[ -\frac{\Delta {\cal U}}{KT} e^{-\frac{\gamma}{2} t}\right]\hspace{0.2cm} \; \; ,$$ where $\Delta {\cal U}$ is a measure of the initial departure from the average energy density distribution due to the preparation of the nonstationary state of the intermediate bath modes as a result of excitation at $t=0$, and $k$ is given by $$k=\frac{\omega_0}{2\pi {\bar{\omega}}_b} \left[ \left \{ \left( \frac{\Gamma}{2} \right )^2 + {\bar{\omega}}_b^2 \right \}^{1/2} - \frac{\Gamma}{2} \right ] \; e^{-E_b/KT} \; \; .$$ The above result (88) illustrates a strong nonexponential relaxation of the system mode undergoing a nonstationary activated rate process. The origin of this is an initial preparation of nonequilibrium mode density distribution (with a deviation $\Delta {\cal U}$) which eventually relaxes to an equilibrium distribution. Eq. (88) implies that the initial transient rate is different from the asymptotic steady state Kramers rate. What is immediately apparent is that the sign of $\Delta {\cal U} [ = {\cal U}(\omega,0)-\frac{1}{2}KT ]$ determines whether the initial rate will be faster or slower than the steady state rate. When $\Delta {\cal U}$ is negative, i.e., the contribution of thermal energy dominates, the initial rate of thermal activation of the reaction co-ordinate gets enhanced as a consequence. On the other hand, when the sudden excitation of the nonequilibrium modes provides a positive deviation $\Delta {\cal U}$, the initial rate of activation becomes slower. This is because there likely to exist some time lag for the nonthermal energy gained by the few nonequilibrium modes by sudden excitation to be distributed over a range before it become available to the reaction co-ordinate as thermal energy for activation. It is also interesting to consider the zero and high temperature limits. When $T\rightarrow 0$ both the steady state Kramers rate $k$ as well as the time-dependent factor $exp[-\frac{\Delta {\cal U}}{KT} e^{-\gamma t/2} ]$ goes to zero. If $T=0$, then $k(t)$ is zero at all time. However, it seems intuitively that there should be a transient period during which the rate is finite. It may be noted that since the relaxation of the nonequilibrium bath modes (following the sudden excitation) is very slow compared to the rate of activation process, the particle undergoing barrier crossing cannot ‘sense’ this transient (ideally if the relaxation to equilibrium is adiabatic, i.e., the thermalization of the initial departure $\Delta{\cal U}$ is very slow, there should be no transient). We believe that the distinct separation of the two time-scales implied in the dynamics makes the transient unobservable. An interplay of overlapping time-scales pertaining to the relaxation of the bath and the activation of the system may give rise to transients in $k(t)$ at $T=0$. Evidently this is outside the scope of the present treatment. When $T$ is very high such that $\frac{1}{2} KT$ far exceeds ${\cal U}(\omega,0)$ the initial rate gets strongly enhanced (since $\Delta {\cal U}$ is negative) and the time-dependent exponential factor becomes roughly independent of temperature. In the limit $t\rightarrow \infty$ or $\Delta {\cal U}\rightarrow 0$ we recover steady state Kramers rate, as expected. The activation rate is thus consequently modified which effectively incorporates a secondary relaxation kinetics. The quasi-thermal excitations decay on the time scale $\frac{1}{\gamma}$, which is well separated from other internal time scales of the thermal bath. The dynamic nature of the coupling between the system and the nonequilibrium modes is responsible for fluctuating barrier. A closer look into the origin of the non-exponential kinetics makes it clear that the spiritual root of $D$-term is essentially the $\Delta$-containing term in Eq.(71) or $\frac{\partial^{2} p}{\partial x\partial v}$ term in Eq.(46) which is a non-Markovian contribution. We thus identify the non-exponential relaxation of the system mode as a typical non-Markovian dynamical feature. In the case of very small $\gamma$ one naturally recovers the exponential relaxation and Arrhenious rate of activation of the usual kinetic scheme. A relevant pertinent point regarding some of the related works need be considered here. Generalized Langevin equation (GLE) has been widely employed in various contexts, e.g., in the description of reactions in liquids. A search for realistic models began with the realization that friction exerted by the solvent on the solute is space dependent. A formally consistent approach to the problem of space and time dependent friction had been introduced early by Lindenberg and co-workers$^{20,21}$. Carmeli and Nitzan$^{22}$ have also derived a stochastic dynamical equation which is a generalization of GLE to the case of space and time dependent friction. Pollak and Berezhkorskii$^{25}$ have demonstrated that the space and time-dependent friction model is identical to a multidimensional anisotropic but Markovian friction problem in which the reaction co-ordinate is coupled to an additional co-ordinate which is governed by a Langevin type equation. A theory for treating spatially dependent friction in the classical activated rate processes has been considered and following the method of Pollak an effective Grote-Hynes reactive frequency for this case has been obtained as a transcendental equation$^{23}$. More recently a general theory for thermally activated rate constants influenced by spatially dependent and time correlated friction$^{24}$ has been proposed. While in the above problems one is concerned with the space and time dependent friction, which is essentially a characteristic of the solvent mode structure, in the present problem we deal with effect of a secondary relaxation of intermediate oscillator modes (following an initial excitation) on the primary kinetics of the system mode. The mode density function due to initial excitation differs from its equilibrium value - a feature which is marked in the nonequilibrium fluctuation-dissipation relation. Thus the exponential relaxation in Eq.(81) is not be confused with the exponential time-dependent friction employed in earlier instances. The origin of these two exponential terms are fundamentally different. The non-exponential kinetics is essentially an offshoot of a dynamic modification of the fluctuation-dissipation theorem appropriately carried over to a nonstationary regime. This nonequilibrium nature of activated process is reflected in the nonstationary kinetics that we derive here. The non - exponential relaxation kinetics had been explored earlier in different occasions in relation to disordered systems$^{13}$, viscous liquids$^{19}$, oxygen binding to h$\mbox{\ae}$moglobin$^{16}$, where phenomenological fluctuating barrier models have been employed (barriers arising from the collective motions of many degrees of freedom). The present model although oversimplified in many respects captures the essential nature of influence of an initial non-thermal mode density distribution on the relaxation kinetics of the system. **[VI.Conclusions]{}** In conclusion, we consider a simple microscopic system-nonequilibrium bath model to simulate nonstationary thermally activated processes. The nonequilibrium bath is effectively realized in terms of a semi-infinite dimensional broad-band reservoir which is subsequently kept in contact with a thermal reservoir which allows the nonthermal bath to relax with a characteristic time. A systematic separation of timescales is then used to construct the appropriate Langevin equation for the particle, which is nonlinear and non-Markovian in character. Based on a strategy of Van Kampen’s expansion in $\epsilon \tau_{c}$ of the relevant physical quantity where $\epsilon$ is the strength and $\tau_{c}$ is the correlation time of fluctuations of the relaxing modes, we show that this Langevin equation can be recast into the form of a generalized Fokker-Planck equation, when the correlation time is short but finite. Adelman’s form of the Fokker-Planck equation \[ Eq.(8) \] as well as the standard Markovian description can be recovered in the appropriate limits. We now summarize the main conclusions of this study: \(i) The model proposed here captures the essential features of Langevin dynamics with a fluctuating barrier. The present approach is equipped to deal with situations both in the non-stationary short time as well as stationary long time regimes. The origin of the short time non-exponential kinetics can be traced back in a non-stationary fluctuation-dissipation theorem. \(ii) We derive the expression for the steady state Kramers escape rate in the non-Markovian case and show that the Grote-Hynes ‘reactive frequency’ can be realized explicitly in terms of the microscopic parameters of the nonequilibrium relaxing modes and their arbitrary dynamic coupling to the system mode. \(iii) The central result of this paper is the derivation of a nonstationary Kramers rate in closed analytic form. This essentially illustrates the influence of an initial excitation and subsequent relaxation of the nonequilibrium bath modes on the system degree of freedom undergoing an activated process. The system mode is shown to follow strong non-exponential kinetics. The model considered in the present paper may be realized in a guest-host system embedded in a lattice where the immediate local neighborhood of the guest comprises intermediate oscillator modes whereas the lattice plays the role of a thermal bath. Appropriately identified reaction co-ordinate coupled to other degrees of freedom in a molecule embedded in a matrix may be another worthwhile candidate for such a scheme. Although simple, the model thus allows us explicit solutions and in view of the prototypical role played by the present model in several earlier investigations, we hope that the conclusions drawn here will find applications in some related experiments of physics and chemistry of complex systems. [**[Acknowledgments]{}**]{} : Partial financial support from the Department of Science and Technology (Govt. of India) is thankfully acknowledged. One of the authors (JRC) is thankful to Prof. J. K. Bhattacharjee (Dept. of Theoretical Physics, I.A.C.S) and to S. K. Banik for helpful discussions and suggestions. 1. H. A. Kramers, Physica [**7**]{}, 284 (1940). 2. See for example, a review by P. Hänggi, P. Talkner and M. Borkovec, Rev. Mod. Phys. [**62**]{}, 251 (1990) and the references given therein. 3. J. T. Hynes and J. M. Dentch, [*Physical Chemistry : An Advanced Treatise*]{}, vol. IIB (Academic, New York, 1975) ; S. A. Adelman and J. D. Doll, Acc. Chem. Res. [**10**]{}, 378 (1997) ; J. M. Dentch and I. Oppenheim, J. Chem. Phys. [**54**]{}, 3547 (1971). 4. R. F. Grote and J. T. Hynes, J. Chem. Phys. [**73**]{}, 2715 (1980). 5. S. A. Adelman, J. Chem. Phys. [**64**]{}, 124 (1976). 6. P. Hänggi and F. Mojtabai, Phys. Rev. A [**26**]{}, 1168 (1982) ; P. Hänggi, J. Stat. Phys. 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ArXiv

--- abstract: 'We investigate theoretically the photoacoustic generation by a gold nanosphere in water in the thermoelastic regime. Specifically, we consider the long-pulse illumination regime, in which the time for electron-phonon thermalisation can be neglected and photoacoustic wave generation arises solely from the thermo-elastic stress caused by the temperature increase of the nanosphere or its liquid environment. Photoacoustic signals are predicted based on the successive resolution of a thermal diffusion problem and a thermoelastic problem, taking into account the finite size of the gold nanosphere, thermoelastic and elastic properties of both water and gold, and the temperature-dependence of the thermal expansion coefficient of water. For sufficiently high illumination fluences, this temperature dependence yields a nonlinear relationship between the photoacoustic amplitude and the fluence. For nanosecond pulses in the linear regime, we show that more than $90\ \%$ of the emitted photoacoustic energy is generated in water, and the thickness of the generating layer around the particle scales close to the square root of the pulse duration. The amplitude of the photoacoustic wave in the linear regime are accurately predicted by the point-absorber model introduced by Calasso et al. \[\], but our results demonstrate that this model significantly overestimates the amplitude of photoacoustic waves in the nonlinear regime. We therefore provide quantitative estimates of a critical energy, defined as the absorbed energy required such that the nonlinear contribution is equal to that of the linear contribution. Our results suggest that the critical energy scales as the volume of water over which heat diffuses during the illumination pulse. Moreover, thermal nonlinearity is shown to be expected only for sufficiently high ultrasound frequency. Finally, we show that the relationship between the photoacoustic amplitude and the equilibrium temperature at sufficiently high fluence reflects the thermal diffusion at the nanoscale around the gold nanosphere.' author: - Amaury Prost - Florian Poisson - Emmanuel Bossy bibliography: - 'PRB\_GNS\_PA.bib' title: 'Photoacoustic generation by a gold nanosphere: From linear to nonlinear thermoelastics in the long-pulse illumination regime' --- Introduction ============ Photoacoustic imaging is a promising modality for biomedical applications that has emerged during the last two decades [@beard2011; @wang2012]. This non-invasive modality is based on the conversion of absorbed light energy into ultrasound via the thermoelastic effect [@gusev1993]. The image contrast therefore depends on the optical absorption properties of the medium. In biological tissues, endogenous optical absorption may be used to form various types of images. For instance imaging of the hemoglobin enables reconstruction of the vascularization network [@zhang2009]. To further enhance the contrast or obtain complementary information, various exogenous contrast agents have been developed for photoacoustics [@wu2014]. In particular, plasmonic noble metal nanoparticles have been introduced as photoacoustic contrast agents in the early 2000s [@oraevsky2001; @eghtedari2003]. Gold nanoparticles (GNP) are very attractive as photoacoustic contrast agents thanks to their large optical absorption cross-section, their resistance to high illumination fluences and their spectral selectivity based on their surface plasmon resonance [@jain2006]. Typically, the optical absorption cross-section of noble metal nanoparticles is a few orders of magnitude larger than that of traditional molecular dyes [@jain2006; @wu2014]. One major consequence of the strong optical cross-section of GNPs is that their temperature can increase significantly when they absorbe pulsed light [@pustovalov2005], leading to various possible phenomena including nano/micro-bubble formation [@egerev2008; @pustovalov2008; @Lapotko2009; @zharov2011] or even modifications of the particle shape. In the context of photoacoustic imaging, bubble formation is interesting as the emitted signals are usually strong and exhibit a nonlinear relationship between the incident fluence and the photoacoustic amplitude [@Sarimollaoglu2014; @zharov2011]. More generally, nonlinear photoacoustic phenomena provide a means of selectively detecting contrast agents from an absorption background that behaves linearly [@Sarimollaoglu2014], similarly to what is done in the field of ultrasound imaging. Several phenomena may induce nonlinear relationships between the photoacoustic signal amplitude and the energy of the incident light in addition to bubble formation, such as optical saturation [@danielli2010; @zharov2011], photochemical reaction [@OConnor1983] and temperature-dependent thermodynamic parameters [@diebold2001; @burmistrova1979; @gusev1993; @inkov2001; @egerev2008; @Danielli2014; @Wang2014; @simandoux2014]. The latter phenomenon is related to the temperature dependence of the thermal expansion coefficient, and is at the core of this paper, in which we investigate theoretically the photoacoustic generation by a gold nanosphere in the thermoelastic regime (absence of bubble formation). In this work, our study is restricted to what we call the long-pulse illumination regime (typically longer than several picoseconds), for which the typical electron-phonon relaxation time in the gold nanoparticle is negligible compared to the illumination duration. In particular, electrons and phonon may be considered not only to have thermalized distributions (such as assumed in the two-temperature model [@eesley1983]), but also to be described by a unique temperature value, a situation very different from that encountered with sub-picosecond light pulses where nonthermal distributions may be involved [@tas1994; @groeneveld1995]. In the long-pulse illumination regime, a single value for temperature may be used to describe thermodynamic properties, and the photoacoustic wave generation arises from the thermo-elastic stress caused by the temperature increase of the nanosphere or its liquid environment. This regime encompasses the nanosecond-pulse regime commonly used in photoacoustic imaging, and to which most of our study of thermal nonlinearity is restricted. Although nanoparticles have now been used for more than a decade as contrast agents for photoacoustic imaging in many research studies, comparatively few studies investigated the physics involved at the scale of the nanoparticle, whether with theoretical or experimental approaches. In the linear regime,  @inkov2001 introduced a three-step model to predict the photoacoustic emission by an absorbing spherical particle, based on solving (1) a light absorption problem, (2) a thermal diffusion problem and (3) an acoustic problem. Analytical solutions were provided in the linear regime, but were limited to thermally small or large particles, assumptions that are not valid for gold nanospheres illuminated with nanosecond pulses, as commonly encountered in photoacoustic imaging. For nanospheres illuminated with nanosecond pulses, the thermal relaxation time is comparable to the pulse duration, making it complex to analytically derive the temperature field. Numerical approaches are usually required to solve the thermal problem in this case, as was done for instance by @baffou2011. More recently, it has been demonstrated experimentally in the linear regime that for gold nanospheres illuminated with nanosecond pulses, it is mostly the liquid surrounding the nanoparticles that emits photoacoustic waves [@chen2012], as was theoretically discussed earlier for thermally small particles [@inkov2001]. A theoretical explanation in the linear regime was proposed by @chen2012, but the temperature field was modeled via a quasi-static thermal field. Other studies provided theoretical expressions for the photoacoustic emission by spherical absorbers, but with assumptions not always valid in this work and more importantly also limited to the linear regime [@diebold1988; @diebold2002]. Two recent studies also reported comparisons between experimental results and theoretical predictions in the linear regime [@shinto2013; @fukasawa2014], but the authors assumed heat and stress confinement at the scale of a whole particle suspension and did therefore not consider the photoacoustic generation at the scale of individual particles. In the nonlinear thermoelastic regime, a few works have been reported with nanoparticles.  @diebold2001 provided analytical expressions of the waveforms emitted by a point-absorber model. In our work, we extensively use the predictions of this model to be compared to our own model that takes into account the finite size of the gold nanosphere, both in the linear and nonlinear regime. Following the physical approach introduced by @inkov2001,  @egerev2008 described and observed the nonlinear photoacoustic generation by gold nanospheres in water in the thermoelastic regime. A simple criteria to assess the significance of nonlinear generation was derived based on scaling arguments, that shows that thermal nonlinearity should be observed at sufficiently high fluences (Eq. (3) of  @egerev2008), as demonstrated experimentally. The validity of this criteria will be discussed in sections \[subsub:PredictionFromNonLinearAnalyticPointModel\] and \[subsubsec:optimal size for nonlinear generation\], in comparison with the quantitative predictions from both the point-absorber model and our model. In the thermoelastic regime, another previous experimental study reported a nonlinear increase of the photoacoustic signal with the laser fluence [@Nam2012]. The origin of the nonlinearity was assumed to be the temperature-dependence of the thermal expansion coefficient, but presumably caused by thermal coupling within aggregated nanoparticles in cells, rather than by temperature elevation around individual nanoparticles as considered in our work. The main objective of our theoretical work is to provide physical insight and quantitative predictions regarding the photoacoustic generation by a single gold nanosphere, for both the linear and nonlinear thermoelastic regimes, beyond the few initial results from previous relevant studies [@diebold2001; @inkov2001; @egerev2008; @chen2012]. In section \[sec:TheoreticalApproach\], we first introduce the physical model used to predict the photoacoustic signal from a gold nanosphere. Our model takes into account the finite size of the nanosphere and its elastic and thermo-elastic properties, as well as the temperature-dependence of the thermal expansion coefficient of the surrounding liquid. Predictions from the point-absorber model are also given for further comparison in the results section. Section \[sec:TheoreticalApproach\] also describes the principles that were used for solving the thermal and acoustic problems, with details further given in Appendixes \[appendix:GreenTh\] and  \[appendix:FDTD\]. Section \[sec:Results\] provides results and discussions in both the linear and nonlinear thermoelastic regimes. In the linear regime, we study the origin of the photoacoustic wave as a function of the size of the gold nanosphere and the pulse duration. From there, the paper focuses on the nanosecond pulse regime for which the generation mostly occurs in the surrounding liquid. A scaling law is found that describes the typical thickness of the water layer that generates the photoacoustic wave. A comparison between our results and those from the point-absorber model [@diebold2001] is then provided, as a preamble to further analysis in the nonlinear regime. In the nonlinear regime, we first derive predictions from the expressions given by the point-absorber model. We then describe quantitatively temperature rises in gold nanospheres, showing that thermal nonlinearity is indeed expected at commonly encountered light fluences. Our predictions for the photoacoustic amplitude emitted by a gold nanosphere in the nonlinear regime are given and compared to those from the point-absorber model, and we study the occurence of nonlinearity as a function of fluence and particle size. Finally, we investigate the influence of the equilibriume temperature on the photoacoustic amplitude, for a given nanosphere and fixed fluence. Theoretical approach {#sec:TheoreticalApproach} ==================== We consider the photoacoustic generation by a single gold nanosphere in water. In this section, we describe the models and computations that are used to produce the theoretical predictions given in Sec. \[sec:Results\]. Throughout all this work, spherical symmetry is assumed, with the center of the gold nanosphere as the center of symmetry. Physical models {#sub:PhysicalModel} --------------- ### Photoacoustic generation by a point absorber in a liquid {#subsub:PAFluidModel} In order to discuss further below the generation of photoacoustic waves by a gold nanosphere and to introduce relevant quantities used throughout our work, we first recall here the basic equations which describe the photoacoustic generation in the simple situation of a mechanically and thermally homogeneous liquid medium. In particular, we provide the analytical expressions derived by @diebold2001 of the photoacoustic pressure wave emitted for the limiting case of a point-absorber, both in the linear and nonlinear thermo-elastic regime. The corresponding analytical predictions are discussed in the results section, in particular in comparison with the theoretical predictions for a gold nanosphere. We will show that the point-absorber model accurately predicts the photoacoustic emission by a gold nanosphere only in the linear regime, which justifies the introduction of our nonlinear model for the gold nanosphere in the next section. #### Model equations. When the physical properties are assumed to be homogeneous and constant in time, the generation of photoacoustic waves in a liquid medium is dictated by the following system of coupled equations [@morse1986; @gusev1993; @diebold2001] $$\begin{aligned} \rho_0 c_p\frac{\partial T}{\partial t}(\mathbf{r},t)-\kappa\Delta T (\mathbf{r},t)=P_v(\mathbf{r},t) \label{eq:ThermalEqHomogeneous}\\ \Delta p(\mathbf{r},t) -\frac{1}{c_s^2}\frac{\partial^2 p}{\partial t^2}(\mathbf{r},t)=-\rho_0\beta_0\frac{\partial^2 T}{\partial t^2} (\mathbf{r},t) \label{eq:PAEqHomogeneous}\end{aligned}$$ where $p(\mathbf{r},t)$ and $T(\mathbf{r},t)$ are respectively the (photo)acoustic pressure field and the temperature field. The relevant physical properties are the mass density $\rho_0$, the coefficient of thermal expansion $\beta_0=-\frac{1}{\rho_0}\left(\frac{\partial \rho}{\partial T}\right)_0$, the acoustic velocity $c_s$, the thermal conductivity $\kappa$ and the specific heat capacity at constant pressure $c_p$. Eq.(\[eq:ThermalEqHomogeneous\]) is a standard heat diffusion equation, with a heat source term $P_v(\mathbf{r},t)$ representing the volumetric density of power converted to heat (with dimensions unit power per unit volume). Eq.(\[eq:ThermalEqHomogeneous\]), as fully decoupled from the pressure field, is only valid for liquid and solid media (as opposed to gases) for which the ratio $\gamma = \frac{c_p}{c_v}$ of the specific heat capacity at constant pressure to the specific heat capacity at constant volume may be considered very close to 1 (see demonstration in Appendix \[appendix:DerivationEquation\] based on @morse1986), an assumption that will be made throughout all this work. The photoacoustic wave equation (\[eq:PAEqHomogeneous\]) is a classical wave equation with a source term given by the second time-derivative of the temperature field. Equations (\[eq:ThermalEqHomogeneous\]) and (\[eq:PAEqHomogeneous\]) indicate that to solve the photoacoustic problem given a source term $P_v(\mathbf{r},t)$, one has to *first* solve a thermal diffusion problem, and *then* solve an acoustic problem once the source term given by the temperature field is known. In the context of photoacoustics, the heat source term arises from optical absorption, and is therefore proportional to some illumination function such as the fluence rate (or intensity) $\Phi_r(\mathbf{r},t)$ (unit power per unit surface). In this work, we will considered a single optical absorber (with an absorption cross-section $\sigma_a$) illuminated with some incident pulsed light described by the following expression $$\Phi_r(\mathbf{r},t)=\Phi_0 \frac{1}{\tau_p} f(\frac{t}{\tau_p})$$ where $\Phi_0$ is the fluence (unit energy per unit surface) and $f$ is a dimensionless peaked function describing the temporal profile of the fluence rate. $f$ verifies $\int_{-\infty}^{+\infty} f(\hat{\tau})\mathrm{d}\hat{\tau}=1$ and is normalized such as $\tau_p$ is defined as the full width at half maximum ($\tau_p$ is further referred to in the text as the pulse duration). Throughout this work, the laser temporal profile is chosen as a Gaussian defined accordingly by $$\label{eq:GaussianPulse} f(\hat{\tau})=\frac{2 \sqrt{\ln(2)}}{\sqrt{\pi}} e^{-4 \ln(2)\hat{\tau}^2}$$ #### Point absorber in the linear regime. For an optical absorber of vanishingly small size, but with a finite optical absorption cross-section $\sigma_a$, @diebold2001 provided an analytical expression of the photoacoustic pressure wave emitted by the “point-absorber” (referred to as a photoacoustic point source in @diebold2001): $$\label{eq:AnalyticDieboldLinear} p(\mathbf{r},t)=E_{abs}\beta_0 \frac{1}{c_p \tau_p^2} \frac{1}{4 \pi r}\frac{{\mathrm{d}}f}{{\mathrm{d}}\hat{\tau} }\left(\hat{\tau}=\frac{t-\frac{r}{c_s}}{\tau_p}\right)$$ where $E_{abs}=\sigma_a \Phi_0 $ is the energy absorbed by the point absorber. As expected from the linearity of Eqs. (\[eq:ThermalEqHomogeneous\]) and (\[eq:PAEqHomogeneous\]), the photoacoustic pressure is proportional to the absorbed optical energy. ![\[fig:Beta\] Thermal expansion coefficient of water as a function of temperature. Data derived from the density of water as a function of temperature [@handbook]](BetaWater.eps) . #### Nonlinear thermo-elastic regime. When significant temperature rises occur, the physical properties involved in (\[eq:ThermalEqHomogeneous\]) and (\[eq:PAEqHomogeneous\]) may vary during the illumination and subsequent photoacoustic generation. It is well known that amongst the relevant thermodynamics properties, the thermal expansion coefficient $\beta$ shows the most significant temperature dependency [@gusev1993]. The temperature dependence of the thermal expansion coefficient $\beta(T)$ of water is shown in Fig. \[fig:Beta\]. Taking into account this temperature dependence, Eq. \[eq:PAEqHomogeneous\] has to be modified as $$\label{eq:NonLinearPAfluid} c_s^2 \Delta p(\mathbf{r},t)-\frac{\partial^2 p}{ \partial t^2}(\mathbf{r},t)=-\rho_0 c_s^2\frac{\partial}{\partial t}\left(\beta(T)\frac{\partial T}{\partial t}(\mathbf{r},t)\right)$$ Whereas the temperature field remains linearly related to the optical illumination (via Eq. \[eq:ThermalEqHomogeneous\]), the photoacoustic pressure wave in Eq. \[eq:NonLinearPAfluid\] is nonlinearly dependent on the temperature field and therefore on the optical illumination. Note that the source term in Eq. \[eq:NonLinearPAfluid\] is slightly different from that given initially in the pioneer work by @burmistrova1979 or in @gusev1993 (see detailed derivation of our equation in Appendix \[appendix:DerivationEquation\]). #### Point absorber in the nonlinear regime. For the point absorber model, under the assumption that the temperature dependence of the thermal expansion coefficient can be linearized as $\beta(T)=\beta_0+\beta_1 (T-T_0)$ (with $\beta_0=\beta(T_0)$ and $\beta_1=\frac{{\mathrm{d}}\beta}{{\mathrm{d}}T}(T_0)$), and for a gaussian temporal profile of the illumination pulse, @diebold2001 also provided an analytic expression of the photoacoustic pressure wave emitted by the point absorber in the nonlinear regime: $$\begin{aligned} \label{eq:CalassoNonlinearPA} p(\mathbf{r},t)=&E_{abs}\beta_0 \frac{1}{c_p \tau_p^2} \frac{1}{4 \pi r} \frac{{\mathrm{d}}f}{{\mathrm{d}}\hat{\tau} }\left(\hat{\tau}=\frac{t-\frac{r}{c_s}}{\tau_p}\right) + \nonumber \\ & E_{abs}^2\beta_1\frac{1}{\rho_0 \chi^{3/2} c_p^2 \tau_p^{7/2}}\frac{1}{4 \pi r}h\left(\hat{\tau}=\frac{t-\frac{r}{c_s}}{\tau_p}\right)\end{aligned}$$ where $h(\hat{\tau})$ is a dimensionless function with a tripolar shape, given in details by Eq. (\[eq:h\_function\]) in Appendix \[appendix:DerivationCalasso\]. Note that the numerical prefactor in the nonlinear term of Eq. \[eq:CalassoNonlinearPA\] is different from that the original equation (25) given in @diebold2001, for various reasons detailed in Appendix \[appendix:DerivationEquation\], including the modification required to take into account the correct source term of Eq. \[eq:NonLinearPAfluid\]. ### Photoacoustic generation by a gold nanosphere in a liquid {#subsub:PAGNSModel} In this section, we present the physical models used to describe the photoacoustic emission by a finite-size gold nanosphere (of radius $R_s$) immersed in water. As opposed to the case of a point absorber, no analytical expression is available for the emitted photoacoustic pressure wave by a finite-size solid sphere, except for very limiting cases in the linear regime [@diebold1988; @diebold2002; @diebold2002; @egerev2009], out of scope here. In @chen2012, the thermal source term was modelled via a quasi-static thermal field, and the solution in the linear regime was approximated in the Fourier domain assuming the sphere was small compared to the ultrasound wavelength. The model used in our work takes into account thermal diffusion around the nanosphere, photoacoustic generation and propagation in both the gold nanosphere and its liquid environment, and any arbitrary temperature-dependence of the thermal expansion coefficient of the liquid environment. #### Thermal model. Because the thermal conductivity of gold is much larger than that of water, the temperature within the nanosphere is considered in this work to be uniform, which is known to be a very accurate approximation for gold spheres of diameter of the order of a few tens of nanometers, and for pulse duration no shorter than a few ps [@baffou2011; @pustovalov2005]. In particular, we emphasize that for the pulse durations considered in our work, longer than a few tens of ps, the electron-phonon thermalization that occurs on a time scale no longer than a few ps can be totally neglected [@baffou2011; @pustovalov2005], as opposed to the situation usually encountered in picosecond acoustics with sub-picosecond illumination [@tas1994; @groeneveld1995]. Under this assumption, the spatio-temporal evolution of the temperature fields $T_s(t)$ inside the solid gold nanosphere and $T_f(\mathbf{r},t)$ in its liquid environment can be described by the following system of differential equations [@baffou2011]: \[eq:ThermalEqs\] $$\begin{aligned} \frac{\partial T_{s}}{\partial t}(t)- \frac{3}{R_s}\frac{\kappa^f}{\rho_0^{s} c_p^{s}}\frac{\partial T_f}{\partial r}(R_s^+,t) = &\ \frac{\sigma_{abs}\Phi_0}{\rho_0^{s} c_p^{s}\frac{4}{3}\pi R_s^3}\frac{1}{\tau_p} f(\frac{t}{\tau_p}) \label{eq:TsEquation}\\ \frac{\partial T_f}{\partial t}(\mathbf{r},t)-\frac{\kappa^f }{\rho_0^f c_p^f }\Delta T_f(\mathbf{r},t)=&\ 0, \; r>R_s \end{aligned}$$ with the following boundary conditions: \[eq:TBoundaryCond\] $$\begin{aligned} T_f(R_s^+,t)&= T_s(t) \label{eq:ContinuityT}\\ T_f(r \to \infty ,t)&=T_{0} \label{eq:Tinfty}\end{aligned}$$ with $r=\|\mathbf{r}\|$, and the subscripts $s$ and $f$ referring respectively to the solid and fluid phases. Eq. \[eq:TsEquation\] states that the variation of the uniform sphere temperature increases via the absorbed optical energy and decreases via thermal conduction at the gold/water interface. This equation takes into account the continuity condition for the thermal flux across the interface. Eqs. \[eq:ContinuityT\] and \[eq:Tinfty\] provide the additional boundary conditions required to solve the problem. The continuity equation Eq. \[eq:ContinuityT\] assumes that any interfacial thermal resistivity is neglected. This assumption is discussed further in section \[subsec:Discussion\]. #### Thermoelastic equations. Under spherical symmetry and for isotropic materials, the thermoelastic equations in both heterogeneous solid and liquid media can be written as a first order velocity-stress system of equations that reads [@royer1999; @chen2012] \[eq:SolidPAEq\] $$\begin{aligned} \frac{\partial v_r}{\partial t}(\mathbf{r},t)=& +\frac{1}{\rho_0(r)}\left[\frac{\partial \sigma_{rr}}{\partial r}(\mathbf{r},t) + \frac{2}{r}(\sigma_{rr}-\sigma_{\theta \theta})\right] \\ \frac{\partial \sigma_{rr}}{\partial t}(\mathbf{r},t)=& \left[ (\lambda(r)+2\mu(r)) \frac{\partial }{\partial r}+2 \lambda(r)\frac{1}{r}\right]v_r(\mathbf{r},t) \nonumber\\ &-(\lambda(r)+\frac{2}{3}\mu(r))\beta(T(\mathbf{r},t)) \frac{\partial T}{\partial t}(\mathbf{r},t) \\ \frac{\partial \sigma_{\theta \theta}}{\partial t}(\mathbf{r},t)=& \left[ \lambda(r)\frac{\partial }{\partial r}+2 (\lambda(r)+\mu(r))\frac{1}{r}\right]v_r(\mathbf{r},t) \nonumber\\ &-(\lambda(r)+\frac{2}{3}\mu(r))\beta(T(\mathbf{r},t)) \frac{\partial T}{\partial t}(\mathbf{r},t)\end{aligned}$$ where $\mathbf{\sigma}$ is the stress tensor, $v_r$ is the radial displacement velocity, and $\lambda$ and $\mu$ are the Lamé coefficients. One can readily verify that if $\mu$ is set to zero in Eqs. \[eq:SolidPAEq\], i.e. the material is a liquid ($\sigma_{rr}=-p$), the system yields Eq. \[eq:NonLinearPAfluid\] (or Eq. \[eq:PAEqHomogeneous\] for constant $\beta$). In the relevant case here of a solid/liquid interface, the following continuity conditions must hold for both the radial velocity and stress at the sphere interface : \[eq:ContinuitySolidPAEq\] $$\begin{aligned} v_r(R_s^-,t)&=v_r(R_s^+,t)\\ \sigma_{rr}(R_s^-,t)&=-p(R_s^+,t)\end{aligned}$$ Computations for a gold nanosphere {#sub:Computations} ---------------------------------- The equations that describe the photoacoustic generation by a solid and optically absorbing sphere (Eqs. \[eq:ThermalEqs\] to \[eq:ContinuitySolidPAEq\]) are much more complex than the equations for a homogeneous liquid (Eqs. \[eq:ThermalEqHomogeneous\] to \[eq:PAEqHomogeneous\]) and cannot be solved analytically. However, their resolution still requires to first compute the temperature field from the thermal problem, and then to use this temperature field as a source term in the thermoelastic problem. The full resolution of both the thermal and thermoelastic problems is referred to further in the text as a numerical simulation, based on the computational approaches described below. ### Temperature computations {#subsub:ThermalSolution} The system of equations \[eq:ThermalEqs\] and \[eq:TBoundaryCond\] may be solved analytically for an impulse excitation, i.e $\frac{1}{\tau_p} f(\frac{t}{\tau_p})\to\delta(t) $, using the Laplace Transform. After tedious but simple algebric manipulations and use of tables of known inverse Laplace transforms, one may obtain the Green’s function $G_{\mathrm{th}}(\mathbf{r},t)$ (solution to a $\delta(t)$ excitation) of the thermal problem, as was done by @egerev2009. The expression of $G_{\mathrm{th}}(\mathbf{r},t)$ is given in Appendix \[appendix:GreenTh\]. The temperature field in water for a pulse excitation can then be calculated by the convolution of the thermal Green’s function with the source term: $$\label{eq:SolutionT} T(\mathbf{r},t)= G_{\mathrm{th}}(\mathbf{r},t)\ast \frac{\sigma_{abs}\Phi_0}{\rho_0^{s} c_p^{s}\frac{4}{3}\pi R_s^3}\frac{1}{\tau_p}f(\frac{t}{\tau_p})$$ For all our results, the convolution in Eq. \[eq:SolutionT\] was performed numerically, with the source function $f$ given by Eq. \[eq:GaussianPulse\]. The temperature field $T(\mathbf{r},t)$ was computed and sampled on a regular grid $T(n\times\Delta r,m\times \Delta t)$ required by the finite-difference in time-domain resolution of the thermoelastic problem described below. ### Acoustic computations {#subsub:ElasticSolution} In this work, we used a finite-difference time-domain (FDTD) algorithm to solve the thermo-elastic problem. We adapted the well-known Virieux’s scheme to our problem with spherical symmetry. In brief, the Virieux’s scheme for elastodynamics [@virieux1986] (analog to the Yee’s scheme for electromagnetism [@yee1966]) is based on a spatio-temporal discretization of the system of continuous equations (Eqs. \[eq:SolidPAEq\]) on staggered grids (spatial grid step $\Delta r$ and temporal grid step $\Delta t$). The solution is computed step by step in time, over the whole spatial domain at each time step. Any known source term may be taken into account, both in the sphere and in water. In particular, it makes it straightforward to take into account the temperature-dependence of the thermal expansion coefficient of water, by simply computing the value of $\beta^f(T_f(\mathbf{r},t))$ at each point in space and time. Another well-known key advantage of the Virieux’s scheme is that boundary conditions such as given by Eqs. \[eq:ContinuitySolidPAEq\] are implicitly verified [@virieux1986]. As a consequence, reflected and transmitted acoustic waves at the water-gold interface were taken into account in our numerical solutions. The discretized equations that were used are detailed in Appendix \[appendix:FDTD\]. The spatial grid step $\Delta r$ was chosen small enough to ensure a proper convergence of the FDTD solution: the convergence was ensured by verifying that results with two different spatial steps showed no significant difference. The values of $\Delta r$ typically ranged from 0.1 nm to 5 nm depending on the sphere radius and the pulse duration. The dimension of the spatial domain was taken sufficiently large (typically several tens of $\mu$m) such that any spurious reflections from the domain boundary would arrive far after the photoacoustic pressure waveforms. The time step $\Delta t$ was derived from $\Delta r$ via the stability condition given in Appendix \[appendix:FDTD\]. ### Values of the physical properties {#subsub:PhysicalProperties} Properties Gold Water Unit -------------------------------------------- ------ ------------------ ------------------------------------------------ Mass density $\rho_0$ 19.3 1.00 $\times 10^3\mathrm{kg.m}^{-3}$ Specific heat capacity $c_p$ 129 4200 $\mathrm{J.kg}^{-1}\mathrm{.K}^{-1}$ Thermal conductivity $\kappa$ 318 0.60 $\mathrm{W.m}^{-1}\mathrm{.K}^{-1}$ Thermal diffusivity $\chi$ 128 0.142 $\times 10^{-6}\mathrm{m}^{2}\mathrm{.s}^{-1}$ Thermal expansion $\beta$ 0.43 Fig.\[fig:Beta\] $\times 10^{-4}\mathrm{K}^{-1}$ First Lam$\acute{e}$ coefficient $\lambda$ 147 2.25 GPa Second Lam$\acute{e}$ coefficient $\mu$ 27.8 2.25 GPa Compressional wave velocity 3.24 1.50 $\mu\mathrm{m.}\mathrm{ns}^{-1}$ Shear wave velocity 1.20 - $\mu\mathrm{m.}\mathrm{ns}^{-1}$ : Physical constants associated with gold and water, at T $\sim 25 ^\circ C$, from [@handbook] \[tab:MaterialProperties\] All the values of the physical properties of gold and water used in the computations are summarized in Table \[tab:MaterialProperties\]. Except for the thermal expansion coefficient whose value may depend on temperature, the values for all other properties (assumed to be constant) were those at room temperature ($\sim 25 ^\circ C$). The absorption cross-section of a gold nanosphere depends on its size, and therefore so does the absorbed energy for a given fluence. The values of absorption cross-sections used in this work were derived from the Mie theory with optical constants from @Johnson1972, for an illumination wavelength $\lambda\ = \ 532 \ \mathrm{nm}$. A few typical values are given in Table \[tab:Cross-Sections\]. For a diameter below typically 50 nm, the absorption cross-section is much larger than the scattering cross-section and scales as the nanoparticle volume [@jain2006]. [m[90pt]{} c c c c c c c]{} $R_s$ ($\mathrm{nm}$) &10 & 20 & 30 & 40 & 50 & 60\ $\sigma_{abs}$ ($\times 10^{-14}~\mathrm{m^2}$) & $3.7 \times 10^{-2}$ & $3.3 \times 10^{-1}$ & $1.1$ & $1.8$ & $2.1$ & $2.3$\ \[tab:Cross-Sections\] Results and discussion {#sec:Results} ====================== Linear regime {#sub:LinearRegime} ------------- In this section, the temperature-dependence of the thermal expansion coefficient of water is neglected, i.e we consider the linear photoacoustic regime. Our first objective is to investigate the relative contribution of the gold nanosphere and its liquid environment to the sound generation. When the photoacoustic wave is predominantly generated from the liquid environment rather than from the solid sphere, we then investigate the typical thickness of the water layer that generates the photoacoustic wave. As a preamble to our results in the nonlinear regime, our results for the gold nanosphere are compared to those predicted by the point absorber model in the linear regime first. Throughout all the paper, all absolute photoacoustic amplitudes are given at 1 mm from the center of the absorber ($r=1$ mm). In the linear regime, all the results are predicted for an equilibrium temperature $T_0=20^\circ C$. ### Origin of the photoacoustic wave {#subsub:OriginPA} The absorption of the laser pulse by a spherical gold nanoparticle creates a transient temperature rise in both the particle and its liquid environment due to heat diffusion. From Eqs. \[eq:SolidPAEq\], it is clear that both the gold nanosphere and its environment may generate photoacoustic waves. Here, we investigate the relative contribution to the photoacoustic signal from the gold nanosphere and from its water environment, as a function of the pulse duration $\tau_p$ and sphere radius $R_s$. The considered radii are on the order of a few nanometers to tens of nanometers, and the pulse durations typically range from tens of picoseconds to tens of nanoseconds. For each pair of parameters ($\tau_p, R_s$), two different numerical simulations were performed. Simulation (S1) computed the photoacoustic wave generated from the whole system, i.e the gold nanosphere and its water environment; simulation (S2) was identical except that the thermal expansion coefficient of gold was set to zero. The results from (S2) therefore only takes into account sound generation from water. Fig.\[fig:PAWavesFromSphereVsEnvironment\] shows a plot of the waveforms from simulations (S1) and (S2) for a 20-nm radius gold nanosphere for three different values of $\tau_p$ (10 ps, 500 ps and 5 ns). It is clear from Fig.\[fig:PAWavesFromSphereVsEnvironment\] that the predominant origin of the photoacoustic generation highly depends on the pulse duration: “short” pulses mostly excite acoustic waves in the nanosphere, which are then radiated into the water, whereas photoacoustic waves with “long” pulses originate mostly from the liquid around the nanosphere. Both regimes have been studied experimentally. Indeed, various investigations have been conducted on acoustic vibration of gold nanoparticles in the short pulse regime (fs or ps excitation), see for instance [@delfatti1999]. In the nanosecond regime, Chen et al. have experimentally demonstrated that the photoacoustic signals originate from the environment rather than the nanosphere itself [@chen2012]. Their demonstration was based on the fact that the photoacoustic signal amplitude followed the properties of the temperature-dependence of the thermal expansion coefficient of the liquid around the particle. Fig.\[fig:PAWavesFromSphereVsEnvironment\] illustrates that our model and simulations encompass these different regimes, from the excitation of vibration modes in the sphere by short pulses (although no shorter than a few picosecond as a requirement of our thermal model) to photoacoustic generation directly in the surrounding liquid. It is therefore adapted to model a variety of different phenomena. ![\[fig:RelativeContribution\] Relative contribution from water to the overall photoacoustic energy (from water and gold), as a function of the laser pulse duration $\tau_p$ and the nanosphere radius $R_s$. For a pulse duration of $\tau_p=5\ \mathrm{ns}$, more than typically $90\ \%$ of the energy is emitted from water.](RelativeContributionSphere.eps) In addition, our model can provide a quantitative assessment of the relative contribution to the generated photoacoustic wave as a function of pulse duration $\tau_p$: to do so, the energy $\epsilon$ of the emitted photoacoustic wave (defined as $\epsilon = \frac{4 \pi r^2}{\rho_0 c_p}\int_{-\infty}^{+\infty} p^2(r,t) {\mathrm{d}}t$, $r> R_s$, independent of $r$) was computed for simulations (S1) and (S2). The relative contribution from water was defined as $\eta=\frac{\epsilon(S2)}{\epsilon(S1)}$. The values of $\eta$ as a function of pulse duration and sphere diameter, plotted on Fig.\[fig:RelativeContribution\], show that both the sphere diameter and the pulse duration affect the relative contribution from water. However, for pulse durations larger than a few nanoseconds, most of the photoacoustic energy comes from the surrounding water, in agreement with the experimental results in [@chen2012]. For sphere diameters up to 40 nm, more than $90 \%$ of the photoacoustic energy is generated in water.\ From this point and throughout the rest of paper, we focus our interest on the nanosecond regime, for which the photoacoustic emission from the nanosphere is negligible compared to that of water around it. Within this context, the following two sections investigate and quantify the typical dimension of the water layer that generates the photoacoustic wave, and compare the photoacoustic amplitude predicted for the nanospheres to those predicted from the point-absorber model [@diebold2001]. ### Typical thickness of the generating water layer {#subsub:ExtentPALayer} ![\[fig:Shell\_dependence\] Typical thickness of the water layer that emits the photoacoustic wave as a function of the pulse duration $\tau_p$, for different values $R_s$ of the gold nanosphere radius.](figure_shell.eps) To quantify the size of the water layer that contributes to the photoacoustic generation, the following approach was implemented. The photoacoustic source term in Eq.\[eq:SolidPAEq\] may be straightforwardly turned off in the simulations by forcing $\beta^f$ to zero at any desired locations. Several simulations were therefore run by limiting the extent of the photoacoustic source term to distances $r\in [R_s;R_s+\rho_{source}]$, with $\rho_{source}$ varied from 0 to $+\infty$. In practice, $\rho_{source}$ was varied up to a maximum value large enough so the photoacoustic signal did not differ significantly from its asymptotic value, corresponding to the case where all source points in water are active. The extent of the generating layer in water was then defined by the value $\rho_{layer}=\rho_{source}$ for which the amplitude of the photoacoustic signal reached $80\ \%$ of the amplitude of the asymptotic signal. This procedure was reiterated for different values of the laser pulse duration and the nanosphere radius, to compute the values of $\rho_{layer}(\tau_p, R_s)$ plotted on Fig.\[fig:Shell\_dependence\]. Fig.\[fig:Shell\_dependence\] shows that the size of the contributing layer is in first approximation independent of the size of the sphere, and that it scales with the pulse duration approximately as $\rho_{layer}(\tau_p)\sim \sqrt{\tau_p}$. This scaling law suggests that the extent of the generating layer is dictated mostly by the diffusion of heat in water, regardless of the nanosphere diameter. As a consequence, each gold nanosphere may be considered as a nanometric absorber which thermally probes its environment within a spatial range driven by the laser pulse duration (longer than nanosecond). As an order of magnitude, a pulse duration $\tau_p = 5\ \mathrm{ns}$ yields $\rho_{layer} \sim 30\ \mathrm{nm}$. ### Comparison with the photoacoustic point-absorber model {#subsub:ComparisonPointSource} It was shown above in Section \[subsub:OriginPA\] that for a nanometric sphere illuminated with a nanosecond pulse, the photoacoustic wave is mostly generated by the liquid surrounding the particle. Within this regime, the analytical model proposed by @diebold2001 for point-absorbers in the linear regime is therefore expected to predict reasonably well the amplitude and shape of photoacoustic waves generated by gold nanospheres. The objective of this section is to quantify the accuracy of this theoretical model by comparing its predictions to our simulations for finite-size absorbers. This comparison will be further developed in the next results section for the nonlinear regime. From the analytical expression given by Eq. \[eq:AnalyticDieboldLinear\], the photoacoustic energy emitted by a point absorber is given by $$\label{eq:CalassoLinearEnergy} \epsilon_{\mathrm{point,linear}}=E_{abs}^2\frac{\beta_0^2}{c_p^3}\frac{1}{4 \pi}\frac{1}{\tau_p^3}\int_{-\infty}^{+\infty}\left[\frac{{\mathrm{d}}f}{{\mathrm{d}}u}\right]^2{\mathrm{d}}u$$ ![\[fig:ComparisonDieboldSimu\_lin\] Ratio of the photoacoustic energy emitted from a gold nanosphere to that emitted from a point-absorber of identical absorption cross-section, as a function of the sphere dimensions, for a pulse duration $\tau_p=5 \mathrm{ns}$.](ComparisonAnalyticalNumerical_linear.eps) For gold nanospheres of different sizes, for a fixed pulse duration $\tau_p = 5~\mathrm{ns}$, we compared the emitted photoacoustic energy predicted by Eq. \[eq:CalassoLinearEnergy\] to that predicted for gold nanosphere in the linear regime, for equivalent absorption cross-sections. Fig. \[fig:ComparisonDieboldSimu\_lin\] shows a plot of $\alpha=\frac{\epsilon_{\mathrm{GNS, linear}}}{\epsilon_{\mathrm{point,linear}}}$ as a function of the nanosphere radius $R_s$. The predictions from the point absorber model and from our model turn out to be identical for vanishingly small diameters, as expected. Incidentally, this provides a validation of our numerical simulations in the linear regime. For finite sizes, the effect of the gold nanosphere, both as an acoustic scatterer and as a photoacoustic source (via $\beta^{s}$), is to marginally decrease the emitted acoustic energy compared to a point absorber of identical absorption cross-section. The effect is small, as expected from the very small ratio of the nanosphere diameter (typically tens of nm) to the acoustic wavelength (typically $20\ \mu\mathrm{m}$ for a 5-ns pulse). Therefore, in the nanosecond pulse regime, the point-absorber model proposed by @diebold2001 provides accurate quantitative predictions for the emission of photoacoustic waves by a gold nanosphere, with overestimation of the acoustic energy less than $10\ \%$ for sphere radii up to 30 nm. nonlinear regime ---------------- In this section, we quantitatively investigate for a gold nanosphere the impact of the temperature dependence of the thermal expansion coefficient, that leads to the so called thermal photoacoustic nonlinearity. In a preamble section, we first discuss the consequences that can be derived from the analytical expression provided by @diebold2001 within the point-absorber model. We then report quantitative predictions obtained for a gold nanosphere in the nonlinear regime. ### Predictions from the analytic point-absorber model {#subsub:PredictionFromNonLinearAnalyticPointModel} #### Existence of a critical absorbed energy. {#par:DefCriticalEnergy} ![\[fig:transition\_figure\] Critical energy $E_c$ as a function of the equilibrium temperature $T_{0}$](CriticalEnergy_fctT.eps) Eq. \[eq:CalassoNonlinearPA\] shows that for fixed physical constants, the relative contribution of the nonlinear term to the photoacoustic pressure wave only depends on the absorbed energy $E_{abs}$ and the pulse duration $\tau_p$. For a fixed pulse duration $\tau_p$= 5 ns, Fig.\[fig:Dieboldprediction\_log\] illustrates the change from the linear regime to the nonlinear regime as a function of the absorbed energy. We define the critical energy $E_c$ as the value of absorbed energy for which the peak amplitudes of the nonlinear and the linear contributions in Eq. \[eq:CalassoNonlinearPA\] are identical. With this definition, the nonlinear contribution is thus significant for $E_{abs}\gtrsim E_c$, and becomes predominant for $E_{abs}\gg E_c$. From Eq. \[eq:CalassoNonlinearPA\], $E_c$ is given by $$\label{eq:CriticalEnergy} E_c=\frac{\mathrm{max}(\frac{{\mathrm{d}}f}{{\mathrm{d}}\hat{\tau}})}{\mathrm{max}(h)} \times \rho_0 c_p \frac{\beta_0}{\beta_1}(T_{0})\sqrt{\tau_p \chi}^3$$ The numerical prefactor can be computed numerically from the function $h$ given by Eq. (\[eq:h\_function\]) in Appendix \[appendix:DerivationCalasso\], and yields $$\label{eq:CriticalEnergy} E_c\simeq 20.2 \times \rho_0 c_p \frac{\beta_0}{\beta_1}(T_{0})\sqrt{\tau_p \chi}^3$$ As indicated on Fig. \[fig:Dieboldprediction\_log\], $E_c=$ 31 fJ for $\tau_p$= 5 ns and $T_{0}\ =\ 20\ ^\circ C$. By coupling Eq. \[eq:CriticalEnergy\] with the evolution of $\beta(T)$ in water shown in Fig. \[fig:Beta\], the temperature-dependence of the critical energy was computed and plotted on Fig. \[fig:transition\_figure\]. As $\beta$ vanishes around $4^\circ C$, so does $E_c$, i.e. only nonlinear generation is predicted for any absorbed energy at that temperature. More importantly, the curve in Fig. \[fig:transition\_figure\] indicates that the critical energy is highly sensitive to the equilibrium temperature in a narrow range of a few degrees around $4^\circ C$. On the contrary, the critical energy is only weakly dependent on temperature at physiological temperatures (variation less than an order of magnitude for several tens of degrees). Interestingly, the critical energy given by the point-absorber model scales as the volume around the absorber over which heat diffuses during $\tau_p$. The criteria given by equation (3) by @egerev2008 for a spherical particle of radius $R_s$ can be restated as a critical energy with a form similar to that of the point-absorber model: $$\label{eq:Ec_egerev} E_c\propto \rho_0 c_p \frac{\beta_0}{\beta_1}(T_{0})R_s^3$$ However, the volume term found by @egerev2008 is the volume of the sphere, whereas it is given from @diebold2001 by the volume of heat diffusion in water during the illumination pulse. These two predictions are incompatible: the critical energy given by Eq. \[eq:Ec\_egerev\] goes to zero for vanishingly small spheres ($R_s \to 0$), whereas the point-absorber model predicts a finite critical energy. Nevertheless, the two different expressions of the critical energy have in common to provide as a strong physical insight that the critical energy scales as some volume which origin remains to be determined. This will be further discussed based on our results for gold nanospheres in section \[subsubsec:optimal size for nonlinear generation\]. ![\[fig:SpectraDiebold\] Photoacoustic amplitude spectra as a function of the absorbed energy, for $\tau_p\ =\ 5 \ \mathrm{ns}$ at $T_{0}\ =\ 20\ ^\circ \mathrm{C}$. Each spectrum was normalized by the corresponding absorbed energy $E_{abs}$ in order to illustrate the nonlinear dependence on $E_{abs}$.](EabsNormalizedSpectrum.eps) #### Frequency considerations. The difference in temporal shapes for the linear and the nonlinear contributions was pointed out by @diebold2001. Here, we investigate the consequences of these different temporal shapes (bipolar for the linear term, tripolar for the nonlinear term) in the frequency domain. For $\tau_p\ =\ 5 \ \mathrm{ns}$, Fig. \[fig:SpectraDiebold\] shows the frequency spectra of the photoacoustic signal as a function of the absorbed energy, with each spectrum normalized by the absorbed energy. Fig. \[fig:SpectraDiebold\] shows that the spectrum amplitude varies nonlinearly with the absorbed energy as expected from Eq. \[eq:CalassoNonlinearPA\] in the time domain. Importantly, it also indicates that the frequency content is shifted towards high frequencies in the nonlinear regime. Therefore, the photoacoustic nonlinearity predicted by @diebold2001 only manifests itself for sufficiently high frequency. This is coherent with the fact that the critical energy decreases with decreasing pulse duration (increasing centre frequency), as indicated by Eq. \[eq:CriticalEnergy\]. As a major consequence from the experimental point of view, Fig. \[fig:SpectraDiebold\] shows that even when the nonlinearity is predominant when considered over the full bandwidth, it remains minor for frequencies below 10 MHz even for $E_{abs}$ as high as $10\times E_c$. ### Temperature rise in a gold nanosphere ![\[fig:TemperatureRise\] Temperature rise inside a 20-nm radius gold nanosphere illuminated with a fluence $\Phi_0=5\mathrm{mJ.cm^{-2}}$ ($E_{abs}\sim 165 \ \mathrm{fJ}$) and $\tau_p\ =\ 5\ \mathrm{ns}$.](Temperature_Rise.eps) It was shown in Section \[subsub:ComparisonPointSource\] in the limits of the linear model that the photoacoustic wave generated by a gold nanosphere is very close to that generated by a point absorber of identical absorption cross-section, and therefore depends only on the absorbed energy $E_{abs}$, with no dependency on thermal properties. In the case of thermal nonlinearity (caused by the temperature-dependence of the thermal expansion coefficient), it is the temperature field that drives the effective value of $\beta(T)$. As a consequence, because the temperature fields are different for a point absorber and a gold nanosphere with the same absorption cross-section, one expects the thermal photoacoustic nonlinearity to be dependent on the size of the absorber. As opposed to the point-absorber model, all the values of the temperature field are finite when finite-size absorbers such as gold nanospheres are considered. Moreover, for a given incident fluence, the temperature rise in a nanosphere is highly dependent on its size, both through the size-dependence of the thermal diffusion (the thermal Green’s function given in Appendix \[appendix:GreenTh\] depends on the particle radius) and the absorption cross-section (see table \[tab:Cross-Sections\]). Fig. \[fig:TemperatureRise\] illustrates the temperature rise in a 40-nm diameter gold nanosphere illuminated with a fluence $\Phi_0=5\mathrm{mJ.cm^{-2}}$ ($E_{abs}\sim 165 \ \mathrm{fJ}$) and $\tau_p\ =\ 5\ \mathrm{ns}$. This plot shows that for a fluence value typical of those used for biomedical applications, the temperature rise in a 40-nm diameter gold nanosphere is significantly larger than the equilibrium temperature. In this case, the photoacoustic nonlinearity is likely to become significant, as demonstrated further below. For a given illumination fluence, Fig. \[fig:PeakTemperatureRise\] shows that the peak temperature rise depends on the sphere radius and turns out to be maximum for a radius value around 35 nm, reflecting the dependence on radius via both thermal diffusion and the size-dependence of the absorption cross-section. ![\[fig:PeakTemperatureRise\] Peak temperature rise as a function of gold nanosphere radius, for a fluence $\Phi_0=5\mathrm{mJ.cm^{-2}}$ ($E_{abs}\sim 165 \ \mathrm{fJ}$) and $\tau_p\ =\ 5\ \mathrm{ns}$. ](maxTemperature_Rise.eps) ### Gold nanosphere vs. point absorber ![\[fig:ComparisonDieboldvsGNS\] Photoacoustic amplitude from a gold nanosphere, normalized to that of the point-absorber model, as a function of nanosphere diameter. Parameters: $T_{0}=20\ ^\circ \mathrm{C}$, $\tau_p\ =\ 5\ \mathrm{ns}$](ComparisonAnalyticalNumerical_nonlinear.eps) Simulations were run to predict the photoacoustic signals generated from gold nanospheres of various diameters illuminated with 5-ns pulses of various fluences, in order to compare the photoacoustic amplitude to that predicted from the point-absorber model [@diebold2001] with matched absorption cross-sections. Fig. \[fig:ComparisonDieboldvsGNS\] shows a plot of the ratio of the photoacoustic peak amplitude predicted for the nanosphere to that predicted for the point-absorber as a function of the absorbed energy. Figure  \[fig:ComparisonDieboldvsGNS\] confirms that in the linear regime, i.e for low enough absorbed energy (or equivalently $E_{abs}\ll E_c$), a gold nanosphere may be considered as a point-absorber, i.e. the predictions from the corresponding point-absorber model are accurate. However, in the nonlinear regime, the point-absorber model significantly overestimates the amplitude of the photoacoustic signals. In other words, the critical energy for a gold nanosphere is significantly higher than that predicted by the point-absorber model. For the gold nanosphere, in order to define the critical energy in agreement with the definition for the point absorber (see sec \[par:DefCriticalEnergy\]), the nonlinear contribution was defined by the difference between the total signal predicted in the nonlinear regime and the signal predicted in the linear regime only (by keeping $\beta$ constant). As an illustration, the photoacoustic amplitude predicted for a 20-nm radius gold nanosphere at equilibrium temperature $T_{0}=20\ ^\circ \mathrm{C}$, illuminated with a 5-ns pulse of fluence $\Phi_0=5\mathrm{mJ.cm^{-2}}$ ($E_{abs}\sim 165 \ \mathrm{fJ}$), is about three times lower than that predicted by the point-absorber model, as illustrated in Fig. \[fig:ComparisonDieboldvsGNS20nm\], and the critical energy is 216 fJ whereas the point-absorber model predicts a value of 31 fJ. On the other hand, the frequency features discussed in the previous section for the point-absorber model remain strictly identical for a gold nanosphere, i.e. the nonlinearity is observed only at high enough frequencies. Fig. \[fig:SpectraDiebold\] for the point-absorber remains valid for a gold nanosphere, provided that the appropriate value of critical energy is used. In conclusion to this comparison between our predictions for gold nanospheres and predictions for the point-absorber model, the finite size of the gold nanosphere must be taken into account to obtain accurate quantitative predictions regarding the occurence of thermal-based photoacoustic nonlinearities. [|c|&gt;p[3 cm]{}|c\*[5]{}[|c]{}|]{} $T_0(^\circ C) \backslash R_s$ (nm)& & 10 & 20 & 30 & 40 & 50 & 60\ & $E_c^{R_s} (\mathrm{fJ})$ & 9.4 & 19 & 38 & 66 & 110 & 170\ & $\Phi_c^{R_s} (\mathrm{mJ}/\mathrm{cm}^2)$ & 2.5 & 0.59 & 0.35 & 0.36 & 0.51 & 0.74\ & $\Delta T_{max} (^\circ C)$ & 17.4 & 13.7 & 13.8 & 14.5 & 15.5 & 17.1\ & $E_c^{R_s} (\mathrm{fJ})$ & 52 & 93 & 150 & 230 & 340 & 490\ & $\Phi_c^{R_s} (\mathrm{mJ}/\mathrm{cm}^2)$ & 14 & 2.8 & 1.4 & 1.3 & 1.6 & 2.1\ & $\Delta T_{max} (^\circ C)$ & 96 & 65 & 55 & 51 & 49 & 49\ & $E_c^{R_s} (\mathrm{fJ})$ & 120 & 220 & 350 & 520 & 740 & 1040\ & $\Phi_c^{R_s} (\mathrm{mJ}/\mathrm{cm}^2)$ & 33 & 6.5 & 3.2 & 2.9 & 3.6 & 4.5\ & $\Delta T_{max} (^\circ C)$ & 230 & 150 & 126 & 114 & 107 & 104\ & $E_c^{R_s} (\mathrm{fJ})$ & 230 & 400 & 630 & 940 & 1350 & 1850\ & $\Phi_c^{R_s} (\mathrm{mJ}/\mathrm{cm}^2)$ & 62 & 12 & 5.8 & 5.1 & 6.4 & 8.1\ & $\Delta T_{max} (^\circ C)$ & 420 & 280 & 230 & 210 & 190 & 190\ \[tab:BigTable\_Ec\_Phic\_Tmax\] ### Optimal size for nonlinear generation {#subsubsec:optimal size for nonlinear generation} In order to quantitatively predict the occurence of nonlinear photoacoustic generation as a function of parameters that are controllable experimentally, simulations were run for gold nanospheres with different radii and equilibrium temperatures. Table. \[tab:BigTable\_Ec\_Phic\_Tmax\] reports the corresponding results as the value of the critical energy $E_c$, critical fluence $\Phi_c=E_c/\sigma_a$ and the peak temperature in the sphere. In particular, these results show that for any equilibrium temperature, there is an optimal sphere radius around 40 nm for which the critical fluence is minimal (as a function of size). In other words, our model predicts that at a given illumination fluence, the thermal-based photoacoustic nonlinearity is maximised for a sphere radius around 40 nm. We note that the value of the gold nanosphere radius that maximizes the photoacoustic nonlinearity (for a given illumination fluence) is close to the one that maximizes the peak temperature rise in the nanosphere (see Fig. \[fig:PeakTemperatureRise\]): because our model assumes the continuity of temperature across the gold/water interface, the peak temperature in water is also maximized for a radius of 40 nm, which is expected to maximize thermal nonlinearity. ![\[fig:Ec\_f\_R\] Critical energy as a function of nanosphere radius (logarithmic axis). ](CriticalEnergy_fctR.eps) The values of critical energy given in Table. \[tab:BigTable\_Ec\_Phic\_Tmax\] can be further analyzed as a function of the particle size, for a given equilibrium temperature, in order to assess the relevance of the scaling law predicted by the point-absorber model ($E_c$ independent of $R_s$, see Eq. \[eq:CriticalEnergy\]) and the model by @egerev2008 ($E_c \propto R_s^3$, see Eq. \[eq:Ec\_egerev\]). For $T_0=20\ ^{\circ}C$, Fig \[fig:Ec\_f\_R\] shows a plot of the values of the critical energy predicted by our model, by the point-absorber model, and for $E_c \propto R_s^3$ and $E_c \propto R_s^2$. This plot is compatible with the hypothesis that the critical energy tends towards the constant value of the point-absorber for small spheres, and scales as the surface $R_s^2$ for large spheres. For large spheres, it is indeed expected that the volume of heat diffusion scales as the surface of the sphere times the radial distance traveled by heat (only dependent on the pulse duration and the diffusivity in water). Beyond providing quantitative values for the critical energy (or equivalently the critical fluence via the absorption cross-section), our model therefore also provides some physical insight into the origin of the critical energy. ### Influence of the equilibrium temperature ![\[fig:effetTeq\]Peak photoacoustic amplitude emitted by a 40-nm diameter gold nanosphere, as a function of equilibrium temperature, for two different fluence $\Phi$. $\Phi=0.1\ \mathrm{mJ/cm^2}$ corresponds to the linear regime, whereas significant nonlinearities occur at $\Phi=10\ \mathrm{mJ/cm^2}$. ](EffetTeq.eps) In this section, we study the influence of the equilibrium temperature on the photoacoustic amplitude, for a fixed illumination fluence. Fig. \[fig:effetTeq\] shows the results obtained in the linear regime (lower black curve) and in the nonlinear regime (upper blue curve). In the linear regime, the photoacoustic amplitude reflects the temperature dependence of the thermal expansion coefficient $\beta_0(T_0)$, as expected from Eq. \[eq:PAEqHomogeneous\]. In the nonlinear regime, at a higher illumination fluence, the temperature-dependence of the signal is much less significant, and essentially reflects the effective value of the thermal expansion coefficient, different from the value at equilibrium because of the significant temperature rise around the particle. As a consequence, the evolution of the peak photoacoustic amplitude as a function of equilibrium temperature reflects temperature rises at the nanoscale. In particular, the strong signal observed at $T_0 = 4 \ ^\circ C$ for which $\beta_0=0$ is purely nonlinear in nature as the linear contribution vanishes at this temperature. This has first been observed experimentally by @hunter1981 in protons experiments and has been more recently reported with gold nanoparticles by @fukasawa2014 and @simandoux2014. Recent experimental measurements of the photoacoustic amplitudes as a function of equilibrium temperature with 40-nm diameter gold nanospheres by @simandoux2014 are in good agreement with the predictions given in Fig. \[fig:effetTeq\]. An important consequence of the results shown in Fig. \[fig:effetTeq\] is that when thermal nonlinearity takes place, the amplitude of photoacoustic waves cannot provide a measurement of the equilibrium temperature through the knowledge or calibration of $\beta_0(T_0)$. While photoacoustic measurements can provide a measurement of the equilibrium temperature in the linear regime, as was originally proposed by @larina2005 and further investigated in subsequent works (see  [@shah2008; @Petrova2013; @Gao2013] for instance), this approach is expected to not work any longer with nanoparticles illuminated with high enough fluences. Discussion {#subsec:Discussion} ---------- The photoacoustic amplitude predicted by the point absorber model (see Eq. \[eq:AnalyticDieboldLinear\]) is independent of thermal diffusion in water, and only depends on the absorbed energy. Therefore our results suggest that when gold nanospheres behave as point-absorbers in the *linear* regime, the photoacoustic amplitude is insensitive to heat transfer from gold to water, although the temperature field does depend on the thermal diffusivity of water (see Appendix \[appendix:GreenTh\]). On the contrary, *nonlinear* photoacoustic signals are strongly dependent on the thermal diffusivity of water (as predicted by the point-absorber model in water by Eq. \[eq:CalassoNonlinearPA\]). This suggest that the occurence of thermal nonlinearity is required to probe thermal diffusion properties at the spatial and temporal nanoscale. In the model described here, the temperature was assumed to be continuous across the gold/water interface, whereas it is known that an interfacial thermal resistivity exists and may have significant impact at the nanoscale [@alper2010; @schmidt2008; @wilson2002]. To check the importance of this effect on our predictions, we solved the thermal problem by taking into account an interfacial thermal resistance at the gold-water interface. The temperature field for a delta impulse illumination (Green’s function) was found to strongly depend on the thermal resistance, in agreement with earlier work [@juve2009]. However, when convolved with a 5-ns pulse illumination, the temperature field for a interfacial resistance of $10^{-8}\ \mathrm{m^2.K.W^{-1}}$, typical of the gold-water interface, was nearly identical to the case with no thermal resistance (less than a few $\%$ relative difference). The results presented in this work with no interfacial thermal resistance therefore remains valid in the few-nanosecond regime. On the other hand, the presence of coating, such as silica coatings or PEG (poly-ethylene glycol) coatings, may have a significant effect, and will be investigated in a future work. Importantly, our work describes the photoacoustic generation by a *single* gold nanosphere. When collections of nanoparticles such as encountered in suspension are considered, the emitted photoacoustic waves arise from the sum of each contribution from individual particles. In this case, the frequency content of the resulting wave is dictated by the spatial distribution of the collection of nanoparticles, which acts as a low-pass filter. Because the thermal nonlinearity was shown to be significant only at high frequencies, it is expected that it may not be observable at low frequencies. In a recent experimental study [@simandoux2014] with a detection frequency of 20 MHz and 40-nm diameter gold nanospheres suspended in $100\ \mathrm{\mu m}$ diameter tube, a linear relationship was observed between the peak-to-peak amplitude and the fluence, up to a fluence of $\Phi=7\ \mathrm{mJ/cm^2}$, yet close to the corresponding critical fluence predicted here $\Phi_c = 6.5\ \mathrm{mJ/cm^2}$. However, as discussed above, the critical fluences predicted here are only valid for a single nanoparticle, and the prediction for ensembles of nanoparticles, beyond the scope of this work, requires taking into account the nanoparticles spatial distribution. It also has to be kept in mind that our model assumes that there is no limitation on the peak temperature in the gold nanosphere and in water. In practice, the model becomes of course irrelevant if the predicted temperatures would lead to phase transitions in either gold or water. In addition, the thermal model used in our study is strictly restricted to the long-pulse illumination regime (pulses typically longer than a few picosecond), and the corresponding modeling cannot be applied to the sub-picosecond illumination regime which usually requires either a two-temperature model [@eesley1983] or nonthermal distributions [@tas1994; @groeneveld1995] for the electrons and phonons in nanoparticles. Summary and conclusion ====================== In summary, we theoretically investigated the photoacoustic generation by a gold nanosphere in water in the thermoelastic regime. Photoacoustic signals were predicted numerically based on the successive resolution of a thermal diffusion problem and a thermoelastic problem, taking into account the finite size of the gold nanosphere, thermoelastic and elastic properties of both water and gold, and the temperature dependence of the thermal expansion coefficient of water. For sufficiently high illumination fluences, this temperature-dependence yields a nonlinear relationship between the photoacoustic amplitude and the fluence. For nanosecond pulses in the linear regime, we showed that more than $90\ \%$ of the emitted photoacoustic energy is generated in water, and the thickness of the generating layer around the particle scales close to the square root of the pulse duration. In the linear regime, we showed that the point-absorber model introduced by @diebold2001 accurately predicts the amplitude of the photoacoustic waves generated by gold nanospheres for diameters up to several tens of nanometers. However, the point-absorber model significantly overestimates the amplitude of photoacoustic waves generated by gold nanospheres in the nonlinear regime. Our model for finite-size particles provided quantitative estimates of the critical energy, defined as the absorbed energy required such that the nonlinear contribution is equal to that of the linear contribution. Our results suggest that the critical energy scales as the volume of water over which heat diffuses during the illumination pulse. A frequency analysis of the nonlinear signals indicated that the thermal nonlinearity from a gold nanosphere is more pronounced at high frequencies dictated by the pulse duration. Finally, we show that the relationship between the photoacoustic amplitude and the equilibrium temperature at sufficiently high fluence reflects the thermal diffusion at the nanoscale around the gold nanosphere. Although our model was limited to the case of a bare nanoparticle, the approach used in this work can be extended to the more general case of a coated nanoparticle, provided that the corresponding temperature field can be predicted, either by analytical or numerical means (such as given by @baffou2011). To limit the computational costs, and for sake of simplicity, this work was limited to nanospheres with central symmetry. However, the same methodology could apply in two dimensions for axisymmetric nanoparticles such a gold nanorods at the cost of more intensive computations. Acknowledgements ================ This work was funded by the French Centre National de la Recherche Scientifique, the Plan Cancer 2009-2013 (Action 1.1, Gold Fever), and the Agence Nationale de la Recherche (Golden Eye, ANR-10-INTB-1003, and the LABEX WIFI within the French Program ’Investments for the Future’, ANR- 10-LABX-24 and ANR-10-IDEX-0001-02 PSL\*). Nonlinear thermoelastic regime {#appendix:NonlinearRegime} ============================== Derivation of Eq. \[eq:NonLinearPAfluid\] {#appendix:DerivationEquation} ----------------------------------------- We consider the linear regime of acoustic propagation, *i.e.* small Mach number ($\frac{\|\mathbf{v}\|}{c_s}\ll 1$) and small density changes ($\frac{\rho-\rho_0}{\rho_0}\ll 1$). In particular, the following approximation holds: $\frac{\delta X}{\delta t}=\frac{\partial X}{\partial t}+(\mathbf{v}.\nabla)X\simeq \frac{\partial X}{\partial t}$. The three fundamental equations of linear acoustics can be written as $$\begin{aligned} \frac{\partial\rho}{\partial t}&=-\rho_0\ \mathrm{div}(\mathbf{v}) \label{eq:ContinuityEq}\\ \rho_0\frac{\partial \mathbf{v}}{\partial t}&= -\nabla p \label{eq:Euler}\\ \rho T\frac{\partial s}{\partial t}&=\kappa\Delta T +P_v \label{eq:EntropicDiffusion}\end{aligned}$$ where $p(\mathbf{r},t)$ is the acoustic pressure, $\mathbf{v}(\mathbf{r},t)$ is the acoustic displacement velocity, $s(\mathbf{r},t)$ is the specific entropy and $T(\mathbf{r},t)$ is the temperature. The thermal conduction coefficient $\kappa$ is assumed to be constant. Moreover, the differentials of the state functions $\rho=\rho(p,T)$ et $s=s(T,p)$ can be written as [@morse1986] $$\begin{aligned} \delta \rho &=\frac{\gamma}{c_s^2}[\delta p -\beta \delta T]\label{eq:evolution drho}\\ \delta s&=\frac{c_p}{T}[\delta T + \frac{\gamma-1}{\gamma \beta}\delta p ]\label{eq:evolution ds}\end{aligned}$$ where the thermodynamical coefficients are defined as $$\begin{aligned} c_s^2=\left(\frac{\partial p}{\partial \rho}\right)_s \\ c_p=T \left(\frac{\partial s}{\partial T}\right)_{p} \\ c_v=T\left(\frac{\partial s}{\partial T}\right)_{\rho} \\ \gamma=\frac{c_p}{c_v}\\ \beta=-\frac{1}{\rho}\left(\frac{\partial \rho}{\partial T}\right)_{p}\end{aligned}$$ with $c_s^2$ the isentropic sound velocity, $c_p$ the specific heat capacity at constant pressure, $c_v$ the specific heat capacity at constant volume and $\beta$ the thermal expansion coefficient. Eqs. \[eq:evolution drho\] and \[eq:evolution ds\] may be written as $$\begin{aligned} \frac{\partial\rho}{\partial t}&=\frac{\gamma}{c_s^2}[\frac{\partial p}{\partial t} -\beta \frac{\partial T}{\partial t}]\label{eq:evolution temporelle rho}\\ \frac{\partial s}{\partial t}&=\frac{c_p}{T}[ \frac{\partial T}{\partial t} + \frac{\gamma-1}{\gamma \beta}\frac{\partial p}{\partial t}] \label{eq:evolution temporelle s}\end{aligned}$$ where all the thermodynamical coefficients may vary with properties such as temperature. In the following, we consider that only the variations of $\beta(T)$ with temperature are significant, and assume all other thermodynamical properties to be constant. By coupling the first two fundamental equations \[eq:ContinuityEq\] and \[eq:Euler\], one gets $\frac{\partial^2 \rho}{\partial t^2}-\Delta p=0$, which can be transformed into the following equation by use of Eq. \[eq:evolution temporelle rho\] $$\frac{\gamma}{c_s^2}\frac{\partial^2 p}{\partial t^2}-\Delta p =\frac{\rho_0 }{c_p}\frac{\partial }{\partial t} \left(\beta(T) T \frac{\partial s}{\partial t} \right)$$ By use of Eq. (\[eq:evolution temporelle s\]) into the evolution equation (\[eq:evolution ds\]) for the entropy, one obtains the following equation for the temperature field: $$\rho_0 c_v \frac{\partial T}{\partial t}=\kappa\Delta T +P_v +\rho_0 c_v \frac{\gamma -1}{\gamma \beta}\frac{\partial p}{\partial t} \label{eq:équation diffusion température avec rho}$$ Under the assumption, valid for liquids, that $\gamma \sim 1$, one eventually obtains Eqs. (\[eq:ThermalEqHomogeneous\]) and (\[eq:CalassoNonlinearPA\]) given in section \[sub:PhysicalModel\]: $$\begin{aligned} \rho_0 c_v \frac{\partial T}{\partial t}&=\kappa\Delta T +P_v\label{eq:équation diffusion température}\\ \frac{1}{c_s^2}\frac{\partial^2 p}{\partial t^2}-\Delta p &=\rho _0\frac{\partial }{\partial t} \left(\beta(T)\frac{\partial T}{\partial t} \right)\label{eq:équation onde pression} \end{aligned}$$ Eq. (\[eq:équation onde pression\]) may also obtained from Eqs. (\[eq:SolidPAEq\]), by considering a liquid as a special type of solid for which $\mu=0$. Consequences from the form of Eq. (\[eq:CalassoNonlinearPA\])/(\[eq:équation onde pression\]) --------------------------------------------------------------------------------------------- The photoacoustic wave equation Eq. (\[eq:CalassoNonlinearPA\])/(\[eq:équation onde pression\]) used in this work and derived above, differs from the equation originally introduced by @burmistrova1979 which is given only for small temperature variations around $T_0$ and reads: $$\label{eq:Burmistrova} \frac{1}{c_s^2}\frac{\partial^2 p}{\partial t^2}-\Delta p =\rho _0\frac{\partial^2 }{\partial t^2}\left[ \beta(T) (T-T_0)\right]$$ Eq. (\[eq:Burmistrova\]) corresponds to that given by @gusev1993, further used by @diebold2001 for the analytic derivation corresponding to the point-absorber model. Although the two forms source term are very different in these two equations, it turns out as demonstrated below that the consequences on the solution are relatively minor when one consider only the first-order dependence of $\beta(T)$ with temperature, as $\beta(T)=\beta_0+\beta_1 (T-T_0)$ (with $\beta_0=\beta(T_0)$ and $\beta_1=\frac{{\mathrm{d}}\beta}{{\mathrm{d}}T}(T_0)$). By using this first-order expansion of $\beta(T)$ around $T_0$, our equation Eq. (\[eq:CalassoNonlinearPA\]) becomes $$\frac{1}{c_s^2}\frac{\partial^2 p}{\partial t^2}-\Delta p =\rho_0\left[\left(\beta_0+\beta_1(T-T_0)\right)\frac{\partial^2 T}{\partial t^2}+\beta_1\left(\frac{\partial T}{\partial t}\right)^2\right]$$ whereas the original equation used by @burmistrova1979 [@gusev1993; @diebold2001] becomes $$\label{eq:equation gusev beta linéarisé} \frac{1}{c_s^2}\frac{\partial^2 p}{\partial t^2}-\Delta p =\rho_0\left[\left(\beta_0+2\beta_1(T-T_0)\right)\frac{\partial^2 T}{\partial t^2}+2\beta_1\left(\frac{\partial T}{\partial t}\right)^2\right]$$ It therefore turns out that the only remaining difference is a factor 2 before $\beta_1$. In consequence, all the analytical calculations based on Eq. (\[eq:Burmistrova\]), as given by @burmistrova1979 [@gusev1993; @diebold2001], may be used straightforwardly by changing $\beta_1$ to $\beta_1/2$. Modified expression for the point-absorber model {#appendix:DerivationCalasso} ------------------------------------------------ Three modifications are needed to go from the original equation (25) given by @diebold2001 to our Eq. (\[eq:CalassoNonlinearPA\]): 1. @diebold2001 erroneously used a term $e^{r^2/2 \chi \xi }$ in their Eq. (19), instead of $e^{r^2/4 \chi \xi }$, as straightforward from their Eq. (6). As can be derived from simple calculations, the only consequence of this minor error is that equation (25) by @diebold2001 has to be multiplied by a factor $2^{3/2}$. 2. In @diebold2001, the derivation of the nonlinear contribution to the photoacoustic signal by a point-absorber was based on Eq. (\[eq:Burmistrova\]) (or stricly speaking its equivalent formulated as a displacement potential), under the assumption that $\beta(T)=\beta_0+\beta_1 (T-T_0)$. From the discussion above, we therefore had to change the expression given in @diebold2001 by replacing $\beta_1$ to $\beta_1/2$ to take into account our equation \[eq:CalassoNonlinearPA\]. (N.B. @diebold2001 used the notation $(\beta1,\beta2)$ whereas we used $(\beta0,\beta1)$). In other words, Eq (25) in @diebold2001 simply has to be multiplied by a factor 1/2 to take into account Eq. (\[eq:CalassoNonlinearPA\]), but all the analytical derivations remain valid except for this numerical prefactor.\ Overall, to take into account the two prefactors above, Eq (25) in @diebold2001 has to be multiplied by a factor $1/2\times2^{3/2}=2^{1/2}$. 3. The pulse duration $\theta$ used by @diebold2001 is related to the pulse duration $\tau_p$ defined here by $\tau_p=2\sqrt{\ln(2)}\ \theta$. Moreover, we use the dimensionless retarded time $\hat{\tau}=(t-r/c_s)/\tau_p$ whereas @diebold2001 used $\hat{\tau}=(t-r/c_s)/\theta$. This change of variable further adds a numerical prefactor to the expression of the nonlinear term. Taking into account the three modifications above, the nonlinear contribution $p^{NL}$ from the point-absorber model can be written as $$p^{NL}(\mathbf{r},t)=E_{abs}^2\beta_1\frac{1}{\rho_0 \chi^{3/2} c_p^2 \tau_p^{7/2}}\frac{1}{4 \pi r}h\left(\hat{\tau}=\frac{t-\frac{r}{c_s}}{\tau_p}\right)$$ where $h(\hat{\tau})$ is given by $$\label{eq:h_function} h(\hat{\tau})=\frac{[\ln(2)]^{3/4}}{8\pi^2} \frac{{\mathrm{d}}^2}{{\mathrm{d}}\hat{\tau}^2}\left[\int_{0}^{\infty}\frac{\mathrm{erf}(\xi/\sqrt{2})}{\xi^{3/2}}e^{-2\left(\xi/2-2\sqrt{\ln(2))}\hat{\tau}\right)^2}{\mathrm{d}}\xi \right]$$ Green’s function of the thermal problem {#appendix:GreenTh} ======================================= The following expression of the Green’s function of the thermal problem may be found in @egerev2009 and written as $$\begin{aligned} &G_{\mathrm{th}}(\mathbf{r},t)=\frac{1}{\alpha_1-\alpha_2}\frac{R_s}{r} \Biggl[ \nonumber \\ &+\alpha_1 \exp\left(\alpha_1(\frac{r}{R_s}-1)+\alpha^2_1\frac{t}{\tau_{th}}\right) \mathrm{erfc}\left(\frac{\frac{r}{R_s}-1}{2\sqrt{\frac{t}{\tau_{th}}}}+\alpha_1\sqrt{\frac{t}{\tau_{th}}}\right)\nonumber \\ & -\alpha_2 \exp\left(\alpha_2(\frac{r}{R_s}-1)+\alpha^2_2\frac{t}{\tau_{th}}\right) \mathrm{erfc}\left(\frac{\frac{r}{R_s}-1}{2\sqrt{\frac{t}{\tau_{th}}}}+\alpha_2\sqrt{\frac{t}{\tau_{th}}}\right) \Biggr]\end{aligned}$$ where: $$\tau_{th}=\frac{R_s^2}{\chi}, \ \alpha_\frac{1}{2}=\frac{\eta}{2}\left(1\pm\sqrt{1-\frac{4}{\eta}}\right), \ \eta=3\frac{\rho_0^w c_p^w}{\rho_0^{Au}c^{Au}_p}$$ FDTD algorithm for the thermoelastic problem {#appendix:FDTD} ============================================ Following the approach introduced for elastodynamics by @virieux1986, Eqs. \[eq:SolidPAEq\] were discretized on staggered grids (see Fig. \[fig:FDTD\]) with the following centered finite-difference approximation: $$\label{eq: discrétisation définition base} \frac{\partial f}{\partial a}(a_i)\approx \frac{f(a_i+\frac{\Delta a}{2})-f(a_i-\frac{\Delta a}{2})}{\Delta a}$$ where $a$ may be either $r$ or $t$. Moreover, whenever needed at locations where they were undefined (either field variables or material properties), variables were approximated by their arithmetic mean with $f(a_i)=\frac{f(a_i+\frac{\Delta a}{2})-f(a_i-\frac{\Delta a}{2})}{2}$. All material properties were defined at locations ${k\Delta r}$. The Eqs. \[eq:SolidPAEq\] accordingly read $$\begin{aligned} \label{eq: Equa PA solide vitesse discretisee} &\frac{v_r(k+\frac{1}{2},m+\frac{1}{2})-v_r(k+\frac{1}{2},m-\frac{1}{2})}{\Delta t}= \frac{2}{\rho_0(k+1)+\rho_0(k)}\times \left[\right.\nonumber \\ &\frac{\sigma_{rr}(k+1,m)- \sigma_{rr}(k,m)}{\Delta r} + \nonumber \\ & \frac{2}{\Delta r(k+\frac{1}{2})} \times \left.\left[\frac{\sigma_{rr}(k+1,m)+\sigma_{rr}(k,m)}{2}-\frac{\sigma_{\theta\theta}(k+1,m)+\sigma_{\theta\theta}(k,m)}{2}\right]\right] \\ \nonumber \\ &\frac{\sigma_{rr}(k,m+1)-\sigma_{rr}(k,m)}{\Delta t}= \nonumber \\ & \lambda(k)\frac{1}{\Delta r}\frac{1}{k^2}\left[\left(k+\frac{1}{2}\right)^2 v_r(k+\frac{1}{2},m+\frac{1}{2}) -\left(k-\frac{1}{2}\right)^2 v_r(k-\frac{1}{2},m+\frac{1}{2}) \right] + \nonumber\\ & 2\mu(k)\frac{1}{\Delta r} \left[v_r(k+\frac{1}{2},m+\frac{1}{2})- v_r(k-\frac{1}{2},m+\frac{1}{2}) \right] - \nonumber\\ & \left[\lambda(k)+\frac{2}{3}\mu(k)\right] \beta(T(k,m+\frac{1}{2})) \frac{T(k,m+1)-T(k,m)}{\Delta t} \\ \nonumber \\ & \frac{\sigma_{\theta\theta}(k,m+1)-\sigma_{\theta\theta}(k,m)}{\Delta t}= \nonumber \\ &\lambda(k)\frac{1}{\Delta r}\frac{1}{k^2}\left[\left(k+\frac{1}{2}\right)^2 v_r(k+\frac{1}{2},m+\frac{1}{2}) -\left(k-\frac{1}{2}\right)^2 v_r(k-\frac{1}{2},m+\frac{1}{2}) \right] + \nonumber\\ &2\mu(k)\frac{1}{\Delta r} \frac{1}{2}\left[\frac{v_r(k+\frac{1}{2},m+\frac{1}{2})}{k+\frac{1}{2}} + \frac{v_r(k-\frac{1}{2},m+\frac{1}{2})}{k-\frac{1}{2}} \right] - \nonumber \\ & \left[\lambda(k)+\frac{2}{3}\mu(k)\right] \beta(T(k,m+\frac{1}{2})) \frac{T(k,m+1)-T(k,m)}{\Delta t}\end{aligned}$$ \[eq:EqPAsolidFDTD\] where we used $\left(\frac{\partial}{\partial r}+\frac{2}{r}\right)v_r=\frac{1}{r^2}\frac{\partial r^2 v_r}{\partial r}$. At the central point, Eqs. \[eq:EqPAsolidFDTD\]b and \[eq:EqPAsolidFDTD\]c cannot be used because of the singularity in $1/r$. At $r=0$, we used $\frac{v_r}{r}\sim r \frac{\partial v_r}{\partial r}$ and $v_r(0)=0$ to obtain the following form $$\begin{aligned} \frac{\partial \sigma_{rr}}{\partial t}(0,t)&= (3\lambda(0)+2\mu(0))\frac{\partial v_r}{\partial r}(0,t) - \left( \lambda(0)+\frac{2}{3}\mu(0)\right)\beta(T) \frac{\partial T}{\partial t}(0,t) \\ \frac{\partial \sigma_{\theta,\theta}}{\partial t}(0,t)&= \frac{\partial \sigma_{rr}}{\partial t}(0,t)\end{aligned}$$ ![\[fig:FDTD\] Spatio-temporal mesh used in FDTD simulations. The stress field $\sigma_{rr}$ (blue points) and the radial velocity displacement (red arrows) $v$ are discretized over staggered grids, both in time and space [@virieux1986]. The grids are staggered such that pressure variables are defined on the central point $r=0$. Values of $\sigma_{\theta\theta}$ and $T$ are defined at the same positions as $\sigma_{rr}$.](FDTD.eps){width="9"} By using $\frac{\partial v_r}{\partial r}(0,t)\sim\frac{v_r(\frac{1}{2}\Delta r)}{\Delta r/2}$, the following discretized equations are finally obtained for the central point: $$\begin{aligned} \frac{\sigma_{rr}(0,m+1)-\sigma_{rr}(0,m)}{\Delta t}=& \left[3\lambda(0)+2\mu(0)\right]\frac{1}{\Delta r/2}v_r(\frac{1}{2},m+\frac{1}{2}) - \nonumber \\ &\left[ \lambda(0)+\frac{2}{3}\mu(0)\right]\beta(T(0,m+1/2)) \frac{T(0,m+1)-T(0,m)}{\Delta t} \\ \sigma_{\theta\theta}(0,m)=\sigma_{rr}(0,m)&\end{aligned}$$ The temporal step was to the spatial step by the following stability condition: $$\Delta t=0.99\times\frac{\Delta r}{\sqrt{3}\; c_{gold}}$$ with $c_{gold} = 3240\ \mathrm{m/s}$ the highest speed of sound involved in the problem.

ArXiv

--- abstract: | #### Morphology and defects: {#morphology-and-defects .unnumbered} Issues of Ge hut cluster array formation and growth at low temperatures on the Ge/Si(001) wetting layer are discussed on the basis of explorations performed by high resolution STM and [*in-situ*]{} RHEED. Dynamics of the RHEED patterns in the process of Ge hut array formation is investigated at low and high temperatures of Ge deposition. Different dynamics of RHEED patterns during the deposition of Ge atoms in different growth modes is observed, which reflects the difference in adatom mobility and their ‘condensation’ fluxes from Ge 2D gas on the surface for different modes, which in turn control the nucleation rates and densities of Ge clusters. Data of HRTEM studies of multilayer Ge/Si heterostructures are presented with the focus on low-temperature formation of perfect films. #### Photo-emf spectroscopy: {#photo-emf-spectroscopy .unnumbered} Heteroepitaxial Si [*p–i–n*]{}-diodes with multilayer stacks of Ge/Si(001) quantum dot dense arrays built in intrinsic domains have been investigated and found to exhibit the photo-emf in a wide spectral range from 0.8 to 5$\mu$m. An effect of wide-band irradiation by infrared light on the photo-emf spectra has been observed. Photo-emf in different spectral ranges has been found to be differently affected by the wide-band irradiation. A significant increase in photo-emf is observed in the fundamental absorption range under the wide-band irradiation. The observed phenomena are explained in terms of positive and neutral charge states of the quantum dot layers and the Coulomb potential of the quantum dot ensemble. A new design of quantum dot infrared photodetectors is proposed. #### Terahertz spectroscopy: {#terahertz-spectroscopy .unnumbered} By using a coherent source spectrometer, first measurements of terahertz dynamical conductivity (absorptivity) spectra of Ge/Si(001) heterostructures were performed at frequencies ranged from 0.3 to 1.2 THz in the temperature interval from 300 to 5K. The effective dynamical conductivity of the heterostructures with Ge quantum dots has been discovered to be significantly higher than that of the structure with the same amount of bulk germanium (not organized in an array of quantum dots). The excess conductivity is not observed in the structures with the Ge coverage less than 8Å. When a Ge/Si(001) sample is cooled down the conductivity of the heterostructure decreases. address: | (1) A M Prokhorov General Physics Institute of RAS, 38 Vavilov Street, Moscow 119991, Russia\ (2)Technopark of GPI RAS, 38 Vavilov Street, Moscow, 119991, Russia\ (3)Moscow Institute of Physics and Technology, Institutsky Per. 9, Dolgoprudny, Moscow Region, 141700, Russia author: - 'Vladimir A Yuryev$^{1,2}$' - 'Larisa V Arapkina$^{1}$' - 'Mikhail S Storozhevykh$^{1}$' - 'Valery A Chapnin$^{1}$' - 'Kirill V Chizh$^{1}$' - 'Oleg V Uvarov$^{1}$' - 'Victor P Kalinushkin$^{1,2}$' - 'Elena S Zhukova$^{1,3}$' - 'Anatoly S Prokhorov$^{1,3}$' - 'Igor E Spektor$^{1}~$ and Boris P Gorshunov$^{1,3}$' bibliography: - 'EMNC2012-article.bib' title: | Ge/Si(001) heterostructures with dense arrays of Ge\ quantum dots: morphology, defects, photo-emf spectra\ and terahertz conductivity --- \[1995/12/01\] Introduction {#introduction .unnumbered} ============ Artificial low-dimensional nano-sized objects, like quantum dots, quantum wires and quantum wells, as well as structures based on them, are promising systems for improvement of existing devices and for development of principally new devices for opto-, micro- and nano-electronics. Besides, the investigation of physical properties of such structures is also of fundamental importance. In both regards, amazing perspectives are provided when playing around with quantum dots that can be considered as artificial atoms with a controlled number of charge carriers that have a discrete energy spectrum [@Pchel_Review-TSF; @Pchel_Review]. Arrays of a *large* number of quantum dots including multilayer heterostructures make it possible to create artificial “solids" whose properties can be controllably changed by varying the characteristics of constituent elements (“atoms") and/or the environment (semiconductor matrix). The rich set of exciting physical properties in this kind of systems originates from single-particle and collective interactions that depend on the number and mobility of carriers in quantum dots, Coulomb interaction between the carriers inside a quantum dot and in neighbouring quantum dots, charge coupling between neighbouring quantum dots, polaron and exciton effects, etc. Since characteristic energy scales of these interactions (distance between energy levels, Coulomb interaction between charges in quantum dots, one- and multiparticle exciton and polaron effects, plasmon excitations, etc.) are of order of several meV [@3-Colomb_interactions-Dvur; @4-Drexler-InGaAs; @5-Lipparini-far_infrared], an appropriate experimental tool for their study is provided by optical spectroscopy in the far-infrared and terahertz bands. To get access to the effects, one has to extend the operation range of the spectrometers to the corresponding frequency domain that is to the terahertz frequency band. Because of inaccessibility of this band, and especially of its lowest frequency part, below 1 THz (that is $\apprle 33$cm$^{-1}$), for standard infrared Fourier-transform spectrometers, correspondent data is presently missing in the literature. In this paper, we present the results of the first detailed measurements of the absolute dynamical (AC) conductivity of multilayer Ge/Si heterostructures with Ge quantum dots, at terahertz and sub-terahertz frequencies and in the temperature range from 5 to 300K. In addition, for at least two tens of years, multilayer Ge/Si heterostructures with quantum dots have been candidates to the role of photosensitive elements of monolithic IR arrays promising to replace and excel platinum silicide in this important brunch of the sensor technology [@Wang-properties; @Wang-Cha; @Dvur-IR-20mcm]. Unfortunately, to date achievements in this field have been less than modest. We believe that this state of affairs may be improved by rigorous investigation of formation, defects and other aspects of materials science of such structures, especially those which may affect device performance and reliability, focusing on identification of reasons of low quantum efficiency and detectivity, high dark current and tend to degrade with time as well as on search of ways to overcome these deficiences. New approaches to device architecture and design as well as to principles of functioning are also desirable. This article reports our latest data on morphology and defects of Ge/Si heterostructures. On the basis of our recent results on the photo-emf in the Si [*p–i–n*]{}-structures with Ge quantum dots, which are also reported in this article, we propose a new design of photovoltaic quantum dot infrared photodetectors. Methods {#methods .unnumbered} ======= Equipment and techniques {#equipment-and-techniques .unnumbered} ------------------------ The Ge/Si samples were grown and characterized using an integrated ultrahigh vacuum instrument [@classification; @stm-rheed-EMRS; @CMOS-compatible-EMRS; @VCIAN2011] built on the basis of the Riber SSC2 surface science center with the EVA32 molecular-beam epitaxy (MBE) chamber equipped with the RH20 reflection high-energy electron diffraction (RHEED) tool (Staib Instruments) and connected through a transfer line to the GPI-300 ultrahigh vacuum scanning tunnelling microscope (STM) [@gpi300; @STM_GPI-Proc; @STM_calibration]. Sources with the electron beam evaporation were used for Ge or Si deposition. A Knudsen effusion cells was utilized if boron doping was applied for QDIP [*p–i–n*]{}-structure formation. The pressure of about $5\times 10^{-9}$Torr was kept in the preliminary sample cleaning (annealing) chamber. The MBE chamber was evacuated down to about $10^{-11}$Torr before processes; the pressure increased to nearly $2\times 10^{-9}$Torr at most during the Si substrate cleaning and $10^{-9}$Torr during Ge or Si deposition. The residual gas pressure did not exceed $10^{-10}$Torr in the STM chamber. Additional details of the experimental instruments and process control can be found in Ref.[@VCIAN2011]. RHEED measurements were carried out [*in situ*]{}, i.e., directly in the MBE chamber during a process [@stm-rheed-EMRS]. STM images were obtained in the constant tunnelling current mode at the room temperature. The STM tip was zero-biased while the sample was positively or negatively biased when scanned in empty- or filled-states imaging mode. Structural properties of the Ge/Si films were explored by using the Carl Zeiss Libra-200 FE HR HRTEM. The images were processed using the WSxM software [@WSxM]. For obtaining spectra of photo-electromotive force (photo-emf) a setup enabling sample illumination by two independent beams was used; one of the beams was a wide-band infrared (IR) radiation, generated by a tungsten bulb, passed through a filter of Si or Ge (bias lighting) and the other was a beam-chopper modulated narrow-band radiation cut from globar emission by an IR monochromator tunable in the range from 0.8 to 20 $\mu$m. The spectra were taken at the chopping frequency of 12.5 Hz at temperatures ranged from 300 to 70 K and a widely varied power of the bias lighting. The measurements of the terahertz dynamic conductivity and absorptivity of Ge/Si heterostructures at room and cryogenic temperatures (down to 5 K) have been performed using the spectrometer based on backward-wave oscillators (BWO) as radiation sources. This advanced experimental technique will be described in detail below in a separate section. Sample preparation procedures {#sample-preparation-procedures .unnumbered} ----------------------------- ### Preparation of samples for STM and RHEED {#preparation-of-samples-for-stm-and-rheed .unnumbered} Initial samples for STM and RHEED studies were $8\times 8$ mm$^{2}$ squares cut from the specially treated commercial boron-doped Czochralski-grown (CZ) Si$(100)$ wafers ($p$-type, $\rho\,= 12~{\Omega}\,$cm). After washing and chemical treatment following the standard procedure described elsewhere [@cleaning_handbook], which included washing in ethanol, etching in the mixture of HNO$_3$ and HF and rinsing in the deionized water [@VCIAN2011], the silicon substrates were loaded into the airlock and transferred into the preliminary annealing chamber where they were outgassed at the temperature of around [565]{} for more than 6h. After that, the substrates were moved for final treatment and Ge deposition into the MBE chamber where they were subjected to two-stages annealing during heating with stoppages at [600]{} for 5min and at [800]{} for 3min [@classification; @stm-rheed-EMRS]. The final annealing at the temperature greater than [900]{} was carried out for nearly 2.5min with the maximum temperature of about [925]{} (1.5min). Then, the temperature was rapidly lowered to about [750]{}. The rate of the further cooling was around 0.4/s that corresponded to the ‘quenching’ mode applied in [@stm-rheed-EMRS]. The surfaces of the silicon substrates were completely purified of the oxide film as a result of this treatment [@our_Si(001)_en; @phase_transition; @stm-rheed-EMRS]. Ge was deposited directly on the deoxidized Si(001) surface. The deposition rate was varied from about $0.1$ to $0.15$/s; the effective Ge film thickness $(h_{\rm Ge})$ was varied from 3 to 18 for different samples. The substrate temperature during Ge deposition $(T_{\rm gr})$ was [360]{} for the low-temperature mode and 600 or [650]{} for the high-temperature mode. The rate of the sample cooling down to the room temperature was approximately 0.4/s after the deposition. ### Preparation of multilayer structures {#preparation-of-multilayer-structures .unnumbered} Ge/Si heterostructures with buried Ge layers were grown on CZ $p$-Si$(100)$:B wafers ($\rho\,= 12~{\Omega}\,$cm) washed and outgassed as described above. Deoxidized Si(001) surfaces were prepared by a process allowed us to obtain clean substrate surfaces (this was verified by STM and RHEED) and perfect epitaxial interfaces with Si buffer layers (verified by HRTEM): the wafers were annealed at 800 under Si flux of $\apprle 0.1$Å/s until a total amount of the deposited Si, expressed in the units of the Si film thickness indicated by the film thickness monitor, reached 30[Å]{}; 2-minute stoppages of Si deposition were made first twice after every 5 and then twice after every 10. Afterwards, a $\sim$100nm thick Si buffer was deposited on the prepared surface at the temperature of $\sim$650. Then, a single Ge layer or a multilayer Ge/Si structure was grown. A number of Ge layers in multilayer structures reached 15 but usually was 5; their effective thickness ($h_{\rm Ge}$), permanent for each sample, was varied from sample to sample in the range from 4 to 18Å; the thickness of the Si spacers ($h_{\rm Si}$) was $\sim$50nm. The Ge deposition temperature was $\sim$360, Si spacers were grown at $\sim$530. A heterostructure formed in such a way was capped by a $\sim$100nm thick Si layer grown at $\sim$530. All layers were undoped. The samples were quenched after the growth at the rate of $\sim$0.4/s. ### [Growth of *p–i–n*]{}-structures {#growth-of-pin-structures .unnumbered} [*p–i–n*]{}-structures were grown on commercial phosphorus-doped CZ $n$-Si(100) substrates ($\rho = 0.1\,\Omega$cm). Si surfaces were prepared for structure deposition in the same way as for the growth of multilayer structures. $i$-Si buffer domains of various thicknesses were grown on the clean surfaces at $\sim$650. Then, a stacked structure of several periods of quantum dot (QD) dense arrays separated by Si barriers was grown under the same conditions as the multilayer structures; $h_{\rm Si}$ was widely varied in different structures reaching 50nm; $h_{\rm Ge}$ always was 10Å. A sufficiently thick undoped Si layer separated the stacked QD array from the Si:B cap doped during the growth, the both layers were grown at $\sim$530. Figure \[fig:p-i-n\_Schematics\] demonstrates two such structures (referred to as R163 and R166) which are in the focus of this article. Their caps were doped to $5\times 10^{18}$ and $10^{19}$cm$^{-3}$ in the R163 and R166 samples, respectively. Buffer layer and barrier thicknesses were 99 and 8nm in the R163 structure and 1690 and 30nm in R166. Mesas were formed on samples for photoelectric measurements. Ohmic contacts were formed by thermal deposition of aluminum. Terahertz BWO-spectroscopy {#terahertz-bwo-spectroscopy .unnumbered} -------------------------- The BWO-spectrometers provide broad-band operation (frequencies $\nu$ ranging from 30 GHz to 2 THz), high frequency resolution ($\Delta \nu/\nu = 10^{-5}$), broad dynamic range (40–50 dB), continuous frequency tuning and, very importantly, the possibility of *direct* determination of spectra of any “optical” parameter, like complex conductivity, complex dielectric permittivity, etc. (‘direct’ means that no Kramers–Kronig analysis—typical for far-infrared Fourier transform spectroscopy—is needed). The principle of operation of BWO-spectrometers is described in details in the literature (see, e.g., [@6-Kozlov-Volkov; @7-Gorshunov-BWO_spectroscpoy]). It is based on measurement of the complex transmission coefficient $Tr^{*} = Tr\exp(i\varphi)$ of a plane-parallel sample with subsequent calculation of the spectra of its optical parameters from those of the transmission coefficient amplitude $Tr(\nu)$ and the phase $\varphi(\nu)$. The corresponding expression can be written as [@8-Born-Wolf; @9-Dressel] $$Tr^{*}=Tr\exp(i\varphi) = \frac{T_{12}T_{21}\exp(i\delta)}{1+ T_{12}T_{21}\exp(2i\delta)}.\label{eqn:THz-Eq1}$$ Here $$\begin{aligned} \nonumber T_{pq}= t_{pq}\exp(i\varphi_{pq}), t^{2}_{pq}=\frac{4(n_{p}^2+k_{p}^2)}{(k_p + k_q)^2+(n_p + n_q)^2}, \varphi_{pq}= \arctan\{\frac{k_pn_p-k_qn_q}{n_p^2+k_p^2+n_pn_q+k_pk_q}\}\end{aligned}$$ are Fresnel coefficients for the interfaces ‘air–sample’, indices $p,~ q = 1,~ 2 $ correspond: ‘1’ to air (refractive index $n_1 = 1$, extinction coefficient $k_1 = 0$) and ‘2’ to the material of the sample $(n_2,~k_2)$, $\delta = \frac{2{\pi}d}{\lambda}(n_2+ik_2)$, $d$ is the sample thickness, $\lambda$ is the radiation wavelength. The sample parameters (for instance, $n_2$ and $k_2$ ) are found for each fixed frequency by solving two coupled equations for the two unknowns, $Tr(n_2, k_2, \nu) = Tr_{\mathrm{exp}}(\nu)$ and $\varphi(n_2, k_2, \nu) = \varphi_{\mathrm{exp}}(\nu)$ \[here $Tr_{\mathrm{exp}}(\nu)$ and $\varphi_{\mathrm{exp}}(\nu)$ are the measured quantities\]. The so-found values of $n_2(\nu)$ and $k_2(\nu)$ can then be used to derive the spectra of the complex permittivity $\varepsilon^*(\nu) = \varepsilon'(\nu) + i \varepsilon''(\nu) = n_2^2 - k^2_2 + 2 i n_2 k_2$, complex conductivity $\sigma^*(\nu) = \sigma_1(\nu) + i \sigma_2(\nu) = \nu n_2 k_2 + i \nu (\varepsilon_{\infty} - \varepsilon')/2$, etc. ($\varepsilon_{\infty}$ is the high-frequency contribution to the permittivity). If the sample is characterized by low enough absorption coefficient, Fabry–Perot-like interference of the radiation within the plane-parallel layer leads to an interference maxima and minima in the transmission coefficient spectra. In this case there is no need to measure the phase shift spectra since the pairs of optical quantities of the sample can be calculated from the transmission coefficient spectrum alone: the absorptive part (like $\varepsilon''$ or $\sigma_1$) is determined from the interferometric maxima amplitudes and the refractive part (like $\varepsilon'$ or $n$) is calculated from their positions [@6-Kozlov-Volkov; @7-Gorshunov-BWO_spectroscpoy]. When measuring the dielectric response of the films (like heterostructures in the present case) on dielectric substrates, first the dielectric properties of the substrate material are determined by standard techniques just described. Next, one measures the spectra of the transmission coefficient and of the phase shift of the film-substrate system, and it is these spectra that are used to derive the dielectric response of the film by solving two coupled equations for two unknowns—“optical” parameters of the film. The corresponding expression for the complex transmission coefficient of a two-layer system can be written as [@8-Born-Wolf; @9-Dressel]: $$Tr^*_{1234}= Tr\exp(i\varphi)= \frac{T_{12}T_{23}T_{34}\exp\{i (\delta_2+\delta_3)\}}{1+T_{23}T_{34}\exp(2i\delta_3)+T_{12}T_{23}\exp(2i\delta_2)+ T_{12}T_{34}\exp\{2i(\delta_2+\delta_3)\}}, \label{eqn:THz-Eq2}$$ where indices 1 and 4 refer to the media on the two sides of the sample, i.e., of the film on substrate, $\delta_p = (n_p + i k_p)$, with $d_p$ being the film and substrate thicknesses ($p = 2,~ 3$). The other notations are the same as in Eq. (\[eqn:THz-Eq1\]). The measurements are performed in a quasioptical configuration, no waveguides are used [@6-Kozlov-Volkov; @7-Gorshunov-BWO_spectroscpoy] and this makes measurement schemes extremely flexible. All measurement and analysis procedures are PC-controlled. Most important parameters of the BWO-spectrometer are summarized in Table \[tab:BWO\_parameters\]. Results and Discussion {#results-and-discussion .unnumbered} ====================== Morphology and defects {#morphology-and-defects-1 .unnumbered} ---------------------- ### STM and RHEED study of Ge/Si(001) QD arrays: morphology and formation {#stm-and-rheed-study-of-gesi001-qd-arrays-morphology-and-formation .unnumbered} Previously, we have shown in a number of [*STM studies*]{} [@classification; @VCIAN2011; @Nucleation_high-temperatures; @Hut_nucleation; @initial_phase; @CMOS-compatible-EMRS] that the process of the hut array nucleation and growth at low temperatures starts from occurrence of two types of 16-dimer nuclei [@Hut_nucleation] on wetting layer (WL) patches of 4-ML height [@initial_phase] giving rise to two known species of $\{105\}$-faceted clusters—pyramids and wedges [@classification]—which then, growing in height (both types) and in length (wedges), gradually occupy the whole wetting layer, coalless and start to form a nanocrystalline Ge film (Figure \[fig:STM-360\]) [@VCIAN2011; @CMOS-compatible-EMRS]. This is a life cycle of hut arrays at the temperatures &lt;600. We refer to cluster growth at these temperatures as the low-temperature mode. At high temperatures (&gt;600), only pyramids represent a family of huts: they were found to nucleate on the WL patches in the same process of 16-dimer structure occurrence as at low temperatures [@Nucleation_high-temperatures]. We failed to find wedges or their nuclei if Ge was deposited at these temperatures and this fact waits for a theoretical explanation. In addition to pyramids, shapeless Ge heaps faceting during annealing have been observed on WL in the vicinity of pits and interpreted as possible precursors of large faceted clusters [@VCIAN2011; @Nucleation_high-temperatures]. Note that a mechanism of Ge hut formation via faceting of some shapeless structures appearing near WL irregularities, which resembles the process described in the current article, was previously considered as the only way of Ge cluster nucleation on Si(001) [@Nucleation; @Goldfarb_2005]. Now we realize that huts nucleate in a different way[@Hut_nucleation] and formation of the faceting heaps at high temperatures is a process competing with appearance of real pyramidal huts which arise due to formation of the 16-dimer nuclei on tops of WL patches [@Hut_nucleation; @CMOS-compatible-EMRS; @initial_phase]. Yet, further evolution of the Ge heaps into finalized faceted clusters, such as domes, in course of Ge deposition is not excluded [@Nucleation_high-temperatures]. During further growth at high temperatures, pyramids reach large sizes becoming much greater than their low-temperature counterparts and usually form incomplete facets or split edges (Figure \[fig:STM-600\]). An incomplete facet seen in Figure \[fig:STM-600\]a and especially a “pelerine” of multiple incomplete facets seen in Figure \[fig:STM-600\]b,c around the pyramid top indicate unambiguously that this kind of clusters grow from tops to bottoms completing facets rather uniformly from apexes to bases, and bottom corners of facets are filled the latest. Sometimes it results in edge splitting near the pyramid base (Figure \[fig:STM-600\]b,d). [*RHEED*]{} has allowed as to carry our [*in-situ*]{} explorations of forming cluster arrays. We have compared RHEED patterns of Ge/Si(001) surfaces during Ge deposition at different temperatures and a dynamics of diffraction patterns during sample heating and cooling. Diffraction patterns of reflected high-energy electrons for samples of thin ($h_{\rm Ge}=$ 4Å) Ge/Si(001) films deposited at high (650 or 600) and low (360) temperatures with equal effective thicknesses are presented in Figure \[fig:rheed\]a,b. The patterns are similar and represent a typical $(2\times 1)$ structure of Ge WL; reflexes associated with appearance of huts (the 3D-reflexes) are absent in both images, that agrees with the data of the STM analysis. Diffraction patterns presented in Figure \[fig:rheed\]a,c,e are related to the samples with $h_{\rm Ge}$ increasing from 4 to 6. The 3D-reflexes are observed only in the pattern of the samples with $h_{\rm Ge}=$ 6[Å]{}, that is also in good agreement with the STM data [@VCIAN2011; @initial_phase]. Influence of the sample annealing at the deposition temperature is illustrated by a complimentary pair of the RHEED patterns given in Figure \[fig:rheed\]c,d. Annealing of specimens at the temperature of growth (650) resulted in appearance of the 3D-reflexes (Figure \[fig:rheed\]d) that also corresponds with the results of our STM studies [@VCIAN2011]. Difference in evolution of diffraction patterns during the deposition of Ge is a characteristic feature of the high-temperature mode of growth in comparison with the low-temperature one. The initial Si(001) surface before Ge deposition is $(2\times 1)$ reconstructed. At high temperatures, as $h_{\rm Ge}$ increases, diffraction patterns evolve as $(2\times 1)\rightarrow$ $(1\times 1)\rightarrow$ $(2\times 1)$ with very weak -reflexes. Brightness of the -reflexes gradually increases (the $(2\times 1)$ structure becomes pronounced) and the 3D-reflexes arise only during sample cooling (Figure \[fig:RHEED\_cool-600\]). At low temperatures, the RHEED patterns change as $(2\times 1)\rightarrow$ $(1\times 1)\rightarrow$ $(2\times 1)\rightarrow$ $(2\times 1)+3$D-reflexes. The resultant pattern does not change during sample cooling. This observation reflects the process of Ge cluster “condensation” from the 2D gas of mobile Ge adatoms. High Ge mobility and low cluster nucleation rate in comparison with fluxes to competitive sinks of adatoms determines the observed difference in the surface structure formation at high temperatures as compared with that at low temperatures [@VCIAN2011; @Nucleation_high-temperatures] when the adatom flux to nucleating and growing clusters predominates and adatom (addimer) mobility is relatively small. ### STM and HRTEM study of Ge/Si heterostructures with QD array: morphology and defects {#stm-and-hrtem-study-of-gesi-heterostructures-with-qd-array-morphology-and-defects .unnumbered} Structures overgrown with Si were examined by means of HRTEM for structural perfection or possible defects, e.g., imperfections induced by array defects reported in Ref. [@defects_ICDS-25]. Data of HRTEM studies evidence that extended defects do not arise at low $h_{\rm Ge}$ on the buried Ge clusters and perfect epitaxial heterostructures with quantum dots form under these conditions that enables the formation of defectless multilayer structures suitable for device applications. Figure \[fig:TEM-6A\] relates to the five-layer Ge/Si structure with [*h$_{\mathrm{Ge}}$*]{}= 6. We succeeded to resolve separate Ge clusters whose height is, according to our STM data [@VCIAN2011; @classification], $\apprle$3ML over WL patches (Figure \[fig:TEM-6A\]a,b). A lattice structure next to the cluster apex is not disturbed (Figure \[fig:TEM-6A\]c,d); its parameters estimated from the Fourier transform of an image taken from this domain (Figure \[fig:TEM-6A\]e,f), $\sim 5.4$ along the \[001\] direction and $\sim 3.8$ along \[110\], within the accuracy of measurements coincide with the parameters of the undisturbed Si lattice. Stacking faults (SF) have been found to arise above Ge clusters at $h_{\rm Ge}$ as large as 10 (Figure \[fig:TEM-10A1L\]). SFs often damage Si structures with overgrown Ge layers at this values of $h_{\rm Ge}$. A high perfection structure is observed around Ge clusters in Figure \[fig:TEM-10A1L\]a although their height is up to 1.5 nm over WL (the typical height of huts is known from both our STM and HRTEM data). Yet, a tensile strained domain containing such extended defects as SFs and twin boundaries forms over a cluster shown in Figure \[fig:TEM-10A1L\]b,c (twinning is clearly observable in Figure \[fig:TEM-10A1L\]d). One can see, however, that this cluster is extraordinary high: its height over WL exceeds 3.5 nm. Such huge clusters have been described by us previously as defects of arrays [@defects_ICDS-25]; we predicted in that article such formations to be able to destroy Ge/Si structures generating high stress fields in Si spacer layers and, as a consequence, introducing extended defects in device structures. As seen in Figure \[fig:TEM-10A1L\]b,c, the stress field spreads under the cluster in the Si buffer layer grown at much higher temperature than the cap. Unfortunately, the huge Ge hut clusters (as we showed in Ref. [@defects_ICDS-25], they are not domes) usually appear in the arrays and their number density was estimated as $\sim$$10^9$cm$^{-3}$ from the STM data. Strain domains are also seen next to Ge clusters in the five-layer structures depicted in Figure \[fig:TEM-9-10A\] ([*h$_{\mathrm{Ge}}$*]{}= 9 or 10). We found that such domains are not inherent to all cluster vicinities but only to some of them (Figure \[fig:TEM-9-10A\]a,d). The disturbed strained domains give a contrast different from that of the undisturbed Si lattice (Figure \[fig:TEM-9-10A\]e). Zoom-in micrographs of the disturbed regions show perfect order of atoms in the crystalline lattice (Figure \[fig:TEM-9-10A\]b,c,e,f) everywhere except for the closest vicinities of the Si/Ge interface where point defects and a visible lattice disordering immediately next to the cluster are registered (Figure \[fig:TEM-9-10A\]b,c,e,f). However, some farther from the interfaces but still near cluster apexes the crystalline order restores (Figure \[fig:TEM-9-10A\]h). We have estimated the lattice parameter in the disturbed regions from the Fourier transforms of the HRTEM micrographs taken in these domains (Figure \[fig:TEM-9-10A\]i). The values we obtained appeared to be vary for different regions. Yet, they usually appreciably exceeded the Si lattice parameter. Moreover, they often reached the Ge parameter of $\sim$5.6–5.7 (along \[001\] and $\sim$4 along \[110\]). This might be explained either by appreciable diffusion of Ge from clusters (previously, we have already reported an appreciable diffusion of Si in Ge clusters in analogous structures from covering Si layers grown at 530 [@Raman_conf; @our_Raman_en]) or by Si lattice stretching under the stress. Likely both factors acts. It is worth while emphasising that the stretched domains usually do not contain extended defects, as it is seen from the HRTEM micrographs, except for the cases of array defects (huge clusters) like that demonstrated in Figure \[fig:TEM-10A1L\]. We suppose that the extended defects in these regions arise because the strain exceeds an elastic limit near huge clusters. Finally, we have tried to find out if huge clusters exists in arrays of [*h$_{\mathrm{Ge}}$*]{}= 9 (Figure \[fig:STM-9A\]). We have been convinced that even in rather uniform arrays large clusters (Figure \[fig:STM-9A\]e), which might generate considerable stress, are abundant and even huge ones (Figure \[fig:STM-9A\]d), which should produce lattice disordering (extended defects), are available. Effect of such defects as huge clusters on device performance and a cause of their appearance in hut arrays await further detailed studies. Photo-emf of Ge/Si [*p–i–n*]{}-structures {#photo-emf-of-gesi-pin-structures .unnumbered} ----------------------------------------- ### Photo-emf spectra {#photo-emf-spectra .unnumbered} We have investigated heteroepitaxial Si [*p–i–n*]{}-diodes with multilayer stacks of Ge/Si(001) QD dense arrays built in intrinsic domains and found them to exhibit the photo-emf in a wide spectral range from 0.8 to 5 $\mu$m [@NES-2011; @photon-2011]. An effect of wide-band irradiation by infrared light on the photo-emf spectra has been observed. Here we describe the most representative data obtained for two radically different structures denoted as R163 and R166 (Figure \[fig:p-i-n\_Schematics\]). Typical photo-emf spectra obtained for R163 and R166 structures are presented in Figure \[fig:r163\_r166\]. In the spectra, we mark out three characteristic ranges which differently respond to bias lighting and differently depend on its power. (i) [*Wavelength range from 0.8 to 1.0 $\mu$m*.]{} The photo-emf response increases with the increase in the bias lighting power, reaches maximum at $P \approx 0.63$ mW/cm$^2$ with a Si filter and at $P \approx 2.6$ mW/cm$^2$ with a Ge filter and decreases with further increase in the power. \(ii) [*Wavelength range from 1.1 to 2.6 $\mu$m*.]{} The photo-emf response decreases monotonously in this range with the increase in the power of the bias lighting with any, Si or Ge, filter. \(iii) [*Wavelength range &gt;2.6 $\mu$m*.]{} The photo-emf response increases with the increase in the bias lighting power and comes through its maximum at $P \approx 0.63$ mW/cm$^2$ if a Si filter is used and at $P \approx 0.25$ mW/cm$^2$ for a Ge filter. The response decreases with further growth of the bias lighting power for a Si filter and remains unchanged when Ge filter is utilized. We propose the following model for explanation of these observations: In the studied structures, all QD layers are located in the $i$-domain (Figure \[fig:bands\]). One can see from these sketches that some QD layers are positively charged (the ground states of QDs is above the Fermi level and hence they are filled by holes) while others are neutral (the QDs’ ground states are below Fermi level and hence empty). Then, one may consider a QD layer as a single ensemble of interacting centers because the average distance between QDs’ apexes is about 13 nm whereas QDs’ bases adjoin. Consequently, one can imagine an allowed energy band with some bandwidth, determined by QDs’ sizes and composition dispersion, and a certain density of states in this band. Let us explore in detail every range of the photo-emf spectra taking into account the proposed model. ### Wavelength range from 0.8 to 1.0$\mu$m {#wavelength-range-from-0.8-to-1.0mum .unnumbered} Without bias lighting, all radiation in the Si fundamental absorption range can be believed to be absorbed in Si (cap-layer, spacers, buffer layer and substrate) and QDs are not involved in the absorption, so the total charge of QD layers remains unaltered. Electron-holes pairs are generated in the intrinsic region of the [*p–i–n*]{}-diode as a result of the absorption and separated by the junction field which converts the radiation to emf. However, carrier separation is hindered because of presence of the potential barriers for holes in the valence band which are produced by the charged QD layers situated in intrinsic domain. Calculated height of these barriers equals 0.1 to 0.2 eV depending on the layer position in the structure. Transitions from QD ensemble states to the valence and conduction bands of Si start under bias lighting. Carriers excited by bias lighting do not contribute to the photo-emf signal measured at the modulation frequency of the narrow-band radiation. QDs captured a photon change their charge state. An effective layer charge decreases as a result of the absorption of the bias lighting radiation that results in reduction of potential barrier height and more efficient carrier separation in the junction field. Increase in the photo-emf response in the fundamental absorption range under bias lighting is explained by this process. ### Wavelength range from 1.1 to 2.6$\mu$m {#wavelength-range-from-1.1-to-2.6mum .unnumbered} This band is entirely below the Si fundamental absorption range. Therefore the response in this region cannot be explained in terms of absorption in bulk Si. One can explain the presence of the photo-emf signal in this region considering the following model: Both hole transitions from the QD ensemble states to the valence band and electron transitions from the QD ensemble states to the conduction band due to absorption of photons with the energy between $\sim 1.12$ and $\sim 0.4$ eV are possible. The probability of every kind of the transitions is determined by the photon energy, the density of states in the QD ensemble and by effective charge of the QD layer. It follows from theoretical studies [@Gerasimenko_Si-mat_nanoelectr; @Brudnyi-Ge-small_QD] and experiments on photoluminescence [@PL-Si/Ge_1.4-1.8mcm; @PL-Si/Ge] that photons with energies ranged from 0.7 to 0.9 eV are required for electron transitions from the QD states to the conduction band. However, it is necessary to mention the research of photoconductivity [@Talochkin-Lateral_photoconductivity_Ge/Si], in which electron transitions for low photon energy ($\sim 0.4$ eV) have been shown to be likely. The availability of these transitions is explained by dispersion of sizes and composition of QDs, effect of diffusion on the hetero-interface and deformation effects. The likelihood of electron transitions drops rapidly with photons energy decrease because of reduction of the density of states in the QDs ensemble when approaching to the conduction band edge. This is the reason of the observed monotonous decrease in the photo-emf signal with the increase in the radiation wavelength in this range. At the same time, bias lighting switching on leads to growth of concentration of the unmodulated (“dark”) carrier, depletion of QDs and as a consequence to the observed reduction of the photo-emf response at the chopping frequency. ### Wavelength range &gt;2.6$\mu$m {#wavelength-range-2.6mum .unnumbered} As mentioned above, electron transitions can happen at low energy of the exciting radiation ($\sim$0.4 eV) which correspond to wavelength of $\sim$3.1 $\mu$m. Yet, the photo-emf signal is observed at the radiation wavelengths up to 5 $\mu$m in our measurements. The presence of the photo-emf response in this range can only be explained if the QD layer is considered as a single ensemble of mutually interacting centers. An effective positive charge in the QD layer forms a potential well for electrons in the conduction band. This leads to reduction of energy needed for electron transitions from the QD ensemble states to the conduction band. Partial emptying of the states makes electron transitions possible and, at the same time, does not lead to significant change in the potential wells depth. As a result, electron transitions can happen at the exciting radiation energies as low as 0.25 eV. Hole transitions also can happen at these energies via a large number of excited states in the QD ensemble. It may be concluded that the likelihood the electron transitions decreases faster than that of the hole transitions as the exciting radiation energy decreases in the considered wavelength range. However, first it is necessary to empty the levels by the electron transitions to make possible hole transitions. This could be achieved by using an additional radiation of the spectral domain where the probability of the electron transitions is high. So, bias lighting stimulates the hole transitions by exciting electrons that leads to emptying the levels. In this case the electron concentration is not modulated as distinct from the hole concentration which is modulated at the chopper frequency. This explains the observed low magnitude of the photo-emf in the wavelength range &gt;2.6 $\mu$m and its increase under bias lighting. ### Influence of buffer layer thickness on photo-emf spectra {#influence-of-buffer-layer-thickness-on-photo-emf-spectra .unnumbered} As seen from Figure \[fig:bands\], the buffer layer thickness determines the QD layers position the in intrinsic domain and thus controls the relative position of the Fermi level and the mini-band of the QD array in the region where the QD layers are situated. The charge of the QD layer is determined by the band occupation of the QD ensemble which, in turn, is controlled by the Fermi level location. For this reason the effect of bias lighting on photo-emf generated by the narrow-band radiation in the fundamental absorption range is much stronger for the R166 structure, which have a thick buffer layer, than for the R163 one. This is clearly seen in Figure \[fig:bias\]. The absolute value of photo-emf in the R166 structure is lower than that in the R163 sample due to higher potential barriers for holes in the valence band. Yet, the photo-emf response increases with the growth of the bias lighting power much stronger in the R166 [*p–i–n*]{}-diode than in the R163 one. ### Prospective photovoltaic IR detectors {#prospective-photovoltaic-ir-detectors .unnumbered} On the basis of our results on the photo-emf in the Si [*p–i–n*]{}-structures with Ge quantum dots, we have recently proposed [@Yur1-patent-Ge] a new design of photovoltaic quantum dot infrared photodetectors which enables detection of variations of photo-emf produced by the narrow-band radiation in the Si fundamental absorption range (a reference beam) under the effect of the wide-band IR radiation inducing changes in the Coulomb potential of the quantum dot ensemble which, in turn, affects the efficiency of the photovoltaic conversion of the reference beam. The quantum dot array resembles a grid of a triode in these detectors which is controlled by the detected IR light. The reference narrow-band radiation generates a potential between anode and cathode of this optically driven quantum dot triode; a magnitude of this voltage depends on the charge of the QD grid (Figure \[fig:bands\]). Such detectors can be fabricated on the basis of any appropriate semiconductor structures with potential barriers, e.g., [*p–i–n*]{}-structures, $p$–$n$-junctions or Schottky barriers, and built-in arrays of nanostructures. There are many ways to deliver the reference beam to the detector, e.g., by irradiating the sensor by laser or LED. We propose, however, surface plasmon polaritons delivered to the detector structures by the plasmonic waveguides [@Bozhevolnyi-waveguides; @Zayats-waveguides] to be applied as the reference beams in the detector circuits. This approach makes such detectors, if based on Si, fully compatible with existing CMOS fabrication processes [@Zayats-Si_waveguides] that, in turn, opens a way to development of plasmonic IR detector arrays on the basis of the monolithic silicon technology. THz conductivity of multilayer Ge/Si QD arrays {#thz-conductivity-of-multilayer-gesi-qd-arrays .unnumbered} ---------------------------------------------- The effective dynamic conductivity of Ge quantum dot layer was determined by measuring the transmission coefficient spectra of heterostructures grown on Si(001) substrates. Characteristics of the substrates were determined beforehand as demonstrated by Figures \[fig:THz-fig1\] and \[fig:THz-fig2\]. In Figure \[fig:THz-fig1\], the interferometric pattern in the transmission coefficient spectrum $Tr(\nu)$ of a plane-parallel Si substrate is clearly seen. Pronounced dispersion of $Tr(\nu)$ peaks and their temperature dependence allow to extract the parameters of the charge carriers (holes) by fitting the spectra with Eq. (\[eqn:THz-Eq2\]) and by modelling the sample properties with the Drude conductivity model where the complex AC conductivity is given by an expression [@9-Dressel; @10-Sokolov] $$\sigma^*(\nu) = \sigma_1(\nu) + i\sigma_2(\nu) = \frac{\sigma_0\gamma^2}{\gamma^2+\nu^2} + i\frac{\sigma_0\nu\gamma}{\gamma^2+\nu^2}. \label{eqn:THz-Eq3}$$ Here $\sigma_1$ is the real part and $\sigma_2=\nu(\varepsilon_{\infty} - \varepsilon')/2$ is the imaginary part of the conductivity, $\varepsilon_{\infty}$ is the high-frequency dielectric constant, $\sigma_0=\nu^2_{\mathrm{pl}}/2\gamma$ is the DC conductivity, $\nu^2_{\mathrm{pl}} = (ne^2 /{\pi}m^*)^{\frac{1}{2}}$ is the plasma frequency of the carriers condensate, $n$, $e$ and $m^*$ are, respectively, their concentration, charge and effective mass and $\gamma$ is their scattering rate. Figure \[fig:THz-fig2\] shows the temperature variation of the plasma frequency and the scattering rate of charge carriers. Lowering of the plasma frequency is mainly connected with the carriers’ freezing out and the $\gamma(T)$ behaviour is well described by a $T^{-\frac{3}{2}}$ dependence, as expected. The values of effective dynamical conductivity and absorption coefficient $\alpha = 4{\pi}k/\lambda$ of the heterostructures with Ge quantum dots were determined basing on the measurements of terahertz transmission coefficient spectra of the Si substrate with the heterostructure on it as compared to the spectra of the same substrate with the heterostructure etched away; this allowed us to avoid influence of (even slight) differences in dielectric properties of substrates cut of a standard commercial silicon wafer. By comparing the so-measured transmissivity spectra we reliably detect, although small, changes in the amplitudes of interference maxima of a bare substrate caused by heterostructures. This is demonstrated by Figure \[fig:THz-fig3\]: at $T = 300$K we clearly and firmly register a 2% lowering of the peak transmissivity introduced by the heterostructure. When cooling down, the difference decreases and we were not able to detect it below about 170K, see Figure \[fig:THz-fig4\]. Correspondingly, as is seen in Figure \[fig:THz-fig4\], the AC conductivity of the heterostructure decreases while cooling, along with the conductivity of the Si substrate. The latter observation might be an indication of the fact that the charges are delivered into the quantum dots array from the substrate; the statement, however, needs further exploration. Measuring the room temperature spectra, we have found that the AC conductivity and the absorption coefficient of the heterostructure do not depend on the effective thickness (measured by the quartz sensors during MBE) of the germanium layer ($h_{\mathrm{Ge}}$) for $h_{\mathrm{Ge}}$ ranging from 8 to 14, see Figure \[fig:THz-fig5\]. For larger coverage, $h_{\mathrm{Ge}}>14$Å, both quantities start to decrease. One of the main findings of this work is that the AC conductivity and absorption coefficient of Ge/Si heterostructures have been discovered to be significantly higher than those of the structure with the same amount of germanium not organized in an array of quantum dots. Crucial role played by quantum dots is supported by a decrease of $\sigma_{\rm AC}$ and $\alpha$ observed for large germanium coverage ($h_{\mathrm{Ge}}>14$Å), when structurization into quantum dots gets less pronounced and the thickness of Ge layer becomes more uniform. On the other hand, it is worth noting that no extra absorption of terahertz radiation was detected in the samples with low coverage, $h_{\mathrm{Ge}}=4.4$ and 6Å. This can be explained either by the absence of quantum dots in that thin Ge layer or by their small sizes, by a large fraction of the free wetting layer or by relatively large distances between the clusters as compared to their sizes, i.e., by the absence or smallness of the effect of quantum dots on the dielectric properties of the heterostructure. As seen from Figure \[fig:THz-fig5\], the values $\sigma_{\rm AC}\approx 100\,\Omega^{-1}\mathrm{cm}^{-1}$ and $\alpha\approx 4000\,\mathrm{cm}^{-1}$ are considerably higher than the values measured for bulk germanium, $\sigma_{\rm AC}({\rm Ge})\approx 10^{-2}\,\Omega^{-1}\mathrm{cm}^{-1}$, by about four orders of magnitude, and $\alpha({\rm Ge})\approx 40\,\mathrm{cm}^{-1}$, by about two orders of magnitude. Assuming that the AC conductivity of heterostructure is connected with the response of (quasi) free carriers, one can express it with a standard formula $\sigma=e\mu n =ne^2(2\pi \gamma m^*)^{-1}$ ($\mu$ is the mobility of charge carriers). Then, the observed increase has to be associated with considerable enhancement either of the mobility (suppression of scattering rate) of charge carriers within a quantum dot array or of their concentration. The second possibility has to be disregarded since the total concentration of charges in the sample (substrate plus heterostructure) remains unchanged. As far as the mobility increase is concerned, we are not aware of a mechanism that could lead to its orders of magnitude growth when charges get localized within the quantum dot array. Another interpretation of the observed excess AC conductivity could be based on some kind of [*resonance*]{} absorption of terahertz radiation. Known infrared experiments exhibit resonances in quantum dot arrays that are caused by the transitions between quantized energy levels, as well as between the split levels and the continuum of the valence or conduction band [@4-Drexler-InGaAs; @11-Heitmann; @12-Boucaud; @13-Weber; @14-Savage]. Carriers localized within quantum dots can form bound states with the carriers in the surrounding continuum (excitons) or with optical phonons (polarons), which can in turn interact with each other and form collective complexes [@3-Colomb_interactions-Dvur; @4-Drexler-InGaAs; @12-Boucaud; @13-Weber; @14-Savage; @15-Hameau]. Plasma excitations generated by electromagnetic radiation in the assembly of conducting clusters or quantum dots also have energies of about 10 meV [@16-Sikorski; @18-Dahl; @17-Demel], i.e., fall into the THz band. It is important that these effects can be observed not only at low, but at elevated temperatures as well, up to the room temperature. At this stage, we are not able to unambiguously identify the origin of the THz absorption seen at $T = 170$ to 300K in Ge/Si heterostructure with Ge quantum dots. Among the aforementioned, the mechanisms involving polaritons or plasma excitations seem to be least affected by thermal fluctuations and could be considered as possible candidates. To get detailed insight into microscopic nature of the observed effect, further investigations of heterostructures with various geometric and physical parameters, as well as in a wider frequency and temperature intervals are in progress. Conclusions {#conclusions .unnumbered} =========== In conclusion of the article, we highlight its main provisions. Using high resolution STM and [*in-situ*]{} RHEED we have explored the processes of Ge hut cluster array formation and growth at low temperatures on the Ge/Si(001) wetting layer. Different dynamics of the RHEED patterns in the process of Ge hut array formation at low and high temperatures of Ge deposition reflects the difference in adatom mobility and their fluxes from 2D gas of mobile particles (atoms, dimers and dimer groups) on the surface which govern the nucleation rates and densities of arising Ge clusters. HRTEM studies of multilayer Ge/Si heterostructures with buried arrays of Ge huts have shown that the domains of stretched lattice occurring over Ge clusters in Si layers at high Ge coverages usually do not contain extended defects. We suppose that the extended defects in these regions arise because the strain exceeds an elastic limit near huge clusters. Silicon [*p–i–n*]{}-diodes with multilayer stacks of Ge cluster arrays built in [*i*]{}-domains have been found to exhibit the photo-emf in a wide spectral range from 0.8 to 5$\mu$m. A significant increase in photo-emf response in the fundamental absorption range under the wide-band IR radiation has been reported and explained in terms of positive and neutral charge states of the quantum dot layers and the Coulomb potential of the quantum dot ensemble. A new type of photovoltaic QDIPs is proposed in which photovoltage generated by a reference beam in the fundamental absorption band is controlled by the QD grid charge induced by the detected IR radiation [@Yur1-patent-Ge]. Using a BWO-spectrometer, first measurements of terahertz dynamical conductivity spectra of Ge/Si heterostructures were carried out at frequencies ranged from 0.3 to 1.2 THz in the temperature interval from 5 to 300K. The effective dynamical conductivity of the heterostructures with Ge quantum dots has been found to be significantly higher than that of the structure with the same amount of Ge not organized in quantum dots. The excess conductivity is not observed in the structures with the Ge coverage less than 8Å. When a Ge/Si sample is cooled down the conductivity of the heterostructure decreases. Abbreviations {#abbreviations .unnumbered} ============= AC, alternating current; BWO, backward-wave oscillator; CMOS, complementary metal-oxide semiconductor; CZ, Czochralski or grown by the Czochralski method; DC, direct current; emf, electromotive force; HRTEM, high resolution transmission electron microscope; IR; infrared; LED, light emitting diode; MBE, molecular beam epitaxy; ML, monolayer; QD, quantum dot; QDIP, quantum dot infrared photodetector; RHEED, reflected high energy electron diffraction; SF, stacking fault; SIMS, secondary ion mass spectroscopy; STM, scanning tunneling microscope; WL, wetting layer; UHV, ultra-high vacuum. Competing interests {#competing-interests .unnumbered} =================== The authors declare that they have no competing interests. Authors contributions {#authors-contributions .unnumbered} ===================== VAY conceived of the study and designed it, performed data analysis, and took part in discussions and interpretation of the results; he also supervised and coordinated the research projects. LVA participated in the design of the study, carried out the experiments, performed data analysis, and took part in discussions and interpretation of the results. MSS investigated the photo-emf spectra; he carried out the experiments, performed data analysis, and took part in discussions and interpretation of the results. VAC participated in the design of the study, took part in discussions and interpretation of the results; he also supervised the researches performed by young scientists and students. KVC took pat in the experiments on investigation of the photo-emf spectra and the terahertz conductivity; he prepared experimental samples and took part in discussions and interpretation of the results. OVU performed the HRTEM studies and took part in discussions and interpretation of the results. VPK participated in the design of the study, took part in discussions and interpretation of the results; he also supervised the research project. ESZ carried out the experiments on the terahertz spectroscopy; she performed measurements and data analysis, and took part in discussions and interpretation of the results. ASP participated in the studies by the terahertz spectroscopy; he took part in discussions and interpretation of the results. IES participated in the studies by the terahertz spectroscopy; he took part in discussions and interpretation of the results. BPG performed the explorations by the terahertz spectroscopy; he participated in the design of the study, performed measurements and data analysis, and took part in discussions and interpretation of the results; he also supervised the research project. Acknowledgements {#acknowledgements .unnumbered} ================ Tables {#tables .unnumbered} ====== Table \[tab:BWO\_parameters\] - Main parameters of the terahertz BWO-spectrometer {#tabletabbwo_parameters---main-parameters-of-the-terahertz-bwo-spectrometer .unnumbered} --------------------------------------------------------------------------------- Figures {#figures .unnumbered} =======  Figure \[fig:STM-360\] - STM images of Ge/Si(001) quantum dot arrays grown at 360: {#figurefigstm-360---stm-images-of-gesi001-quantum-dot-arrays-grown-at-360 .unnumbered} ----------------------------------------------------------------------------------- $h_{\rm Ge}$ (Å) is (a) 6, (b) 8, (c) 10, (d) 14, (e) 15, (f) 18.  Figure \[fig:STM-600\] - STM empty-state images of high-temperature pyramids: {#figurefigstm-600---stm-empty-state-images-of-high-temperature-pyramids .unnumbered} ------------------------------------------------------------------------------ $T_{\rm gr}=650$; (a) $87\times87$nm, steps of the incomplete upper left facet, running normal to the base side, are seen near the left corner of the pyramid; (b) $87\times87$nm, a cluster with edges split near the base and an apex formed by a set of incomplete {105} facets; (c) $57\times57$nm, a magnified image of a facet with several {105} incomplete facets near an apex; (d) $22\times22$nm, a split edge near a base.  Figure \[fig:rheed\] - *In situ* RHEED patterns of Ge/Si(001) films: {#figurefigrheed---in-situ-rheed-patterns-of-gesi001-films .unnumbered} --------------------------------------------------------------------- *E* = 10keV, \[110\] azimuth; (a) $T_{\rm gr} =$ 650, $h_{\rm Ge}=$ 4Å; (b) $T_{\rm gr} =$ 360, $h_{\rm Ge}=$ 4Å; (c) $T_{\rm gr} =$ 650, $h_{\rm Ge}=$ 5Å; (d) $T_{\rm gr} =$ 650, $h_{\rm Ge}=$ 5Å, annealing at the deposition temperature for 7 min; (e) $T_{\rm gr} =$ 650, $h_{\rm Ge}=$ 6Å, the similar pattern is obtained for $T_{\rm gr} =$ 600; the patterns were obtained at room temperature after sample cooling.  Figure \[fig:RHEED\_cool-600\] - RHEED patterns of Ge/Si(001) deposited at 600 obtained during sample cooling: {#figurefigrheed_cool-600---rheed-patterns-of-gesi001-deposited-at-600-obtained-during-sample-cooling .unnumbered} --------------------------------------------------------------------------------------------------------------- $h_{\rm Ge}=$ 6Å; *E* = 10keV, \[110\] azimuth; cooling rate is $\sim 0.4$/s (see the cooling diagram in Ref. [@stm-rheed-EMRS]); (a) $T=600$, before cooling; (b)–(d) during cooling, time from beginning of cooling (min.): (b) 1, (c) 2, (d) 3; (e) room temperature, after cooling; arrows indicate the arising -reflexes to demonstrate a process of the $(2\times 1)$ pattern appearance; the images were cut from frames of a film.  Figure \[fig:TEM-6A\] - HRTEM data for the five-layer Ge/Si heterostructure with buried Ge clusters: {#figurefigtem-6a---hrtem-data-for-the-five-layer-gesi-heterostructure-with-buried-ge-clusters .unnumbered} ----------------------------------------------------------------------------------------------------- [*h$_{\mathrm{Ge}}$*]{}= 6 (see Figure \[fig:STM-360\]a); (a) a long shot, the mark is 100 nm; (b) Ge clusters resolved in a layer, figure ‘1’ indicates one of the clusters, ‘2’ shows a WL segment; the mark is 50 nm; (c),(d) magnified images of a Ge cluster, the panel (d) corresponds to the light square in the panel (c); the marks are 10 and 5 nm, respectively; (e) a close-up image of a domain next to the top of the cluster imaged in (d); (f) the Fourier transform of the image (e), the measured periods are $\sim 5.4$ along \[001\] and $\sim 3.8$ along \[110\]; arrows in panels (c) to (f) indicate the \[001\] direction.  Figure \[fig:TEM-10A1L\] - HRTEM images of the one-layer Ge/Si structures with buried Ge clusters: {#figurefigtem-10a1l---hrtem-images-of-the-one-layer-gesi-structures-with-buried-ge-clusters .unnumbered} --------------------------------------------------------------------------------------------------- [*h$_{\mathrm{Ge}}$*]{}= 10 (see Figure \[fig:STM-360\]c); (a) a perfect epitaxial structure of Ge and Si layers; the mark is 10 nm; (b), (c) a huge cluster (&gt;3,5 nm high) gives rise to tensile strain generating point and extended defects in the Si cap, the stress field spreads under the cluster \[the mark is 10 nm in (b)\]; (d) a magnified image obtained from the tensile domain, extended defects are seen; ‘1’ denotes Ge clusters, ‘2’ is a domain under tensile stress, ‘3’ indicates a twin boundary.  Figure \[fig:TEM-9-10A\] - TEM data for the five-layer Ge/Si heterostructures, [*T$_{\mathrm{gr}}$*]{}= 360: {#figurefigtem-9-10a---tem-data-for-the-five-layer-gesi-heterostructures-t_mathrmgr-360 .unnumbered} ------------------------------------------------------------------------------------------------------------- \(a) to (c) [*h$_{\mathrm{Ge}}$*]{}= 9; (d) to (i) [*h$_{\mathrm{Ge}}$*]{}= 10; (a) domains of tensile strain in Si over Ge clusters are observed more or less distinctly near most clusters, but not around all; the surface is down; the mark equals 20 nm; (b), (c) zoom in two strained domains, no extended defects are observed; (d) strained domains are more pronounced, the strain is well recognized even under some clusters; (e) a magnified image of a strained domain; a strained lattice is well contrasted with the normal one; (f) zoom in the dilated lattice, a perfectly ordered lattice is observed; (g) a Si domain next to the Ge/Si interface near the cluster apex, a vacancy (‘V’) and disordered lattice (upper right corner) are revealed; letter ‘I’ indicates the direction to the interface along [[&lt;]{}11$\overline{1}$&gt;]{}; (h) the same as in (g) but some farther from the interface, the lattice is perfect; (i) the Fourier transform of an image obtained from a strained domain demonstrates an enhanced lattice parameter (the strain varies from domain to domain, the estimated lattice period in the \[001\] direction sometimes reaches $\sim$5.6Å).  Figure \[fig:STM-9A\] - STM images of Ge/Si(001), [*h$_{\mathrm{Ge}}$*]{}= 9, [*T$_{\mathrm{gr}}$*]{}= 360: {#figurefigstm-9a---stm-images-of-gesi001-h_mathrmge-9-t_mathrmgr-360 .unnumbered} ------------------------------------------------------------------------------------------------------------ \(a) to (d) array top views with different magnifications; (e) a large cluster in the array, $\sim$2,5 nm high; (f) a huge cluster (&gt;3,5 nm high) interpreted as an array defect. (a)\ (b) Figure \[fig:p-i-n\_Schematics\] - Schematics of the [*p–i–n*]{}-structures: {#figurefigp-i-n_schematics---schematics-of-the-pin-structures .unnumbered} ----------------------------------------------------------------------------- \(a) R163, (b) R166.  Figure \[fig:r163\_r166\] - Photo-emf spectra of the [*p–i–n*]{} structures: {#figurefigr163_r166---photo-emf-spectra-of-the-pin-structures .unnumbered} ----------------------------------------------------------------------------- (a) R163: (1) without bias lighting; (2)–(5) under bias lighting (Ge filter): (2) $W=0.25$mW/cm$^2$; (3) $W=0.77$mW/cm$^2$; (4) $W=1.5$mW/cm$^2$; (5) $W=2.16$mW/cm$^2$; (b) R166: (1) without bias lighting; (2)–(6) under bias lighting (Si filter): (2) $W=0.63$mW/cm$^2$; (3) $W=3.3$mW/cm$^2$; (4) $W=5.3$mW/cm$^2$; (5) $W=12$mW/cm$^2$; (6) $W=17.5$mW/cm$^2$. (a)\ (b) Figure \[fig:bands\] - Schematics of band structures of [*p–i–n*]{}-diodes: {#figurefigbands---schematics-of-band-structures-of-pin-diodes .unnumbered} ---------------------------------------------------------------------------- \(a) R163, (b) R166; figure ‘1’ indicates potential barriers for holes in the valence band.  Figure \[fig:bias\] - Dependence of photo-emf response of the R163 and R166 [*p–i–n*]{}-structures on bias lighting power density. {#figurefigbias---dependence-of-photo-emf-response-of-the-r163-and-r166-pin-structures-on-bias-lighting-power-density. .unnumbered} -----------------------------------------------------------------------------------------------------------------------------------  Figure \[fig:THz-fig1\] - Spectra of transmission coefficient of a silicon substrate (a commercial wafer, $\rho = 12\,\Omega$cm), measured at two temperatures using two different BWO working in spectral ranges from 11 cm$^{-1}$ to 24 cm$^{-1}$ and from 29 cm$^{-1}$ to 39 cm$^{-1}$: {#figurefigthz-fig1---spectra-of-transmission-coefficient-of-a-silicon-substrate-a-commercial-wafer-rho-12omegacm-measured-at-two-temperatures-using-two-different-bwo-working-in-spectral-ranges-from-11-cm-1-to-24-cm-1-and-from-29-cm-1-to-39-cm-1 .unnumbered} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Dots show the measurement results, lines are least-square fits based on the Drude conductivity model, as described in the text.  Figure \[fig:THz-fig2\] - Temperature dependences of the silicon substrate parameters obtained by fitting the transmission coefficient spectra as shown in Figure \[fig:THz-fig1\] and described in the text: {#figurefigthz-fig2---temperature-dependences-of-the-silicon-substrate-parameters-obtained-by-fitting-the-transmission-coefficient-spectra-as-shown-in-figurefigthz-fig1-and-described-in-the-text .unnumbered} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \(a) plasma frequency of charge carriers and (b) scattering rate. Solid line in (b) shows the $T^{-3/2}$ behavior.  Figure \[fig:THz-fig3\] - Spectra of transmission coefficient of Ge/Si heterostructure on Si substrate (solid symbols) and of bare substrate (open symbols) measured at two different temperatures: {#figurefigthz-fig3---spectra-of-transmission-coefficient-of-gesi-heterostructure-on-si-substrate-solid-symbols-and-of-bare-substrate-open-symbols-measured-at-two-different-temperatures .unnumbered} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Horizontal lines show the difference in peak transmissivity that is observed at 300 K and disappears at $\sim$170 K. The peaks positions are shifted due to slight difference in the Si substrate thickness.  Figure \[fig:THz-fig4\] - Temperature dependences of dynamical conductivity of Ge/Si heterostructure and of Si substrate: {#figurefigthz-fig4---temperature-dependences-of-dynamical-conductivity-of-gesi-heterostructure-and-of-si-substrate .unnumbered} -------------------------------------------------------------------------------------------------------------------------- Frequency is around 1 THz.  Figure \[fig:THz-fig5\] - Terahertz conductivity and absorption coefficient of Ge/Si heterostructure with Ge quantum dots versus Ge coverage: {#figurefigthz-fig5---terahertz-conductivity-and-absorption-coefficient-of-gesi-heterostructure-with-ge-quantum-dots-versus-ge-coverage .unnumbered} ---------------------------------------------------------------------------------------------------------------------------------------------- \(a) terahertz conductivity, (b) absorption coefficient; lines are guides to the eye.

ArXiv

--- abstract: 'In this paper, we study the entanglement properties of a spin-1 model the exact ground state of which is given by a Matrix Product state. The model exhibits a critical point transition at a parameter value $a=0$. The longitudinal and transverse correlation lengths are known to diverge as $a\rightarrow0$. We use three different entanglement measures $S(i)$ (the one-site von Neumann entropy), $S(i,j)$ (the two-body entanglement) and $G(2,n)$ (the generalized global entanglement) to determine the entanglement content of the MP ground state as the parameter $a$ is varied. The entanglement length, associated with $S(i,j)$, is found to diverge in the vicinity of the quantum critical point $a=0$. The first derivative of the entanglement measure $E$ $(=S(i),\: S(i,j))$ w.r.t. the parameter $a$ also diverges. The first derivative of $G(2,n)$ w.r.t. $a$ does not diverge as $a\rightarrow0$ but attains a maximum value at $a=0$. At the QCP itself all the three entanglement measures become zero. We further show that multipartite correlations are involved in the QPT at $a=0$.' author: - Amit Tribedi and Indrani Bose title: Quantum Critical Point and Entanglement in a Matrix Product Ground State --- Department of Physics Bose Institute 93/1, Acharya Prafulla Chandra Road Kolkata - 700 009, India I. INTRODUCTION {#i.-introduction .unnumbered} =============== The entanglement characteristics of the ground states of many body Hamiltonians describing condensed matter systems constitute an important area of study in quantum information theory. Entanglement is an essential resource in quatum computation and communication protocols. Condensed matter, specially, spin systems have been proposed as candidate systems for the realization of some of the protocols. Entanglement provides a measure of non-local quantum correlations in the system and it is of significant interest to determine how the correlations associated with the ground state of the system change as one or more than one parameter of the system is changed. The focus on ground state characteristics arises from the possibility of quantum phase transitions (QPTs) which occur at temperature $T=0$ (when the system is in its ground state) and are driven solely by quantum fluctuations [@key-1]. A QPT is brought about by tuning a parameter, either external or intrinsic to the Hamiltonian, to a special value termed the transition point. In thermodynamic critical phenomena, the thermal correlation length diverges and the thermodynamic quantities become singular as the critical point is approached. In the quantum case, the correlation length diverges in the vicinity of the QCP and the ground state properties develop non-analytic features. An issue of considerable interest is whether the quantum correlations, like the usual correlation functions, become long-ranged near the QCP. In a wider perspective, the major goal is to acquire a clear understanding of the variation in entanglement characteristics as a tuning parameter is changed. QPTs have been extensively studied in spin systems both theoretically and experimentally. In recent years, several theoretical studies have been undertaken to elucidate the relationship between QPTs and entanglement in spin systems [@key-2; @key-3; @key-4; @key-5; @key-6; @key-7]. In particular, a number of entanglement measures have been identified which develop special features close to the transition point. One such measure is concurrence which quantifies the entanglement between two spins $(S=\frac{1}{2})$. At a QCP, as illustrated by a class of exactly-solvable spin models $(S=\frac{1}{2})$, the derivative of the ground state concurrence has a logarithmic singularity though the concurrence itself is non-vanishing upto only next-nearest-neighbour-distances between two spins [@key-2; @key-3]. Discontinuities in the ground state concurrence have been shown to characterize first order QPTs [@key-8; @key-9; @key-10]. Later, Wu et al. [@key-5] showed that under some general assumptions a first order QPT, associated with a discontinuity in the first derivative of the ground state energy, gives rise to a discontinuity in a bipartite entanglement measure like concurrence and negativity. Similarly, a discontinuity or a divergence in the first derivative of the same entanglement measure is the signature of a second order phase transition with a discontinuity or a divergence in the second derivative of the ground state energy. Another measure of entanglement, studied in the context of QPTs, is the entropy of entanglement between a block of $L$ adjacent spins in a chain with the rest of the system [@key-4]. At the QCP, the entropy of entanglement diverges logarithmically with the length of the block. There is, however, no direct relation with the long range correlations in the system. A number of entanglement measures have recently been proposed which are characterized by a diverging length scale, the entanglement length, close to a QCP. The localizable entanglement (LE) between two spins is defined as the maximum average entanglement that can be localized between them by performing local measurements on the rest of the spins [@key-11]. The entanglement length sets the scale over which the LE decays. The two-body entanglement $S(i,j)$ is a measure of the entanglement between two separated spins, at sites $i$ and $j$, and the rest of the spins [@key-7]. Let $\rho(i,j)$ be the reduced density matrix for the two spins, obtained from the full density matrix by tracing out the spins other than the ones at sites $i$ and $j$. The two body entanglement $S(i,j)$ is given by the von Neumann entropy $$S(i,j)=-Tr\,\rho(i,j)\, log_{2}\,\rho(i,j)\label{1}$$ In a translationally invariant system, $S$ depends only on the distance $n=\mid j-i\mid$. As pointed out in [@key-7], the spins that are entangled with one or both the spins at sites $i$ and $j$ contribute to $S$. The following results have been obtained in the case of the $S=\frac{1}{2}$ exactly solvable anisotropic $XY$ model in a transverse magnetic field. The model, away from the isotropic limit, belongs to the universality class of the transverse Ising model. The two-body entanglement $S(i,j)$ has a simple dependence on the spin correlation functions in the large $n$ limit. Away from the critical point, $S(i,j)$ is found to saturate over a length scale $\xi_{E}$ as $n$ increases. Near the QCP, one obtains $$S(i,j)-S(\infty)\sim n^{-1}\, e^{-\frac{n}{\xi_{E}}}\label{2}$$ The entanglement length (EL), $\xi_{E}$, has an interpretation similar to that in the case of LE. The EL diverges with the same critical exponent as the correlation length at the QCP. $S(i,j)$ thus captures the long range correlations associated with a QPT. At the critical point itself, $S(i,j)-S(\infty)$ has a power-law decay, i.e., $S(i,j)-S(\infty)\sim n^{-\frac{1}{2}}$. In the limit of large $n$, the first derivative of $S(i,j)$ w.r.t. a Hamiltonian parameter develops a $\lambda-$like cusp at the critical point. The universality and a finite-size scaling of the entanglement have also been demonstrated. The one-site von Neumann entropy $$S(i)=-Tr\,\rho(i)\, log_{2}\,\rho(i)\label{3}$$ is also known to be a good indicator of a QPT [@key-3]. It provides a measure of how a single spin at the site $i$ is entangled with the rest of the system. The reduced density matrix $\rho(i)$ is obtained from the full density matrix by tracing out all the spins except the one at the site $i$. Oliveira et al. [@key-6] have proposed a generalized global entanglement (GGE) measure $G(2,n)$ which quantifies multipartite entanglement (ME). $G(2,n)$ for a translationally symmetric system is given by $$G(2,n)=\frac{d}{d-1}[1-\sum_{l,m=1}^{d^{2}}\mid[\rho(j,j+n)]_{lm}\mid^{2}]\label{4}$$ where $\rho(j,j+n)$ is the reduced density matrix of dimension $d$. The factor 2 in $G(2,n)$ indicates that the reduced density matrix is that for a pair of particles. Wu et al. [@key-5] considered QPTs characterized by non-analyticities in the derivatives of the ground state energy. These arise from the non-analyticities in one or more of the elements of the reduced density matrix. In terms of the GGE, a discontinuity in $G(2,n)$ signals a first order QPT, brought about by a discontinuity in one or more of the elements, $[\rho_{j,j+n}]_{lm}$ of the reduced density matrix [@key-6]. A discontinuity or divergence in the first derivative of $G(2,n)$ w.r.t. the tuning parameter occurs due to a discontinuity or divergence in the first derivetives of one or more of the elements of the reduced density matrix. The associated QPT is of second order. Non-analyticities in $G(2,n)$ and its derivatives thus serve as indicators of QPTs. In the case of the $XY$ $S=\frac{1}{2}$ spin chain, the GGE measure shows a diverging EL as the QCP is approached. The EL $\xi_{E}=\frac{\xi_{C}}{2}$ where $\xi_{C}$ is the usual correlation length. Thus, both the length scales diverge with the same critical exponent near the QCP. The relationship between entanglement and QPTs has mostly been explored for spin-$\frac{1}{2}$ systems. The entanglement properties of the ground states of certain spin$-1$ Hamiltonians have been studied using different measures [@key-11; @key-12; @key-13]. Numerical studies show that the LE has the maximal value for the ground state of the spin-1 Heisenberg antiferromagnet with open boundary conditions (OBC) [@key-13]. In the case of the spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) model [@key-14], the result can be proved exactly. A class of spin-1 models, the $\phi$-deformed AKLT models, is characterized by an exponentially decaying LE with a finite EL $\xi_{E}$. The length $\xi_{E}$ diverges at the point $\phi=0$ though the conventional correlation length remains finite [@key-13]. A recent study [@key-15] shows that in the case of spin-1 systems, the use of LE for the detection of QPTs is not feasible. An example is given by the $S=1$ XXZ Heisenberg antiferromagnet with single-ion anisotropy. The model has a rich phase diagram with six different phases. The LE is found to be always 1 in the entire parameter region and hence is insensitive to QPTs. The ground states of certain spin-1 models have an exact representation in terms of matrix product states (MPS) [@key-16; @key-17; @key-18]. The ground state of the spin-1 AKLT model, termed a valence bond solid (VBS) state, is an example of an MPS. The ground state is characterized by short-ranged spin-spin correlations and a hidden topological order known as the string order. The excitation spectrum of the model is further gapped. In the MPS formalism, ground state expectation values like the correlation functions are easy to calculate. This has made it particularly convenient to study phase transitions in spin models with MP states as exact ground states [@key-17]. The transitions identified so far include both first and second order transitions and are brought about by the tuning of the Hamiltonian parameters. The second order transition in the class of finitely correlated MP states, however, differs from the conventional QPT in one important respect. The spin correlation function is always of the form $A_{C}\, e^{-\frac{n}{\xi_{C}}}$ for large $n$. The correlation length $\xi_{C}$ diverges as the transition point is approached. The pre-factor $A_{C}$, however, vanishes at the transition point [@key-17]. This is in contrast to the power-law decay of the correlation function at a conventional QCP. Some distinct features of QPTs in MP states have recently been identified [@key-19]. One of these relates to the analyticity of the ground state energy density for all values of the tuning parameter. In a conventional QPT, the energy density becomes non-analytic at the QCP. The MP states appear to provide an ideal playground for exploring novel types of QPTs. In this paper, we consider a spin-1 model, the exact ground state of which is given by an MP state [@key-20]. The model has a rich phase diagram with a number of first order phase transitions and a critical point transition. We study the entanglement properties of the ground state with a view to pinpoint the special features which appear close to the critical point. This is done by using three different entanglement measures, namely, the single-site, two-body and generalized global entanglement defined earlier. II. REDUCED DENSITY MATRIX OF MP {#ii.-reduced-density-matrix-of-mp .unnumbered} ================================= GROUND STATE {#ground-state .unnumbered} ============ We consider a spin-1 chain Hamiltonian proposed by Klümper et al. [@key-20] which describes a large class of antiferromagnetic (AFM) spin-1 chains with MP states as exact ground states. The Hamiltonian satisfies the symmetries : (i) rotational invariance in the $x-y$ plane, (ii) invariance under $S^{z}\rightarrow-S^{z}$ and (iii) translation and parity invariance. The Hamiltonian has the general form $$H=\sum_{j=1}^{L}h_{j,\, j+1}$$ $$h_{j,\, j+1}=\alpha_{0}A_{j}^{2}+\alpha_{1}(A_{j}B_{j}+B_{j}A_{j})+\alpha_{2}B_{j}^{2}+\alpha_{3}A_{j}+\alpha_{4}B_{j}(1+B_{j})+$$ $$+\alpha_{5}((S_{j}^{z})^{2}+(S_{j+1}^{z})^{2}+C\label{5}$$ where $L$ is the number of sites in the chain and periodic boundary conditions (PBC) hold true. The parameters $\alpha_{j}$ are real and $C$ is a constant. The nearest-neighbour (n.n.) interactions are $$\begin{array}{c} A_{j}=S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}\\ B_{j}=S_{j}^{z}S_{j+1}^{z}\end{array}\label{6}$$ The constant $C$ in Eq. (5) may be adjusted so that the ground state eigenvalue of $h_{j,\, j+1}=0$. Hence $$h_{j,\, j+1}\geq0\quad\Rightarrow H\geq0\label{7}$$ i.e., $H$ has only non-negative eigenvalues. In the AFM case, the $z$-component of the total spin of the ground state $S_{tot}^{z}=0$. Klümper et al. showed that in a certain subspace of the $\alpha_{j}-$parameter space , the AFM ground state has the MP form. Let $\left|0\right\rangle $ and $\left|\pm\right\rangle $ be the eigenstates of $S^{z}$ with eigenvalues $0$, $+1$ and $-1$ respectively. Define a $2\times2$ matrix at each site $j$ by $$g_{j}=\left(\begin{array}{cc} \left|0\right\rangle & -\sqrt{a}\left|+\right\rangle \\ \sqrt{a}\left|-\right\rangle & -\sigma\left|0\right\rangle \end{array}\right)\label{8}$$ with non-vanishing parameters $a,\,\sigma\neq0$. The global AFM state is written as $$\left|\psi_{0}\,(a,\,\sigma)\right\rangle =Tr\,(g_{1}\otimes g_{2}\otimes......\otimes g_{L})\label{9}$$ where ‘$\otimes$’ denotes a tensor product. One can easily check that $S_{tot}^{z}\left|\psi_{0}\right\rangle =0$, i.e., the state is AFM. One now demands that the state $\left|\psi_{0}\,(a,\,\sigma)\right\rangle $ is the exact ground state of the Hamiltonian $H$ with eigenvalues $0$. For this, it is sufficient to show that $$h_{j,\, j+1}\:(g_{j}\otimes g_{j+1})=0\label{10}$$ Eq. (3) and (10) are satisfied provided the following equalities $$\begin{array}{cc} 1)\,\sigma=sign(\alpha_{3}), & 2)\, a\,\alpha_{0}=\alpha_{3}-\alpha_{1},\\ 3)\,\alpha_{5}=\mid\alpha_{3}\mid+\alpha_{0}(1-a^{2}), & 4)\,\alpha_{2}=\alpha_{0}a^{2}-2\mid\alpha\mid\end{array}\label{11}$$ and inequalities $$a\neq0,\:\alpha_{3}\neq0,\:\alpha_{4}>0,\:\alpha_{0}>0\label{12}$$ hold true. The state $\left|\psi_{0}\,(a,\,\sigma)\right\rangle $ is the ground state of the Hamiltonian $(5)$ with ground state energy zero provided the equalities in $(8)$ are satisfied. The inequalities constrain the other eigenvalues of $h_{j,j+1}$ to be positive. If the inequalities are satisfied, the ground state can be shown to be unique for any chain length $L$. Also, in the thermodynamic limit $L\rightarrow\infty$, the excitation spectrum has a gap $\Delta$. With equality signs in the inequalities $(12)$, the state $\left|\psi_{0}\,(a,\,\sigma)\right\rangle $ is still the ground state but is no longer unique. The spin$-1$ model has the typical feature of a Haldane-gap (HG) antiferromagnet. In fact, the AKLT model is recovered as a special case with $a=2,\:\sigma=1,\:\alpha_{3}=3\alpha_{0}>0,\:\alpha_{2}=-2\alpha_{0}$ and $\alpha_{4}=3\alpha_{0}$. The state $(9)$ now represents the VBS state. Using the transfer matrix method [@key-16], the ground state correlation functions can be calculated in a straightforward manner. The results are $(L\rightarrow\infty,\, r\geq2)$ : Longitudinal correlation function $$\left\langle S_{1}^{z}\, S_{r}^{z}\right\rangle =-\frac{a^{2}}{(1-|a|)^{2}}\left(\frac{1-|a|}{1+|a|}\right)^{r}\label{13}$$ Transverse correlation function$$\left\langle S_{1}^{x}\, S_{r}^{x}\right\rangle =-|a|\,[\sigma+sign\, a]\left(\frac{-\sigma}{1+|a|}\right)^{r}\label{14}$$ The correlations $(13)$ and $(14)$ decay exponentially with the longitudinal and transverse correlation lengths given by$$\xi_{l}^{-1}=ln\left|\frac{1+|a|}{1-|a|}\right|,\;\;\xi_{t}^{-1}=ln(1+|a|)\label{15}$$ Furthermore, the string order parameter has a non-zero expectation value in the ground state. One finds that the correlation lengths diverge as $a\rightarrow0$. At the point $a=0$, the correlation functions given by Eq. $(13)$ and $(14)$ are zero. At a conventional QCP, the correlation functions have a power-law decay. We will, however, refer to the point as a QCP since the correlation lengths diverge as the point is approached. A consequence of the diverging correlation length is that the excitation spectrum of the spin-1 model, which is gapped (the Haldane phase) for $a>0$, becomes gapless at the critical point $a=0$ [@key-20]. The presence or absence of a gap in the excitation spectrum of a system is reflected in the low temperature thermodynamic properties of the system. Furthermore, the string order parameter has a non-zero expectation value in the ground state for $a>0$ and becomes zero at $a=0$ indicating the appearance of a new phase. Refs. [@key-16; @key-17] provide several other examples of spin-1 models with finitely correlated MP states as exact ground states. All these models exhibit critical point transitions with features similar to those in the case of the spin-1 model described by the Hamiltonian in Eq. $(5)$. We now focus on the entanglement properties of the MP ground state (Eq. $(9)$). We consider $a$ to be $\geq0$ and $\sigma=+1$ in Eq. $(8)$. The one-site reduced density matrix $\rho(i)$ (Eq. $(3)$) obtained by tracing out all the spins except the $i$-th spin from the ground state density matrix $\rho=\left|\psi_{0}\right\rangle $$\left\langle \psi_{0}\right|$, can be calculated using the transfer matrix method [@key-16]. The density matrix, from Eq. $(9)$, is$$\rho=\left|\psi_{0}\right\rangle \left\langle \psi_{0}\right|=\sum_{\{ n_{\alpha},m_{\alpha}\}}g_{n_{1}n_{2}}\, g_{n_{2}n_{3}}........g_{n_{L}n_{1}}\: g_{m_{1}m_{2}}^{\dagger}\, g_{m_{2}m_{3}}^{\dagger}......g_{m_{L}m_{1}}^{\dagger}\label{16}$$ The summation is over all the indices, $n_{i}$, $m_{i}$, $i=1,2,.....L$. We define a $4\times4$ matrix $f$ (the elements of which are operators) at any lattice site as $$f_{\mu_{1}\mu_{2}}\Rightarrow f_{(n_{1},m_{1})(n_{2},m_{2})}\equiv g_{n_{1}n_{2}}\, g_{m_{1}m_{2}}^{\dagger}\label{17}$$ The convention of the ordering of the multi-indices is $\mu=1,2,3,4\:\leftrightarrow(11),(12),(21),(22)$. Thus, $f$ can be written as $$f=\left(\begin{array}{cccc} \left|0\right\rangle \left\langle 0\right| & -\sqrt{a}\left|0\right\rangle \left\langle 1\right| & -\sqrt{a}\left|1\right\rangle \left\langle 0\right| & a\left|1\right\rangle \left\langle 1\right|\\ \sqrt{2}\left|0\right\rangle \left\langle -1\right| & -\left|0\right\rangle \left\langle 0\right| & -a\left|1\right\rangle \left\langle -1\right| & \sqrt{a}\left|1\right\rangle \left\langle 0\right|\\ \sqrt{a}\left|-1\right\rangle \left\langle 0\right| & -a\left|-1\right\rangle \left\langle 1\right| & -\left|0\right\rangle \left\langle 0\right| & \sqrt{a}\left|0\right\rangle \left\langle 1\right|\\ a\left|-1\right\rangle \left\langle -1\right| & -\sqrt{a}\left|-1\right\rangle \left\langle 0\right| & -\sqrt{a}\left|0\right\rangle \left\langle -1\right| & \left|0\right\rangle \left\langle 0\right|\end{array}\right)\label{18}$$ Also,$$\rho(i)=Tr_{1,..L}^{i}\left|\psi_{0}\right\rangle \left\langle \psi_{0}\right|\label{19}$$ where the trace is over all the spins except the $i$-th one. The transfer matrix $F$ at a site $m$ is obtained by taking the trace over $f$ at the same site, i.e., $$F_{m}=\sum_{k}\left\langle k\right|f_{m}\left|k\right\rangle \label{20}$$ where the states $\left|k\right\rangle $ are the states $\left|0\right\rangle $, $\left|\pm1\right\rangle $. The transfer matrix $F$ is obtained as$$F=\left(\begin{array}{cccc} 1 & 0 & 0 & a\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ a & 0 & 0 & 1\end{array}\right)\label{21}$$ The eigenvalues are $$\varepsilon_{1}=1+a,\:\varepsilon_{2}=1-a,\:\varepsilon_{3}=-1,\:\varepsilon_{4}=-1\label{22}$$ The corresponding eigenvectors are$$\begin{array}{cc} \left|e_{1}\right\rangle =\frac{1}{\sqrt{2}}\left(\begin{array}{c} 1\\ 0\\ 0\\ 1\end{array}\right),\quad & \left|e_{2}\right\rangle =\frac{1}{\sqrt{2}}\left(\begin{array}{c} -1\\ 0\\ 0\\ 1\end{array}\right)\\ \left|e_{3}\right\rangle =\left(\begin{array}{c} 0\\ 1\\ 0\\ 0\end{array}\right),\quad & \left|e_{4}\right\rangle =\left(\begin{array}{c} 0\\ 0\\ 1\\ 0\end{array}\right)\end{array}\label{23}$$ From Eq. $(20)$,$$\rho(i)=\frac{\sum_{\alpha=1}^{4}\left\langle e_{\alpha}\right|F^{L-1}f\left|e_{\alpha}\right\rangle }{\sum_{\alpha=1}^{4}\left\langle e_{\alpha}\right|F^{L}\left|e_{\alpha}\right\rangle }\label{24}$$ The factor in the denominator takes care of the condition $Tr\,\rho=1$. On taking the thermodynamic limit $L\rightarrow\infty$, we get $$\rho(i)=\varepsilon_{1}^{-1}\,\left\langle e_{1}\right|f\left|e_{1}\right\rangle \label{25}$$ In the $\left|0,\pm1\right\rangle $ basis, the reduced density matrix becomes$$\rho(i)=\left(\begin{array}{ccc} \frac{1}{1+a} & 0 & 0\\ 0 & \frac{a}{2(1+a)} & 0\\ 0 & 0 & \frac{a}{2(1+a)}\end{array}\right)\label{26}$$ The calculation of the two-site reduced density matrix $\rho(i,j)$ follows in the same manner. $\rho(i,j)$ is given by$$\rho(i,j)=Tr_{1,..L}^{i,j}\left|\psi_{0}\right\rangle \left\langle \psi_{0}\right|\label{27}$$ where the trace is taken over all the spins except the $i$-th and $j$-th ones. $$\rho(i,j)=\frac{\sum_{\alpha=1}^{4}\left\langle e_{\alpha}\right|F^{i-1}f\, F^{j-i-1}f\, F^{L-j}\left|e_{\alpha}\right\rangle }{\sum_{\alpha=1}^{4}\left\langle e_{\alpha}\right|F^{L}\left|e_{\alpha}\right\rangle }\label{28}$$ In the thermodynamic limit $L\rightarrow\infty$, $\rho(i,j)$ reduces to $$\rho(i,j)=\sum_{\alpha=1}^{4}\varepsilon_{\alpha}^{-2}\,\left(\frac{\varepsilon_{\alpha}}{\varepsilon_{1}}\right)^{n+1}\,\left\langle e_{1}\right|f\left|e_{\alpha}\right\rangle \left\langle e_{\alpha}\right|f\left|e_{1}\right\rangle \label{29}$$ where $n=|j-i|$. The matrix $\rho(i,j)$ is a $9\times9$ matrix and defined in the two-spin basis states $\left|lm\right\rangle $ with the ordering $$\left|lm\right\rangle \equiv\left|11\right\rangle ,\left|10\right\rangle ,\left|01\right\rangle ,\left|1-1\right\rangle ,\left|-11\right\rangle ,\left|00\right\rangle ,\left|0-1\right\rangle ,\left|-10\right\rangle ,\left|-1-1\right\rangle \label{30}$$ The non-zero matrix elements, $b_{pq}$ $(p=1,...,9,\, q=1,...,9)$, of $\rho(i,j)$ are : $$\begin{array}{c} b_{11}=b_{99}=\frac{a^{2}}{4(1+a)^{2}}-\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}\\ b_{22}=b_{33}=b_{77}=b_{88}=\frac{a}{2(1+a)^{2}}\\ b_{44}=b_{55}=\frac{a^{2}}{4(1+a)^{2}}+\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}\\ b_{23}=b_{32}=b_{46}=b_{64}=b_{56}=b_{65}=b_{78}=b_{87}=\frac{a}{2(1+a)}\left(-\frac{1}{1+a}\right)^{n}\\ b_{66}=\frac{1}{(1+a)^{2}}\end{array}\label{31}$$ It is easy to check that $\rho(i,j)$ has a block-diagonal form. III. ENTANGLEMENT MEASURES $\; S(i)$, $S(i,j)$, $G(2,n)$ {#iii.-entanglement-measures-si-sij-g2n .unnumbered} ======================================================== We now determine the entanglement content of the ground state $\left|\psi_{0}\right\rangle $ (Eq. $(9)$) of the Hamiltonian (Eq. $(5)$) using the entanglement measures $S(i)$, $S(i,j)$, and $G(2,n)$. The calculations are carried out for different values of the parameter $a$ in Eq. $(8)$. The ultimate aim is to probe the special features, if any, of entanglement in the vicinity of the QCP at $a=0$. From Eq. $(3)$ and $(26)$, the one-site entanglement $$S(i)=\frac{1}{1+a}[(1+a)log_{2}\,(1+a)-a\, log_{2}\, a+a]\label{32}$$ Figure $1$ (top) shows the variation of $S(i)$ w.r.t. $a$. The one-site entanglement has the maximum possible value $log_{2}\,3$. This is attained at the AKLT point $a=2$. The VBS state is in this case the exact ground state. In the VBS state, each spin-1 at a specific lattice site can be considered as a symmetric combination of two spin-$\frac{1}{2}$’s [@key-14]. In the VBS state, each spin-$\frac{1}{2}$ at a particular lattice site forms a spin singlet with a spin-$\frac{1}{2}$ at a neighbouring lattice site. $S(i)$ has the value zero at the QCP $a=0$. Figure $1$ (bottom) shows the variation of $\frac{\partial S(i)}{\partial a}$ with the parameter $a$. The derivative diverges as the QCP is approached. This is the expected behaviour at the QCP of a conventional QPT. In the latter case, however, $S(i)$ has the maximum value at the QCP [@key-3]. From Eq. $(1)$ and $(31)$, the two-body entanglement $S(i,j)$ is $$S(i,j)=-\sum_{i=1}^{9}\lambda_{i}\, log_{2}\lambda_{i}\label{33}$$ Where $\lambda_{i}$’s are the eigenvalues of the reduced density matrix $\rho(i,j)$. These are given by$$\begin{array}{c} \lambda_{1}=\lambda_{2}=\frac{a}{2(1+a)^{2}}-\frac{a}{2(1+a)}\left(-\frac{1}{1+a}\right)^{n}\\ \lambda_{3}=\lambda_{4}=\frac{a}{2(1+a)^{2}}+\frac{a}{2(1+a)}\left(-\frac{1}{1+a}\right)^{n}\\ \lambda_{5}=\frac{a^{2}}{4(1+a)^{2}}+\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}\\ \lambda_{6}=\lambda_{7}=\frac{a^{2}}{4(1+a)^{2}}-\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}\\ \lambda_{8}=\frac{1}{2}\left(\frac{a^{2}}{4(1+a)^{2}}+\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}+\frac{1}{1+a^{2}}\right)-\frac{1}{2(1+a)}\\ \left(\frac{(a^{2}-4)^{2}}{16(1+a)^{2}}+2a^{2}\left(-\frac{1}{1+a}\right)^{2n}+\frac{a^{4}}{16(1-a)^{2}}\left(\frac{1-a}{1+a}\right)^{2n}+\frac{a^{2}(a^{2}-4)}{8(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{2n}\right)^{\frac{1}{2}}\\ \lambda_{9}=\frac{1}{2}\left(\frac{a^{2}}{4(1+a)^{2}}+\frac{a^{2}}{4(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{n}+\frac{1}{1+a^{2}}\right)+\frac{1}{2(1+a)}\\ \left(\frac{(a^{2}-4)^{2}}{16(1+a)^{2}}+2a^{2}\left(-\frac{1}{1+a}\right)^{2n}+\frac{a^{4}}{16(1-a)^{2}}\left(\frac{1-a}{1+a}\right)^{2n}+\frac{a^{2}(a^{2}-4)}{8(1-a^{2})}\left(\frac{1-a}{1+a}\right)^{2n}\right)^{\frac{1}{2}}\end{array}\label{34}$$ Knowing the reduced density matrix $\rho(i,j)$, the correlation functions $\left\langle S_{i}^{\alpha}\, S_{j}^{\alpha}\right\rangle \;$ ($\alpha=x,y,z$) can be calculated in the usual manner. One then recovers the expressions in Eq. $(13)$ and $(14)$ ($r=n+1$, where $n=|j-i|$). Figure $2$ (top) shows the variation of $S(i,j)$ as a function of $a$ for $n=1000$. Figure $2$ (bottom) shows the variation of the derivative $\frac{\partial S(i,j)}{\partial a}$ w.r.t. $a$ for the same value of $n$. The maximum of $S(i,j)$ is at the AKLT point $a=2$ and has the value zero at $a=0$. For large $n$, the derivative $\frac{\partial S(i,j)}{\partial a}$ diverges near the QCP at $a=0$. The last fearure is characteristic of a conventional QPT [@key-7]. We next calculate the GGE $G(2,n)$ (Eq. $(4)$). This is easily done as the matrix elements of the reduced density matrix (Eq. $(3)$) are known. Figure $3$ (top) shows the variation of $G(2,n)$ versus $a$ for $n=1000$. Figure $3$ (bottom) shows the plot of $\frac{\partial G(2,n)}{\partial a}$ against $a$. Again $G(2,n)$ has the maximum value at the AKLT point and is zero at $a=0$. The derivative $\frac{\partial G(2,n)}{\partial a}$ does not diverge as $a\rightarrow0$ in contrast to the case of a conventional QPT [@key-6]. The derivative, however, attains the maximum value at the QCP $a=0$. Figure $4$ (top) shows the plots of $S(i)$, $S(i,j)$, and $G(2,n)$ versus $a$ for $n=1000$. Figure $4$ (bottom) shows the plots of the first derivatives of the same quantities w.r.t. $a$ for $n=1000$. The plots are shown for comparing the different entanglement measures. We next determine the EL $\xi_{E}$ and its variation w.r.t. the parameter $a$. We consider the entanglement measure $S(i,j)$ for this purpose. Close to the QCP $a=0$ and in the limit of large $n$, one can write$$S(n=|j-i|)-S(\infty)\sim A_{e}\, e^{-\frac{n}{\xi_{E}}}\label{35}$$ The longitudinal and transverse correlation functions, $p_{n}^{z}=\left\langle S_{1}^{z}S_{n+1}^{z}\right\rangle $ and $p_{n}^{x}=\left\langle S_{1}^{x}S_{n+1}^{x}\right\rangle $ are given by Eq. $(13)$ and $(14)$ with $r=n+1$. For $a<1$, $p_{n}^{z}$ decays faster than $p_{n}^{x}$ with $n$. The eigenvalues $\lambda_{i}$’s, $i=1,...9$, can be expressed in terms of the correlation functions $p_{n}^{z}$ and $p_{n}^{x}$. For large $n$, the contributions from $p_{n}^{z}$ can be ignored. The eigenvalues now reduce to the expressions $$\begin{array}{c} \lambda_{1}=\lambda_{2}=\frac{a}{2(1+a)^{2}}-4\, p_{n}^{x}\\ \lambda_{3}=\lambda_{4}=\frac{a}{2(1+a)^{2}}+4\, p_{n}^{x}\\ \lambda_{5}=\frac{a^{2}}{4(1+a)^{2}}\\ \lambda_{6}=\lambda_{7}=\frac{a^{2}}{4(1+a)^{2}}\\ \lambda_{8}=\frac{1}{2}\left(\frac{a^{2}}{4(1+a)^{2}}+\frac{1}{1+a^{2}}\right)-\frac{1}{2}\left(\frac{(a^{2}-4)^{2}}{16(1+a)^{2}}+2\,\left(p_{n}^{x}\right)^{2}\right)^{\frac{1}{2}}\\ \lambda_{9}=\frac{1}{2}\left(\frac{a^{2}}{4(1+a)^{2}}+\frac{1}{1+a^{2}}\right)+\frac{1}{2}\left(\frac{(a^{2}-4)^{2}}{16(1+a)^{2}}+2\,\left(p_{n}^{x}\right)^{2}\right)^{\frac{1}{2}}\end{array}\label{36}$$ From Eq. $(1)$ and $(36)$, one ultimately arrives at the expressions$$S(n=|j-i|)-S(\infty)\sim A_{e}^{'}\,\left(p_{n}^{x}\right)^{2}\sim A_{e}\, e^{-\frac{n}{\xi_{E}}}\label{37}$$ The pre-factor $A_{e}=0$ at the QCP $\: a=0$ . The EL $\:\xi_{E}$ is given by $$\xi_{E}=\frac{\xi_{t}}{2}=\frac{1}{2\, ln(1+a)}\label{38}$$ where $\xi_{t}$ is the transverse correlation length (Eq. $(15)$). In the case of the $S=\frac{1}{2}$ anisotropic XY model in a transverse field, an expression similar to that in Eq. $(37)$ is obtained close to the QCP in the limit of large $n$ [@key-7]. The pre-factor in this case, however, does not vanish at the QCP but has a power-law dependence on $n$. Figure $5$ shows the variation of $\xi_{E}$ w.r.t. $a$ based on the entanglement measure $S(i,j)$. The total correlations, with both classical and quantum components, between two sites $i$ and $j$ are quantified in terms of the quantum mutual information [@key-21; @key-22]$$I_{ij}=S(i)+S(j)-S(i,j)\label{39}$$ As explained in [@key-21], a comparison of the singular behaviour of $S(i)$ with that of $I_{ij}$ allows one to determine whether two-point ($Q2$) or multipartite ($QS$) quantum correlations are important in a QPT. Figure $6$ (top) shows a plot of $I_{ij}$ versus $a$ for $n=1000$. Figure $6$ (bottom) shows the variation of $\frac{\partial I_{ij}}{\partial a}$ versus $a$ for $n=1000$. The derivative does not diverge as $a\rightarrow0$, a behaviour distinct from that of $\frac{\partial S(i)}{\partial a}$ close to $a=0$. The difference in the singular behaviour of quantities associated with $S(i)$ and $I_{ij}$ indicates that multipartite quantum correlations are involved in the QPT.  **FIG. 1:** Plot of $S(i)$ (top) and $\frac{\partial S(i)}{\partial a}$ (bottom) vs. $a$ .  **FIG.** **2:** Plot of $S(i,j)$ (top) and $\frac{\partial S(i,j)}{\partial a}$ (bottom) as a function of $a$.  **FIG. 3:** Plot of $G(2,n)$ (top) and $\frac{\partial G(2,n)}{\partial a}$ (bottom) as a function of $a$.  **FIG. 4:** Plots of $S(i)$ ($q$), $S(i,j)$ ($r$), and $G(2,n)$ ($p$) (top) and the corresponding first derivatives w.r.t. $a$ (bottom) as a function of $a$. $E$ represents the entanglement measure.  **FIG.** **5:** Plot of EL as a function of $a$.  **FIG.** **6:** Plots of $I_{ij}$ (top) and $\frac{\partial I_{ij}}{\partial a}$ (bottom) as a function of $a$. IV. DISCUSSIONS {#iv.-discussions .unnumbered} =============== The MP states provide exact representations of the ground states of several spin models in low dimensions [@key-17]. The remarkable features of such states arises from the fact that complicated many body states have a simple factorized form. The simplicity in structure makes the calculation of the ground state expectation values particularly easy to perform. The spin-1 AKLT model is a well-known example of a spin model in 1d the exact ground state of which (a VBS state) has an MP representation. The AKLT model and the spin-1 Heisenberg AFM chain belong to the same universality class [@key-23]. The insight gained from the study of models in the MP formalism is expected to be of relevance in understanding the properties of more physical systems. The MP states also serve as good trial wave functions for standard spin models. The MP representation lies at the heart of the powerful density matrix renormalization group (DMRG) method and provides the basis for several interesting developments in quantum information [@key-24]. Studies of the entanglement characteristics of the MP states have begun only recently. The QPTs which occur in such states have characteristics different from those of conventional QPTs. It is thus of considerable interest to determine whether the entanglement content of MP states develops special features close to a QCP. In this paper, we consider a spin-$1$ model the exact ground state of which is of the MP form over a wide range of parameter values. The model exhibits a novel QPT in that the longitudinal and transverse correlation lengths diverge as the QCP is approached but the correlation functions vanish at the QCP. In a conventional QPT, the correlation functions have a power-law decay at the QCP. In the spin-1 model, the divergence of correlation lengths is accompanied by the excitation gap going to zero. The string order parameter, which has a non-zero expectation value in the MP state for $a>0$, vanishes at the QCP $a=0$. The distinctive signatures indicate the appearance of a new phase. We study the entanglement properties of the MP state for different values of the parameter $a$. The measures used are $S(i)$ (one-site von Neumann entropy), $S(i,j)$ (two-body entanglement) and $G(2,n)$ (GGE). All the entanglement measures have zero value at the QCP so that the ground state is disentangled at that point. As seen from the different plots, figures $(1)$, $(2)$, $(3)$, and $(4)$, the entanglement content, as measured by $E=$$S(i)$, $S(i,j)$ and $G(2,n)$, has a slow variation w.r.t. $a$ for $a>2$. At the AKLT point $a=2$, $E$ reaches its maximum value and as $a$ is reduced further, the magnitude of $E$ falls rapidly to approach zero value at $a=0$. The study of conventional QPTs shows that $E$ is maximum at a QCP [@key-3; @key-6; @key-7]. Also, $\frac{\partial E}{\partial a}$ diverges as the QCP is approached. The EL, $\xi_{E}$, as calculated from $S(i,j)$ and $G(2,n)$ for large $n$, also diverges with $\xi_{E}=\frac{\xi_{C}}{2}$ where $\xi_{C}$ is the usual correlation length. In the case of the spin-1 model under consideration, $\frac{\partial E}{\partial a}$ diverges as $a\rightarrow0$ when $E=S(i)$ and $S(i,j)$. The EL, $\xi_{E}$, as calculated from $S(i,j)$ in the large $n$ limit, also diverges with $\xi_{E}=\frac{\xi_{C}}{2}$. One now has the interesting situation that the entanglement content of the MP state decreases as $a\rightarrow0$ but the entanglement is spread over larger distances. The derivative $\frac{\partial G(2,n)}{\partial a}$, however, does not diverge as $a\rightarrow0$ but attains a maximum value at the QCP. The results can be understood by noting that in all the three cases, $E=S(i),\: S(i,j)$ and $G(2,n)$, the reduced density matrices $\rho(i)$ and $\rho(i,j)$ smoothly approach the forms associated with pure states as the parameter $a$ tends to zero. The matrix elements of the reduced density matrices do not develop non-analyticities in the parameter region of interest. Thus, the energy density, calculated from the reduced density matrix $\rho(i,j)$, does not develop a non-analyticity at the QCP. The derivative $\frac{\partial G(2,n)}{\partial a}$ depends upon the first derivatives of the matrix elements of $\rho(i,j)$. Since the latter is analytic for $a\geq0$, $\frac{\partial G(2,n)}{\partial a}$ does not diverge in the whole parameter regime including the point $a=0$. In the cases of the entanglement measures $E=S(i)$ and $S(i,j)$, the derivative $\frac{\partial E}{\partial a}$ diverges as $a\rightarrow0$ due to the divergence of $log_{2}\, a$ in the same limit. The divergence is thus due to the special form of the von Neumann entropy and occurs for $n\geq1$. A recent work [@key-25] provides another example of such a singularity. Though $\frac{\partial G(2,n)}{\partial a}$ does not diverge or become discontinuous at the QCP $a=0$, it attains its maximum value at the point. The first derivative of the string order parameter w.r.t. $a$ also attains its maximum value at $a=0$ though the order parameter itself vanishes at the point. Figure $(4)$ shows that amongst the three entanglement measures $E=$$S(i)$, $S(i,j)$ and $G(2,n)$, used in this study to obtain a quantitative estimate of the entanglement content of the MP ground state, the measure $S(i,j)$ yields the largest value of the entanglement at different values of $a$. The difference in the singular behaviour of the measures $S(i)$ and the mutual information entropy $I_{ij}$ as $a\rightarrow0$ indicates that multipartite quantum correlations are involved in the QPT. 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ArXiv

--- abstract: 'We show how the recently proposed Taylor-Lagrange renormalization scheme can lead to extensions of singular distributions which are reminiscent of the Pauli-Villars subtraction. However, at variance with the Pauli-Villars regularization scheme, no infinite mass limit is performed in this scheme. As an illustration of this mechanism, we consider the calculation of the self-energy in second order perturbation theory in the Yukawa model, within the covariant formulation of light-front dynamics. We show in particular how rotational invariance is preserved in this scheme.' author: - 'P. Grangé' - 'J.-F. Mathiot' - 'B. Mutet' - 'E. Werner' title: | Taylor-Lagrange renormalization scheme,\ Pauli-Villars subtraction and Light-Front dynamics --- Introduction ============ The understanding of the structure of bound state systems in nuclear and particle physics requires the development of a relativistic nonperturbative framework. For obvious practical reasons, any calculation of this type relies on approximations, and one thus needs a systematic strategy in order to improve the approximations which are made, in complete analogy with perturbation theory. Light-front dynamics (LFD) is a very powerful tool to calculate bound state properties. It is one of the three forms of dynamics proposed in 1949 by Dirac [@dirac]. In order to perform systematic calculations of physical observables on a large scale, one should however be able to solve three important problems. The first one is the explicit violation of rotational invariance by the choice of a given light-front plane. The control of this violation is important in order to define unambiguously all physical observables. This can be done simply using the covariant formulation of light-front dynamics (CLFD) [@karmanov; @cdkm]. The second one is the possible appearance of uncanceled divergences when the Fock space is truncated, in any approximate nonperturbative calculation. One thus should make sure that no divergences are left uncanceled. This is enforced using the Fock sector dependent renormalization scheme [@kms_08]. Finally, one should develop a regularization/renormalization scheme which preserves all symmetries, and which is well adapted to extended numerical calculations. We have advocated in a previous study [@grange] the use of the Taylor-Lagrange renormalization scheme (TLRS) [@GW]. We have shown in particular that this scheme is very well adapted to any calculation in LFD. It is systematic, can treat singularities of any type on the same footing, and moreover does not require to perform any infinite scale limit. Many other regularization methods are available in the literature. Let us mention here the most important ones. [*i) The cut-off method*]{}. This is a simple, but to some extent brutal, way to regularize divergent amplitudes. It however violates gauge invariance and, in LFD, rotational invariance [@kms_04]. It should thus be avoided in any realistic calculation. [*ii) The Pauli-Villars (PV) subtraction method*]{}. In LFD, the PV regularization scheme amounts to extend the Fock space to include PV fields with negative norm [@kms_08]. While this method is attractive for its simplicity and immediate use, it has some serious disadvantages in any systematic calculations. The number of PV fields may be large if singularities are of high order, as in effective field theories. This may imply a large number of PV components which are not easy to implement in systematic calculations. Moreover, the limit of large PV masses has to be performed numerically. This may also be rather delicate to achieve in large systematic numerical calculations. [*iii) The dimensional regularization method*]{}. This method is largely used in perturbation theory in the covariant Feynman approach. In LFD however, it has never been used since one would have to reformulate LFD in arbitrary D dimensions. This is also not in the spirit of LFD which deals only with physical degrees of freedom. [*iiv) The Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) method.*]{} In this method each Feynman graph is treated separately. The contribution is made finite by subtracting as many terms as necessary from the Taylor expansion at zero external momenta of the integrand. All Feynman integrals being convergent, no intermediate regularization is required, thereby showing that all $n$-point functions are effectively regularization independent. They only depend on the renormalisation conditions. However there are some complications for zero masses and despite its theoretical importance, the BPHZ scheme is not easy to deal with in practical calculations. The link between the TLRS and BPHZ schemes resides in the use of specific test functions equal to their Taylor remainder of any order.\ We shall exhibit in this study the relationship between TLRS and the PV-type regularization procedure. We shall illustrate this relationship in the simple, but instructive, calculation of the various components of the self-energy of a fermion in second order perturbation theory in the Yukawa model, paying particular attention to the restoration of rotational invariance. The plan of the article is the following. We recall in Sec. \[TLRS\] the main features of the TLRS renormalization scheme. We then calculate in Sec. \[CLFD\] the self-energy in second order perturbation theory in the Yukawa model. We draw our conclusions in Sec. \[conc\]. The Taylor-Lagrange renormalization scheme and Pauli-Villars subtraction {#TLRS} ======================================================================== It is a common lore [@collins] that any field $\phi(x)$ (taken here as a scalar field for simplicity) should be considered as an operator-valued distribution. This means that it should be defined by its application on test functions, denoted by $\rho$, with well identified mathematical properties. In flat space, the physical field ${\varphi}(x)$ is thus given by [@grange] $$\label{conv} \varphi(x) \equiv \int d^Dy \phi(y) \rho(x-y)\ ,$$ in $D$ dimensions. If we denote by $f$ the Fourier transform of the test function, we can further write ${\varphi}(x)$ in terms of creation and destruction operators, leading to $$\!\varphi (x)\!=\!\!\int\!\frac{d^{D-1}{\bf p}}{(2\pi)^{{\bf (D-1)}}}\frac{f(\varepsilon_p^2,{\bf p}^2)}{2\varepsilon_p} \left[a^+_{\bf p} e^{i{\bf p.x}}+a_{\bf p}e^{-i{\bf p.x}}\right],$$ with $\varepsilon^2_p = {\bf p}^2+m^2$. From this decomposition, it is apparent that test functions should be attached to each fermion and boson field. Each propagator being the contraction of two fields should be proportional to $f^2$. In order to have a dimensionless argument for $f$, we shall introduce an arbitrary scale $\Lambda$ to “measure” all momenta. $\Lambda$ can be any of the masses of the constituents. To deal with massless theories, we shall take some arbitrary value. The final expression of any amplitude should be independent of $\Lambda$. In CLFD, the test function is thus a function of $\frac{{\bf p}^2}{\Lambda^2}$ only. As recalled in [@grange], the test function $f$ should have two important properties: [*i)*]{} The physical field $\varphi(x)$ should be independent of the choice of the test function. This later should therefore be chosen among the partitions of unity (PU). It is a function of finite support which is $1$ everywhere except at the boundaries. This choice is also necessary in order to satisfy Poincaré invariance since, if $f$ is a PU, any power of $f$, $f^n$, is also a PU. In the limit where the test function goes to $1$ over the whole space, we then have $f^n \to f$ and Poincaré invariance is recovered. [*ii)*]{} In order to be able to treat in a generic way singular distributions of any type, the test function is chosen as a super regular test function (SRTF). It is a function of finite extension - or finite support - vanishing with all its derivatives at its boundaries, either in the ultraviolet (UV) or infrared (IR) domain. Any physical amplitude is thus written in a schematic way like $$\label{ampli1} {\cal A}=\int_0^\infty dX\ T(X)\ f(X)\ ,$$ for a one-dimensional distribution. In this form, the amplitude ${\cal A}$ does not differ from the calculation using a cut-off procedure. In the UV domain for example, the cut-off, denoted by $H$, would correspond to the support of $f$, with $f(X \ge H)=0$. In order to go beyond the use of a naive cut-off, we should investigate the scaling properties inherent to the limit $X \to \infty$ since in this limit $\eta^2 X$ goes also to $\infty$, where $\eta^2$ is an arbitrary dimensionless scale. To do that, we shall consider as an example a distribution $T(X)$ leading to an $X$ integral diverging like $\log(H)$ in the absence of $f(X)$ and use the Lagrange formula written in the following form, in the UV domain : $$\label{la3a} f(aX)=-X\int_a^\infty\frac{dt}{t} \partial_X \left[f(Xt)\right] \ ,$$ for an arbitrary intrinsic scale $a$ which can be chosen positive if $T$ has no other singularity at finite $X$ . This formula is an identity for any function $f$ which is a SRTF. In order to introduce the arbitrary scale $\eta^2$, we shall consider a running boundary condition, i.e. a boundary condition which depends on the given variable $X$ using $$\label{running} H(X)\equiv \eta^2 X g(X)\ ,$$ up to an additive arbitrary finite constant irrelevant in the UV domain. Note that the support of the test function is the same in the right- and left-hand sides of Eq. (\[la3a\]). This implies that $Xt\le H(X)$ for any argument $Xt$ of the test function (see [@grange] for more details). The variable $X$ on which the running condition (\[running\]) is applied should not be linked to any intrinsic scale. This is necessary in order to make sure that the limit $X \to \infty$ is properly done, i.e. that $X$ should be larger than any other physical scale present in the amplitude. Once this is done, the test function depends generally on $aX$, where $a$ is a priori a function of the kinematical variables of the system under consideration, and one should consider the Lagrange formula in the form (\[la3a\]). In order to extend the test function to $1$ over the whole space, we shall consider a set of function $g(X)$, denoted by $g_\alpha(X)$, where by construction $\alpha$ is a real positive number less than $1$. A typical example of $g_\alpha(X)$ is given in [@grange], where it is shown that in the limit $\alpha \to 1^-$, with $\eta^2>1$, the running support of the PU test function stretches then over the whole integration domain, and $f \to 1$. In this limit $g_\alpha(X) \to 1$. After integration by parts, the amplitude ${\cal A}$ writes $$\label{afind} {\cal A}=\int_0^\infty dX \ \partial_X \left[ X T(X)\right] \int_a^{\eta^2 g_\alpha(X)} \frac{dt}{t} f(Xt)\ .$$ In the limit $\alpha \to 1^-$, the requirements are such that $g_\alpha(X) \to 1$ and $f \to 1$. One can thus define the extension in the UV domain, denoted by $\widetilde T^>$, of the singular distribution $T$ by $$\label{Ato} {\cal A}\equiv \int_0^\infty dX \ \widetilde T^>(X)\ ,$$ with[^1] $$\widetilde T^>(X)\equiv \partial_X \left[ X T(X)\right] \mbox{Log}\left[\frac{\eta^2}{a}\right] \ ,$$ where the derivative should be understood in the sense of distributions [@grange]. It depends logarithmically on the arbitrary scale $\eta^2$, with $\eta^2>1$. The amplitude (\[Ato\]) is now completely finite. Note that we do not need the explicit form of the test function in the derivation of the extended distribution $\widetilde T^>(X)$. We only rely on its mathematical properties and on the running construction of the boundary conditions. The running boundaries of the test functions are essential for the preservation of symmetries which would otherwise be destroyed with the usual cut-off test functions. Qualitatively, the reason for this property is the following: the cut-off functions are equal to $1$ up to a point $x_c$ and fall then down to zero over a finite interval with some shape which does not change when $x_c$ is sent to infinity. On the contrary, with running boundaries, the width of the region where the test function falls from $1$ to zero increases proportionally to $x_c$ when $x_c$ goes to infinity, implying an infinitesimal drop-off of the test functions in the asymptotic limit. We call this behavior an ultrasoft cut-off. The extension of singular distributions in the IR domain can be done similarly [@grange]. For an homogeneous distribution in one dimension, with $T[X/t]=t^{k+1} T(X)$, the extension of the distribution $T$ in the IR domain writes [^2] $$\label{TIR2} \widetilde T^<(X)=(-1)^{k}\partial_{X}^{k+1} \left[ \frac{X^{k+1}}{k!} T(X) \mbox{Log} (\tilde \eta X)\right] \ .$$ The extension $\widetilde T^<(X)$ differs from the original distribution $T(X)$ only at the singularity at $X=0$. The amplitude (\[afind\]) can also be transformed alternatively in order to exhibit a PV-type subtraction. Using the Lagrange formula in the form $$\label{lat} f\left[a X \right]=-\int_a^\infty dt \ \partial_t f\left[X t\right] \ ,$$ we can rewrite the physical amplitude $\cal A$ in the following form, after the change of variable $Z=Xt$ and in the limit $\alpha \to 1^-$ $${\cal A}=-\int_0^\infty dZ \int_a^{\eta^2} dt \ \partial_t \ \left[ \frac{1}{t} \ T\left( \frac{Z}{t} \right) \right] \ .$$ With a typical distribution $T(X)=\frac{1}{X+a}$ with an intrinsic scale $a$, one thus gets immediately $$\widetilde T^>(X) = \frac{1}{X+a} - \frac{1}{X+\eta^2}\ .$$ We recover here a PV-type subtraction, with a scale $\eta^2$. This scale is completely arbitrary, with $\eta^2>1$ and not compulsory infinitely large as required for the PV masses. This extension of $T(X)$ leads to a well defined amplitude (\[Ato\]). Application to the calculation of the self-energy in light-front dynamics {#CLFD} ========================================================================= CLFD was first proposed in Ref. [@karmanov] and detailed in the case of few-body systems in Ref. [@cdkm]. Any physical system of momentum $p$ is described in LFD by a state vector projected onto the plan $t^+ = t + \frac{z}{c}$. In CLFD, the state vector is defined on the general plane determined by the equation $\sigma = \omega {\makebox[0.08cm]{$\cdot$}}x$, with $\omega ^2=0$. The covariance of our approach is due to the invariance of the light-front plane equation. This implies that $\omega$ is not the same in any reference frame, but varies according to Lorentz transformations, like the coordinate $x$. It is not the case in the standard formulation of LFD where $\omega$ is fixed to $\omega=(1,0,0,-1)$ in any reference frame. The evolution of the system is thus defined in terms of the light-front time $\sigma$. We shall consider in the following the simple case of the self-energy of a fermion in the Yukawa model, in second order perturbation theory. From a practical point of view, any amplitude is calculated using the equivalence of Feynman rules, as detailed in Ref. [@cdkm]. In second order of perturbation theory, the self-energy $\Sigma(p)$ (up to a conventional minus sign) is determined by the sum of the two diagrams shown in Fig. \[fig0a\], $$\label{selfenLFD} \Sigma(p)=\Sigma_{2b}(p)+\Sigma_{fc}(p)\ .$$ They correspond to the two-body contribution and the fermion contact term, respectively. These diagrams correspond to time ordered diagrams in the light-front time $\sigma$. Analytical expressions for the corresponding amplitudes read $$\begin{gathered} \Sigma_{2b}(p)=-\frac{g^2}{(2\pi)^3}\int \delta^{(4)}(p+\omega\tau-k_1-k_2) \frac{d\tau}{\tau} \\ ({\not\!}{k_1}+m)\theta(\omega {\makebox[0.08cm]{$\cdot$}}k_1)\delta(k_1^2-m^2) f^2\left[\frac{{\bf k_1}^2}{\Lambda^2} \right] d^4k_1 \\ \theta(\omega{\makebox[0.08cm]{$\cdot$}}k_2)\delta(k_2^2-\mu^2) f^2\left[\frac{{\bf k_2}^2}{\Lambda^2} \right] d^4k_2 \ , \label{2b}\end{gathered}$$ $$\label{2c} \Sigma_{fc}(p)=\frac{g^2}{(2\pi)^3} {\not\!}{\omega} \int \frac{\theta(\omega{\makebox[0.08cm]{$\cdot$}}k_2)\delta(k_2^2-\mu^2)}{2\omega{\makebox[0.08cm]{$\cdot$}}(p-k_2)} \\f^2\left[\frac{{\bf k_2}^2}{\Lambda^2} \right]d^4k_2\ ,$$ where $g$ is the coupling constant of the fermion-boson interaction, and $m$ and $\mu$ are the fermion and boson masses, respectively. Since the test functions are PU, we shall identify in the following $f^2$ with $f$. All the particles are on their mass shell in LFD, but off energy shell, so that the momentum $\tau$ represents the off-shell energy of the intermediate fermion-boson state [@cdkm]. The momentum $p$ corresponds to $p=p_1-\omega \tau_1$ where $\tau_1$ is proportional to the off-shell energy of the initial, or final, fermion, and $p_1$ is the on-shell four-momentum of the fermion. Note that in LFD, the self-energy may depend a priori on the position of the light-front characterized by $\omega$. In an exact calculation, or in perturbation theory, we should check explicitly that this dependence disappears in order to recover the well known results in the 4D Feynman approach. The self-energy can therefore be decomposed in the most general spin structures according to \[sigdec\] $$\begin{aligned} \Sigma_{2b}(p)\!\!\!&=&\!\!\!g^2\left[{\cal A}(M^2)+{\cal B}(M^2)\frac{{{\not\!}p}}{M}+{\cal C}(M^2)\frac{M {{\not\!}\omega}}{\omega{\makebox[0.08cm]{$\cdot$}}p}\right] \label{Sigdecomp2b} \ ,\nonumber \\ \\ \Sigma_{fc}(p)\!\!\!&=&\!\!\!g^2C_{fc}\frac{M{{\not\!}\omega}}{\omega{\makebox[0.08cm]{$\cdot$}}p}\ , \label{Sigdecompfc}\end{aligned}$$ where the coefficients ${\cal A}$, ${\cal B}$, and ${\cal C}$ are scalar functions which depend on $p^2$ only. They are independent of $\omega$. We denote $p^2$ by $M^2$. The scale $M$ in (\[sigdec\]) is just introduced for convenience in order to have the same dimension for all the coefficients ${\cal A},{\cal B},{\cal C}$ and $C_{fc}$. These coefficients can easily be calculated according to \[eq4sen\] $$\begin{aligned} g^2{\cal A}(M^2)&=&\frac{1}{4}\mbox{Tr}[\Sigma(p)]\ , \\ g^2{\cal B}(M^2)&=&\frac{M}{4\omega{\makebox[0.08cm]{$\cdot$}}p} \mbox{Tr}[\Sigma(p){\not\!}{\omega }]\ , \\ g^2{\cal C}(M^2)&=&\frac{1}{4M}\mbox{Tr}\left[\Sigma(p)\left({\not\!}{p} -\frac{M^2{\not\!}{\omega}}{\omega{\makebox[0.08cm]{$\cdot$}}p}\right)\right]\ .\end{aligned}$$ The coefficient $C_{fc}$ is a constant. It can be extracted directly from Eq. (\[2c\]). In order to transform (\[2b\]) using TLRS, we shall use the Lagrange formula (\[la3a\]) in a slightly different form $$\label{latra} f\left[a^2X^2 \right]=-\int_a^\infty dt \ \partial_t f\left[X^2 t^2\right] \ ,$$ where the scale $a$ should be identified later on. With the change of variable $\bar k_1 = k_1 s t$, $\bar k_2 = k_2 s t$, and $ \bar \tau = \tau s t$, we have $$\begin{gathered} \label{s2b} \Sigma_{2b}(p)=-\frac{g^2}{(2\pi)^3} \int d^4\bar k_1 \int d^4 \bar k_2 \int \frac{d \bar \tau}{ \bar \tau} \int_a^\infty dt \ \partial_t \int_a^\infty ds \ \partial_s \frac{1}{st} \ \delta^{(4)}(p s t +\omega \bar \tau-\bar k_1-\bar k_2) \\ \theta(\omega{\makebox[0.08cm]{$\cdot$}}\bar k_2)\delta(\bar k_2^2-\mu^2 s t) ({\not\!}\bar k_1 +m s t)\ \theta(\omega{\makebox[0.08cm]{$\cdot$}}\bar k_1)\delta(\bar k_1^2-m^2 s t )\ f\left[\frac{{\bf \bar k_2}^2}{s^2 a^2\Lambda^2} \right] f\left[\frac{{\bf \bar k_1}^2}{t^2 a^2\Lambda^2} \right] \ .\end{gathered}$$ The four-momentum conservation law, and the on-mass shell conditions in Eq. (\[s2b\]) are equivalent to the original ones in (\[2b\]) after the transformation $$\label{mst} m,\mu,p \to mst, \mu st,pst \ .$$ Using the kinematical variables defined by $$\label{lf} x=\frac{\omega {\makebox[0.08cm]{$\cdot$}}\bar k_2}{\omega {\makebox[0.08cm]{$\cdot$}}\ p s t} \ \ , \ \ R=\bar k_2-x p s t \ \ \mbox{with} \ \ R=(R^0, {\bf R}_\perp, R^\parallel) \ ,$$ we have, in the reference frame where ${\bf p} = 0$, ${\bf R}_\perp={\bf k}_\perp$. Since $\omega {\makebox[0.08cm]{$\cdot$}}R=0$, we also have $R^0=R^\parallel$. The momentum ${\bf k}_\perp$ is the perpendicular component of the four-momentum $\bar k_2$ with respect to the position $\omega$ of the light-front. With the transformations (\[mst\]), we have thus, in the limit of large momenta (UV regime) and using the kinematics detailed in Appendix B.1 of [@grange] $$\begin{aligned} \label{k2inf} {\bf \bar k}_2^2 &\approx& \frac{{\bf k}_\perp^4}{4x^2M^2 s^2 t^2}\ , \\ \label{k1inf} {\bf \bar k}_1^2 &\approx &\frac{{\bf k}_\perp^4}{4(1-x)^2M^2 s^2 t^2}\ .\end{aligned}$$ To simplify the notation, we shall define the dimensional scale $$\alpha(x)=\frac{m^2x+\mu^2(1-x)-M^2x(1-x)}{m^2} \ .$$ With the change (\[mst\]), the coefficients ${\cal A}, {\cal B}$ and ${\cal C}$ thus write [@kms_04] \[ABC2\] $$\begin{aligned} \label{Ap2} {\cal A}(M^2)&=&-\frac{m g^2}{16\pi^2}\int_0^{\infty}d{\bf k}_{\perp}^2\int_0^1 dx\int_a^\infty \! \! \!dt \ \partial_t \!\int_a^\infty \!\! \! \!ds \ \partial_s \,\frac{1}{{\bf k}_{\perp}^2+m^2 s^2t^2\alpha(x)} f[] f[],\nonumber \\ \\ \label{Bp2} {\cal B}(M^2)&=&-\frac{M g^2}{16\pi^2}\int_0^{\infty}d{\bf k}_{\perp}^2\int_0^1 dx\int_a^\infty \! \! \!dt \ \partial_t \!\int_a^\infty \!\! \! \!ds \ \partial_s\,\frac{(1-x)}{{\bf k}_{\perp}^2+m^2 s^2t^2\alpha(x)}f[] f[]\ ,\nonumber \\ \\ \label{Cp2} {\cal C}(M^2)&=&-\frac{g^2}{32\pi^2M}\int_0^{\infty}d{\bf k}_{\perp}^2\int_0^1 dx\int_a^\infty \! \! \!dt \ \partial_t \!\int_a^\infty \!\! \! \!ds \ \partial_s \ \frac{1}{s^2t^2}\,\frac{k_{\perp}^2+[m^2-M^2(1-x)^2]s^2t^2}{(1-x)[{\bf k}_{\perp}^2+m^2 s^2t^2\alpha(x)]} f[] f[] , \nonumber \\\end{aligned}$$ where the notation $f[] f[]$ stands for $$\begin{gathered} f[] f[]=\ f\left[\frac{{\bf k}_\perp^4}{4x^2M^2 a^2 \Lambda^2 s^4 t^2} \right] \\ f\left[\frac{{\bf k}_\perp^4}{4(1-x)^2M^2 a^2 \Lambda^2 s^2 t^4}\right]\ .\end{gathered}$$ The additional factors $st$ and $\frac{1}{st}$ in (\[Bp2\]) and (\[Cp2\]) respectively, as compared to (\[s2b\]), originate from the momentum dependence of the self-energy in the decomposition (\[Sigdecomp2b\]) with the replacem�ent $p \to p st$. In order to calculate the coefficient ${\cal A}$, we can separate the integration over $x$ in two domains, for $x<\frac{1}{2}$ and $x \ge \frac{1}{2}$, as already done in [@grange]. In the first domain, we can first eliminate the integral in $s$ by redefining $\tilde {\bf k}_\perp = \frac{ {\bf k}_\perp}{s}$, and using the Lagrange formula (\[latra\]) on $s$ backward. It thus reads $$\begin{gathered} \label{afinal} {\cal A}(M^2)=\frac{m g^2}{16\pi^2} \int_0^{\infty}d{\bf \tilde k}_{\perp}^2 \int_0^\frac{1}{2} dx \\ \int_a^\infty \! \! \!dt \ \partial_t \frac{1}{{\bf \tilde k}_{\perp}^2+m^2 t^2 \alpha(x)}\ f\left[\frac{{\bf \tilde k}_\perp^4}{4x^2M^2 a^2 \Lambda^2 t^2} \right] \ ,\end{gathered}$$ where the second test function has been put to $1$ since its argument is always smaller than the one retained in (\[afinal\]). With the change of variable $Z=\frac{{\bf \tilde k}_\perp^2}{2xM\Lambda t}$, we get $$\begin{gathered} {\cal A}(M^2)=\frac{m g^2}{16\pi^2} \int_0^{\infty}dZ \int_0^\frac{1}{2} dx\int_a^\infty \! \! \!dt \ \partial_t \\ \frac{2xM\Lambda}{m^2}\frac{1}{\frac{2xM\Lambda}{m^2} Z+ t \alpha(x)}\ f\left[\frac{Z^2}{a^2}\right] \ .\end{gathered}$$ By eliminating the intrinsic scale $\alpha(x)$ with the identification $a=\alpha(x)$, and with $Z=a Y$, we get $$\begin{gathered} {\cal A}(M^2)=\frac{m g^2}{16\pi^2} \int_0^{\infty}dY \int_0^\frac{1}{2} dx \int_{\alpha(x)}^\infty \! \! \!dt \ \partial_t \\ \frac{2xM\Lambda}{m^2}\frac{1}{\frac{2xM\Lambda}{m^2} Y+ t }\ f[Y^2] \ .\end{gathered}$$ The integration over $t$ gives simply, with $f\to 1$ and the upper limit fixed by the running condition $Y^2t^2 \le H(Y^2)$, i.e. $t \le \eta$ $$\begin{gathered} {\cal A}(M^2)=\frac{m g^2}{16\pi^2} \int_0^{\infty}dY \int_0^\frac{1}{2} dx \\ \frac{2xM\Lambda}{m^2}\left[ \frac{1}{\frac{2xM\Lambda}{m^2}Y+\eta} - \frac{1}{\frac{2xM\Lambda}{m^2}Y+\alpha(x)}\right]\ .\end{gathered}$$ The calculation in the interval $\frac{1}{2} \le x < 1$ and a first integration over $t$ instead of $s$ gives the same integrand, so that we finally get $$\begin{aligned} {\cal A}(M^2)&=&\frac{m g^2}{16\pi^2} \int_0^{\infty}dY \int_0^1 dx \frac{2xM\Lambda}{m^2} \left[ \frac{1}{\frac{2xM\Lambda}{m^2}Y+\eta} - \frac{1}{\frac{2xM\Lambda}{m^2}Y+\alpha(x)}\right]\nonumber \\ &=&-\frac{m g^2}{16\pi^2} \log \eta+\frac{m g^2}{16\pi^2} \int_0^1 dx \log \left[ \frac{m^2x+\mu^2(1-x)-M^2x(1-x)}{m^2}\right] \ .\end{aligned}$$ This result is the same as the one already given in [@grange]. The calculation of ${\cal B}$ proceeds in exactly the same manner, with just an extra factor $(1-x)$ in the integrand. We shall now concentrate on the calculation of the coefficient ${\cal C}$. Using (\[Cp2\]), we can decompose ${\cal C}$ in two parts $$\begin{aligned} \label{C1} {\cal C}(M^2)&=&-\frac{g^2}{32\pi^2 M}\int_0^{\infty}d{\bf k}_{\perp}^2 \int_0^{1} dx\int_a^\infty \! \! \!dt \ \partial_t \!\int_a^\infty \!\! \! \!ds \ \partial_s \,\frac{m^2 \partial_x \alpha(x)}{{\bf k}_{\perp}^2+m^2 s^2t^2\alpha(x)} f[] f[] \nonumber \\ &&-\frac{g^2}{32\pi^2M}\int_0^{\infty}d{\bf k}_{\perp}^2 \int_0^{1} \frac{dx}{1-x}\int_a^\infty \! \! \!dt \ \partial_t \!\int_a^\infty \!\! \! \!ds \ \partial_s \ \frac{1}{s^2t^2} f[] f[] \\ &\equiv&{\cal C}_1 + {\cal C}_2 \ . \nonumber\end{aligned}$$ Let us first calculate ${\cal C}_1$. With the change of variable $\alpha(x) s^2 t^2 =u$, we have $$\begin{gathered} {\cal C}_1=-\frac{g^2}{32\pi^2M}\int_a^\infty \! \! \!dt \ \partial_t \!\int_a^\infty \!\! \! \!ds \ \partial_s \ \frac{1}{s^2t^2} \\ \int_0^{\infty}d{\bf k}_{\perp}^2\int_{\frac{\mu^2s^2t^2}{m^2}}^{s^2t^2} \frac{m^2 du}{ {\bf k}_{\perp}^2+m^2 u} f[] f[] \ .\end{gathered}$$ The test functions provide the convergence of the integral in ${\bf k}_\perp^2$, as well as an upper limit, $\eta$, in the $t$ and $s$ integrations from the running condition on the test functions. The order of integrations can be changed at will and we remark that it is legitimate at this stage to set the test functions to $1$ since, after integration over $u$ the integral in ${\bf k}_\perp^2$ is henceforth finite. We thus get $$\begin{gathered} {\cal C}_1=-\frac{g^2}{32\pi^2M}\int_a^\eta \! \! \!dt \ \partial_t \!\int_a^\eta \!\! \! \!ds \ \partial_s \ \frac{1}{s^2t^2} \\ \int_0^{\infty}d{\bf k}_{\perp}^2\left[ \log\left[\frac{{\bf k}_{\perp}^2}{m^2}+s^2t^2\right]-\log\left[\frac{{\bf k}_{\perp}^2}{m^2}+\frac{\mu^2}{m^2}s^2t^2\right] \right]\ .\end{gathered}$$ After a change of variable $X=\frac{{\bf k}_{\perp}^2}{m^2}+s^2t^2$ in the first term, and $X=\frac{{\bf k}_{\perp}^2}{m�^2}+\frac{\mu^2}{m^2}s^2t^2$ in the second, we get $${\cal C}_1=-\frac{g^2 m^2}{32\pi^2M}\int_a^\eta \! \! \!dt \ \partial_t \!\int_a^\eta \!\! \! \!ds \ \partial_s \ \frac{1}{s^2t^2} \int_{\frac{\mu^2s^2t^2}{m^2}}^{s^2t^2} dX \log X \ .$$ which finally gives, with $X=Y s^2t^2$ $${\cal C}_1=-\frac{g^2 m^2}{32\pi^2M}\int_a^\eta \! \! \!dt \ \partial_t \!\int_a^\eta \!\! \! \!ds \ \partial_s \int_{\frac{\mu^2}{m^2}}^{1}dY \log [Ys^2t^2] \ .$$ The final integrations over $s$ and $t$ lead to a double PV-type subtraction $$\begin{gathered} {\cal C}_1=-\frac{g^2 m^2}{32\pi^2M} \int_{\frac{\mu^2}{m^2}}^{1}dY \left[\log [Y] - \log \left[Y\frac{\eta^2}{a^2}\right] \right. \\ \left. - \log \left[Y\frac{\eta^2}{a^2}\right]+ \log \left[Y\frac{\eta^4}{a^4}\right]\right] \equiv 0\ .\end{gathered}$$ This result is very similar to the calculation of the coefficient ${\cal C}$ in [@kms_04] where it was shown that it is zero with one PV fermion and one PV boson subtraction. These subtractions are here provided by the integration over the variables $s$ and $t$. Note that we indeed need both subtractions to get the final result. This result relies on the property that the numerator of the integrand in ${\cal C}_1$ in (\[C1\]) is just the derivative of $\alpha(x)$. This is not the case for any other components. Because of this peculiarity, the calculation of ${\cal C}_1$ proposed in [@grange] (coefficient ${\cal J}_2$ in Appendix B.3) is only correct for this case since otherwise it would lead to a null result independently of the form of the singular distribution. In the calculation of [@grange], the identification of the product of the two test functions with only one symmetric in the change $x \to 1-x$ is legitimate since under this change only the boundaries of the $u$-integral are interchanged thereby giving zero for ${\cal J}_2$. We turn now to the calculation of the coefficient ${\cal C}_2$. Since the integrand has no intrinsic scale, it is natural to calculate ${\cal C}_2$ using the IR extension given in (\[TIR2\]) for an homogeneous distribution. With the change of variable $\frac{{\bf k}_\perp^2}{s^2t^2}=\frac{1}{Y}$, we can integrate over $s$ and $t$ using the Lagrange formula (\[la3a\]) back. We get $$\begin{gathered} {\cal C}_2(M^2)=-\frac{g^2}{32\pi^2M} \int_0^{\infty}\frac{dY}{Y^2} \int_0^{1} \frac{dx}{1-x} \\ f\left[ \frac{1}{4x^2Y^2M^2\Lambda^2}\right] f\left[ \frac{1}{4(1-x)^2Y^2M^2\Lambda^2}\right]\ .\end{gathered}$$ To keep the symmetry $x \to (1-x)$ in the integrand, we shall rewrite ${\cal C}_2$ as $$\begin{gathered} {\cal C}_2(M^2)=-\frac{g^2}{32\pi^2M} \int_0^{\infty}\frac{dY}{Y^2} \int_0^{\frac{1}{2}} \frac{dx}{x(1-x)} \\ f\left[ \frac{1}{4x^2Y^2M^2\Lambda^2}\right] f\left[ \frac{1}{4(1-x)^2Y^2M^2\Lambda^2}\right]\ .\end{gathered}$$ In the domain $x \le 1/2$, the first test function only matters, i.e. is different from $1$ within the relevant boundary. We have therefore, with the change of variable $2YxM\Lambda = u$ $${\cal C}_2(M^2)=-\frac{g^2 \Lambda}{16\pi^2}\int_0^{\frac{1}{2}} \frac{dx}{1-x} \int_0^{\infty}\frac{du}{u^2} f\left[ \frac{1}{u^2}\right]\ .$$ The test function just provides the extension of the distribution $\frac{1}{u^2}$ at $u=0$ from (\[TIR2\]), which is the pseudo-function of $1/u^2$ [@grange]. We finally have $${\cal C}_2(M^2)=-\frac{g^2 \Lambda}{16\pi^2} \log 2 \int_0^{\infty}du \ \mbox{Pf}\left[\frac{1}{u^2} \right] \equiv 0\ .$$ We find that the coefficient $C(M^2)$ is identically zero, as it should be. The calculation of the contribution from the contact interaction can be done very similarly to the calculation of $C_2$. We have, from (\[2c\]) and (\[Sigdecompfc\]) [@kms_04] $$C_{fc}=\frac{g^2}{32\pi^2M} \int_0^{\infty}d {\bf k}_{\perp}^2 \int_0^{\infty} \frac{dx}{x(1-x)} \\ f\left[\frac{{\bf k}_\perp^4}{4x^2M^2 s^4 t^2} \right] \ .$$ The integration over $x$ can be decomposed in two parts, for $x\le \frac{1}{2}$ and for $x > \frac{1}{2}$. In the first one, with ${\bf k}_{\perp}^2 =\frac{1}{Y}$, we get $C_{fc}=-{\cal C}_2(M^2)=0$. The second one is identically zero with a principal value prescription to calculate the integral at x=1. This insures that the self-energy $\Sigma(p)$ is indeed independent of the arbitrary position of the light-front. Conclusions {#conc} =========== We have shown in this study that the recently proposed TLRS [@grange] leads naturally to a regularization of any amplitude very similar to a PV-type subtraction. However, contrary to the original PV regularization method, the TLRS does not necessitate to perform any infinite mass scale limit. The application of our formalism to the calculation of the self-energy in second order perturbation theory in the Yukawa model, using LFD, is very instructive. It is very similar to the standard calculation using PV fields with a negative norm [@kms_04]. The coefficients ${\cal A}$ and ${\cal B}$ in the spin decomposition (\[Sigdecomp2b\]) depends in TLRS on an arbitrarily dimensionless scale $\eta$, while it depends on $\frac{\Lambda_{PV}^2}{m^2}$ in the PV method, where $\Lambda_{PV}$ is the PV boson mass. This mass must be taken very large compared to any physical mass scale present in the amplitude. In both methods, the regularization of the amplitude is achieved by using a single PV-type subtraction. On the other hand, the coefficient ${\cal C}$ is identically zero, as required by rotational invariance. It is regularized in both methods by using two PV-type subtractions, involving both the fermion and boson propagators. This close connection between TLRS and PV-type regularization is of particular interest in further nonperturbative calculations in LFD [@kms_08]. In the TLRS scheme indeed, there is no need for additional non physical components in the state vector describing any physical system. Moreover, there is no large mass scale limit to perform numerically. This may render possible large scale calculations of nonperturbative relativistic bound state systems in LFD. [10]{} P.A.M. Dirac, Rev. Mod. Phys. [**21**]{}, 392 (1949). V.A. Karmanov, Zh. Eksp. Teor. Fiz. [**71**]{} (1976) 399; \[transl.: Sov. Phys. JETP 44 (1976) 210\]. J. Carbonell, B. Desplanques, V.A. Karmanov and J.-F. Mathiot, Phys. Rep. [**300**]{}, 215 (1998). V.A. Karmanov, J.-F. Mathiot and A.V. Smirnov, Phys. Rev. [**D77**]{}, 085028 (2008). P. Grangé, J.-F. Mathiot, B. Mutet, E. Werner, Phys. Rev. [**D80**]{}, 105012 (2009). P. Grangé and E. Werner, [*Quantum fields as Operator Valued Distributions and Causality*]{}, arXiv: math-ph/0612011 and Nucl. Phys. B, Proc. Supp. 161 (2006) 75 V.A. Karmanov, J.-F. Mathiot and A.V. Smirnov, Phys. Rev. [**D69**]{}, 045009 (2004). S.S. Schweber, “[*An Introduction to Relativistic Quantum Field Theory*]{}”, Ed. Harper and Row (1964), p.721;\ J. Collins, “ [*Renormalization*]{}”, (Ed. Cambridge University Press, Cambridge, England, 1984), p.4.\ R. Haag, “[*Local Quantum Physics: Fields, Particules, Algebras*]{}”, Texts and Monographs in Physics, Springer-Verlag, Berlin, Heidelberg, New York,(2nd Edition,1996). [^1]: Here and in Eq.(\[TIR2\]) below the extensions for $\widetilde{T}(X)$ are only valid when taken under the $X$-integral symbol, see [@grange] for discussions [^2]: At $d$ dimension, $k$ is defined by $T[X/t]=t^{k+d} T(X)$. This corrects a misprint in [@grange].

ArXiv

--- abstract: 'A generalization of the Dirac’s canonical quantization theory for a system with second-class constraints is proposed as the fundamental commutation relations that are constituted by all commutators between positions, momenta and Hamiltonian so they are simultaneously quantized in a self-consistent manner, rather than by those between merely positions and momenta so the theory either contains redundant freedoms or conflicts with experiments. The application of the generalized theory to quantum motion on a torus leads to two remarkable results: i) The theory formulated purely on the torus, i.e., based on the so-called the purely intrinsic geometry, conflicts with itself. So it provides an explanation why an intrinsic examination of quantum motion on torus within the Schrödinger formalism is improper. ii) An extrinsic examination of the torus as a submanifold in three dimensional flat space turns out to be self-consistent and the resultant momenta and Hamiltonian are satisfactory all around.' author: - 'D. M. Xun' - 'Q. H. Liu' - 'X. M. Zhu' title: 'Quantum motion on a torus as a submanifold problem in a generalized Dirac’s theory of second-class constraints' --- Introduction ============ The embedding problem of quantum motion of a particle on a two-dimensional curved surface $\Sigma ^{2}$ in the flat space $R^{3}$ has attracted much attention, including theoretical explorations [jk,dacosta,CB,FC,liu07,liu11,japan1990,japan1992,japan1993]{} and experimental investigations [@Szameit; @onoe]. Fundamentally, there are two formalisms to investigate the quantum motion on $\Sigma ^{2}$. One is within the Schrödinger formalism that needs a wave function and another is within the Dirac one that purely deals with operators, but they usually give different predictions. In this section, we will mainly review these two formalisms, and present a generalization of the Dirac’s canonical quantization theory for a system of the second-class constraints. Schrödinger and Dirac formalism: discrepancies in curvature dependent quantum potentials ---------------------------------------------------------------------------------------- By the *Schrödinger formalism* we mean that the Schrödinger equation is first formulated in $R^{3}$, actually in a curved shell of an equal and finite thickness $\delta $ whose intermediate surface coincides with the prescribed one $\Sigma ^{2}$ (or equivalently, the particle moves within the range of the same width $\delta $ due to a confining potential around the surface), and an effective Schrödinger equation on the curved surface $\Sigma ^{2}$ is then derived by taking the squeezing limit $\delta \rightarrow 0$ to confine the particle to the $\Sigma ^{2}$ [jk,dacosta,CB,liu11]{}. It leads to a unique form of the so-called geometric potential [@Szameit; @liu11] $$V_{g}=-\frac{\hbar ^{2}}{2m}\left( M^{2}-K\right) \label{gp}$$that depends on both the mean and the gaussian curvature $M$ and $K$ which are, respectively, the extrinsic and the intrinsic curvature. This amounts to an extrinsic examination of the quantum motion on $\Sigma ^{2}$ within the Schrödinger formalism. The potential (\[gp\]) has been experimentally confirmed [@Szameit; @onoe]. To note that the extrinsic curvature $M$ is a geometric consequence of embedding the system on $\Sigma ^{2}$ in $R^{3}$ and is inaccessible with purely intrinsic description. However, for this formalism, we do not know why the Schrödinger equation can not be entirely formulated on $\Sigma ^{2}$ without considering any embedding. We are familiar with a fact an intrinsic examination of the quantum motion on $\Sigma ^{2}$ within the Schrödinger formalism that predicts no curvature dependent quantum potential, which is contrary to the experiments [@Szameit; @onoe]. By the *Dirac formalism* we mean to use the Dirac’s canonical quantization theory on systems with the second-class constraints [dirac1,dirac2]{}, with an understanding that Dirac formalism can also be applied to the system* *that is considered either within purely intrinsic geometry on $\Sigma ^{2}$ or as a submanifold in $R^{3}$, predicting a curvature dependent potential $V_{D}$ with two real parameters $% \alpha $ and $\beta $ [@japan1990; @japan1992], $$V_{D}=-\frac{\hbar ^{2}}{2m}\left( \alpha M^{2}-\beta K\right) . \label{vd}$$This form of the potential (\[vd\]) can also be easily constructed by dimensional analysis for two geometric invariants $M$ and $K$ have dimension of *length*$^{-1}$ and *length*$^{-2}$, respectively. In comparison with the Schrödinger formalism, we have one more unknown associated with the Dirac one, that is, once taking the $\Sigma ^{2}$ as a submanifold in $R^{3}$ we do not know what form of the potential can be singled out among a family of it (\[vd\]). However, Schrödinger’s theory gives an unambiguous choice with $\alpha =\beta =1$ [jk,dacosta,FC,liu11]{}. So far, we find that both formalisms suffer from shortcomings. Since the extrinsic examination of the torus within the Schrödinger formalism has experimental supports, an immediate question is whether there is a possible theoretical framework from which we can fix the parameters $\alpha $ and $% \beta $ within a possibly generalized Dirac’s theory, so rendering it compatible with Schrödinger’s and also the experimental results. This question will be partially answered in this paper. Schrödinger and Dirac formalism: discrepancies in momentum operators -------------------------------------------------------------------- In addition to the unique form of the geometric potential $V_{g}=-\hbar ^{2}\left( M^{2}-K\right) /2m$, Schrödinger’s theory also leads to a unique definition of the geometric momentum $\mathbf{p}$ [@liu07; @liu11], $$\mathbf{p}=-i\hbar (\mathbf{r}^{\mu }\partial _{\mu }+M\mathbf{n}), \label{gm}$$where $\mathbf{r=(}x(x^{1},x^{2}),y(x^{1},x^{2}),z(x^{1},x^{2})\mathbf{)}$ is the position vector in $R^{3}$ on the surface $\Sigma ^{2}$ whose local coordinates are $x^{\mu }\equiv (x^{1},x^{2})$ and $\mathbf{r}^{\mu }=g^{\mu \nu }\mathbf{r}_{\nu }=g^{\mu \nu }\partial \mathbf{r/}x^{\nu }$, and at this point $\mathbf{r}$, $\mathbf{n=(}n_{x},n_{y},n_{z}\mathbf{)}$ denotes the normal and $M\mathbf{n}$ symbolizes the mean curvature vector field, another geometric invariant. Throughout the paper, the Einstein summation convention over repeated indices is used.  However, the present formulation of Dirac’s theory opens a wide door to permit various definitions of the generalized momenta, including i) the well-known generalized ones $p_{\mu }=-i\hbar (\partial _{\mu }+\Gamma _{\mu }/2)$ which satisfy quantum commutator $[x^{\nu },p_{\mu }]=i\hbar \delta _{\mu }^{\nu }$, where $\Gamma _{\mu }$ is the once-contracted Christoffel symbol $\Gamma _{\mu \nu }^{\sigma }$ constructed with Riemannian metric $% g^{\mu \nu }$ [@japan1990], where greek letters $\mu $, $\nu $, $\sigma $, etc. run between $1$ to $2$, and ii) geometric momentum (\[gm\]), and etc. [@liu11; @japan1992]. *It is very important to note that in the extrinsic examination of quantum motion on* $\Sigma ^{2}$* in* $% R^{3}$*, the local coordinates* $x^{\mu }\equiv (x^{1},x^{2})$* are no longer position operators but parameters, and the position operators are* $\mathbf{r=(}x(x^{1},x^{2}),y(x^{1},x^{2}),z(x^{1},x^{2})% \mathbf{)}$*.* A framework based on the purely intrinsic geometry implies that every quantity solely relies on the Riemannian metric $g^{\mu \nu }$ and its various constructions such as Christoffel symbol $\Gamma _{\mu \nu }^{\sigma }$ and the gaussian curvature $K$. Consequently, neither momentum nor Hamiltonian in quantum mechanics depends on the extrinsic curvature. When the curvature dependent potential with$\ $(\[vd\]) $\alpha \neq 0$ and geometric momentum (\[gm\]) appear in a formulation of quantum mechanics for a system on $\Sigma ^{2}$, we in fact take the system under study to be embedded in $R^{3}$, which is beyond the purely intrinsic geometry. A generalization Dirac’s theory for a system of the second-class constraints ---------------------------------------------------------------------------- We are deeply impressed by the very success of Schrödinger’s theory that produces unique result of the geometric potential (\[vd\]) and momentum (\[gm\]), and also by the disturbing arbitrariness associated with Dirac’s theory of the second-class constraints. As we know, Dirac’s theory postulates that a quantum commutator $[A,B]$ of two variables $A$ and $B$ in quantum mechanics is achieved by direct correspondence of the Dirac’s brackets $\{A{,B\}}_{D}$ as $\{A{,B\}}_{D}\rightarrow \lbrack A,B]\ $which is defined by $[A,B]=i\hbar O(\{A{,B\}}_{D})$ where $O(F)$ is used to emphasize the operator form of the classical quantity $F$ in order to avoid possible confusion. When all constraints are removed, the Dirac bracket $\{A{% ,B\}}_{D}$ assumes its usual* *form, the Poisson* *bracket $% \{A{,B\}}$. However, Dirac himself states that *fundamental commutation relations involve only those between canonical positions* $x_{{i}% }$* and canonical momenta* ${p}_{{i}}$* *[@dirac1; @dirac2]. One can ask a curious question: when there is no constraint, why there is no such a *fundamental* canonical quantization rule between $f$ $(=x_{{i}% }$, $p_{{i}})$ and the Hamiltonian $H$ as $[f,H]=i\hbar O(\{f,H\})$? This is because the direct quantization $[f,H]=i\hbar O(\{f,H\})$ might be redundant, or meaningless, or practically useless, etc. For instance, when the system has a classical analogue, the Hamiltonian is the same function of the positions and momenta in the quantum theory as in the classical theory, provided that the Cartesian system of axes is used [dirac2,dirac3,Greiner]{}. In this case the rule $[f,H]=i\hbar O(\{f,H\})$ turns out to be redundant. When a quantum Hamiltonian has no classical analogy, the canonical quantization rule $[f,H]=i\hbar O(\{f,H\})$ is meaningless. In many other cases, e.g., to quantize a classical Hamiltonian $% H=\gamma x^{3}p^{3}$ with $\gamma $ being a real parameter, the rule should be imposed but is practically useless. Thus, it appears unacceptable to include the canonical quantization rule $[f,H]=i\hbar O(\{f,H\})$ as a fundamental element of a theory. For systems with the second-class constraints, the situation is totally different. Discrepancies between either curvature dependent quantum potentials or momentum operators present when different formalisms, or different geometric points, are utilized. It strongly implies that, while the quantization of the system is performed, the proper operator form of positions, momenta and Hamiltonian are simultaneously determined in a self-consistent way. Therefore we have attempted to generalize the Dirac’s theory so as to add $[f,H]$ into the category of the *fundamental commutation relations* which should also be directly achieved via following quantization rule [@liu11], $$\lbrack f,H]=i\hbar O(\{f,H\}_{D}),\text{ }f=x_{i}\text{ and }p_{j}. \label{generalized}$$In rest part of the paper, the convention $O(F)=F$ in quantum mechanics assumes without no longer emphasizing it an operator with the symbol $O$. These commutation relations (\[generalized\]) may not be applicable when the system has no constraint. So we would like to call them the second category of fundamental* *ones [@liu11], whereas the existing ones between positions and momenta, the first. This generalized Dirac’s theory reproduces the usual form for the system that has a classical analogue but has not a constraint, together with the necessary utilization of the Cartesian system of axes, therefore enriches the Dirac formalism of quantum mechanics. We will call it the *general theory of the canonical quantization* (GTCQ). Purpose and organization of the paper ------------------------------------- As an application of the GTCQ to quantum motion on a sphere [@liu11], we find that, on one hand, an attempt of trying a proper description within the purely intrinsic geometry proves problematic, and one the other hand, an account of embedding the sphere in three-dimensional space is very coherent. Notice that the classification theorem for compact surfaces states that [class]{}, every compact orientable surface is homeomorphic either to a sphere or to a connected sum of tori, implying that if there is any difficulty associated with quantum mechanics for a particle constrained on a sphere or a torus, enormous theoretical problems would arise from dealing with an arbitrary two-dimensional curved surface in quantum mechanics. It forms one of the reasons that the sphere [@liu11] and the torus [torus1,torus2,torus3,torus4]{} are used to test various theories. The main purpose of the present study is to take the torus to show that Dirac formalism is complementary to the Schrödinger one. The former eliminates the purely intrinsic description, and the latter gives the unique form of the geometric potential, while both define the identical form of the geometric momentum. This paper is organized as follows. In following section II, we present the GTCQ for quantum motion on the torus within purely intrinsic geometry. Results show that the theory can never be consistently set up. In section III, we revisit the same problem as a submanifold in flat space $R^{3}$ with the GTCQ. Results show that the theory turns out to be self-consistent all around, and the obtained geometric momentum (\[gm\]) and potential ([gp]{}) are also satisfactory. Section IV briefly remarks and concludes this study. GTCQ for a torus within intrinsic geometry ========================================== The toroidal surface is with two local coordinates $\theta \in \lbrack 0,2\pi ),\varphi \in \lbrack 0,2\pi )$$$\mathbf{r}=((a+r\sin \theta )\cos \varphi ,(a+r\sin \theta )\sin \varphi ,r\cos \theta ),\text{ }a>r\neq 0, \label{rr}$$where $\varphi $ is the azimuthal angle and $\theta $ the polar angle, and $% a $ and $r$ are the outer and inner radii of the torus, respectively. The constraint is $r=b\neq 0$. In this section, we will first give the classical mechanics for motion on the torus, and then turn into the Dirac formalism of quantum mechanics. In classical mechanics, the theory appears nothing surprising, but after transition to quantum mechanics, it becomes contradictory to itself. Classical mechanical treatment ------------------------------ The Lagrangian $L$ in the toric coordinate system is, $$L=\frac{m}{2}(\dot{r}^{2}+r^{2}\dot{\theta}^{2}+(a+r\sin \theta )^{2}\dot{% \varphi}^{2})-\lambda (r-b), \label{lag}$$where $\lambda $ is the Lagrangian multiplier enforcing the constrained of motion on the surface. The Lagrangian is singular because it does not contain the “velocity” $\dot{\lambda}$. Hence we need the Dirac formalism of the classical mechanics for a system with the second-class constraints, which gives the canonical momenta conjugate to $r,\theta ,\varphi $ and $% \lambda $ in the following,$$\begin{aligned} p_{r} &=&\frac{\partial L}{\partial \dot{r}}=m\dot{r}, \\ p_{\theta } &=&\frac{\partial L}{\partial \dot{\theta}}=mr^{2}\dot{\theta}, \\ p_{\varphi } &=&\frac{\partial L}{\partial \dot{\varphi}}=m(a+r\sin \theta )^{2}\dot{\varphi}, \\ p_{\lambda } &=&\frac{\partial L}{\partial \dot{\lambda}}=0. \label{plamb}\end{aligned}$$Eq. (\[plamb\]) represents the primary constraint:$$\varphi _{1}\equiv p_{\lambda }\approx 0, \label{prim}$$hereafter symbol “$\approx $” implies a weak equality [@dirac2]. After all calculations are finished, the weak equality takes back the strong one. By the Legendre transformation, the primary Hamiltonian $H_{p}$ is [dirac2]{},$$H_{p}=\frac{1}{2m}(p_{r}^{2}+\frac{p_{\theta }^{2}}{r^{2}}+\frac{p_{\varphi }^{2}}{(a+r\sin \theta )^{2}})+\lambda \left( r-b\right) +\dot{\lambda}% p_{\lambda }, \label{hami}$$where $\dot{\lambda}$ is also a Lagrangian multiplier guaranteeing that this Hamiltonian is defined on the symplectic manifold. The secondary constraints (not confusing with second-class constraints) are generated successively, then determined by the conservation condition [@dirac2],$$\varphi _{i+1}\equiv \left\{ \varphi _{i},H_{p}\right\} \approx 0,\text{\ }% (i=1,2,....),$$where $\left\{ f,g\right\} $ is the Poisson bracket with $% q_{1}=r,q_{2}=\theta ,q_{3}=\varphi $, and $p_{1}=p_{r},p_{2}=p_{\theta },p_{3}=p_{\varphi }$, $$\left\{ f,g\right\} \equiv \frac{\partial f}{\partial q_{k}}\frac{\partial g% }{\partial p_{k}}+\frac{\partial f}{\partial \lambda }\frac{\partial g}{% \partial p_{\lambda }}-(\frac{\partial f}{\partial p_{k}}\frac{\partial g}{% \partial q_{k}}+\frac{\partial f}{\partial p_{\lambda }}\frac{\partial g}{% \partial \lambda }). \label{possi}$$The complete set of the secondary constraints is, $$\begin{aligned} \varphi _{2} &\equiv &\left\{ \varphi _{1},H_{p}\right\} =-(r-b)\approx 0, \label{db1} \\ \varphi _{3} &\equiv &\left\{ \varphi _{2},H_{p}\right\} =-\frac{p_{r}}{m}% \approx 0, \label{db2} \\ \varphi _{4} &\equiv &\left\{ \varphi _{3},H_{p}\right\} =\frac{\lambda }{m}-% \frac{1}{m^{2}}(\frac{p_{\theta }^{2}}{r^{3}}+\frac{p_{\varphi }^{2}\sin \theta }{(a+r\sin \theta )^{3}})\approx 0, \label{thi} \\ \varphi _{5} &\equiv &\left\{ \varphi _{4},H_{p}\right\} =\frac{\dot{\lambda}% }{m}-\frac{3ap_{\theta }p_{\varphi }^{2}\cos \theta }{m^{3}r^{2}(a+r\sin \theta )^{4}}\approx 0. \label{for}\end{aligned}$$Eqs. (\[db1\]) and (\[db2\]) show, respectively, that on the surface of torus $r=b$, no motion along the normal direction is possible $p_{r}=0$, while Eqs. (\[thi\]) and (\[for\]) determine, respectively, the Lagrangian multipliers $\lambda $ and $\dot{\lambda}$. The Dirac bracket instead of the Poisson one for two variables $A$ and $B$ is defined by,$$\left\{ A,B\right\} _{D}\equiv \left\{ A,B\right\} -\left\{ A,\varphi _{u}\right\} C_{uv}^{-1}\left\{ \varphi _{v},B\right\} ,$$where the $4\times 4$ matrix $C\equiv \left\{ C_{uv}\right\} $ whose elements are defined by $C_{uv}\equiv \left\{ \varphi _{u},\varphi _{v}\right\} $ with $u,v=1,2,3,4$ from Eqs. (\[prim\]) and (\[db1\])-(\[thi\]). The inverse matrix $C^{-1}$ is,$$C^{-1}=\left\{ \begin{array}{cccc} 0 & C_{12}^{-1} & 0 & m \\ -C_{12}^{-1} & 0 & -m & 0 \\ 0 & m & 0 & 0 \\ -m & 0 & 0 & 0% \end{array}% \right\} ,$$where$$C_{12}^{-1}=\frac{3}{m}\left( \frac{p_{\theta }^{2}}{b^{4}}+\frac{p_{\varphi }^{2}\sin ^{2}\theta }{\left( a+b\sin \theta \right) ^{4}}\right) .$$Thus, the generalized positions $q^{\mu }$ $(=\theta ,\varphi )$ and momenta $p_{\mu }$ satisfy the following Dirac brackets,$$\{q^{\mu },q^{\nu }\}_{D}=0,\text{ }\{p_{\mu },p_{\nu }\}_{D}=0,\text{ }% \{q^{\mu },p_{\nu }\}_{D}=\delta _{\nu }^{\mu }. \label{xp1}$$By use of the equation of motion,$$\dot{f}=\left\{ f,H_{c}\right\} _{D},$$we obtain those for the positions $\theta $, $\varphi $ and the momenta $% p_{\theta }$, $p_{\varphi }$, respectively,$$\begin{aligned} \dot{\theta} &\equiv &\left\{ \theta ,H_{c}\right\} _{D}=\frac{p_{\theta }}{% mb^{2}},\text{\ \ }\dot{\varphi}\equiv \left\{ \varphi ,H_{c}\right\} _{D}=% \frac{p_{\varphi }}{m(a+b\sin \theta )^{2}}, \label{xh} \\ \dot{p}_{\theta } &\equiv &\left\{ p_{\theta },H_{c}\right\} _{D}=\frac{% b\cos \theta p_{\varphi }^{2}}{m(a+b\sin \theta )^{3}},\text{ \ }\dot{p}% _{\varphi }\equiv \left\{ p_{\varphi },H_{c}\right\} _{D}=0. \label{ph}\end{aligned}$$In these calculations (\[xh\]) and (\[ph\]), we in fact need only the usual form of Hamiltonian, $H_{p}\rightarrow H_{c}$,$$H_{c}=\frac{1}{2m}\left( \frac{p_{\theta }^{2}}{b^{2}}+\frac{p_{\varphi }^{2}% }{\left( a+b\sin \theta \right) ^{2}}\right) .$$ So far, the classical mechanics for the motion on the torus is complete and coherent in itself.  Quantum mechanical treatment ---------------------------- In quantum mechanics, we assume that the Hamiltonian takes the following general form,$$\begin{aligned} H &=&-\frac{\hbar ^{2}}{2m}\left[ \nabla ^{2}+\left( \alpha M^{2}-\beta K\right) \right] \notag \\ &=&-\frac{\hbar ^{2}}{2m}\left[ \frac{1}{b^{2}}\frac{\partial ^{2}}{\partial \theta ^{2}}+\frac{\cos \theta }{b\left( a+b\sin \theta \right) }\frac{% \partial }{\partial \theta }+\frac{1}{\left( a+b\sin \theta \right) ^{2}}% \frac{\partial ^{2}}{\partial \varphi ^{2}}\right. \notag \\ &&+\left. \alpha \frac{1}{4}\left( \frac{a+2b\sin \theta }{ab+b^{2}\sin \theta }\right) ^{2}-\beta \frac{\sin \theta }{ab+b^{2}\sin \theta }\right] , \label{h}\end{aligned}$$where, $$M=-\frac{1}{2}\frac{a+2b\sin \theta }{ab+b^{2}\sin \theta },\text{ }K=\frac{% \sin \theta }{ab+b^{2}\sin \theta }.$$We are ready to construct commutator $[A,B]$ of two variables $A$ and $B$ in quantum mechanics, which can be straightforwardly realized by a direct correspondence of the Dirac’s brackets as $\{A,B\}_{D}\rightarrow \left[ A,B% \right] /i\hbar $. From the Dirac’s brackets (\[xp1\]), the first category of the fundamental commutators between operators $q^{\mu }$ and $p_{\nu }$ are given by,$$\lbrack q^{\mu },q^{\nu }]=0,\text{ }[p_{\mu },p_{\nu }]=0,\text{ }[q^{\mu },p_{\nu }]=i\hbar \delta _{\nu }^{\mu }. \label{xp2}$$In light of the GTCQ, we have the second category of fundamental commutators between $q^{\mu }$ and $H$ from Eq. (\[xh\]),$$\begin{aligned} \left[ \theta ,H\right] &=&\frac{\hbar ^{2}}{mb^{2}}\left( \frac{\partial }{% \partial \theta }+\frac{b\cos \theta }{2\left( a+b\sin \theta \right) }% \right) =i\hbar \frac{p_{\theta }}{mb^{2}}, \label{qxh1} \\ \left[ \varphi ,H\right] &=&\frac{\hbar ^{2}}{m(a+b\sin \theta )^{2}}\frac{% \partial }{\partial \varphi }=i\hbar \frac{p_{\varphi }}{m(a+b\sin \theta )^{2}}. \label{qxh2}\end{aligned}$$From these quantum commutators, the operators $p_{\theta }$ and $p_{\varphi } $ are, respectively,$$p_{\theta }=-i\hbar \left[ \frac{\partial }{\partial \theta }+\frac{b\cos \theta }{2\left( a+b\sin \theta \right) }\right] ,\text{ }p_{\varphi }=-i\hbar \frac{\partial }{\partial \varphi }\text{.} \label{cmom}$$Using these operators, we can directly calculate two quantum commutators $% \left[ p_{\theta },H\right] $ and $\left[ p_{\varphi },H\right] $ with quantum Hamiltonian (\[h\]), and the results are, respectively,$$\begin{aligned} \left[ p_{\theta },H\right] &=&i\hbar \frac{b\cos \theta }{m(a+b\sin \theta )^{3}}p_{\varphi }^{2}+i\hbar \frac{\hbar ^{2}\cos \theta \left( a^{2}(\alpha -2\beta +1)+2ab(\alpha -\beta )\sin \theta -b^{2}\right) }{% 4bm(a+b\sin \theta )^{3}}, \label{qph1} \\ \left[ p_{\varphi },H\right] &=&0. \label{qph2}\end{aligned}$$The second equation (\[qph2\]) is satisfactory, whereas the first one ([qph1]{}) can hardly hold true. In the GTCQ, the quantum commutator $\left[ p_{\theta },H\right] $ (\[qph1\]) must be the canonical quantization of the Dirac bracket (\[ph\]). We get, with noting the mutual commutabiliy between two observables $p_{\varphi }$ and $\theta $, $$i\hbar \left\{ p_{\theta },H\right\} _{D}=\frac{i\hbar b\cos \theta p_{\varphi }^{2}}{m(a+b\sin \theta )^{3}}. \label{qph1-1}$$In comparison with the right-handed sides of the Eqs. (\[qph1\]) and ([qph1-1]{}), we obtain a unique solution, $$\alpha =\beta =\frac{a^{2}-b^{2}}{a^{2}}(\neq 1),$$which leads an unacceptable curvature dependent quantum potential that includes the extrinsic curvature $M$,$$V_{D}=-\frac{\hbar ^{2}}{2m}\frac{a^{2}-b^{2}}{a^{2}}\left( M^{2}-K\right) =-% \frac{\hbar ^{2}}{2m}\frac{a^{2}-b^{2}}{4b^{2}\left( a+b\sin \theta \right) ^{2}}.$$However, no matter what other values of $\alpha $ and $\beta $ are chosen, there is a manifest breakdown of the canonical quantization rule between Dirac bracket $\left\{ p_{\theta },H\right\} _{D}$ (\[ph\]) and the quantum commutator $\left[ p_{\theta },H\right] $ (\[qph1\]). So we see that the intrinsic geometry is insufficient for the GTCQ to be self-consistent. If using original form of the Dirac’s theory instead, we still have results (\[cmom\])-(\[qph2\]) but we can never require them as the canonical quantization of the relevant Dirac brackets (\[xh\])-(\[ph\]). It is sheer nonsense for we neither are able to exclude the extrinsic curvature $M$, nor give a unambiguous prediction of the curvature dependent potential to be testable by experiment. One should be noted that we have not introduced additional assumptions such as “dummy factors” techniques [@Kleinert] etc. in passing from Dirac’s brackets to the quantum commutators. They mean further generalizations of the Dirac’s theory. In classical limit $\hbar \rightarrow 0$, all inconsistency vanishes, as expected. Summary ------- From the studies in this section, we see that the GTCQ of second-class constraints for quantum motion on the torus can not be consistently formulated. We therefore need to invoke an extrinsic examination of the same problem, as will be done in next section. GTCQ for a torus as a submanifold ================================= The surface equation of the torus (\[rr\]) in Cartesian coordinates $% \left( x,y,z\right) $ is given by,$$f\left( \mathbf{x}\right) \equiv a^{2}-b^{2}+(x^{2}+y^{2}+z^{2})-2a\sqrt{% x^{2}+y^{2}}=0.$$In this section, we will also first give the classical mechanics for motion on the torus within the Dirac formalism of the classical mechanics for a system with the second-class constraints, and then turn into quantum mechanics. The GTCQ proves to be self-consistent all around and the resultant momenta and Hamiltonian are exactly those given by the Eq. ([gm]{}) and (\[gp\]), respectively. Classical mechanical treatment ------------------------------ The Lagrangian $L$ in the Cartesian coordinate system is,$$L=\frac{m}{2}\left( \dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right) -\lambda f\left( \mathbf{x}\right) . \label{lagca}$$The generalized momentum $\mathbf{p}$ whose three components $p_{i}$ $% (i=x,y,z)$ and $p_{\lambda }$ canonically conjugate to variables $x_{i}$ $% (x_{1}=x,x_{2}=y,x_{3}=z,)$ and $\lambda $, are given by, respectively,$$\begin{aligned} p_{i} &=&\frac{\partial L}{\partial \dot{x}_{i}}=m\dot{x}_{i},(i=1,2,3), \\ p_{\lambda } &=&\frac{\partial L}{\partial \dot{\lambda}}=0. \label{plambca}\end{aligned}$$Eq. (\[plambca\]) represents the primary constraint,$$\varphi _{1}\equiv p_{\lambda }\approx 0. \label{prim2}$$By the Legendre transformation, the primary Hamiltonian $H_{p}$ is,$$H_{p}=\frac{1}{2m}p_{i}^{2}+\lambda f\left( \mathbf{x}\right) +\dot{\lambda}% p_{\lambda }.$$The secondary constraints are determined by successive use of the Poisson brackets,$$\begin{aligned} \varphi _{2} &\equiv &\left\{ \varphi _{1},H_{p}\right\} =-(a^{2}-b^{2}+x_{i}^{2}-2a\sqrt{x^{2}+y^{2}})\approx 0, \label{1st2} \\ \varphi _{3} &\equiv &\left\{ \varphi _{2},H_{p}\right\} =-\frac{2\left( \sqrt{x^{2}+y^{2}}(p_{x}x+p_{y}y+p_{z}z)-a(p_{x}x+p_{y}y)\right) }{m\sqrt{% x^{2}+y^{2}}}\approx 0, \\ \varphi _{4} &\equiv &\left\{ \varphi _{3},H_{p}\right\} =\frac{4\lambda \left( a^{2}-2a\sqrt{x^{2}+y^{2}}+x_{i}^{2}\right) }{m}+\frac{% 2a(p_{y}x-p_{x}y)^{2}}{m^{2}\left( x^{2}+y^{2}\right) ^{3/2}}-\frac{% 2p_{i}^{2}}{m^{2}}\approx 0, \label{thica} \\ \varphi _{5} &\equiv &\left\{ \varphi _{4},H_{p}\right\} =\frac{4\dot{\lambda% }\left( a^{2}-2a\sqrt{x^{2}+y^{2}}+x_{i}^{2}\right) }{m}-\frac{% 6a(p_{x}x+p_{y}y)(p_{y}x-p_{x}y)^{2}}{m^{3}\left( x^{2}+y^{2}\right) ^{5/2}}% \approx 0. \label{forca}\end{aligned}$$Similarly, the Dirac bracket between two variables $A$ and $B$ is defined by,$$\left\{ A,B\right\} _{D}=\left\{ A,B\right\} -\left\{ A,\varphi _{u}\right\} D_{uv}^{-1}\left\{ \varphi _{v},B\right\} ,$$where the $4\times 4$ matrix $D\equiv \left\{ D_{uv}\right\} $ whose elements are defined by $D_{uv}\equiv \left\{ \varphi _{u},\varphi _{v}\right\} $ with $u,v=1,2,3,4$ from Eqs. (\[prim2\]) and (\[1st2\])-(\[thica\]). The inverse matrix $D^{-1}$ is easily carried out,$$D^{-1}=\left( \begin{array}{cccc} 0 & D_{12}^{-1} & 0 & \kappa \\ -D_{12}^{-1} & 0 & -\kappa & 0 \\ 0 & \kappa & 0 & 0 \\ -\kappa & 0 & 0 & 0% \end{array}% \right) ,$$where,$$D_{12}^{-1}=\frac{\left( 3a^{2}-7a\sqrt{x^{2}+y^{2}}\right) (p_{y}x-p_{x}y)^{2}+4\left( x^{2}+y^{2}\right) ^{2}p_{i}^{2}}{4b^{4}m\left( x^{2}+y^{2}\right) ^{2}},\text{ }\kappa =\frac{m}{4b^{2}}.$$Then primary Hamiltonian $H_{p}$ assumes its usual one: $H_{p}\rightarrow H_{c},$$$H_{c}=\frac{p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2m}. \label{HP}$$All fundamental Dirac’s brackets are as follows,$$\begin{aligned} \{x_{i},x_{j}\}_{D} &=&0, \label{xxca1} \\ \{x_{i},p_{j}\}_{D} &=&\delta _{ij}-\frac{1}{b^{2}}f_{i}f_{j}, \label{xpca1} \\ \{p_{i},p_{j}\}_{D} &=&-\frac{1}{b^{2}}\left[ f_{i}\left( p_{j}+\frac{% a\left( xp_{y}-yp_{x}\right) }{\left( x^{2}+y^{2}\right) ^{3/2}}\left( y\delta _{1j}-x\delta _{2j}\right) \right) -f_{j}\left( p_{i}+\frac{a\left( xp_{y}-yp_{x}\right) }{\left( x^{2}+y^{2}\right) ^{3/2}}\left( y\delta _{1i}-x\delta _{2i}\right) \right) \right] , \label{xhca1} \\ \{x_{i},H_{c}\}_{D} &=&\frac{p_{i}}{m}=\dot{x}_{i}, \\ \{p_{i},H_{c}\}_{D} &=&-\frac{1}{mb^{2}}\left[ f_{i}\left( p_{x}^{2}+p_{y}^{2}+p_{z}^{2}-\frac{a\left( xp_{y}-yp_{x}\right) ^{2}}{% \left( x^{2}+y^{2}\right) ^{3/2}}\right) \right] =\dot{p}_{i}, \label{phca1}\end{aligned}$$ where $f_{i}=x_{i}-a\left( x\delta _{1i}+y\delta _{2i}\right) /\sqrt{% x^{2}+y^{2}}$. Quantum mechanical treatment ---------------------------- Now let us turn to quantum mechanics. The first category of the fundamental commutators between operators $x_{i}$ and $p_{i}$ are, by quantization of (\[xxca1\])-(\[xhca1\]),$$\begin{aligned} \left[ x_{i},x_{j}\right] &=&0,\text{ \ }\left[ x_{i},p_{j}\right] =i\hbar \left( \delta _{ij}-\frac{1}{b^{2}}f_{i}f_{j}\right) , \label{xx-xp} \\ \left[ p_{i},p_{j}\right] &=&-\frac{i\hbar }{b^{2}}\left[ f_{i}\left( p_{j}+a% \frac{L_{z}\left( y\delta _{1j}-x\delta _{2j}\right) +\left( y\delta _{1j}-x\delta _{2j}\right) L_{z}}{2\left( x^{2}+y^{2}\right) ^{3/2}}\right) \right. \notag \\ &&\left. -f_{j}\left( p_{i}+a\frac{L_{z}\left( y\delta _{1i}-x\delta _{2i}\right) +\left( y\delta _{1i}-x\delta _{2i}\right) L_{z}}{2\left( x^{2}+y^{2}\right) ^{3/2}}\right) \right] , \label{ppca2}\end{aligned}$$where $L_{z}=xp_{y}-yp_{x}$. It seems that we have complicated operator-ordering problem as passing from the Dirac bracket Eq. (\[xhca1\]) to the quantum commutator (\[ppca2\]). In fact, only one pair between the noncommuting observables $x_{i}$ (precisely, $\left( y\delta _{1j}-x\delta _{2j}\right) $) and $L_{z}$ matters, and the product of $% \left( y\delta _{1j}-x\delta _{2j}\right) $ and $L_{z}$ can be made Hermitian by a symmetric construction, $\left( \left( y\delta _{1j}-x\delta _{2j}\right) L_{z}+L_{z}\left( y\delta _{1j}-x\delta _{2j}\right) \right) /2$. Other products of factors $f_{i}$ (or $f_{j}$) and $L_{z}$ impose no operator-ordering problem because of the Jacobi identity. There is a family of the momenta $p_{i}$ all of them are solutions to the Eq. (\[ppca2\]), as explicitly shown in [@japan1992]. With these momenta $p_{i}$ at hand, we completely do not know the correct form of the quantum Hamiltonian, as suggested by Eq. (\[HP\]). It is therefore understandable that the quantum Hamiltonian would contain arbitrary parameters. However, the GTCQ requires the second category of the fundamental commutators as $\left[ x_{i},H\right] $ and $\left[ p_{i},H\right] $. We immediately find that the momenta $p_{i}$ from following commutators, $$\left[ x_{i},H\right] =i\hbar \frac{p_{i}}{m}. \label{xhca2}$$The obtained momenta $p_{i}$ are nothing but three components of the geometric momentum (\[gm\]) on the torus [@torus4], $$\begin{aligned} p_{x} &=&-i\hbar \left( \frac{\cos \theta \cos \varphi }{b}\frac{\partial }{% \partial \theta }-\frac{\sin \varphi }{a+b\sin \theta }\frac{\partial }{% \partial \varphi }-\frac{a+2b\sin \theta }{2b(a+b\sin \theta )}\sin \theta \cos \varphi \right) , \\ p_{y} &=&-i\hbar \left( \frac{\cos \theta \sin \varphi }{b}\frac{\partial }{% \partial \theta }+\frac{\cos \varphi }{a+b\sin \theta }\frac{\partial }{% \partial \varphi }-\frac{a+2b\sin \theta }{2b(a+b\sin \theta )}\sin \theta \sin \varphi \right) , \\ p_{z} &=&i\hbar \left( \frac{\sin \theta }{b}\frac{\partial }{\partial \theta }+\frac{a+2b\sin \theta }{2b\left( a+b\sin \theta \right) }\cos \theta \right) .\end{aligned}$$ As to the form of quantum Hamiltonian, we also start from the general form (\[h\]), and now resort to the following complicated operator-ordering arrangement with $w_{\pm }=(x\pm iy)^{3/2}$, $$\begin{aligned} \left[ p_{i},H\right] &=&-\frac{i\hbar }{mb^{2}}\{mHf_{i}+mf_{i}H \notag \\ &&-\frac{a}{4}\alpha _{1}[f_{i}(L_{z}\frac{1}{w_{+}}L_{z}\frac{1}{w_{-}}+% \frac{1}{w_{-}}L_{z}\frac{1}{w_{+}}L_{z})+(L_{z}\frac{1}{w_{+}}L_{z}\frac{1}{% w_{-}}+\frac{1}{w_{-}}L_{z}\frac{1}{w_{+}}L_{z})f_{i}] \notag \\ &&-\frac{a}{4}\alpha _{2}[f_{i}(L_{z}\frac{1}{w_{+}}\frac{1}{w_{-}}L_{z}+% \frac{1}{w_{-}}L_{z}L_{z}\frac{1}{w_{+}})+(L_{z}\frac{1}{w_{+}}\frac{1}{w_{-}% }L_{z}+\frac{1}{w_{-}}L_{z}L_{z}\frac{1}{w_{+}})f_{i}] \notag \\ &&-\frac{a}{4}\alpha _{3}[f_{i}(\frac{1}{w_{+}}L_{z}L_{z}\frac{1}{w_{-}}% +L_{z}\frac{1}{w_{-}}\frac{1}{w_{+}}L_{z})+(\frac{1}{w_{+}}L_{z}L_{z}\frac{1% }{w_{-}}+L_{z}\frac{1}{w_{-}}\frac{1}{w_{+}}L_{z})f_{i}] \notag \\ &&-\frac{a}{4}\alpha _{4}[f_{i}(\frac{1}{w_{+}}L_{z}\frac{1}{w_{-}}% L_{z}+L_{z}\frac{1}{w_{-}}L_{z}\frac{1}{w_{+}})+(\frac{1}{w_{+}}L_{z}\frac{1% }{w_{-}}L_{z}+L_{z}\frac{1}{w_{-}}L_{z}\frac{1}{w_{+}})f_{i}] \notag \\ &&-\frac{a}{2}\alpha _{5}\frac{1}{w_{+}w_{-}}(f_{i}L_{z}^{2}+L_{z}^{2}f_{i})% \}, \label{op-ord}\end{aligned}$$where $\alpha _{k}$, $(k=1,2,...5)$ are five real parameters satisfying $% \sum \alpha _{k}=1$. In comparison of both sides of the this equation, we find that the solution $\alpha =\beta =1$, and two of the five real parameters $\alpha _{k}$ are freely to be specified, $$\alpha _{1}=\frac{11}{9}-\alpha _{4}-\alpha _{5},\alpha _{2}=\alpha _{3}=-% \frac{1}{9}.$$We see that free parameters remain, but they are irrelevant to observable quantities such as momentum and potential. In fact, with $\alpha =\beta =1$ in (\[vd\]), a much simpler choice of the operator-ordering without free parameters is possible,$$\begin{aligned} \left[ p_{i},H\right] &=&-\frac{i\hbar }{mb^{2}}\{mHf_{i}+mf_{i}H+\frac{1}{9}% \frac{a}{4}[(\frac{1}{w_{+}}f_{i}L_{z}^{2}\frac{1}{w_{-}}+\frac{1}{w_{-}}% f_{i}L_{z}^{2}\frac{1}{w_{+}}) \notag \\ &&+(\frac{1}{w_{+}}L_{z}^{2}\frac{1}{w_{-}}f_{i}+\frac{1}{w_{-}}L_{z}^{2}% \frac{1}{w_{+}}f_{i})]-\frac{10}{9}\frac{a}{2}\frac{1}{w_{+}w_{-}}% (f_{i}L_{z}^{2}+L_{z}^{2}f_{i})\}. \label{ph2}\end{aligned}$$Even we can by no means exhaust all possible forms of the operator-ordering, from Eqs. (\[op-ord\]) and (\[ph2\]), we can at least conclude that the curvature dependent potential (\[vd\]) given by the Dirac formalism converges to the geometric potential (\[gp\]) given by the Schrödinger one. Summary ------- An examination of the motion on torus as a submanifold problem in GTCQ ensures a highly self-consistent description, and this formalism comes compatible with the Schrödinger one. Remarks and conclusions ======================= It is long known that Dirac’s theory of second-class constraints, in which the fundamental commutation relations involve only those between canonical positions and canonical momenta, contains redundant freedoms and causes difficulty sometimes. To overcome the problems, we recently put forward a proposal that the commutators between the positions, momenta, and Hamiltonian form a full set of the fundamental commutation relations to construct a self-consistent quantum theory, the so-called GTCQ. Then the GTCQ produces a unique form of the geometric momentum, and imposes additional requirement on the form of the Hamiltonian via the curvature dependent potential that has no direct analogy. We see that the geometric potential comes as the consequence of the extrinsic examination of the constrained motion. Through a careful analysis of the quantum motion on a torus, we demonstrate that the purely intrinsic geometry does not suffice for the GTCQ to be self-consistently formulated, but an extrinsic examination of the torus in three dimensional flat space does. Our study implies that the Dirac formalism is complementary to the Schrödinger one. The former can be helpful to eliminate the intrinsic description, and the latter gives the unique form of the geometric potential, while both define the identical form of the geometric momentum. This work is financially supported by National Natural Science Foundation of China under Grant No. 11175063. [99]{} H. Jensen and H. Koppe, *Ann. Phys*. **63**, 586(1971). R. C. T. da Costa, *Phys. Rev. A* **23**, 1982(1981). A. V. Chaplik and R. H. Blick, *New J. Phys.* **6**, 33(2004). G. Ferrari and G. Cuoghi, *Phys. Rev. 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ArXiv

--- author: - 'S. Afach , C. A. Baker , G. Ban , G. Bison , K. Bodek , Z. Chowdhuri , M. Daum , M. Fertl [^1], B. Franke [^2], P. Geltenbort , K. Green , M. G. D. van der Grinten , Z. Grujic , P. G. Harris , W. Heil , V. Hélaine [^3], R. Henneck , M. Horras [^4], P. Iaydjiev [^5], S. N. Ivanov [^6], M. Kasprzak , Y. Kermaïdic , K. Kirch , P. Knowles [^7], H.-C. Koch , S. Komposch , A. Kozela , J. Krempel , B. Lauss , T. Lefort , Y. Lemière , A. Mtchedlishvili , O. Naviliat-Cuncic [^8], J. M. Pendlebury , F. M. Piegsa , G. Pignol , P. N. Prashant , G. Quéméner , D. Rebreyend , D. Ries , S. Roccia , P. Schmidt-Wellenburg , N. Severijns , A. Weis , E. Wursten , G. Wyszynski , J. Zejma , J. Zenner , G. Zsigmond' date: 'Received: date / Revised version: date' title: 'Measurement of a false electric dipole moment signal from $^{199}$Hg atoms exposed to an inhomogeneous magnetic field' --- =1 Introduction {#sec:intro} ============ Recent investigations characterizing frequency shifts for spins contained in vessels permeated with magnetic and electric fields $B$, $E$ have been motivated principally by the search for electric dipole moments (EDMs) of simple non-degenerate systems (neutron, atoms, molecules) and the potential discovery of new sources of CP violation [@ram2013]. Such experiments look for shifts, proportional to an applied electric field, of the Larmor precession frequency of stored particles. Any additional such shift is therefore a potential source of systematic errors. Among the few magnetic-field related spurious shifts, one is of particular concern: due to the motional magnetic field ${\mathbf E \times \mathbf v/{c^2} }$, a shift arises that is proportional to the electric-field strength and therefore mimics an EDM signal. Interestingly enough, ${\mathbf E \times \mathbf v/{c^2} }$ effects were already the main limiting factor for the early neutron beam experiments [@ramsey1982]. Then, with the advent of the storable ultra-cold neutrons (UCN), it was erroneously assumed for many years that this false EDM signal would vanish, based on the argument that the velocity of trapped particles averages to zero. The first correct and comprehensive calculation of this effect was given in Ref. [@pendlebury2004], in the context of an EDM experiment with stored particles. For completeness, it should be mentioned that Stark interference effects, such as the one reported for $^{199}{\rm Hg}$ in Ref. [@loftus2011], are also known to produce false EDM signals for atoms. The effect discussed in the present article is of a different nature, and to make the distinction we will refer to it as the motional false EDM. Our collaboration is conducting a program to search for the neutron EDM [@baker2011], using the new ultracold neutron (UCN) source [@Lauss2014] at the Paul Scherrer Institute (PSI). We are currently working with an upgraded version of the spectrometer [@baker2013] that was used to establish the best nEDM limit, $$\left| d_{{\rm n}}\right| < 2.9 \times 10^{-26}\, e\,\text{cm} \, (90\% \, \text{C.L.}),$$ at the Laue Langevin Institute (ILL) [@baker2006]. One distinct feature of this device is a mercury co-magnetometer [@green1998] using a spin-polarized vapor of $^{199}$Hg atoms that precess in the same volume as the neutrons. The nEDM analysis is then based on the ratio of the Larmor precession frequencies, $R=f_{{\rm n}}/f_{{{\rm Hg}}}$, which to first order is free of magnetic field fluctuations. However, both neutrons and mercury atoms are subject to a frequency shift that is proportional to the electric field, due to the unavoidable presence of magnetic-field gradients. As will be shown, the motional false neutron EDM, $d_{{\rm n}}^{\rm false}$, is negligible, at least at the current level of sensitivity. In contrast, the mercury-induced false nEDM $$d_{{\rm n}}^{{\rm false}, {{\rm Hg}}} = \frac{\gamma_{{\rm n}}}{\gamma_{{{\rm Hg}}}} d_{{{\rm Hg}}}^{\rm false} \approx 3.8\, d_{{{\rm Hg}}}^{\rm false},$$ where $d_{{{\rm Hg}}}^{\rm false}$ is the motional mercury false EDM and $\gamma_{{\rm n}}$, $\gamma_{{{\rm Hg}}}$ are the gyromagnetic ratios of the neutron and $^{199}$Hg respectively, is a major systematic effect that must be precisely controlled. One of the main improvements accomplished recently within the experiment is the installation of an array of cesium magnetometers that surrounds the precession chamber. This new device has made it possible to measure the magnetic field distribution, and thus to calculate the vertical gradient in the trap, which underlies the false EDM discussed here. In this article, we report on the first direct measurement of a motional false EDM signal for stored mercury atoms. A comparison to theoretical expectations is also presented. Theory of frequency shifts induced by magnetic field gradients: a brief reminder {#sec:theory} ================================================================================ Particles with a magnetic moment exposed to a magnetic field, ${\bf B_\textnormal{0}} = B_0 {\bf \hat{z}}$, precess at the Larmor frequency $f_{{\rm L}} = \gamma \, B_0 / 2 \pi$ where $\gamma$ is the gyromagnetic ratio. Because of experimentally unavoidable magnetic field gradients, the Larmor frequency of a particle moving through this field will be subject to a shift, known as the Ramsey-Bloch-Siegert (RBS) shift [@ramsey1955]. If an electric field ${\bf E}$ (parallel or anti-parallel to ${\bf B_\textnormal{0}}$) is applied – as is the case in experiments searching for EDMs – the moving particle will experience an additional motional magnetic field ${\mathbf B_v = \mathbf E \times \mathbf v/{c^2} }$. It is the interplay between this field and the magnetic field gradients that lies at the origin of a frequency shift proportional to the electric field strength, thus inducing a false EDM. As mentioned above, the first detailed calculation of such false EDMs for stored particles was given in Ref. [@pendlebury2004] in the context of the RAL-Sussex-ILL neutron EDM experiment [@baker2006]. The authors derived expressions for the two limiting cases: non adiabatic and adiabatic, corresponding to $2\pi f_{\rm L} \tau \gg 1$ and $2\pi f_{\rm L} \tau \ll 1$ respectively, where $\tau$ is the typical time particles take to cross the trap. Both regimes are of interest, since $^{199}{\rm Hg}$ atoms fall into the first category whereas UCNs fall into the second. More general results, valid for a broad range of frequencies, were obtained only for cylindrical symmetry and specular reflections. The expressions of the frequency shifts for the two limiting regimes are : $$\begin{aligned} \delta f_\textrm{L} &= \frac{\gamma^2 D^2}{32 \pi \, c^2} \frac{\partial B_0}{\partial z} E & \quad \textrm{(non adiabatic)} \label{eq_deltaOmegaNonAdiabatic}\\ \delta f_\textrm{L} &= \frac{v_{xy}^2}{4\pi\, B_0^2\, c^2} \frac{\partial B_0}{\partial z} E & \quad \textrm{(adiabatic),} \label{eq_deltaOmegaAdiabatic}\end{aligned}$$ where $\gamma$ is the gyromagnetic ratio, $D$ is the diameter of the trap, $c$ is the velocity of light and $v_{xy}$ is the particle velocity transverse to $B_0$. Note the absence of the gyromagnetic ratio in Eq. (\[eq\_deltaOmegaAdiabatic\]). Indeed, in the adiabatic case, the frequency shift can be interpreted as originating from a phase of purely geometric nature, or Berry’s phase [@ber1984; @commins1991], and is therefore independent of the coupling strength to the magnetic field. These results were then complemented and extended using the general theory of relaxation (Redfield theory) [@lamoreaux2005; @pignol2012], and then by solving the Schrödinger equation directly [@steyerl2014]. In Ref. [@pignol2012], an expression valid for arbitrary field distributions or trap shapes was obtained in the non-adiabatic limit : $$\begin{aligned} \delta f_\textrm{L} &= \frac{\gamma^2}{2 \pi c^2} \left\langle x B_x\, + yB_y \right\rangle E \quad \textrm{(non adiabatic),} \label{eq_deltaOmegaNonAdiabaticGeneralized}\end{aligned}$$ where the brackets refer to the average over the storage volume. For a cylindrical uniform gradient and a trap with cylindrical symmetry, Eq. (\[eq\_deltaOmegaNonAdiabaticGeneralized\]) reduces to Eq. (\[eq\_deltaOmegaNonAdiabatic\]). Using the relationship between the frequency shift and the false EDM, $${d}^{\rm false} = \frac{h}{2E} \delta f_{\rm L} (E)$$ where $h$ is Planck’s constant, together with Eqs. (\[eq\_deltaOmegaNonAdiabatic\]) and (\[eq\_deltaOmegaAdiabatic\]), one can now readily calculate the magnitude of the false EDMs for the mercury and for the neutron (both direct and mercury induced). Given our experimental conditions (see section \[sec:setup\]) and assuming a neutron velocity of 3 m/s, one obtains: $$\begin{aligned} &d_{{\rm n}}^{\rm false} = \frac{\partial B_0}{\partial z} \, 1.490 \times 10^{-29}\, e \, \text{cm}/\text{(pT/cm)} \\ & \nonumber\\ &d_{{{\rm Hg}}}^{\rm false} = \frac{\partial B_0}{\partial z} \, 1.148 \times 10^{-27}\, e \, \text{cm}/ \text{(pT/cm)} \label{falseHgEDMTheory}\\ & \nonumber \\ &d_{{\rm n}}^{\rm false, {{\rm Hg}}} = \frac{\partial B_0}{\partial z} \, 4.418 \times 10^{-27}\, e \, \text{cm}/\text{(pT/cm).}\end{aligned}$$ Considering a typical value of 10 pT/cm for the vertical ($z$ direction) gradient in our setup, we can conclude on the one hand that the direct false neutron EDM is negligible, at least at the current level of sensitivity. On the other hand, the mercury-induced false neutron EDM is a major systematic error that must be properly taken into account. Experimental apparatus {#sec:setup} ====================== The experimental study was performed with the nEDM spectrometer installed at the PSI UCN source. This room-temperature apparatus uses the Ramsey method of separated oscillatory fields [@green1998; @ramsey1950] to search for a shift, proportional to the strength of an applied electric field, in the neutron Larmor precession frequency. Under normal operation, polarized UCNs are stored in a $\sim20$ liter chamber (internal diameter [*D*]{} = 47 cm, height [*H*]{} = 12 cm), composed of a hollow polystyrene cylinder (coated with deuterated polystyrene) [@bodek2008; @kuzniak2008] and two disk-shaped aluminum electrodes coated with diamond-like carbon (Fig. \[fig:oILL\]). A cos$\theta$ coil produces a highly homogeneous magnetic field, $B_0 \approx 1 \, \mu\text{T}$, in the vertical direction while the two electrodes – the top one being connected to a high voltage (HV) source and the bottom one to ground potential – generate a strong electric field ($E \approx 10\, \text{kV/cm}$), either parallel or anti-parallel to ${\bf B_\textnormal{0} }$. In addition, a set of trim coils permits an optimization of the magnetic field uniformity at the $10^{-3}$ level. ![Schematic view of the precession chamber of the nEDM@PSI experiment.[]{data-label="fig:oILL"}](fig1){width="\linewidth"} The key to such experiments relies on the ability to control the magnetic field both in terms of stability and homogeneity. To this end, we use two highly sensitive and complementary atomic magnetometers based on mercury ($^{199}{\rm Hg}$) and cesium ($^{133}$Cs) atoms, respectively. Mercury is used in a co-magnetometer mode: polarized mercury atoms precess in the same volume as the neutrons, hence probing approximately the same space- and time-averaged magnetic field. Cesium is used in a set of external magnetometers surrounding the storage chamber. The former is an ideal tool to correct for field drifts, while the latter gives access to the spatial field distribution. The mercury co-magnetometer --------------------------- To date, $^{199}{\rm Hg}$ is the only atomic element that has been used as a co-magnetometer for a neutron EDM experiment. Thanks to its nuclear polarization, it benefits from long wall collision relaxation times, and polarization lifetimes larger than 100 s can be achieved. Moreover, it is one of the rare elements in which nuclear spin polarization can be created and monitored by optical means. It is worth noting that the best absolute EDM limit comes from an experiment using $^{199}{\rm Hg}$[@griffith2009][^9]: $$\left| d({\rm ^{199}Hg}) \right| < 3.1 \times 10^{-29}\, e\,\text{cm} \, (95\% \, {\rm CL}).$$ In our experiment, a vapor of mercury atoms is spin-polarized by optical pumping in a polarization chamber located underneath the precession chamber (Fig. \[fig:oILL\]). The operation of the co-magnetometer is synchronous with the nEDM measurement, and follows cycles about 300 s long. During neutron counting and filling, mercury atoms are continuously injected and optically pumped in the polarization chamber. Once the precession chamber is filled with UCNs, we let the vapor diffuse into the precession chamber where, after the application of a $\pi / 2$ pulse, the atoms freely precess around ${\bf B_\textnormal{0}}$ at a frequency of about 8 Hz. The interaction of the precessing atoms with a circularly polarized resonant probe beam produces a light-intensity modulation whose analysis yields the Larmor frequency of the atoms. One of the major drawbacks of the $^{199}{\rm Hg}$ co-magnetometer is its sensitivity to high voltage. As illustrated in Fig. \[fig:tauHg\], which displays the transverse polarization relaxation time T$_2$ versus the cycle number, sudden T$_2$ drops are systematically observed after each HV polarity reversal. The corresponding reduction of the signal amplitude directly affects the precision of the magnetometer. Fortunately, optimal performance can be recovered via discharge cleaning in an oxygen atmosphere. On average, the precision of the mercury co-magnetometer is of the order of 100 fT, equivalent to a magnetometric precision at the 0.1 ppm level per cycle. ![Transverse relaxation time T$_2$ of $^{199}\text{Hg}$ atoms (green points) together with the high voltage value (blue line) versus cycle number. Sudden drops of T$_2$ are observed after each polarity reversal.[]{data-label="fig:tauHg"}](fig2){width="\linewidth"} The array of cesium magnetometers {#sec:CsM} --------------------------------- An array of 16 cesium magnetometers (CsM) [@Knowles2009] allows measurement of the magnetic field distribution in the region of interest and, in particular, it gives us knowledge of the vertical gradient $\partial B_0 / \partial z$. Six HV-compatible (i.e. fully optically coupled) magnetometers were placed on top of the precession chamber, and ten standard ones below (Fig. \[fig:oILL\] and \[fig:HV-CsM\]). These laser-pumped magnetometers use a vapor of $^{133}$Cs atoms (gyromagnetic ratio $\gamma = 2 \pi \times 3.5\, \text{kHz}/\mu \text{T}$) and are operated in a phase-stabilized mode. They have a high statistical sensitivity ($\sim$ 100 fT for 40 s long measurements); however, they suffer from inaccurracies of their absolute field readings, with offsets that can be as high as 100 pT. They are therefore precise but not accurate. Finally, it is important to note that these magnetometers – like the mercury co-magnetometer – are scalar: they measure the magnitude of the magnetic field at the center of the bulb containing the cesium vapor. ![Picture of the six HV-compatible Cs magnetometers installed on the top HV electrode in Al enclosures. Optical fibers are also visible.[]{data-label="fig:HV-CsM"}](fig3){width="\linewidth"} Measurement and data analysis {#sec:analysis} ============================= A preliminary measurement with a limited number of CsM was performed in 2011, and led to a first result [@marlonThesis]. The present analysis is based on a dedicated data-taking period of 2 weeks’ duration in December 2013, where eight different gradient settings were explored: four with the magnetic field pointing upwards ($B_0^{\uparrow}$), and four downwards ($B_0^{\downarrow}$). Two trim coils were used to set a vertical gradient in addition to the $B_0$ field generated by the main coil. For each field configuration, about 500 cycles were recorded with a basic HV polarity pattern $(+\,-\,-\,+)$ and polarity changes every 20 cycles. The voltage was set to 120 kV, i.e. as high as possible to maximize the frequency shift while preserving a smooth operation (limited number of electrical breakdowns). As discussed above, the frequent polarity reversals induced a significant degradation of the mercury magnetometer’s sensitivity. Consequently, we decided to limit the free precession time to 40 s, a good compromise between sensitivity and the number of cycles. The mercury frequency was extracted using our standard “two windows” method [@chibane1995]. It consists of fitting the signal phase at either end of the signal, using data in two 15 s windows at the beginning and end of the time series. This method optimally takes into account possible frequency drifts during the precession time. During data taking, the mercury frequency uncertainty varied in the range 1-2 $\mu \text{Hz}$. Outputs from all 16 CsM were continuously recorded at a rate of 1 Hz, and a mean value of the magnetic field was calculated for time periods having an exact overlap with the mercury precession. We further made the approximation $$B_{\rm CsM} = \sqrt[2]{B_z^2 + B_T^2} \approx B_z(\vec{r}_{\rm CsM}),$$ where $B_T$ is a small transverse component. From several 3D mapping campaigns during which all coils (main and trim) were mapped, we know that this approximation is valid at the 10$^{-4}$ level. Gradient extraction {#subsec:gzExtrac} ------------------- We extracted the vertical gradient by fitting a harmonic polynomial expansion of the magnetic field to the CsM array data. The choice of harmonic polynomials ensures that the resulting expressions satisfy Maxwell’s equations. Due to the limited number of magnetometers the expansion was limited to the next-to-linear order (NLO), which involves 9 parameters: $$\begin{aligned} B_z(x,y,z) =\, &b_0 + g_x \, x + g_y \, y + g_z \, z + \nonumber \\ &g_{xx} (x^2 - z^2) + g_{yy} (y^2 - z^2)+ \nonumber \\ &g_{xy} xy + g_{xz} xz + g_{yz} yz. \label{eqn:fit_NLO} \end{aligned}$$ From expression (\[eqn:fit\_NLO\]), one can easily calculate the volume average of $B_z$ and of its vertical gradient, assuming a trap with cylindrical symmetry: $$\begin{aligned} B_0 \equiv \left\langle B_z \right\rangle &= b_0 +(g_{xx} + g_{yy})\left(\frac{D^2}{16}-\frac{H^2}{6}\right) \\ \left\langle \frac{\partial B_z}{\partial z}\right\rangle &= g_z. \label{eq:B_0}\end{aligned}$$ Let us now turn to the delicate task of estimating gradient uncertainties. The two main sources that have to be taken into account are the error on the magnetic field, and the extraction procedure. To assess their respective effects, extensive studies have been carried out using a toy model to generate known field distributions and check the extracted parameters [@victorThesis]. It was found that the errors coming both from the magnetometer offsets and from the expansion truncation never exceed 5 pT/cm. In addition, we used a technique known as the jackknife method to get an error directly from the data. It involves performing a series of $\chi^2$ minimizations (unweighted in our case) by removing one out of the 16 magnetometers at a time. The dispersion of the extracted parameters provides an estimate of the error. For the different field configurations, we systematically obtained errors in the range $10 \pm 5 \, \text{pT/cm}$, consistent with the model outcome. These jackknife errors were used subsequently in the analysis. Frequency shift measurement {#subsec:freqShift} --------------------------- A sample of a raw data time series $f_{{\rm Hg}}$ against cycle number is displayed in Fig. \[fig:fHg\_raw\], together with the corresponding high-voltage values. Despite the large point-to-point fluctuations and a slow linear drift, one can clearly observe a small but systematic correlation of the frequency shift with the electric field polarity. To correct for the slow magnetic field drift, we sliced the data relative to the electric field polarity and analyzed data sets corresponding to the $(+\,-\,-\,+)$ HV pattern. By doing so, any linear drift is exactly cancelled and higher orders are attenuated. For a data slice $(+\,-\,-\,+)$, corresponding to 40 cycles, the extracted frequency shift and its uncertainty are given by $$\delta f_{{{\rm Hg}}} = \left\langle f_{{{\rm Hg}}}^+ \right\rangle - \left\langle f_{{{\rm Hg}}}^- \right\rangle$$ and $$\Delta\delta f_{{{\rm Hg}}} = \sqrt{\Delta\left\langle f_{{{\rm Hg}}}^+ \right\rangle^2 + \Delta\left\langle f_{{{\rm Hg}}}^- \right\rangle^2},$$ where $\left\langle f_{{{\rm Hg}}}^{+(-)} \right\rangle$ and $\Delta\langle f_{{{\rm Hg}}}^{+(-)} \rangle$ stand for the mean frequency and its uncertainty as derived from the frequency distribution for the given HV polarity ($+$ or $-$). Finally, a weighted mean over the whole set of data slices was performed to estimate the electric-field induced frequency shift $\delta f_{\rm L} (E)$ for a given vertical gradient. ![Mercury frequency versus cycle number. The blue line shows the value of the applied high voltage.[]{data-label="fig:fHg_raw"}](fig4){width="\linewidth"} Results and discussion {#results} ====================== The final result is displayed in Fig. \[fig:dFalseVSgz\]. The motional false mercury EDM is plotted against the extracted vertical gradient $g_z$. The solid lines (red for $B_0^{\uparrow}$, blue for $B_0^{\downarrow}$) correspond to a global linear fit with a single free parameter, namely the slope $a$ ($\chi^2/\nu = 2.1/7$). ![Motional false mercury EDM versus the vertical gradient $g_z$ for $B_0^{\uparrow}$ (red up triangles) and $B_0^{\downarrow}$ (blue down triangles). The solid lines correspond to a linear fit, and the dashed line to the theory discussed in section \[sec:theory\]. The horizontal error bars are smaller than the symbol size. []{data-label="fig:dFalseVSgz"}](fig5){width="\linewidth"} We can now compare the measured slope to its theoretical expectation from Eq. 6: $$\begin{aligned} \left| a_{\rm exp} \right| &= 1.122(35) \times 10^{-27} \, e \,\text{cm}/(\text{pT/cm}),\\ \mathrm{and} \left| a_{\rm th} \right| &= 1.148 \times 10^{-27} \, e \,\text{cm}/(\text{pT/cm}).\end{aligned}$$ The agreement at the 1$\sigma$ level makes us confident that our magnetic gradient extraction procedure is reliable. This encouraging result is nonetheless not sufficient to directly control the mercury-induced false neutron EDM at the required level of sensitivity. Indeed, for an error of 10 pT/cm on the vertical gradient, Eq. (\[eq\_deltaOmegaNonAdiabatic\]) translates to a systematic error of $4.4 \times 10^{-26} e \,\text{cm}$ on the neutron EDM, which is already larger than the current limit. There is fortunately a way to circumvent this issue. In their last nEDM paper [@baker2006], the authors describe an analysis technique that enables one to find experimentally the working point with no vertical gradient and therefore no motional false EDM. This method, based on a tiny center of mass offset between the cold neutrons and the warmer mercury atoms, nevertheless induces some additional systematic errors. These errors were carefully assessed and found subdominant with a final result statistically limited. Whereas the use of the Hg comagnetometer is essential and does not limit our nEDM sensitivity for the time being, with a foreseen sensitivity of a few $10^{-27} e \,\text{cm}$ in the coming years, new magnetometry solutions will be needed in the future. We pursue an intensive R&D program on magnetometry using $^{133}$Cs but also $^3$He atoms [@koch2015]. In particular, efforts towards improving the absolute accuracy of the Cs magnetometers are currently underway [@grujic2015] as well as the implementation of Cs vector magnetometers [@afach2015]. In parallel, we have started design and construction of a next generation nEDM spectrometer [@baker2011] which, among other improvements, will benefit from a much better magnetic field control (passive and active). Advanced magnetometry and improved magnetic shielding will be combined with co-magnetometry or could even allow operation with only external magnetometers. Any possible mercury-induced false motional nEDM will therefore be much further suppressed or completely avoided. Conclusions {#sec:conclusions} =========== We have performed a measurement of a frequency shift proportional to the electric field strength for stored $^{199}$Hg atoms[^10], using a spectrometer devoted to the search for the neutron electric dipole moment at PSI. This shift, which we call the motional false EDM, originates from the combination of vertical magnetic field gradients with the motional magnetic field and could be measured for the first time thanks to the unique combination of a mercury co-magnetometer and an array of external cesium magnetometers. The agreement with a prediction based on the general Redfield theory of relaxation provides additional confidence in the validity of our gradient-extraction procedure as well as in our capability to measure and control the vertical gradient. The same method was used in a recent measurement of the neutron magnetic moment [@afach2014]. Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful to the PSI staff (the accelerator operating team and the BSQ group) for providing excellent running conditions, and we acknowledge the outstanding support of M. Meier and F. Burri. Support by the Swiss National Science Foundation Projects 200020-144473 (PSI), 200021-126562 (PSI), 200020-149211 (ETH) and 200020-140421 (Fribourg) is gratefully acknowledged. The LPC Caen and the LPSC acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-09–BLAN-0046. The Polish partners acknowledge The National Science Centre, Poland, for the grant No. UMO-2012/04/M/ST2/00556. This work was partly supported by the Fund for Scientific Research Flanders (FWO), and Project GOA/2010/10 of the KU Leuven. The original apparatus was funded by grants from the UK’s PPARC (now STFC). J. Engel, M. J. Ramsey-Musolf and U. van Kolck, Prog. in Particle and Nuclear Phys. **71**,(2013)21-74. N. F. Ramsey, Rep. Prog. Phys., Vol. 45 (1982) 95. J. M. Pendlebury [*et al.*]{}, Phys. Rev. A **70** (2004) 032102. T. H. Loftus [*et al.*]{}, Phys. Rev. Letters [**106**]{} (2011) 253002. C.A. Baker [*et al.*]{}, Physics Procedia [**17**]{} (2011) 159. 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Technol. **6** (1995) 11671. W. C. Griffith, M. D. Swallows, T. H. Loftus, [*et al.*]{}, Phys. Rev. Lett. [**102**]{} (2009) 101601. V. Helaine, PhD thesis, University of Caen Basse-Normandie (2014). H.-C. Koch [*et al.*]{}, submitted to Eur. Phys. J. D, arxiv.org/abs/1502.06366. Z. D. Grujić [*et al.*]{}, Eur. Phys. J. D [**69**]{} (2015). S. Afach [*et al.*]{}, submitted to Optics Express, arxiv.org/abs/1507.08523. S. Afach [*et al.*]{}, Phys Lett B [**739**]{} (2014) 128. [^1]: Now at University of Washington, Seattle WA, USA. [^2]: Now at Max-Planck-Institute of Quantum Optics, Garching, Germany. [^3]: Now at LPSC, Grenoble, France. [^4]: Present address: Hauptstrasse 60, CH-4455 Zunzgen, Switzerland. [^5]: On leave from INRNE, Sofia, Bulgaria. [^6]: On leave from PNPI, St. Petersburg, Russia. [^7]: Present address: Rilkeplatz 8/9, A-1040 Vienna, Austria. [^8]: Now at Michigan State University, East-Lansing, USA. [^9]: One may wonder why the effect discussed in the present article was not observed in that experiment. They actually use spectroscopy cells filled with 475 Torr of CO buffer gas acting as a UV quencher. Consequently, mercury atoms move in the diffusive regime where the motional false EDM essentially vanishes – in contrast to the ballistic regime of our mercury co-magnetometer. [^10]: It should be noted that, strictly speaking, we have only observed that frequency shifts were E-odd (measurements were done at different gradients but at a single HV value). However, the absence of physical justification for higher-order odd terms together with the excellent agreement with theory led us to disregard this possibility.

ArXiv

--- abstract: 'The new theoretical input to the analysis of the experimental data of the CCFR collaboration for $F_3$ structure function of $\nu N$ deep inelastic scattering is considered. This input comes from the next-to-next-to-leading order corrections to the anomalous dimensions of the Mellin moments of the $F_3$ structure function and N$^3$LO corrections to the related coefficient funtions. The QCD scale parameter $\Lambda_{\overline{MS}}^{(4)}$ is extracted from higher-twist independent fits. The results obtained demonstrate the minimization of the influence of perturbative QCD contributions to the value of $\Lambda_{\overline{MS}}^{(4)}$.' --- [CERN-TH/2000-343]{}\ hep-ph/0012014 [**Application of new multiloop QCD input\ to the analysis of $xF_3$ data**]{}\ $^{(a)}$, [**G. Parente**]{}$^{(b,1)}$ and [**A.V. Sidorov**]{}$ ^{(c,2)}$\ (a) Theoretical Physics Division, CERN CH - 1211 Geneva 23 and\ Institute for Nuclear Research of the Academy of Sciences of Russia, 117312 Moscow, Russia\ (b) Department of Particle Physics, University of Santiago de Compostela,\ 15706 Santiago de Compostela, Spain\ (c) Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia [**ABSTRACT**]{} The new theoretical input to the analysis of the experimental data of the CCFR collaboration for $F_3$ structure function of $\nu N$ deep inelastic scattering is considered. This input comes from the next-to-next-to-leading order corrections to the anomalous dimensions of the Mellin moments of the $F_3$ structure function. The QCD scale $\Lambda_{\overline{MS}}^{(4)}$ is extracted from higher-twist independent fits. The results obtained demonstrate the minimization of the influence of perturbative QCD contributions to the value of $\Lambda_{\overline{MS}}^{(4)}$. [*Based on Contributed to the Proceedings of Quarks-2000 International Seminar, Pushkin, May 2000, Russia and of ACAT’2000 Workshop, Fermilab, October 2000, USA*]{} $^{1}$ Supported by Xunta de Galicia (PGIDT00PX20615PR) and CICYT (AEN99-0589-C02-02)\ $^{2}$ Supported by RFBI (Grants N 99-01-00091, 00-02-17432) and by INTAS call 2000 (project N587) CERN-TH/2000-343\ November 2000 [**Application of new multiloop QCD input\ to the analysis of $xF_3$ data** ]{} [**A.L. Kataev$^{a}$, G. Parente$^{b}$ and A.V. Sidorov$^{c}$**]{} [$^{a}$Theoretical Physics Division, CERN, CH-1211 Geneva, Switzerland and\ Institute for Nuclear Research of the Academy of Sciences of Rusia,\ 117312 Moscow, Russia\ $^{b}$Department of Particle Physics, University of Santiago de Compostela,\ 15706 Santiago de Compostela, Spain\ $^{c}$ Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,\ 141980 Dubna, Russia]{} Introduction ============ One of the most important current problems of symbolic perturbative QCD studies is the analytical evaluation of the next-to-next-to-leading order (NNLO) QCD corrections to the kernels of the DGLAP equations  [@DGLAP] for different structure functions of the deep-inelastic scattering (DIS) process. In this note we will apply the related information for the fixation of definite uncertainties of the NNLO analysis [@KKPS; @KPS1] of experimental data for $F_3$ structure function (SF) data of $\nu N$ DIS, provided by the CCFR collaboration [@CCFR] at the Fermilab Tevatron and present preliminary results of our improved fits which will be described elsewhere [@KPS2]. Methods of analysis of DIS data =============================== There are several methods of analysis of the experimental data of DIS in the high orders of perturbation theory. The traditional method is based on the solution of the DGLAP equation, which in the case of the $F_3$ SF has the following form: $$Q^2\frac{d}{dQ^2}F_3(x,Q^2)=\frac{1}{2}\int_x^{1}\frac{dy}{y} \bigg[V_{F_3}(y,A_s)+\beta(A_s)\frac{\partial{\rm ln}C_{F_3}(y,A_s)} {\partial A_s}\bigg]F_3\bigg(\frac{x}{y},Q^2\bigg)$$ where $A_s=\alpha_s/(4\pi)$, $\mu\partial A_s/\partial\mu=\beta(A_s)$ is the QCD $\beta$-function and $C_{F_3}(y,A_s)$ is the coefficient function, defined as $$C_{F_3}(y,A_s)=\sum_{n\geq 0} C_{F_3,n}(y) \bigg(\frac{\alpha_s}{4\pi}\bigg)^{n}$$ and $V_{F_3}(z)$ is the DGLAP kernel, related to a non-singlet (NS) $F_3$ SF. The solution of Eq.(1) is describing the predicted by perturbative QCD violation of scaling [@Bj] or automedeling [@BVT] behaviour of the DIS SFs by the logarithmically decreasing order $\alpha_s$-corrections. The coefficient function we are interested in has been known at the NNLO for quite a long period. The term $C_{F_3,2}(y)$ was analytically calculated in Ref.[@VZ]. The results of these calculations were confirmed recently [@MV] using a different technique. The kernel $V_{F_3}(z,\alpha_s)$ is analytically known only at the NLO. However, since there exists a method of symbolic evaluation of multiloop corrections to the renormalization group functions in the $\overline{MS}$-scheme [@T] and its realization at the FORM system, it became possible to calculate analytically the NNLO corrections to the $n=2,4,6,8,10$ Mellin moments of the NS kernel of the $F_2$ SF [@Larin]. They have the following expansion: $$-\int_0^{1} z^{n-1}V_{NS,F_2}(z,\alpha_s)dz = \sum_{i\geq 0}\gamma_{NS,F_2}^{(i)}(n) \bigg(\frac{\alpha_s}{4\pi}\bigg)^{i+1}$$ and are related to the anomalous dimension of NS renormalization group (RG) constants of $F_2$ SF[^1] : $$\mu\frac{\partial\ln Z_n^{NS,F_2}}{\partial\mu} =\gamma_{NS,F_2}^{(n)}(\alpha_s)~~~~.$$ These results were used in the process of the fits of Refs.[@KKPS; @KPS1] of the CCFR data for the $F_3$ SF with the help of the Jacobi polynomial method [@Jacobi]. It allows the reconstruction of the SF $F_3$ from the [**finite**]{} number of Mellin moments $M_{j,F_3}(Q^2)$ of the $xF_3$ SF: $$F_3^{N_{max}}(x,Q^2)=w\sum_{n=0}^{N_{max}} \Theta_n^{\alpha,\beta}(x)\sum_{j=0}^{n}c_j^{(n)}(\alpha,\beta) M_{j+2,F_3}^{TMC}(Q^2)$$ where $w=w(\alpha,\beta)=x^{\alpha-1}(1-x)^{\beta}$, $\Theta_n^{\alpha,\beta}$ are the orthogonal Jacobi polynomials and $c_j^{(n)}(\alpha,\beta)$ is the combination of Euler $\Gamma$-functions, which is factorially increasing with increasing of $N_{max}$ and thus $n$. The expressions for $M_{j+2,F_3}^{TMC}(Q^2)$ include the information about Mellin moments of the coefficient function $$C_{n,F_3}(Q^2)=\int_0^{1}x^{n-1}C_{F_3}(x,\alpha_s)dx =\sum_{i\geq 0}C^{(i)}(n)\bigg(\frac{\alpha_s}{4\pi}\bigg)^{i}$$ where $C^{(0)}(n)=1$. The target mass corrections, proportional to $(M_N^2/Q^2)M_{j+4,F_3}(Q^2)$, are also included into the fits. Therefore, the number of the Jacobi polynomials $N_{max}=6$ corresponds to taking into account the information about RG evolution of 10 moments, and $N_{max}=9$ presumes that the evolution of $n=13$ number of Mellin moments is considered. The procedure of reconstruction of $F_3(x,Q^2)$ from the finite number of Mellin moments and the related fits of the experimental data were implemented in the form of FORTRAN programs. The details of the fits of the CCFR data, based on RG evolution of 10 moments, are desribed in Refs.[@KKPS; @KPS1] (for the brief review see Ref.[@KPSB]). In the process of these analyses the following approximations were made: a) it was assumed that for a large enough number of moments, $\gamma_{NS,F_3}^{(n)}(\alpha_s)\approx\gamma_{NS,F_2}^{(n)}(\alpha_s)$; b) since the odd NNLO terms of $\gamma_{NS,F_2}^{(n)}$ are explicitly unknown, they were fixed using the smooth interpolation procedure proposed in Ref.[@PKK]. It was known that the additional contributions, proportional to the $d^{abc}d^{abc}$ structure of the colour gauge group $SU(N_c)$ are starting to contribute to the coefficients of $\gamma_{NS,F_3}^{(n)}(\alpha_s)$ from the NNLO [@KPS1]. In the process of the analysis of Refs.[@KKPS; @KPS1] it was assumed that they were not dominating and therefore were not taken into account. New inputs for the fits ======================= After recent explicit analytical evaluation of the NNLO coefficients of $\gamma_{NS,F_3}^{(n)}(\alpha_s)$ at $n=$3,5,7\ ,9,11,13 (see Ref.[@RV]) it became possible to fix this uncertainty (it is worth noting that the NNLO contribution to $\gamma_{NS,F_2}^{(n)}(\alpha_s)$ for $n=$12 was analytically evaluated in Ref.[@RV] also). To estimate the NNLO terms of $\gamma_{NS,F_3}^{(n)}(\alpha_s)$ at $n=$4,6,8,10,12 we applied the smooth interpolation procedure, identical to the one used to estimate the odd NNLO terms of $\gamma_{NS,F_2}^{(n)}(\alpha_s)$, while the numerical value of $\gamma_{NS,F_3}^{(2)}(2)$ was fixed with the help of an extrapolation procedure, where we have not used the value at $n=1$. The justification and more details of this procedure will be given elsewhere [@KPS2]. The used numerical results of the NNLO contributions $\gamma_{NS,F_3}^{(2)}(n)$ with and without $d^{abc}d^{abc}$-factors are presented in Table 1, where we marked in parenthesis the estimated even terms. The expressions for the NNLO contributions to the NS anomalous dimensions terms $\gamma_{NS,F_2}^{(2)}(n)$ are also given for comparison. They include the numerical results of the explicit analytical calculations of Refs.[@Larin; @RV], normalized to $f=4$ numbers of active flavours, and the results of the smooth interpolation procedure, in parenthesis, applied for estimating explicitly uncalculated odd terms. The satisfactory agreement between the numbers in the second and third columns supports the assumptions a) and b) mentioned above. $n$ $\gamma_{NS,F_3}^{(2)}(n)$ $d^{abc}d^{abc}$ neglected in $\gamma_{NS,F_3}^{(2)}(n)$ $\gamma_{NS,F_2}^{(2)}(n)$ ----- ---------------------------- ---------------------------------------------------------- ---------------------------- 2 (631) (585) 612.06 3 861.65 836.34 (838.93) 4 (1015.37) (1001.42) 1005.82 5 1140.90 1132.73 (1135.28) 6 (1247) (1241.21) 1242.01 7 1338.27 1334.32 (1334.65) 8 (1420) (1416.73) 1417.45 9 1493.47 1491.13 (1492.02) 10 (1561) (1558.85) 1559.01 11 1622.28 1620.73 (1619.83) 12 (1679.81) (1677.70) 1678.40 : The numerical expressions of the NNLO coefficients of anomalous dimensions of the $n$-th NS moments of the $F_3$ and $F_2$ SFs at $f=4$. The numbers in parenthesis are the estimated results. []{data-label="tab:a"} $n$ $C^{(1)}(n)$ $C^{(2)}(n)$ $C^{(3)}(n)$ $C^{(3)}(n)_{[1/1]}$ ----- -------------- -------------- -------------- ---------------------- 1 $-$4 $-$52 $-$644.35 $-$676 2 $-$1.78 $-$47.47 ($-$1127.45) $-$1268 3 1.67 $-$12.72 $-$1013.17 97 4 4.87 37.12 ($-$410.66) 283 5 7.75 95.41 584.94 1175 6 10.35 158.29 (1893.58) 2421 7 12.72 223.90 3450.47 3940 8 14.90 290.88 (5205.39) 5679 9 16.92 358.59 7120.99 7602 10 18.79 426.44 (9170.21) 9677 11 20.55 494.19 11332.82 11884 12 22.20 561.56 (13590.97) 14205 13 22.76 628.45 15923.91 17353 : The numerical expressions for the coefficients of the coefficient functions for $n$-th Mellin moments of the $F_3$ SF up to N$^3$LO and their \[1/1\] Padé estimates. []{data-label="tab:b"} In Table 2 the numerical expressions for the coefficients of Eq.(6) for $f=4$ numbers of active flavours are given. They include the results of explicit calculations of N$^3$LO corrections of odd moments [@RV], supplemented with the information about the coefficients of the Gross–Llewellyn Smith sum rule [@GL; @LV], defined by the $n=1$ Mellin moment of the $xF_3$ SF. The numbers in parenthesis are the results of the interpolation procedure. In the last column we present the values of $C^{(3)}(n)$, obtained with the help of the \[1/1\] Padé estimates approach. One can see that the agreement of Padé estimates with the used N$^3$LO results is good in the case of the Gross–Llewellyn Smith sum rule (this fact was already known from the considerations of Ref.[@Samuel]). In the case of $n=2$ and $n\geq 6$ moments the results are also in satisfactory agreement. Indeed, one should keep in mind that the difference between the results of column 3 and 4 of Table 2 should be devided by the factor $(1/4)^3$, which comes from our definition of expansion parameter $A_s=\alpha_s/(4\pi)$. Note, that starting from $n\geq 6$ the results of application of \[0/2\] Padé approximants, which in accordance with analysis of Ref.[@Gardi] are reducing scale-dependence uncertainties, are even closer to the the results of the interploation procedure (for the comparison of the estimates, given by \[1/1\] and \[0/2\] Padé approximants in the case of moments of $xF_3$ SF see Ref.[@KPS1], while in Ref.[@PP] the similar topic was analysed within the quantum mechanic model). For $n=$3,4 the interpolation method gives completely different results. The failure of the application of the Padé estimates approach in these cases might be related to the irregular sign structure of the perturbative series under consideration. $N_{max}$ $\Lambda_{\overline{MS}}^{(4)}$ (MeV) ---------------------------------- ----------- --------------------------------------- result of Ref.[@KPS1]: NLO 6 339$\pm$36 7 340$\pm$37 8 343$\pm$37 9 345$\pm$37 10 339$\pm$36                        NNLO 6 326$\pm$35 NNLO results with 6 325$\pm$35 inclusion of NNLO 7 326$\pm$31 terms of $\gamma_{NS,F_3}^{(n)}$ 8 329$\pm$36 9 332$\pm$36 N$^3$LO approximate results with 6 324$\pm$33 inclusion of the interpolated 7 322$\pm$33 values of $C^{(3)}$(n)-terms 8 325$\pm$34 9 326$\pm$33 : The results of the fits of the CCFR data for $xF_3$ SF, taking into account the NNLO approximation for $\gamma_{F_3,NS}^{(n)}$. The initial scale of RG evolution is $Q_0^2$=20 GeV$^2$. []{data-label="tab:c"} Some results of the fits ======================== In Table 3 we present the comparison of the results of the determination of the $\Lambda_{\overline{MS}}^{(4)}$ parameter, made in Ref.[@KPS1], with the new ones, obtained by taking into account more definite theoretical information. Since NNLO corrections to the anomalous dimensions and N$^3$LO contributions to the coefficient functions of odd moments of the $xF_3$ SF are now known up to $n=$13, it became possible to study the dependence of the results of the fits from the value of $N_{max}$, which we can now vary from $N_{max}=6$ to $N_{max}=9$. It should be mentioned that for $N_{max}=6$ the new NNLO result and its $Q_0^2$ dependence are in agreement with the results of Ref.[@KPS1]. However, the incorporation of higher number of moments, and thus the increase of $N_{max}$, make the NNLO (and approximate N$^3$LO ) results almost independent from the variation of $Q_0^2$ in the interval 5 GeV$^2$–100 GeV$^2$. This is the welcome feature of including into the fits the results of the new analytical calculations of the NNLO corrections to anomalous dimensions and N$^3$LO corrections to the coefficient functions of odd moments of the $xF_3$ SF [@RV]. Comparing now the central values of the results of the stable NLO fits of Ref.[@KPS1] with the new NNLO and N$^3$LO results, we observe the decrease of the theoretical uncertainties and, probably, the saturation of the predictive power of the corresponding perturbative series at the 4-loop level. More detailed results of our fits, including extraction of $\alpha_s(M_Z)$, its scale dependence and the information about the behaviour of twist-4 corrections at the NNLO and N$^3$LO, in the case of $N_{max}=9$, will be described elsewhere [@KPS2]. [**Acknowledgements**]{} We present here some of the results, reported at Quarks-2000 International Seminar, Pushkin, May 2000, together the first results from Ref.[@KPS2]. We wish to thank the participants of this productive workshop, and especially A. N. Tavkhelidze and F. J. Ynduráin for their interest and inspiring discussions. One of us (GP) would like to thank the Organizing Committee of Quarks-2000 Seminar for their hospitality in Pushkin and St.Petersburg. We are also grateful to S. A. Larin for constructive comments on the outcome of our previous research [@KKPS; @KPS1], summarized in the talk at Quarks-2000. We are grateful to D. V. Shirkov and V. A. Ilyin for presnting the results of our previous research in the Plenary Meeting talk of the ACAT’2000 International Workshop, Fermilab, 16-20 October 2000 and M. Fischler, who supported the submission of the summary of our previous results to the Poster Session of ACAT’2000. It is also a pleasure to thank S. Catani and A. Peterman for discussions of subjects related to the material of this contribution and of our continuing research. [1]{} Gribov, V. N., and Lipatov L. N., *Sov. J. Nucl. 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ArXiv

--- abstract: | We investigate the nonlinear current-voltage characteristic of mesoscopic conductors and the current generated through rectification of an alternating external bias. To leading order in applied voltages both the nonlinear and the rectified current are quadratic. This current response can be described in terms of second order conductance coefficients and for a generic mesoscopic conductor they fluctuate randomly from sample to sample. Due to Coulomb interactions the symmetry of transport under magnetic field inversion is broken in a two-terminal setup. Therefore, we consider both the symmetric and antisymmetric nonlinear conductances separately. We treat interactions self-consistently taking into account nearby gates. The nonlinear current is determined by different combinations of second order conductances depending on the way external voltages are varied away from an equilibrium reference point (bias mode). We discuss the role of the bias mode and circuit asymmetry in recent experiments. In a photovoltaic experiment the alternating perturbations are rectified, and the fluctuations of the nonlinear conductance are shown to decrease with frequency. Their asymptotical behavior strongly depends on the bias mode and in general the antisymmetric conductance is suppressed stronger then the symmetric conductance. We next investigate nonlinear transport and rectification in chaotic rings. To this extent we develop a model which combines a chaotic quantum dot and a ballistic arm to enclose an Aharonov-Bohm flux. In the linear two-probe conductance the phase of the Aharonov-Bohm oscillation is pinned while in nonlinear transport phase rigidity is lost. We discuss the shape of the mesoscopic distribution of the phase and determine the phase fluctuations. author: - 'M. L. Polianski' - 'M. Büttiker' title: Rectification and nonlinear transport in chaotic dots and rings --- Introduction {#sec:intro} ============ A large part of modern physics is devoted to nonlinear classical and quantum phenomena in various systems. Such effects as the generation of the second harmonic or optical rectification are known from classical physics, while quantum electron pumping through a small sample due to interference of wave functions is a quantum nonlinear effect. Experiments on nonlinear electrical transport often combine classical and quantum contributions. A macroscopic sample without inversion center [@UFN] exhibits a current-voltage characteristic which with increasing voltage departs from linearity due to terms proportional to the square of the applied voltage. If now an oscillating (AC) voltage is applied, a zero-frequency current (DC) is generated. If the sample is sufficiently small, quantum effects can appear due to the wave nature of electrons. The uncontrollable distribution of impurities or small variations in the shape of the sample result in quantum contributions to the DC which are random. For a mesoscopic conductor with terminals $\a ,\b, ... $ we can describe the quadratic current response in terms of second order conductances $\G_{\a\b\g}$. They relate voltages $V_{\b,\oo}$ applied at contacts or neighboring gates $\b$ at frequency $\oo$ to the current at zero frequency at contact $\a$, $$\begin{aligned} \label{eq:IV} I_\a &=&\sum_{\b\g} \G_{\a\b\g} |V_{\b,\oo}-V_{\g,\oo}|^2.\end{aligned}$$ The second order conductances include in detail the role of the shape and the nearby conductors (gates). They depend on external parameters like the frequency of the perturbation, temperature, magnetic field or the connection of the sample to the environment. We concentrate here on the quantum properties of nonlinear conductance through coherent chaotic samples. Chaos could result from the presence of impurities (disorder) or random scattering at the boundaries (ballistic billiard). Due to electronic interference the sign of this effect is generically random even for samples of macroscopically similar shape. [@WW; @AK; @KL] When averaged over an ensemble, the second order conductances vanish. As a consequence, for a fully chaotic sample there is no classical contribution to the DC and the nonlinear response is the result of the sample-specific quantum fluctuations. Interestingly enough, from a fundamental point of view these fluctuations of nonlinear conductance are sensitive to the presence of Coulomb interactions and magnetic field. While interactions strongly affect the fluctuations’ amplitude, their sign is easily changed by a small variation of magnetic flux $\Phi$, similarly to universal conductance fluctuations (UCF) in linear transport. More importantly, without interactions the current (\[eq:IV\]) through a two-terminal sample is a symmetric function of magnetic field, just like linear conductance. However, the idea that Coulomb interactions are responsible for magnetic-field asymmetry in nonlinear current was recently proposed theoretically [@SB; @SZ] and demonstrated experimentally in different mesoscopic systems. [@wei; @Zumbuhl; @marlow; @ensslin; @Bouchiat; @Bouchiat_preprint] (Various aspects of nonlinear quantum [@PB; @Coulomb; @Tsvelik; @PhysicaE] and classical [@AG] charge and spin transport [@Feldman] have been discussed later on.) It is useful to consider (anti) symmetric second order conductance $\Ga,\Gs$ defined as $$\begin{aligned} \label{eq:IVfield} {\genfrac{\{}{\}}{0pt}{}{{\mathcal G}_{s}(\Phi)}{{\mathcal G}_{a}(\Phi)}} &= &\frac{h}{\nu_s e^3}\frac{\DD^2}{2\DD \tilde V^2}\left(\frac{I(\Phi)\pm I(-\Phi)}{2}\right)_{\tilde V\to 0},\end{aligned}$$ where $\tilde V$ is a combination of voltages at the gates and contacts varied in the experiment and $\nu_s$ accounts for the spin degeneracy. We emphasize that, depending on the way voltages are varied, experiments probe different linear combinations of second order conductance elements $\G_{\a\b\g}$ of Eq. (\[eq:IV\]). From now on we will simply call $\Gs ,\Ga$ conductances and if no confusion is possible leave out the expression “second order”. In the presence of a DC perturbation the mesoscopic averages of antisymmetric [@SB; @SZ] and symmetric [@PB; @PhysicaE] conductances vanish, and it is their sample-to-sample fluctuations that are measured. Experiments are usually performed for strongly interacting samples and the magnetic-field components $\Gs,\Ga$ allow one to evaluate the strength of interactions. [@Zumbuhl; @Bouchiat] In previous theoretical works on nonlinear transport through chaotic dots several important issues have been discussed using Random Matrix Theory (RMT). [@SB; @PB; @PhysicaE] Sánchez and Büttiker [@SB] found the fluctuations of $\Ga$ in a dot with arbitrary interaction strength at zero temperature and broken time-reversal symmetry due to magnetic field. Polianski and Büttiker considered the statistics of both $\Ga$ and $\Gs$ for arbitrary flux $\Phi$, the temperature $T$, and the dephasing rate. [@PB] The fluctuations of relative asymmetry $\A=\Ga/\Gs$ and the role of the contact asymmetry on this quantity were discussed in Ref.. The results of RMT approach were compared with experimental data of Zumbühl [@Zumbuhl] and Angers [@Bouchiat] Previously we considered statistics of $\Ga,\Gs$ for the dots where only one DC voltage was varied. However, to avoid parasitic circuit effects some experiments are performed varying several voltages simultaneously. Surprisingly, the importance of the chosen combination of varied voltages (bias mode) was not addressed before in the literature. It turns out that an experiment where only one of the voltages is varied [@marlow; @Lofgren; @Bouchiat] or two voltages are asymmetrically shifted [@Zumbuhl; @ensslin] measure different combinations of nonlinear conductances $\G_{\a\b\g}$. For example, in a weakly interacting dot in the first mode we found that $\Gs\gg \Ga$, [@PhysicaE] but in the second bias mode the fluctuations of nonlinear current are strongly reduced, so that $\Gs\sim \Ga$. It is also important to generalize the previous treatment of the nonlinear current to mesoscopic systems biased by an AC-voltage at [*finite*]{} frequency. The resulting DC is sometimes called “photovoltaic current”. We expect that in such mesoscopic AC/DC converters the interactions lead to significant magnetic field-asymmetry in the DC-signal. The rectification effect of mesoscopic diffusive metallic microjunctions was theoretically considered by Falko and Khmelnitskii [@FK] assuming that electrons do not interact. Therefore, a magnetic-field asymmetry was not predicted and was also not observed in subsequent experiments. [@Bykov; @BykovAB; @Bartolo; @Lin; @Liu] The fact that the interactions induce a magnetic field-asymmetry of the photovoltaic current when the size of the sample is strongly reduced was recently demonstrated in Aharonov-Bohm rings by Angers [@Bouchiat_preprint] However, it turns out that for an AC perturbation another quantum interference phenomenon, also quadratic in voltage, random in sign and magnetic field-asymmetric, contributes to the DC. Due to [*internal*]{} AC- perturbations of the sample, the energy levels are randomly shifted and a phenomenon commonly referred to as “quantum pumping” [@pump; @SAA] appears. Brouwer demonstrated that two voltages applied out of phase generate pumped current linear in frequency, while a single voltage pumps current quadratic in frequency $\oo$. [@pump] Although theory usually considers small (adiabatic) frequencies, a photovoltaic current could be induced by voltages applied at arbitrary frequency. At small $\oo$ the pumping contribution vanishes and only the rectification effect survives. In contrast, it is not clear what the ratio of pumping current to rectification current is at large $\oo$. To distinguish between different mechanisms it is therefore important to consider rectification in a wide range of frequencies in detail. [![Top: Quantum pumping sources include oscillating voltage $V_{\rm p}(\oo)$ at the locally applied gate, which slightly changes the shape of the dot (shown dashed), or microwave antenna emitting photons with energy $\hbar\oo$ into the dot. Bottom: Rectification sources include external bias $V_{1,2}(\oo)$, top gate voltage $V_{0}(\oo)$ with capacitance $C_{g0}$, and parasitic coupling of $V_{\rm p}(\oo)$ due to stray capacitances $C_{\rm stray}$. Microwave antenna can emit photons to the contacts and lead not only to photon-assisted AC transport but also to a rectified DC.[]{data-label="fig:dot"}](3dot3d.eps "fig:"){width="9.cm"}]{} We point here to a crucial difference between rectification and pumping contributions to the photovoltaic effect. Rectification results from external perturbations or the perturbations that can be reduced to the exterior by a gauge transformation. Typical examples are external AC-bias, or gate voltage which shifts all levels uniformly, [@pedersen] or a bias induced by parasitic (stray) capacitance which connects sources of possible internal perturbations to macroscopic reservoirs, [@pump_rectif] see the bottom panel in Fig. \[fig:dot\]. Pumping, on the other hand, is due to internal perturbations like those of a microwave antenna [@VAA] or a locally applied gate voltage, [@pump] see the top panel in Fig. \[fig:dot\]. Internal and external sources affect the Schrödinger equation and its boundary conditions, respectively. In experiment pumping and rectification, often considered together under the name of photovoltaic effect, [@Bykov; @Liu; @Lin; @Bartolo; @BykovAB; @Kvon] are hard to distinguish. Can one clearly separate quantum pumping from rectification effects? To distinguish them it was proposed to use magnetic field asymmetry of DC as a signature of a true quantum pump effect. In Refs. and rectification by (non-interacting) quantum dot was due to stray capacitances of reservoirs with pumping sources. The rectified current was found to be symmetric with respect to $\Phi\to -\Phi$.[@pump_rectif] While such field-symmetric rectification dominated in the experiments of Switkes [@Switkes] and DiCarlo [@DiCarlo] at MHz frequencies, an asymmetry $\Phi\to -\Phi$ observed at larger GHz frequencies seemed to signify a quantum pump effect. [@DiCarlo] It was noted that the Coulomb interactions treated self-consistently do not lead to any drastic changes in the mesoscopic distribution of a pumped current.[@pump] Probably, that is why the effect of interactions on the rectification have not been considered yet, even though the Coulomb interaction in such dots is known to be strong.[@Zumbuhl] However, as it turned out later, Coulomb interactions are responsible for magnetic-field asymmetry in nonlinear transport through quantum dots. [@SB] Similarly this could be expected for rectification as well. Then the magnetic field asymmetry alone can not safely distinguish pumping from rectification. Therefore we thoroughly examine the frequency dependence of the magnetic-field (anti)symmetric conductances $\Ga,\Gs$. Here we neglect any quantum pumping effects and their interference with rectification. [@Vavilov05; @Moskalets_AC] While the role of Coulomb interactions and the full frequency dependence in quantum pumping are yet to be explored, here we answer two important questions concerning a competing mechanism, rectification: (1) In the DC limit $\oo\to 0$ for a strongly interacting quantum dot $\Ga$ and $\Gs$ are of the same order. Is this also the case at finite frequencies? (2) How are the experimental data affected by the bias mode for alternating voltages? A number of very recent experiments on nonlinear DC transport [@ensslin; @Bouchiat] and AC rectification [@Bouchiat_preprint] have used submicron ring-shaped samples with a relatively large aspect ratio. In this work we develop a model of a ring which includes chaotic dynamics due to possible roughness of its boundary and/or the presence of impurities. Similarly to quantum dots, the two-terminal nonlinear conductance of such a ring is field-asymmetric because field-asymmetry exists in each arm. In particular, this leads to deviations of the phase in AB oscillations from $0\mbox{(mod) }\pi$ which characterizes linear conductance obeying Onsager symmetry relations. Experiments find that the amplitude and phase of AB oscillations exhibit rather curious properties. For example, the DC experiment of Leturcq [@ensslin] finds that during many AB oscillations with period $hc/e$ the phase is well-defined. The experiment demonstrates that a nearby gate can vary the phase of the AB oscillations over the full circle. The amplitude of the second harmonic $hc/2e$ is strongly suppressed. On the other hand, the DC experiment [@Bouchiat] and AC experiment [@Bouchiat_preprint] of Angers find that the phase can be defined only for few oscillations at low magnetic fields. For high frequencies, the phase fluctuates strongly as function of frequency. Both in the nonlinear and the rectified current the amplitude of the second harmonic $hc/2e$ in AB oscillations is always comparable with the first harmonic $hc/e$. This is in contrast with the experiments in Ref.. Although we do not fully address all these questions here, our model of a chaotic ring allows us to consider them at least on a qualitative level. Principal results ================= To introduce the reader to the problem of nonlinear transport in Sec. \[sec:bias\] we first qualitatively discuss the Coulomb interaction effect in the simplest DC problem. In reality the statistical properties of conductances $\G_{\a\b\g}$ in Eq. (\[eq:IV\]) are sensitive to electronic interference but to assess the role of Coulomb interactions we can consider a specific sample. In contrast to linear transport, it turns out that the nonlinear current strongly depends on the way voltages at the contacts and/or nearby conductors are varied from their equilibrium values (bias mode). For example, we find that the experiments when only one voltage at the contact is varied [@Lofgren; @marlow; @Bouchiat] or when two contact voltages are shifted oppositely [@Zumbuhl; @ensslin] measure different nonlinear currents. Indeed, for a current $I(\{V_i\})$, bilinear in voltages, its second derivative should depend on the chosen direction in the space of voltages $\{V_i\}$. Interestingly, a sample with weak interactions is very sensitive to the choice of the bias mode, which we attribute to the strong effect of capacitive coupling of the sample with nearby conductors. To make our arguments quantitative and consider the role of magnetic flux $\Phi$ for a quantum dot which is (generally) AC-biased at arbitrary frequency $\oo$, in Sec. \[sec:DC\] we take electronic interference into account. Having done that, we illustrate the interplay between interactions and interference on several important examples. First, we consider nonlinear transport due to a constant applied voltage and then consider rectification of AC voltages. For a two-terminal dot, in a generally asymmetric circuit (capacitive couplings included), in Sec. \[sec:2terminal\] we find the statistics of (anti) symmetric conductances $\Ga,\Gs$ defined in Eq. (\[eq:IVfield\]). Both $\Ga$ and $\Gs$ vanish on average. Quantum fluctuations of $\Gs$ strongly depend on the interaction strength, circuit asymmetry and bias mode. This is in accordance with our qualitative picture. On the other hand, the antisymmetric component $\Ga$ depends only on interactions. Our arguments agree with recent experiments in quantum dots:[@Lofgren; @Zumbuhl] depending on the bias mode different features of the nonlinear conductance tensor are probed. The fluctuations of nonlinear current can be minimized or maximized (on average), which becomes important for weakly interacting electrons. Curiously, for symmetric coupling (transmission and capacitance) of contacts and dot the bias mode in which the voltages at the contacts are changed in opposite directions generally [*minimizes*]{} fluctuations of $\Gs$. Consequently, such a mode is more advantageous for the observation of $\Ga$ or a cleaner linear signal. Near the end of Sec. \[sec:2terminal\] we also demonstrate how to take into account possible classical circuit-induced asymmetry [@Lofgren] due to the finite classical resistance of the wires. In Sec. \[sec:rectify\] we present results elucidating the role of interaction in rectification through two-terminal dots. Usually there are two important time-scales: the dwell time $\dwell$ an electron spends inside the dot and the charge relaxation time $\RC\leq\dwell $ of the dot. For a given geometry, the dwell time depends on the coupling of the dot with reservoirs, but the charge relaxation time is also sensitive to the interaction strength. We have $\RC\ll\dwell$ for strong interactions and $\RC=\dwell$ in the weak interaction limit. Our results for fluctuations of $\Ga(\oo),\Gs(\oo)$ are obtained for arbitrary frequency $\oo$. Although the fluctuations of both $\Ga(\oo)$ and $\Gs(\oo)$ monotonically decrease when $\oo\to\infty$, as functions of frequency $\oo$ they behave differently. At nonadiabatic frequencies $\oo\dwell\gg 1$ the nearby gate short-circuits currents. This effect is even in magnetic field and thus affects only $\Gs$. As a result, for a high-frequency voltage the asymptotes of $\Ga$ and $\Gs$ are generally different and strongly depend on the bias mode. Since the regime of parameters is quite realistic, we expect that the predicted difference of $\Gs(\oo)$ and $\Ga(\oo)$ should be experimentally observable. In the noninteracting limit our results qualitatively agree with those in diffusive metallic junctions. Our model of a ring consisting of a chaotic dot with a ballistic arm which encloses an AB flux is presented in Section \[sec:phase\]. Although it is impossible to find the full mesoscopic distribution of the AB phase $\delta$, its shape can be discussed qualitatively. Since $\tan\d$ is similar to the asymmetry parameter $\A=\Ga/\Gs$ in quantum dots, its distribution can become very wide for a particular choice of the bias mode. On average $\la\d\mbox{(mod)}\pi\ra=0$ in our model, and we find the dependence of the fluctuations of $\d$ on temperature, interactions, and number of channels of the contacts and the arm. Our treatment allows a straightforward generalization to treat AC voltages applied to the ring. The technical calculations are presented in the Appendix. \[\]\[\]\[0.8\][$C_{g0}$]{}\[\]\[\]\[0.8\][$C_{g1}$]{} [![(Left) Rectified current is measured through a coherent quantum dot biased by voltages with (AC) amplitude $V_{i,\oo},i=1,2$ at reservoirs connected by $N_{i}$ ballistic channels and capacitances $C_{i}$ and by voltages $V_{gi,\oo}$ applied at additional gates with capacitances $C_{gi}$. Transport through the dot is sensitive to the total magnetic flux $\Phi$ through the area of the dot. (Center and right) Forward and reverse connection of Ref. exchange voltages at the contacts and classical resistors $r_{1,2}$.[]{data-label="fig:3dot"}](3dotnew.eps "fig:"){width="9.cm"}]{} Model {#sec:model} ===== The 2D quantum dot, see the left panel in Fig. \[fig:3dot\], is biased with several voltages $\{V_{i}\}$ at $M$ ballistic quantum point contacts (QPCs) with $N_{i}, i=1,...,M$ orbital channels. The reservoirs can be capacitively coupled to the dot via capacitances $C_i$. An additional set of voltages $\{V_{gi}\}$ is applied to (several) gates with capacitances $C_{gi}$. All perturbations are assumed to be at the same frequency $\oo$, which is not necessarily small (adiabatic). The dot is in the universal regime, [@Beenakker] when the Thouless energy $E_{\rm Th}=\hbar/\erg$ is large. The dots with area $A=\pi L^2$ (taken circular) are either diffusive with mean free path $l\ll L$, or ballistic, with $l\gg L$ and chaotic classical dynamics (in the latter case the substitution $l\to \pi L/4$ should be used). The mean level spacing (per spin direction) $\Delta=2\pi\h^2/(m^*A)$ and the total number of ballistic channels $N$ together define the dwell time $\dwell=h/(N\Delta)\gg \erg$. We also require that $eV\ll N\Delta$ when we can treat the nonlinearity only to $(eV)^2$. Scattering is spin-independent and this spin degeneracy is accounted for by the coefficient $\nu_s$. The noninteracting electrons are treated using the scattering matrix approach and Random Matrix Theory (RMT) for the energy-dependent scattering matrix $\S(\e)$. For details we refer the reader to reviews. [@Beenakker; @ABG] In this approach the fundamental property of a dot is its scattering matrix $\S$ distributed over circular ensembles of proper symmetry, see Ref. (An alternative method is the Hamiltonian approach based on the properties of the dot’s Hamiltonian $\cal H$ taken from a Gaussian Ensemble. [@ABG]) Transport properties of chaotic dots in RMT for matrices $\S$ or $\cal H$ are usually expressed in terms of an effective, magnetic field-dependent number of channels. Predictions based on this approach are in good agreement with experiment. For multichannel samples with $N\gg 1$ we use the diagrammatic technique described in Refs. and . However, when interactions are present, this treatment should be modified. The approach which assumes that in a pointlike scatterer the interactions appear in the form of a self-consistent potential was introduced by Büttiker and co-authors [@buttiker1] on the basis of gauge-invariance and charge conservation. This (Hartree) approach neglects contributions leading to Coulomb blockade (Fock terms), but is a good approximation for open systems. If the screening in the dot inside the medium with dielectric constant $\e$ is strong, $r_s=(k_{\rm F}a_B)^{-1}=e^2/(\e\hbar v_{\rm F})\lesssim 1$, an RPA treatment of Coulomb interactions is sufficient. For large dots, $L\gg a_B$, the details of screening potential on the scale $\sim a_B$ are not important and we can assign an electric potential $U(\vec r,t)$ defined by excess electrons at $\vec r,t$ at any point $\vec r$ of the sample. If additionally the number of ballistic channels $N$ is much smaller than the dimensionless conductance of a closed sample, $g_{\rm dot}=\Thou/\Delta\gg N$, the potential drops over the contacts and therefore in the interior of the dot it can be taken uniform (“zero-mode approximation”). [@ABG] This potential shifts the bottom of the energy band in the dot and thus modifies the $\S$-matrix. As a consequence, electrons with kinetic energy $E$ have an electro-chemical potential $\tilde E_\a=E-eV_\a$ in the contact $\a$ and $\tilde E=E-eU$ in the dot. (We point out that we neglect the quantum pumping in the dot and consequently the $\S$-matrix depends only on one energy.) Recently, Brouwer, Lamacraft, and Flensberg demonstrated that this self-consistent approach gives the leading order in an expansion in the inverse number of channels $1/N\ll 1$. [@BLF] Therefore, our analytical results present the leading order effect, valid for $1/N\ll 1$. In the self-consistent approach the influx of charge changes the internal electrical potential of the dot $U(t)$, which in turn affects the currents incoming through each conducting lead and/or redistributes charges among the nearby conductors (gates). Such capacitive coupling can often be estimated simply from the geometrical configuration. For example, the capacitance of a dot covered by a top gate at short distance $d\ll L$ is $C\sim \e L^2/d$ and a single quantum dot has $C\sim \e L$. The ratio of charging energy $E_c\sim e^2/C$ to mean level spacing $\Delta$ characterizes the interaction strength. It is proportional to the ratio of the smallest geometrical scale to the effective Bohr’s radius, $E_c/\Delta\sim \mbox{min }\{d,L\}/a_B$. We refer to interactions as strong if $E_c\gg \Delta$ and weak if $E_c\ll \Delta$. Importance of bias mode {#sec:bias} ======================= [![Depending on the bias mode, the experiment probes different transport properties. Plots present (left) linear and (right) nonlinear components of the current as functions of $x=V_1-V_0$ and $y=V_2-V_0$, the dashed curves correspond to equal currents. Thin line shows fixed $V_2$ and $I(\tilde V)$ is a function of source voltage $\tilde V=V_1$. Thick line corresponds to fixed $V_1+V_2$, such that $I(\tilde V)$ depends only on $\tilde V=(V_1-V_2)/\sqrt{2}$. Full and empty dots on the right figure correspond to the forward or reverse configurations shown in Fig. \[fig:dot\].[]{data-label="fig:Vaxes"}](Vaxes.eps "fig:"){width="9.cm"} ]{} We suppose for simplicity that at equilibrium the voltages $V_1=V_2=V_0$ are set. In the following we consider the situation when the (single) gate voltage $V_0$ is held fixed at its equilibrium value. Experiments can be performed in different [*bias modes*]{}, usually either (i) with fixed drain voltage $V_2$ or (ii) at fixed $V_1+V_2$ (the variations of the voltages at the contacts are equal in magnitude but opposite in sign). These different modes correspond to straight lines in the $\{V_1 , V_2\}$ plane shown in Fig. \[fig:Vaxes\]. Let us consider the nonlinear current as a function $I(x,y)$, where $x=V_1-V_0$ and $y=V_2-V_0$ are deviations of contacts voltages from equilibrium. For generality we consider below a situation when the linear combination $-x\sin(\eta-\pi/4)+y\cos(\eta-\pi/4)=0$ is held fixed and the only variable is $$\begin{aligned} \label{eq:tildeV} \tilde V=x\cos(\eta-\pi/4)+y\sin(\eta-\pi/4).$$ This corresponds to a rotation of the original $x , y$ axes such that the new coordinate axis $\tilde V$ makes an angle $\eta$ with the $y=-x$ line, as illustrated in Fig. \[fig:Vaxes\]. The value of $\eta$ fully characterizes the bias mode. Now the two modes introduced above are simply (i) $\eta=\pi/4$ which implies $\tilde V=x$; and (ii) $\eta=0$, which implies $\tilde V=(x-y)/\sqrt{2}$ and corresponds to an asymmetric variation of the voltages. The linear current depends only on $x-y$ (dashed lines on the left panel in Fig. \[fig:Vaxes\] correspond to the lines of equal currents) and in any bias mode the measured linear current $I_{\rm lin}$ is the same for a given $x-y$. If we consider the nonlinear current $I$ as a function of $x,y$, it is by construction a bilinear function of $x,y$. As in the linear case the current must vanish if the voltages are the same and thus $I=0$ for $x-y=0$. Therefore, the bilinear function must be of the form $$\begin{aligned} \label{eq:simpleI} I=I_0 \,\left[ (x+y)\cos\phi+(x-y)\sin\phi\right]\, (x - y)\end{aligned}$$ with unknown (generally fluctuating) parameters $I_0$ and $\phi\in(-\pi/2,\pi/2]$. It is important that the qualitative behavior of $I(x,y)$ depends on the interaction strength: one could expect that transport depends not only on voltages in the leads, but also on the internal nonequilibrium potential $U$ of the sample. This potential can be found if potentials in all reservoirs and the nearby gate are known. In the limit of weak interactions the equilibrium point $V_0$ is important, and if we reverse the bias voltage, $(V,0)\to (0,V)$ the current is fully reversed, that is $\DD^2_{xx}I=-\DD^2_{yy}I$. For the current defined in Eq. (\[eq:simpleI\]) it is possible only when $I\propto (x-y)(x+y)\Rightarrow \phi=0$. Another way to see this is to use the usual expression for the total current in terms of scattering matrices. In this formula the current depends on the difference between Fermi distributions in the leads $\propto f(\e-ex)-f(\e-ey)$, and its expansion up to the second order yields $f''(\e)(x^2-y^2)$. The lines of equal current are curved and directions $\eta=0,\pm \pi/2$ correspond to zero current directions. Thus the dependence of current on the angle $\eta$ is strong. In addition this approach predicts that the current through a two-terminal sample is symmetric with respect to the magnetic flux inversion. In contrast, for strong interactions, the value of $V_0$ is irrelevant and the nonequilibrium electrical potential $U$ is independent of $V_0$. In this case current depends only on the voltage difference $x-y$ and thus $I\propto (x-y)^2\Rightarrow |\phi|=\pi/2$. The equal-current lines are straight and the picture is similar to the left plot in Fig. \[fig:Vaxes\] for linear transport. Therefore we do not expect any nontrivial dependence of the nonlinear current on the choice of the bias mode. It is noteworthy that qualitative considerations can predict neither the sign, nor the magnitude of $I_0$. The only general conclusion which we can make for a weakly or strongly interacting dot is $I(x,y)\propto x^2-y^2$ and $I(x,y)\propto (x-y)^2$, respectively. Experiments extract derivatives of $I$ with respect to the applied voltages. Importantly, this derivative depends on the chosen direction $\eta$. The nonlinear current measured in this bias mode is $$\begin{aligned} \label{eq:IVangle} \label{eq:Iexample} I(\eta)&=& I_0 \tilde V^2\cos\eta\sin (\phi+\eta).\end{aligned}$$ The current is zero when $\eta=-\phi$ and $\eta=\pm\pi/2$, and the bisectrix of the angle between the two zero-current directions at $\eta=-\phi/2+(\pi/4) \mbox{sgn }\phi$ maximizes $\DD^2 I/\DD\tilde V^2$. Sometimes experiments extract information on nonlinearity from measurements in different connections schematically shown in the central and right panels in Fig. \[fig:3dot\]: “Forward” connection corresponds to $x=\pm V,y=0$, while “reverse” connection for the same voltage configuration corresponds to $x=0,y=\pm V$. The gate voltages $V_g$ are kept fixed. In Fig. \[fig:Vaxes\] these forward and reverse points are indicated by black and white dots, respectively. To find the nonlinear conductance Marlow [@marlow] and Löfgren [@Lofgren] determine the difference of conductance at these measurement points. Löfgren [@Lofgren_2004; @Lofgren] use the term “rigidity” for samples for which $G_f(V)=G_r(-V)$ in the points $f^+ =(V,0)$ and $r^- =(0,-V)$. [@Lofgren_2004] Equation (\[eq:simpleI\]) gives the nonlinear contribution $\G$ to the full conductance $G_{f,r}(\pm V)$: $$\begin{aligned} \G\propto I_0\left[(x-y)\sin\phi+(x+y)\cos\phi\right]\,.\end{aligned}$$ Thus for a sample which is called rigid this implies $I_0\cos\phi\to 0$. Since $I_0 = 0$ would mean that there is no second-order response, we must have $\cos\phi\to 0$ which is the case for samples with strong interaction. In other words, “rigidity” in samples which exhibit $O(V^2)$ current is equivalent to strong Coulomb interactions. On the other hand, comparison of data at another pair of points $f^+=(V,0)$ and $r^+=(0,V)$ gives $G_f(V)-G_r(V)\propto I_0\sin\phi$ and provides [*additional*]{} information about the two fluctuating quantities $I_0,\phi$. Reference expects that a Left-Right (LR)-symmetric system has $G_f(V) = G_r(V)$. Therefore rigid and LR-symmetric sample should necessarily have $I_0 \to 0$ and thus could not exhibit a second-order current $O(V^2)$. This point is discussed more quantitatively in Sec. \[sec:2terminal\]. It is important to note that to find the linear DC current one needs to know only $x-y=V_1-V_2$, while for the nonlinear current in general one needs two variables $x=V_1-V_0,y=V_2-V_0$ or any independent pair of their linear combinations. The projection of the vector $(V_1,V_2,V_0)$ on the $V_1+V_2+V_0=$const plane uniquely defines the nonlinear current. This projection can be parametrized by the pair of Cartesian $(x,y)$ or axial coordinates $(\tilde V,\eta)$. However, if in the experiment the voltages $V_{1,2}$ were fixed, this would not be enough to define $(x,y)$ uniquely. In this case Ref. points to the importance of the reference point $V_0$. Indeed, one could arrive at the point with a given $(V_1,V_2)$ from any equilibrium point and the measured current would depend on $V_0$. We prefer to characterize the measurement by the pair $(\tilde V,\eta)$ instead of three variables $(V_1,V_2,V_0)$ because of the simplicity of the final results. The weaker the interaction (or the stronger the capacitive coupling of the sample to the nearby gate) the more important the role of $\eta$ chosen in experiment. We illustrate this important conclusion by quantitative results for nonlinear conductance $\G\propto\DD^2 I/\DD\tilde V^2$ in the following sections. We point out that conductance with respect to the voltage difference $V=V_1-V_2$ is often used, even when a linear combination $\tilde V$ is actually varied in experiment. Voltages $\tilde V$ and $V$ are related, $\tilde V=V/\sqrt{2}\cos\eta$, and one can straightforwardly find $\DD^2 I/\DD V^2$. Generation of DC in quantum dots {#sec:DC} ================================ Now we quantify the qualitative arguments of Sec. \[sec:bias\] and consider the more general situation of a DC current generated by an AC bias. If at first we neglect Coulomb interactions, the nonlinear DC current $I_\a$ in response to the Fourier components $V_{\b,\oo}=V_\b e^{i\phi_\b}$ of the AC voltages applied at the contacts $\b=1,...,M$, can be expressed with the help of the DC-conductance matrix $g_{\a\b}(\e)$ of the dot at the energy $\e$ [@pedersen] $$\begin{aligned} \label{eq:Pedersen} I_\a &=&\frac{\nu_s e^3}{h}\int d\e \frac{f(\e+\hbar\oo)+f(\e-\hbar\oo)-2f(\e)}{(\hbar\oo)^2} \nonumber \\ &\times& \sum_{\b=1}^M g_{\a\b}(\e)|V_{\b,\oo}|^2,\\ \label{eq:gDC} g_{\a\b}(\e)&=&\Tr [{1\!\! 1}_\a\d_{\a\b}-\S^\dagger(\e) {1\!\! 1}_\a\S(\e){1\!\! 1}_\b].$$ If we now include interactions using a self-consistent potential $U_{\oo}$ this formula is modified: [@pedersen] in Eq. (\[eq:Pedersen\]) the Fourier components of the voltages at [*all*]{} contacts are shifted down by the Fourier component of the internal potential $-U_\oo$ $$\begin{aligned} U_\oo &=&\sum_\g u_\g V_{\g,\oo}, \,\,u_\g= \frac{\sum_{\b}G_{\b\g}(\oo)-i\oo C_\g}{\sum_{\b\g}G_{\b\g}(\oo)-i\oo C_\Sigma}\label{eq:uomega},\\ G_{\b\g}(\oo)&=& \frac {\nu_s e^2}{h}\int d\e\,\Tr\left[{1\!\! 1}_\b{1\!\! 1}_\g - {1\!\! 1}_\g{\cal S}^\dagger(\e){1\!\! 1}_\b{\cal S}(\e+\hbar\oo)\right]\nonumber \\ &\times& \frac{f(\e)-f(\e+\hbar\oo)}{\hbar\oo}.\label{eq:sumG}\end{aligned}$$ In Eq. (\[eq:uomega\]) the index $\g$ runs not only over real leads $1,...,M$, but also over all gates $gi$. However, when $\g\in\{gi\}$ the AC conductance $G_{\b\g}(\oo)$ is absent and only capacitive coupling $i\oo C_\g$ remains in the numerator. We point out that the matrix $ G(\oo)$ of dynamical AC conductance at frequency $\oo$ given in Eq. (\[eq:sumG\]) should not be confused with the degenerate matrix $g(\e)$ of energy-dependent DC conductances of electrons with kinetic energy $\e$ given in Eq. (\[eq:gDC\]). The results of Ref. can be expressed in terms of the DC conductances $g_{\a\b}$ and frequency-dependent characteristic potentials $u_\g$, $$\begin{aligned} \label{eq:current4omega} I_\a &=&\frac{\nu_s e^3}{h}\int d\e \frac{f(\e+\hbar\oo)+f(\e-\hbar\oo)-2f(\e)}{(\hbar\oo)^2} \nonumber \\ &\times &\sum_{\b\g} g_{\a\b}(\e)\mbox{Re }\, u_\g |V_{\b,\oo}-V_{\g,\oo}|^2.\end{aligned}$$ Here $\mbox{Re }u_\g$ stands for the real part of $u_\g$, which is in general a complex quantity. In contrast to Eq. (\[eq:Pedersen\]), Eq. (\[eq:current4omega\]) is expressed via differences of voltages applied to all present conductors. Therefore, the current is gauge-invariant. The charge conservation, $\sum_\a I_\a=0$, is obvious from Eq. (\[eq:gDC\]). From this point on we consider Eq. (\[eq:current4omega\]), a specific expression of Eq. (\[eq:IV\]), in detail for several regimes. In Sec. \[sec:2terminal\] we discuss the nonlinear current due to DC applied voltages (previously considered in Ref. ) and the importance of different bias modes in experiments in two-terminal quantum dots. In Sec. \[sec:rectify\] we consider the frequency dependence of $\Gs(\oo)$ and $\Ga(\oo)$. Nonlinearity in quantum dots {#sec:2terminal} ---------------------------- In the static limit [@ChristenButtiker] $\hbar\oo/T\to 0$ the integrand in the first line of Eq. (\[eq:current4omega\]) simplifies to $f''(\e)$ and for $\hbar\oo/N\Delta\to 0$ the derivatives $u_\g$ are real and expressed via subtraces of the Hermitian Wigner-Smith matrix $\S^\dagger \DD_\e\S/(2\pi i)$ [@WignerSmith; @BP] $$\begin{aligned} \label{eq:current4} I_\a &=&\frac {-\nu_s e^3}{h}\sum_{\b\g}\int f'(\e)d\e g'_{\a\b}(\e) u_{\g}(V_\b-V_\g)^2,\\ \label{eq:u} u_\g &=&\frac{ C_\g/\nu_s e^2-\int d\e f'(\e)\Tr {1\!\! 1}_\g\S^\dagger \DD_\e\S/(2\pi i) }{C_\Sigma/\nu_s e^2 -\int d\e f'(\e)\Tr \S^\dagger \DD_\e\S/(2\pi i)}\label{eq:u0}.\end{aligned}$$ For a two-terminal sample the nonlinear current through the first lead is $$\begin{aligned} \label{eq:current} I_1 &=&\frac {-\nu_s e^3}{h}\int f'(\e)g'_{11}(\e)d\e\left[\sum_i u_{gi}\left[(V_1-V_{gi})^2 \right.\right.\nonumber \\ && \mbox{} \left.\left. -(V_2-V_{gi})^2\right]+(u_2-u_1)(V_1-V_2)^2\right].\end{aligned}$$ The characteristic potentials in the last term of Eq. (\[eq:current\]) are sensitive to the asymmetry of the contacts. Indeed, in a strongly interacting dot $u_{gi}=0$ and $u_2-u_1\approx (N_2-N_1)/N$. The current magnitude grows with asymmetry due to the last term in Eq. (\[eq:current\]). On the other hand, the sign of $I_1$ is random because of quantum fluctuations of $g'_{11}$ around zero. [@deriv] As a consequence, if in an experiment the Fermi level is shifted by $\d\m_{\rm F}\sim N\Delta/2\pi$ (or the shape of the dot is changed) the sign of nonlinearity can be inverted. Different modes of bias having been discussed in Sec. \[sec:bias\], we concentrate here on the (anti)symmetric conductances through the quantum dot at fixed gate voltages. When the reservoir voltages are varied in the $\eta$ direction, the nonlinear current is given by the expression $$\begin{aligned} \label{eq:derivIV} I &=&\frac {-2\nu_s e^3}{h}\int f'(\e)g'_{11}(\e)d\e\left[(1-u_1-u_2)\sin \eta\right.\nonumber \\ &&\left.+(u_2-u_1)\cos\eta\right]\cos\eta\tilde V^2,\end{aligned}$$ and one can define exactly the unknown parameters $I_0,\phi$ which we introduced in the qualitative argument leading to Eq. (\[eq:Iexample\]). Depending on $\eta$ one measures different linear combinations of conductances. If we consider conductances $\DD^2 I/2\DD \tilde V^2$ in units of $\nu_s e^3/h$, Eqs. (\[eq:IVfield\]) and (\[eq:derivIV\]) yield $$\begin{aligned} \label{eq:defG} {\cal G}_{a,s}=\frac{2\pi\cos^2\eta\int d\e d{\tilde \e}f'(\e)f'(\tilde \e)\chi_1(\e)\chi_{2,a(s)}(\tilde \e)}{\Delta^2[C_\Sigma/(e^2\nu_s)-\int d\e f'(\e)\Tr\S^\dagger \DD_\e\S/(2\pi i)]}\end{aligned}$$ expressed in terms of fluctuating functions $\chi$ and a traceless matrix $\Lambda= (N_2/N){1\!\! 1}_1-(N_1/N){1\!\! 1}_2$: $$\begin{aligned} \label{eq:chi1} \chi_1(\e) &=& (\Delta/2\pi)\DD_\e \Tr\Lambda {\cal S}^\dagger\Lambda{\cal S}, \\ \label{eq:chi2a}\chi_{2,a}(\e) &=&(i\Delta/2\pi) \Tr\Lambda[\S^\dagger,\DD_\e\S],\\ \label{eq:chi2s} \chi_{2,s}(\e) &=&\Delta\left(\frac{C_0\tan\eta+C_2-C_1}{e^2\nu_s} +\frac{N_2-N_1}{N}\frac{\Tr \S^\dagger\DD_\e\S}{2\pi i }\right.\nonumber \\ &&\left. +\frac{1}{2\pi i} \Tr\Lambda\{\S^\dagger,\DD_\e\S\}\right).\end{aligned}$$ Standard calculations using the Wigner-Smith and/or $\S$-matrix averaging [@waves; @PietBeen; @iop] yield $\la\Ga\ra=\la\Gs\ra=0$. This result signifies that the nonlinear current through a quantum dot is indeed a quantum effect. As a consequence the size of the measured nonlinearity must be evaluated from correlations of $\Ga,\Gs$. The functions $\chi_1(\e,\Phi)$ and $\chi_{2,a/s}(\e',\Phi')$ are uncorrelated, and their autocorrelations [@PhysicaE] readily allow one to find statistical properties of ${\cal G}_{a,s}$. Our results can be expressed in terms of diffuson $\Diff$ or cooperon $\Coop$ in a time representation, $\exp(-\tau/\tau_{\Diff})$ and $\exp(-\tau/\tau_{\Coop})$. Both can be introduced using the $\S$-matrix correlators [@PVB] (correlations of retarded and advanced Green functions lead to the same expression up to a normalization constant [@ABG]). We have $$\begin{aligned} {\cal S}(\tau,\Phi) &=& \int\frac{d\e}{2\pi\hbar} \, {\cal S}(\e,\Phi)e^{i \e\tau/\hbar},\nonumber \\ \langle {\cal S}_{i j} (\tau,\Phi) {\cal S}^{*}_{k l} (\tau',\Phi')\rangle &=& (e^{-\tau/\tau_\Diff}\delta_{ik} \delta_{jl} + e^{-\tau/\tau_\Coop} \delta_{il} \delta_{jk})\nonumber \\ &\times &\frac{\Delta}{2\pi\hbar}\delta(\tau-\tau') \theta(\tau), \label{eq:cum1t}\\ \label{eq:channels} \tau_{\Coop,\Diff}=\frac{h}{N_{\Coop,\Diff}\Delta},\, {\genfrac{\{}{\}}{0pt}{}{N_{\cal C}}{N_{\cal D}}} &=& N+\frac{(\Phi\pm\Phi')^2}{4\Phi_0^2}\frac{h v_F l}{L^2\Delta}.\end{aligned}$$ We also introduce the electrochemical capacitance $C_\m$ [@PietMarkus] which relates the non-quantized mesoscopically averaged excess charge $\la Q\ra$ in the dot in response to small shift of the voltages $\d V$ at all gates. In addition the charge relaxation time $\RC$ of the dot is conveniently introduced by this electrochemical capacitance and the total contact resistance, $$\begin{aligned} \label{eq:excess} C_\m= \frac{\la \d Q\ra}{\d V}=\frac{C_\Sigma}{1+C_\Sigma\Delta/(\nu_s e^2)},\,\,\,\RC= \frac{hC_\m}{\nu_s Ne^2}.\end{aligned}$$ The denominator of Eq. (\[eq:defG\]) is a self-averaging quantity, $\la(...)^2\ra=\la (...)\ra^2=\Delta^2(C_\Sigma/C_\m)^2$. Using the diffusons and cooperons defined in Eq. (\[eq:cum1t\]) we find the following correlations of $\Ga$ and $\Gs$: $$\begin{aligned} \label{eq:main} &&{\genfrac{\{}{\}}{0pt}{}{\la\Ga(\Phi)\Ga(\Phi')\ra}{\la\Gs(\Phi)\Gs(\Phi')\ra}} = {\genfrac{\{}{\}}{0pt}{}{{\mathcal F}_\Diff-{\mathcal F}_\Coop}{{\mathcal F}_\Diff+{\mathcal F}_\Coop+X}}({\mathcal F}_\Diff+{\mathcal F}_\Coop)\nonumber \\ &&\times \left(2\cos^2\eta\frac{2\pi} {\Delta}\frac{C_\m}{C_\Sigma}\right)^2\frac{N_1^3 N_2^3}{N^6},\\ && {\cal F}_{\l}=\left(\frac{\Delta T }{2\hbar^2}\right)^2\int\frac {\tau_\l\tau^2 e^{-\tau/\tau_\l}}{\sinh^2 \pi T \tau/\hbar}d\tau, \label{eq:Fraw}\\ &&\label{eq:X}X= \frac{N^2}{2N_1N_2} \left(\frac{C_0\tan\eta+C_2-C_1}{\nu_s e^2/\Delta} +\frac{N_2-N_1}{N}\right)^2.\end{aligned}$$ There are two very different contributions to Eq. (\[eq:main\]), ${\mathcal F}_{\Coop,\Diff}$ due to quantum interference and $X$ defined by the classical response of the internal potential to external voltage. The terms denoted by ${\cal F}_{\Coop,\Diff}$ are sensitive to temperature, magnetic field, and decoherence. Asymptotical values of ${\cal F}$ in the low temperature, $T\ll \hbar/\tau_{\l}$, or high temperature limits, $T\gg \hbar/\tau_{\l}$, are ${\cal F}_\l \to 1/N_\l^2=(\tau_\l\Delta/h)^2$ and ${\cal F}_\l \to \Delta/(12 T N_\l)=\tau_\l\Delta^2/ (12 hT)$, respectively. The term denoted by $X$ and given by Eq. (\[eq:X\]) contains only quantities specifying the geometry of the sample and gates and the bias mode. In a real experiment the coupling due to capacitances $C_{1,2}$ is usually stronger then that of the external gates, $C_{1,2}\gg C_0$. Symmetrization of the circuit $C_1=C_2$ can diminish the value of $X$. If in addition $N_1=N_2$ and $\eta=0$ (used in the experiments [@Zumbuhl; @ensslin]) we have $X\to 0$. Thus such a symmetric setup and bias mode minimize the fluctuations of the nonlinear current and actually would be best for an accurate measurement of [*linear*]{} transport. Indeed, this regime is not affected by the fluctuations of capacitive coupling $u_0$ of the dot with the nearby gate and thus minimizes fluctuations of $\Gs$ around 0. Fluctuations of $\Ga,\Gs$ are given by different expressions, see the first line of Eq. (\[eq:main\]), where the first term is due to $\la\chi_{2,a}^2\ra$ or $\la\chi_{2,s}^2\ra$. Importantly, $\la\chi_{2,s}^2\ra$ contains both quantum ${\mathcal F}_{\l}\lesssim 1/N_\l^2$ and classical $X$ contributions. If the classical term dominates, $X\gg 1/N^2$, the current is mostly symmetric, $\Gs^2\gg\Ga^2$. This could be expected either for a weakly interacting dot or a very asymmetric setup, $N_1\neq N_2$ .[@PhysicaE] However, if the classical term is reduced due to, e.g., the bias mode, the fluctuations of $\Ga$ and $\Gs$ become comparable. This experimentally important conclusion remains valid for [*any interaction strength*]{}. (Particularly, it leads to a very wide distribution of the Aharonov-Bohm phase considered in Sec. \[sec:phase\].) Experiments of Zumbühl [@Zumbuhl] and Leturcq [@ensslin] are performed in this regime when $\eta=0$ and $X\to 0$. Data in Ref. demonstrate that the part of the total current symmetrized with respect to magnetic field is by far dominated by linear conductance. From Eq. (\[eq:main\]) we expect mesoscopic fluctuations in linear conductance to be $\sim N^2$ times larger then those of $\Gs\Delta$. Thus only when the number of channels is decreased will the nonlinear $\Gs$ become noticeable. A clear observation of $\Gs$ without linear transport contribution was performed in a DC Aharonov-Bohm experiment by Angers [@Bouchiat] in the mode $\eta=\pm \pi/4$ (only one contact voltage was varied). This allowed to evaluate the interaction strength from the ratio of $\Gs/\Ga$. Experiments of Marlow [@marlow] and Löfgren [@Lofgren] measure the full two-terminal conductance and extract nonlinear conductance properties related to various spatial symmetries of the dot. Although the current through a weakly interacting sample is field-symmetric, this is not true in general. Samples of Ref. differ in “rigidity” and degree of symmetry. Rigid samples, $u_0\to 0$, with Left-Right(LR) and Up-Down (UD)-symmetry should have $(u_2-u_1)_s=0$ and $(u_2-u_1)_a=0$ respectively, according to the expectations of Löfgren [@Lofgren] (indices $s$ and $a$ mean the symmetric and antisymmetric part in magnetic field). Due to quantum fluctuations, in experiment none of these symmetry-relations can be exactly fulfilled, see Eq. (\[eq:main\]). According to Eq. (\[eq:derivIV\]), the difference in the full conductances $g=(h/\nu_s e^2)I/V$ measured between different points probes different characteristic potentials. Reference defines three differences $g_{\rm i,ii,iii}$ for three pairs of points in the forward and reverse connection discussed after Eq. (\[eq:IVangle\]). Using Eq. (\[eq:main\]) we find (i) $g_{\rm i}\equiv g_f(V,B)-g_r(-V,B)\propto u_0$, (ii) $g_{\rm ii}\equiv g_f(V,B)-g_f(V,-B)\propto (u_2-u_1)_a$, and (iii) $g_{\rm iii}\equiv g_f(V,B)-g_f(-V,-B)\propto (u_0+u_2-u_1)_s$. The ensemble average of these differences vanishes and their fluctuations for $C_{1,2}=0,N_1=N_2$ are given by $$\begin{aligned} \left\{\begin{array}{c} g_{\rm i}^2 \\ g_{\rm ii}^2 \\ g_{\rm iii}^2\\ \end{array}\right\}=\left\{\begin{array}{c} X \\ {\mathcal F}_\Diff-{\mathcal F}_\Coop\\ X+{\mathcal F}_\Diff+{\mathcal F}_\Coop\\ \end{array}\right\}({\mathcal F}_\Diff+{\mathcal F}_\Coop) \left(\frac{\pi e V} {2\Delta}\frac{C_\m}{C}\right)^2,\nonumber\end{aligned}$$ where $X=2(C/C_\m-1)^2$ is found from Eq. (\[eq:X\]) at $\eta=\pm \pi/4$. In weakly interacting dots $C_\m/C\to 0$ and only magnetic-field symmetric signals $g_{\rm i}$ and $g_{\rm iii}$ survive. In strongly interacting (“rigid”) dots $C_\m/C\to 1$ and $g_{\rm ii}$ becomes similar to $g_{\rm iii}$. We point out that even if the rigid samples are made symmetric with respect to Left-Right inversion, the quantum fluctuations of the sample properties are unavoidable and $g_{\rm ii}^2\neq 0$ at $\Phi\neq 0$. For high magnetic fields and arbitrary interactions ${\mathcal F}_\Coop\to 0$ and experiment should observe $g_{\rm i}^2+g_{\rm ii}^2=g_{\rm iii}^2$. Clearly fluctuations exist also for large magnetic fields beyond the range of applicabilty of RMT. Experimental data (see inset of Fig. 6 in Ref. ) show that $g_{\rm i}^2+g_{\rm ii}^2\sim g_{\rm iii}^2$. It is hard to make a quantitative comparison with Refs. and , since the quantum fluctuations in the nonlinear conductance exist possibly on the background of classical effects due to macroscopic symmetries. We expect that quantum effects become more pronounced as contacts are narrowed. To conclude this subsection we briefly discuss here the case of a macroscopically asymmetric setup. If the experiment were aimed to measure large $\Ga$ compared to $\Gs$, one would try to minimize $\Gs$ by adjusting the setup. Such a procedure minimizes the value of $X$ in Eq. (\[eq:X\]). For $C_{1,2}=0, \eta=\pi/4$ the role of asymmetric contacts $N_1\neq N_2$ was discussed in Ref.. Analogously, one could consider a more general case of $C_{1,2}$ and an arbitrary bias mode $\eta$. This is especially important if the difference $C_1\neq C_2$ can not be neglected due to occasional loss of contact symmetry. The results of an experiment could also be affected by the presence of classical resistance loads $r_{1,2}$ between macroscopic reservoirs and the dot (shown in Fig. \[fig:3dot\]). Swapping of such resistances in the experiment, when connection is switched between “forward” and “reverse” [@Lofgren] affects the voltage division between loads. If we assume the capacitive connection of the dot and reservoirs is still the same, the modification of the expression for $u_\g$ in Eq. (\[eq:uomega\]) is straightforward, $\sum_{\b}G_{\b\g}\to \sum_{\b}G_{\b\g}/(1+r_\g \sum_{\b}G_{\b\g})$. Naturally, at large $r_\g$, $(2 e^2 N_\g/h)r_\g \gg 1$, the main drop of the voltage occurs over the resistor $r_\g$ and not over the QPCs. As a consequence, if $r_{1,2}\neq 0$, values of $u_{1,2}$ can become unequal due to $r_1\neq r_2$ and this leads to the classical circuit asymmetry which we do not consider here. Rectification in quantum dots {#sec:rectify} ----------------------------- Here we consider the DC generated by a quantum dot subject to an AC bias at the frequency $\oo$. In experiment at high bias frequency $\oo\dwell\gtrsim 1$ current is usually measured at zero frequency. In contrast, at small bias frequency $\oo\dwell\ll 1$ higher harmonics (for instance the second harmonic $2\oo$) can be measured. However, up to corrections small due to $\oo\dwell\ll 1$, the second harmonic is just equal to the rectified current, $I_{2\oo}\approx I_0$. Therefore, to leading order, our results for the rectified current describe both experiments. Generally, there are several important time-scales characteristic for time-dependent problems in chaotic quantum dots. To see how they appear let us first consider frequency-dependent linear transport of noninteracting electrons. Its statistics usually depend only on the flux-dependent time scales $\tau_{\Coop,\Diff}$, see Eq. (\[eq:channels\]). If we consider an analog of UCF $\la G^2(\Phi)\ra$ for the frequency-dependent conductances introduced in Eq. (\[eq:sumG\]), we find $$\begin{aligned} \label{eq:linear} \la G(\oo,\Phi)G(\oo',\Phi')\ra=\left(\frac{\nu_s e^2}{h}\frac{N_1N_2}{N}\right)^2\sum_{\l=\Coop,\Diff}&& \nonumber \\ \left(\frac{\Delta T}{2\hbar^2}\right)^2\int \frac{\tau_\l d\tau}{\oo\oo'}\frac{ e^{-\tau/\tau_\l}(e^{i\oo\tau}-1)(1-e^{i\oo'\tau})} {[1-i(\oo+\oo')\tau_\l/2]\sinh^2 \pi T\tau/\hbar}.&&\end{aligned}$$ The presence of $i\oo\tau_\l$ in the diffuson and cooperon contributions in the second line of Eq. (\[eq:linear\]) is due to the energy dependence of the scattering matrix $\S(\e)$, which usually brings up imaginary corrections to the matrix-element correlators. In a DC-problem $\oo\to 0$ it is usually useful to introduce a dimensionless number of channels $N_{\Coop,\Diff}$ modified by the magnetic field, see Eq. (\[eq:channels\]). In this limit at $T\to 0$ the integration in Eq. (\[eq:linear\]) becomes straightforward and summation is then performed over $N_\l^{-2}$. For equal magnetic fields, $\Phi=\Phi'$, we have $N_\Diff=N$, but $N_\Coop$ is strongly modified by large fields, $N_\Coop\to \infty$, which suppresses the weak localization correction and diminishes UCF. However, for an AC problem (especially for $\oo\dwell\gtrsim 1$) it is more convenient to express results in terms of dimensionless quantities $\oo\tau_{\Coop},\oo\tau_{\Diff}$. For example, from Eq. (\[eq:linear\]) the statistics of conductance can be easily evaluated: $\la |G(\oo,\Phi)|^2\ra/\la G(0,\Phi)^2\ra\sim 1/(\oo\dwell)$ and the real and imaginary parts of conductance are similar and uncorrelated at high frequency $\oo\dwell\gg 1$. Inclusion of interactions introduces an (additional) dependence on $\RC$, the charge-relaxation time defined in Eq. (\[eq:excess\]). To leading order in $1/N\ll 1$ the effect of interactions is often to substitute $\dwell\to \RC$ in the noninteracting results, e.g., for the linear conductance [@PietMarkus] or shot noise. [@BP; @Hekking] Interestingly, the subleading corrections depend on both $\RC$ and $\dwell$, e.g., in the weak localization correction in the absence of magnetic field.[@PietMarkus] When the magnetic field is increased to values which finally break time-reversal symmetry, the appearance of different time scales $\tau_{\Coop,\Diff}$ is expected, see e.g., Eq. (\[eq:linear\]). Therefore at intermediate magnetic fields, when $\tau_\Diff\neq \tau_\Coop$, and the interactions taken into account, $\RC\neq \dwell$, the solution of an AC problem is expected to show a complicated dependence on all these time scales. Indeed, if we consider the rectified current such an interplay between $\tau_{\Coop,\Diff}$ and $\RC$ does appear. We find $\la\Ga\ra=\la\Gs\ra=0$ and present below results for correlations of $\Ga$ and $\Gs$: $$\begin{aligned} {\genfrac{\{}{\}}{0pt}{}{\la\Ga(\Phi)\Ga(\Phi')\ra}{\la\Gs(\Phi)\Gs(\Phi')\ra}}= {\genfrac{\{}{\}}{0pt}{}{{\cal F}_{U,\Diff}(\oo)-{\cal F}_{U,\Coop}(\oo)}{{\cal F}_{U,\Diff}(\oo)+{\cal F}_{U,\Coop}(\oo)+X(\oo)}}\nonumber \\ \times [{\cal F}_{G,\Diff}(\oo)+{\cal F}_{G,\Coop}(\oo)]\left( \frac{4\pi\cos^2\eta} {\Delta}\frac{C_\m}{C_\Sigma}\right)^2\frac{N_1^3 N_2^3}{N^6} \label{eq:varGs}.\end{aligned}$$ Here the functions ${\cal F}_{U}(\oo),{\cal F}_{G}(\oo)$ are finite-frequency generalizations of Eq. (\[eq:Fraw\]) $$\begin{aligned} \label{eq:FU}{\cal F}_{U,\l}(\oo)&=&\left(\frac{\Delta T}{\hbar^2 \oo}\right)^2\int \tau_\l d\tau\frac{e^{- \tau/\tau_{\l}}\sin^2 \oo\tau/2 }{2\sinh^2 \pi T\tau/\hbar}\frac{1}{1+\oo^2\RC^2} \nonumber \\&\times&\left(1+ \mbox{Re }\frac{1+i\oo\RC}{1-i\oo\RC} \frac{e^{i\oo\tau}}{1-i\oo\tau_\l}\right) ,\\ {\cal F}_{G,\l}(\oo)&=& \left(\frac{\Delta T}{\hbar^2 \oo^2}\right)^2\int \frac{2d\tau}{\tau_\l}\frac{e^{- \tau/\tau_{\l}}\sin^4 \oo\tau/2}{\sinh^2 \pi T\tau/\hbar} \label{eq:FG}.\end{aligned}$$ The subscripts $U(G)$ of ${\mathcal F}_{U(G)}(\oo)$ illustrate the origin of these functions: they result from averaging of different scattering properties over the energy band defined by $\mbox{max }\{\hbar\oo, T,\hbar/\tau_{\Coop(\Diff)}\}$. The function ${\mathcal F}_{U}(\oo)$ is a characteristic of the internal potential $U_\oo$, see Eq. (\[eq:uomega\]). The function ${\mathcal F}_{G}(\oo)$ results from the energy averaging of the DC conductance $g(\e)$. Such averaging appears because both $G(\oo)$ defined in Eq. (\[eq:sumG\]) and $g(\e)$ in Eq. (\[eq:current4omega\]) are coupled to the Fermi distribution. The function $X(\oo)$ is $$\begin{aligned} \label{eq:Xgeneral} X(\oo) &=&\frac{N^2}{2N_1N_2} \left(\frac{(C_0\tan\eta+C_2-C_1)(1+\oo^2\dwell\RC)}{(1+\oo^2\RC^2)\nu_s e^2/\Delta}\right.\nonumber \\ && \mbox{} \left. +\frac{N_2-N_1}{N(1+\oo^2\RC^2)}\right)^2,\end{aligned}$$ and in the static limit $\oo\to 0$ it is given by Eq. (\[eq:X\]). We point out that when the interactions are negligible, $E_c\sim e^2/C\ll\Delta$, the role of the bias mode is significant. A quantum dot with (fully broken) time-reversal symmetry can be labeled by Dyson symmetry parameter ($\b=2$) $\b=1$. When the setup is ideal, $C_{1,2}=0$, and $\eta\neq 0$, the fluctuations of $\Ga,\Gs$ at large frequencies $\oo\dwell\gg 1$ are $$\begin{aligned} \label{eq:FK} \d\Gs &=& \la\Gs^2\ra^{1/2}=\frac{N_1 N_2}{N^2} \left(\frac 2\beta\frac{\pi}{\oo\dwell}\right)^{1/2}\frac{2\sin 2\eta}{\hbar\oo},\\ \d\Ga &=&\left(\frac{N_1 N_2}{N^2}\right)^{3/2}\frac{\nu_s e^2}{2C}\frac{\cos^2\eta}{\hbar^2\oo^3\dwell},\,\,\b=2.\label{eq:FKasym}\end{aligned}$$ In chaotic quantum dots the role of the Thouless energy $\Thou$ of the open systems is often played by the escape rate $\hbar/\dwell$. If we take this into account, our result (\[eq:FK\]) qualitatively agrees with that of Falko and Khmelnitskii [@FK] obtained for open diffusive metallic junctions. However, when $\eta\to 0$, the fluctuations of $\Gs$ are much smaller and for $|N_1-N_2|\ll\oo\dwell$ they become comparable with those of the antisymmetric conductance (\[eq:FKasym\]). However, very often experiments are performed in samples, where the interaction is not weak. Since $\Ga$ and $\Gs$ are comparable for strong Coulomb interactions in the static limit $\oo\to 0$,[@PhysicaE] we concentrate here on this experimentally relevant regime of $\Delta/E_c\sim \RC/\dwell\ll 1$ and take an ideal symmetric setup, $N_1=N_2$ and $C_{1,2}=0$. Then we have $$\begin{aligned} \label{eq:Xomega} X(\oo)\approx 2\tan^2\eta\left(\frac{\dwell^{-1}+\oo^2\RC}{\RC^{-1}+\oo^2\RC}\right)^2.\end{aligned}$$ Below we consider in detail the case $\eta\neq 0$ and how this bias mode affects the behavior of $\Gs^2(\oo)$. Several frequency regimes can be separated: adiabatic $\oo\tau_\l\ll 1$, intermediate, where $1/\tau_\l\ll \oo\ll 1/\RC$, and high frequencies $\oo\RC\gtrsim 1$. The asymptotes of the functions defined in Eqs. (\[eq:FU\]), (\[eq:FG\]), and (\[eq:Xomega\]) in these regimes are presented in Table \[tab:table\] for reference. For adiabatic frequencies $\oo\tau_\l\ll 1$ the integrands in Eqs. (\[eq:FU\]) and (\[eq:FG\]) do not oscillate on the short time scale $\tau_\l$. At such small frequencies ${\cal F}_U(\oo)={\cal F}_G(\oo)$ are equal to ${\cal F}$ of Eq. (\[eq:Fraw\]) and $X(\oo)\propto (\RC/\dwell)^2\ll 1$ can be neglected. This is essentially the zero frequency regime considered before for nonlinear DC transport. As the frequency grows, an intermediate regime is reached when max $\{T,\hbar/\tau_\l\}\ll \hbar\oo\ll \hbar/\RC$ and ${\cal F}_{U}(\oo), {\cal F}_{G}(\oo)$ start to differ. The scattering properties at large energy difference $\hbar\oo\gg\hbar/\tau_\l$ are uncorrelated and the response of the dot is randomized. Therefore both the conductance averaged over a large energy window $\hbar \oo$ and the response of the internal potential $U_\oo$ to the AC voltage at $\oo\dwell \gg 1$ are strongly suppressed, see Table \[tab:table\]. As a result, if $X(\oo)$ is still negligible, both $\Ga^2$ and $\Gs^2$ decrease with growing frequency as $1/\oo^4$. ------------------------------------ ---------------------- ---------------------------------- ---------------------------- Adiabatic Intermediate High $\mbox{Function}$ $\oo\ll\tau_\l^{-1}$ $\tau_\l^{-1}\ll\oo\ll \RC^{-1}$ $\oo\RC\gg 1$ ${\mathcal F}_U(\oo)\times N_\l^2$ 1 $\pi/(4\oo\tau_\l)$ $\pi/(4\oo^3\tau_\l\RC^2)$ ${\mathcal F}_G(\oo)\times N_\l^2$ 1 $X(\oo)$ $2\tan^2\eta$ ------------------------------------ ---------------------- ---------------------------------- ---------------------------- : Asymptotes of ${\mathcal F}_{U,\l}(\oo),{\mathcal F}_{G,\l}(\oo),X(\oo)$ at $T\to 0$ \[tab:table\] One could expect that interactions qualitatively change the behavior of $\Ga,\Gs$ when the frequencies become comparable to $1/\RC\sim NE_c/\hbar$, the scale defined by the interaction strength. At such frequencies the response of a dot to the potentials at the contacts is not resistive as occurs at low frequencies, but mostly capacitive. If the frequency is high, $\oo\RC\gtrsim 1$, we have $\mbox{Re }u_{1,2}\to 0$ and the function ${\mathcal F}_{U}$ in Eq. (\[eq:FU\]) is suppressed $\sim 1/(\oo\RC)^{2}$. As a consequence, $\Ga^2$ is suppressed stronger then $1/\oo^4$ and goes as $1/\oo^6$ at $\oo\RC\gtrsim 1$. However, a more important signature of this capacitive coupling is the growth of $X(\oo)$ in Eq. (\[eq:Xomega\]), which affects $\Gs^2$. To see the role of this growth we consider now sufficiently large fields $\Phi=\Phi'$ when only the diffuson contribution survives. The growth of $X(\oo)$ in Eq. (\[eq:Xomega\]) reflects enhanced sensitivity of the internal potential $U_\oo$ to the gate voltage, $X(\oo)\propto (\tan\eta \mbox{Re }u_{0})^2$. At high frequencies the impedance of the capacitor $C$ becomes negligible and therefore the internal potential follows the gate voltage and not the reservoir voltages, $u_0\to 1,u_{1,2}\to 0$. Enhanced from its small static value $\RC/\dwell$ to 1 at large frequencies, such coupling affects the fluctuations of $\Gs(\oo)$ if $\eta\neq 0$. The situation is somewhat similar to the weak interaction limit, when the coupling with nearby gates was strong, $u_0\to 1,u_{1,2}\ll 1$, and lead to $\Gs\gg\Ga$. [ ![Zero-temperature large-field fluctuations of $\Ga(\oo)$ (dashed) and $\Gs(\oo)$ (solid curve) in units of $(\pi/4\Delta N^2)^2$ for the bias mode $\eta=\pi/4$. Data are presented in the log-log scale at $N_{1,2}=5$ and $\RC/\dwell=0.05$. The asymptotes $\Ga^2\propto \oo^{-6}$ and $\Gs^2\propto \oo^{-3}$ are different due to $\eta\neq 0$, see Eqs. (\[eq:Gainter\]) and (\[eq:Gsinter\]). []{data-label="fig:GsGaMath"}](GsGaMath.eps "fig:"){width="9cm"}]{}\ The fluctuations of $\Ga(\oo),\Gs(\oo)$ for $\oo\dwell\gg 1$ can be evaluated: $$\begin{aligned} \label{eq:Gainter} &&\Ga^2(\oo)\sim \frac{\Delta^2}{(\hbar\oo)^4(1+\oo^2\RC^2)},\\ &&\Gs^2(\oo)\sim\Ga^2(\oo)+ \frac{(\RC\tan\eta[1+\oo^2\dwell\RC])^2}{\hbar^2\oo\dwell(1+\oo^2\RC^2)^3}. \label{eq:Gsinter}\end{aligned}$$ Fluctuations of $\Ga^2(\oo)$ and $\Gs^2(\oo)$ demonstrate qualitatively different behavior, which we illustrate in Fig. \[fig:GsGaMath\]. Indeed, at sufficiently high frequencies, the dependence of $X(\oo)$ on $\omega$ makes the last term in Eq. (\[eq:Gsinter\]) dominant. At $\oo\RC\gg 1$ the asymptotes of $\Ga^2\propto 1/\oo^6$ and $\Gs^2\propto 1/\oo^3$ become different due to the presence of the second term in Eq. (\[eq:Gsinter\]). These results show that for nonadiabatic frequencies of the external bias the DC current strongly depends on the bias mode $\eta$. We predict that the magnetic field asymmetry of the rectified current, noticeable at small frequencies, might become suppressed for large frequencies, when the symmetrized component dominates due to the presence of capacitive coupling. For convenience, the low-temperature estimates for $\la\Ga^2\ra$ and $\la\Gs^2\ra$ for $\eta\neq 0$, $\Phi\gg \Phi_c$ are collected in Table \[tab:GaGs\]. ----------------------------------------- ----------- ------------------------------------ ------------------------------------- Adiabatic Intermediate High $\mbox{Function}$ $z\ll 1$ $1\ll z\ll \dwell/\RC$ $z\gg \dwell/\RC$ $\frac{\hbar^4}{\Delta^2\dwell^4}\Ga^2$ $1$ $z^{-4}$ $ \frac{\dwell^{2}}{\RC^{2}}z^{-6}$ $\frac{\hbar^2}{\RC^2}(\Gs^2-\Ga^2)$ $1$ $(1+\frac\RC\dwell z^2)^2 z^{-1} $ $\frac{\dwell^4}{\RC^4}z^{-3}$ ----------------------------------------- ----------- ------------------------------------ ------------------------------------- : Estimates for $\la\Ga^2\ra$ and $\la\Gs^2\ra$, ($z=\oo\dwell$) \[tab:GaGs\] It is noteworthy that a recent experiment in AB rings [@Bouchiat_preprint] finds that $\G(\oo,\Phi=0)$ grows with frequency until $\oo\sim 2\Thou$ and then decreases $\sim 1/\oo^{3/2},\oo\to\infty$. While we predict a monotonic decrease of $\la\Gs^2(\oo)\ra$, this growth could be the result of quantum pumping or an interference of the pumping and rectification (both effects were neglected here). Phase of Aharonov-Bohm oscillations {#sec:phase} =================================== In this section we consider nonlinear transport through a chaotic Aharonov-Bohm (AB) ring. The nonlinear conductance $\G$ exhibits periodic AB oscillations and non-periodic fluctuations, similarly to the linear conductance $G$. However, since Coulomb interactions produce asymmetry of $\G$ with respect to magnetic field inversion, the phase of these oscillations is not pinned to $0\mbox{ (mod)}\pi$. As a quantum effect this AB phase is characterized by a mesoscopic distribution. The width of this distribution represents a typical fluctuation. We first discuss what kind of distribution could be expected in a chaotic AB ring and then calculate the fluctuation of the AB phase. Let us assume that $\G$ as a function of magnetic flux $\Phi$ can be expanded into the series of well-defined Fourier harmonics similarly to the linear conductance $G$: $$\begin{aligned} \label{eq:expansion} {\genfrac{\{}{\}}{0pt}{}{ G(\Phi)}{\G(\Phi)}}&=&\sum_{n=0}^\infty {\genfrac{\{}{\}}{0pt}{}{G_n}{\G_n}}\cos\left( \frac{2\pi n\Phi}{\Phi_0}+{\genfrac{\{}{\}}{0pt}{}{0}{\delta_n}}\right).\end{aligned}$$ The phase $\d$ of the main (first) harmonic $\Phi_0=hc/e$ is obtained from the ratio of the (anti) symmetrized conductances defined in Eq. (\[eq:IVfield\]) $$\begin{aligned} \label{eq:tan} \tan\delta=\frac {\int d\Phi \exp(2\pi i \Phi/\Phi_0)\Ga(\Phi)}{\int d\Phi \exp(2\pi i \Phi/\Phi_0)\Gs(\Phi)}.\end{aligned}$$ We can not find the full mesoscopic distribution of the phase $P(\d)$. We can gain some insight in the behavior of this phase by investigating a similar quantity, namely, the asymmetry parameter $\A=\Ga/\Gs$ considered previously for chaotic dots. [@PhysicaE] Based on Eq. (\[eq:tan\]) we argue that the statistical properties of $\arctan \A$ and the AB phase $\d$ should be similar. In quantum dots the parameter $\A$ is given by the ratio $\A=\Ga/\Gs=\chi_{2a}/\chi_{2s}$, see Eqs. (\[eq:defG\]), (\[eq:chi2a\]), and (\[eq:chi2s\]). The functions $\chi_{2a,2s}$ at $T\neq 0$ are convolved separately with $f'(\e)$, and at $T=0$ (which we consider below) they are evaluated at the Fermi energy. The properties of $\chi_{2a,2s}$ and the dependence of $\chi_{2s}$ on the bias mode were described after Eq. (\[eq:X\]). The function $\chi_{2s}$ can have a nonzero (classical) average $\la\chi_{2s}\ra\sim X^{1/2}$ defined by the interaction strength, geometry of the setup, and the bias mode $\eta$. Since $\la\chi_{2,a}\ra=0$ and the fluctuations of $\chi_{2a,2s}$ are small as $1/N^2$, the mesoscopic distribution of $\arctan\A$ is narrow and concentrated close to 0. However, $\la\chi_{2s}\ra=0$ is possible if $X\to 0$, e.g., for symmetric contacts and the bias mode $\eta=0$. In this case, the distribution of $\arctan \A$ becomes wide regardless of the interaction strength. [\[t\][$\phi$]{}\[l\][$P(\phi)$]{} \[l\]\[\]\[0.8\][$N=16,N_L=8$]{} \[l\]\[\]\[0.8\][$N=16,N_L=4$]{} \[l\]\[\]\[.7\][$N=24$]{} \[l\]\[\]\[.7\][$N=2$]{} ![Mesoscopic distribution $P(\phi)$ of $\phi=\arctan \Ga/\Gs$. (Main plot) If the contacts are asymmetric (bold curve, $N=16,N_L=4$) the distribution is narrow, while for symmetric contacts (dashed, $N=16,N_L=8$) it is almost uniform. As shown in the inset, for symmetric contacts at large $N$ the distribution becomes uniform, compare bold curve for $N=2$ and dashed for $N=24$.[]{data-label="fig:delta"}](Angle.eps "fig:"){width="8cm"}]{}\ The role of the classical contribution on the shape of $P(\arctan\A)$ is demonstrated in the main plot in Fig. \[fig:delta\] for $\eta=0$, where the distributions for asymmetric, $N_L=4,N=16$, and symmetric contacts, $N_L=8,N=16$, are presented. While the distribution is almost uniform, when the classical contribution $X$ is absent, it is highly peaked near zero when $X$ dominates. If $X$ is absent, the correlations between $\Ga$ and $\Gs$ are significant at small $N$. This leads to a nonuniform distribution of $P(\arctan\A)$, which is peaked at $0$ and $\pi/2$ when $N=2$, see the inset in Fig. \[fig:delta\]. When $N$ grows, the correlations between $\Ga$ and $\Gs$ vanish and therefore the distribution becomes uniform. Such a distribution could be easily obtained if we make the natural assumption that $\Ga,\Gs$ are independent and distributed by the Gaussian law with the same width. These numerics were performed for $\eta=0$, when the mesoscopic distribution of $\A=\Ga/\Gs$ becomes insensitive to the interaction strength. The role of interactions appears only if $\eta\neq 0$, when the classical contribution $X$ becomes dominant. Similarly, we expect that the distribution of the phase of AB oscillations is also strongly affected by the bias mode. If the bias mode is chosen such that the classical contribution $X$ vanishes, the phase $\d$ strongly fluctuates [*even for weak interactions*]{}. It would be very interesting to check this surprising conclusion experimentally. Let us now consider the fluctuations of the AB phase. Since the scattering theory turned out to be very useful for the discussion of the nonlinear/ rectified current through a chaotic quantum dot, we extend this theory to rings. We make two key assumptions (discussed in the Appendix in more detail) that the magnetic flux through the annulus of the ring is smaller then the flux quantum $\Phi_0$ and that the mean free path $l$, the radius $R$, the width of the ring $W$ and the contacts $W_c$ satisfy the condition $\pi^2 l W\gg 2R W_c$. In this case the RMT can be applied to such chaotic rings as well. Unlike the experiments on large open rings with high aspect ratio $R/W\gg 1$,[@WW; @Liu; @Lin; @Bartolo; @BykovAB] the recent experiments [@ensslin; @Bouchiat; @Bouchiat_preprint] are performed in rings of submicron size, which are effectively zero-dimensional. The treatment of such rings is similar to chaotic quantum dots, and the fluctuations of $\Gs,\Ga$ can be expressed in terms of the diffuson $\Diff$ and the cooperon $\Coop$, see Eq. (\[eq:main\]). The only problem is to find the expression for the effective number of channels as a function of magnetic field, similar to Eq. (\[eq:channels\]). [ ![Model of a chaotic Aharonov-Bohm ring with $N=N_1+N_2$ channels. The model consists of a quantum dot with $M$ channels combined with a ballistic arm with $N_3=N_4=(M-N)/2$ channels.[]{data-label="fig:model"}](ABdotmodel.eps "fig:"){width="6cm"}]{}\ The model we propose for a chaotic AB ring combines chaos and a ring geometry: a chaotic dot is attached to a long ballistic arm which serves to include an AB flux large compared to the fraction of the flux through the sample. This model is shown in Fig. \[fig:model\], where the ring with $N=N_1+N_2$ ballistic channels in the contacts 1,2 is modeled by a dot with $M>N$ channels and a ballistic arm with $N_3=N_4=(M-N)/2$ channels in contacts 3,4. The parameter $\rho=1-N/M$, the ratio of $N_3+N_4$ to the total number of channels $M$, can vary between 0 when the arm is much narrower then the contacts and 1 in the opposite limit. The electronic phase is randomized in the quantum dot, but when electrons propagate along the arm their phase is determined by the geometry and applied magnetic field. This model is a reasonable approximation for the real experiment, it takes into account the long time spent by electron inside the ring and the randomness of its motion. The discussion of the model and the details of calculation of $\Coop,\Diff$ are presented in the Appendix. In experiment the Fourier transform is often taken over the total flux (or applied magnetic field) and the flux through the hole $\Phi_{h}$ cannot be separated from the flux through the dot $\Phi_{d}$. Then the dependence of the diffuson and cooperon on magnetic field is non-periodic, which is indeed observed in the form of nonperiodic fluctuations in the (non-)linear conductance and phase slips of AB oscillations. A possible weakness of this model is in its spatial separation of chaotic scattering and the main part of magnetic field, but in the limit when the arm is much wider then the contacts $1$ and $2$ such a separation is not important and the averaged properties of AB oscillation phase become independent of the arm’s width. If the flux $\Phi_d$ through the dot is much smaller then the flux $\Phi_h$ through the hole, the nonperiodic fluctuations and the periodic AB oscillations are well-separated, which is usually the case in experiment.[@ensslin; @Bouchiat] In view of this separation we can neglect the flux through the chaotic dot, $\Phi_d\ll\Phi_h$, to find the statistics of the AB phase. We assume that the averaging is taken over a magnetic field range containing many AB oscillations but still small compared to the characteristic field of the nonperiodic fluctuations. In such a simplified model of a chaotic AB ring $N_{\Coop,\Diff}$ are given by Eq. (\[eq:DCfinal\]) with $\Phi_d=0,\Phi_h=\Phi$ and the parameter $\rho=(M-N)/M$. The effective number of channels is $$\begin{aligned} \label{eq:DCsimple} {\genfrac{\{}{\}}{0pt}{}{N_{\cal C}}{N_{\cal D}}} &=& M\left(1-\rho\cos \frac{2\pi(\Phi\pm\Phi')}{\Phi_0}\right).\end{aligned}$$ Using this expression for the $N_{\Coop,\Diff}$ we can evaluate the quantum fluctuations of linear conductance in AB rings. At low temperature a typical fluctuation of $G$ at $\Phi=0$ is $\d G=\sqrt{2}(N_1 N_2/N^2)(\nu_s e^2/h)$ and the amplitude of AB oscillations is $\d G_{\rm AB}^2\sim\d G^2 \rho(1-\rho)^{1/2}/(1+\rho)^{3/2}$, which reaches maximum when the widths of the arm and the contacts are equal, $\rho=1/2$. The first two moments of $\tan\d$ can be found analytically. It is zero on average, $\la\tan\d\ra=0$, since the numerator and denominator in Eq. (\[eq:tan\]) are independent random quantities. Equation (\[eq:DCsimple\]) show that $\Diff,\Coop$ are the same functions of $\Phi\pm\Phi'$, so that all necessary ingredients can be expressed in terms of the functions ${\mathcal F}_{U,\Diff}$ and ${\mathcal F}_{G,\Diff}$ of $\Phi-\Phi'$ defined in Eqs. (\[eq:FU\]) and (\[eq:FG\]). We omit now the index $\Diff$ for brevity and denote the average over magnetic field by $\overline{ (...)}=\int_0^{\Phi_0}(...) d\Phi/\Phi_0 $, to find $$\begin{aligned} \label{eq:tandelta} &&\la\tan ^2\delta\ra = \left[\overline{{\cal F}_{U }(\Phi){\cal F}_{G }(\Phi)\cos\Phi} +\overline{ {\cal F}_{G }}\cdot\overline{{\cal F}_{U }(\Phi)\cos\Phi}\right.\nonumber \\ && \mbox{} \left.-\overline{ {\cal F}_{U }}\cdot\overline{{\cal F}_{G }(\Phi)\cos\Phi}\right]/ \left[\overline{({\cal F}_{U }(\Phi)+X(\oo)){\cal F}_{G }(\Phi)\cos\Phi} \right.\nonumber \\ && \mbox{} \left.+ \overline{\frac{{\cal F}_{U }(\Phi)}{2}}\overline{{\cal F}_{G }(\Phi)\cos\Phi}+ \overline{\frac{{\cal F}_{G }(\Phi)}{2}}\overline{{\cal F}_{U }(\Phi)\cos\Phi}\right],\end{aligned}$$ where the function $X(\oo)$ is defined by the setup geometry in Eq. (\[eq:Xomega\]). Again, in the static limit $\oo\to 0$ we have ${\cal F}_{U}={\cal F}_{G}={\cal F}$ and $X$ defined by Eqs. (\[eq:Fraw\]) and (\[eq:X\]). In this case Eq. (\[eq:tandelta\]) can be rewritten as $$\begin{aligned} \label{eq:tan^2result} \frac{1}{\la\tan ^2\delta\ra} &=&1+\frac{[\overline{{\cal F}(\Phi)}+X]\overline{\cos\Phi{\cal F}(\Phi)}}{\overline{\cos\Phi{\cal F}^2(\Phi)}}.\end{aligned}$$ In the limits of high, $T\gg N\Delta/2\pi$, and low temperature, $T\ll N\Delta/2\pi$ the asymptotical values of $\la\tan^2\d\ra$ are $$\begin{aligned} \label{eq:tan^2asymp} \frac{1}{\la\tan ^2\delta\ra}&=&1+\left\{\begin{array}{cc} \frac{\sqrt{1-\rho^2}}{1+\sqrt{1-\rho^2}}+ \frac{12 T}{ \Delta}\frac{XM(1-\rho^2)}{1+\sqrt{1-\rho^2}}, & T\gg \frac{N\Delta}{2\pi}\\ \frac{2\sqrt{1-\rho^2}}{4+\rho^2} + \frac{2XM^2(1-\rho^2)^2}{4+\rho^2}, &T\ll\frac{N\Delta}{2\pi}\end{array}\right.\nonumber\end{aligned}$$ Very important is the case of symmetric contacts, $N_1=N_2$, and antisymmetric bias mode, $\eta=0$, which is used in Ref.. Then $X$ vanishes and the average $\tan^2\d$ becomes independent of interaction strength and as a function of $T$ it is very weak. That is not the case if $\eta\neq 0$, for example, when only one of the voltages changes, $\eta=\pm \pi/4$.[@Bouchiat; @Bouchiat_preprint] Then the statistics of the AB phase becomes temperature and interaction dependent due to the presence of $X$. The limit $M\gg N$ corresponds to a uniformly chaotic ring, which we suppose to be closer to the experimental situation. Then the dependence on $M$ drops out and the high/low temperature asymptotic read $$\begin{aligned} \label{eq:ANGLEwidearm} \frac{1}{\la\tan ^2\delta\ra}&=&1+8X\left\{\begin{array}{cc} 3NT/\Delta, & T\gg N\Delta/2\pi,\\ N^2/5, &T\ll N\Delta/2\pi\end{array}\right. .\end{aligned}$$ This result clearly demonstrates that the phase of the oscillations is expected to deviate strongly from 0, especially if the temperature is low and the number of channels in the contacts is diminished. The temperature is taken into account only in the form of temperature-averaging and the dephasing (previously considered for nonlinear transport of noninteracting electrons in Refs.) is not included. We expect our model for chaotic AB rings to work both for experiments at small frequencies [@ensslin; @Bouchiat] and for large frequencies.[@Bouchiat_preprint] Similarly to quantum dots, the generalization on the finite-frequency case is obvious, if we use Eq. (\[eq:Xomega\]). Even in cases where RMT cannot be assured to be valid for open diffusive rings, the dependence of the AB phase on interaction strength, temperature, and number of external channels given by Eq. (\[eq:ANGLEwidearm\]) should be correct qualitatively. The experiment of Leturcq [@ensslin] is performed in a bias mode $\eta=0$ when $X=0$. Then Eq. (\[eq:ANGLEwidearm\]) gives $\la\tan^2\d\ra=1$. The phase of the oscillations is evaluated from data according to Eq. (\[eq:tan\]) over a large range of fields. In experiment the AB phase is varied continuously as a function of the gate voltage at one of the arms of the ring. The data demonstrate that the phase $\d$ indeed changes in a wide range and is usually far from 0. This substantiates our conclusion that in the mode when the classical contribution is minimized, $X\to 0$, the mesoscopic distribution of $\d$ is very wide. Experiment of Angers [@Bouchiat] varies voltage in a different way, $\eta=\pi/4$, and therefore has $X\neq 0$. We would expect the phase $\d\mbox{(mod)}\pi$ to take values closer to $0$ and the antisymmetric component of the oscillations be relatively smaller even for large fields. Although phases close to 0 are indeed observed, the field averaging is taken only over first few oscillations. In this range $\Ga$, the numerator in Eq. (\[eq:tan\]) is still small and grows linearly with magnetic field. Averaging over a larger field-range similar to Ref. could not be performed because of the phase slips. Another interesting question is a difference in data [@ensslin; @Bouchiat] for the relative magnitudes $\G_2/\G_1$ of the second $hc/2e$ and main harmonic $hc/e$, see Eq. (\[eq:expansion\]). In the nonlinear transport regime this harmonic is small compared to its contribution in the linear transport, $\G_2/\G_1\ll G_2/G_1$,[@ensslin] while in Ref. they were comparable, $\G_2/\G_1\approx G_2/G_1$. Our model also predicts the mesoscopically averaged contribution of $hc/2e$ into linear and nonlinear conductance to be comparable with that of $hc/e$. Our approach assumes full quantum coherence of the ring, and probably the difference in data is due to decoherence. Conclusions {#sec:conclusions} =========== In this paper, we consider mesoscopic chaotic samples (quantum dots or rings) and find the statistics of their nonlinear conductance $\G$. This transport coefficient characterizes nonlinear DC current due to DC-bias or a rectified current due to AC bias or photon-assisted transport. For chaotic samples, the nonlinear effect is of quantum origin, which is clear from the fact that its ensemble average over similar samples vanishes. The linear response of the sample in two-terminal measurements is always symmetric with respect to magnetic field inversion. However, the Coulomb interactions lead to magnetic field asymmetry of the nonlinear DC response, which fluctuates due to the electronic interference. For the quantum dots we consider the fluctuations of (anti) symmetrized components $\Ga,\Gs$ of the nonlinear conductance. In chaotic rings the statistics of the phase of AB oscillations in the nonlinear transport regime, closely related to the ratio $\Ga/\Gs$, is of interest. Unlike the linear conductance measurements, in mesoscopic nonlinear transport experiments the way voltages are varied (“bias mode”) turns out to be important, especially for a weakly interacting sample. We demonstrate this fact qualitatively and discuss the role of Coulomb interactions. Quantitative self-consistent treatment of interactions allows us to consider magnetic-field asymmetry in chaotic quantum dots with many channels. Using Eqs. (\[eq:main\])–(\[eq:X\]) we show that the fluctuations of $\Gs$ are strongly affected by the geometry of the setup and discuss how the bias mode influences data of recent DC experiments. Another important issue is rectification of AC bias, which is quadratic in applied voltage, random, and asymmetric with respect to the magnetic flux inversion. The photovoltaic DC current can be due to rectification of external perturbations or quantum pumping by internal perturbations. Both rectification and quantum pumping share the aforementioned properties, and it is important to clearly separate them especially when the frequency of perturbations is high (nonadiabatic). We consider here only the effects of the external perturbations and discuss the dependence of the fluctuations of $\Ga,\Gs$ on frequency $\oo$. We show that the fluctuations of both $\Ga$ and $\Gs$, presented in Eqs. (\[eq:varGs\])–(\[eq:Xgeneral\]), decrease monotonically as $\oo\to\infty$. However, contrary to naive expectations, their asymptotical behavior can be very different. Since at high frequencies the response of the dot to the external bias becomes rather capacitive then resistive, the coupling to the nearby gates can be strongly enhanced. If the experiment is performed in a bias mode where such coupling contributes, the symmetrized $\Gs^2(\oo)\propto 1/\oo^3$ can become much larger then $\Ga^2(\oo)\propto 1/\oo^6$ valid for a strongly interacting quantum dot. The same conclusion holds in the weakly interacting limit, when $\Gs^2\propto 1/\oo^{3/2}$ and $\Ga^2\propto 1/\oo^{3}$. In addition, we show that recent experiments in chaotic Aharonov-Bohm rings might be considered similarly to quantum dots. The multiply connected geometry alone leads to AB oscillations, yet the mesoscopic distribution of their phase is expected to be qualitatively similar to that of $\arctan \Ga/\Gs$ in quantum dots. Therefore, the bias mode should strongly affect the shape of mesoscopic distribution of the AB phase. The model of an AB ring, which we develop, consists of a dot and a long ballistic arm and takes into account both chaos and a ring geometry. As an application of our model we consider fluctuations of the AB phase. Unlike the AB phase in the linear conductance, pinned to $0\mbox{(mod)}\pi$ by the Onsager symmetry relations, the fluctuations of the AB phase in nonlinear transport are shown to depend on the bias mode, interaction strength, and temperature. Acknowledgements ================ We thank Hélène Bouchiat, Piet Brouwer, Renaud Leturcq, David Sánchez, Maxim Vavilov, and Dominik Zumbühl for valuable discussions. We also thank the authors of Ref. for sharing their results with us before publication. This work was supported by the Swiss National Science Foundation, the Swiss Center for Excellence MaNEP, and the STREP project SUBTLE. diffuson and cooperon for chaotic ring {#sec:appendix} ====================================== In this Appendix we determine the diffuson and cooperon contributions to the $\S$-matrix correlators of the random scattering matrix of a chaotic Aharonov-Bohm (AB) ring. This calculation is performed using Random Matrix Theory (RMT). First we explain what approximations should be made to ensure validity of RMT. Our starting point is the assumption that the $\S$-matrix of the ring is uniformly distributed over the unitary group. This means that the ring is essentially zero-dimensional, similarly to quantum dots. RMT is applicable if all energy-scales are much smaller then the Thouless energy $\Thou$ and the total flux through the annulus of the ring is much smaller then $\Phi_0$. Assume the ring of radius $R$ and width $W\ll R$ to be diffusive with diffusion coefficient $D=l v_{\rm F}/2$. To evaluate $\Thou$ we neglect with transversal motion of an electron and find $\Thou=\hbar/\erg=(\hbar l v_{\rm F})/2R^2$ as a solution to Laplace equation along the circumference of the ring. RMT can be applied to a closed ring if the dimensionless conductance is large, $g=\Thou/\Delta=k_{\rm F}l W/2R\gg 1$, which is usually satisfied for a weak disorder even if $W\ll R$. ![Chaotic dot combined with long ballistic multichannel arm.[]{data-label="fig:Abdot"}](ABdot.eps "fig:"){width="4cm"}\ An open ring with ballistic contacts of the width $W_c$ gains a new energy parameter, the escape rate $\hbar/\dwell=N\Delta/2\pi$, where $N$ is the total number of ballistic channels. The scattering matrix $\S$ is uniformly distributed and independent of the exact positions of the contacts (and therefore the length of the arms) if $\hbar/\dwell\ll\Thou\Rightarrow \pi^2 l W\gg 2R W_c$. In this case the main drop of the potential occurs in the contacts. If a magnetic field is applied, the RMT is valid if the total flux through the annulus of the ring is much less then the flux quantum, $\Phi\ll\Phi_0$. Due to narrow contacts the time-reversal symmetry (TRS) of the $\S$-matrix can be broken at a much smaller scale, $\Phi\sim \Phi_0\sqrt{\erg/\dwell}$. Since in our rings $\erg\ll\dwell$, a full crossover to the broken TRS can be considered. How well are these conditions fulfilled in the experiment? In Ref. chaos was mainly due to diffusive scattering on the boundary and $l\approx R$. The width of the arm is 2-4 channels, while the number of channels in the contacts is $N\sim 2$, estimated from the linear conductance measurements, so $\dwell/\erg\sim 5\div 10\gg 1$. In semiballistic samples of Ref. (obtained by etching, and therefore having diffusive boundary scattering) $W=W_c$ and the mean free path is estimated $l\sim 1\div 2\mu$m$\sim L= 1.2 \mu$m, the side length. Therefore, we have a similar estimate for the ratio $\dwell/\erg$. Although this ratio is not parametrically large due to, e.g., weak disorder $k_{\rm F}l\gg 1$, we believe that such AB rings still can be assumed zero-dimensional due to their good conducting properties together with relatively narrow contacts. In our calculations we make a further simplification by spatially separating chaotic scattering which randomizes the electronic phase and the long ballistic arm attached to it. To find the correlators of the $\S$-matrix elements we use a simplified model, see Fig. \[fig:Abdot\], which combines chaos and a ring geometry. A chaotic $M$-channel dot is attached to a long multi-channel ballistic arm with $(M-N)/2$ orbital channels. We assume that the size of the dot $L$ and the length of the arm $L_a$ are such that $L_a\gg L\gg \sqrt {(M-N)\lambda_{\rm F} L_a}$ to ensure that in the hierarchy of different fluxes the main flux $\Phi_h$ is concentrated in the region embraced by the arm, the flux through the dot $\Phi_d$ is much smaller, but still much larger then the flux through the cross section of the arm. The amplitude of AB oscillations depends on the width of the arm $\propto (M-N)$. The wider the arm (relatively to the contacts) the closer the results should be to a uniformly chaotic ring. For the case when $M\gg N$ we expect it to be valid for the chaos uniformly distributed over the ring. Indeed, in this case an electron makes $\sim M/N\gg 1$ windings around the arm before exiting. In this appendix it is more convenient for us to work with an energy-dependent matrix $\S(\e)$, and the final transformation to time-representation is rather obvious. The total scattering matrix $\S$ is of size $N\times N$ due to scattering channels in the contacts 1 and 2. Chaotic scattering in the $M$-channel quantum dot is characterized by the $M\times M$ matrix $\U$. The scattered electron can either exit the sample through the $N=N_1+N_2$ channels (projection operator $P_0=1_1\oplus 1_2$) or propagate into the arm with $N_3=N_4=(M-N)/2$ channels. Electrons propagate through this arm ballistically and gain phases which depend on the flux through the hole. In the absence of backscattering the electronic amplitudes at energy $\e$ are related to the path length $L_a$ and magnetic field phase $\phi$: $$\begin{aligned} \label{eq:Pmatrix} \binom{b_3}{b_4}=e^{-ik(\e)L_a}\left(\begin{array}{cc} 0 & e^{-i\phi} \\ e^{i\phi}& 0 \\ \end{array}\right)\binom{a_3}{a_4}=P\binom{a_3}{a_4}.\end{aligned}$$ The scattering matrix of the arm is $\P(\e)=0_1\oplus 0_2\oplus P$. Each time an electrons enters the arm either through the third or fourth lead, the matrix $\P$ contributes to the scattering amplitude of the process. The total scattering matrix $\S$ is determined from the following equation: $$\begin{aligned} \S=P_0\sum_{n=0}^\infty\U(\P \U)^n P_0=P_0\U\frac{1}{1-\P\U}P_0,\end{aligned}$$ where multiple $n\geq 0$ windings are taken into account. Both $\U(\e,B)$ and $\P(\e,B)$ are field and energy dependent. Once we are interested only in pair correlators of $\S(\e),\S^\dagger(\e')$ for $N,M\gg 1$, the diffuson $\Diff$ and cooperon $\Coop$ of our scattering matrix are expressed via correlators of the dot, $\Diff_{\cal U}$,$\Coop_{\cal U}$, and tr $\P(\e)\P^{*(\dagger)}(\e')$. The correlators of the $\U$-matrix are known, see Eq. (\[eq:cum1t\]) for their time representation, and for $\Diff$ and $\Coop$ we derive $$\begin{aligned} \label{eq:DC} {\genfrac{\{}{\}}{0pt}{}{\Coop}{\Diff}}^{-1}&=&{\genfrac{\{}{\}}{0pt}{}{{\cal C}_{\cal U}^{-1}-\mbox{tr }\P(\e)\P^*(\e')}{{\cal D}_{\cal U}^{-1}-\mbox{tr }\P(\e)\P^\dagger(\e')}}, \\ {\genfrac{\{}{\}}{0pt}{}{\Coop}{\Diff}}_{\cal U}^{-1}&=& M-2\pi i\frac{\e-\e'}{\Delta}+\frac{h v_F l}{L^2\Delta}\left(\frac{\Phi_d\pm \Phi_d'}{2\Phi_0}\right)^2.\label{eq:DCU}\end{aligned}$$ The flux penetrating the dot is denoted as $\Phi_d$ and the phase $\phi\approx 2\pi\Phi_h/\Phi_0$ gained in the arm depends on the flux $\Phi_h$ through the hole. The traces read $$\begin{aligned} \label{eq:tr} \mbox{ tr }{\genfrac{\{}{\}}{0pt}{}{\P(\e,\Phi)\P^*(\e',\Phi')}{\P(\e,\Phi)\P^\dagger(\e',\Phi')}}&=&(M-N) \cos{\genfrac{\{}{\}}{0pt}{}{\phi+\phi'}{\phi-\phi'}}\nonumber \\ &\times &e^{iL_a [k(\e)-k(\e')]}.\end{aligned}$$ Since we assumed that the area of the arm is small compared to that of the dot, the energy-dependence of Eq. (\[eq:tr\]) can be neglected compared to that of $\Diff_{\cal U},\Coop_{\cal U}$ in Eq. (\[eq:DCU\]). We also assumed that since the arm is much longer than the size of the dot, $L_a\gg L$, the phases $\phi,\phi'$ of open trajectories in the arm correspond to the flux $\Phi_h,\Phi_h'$ through the hole. Therefore, the effective number of channels $N_{\Coop,\Diff}$, similar to Eq. 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ArXiv

--- address: | Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3,\ 53 avenue des Martyrs,\ 38026 Grenoble Cedex, France author: - SELIM TOUATI title: ELECTRIC DIPOLE MOMENTS AND NEUTRINO MASS MODELS --- EDMs generated by the CKM phase =============================== In the standard model (SM), the only source of weak CP-violation is the complex phase of the CKM matrix. In order to measure the strength of CP-violation, one can construct a flavor invariant (basis-independent) which is sensitive to this phase, called the Jarlskog invariant [@Jarl]. A non-vanishing Jarlskog invariant is a necessary condition for having CP-violation. In the SM, all CP-violating effects are proportional to this invariant. However, this invariant is adequate for estimating CP-violation from closed fermion loops. For example, let us consider the CKM-induced lepton EDMs. Because the leptons cannot feel directly the complex phase of the CKM matrix, we need to go through a closed quark loop. The dominant diagram is: ![CKM-induced lepton EDM[]{data-label="fig:CKMleptonEDM"}](figures/FigCKMLeptonEDM) This EDM is tuned by the Jarlskog invariant $\det[Y_{u}^{\dagger}Y_{u},Y_{d}^{\dagger}Y_{d}]$ which is proportional to the imaginary part of a quartet $Im(V_{us}V_{cb}V_{ub}^{\ast}V_{cs}^{\ast})$. As for the quarks, they can feel directly the complex phase of the CKM matrix and then there are non-invariants structures which arise from rainbow-like processes. Indeed, the dominant diagrams for the CKM-induced quark EDMs have a rainbow topology. For instance, for the d-quark EDM: ![CKM-induced d-quark EDM[]{data-label="fig:CKMquarkEDM"}](figures/FigCKMQuarkEDM) This EDM is tuned by the imaginary part of the 1-1 entry of a non-invariant commutator $Im(\textbf{X}^{dd}_{q})$, where: $$\textbf{X}_{q}=\mathbf{[Y}_{u}^{\dagger}\mathbf{Y}_{u},\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}\mathbf{Y}_{d}^{\dagger}\mathbf{Y}_{d}\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}], \label{eq:Xq}$$ which is also proportional to $Im(V_{us}V_{cb}V_{ub}^{\ast}V_{cs}^{\ast})$ as for the lepton EDMs (because we are in the SM), but not with the same proportionality factor. It turns out to be much larger by 10 orders of magnitude: $$Im\mathbf{[Y}_{u}^{\dagger}\mathbf{Y}_{u},\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}\mathbf{Y}_{d}^{\dagger}\mathbf{Y}_{d}\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}]^{dd}\gg\det[Y_{u}^{\dagger}Y_{u},Y_{d}^{\dagger}Y_{d}].$$ In the SM, the rainbow-like flavor structures are typically much larger than the invariant determinants and they are correlated (strictly proportional). Now, let us turn on neutrino masses (beyond the SM) and check whether this behavior is confirmed or not. As we do not know yet the nature of the neutrino (Dirac or Majorana particle), we will consider both scenarios for generating neutrino masses. EDMs in the presence of neutrino masses ======================================= Dirac neutrino masses --------------------- The simplest way of including neutrino masses to the SM is to extend its particles content by adding three right-handed (RH) fully neutral neutrinos (one for each generation). They belong to the trivial representation of the SM gauge group: $N=\nu_{R}^{\dagger}\sim(1,1)_{0}$. We add to the SM Yukawa Lagrangian an extra Yukawa interaction for the neutrinos: $$\mathcal{L}_{Yukawa}=\mathcal{L}_{Yukawa}^{SM}-N^{I}Y_{\nu}^{IJ}L^{J}H^{\dagger C}+h.c.$$ We have a new neutrinos-related flavor structure $Y_{\nu}$ ($3\times3$ matrix in flavor space). In the presence of neutrino masses, we get an additional source of weak CP-violation coming from the complex phase of the PMNS matrix. In complete analogy with the quark sector, we can construct new CP-violating flavor structures which tune the PMNS-induced quark and lepton EDMs. In this case, quark EDMs have a bubble topology whereas lepton EDMs have a rainbow topology. For instance, the dominant diagrams for the PMNS-induced quark and lepton EDMs are shown in figure \[fig:DiracEDMs\]. ![PMNS-induced quark (on the left) and lepton (on the right) EDMs[]{data-label="fig:DiracEDMs"}](figures/FigDiracEDMs) They are tuned respectively by $J_{\mathcal{CP}}^{Dirac}$ and $Im(\textbf{X}_{e}^{Dirac})^{11}$, where $$\begin{aligned} J_{\mathcal{CP}}^{Dirac}= & \frac{1}{2i}\det\left[Y_{\nu}^{\dagger}Y_{\nu},Y_{e}^{\dagger}Y_{e}\right]\\ \textbf{X}_{e}^{Dirac}= & \left[Y_{\nu}^{\dagger}Y_{\nu},Y_{\nu}^{\dagger}Y_{\nu}Y_{e}^{\dagger}Y_{e}Y_{\nu}^{\dagger}Y_{\nu}\right]. \label{eq:XeDirac}\end{aligned}$$ In this scenario, $Im(\textbf{X}_{e}^{Dirac})^{11}$ is 11 orders of magnitude larger than $J_{\mathcal{CP}}^{Dirac}$ and they are correlated (strictly proportional). Majorana neutrino masses ------------------------ Another way for generating neutrino masses is possible if we consider Majorana masses. In this mechanism, there is no additional RH neutrinos, we get directly a gauge-invariant but lepton-number violating mass term for the left-handed (LH) neutrinos. Indeed, we add to the SM Yukawa Lagrangian the effective dimension-five Weinberg operator: $$\mathcal{L}_{Yukawa}=\mathcal{L}_{Yukawa}^{SM}-\frac{1}{2v}(L^{I}H)(\Upsilon_{\nu})^{IJ}(L^{J}H)+h.c,$$ which after spontaneous symmetry breaking collapses to a Majorana mass term for the LH neutrinos: $$\frac{1}{2v}(L^{I}H)(\Upsilon_{\nu})^{IJ}(L^{J}H)\overset{SSB}{\longrightarrow}\frac{v}{2}(\Upsilon_{\nu})^{IJ}\nu_{L}^{I}\nu_{L}^{J}.$$ $\Upsilon_{\nu}$ (3$\times$3 matrix in flavor space) is a new flavor structure purely of the Majorana type. In this model, we must redefine the PMNS matrix in order to add two new CP-violating phases, called Majorana phases, $$U_{PMNS}\rightarrow U_{PMNS}\cdot diag(1,e^{i\alpha_{M}},e^{i\beta_{M}}).$$ Let us consider the PMNS-induced quark and lepton EDMs in this scenario. The dominant diagrams are: ![PMNS-induced quark (on the left) and lepton (on the right) EDMs[]{data-label="fig:MajoEDMs"}](figures/FigMajoEDMs) The CP-violating flavor structures which tune these EDMs are $J_{\mathcal{CP}}^{\mathrm{Majo}}$ [@Branco] and $Im(\mathbf{X}_{e}^{\mathrm{Majo}})^{11}$, where: $$\begin{aligned} J_{\mathcal{CP}}^{\mathrm{Majo}}= & \frac{1}{2i}Tr[\mathbf{\Upsilon}_{\nu}^{\dagger}\mathbf{\Upsilon}_{\nu}\cdot\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e}\cdot\mathbf{\Upsilon}_{\nu}^{\dagger}(\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e})^{T}\mathbf{\Upsilon}_{\nu}-\mathbf{\Upsilon}_{\nu}^{\dagger}(\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e})^{T}\mathbf{\Upsilon}_{\nu}\cdot\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e}\cdot\mathbf{\Upsilon}_{\nu}^{\dagger}\mathbf{\Upsilon}_{\nu}]\\ \mathbf{X}_{e}^{\mathrm{Majo}}= & [\mathbf{\Upsilon}_{\nu}^{\dagger}\mathbf{\Upsilon}_{\nu},\mathbf{\Upsilon}_{\nu}^{\dagger}(\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e})^{T}\mathbf{\Upsilon}_{\nu}].\end{aligned}$$ We find that $Im(\textbf{X}_{e}^{Majo})^{11}$ is 4 orders of magnitude larger than $J_{\mathcal{CP}}^{Majo}$ but in this scenario they are not correlated. In figure \[fig:CorrelationMajo\], we can see the values that can take the PMNS-induced quark and lepton EDMs (tuned respectively by $J_{\mathcal{CP}}^{Majo}$ and $Im(\textbf{X}_{e}^{Majo})^{11}$). The lightest neutrino mass $m_{\nu 1}$ is set to $1eV$ and the CP-violating phases (PMNS phase $\delta_{13}$ and the Majorana phases $\alpha_{M}$ and $\beta_{M}$) are allowed to take on any values. ![Area spanned by $J_{\mathcal{CP}}^{Majo}$ and $Im(\textbf{X}_{e}^{Majo})^{11}$[]{data-label="fig:CorrelationMajo"}](figures/CorrelationMajo) The lines show the strict correlation occuring when only one phase is non-zero. When the three phases are into action, because the flavor structures have different dependences in these phases, the result is that the quark and lepton EDMs become decorrelated. Conclusion ========== In the paper [@SmithTouati], we developped a systematic method to study the flavor structure behind the quark and lepton EDMs which can be extended easily to other more complicated models (Sterile neutrinos, SUSY etc...). The rainbow-like non-invariant flavor structures are found to be typically much larger than the Jarlskog-like flavor invariants. Interestingly, we find a different behavior for Dirac and Majorana neutrinos. Quark and lepton EDMs are proportional in the former case whereas they are completely independent in the latter case. Indeed, quark and lepton EDMs have different dependences on Majorana phases. Finally, by studying the flavor structures behind the quark and lepton EDMs, we get the relations shown in table \[tab:SumRules\] between EDMs of different generations. CKM-induced EDMs PMNS-induced EDMs -- ------------------------------------------------------------------------- --------------------------------------------------------------------------- $\frac{d_{d}}{m_{d}}+\frac{d_{s}}{m_{s}}+\frac{d_{b}}{m_{b}}=0$ $\frac{d_{d}}{m_{d}}=\frac{d_{s}}{m_{s}}=\frac{d_{b}}{m_{b}}$ $\frac{d_{e}}{m_{e}}=\frac{d_{\mu}}{m_{\mu}}=\frac{d_{\tau}}{m_{\tau}}$ $\frac{d_{e}}{m_{e}}+\frac{d_{\mu}}{m_{\mu}}+\frac{d_{\tau}}{m_{\tau}}=0$ : Sum rules[]{data-label="tab:SumRules"} For example, the CKM-induced quark EDMs and the PMNS-induced lepton EDMs are tuned by the non-invariant commutators \[eq:Xq\] and \[eq:XeDirac\] and because a commutator is traceless, we get these sum rules. References {#references .unnumbered} ========== [99]{} C. Smith, S.Touati, . C. Jarlskog, . G. C. Branco, R. G. Felipe, F. R. Joaquim, .

ArXiv

--- author: - | Sok Jérémy\ Ceremade, UMR 7534, Université Paris-Dauphine,\ Place du Maréchal de Lattre de Tassigny,\ 75775 Paris Cedex 16, France.\ \ bibliography: - 'bibliothese.bib' title: '**The positronium and the dipositronium in a Hartee-Fock approximation of quantum electrodynamics**' --- Introduction and main results ============================= The Dirac operator ------------------ Relativistic quantum mechanics is based on the *Dirac operator* $D_0$, which is the Hamiltonian of the free electron. Its expression is [@Th]: $$\label{di_dirac_op} D_0:=m_ec^2\beta-i\hbar c{\ensuremath{\displaystyle\sum}}_{j=1}^3\alpha_j \partial_{x_j}$$ where $m_e$ is the (bare) mass of the electron, $c$ the speed of light and $\hbar$ the reduced Planck constant and $\beta$ and the $\alpha_j$’s are $4\times 4$ matrices defined as follows: $$\label{di_beta_alpha} \beta:=\begin{pmatrix} \mathrm{Id}_{\mathbb{C}^2} & 0\\ 0 & -\mathrm{Id}_{\mathbb{C}^2} \end{pmatrix},\ \alpha_j:= \begin{pmatrix} 0 & \sigma_j \\ \sigma_j & 0 \end{pmatrix},\ j\in\{1,2,3\}$$ $$\sigma_1:=\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix},\ \sigma_2:=\begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix},\ \sigma_3\begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix}.$$ The operator $D_0$ acts on the Hilbert space $ \mathfrak{H}$: $$\label{di_space_one_electron} \mathfrak{H}:=L^2\big({\ensuremath{\mathbb{R}^3}},{\ensuremath{\mathbb{C}^4}}\big);$$it is self-adjoint on $\mathfrak{H}$ with domain $H^1({\ensuremath{\mathbb{R}^3}},{\ensuremath{\mathbb{C}^4}})$. Its spectrum is $\sigma(D_0)=(-\infty,m_ec^2]\cup[m_e c^2,+\infty)$, which leads to the existence of states with arbitrary small energy. Dirac postulated that all the negative energy states are already occupied by “virtual electrons”, with one electron in each state: by Pauli’s principle real electrons can only have a positive energy. In this interpretation the Dirac sea, composed by those negatively charged virtual electrons, constitutes a polarizable medium that reacts to the presence of an external field. This phenomenon is called the *vacuum polarization*. After the transition of an electron of the Dirac sea from a negative energy state to a positive, there is a real electron with positive energy plus the absence of an electron in the Dirac sea. This hole can be interpreted as the addition of a particle with same mass, but opposite charge: the so-called positron. The existence of this particle was predicted by Dirac in 1931. Although firstly observed in 1929 independently by Skobeltsyn and Chung-Yao Chao, it was recognized in an experiment lead by Anderson in 1932. Positronium and dipositronium ----------------------------- The positronium is the bound state of an electron and a positron. This system was independently predicted by Anderson and Mohorovi$\check{\mathrm{c}}$ić in 1932 and 1934 and was experimentally observed for the first time in 1951 by Martin Deutsch. It is unstable: depending on the relative spin states of the positron and electron, its average lifetime in vacuum is 125 ps (para-positronium) or 142 ns (ortho-positronium) [@karsh]. Here we are interested in positronium states in the Bogoliubov-Dirac-Fock (BDF) model. In a previous paper we have proved the existence of a state that can be interpreted as the ortho-positronium. Our aim in this paper is to find another one that can be interpreted as the para-positronium and to find another state that can be interpreted as the dipositronium, the bound state of two electrons and two positrons. To find these states, we use symmetric properties of the Dirac operator. Symmetries ---------- – Following Dirac’s ideas, the free vacuum is described by the negative part of the spectrum $\sigma(D_0)$: $$P^0_-=\chi_{(-\infty,0)}(D_0).$$ A correspondence between negative energy states and positron states is given by the *charge conjugation* ${\ensuremath{\mathrm{C}}}$ [@Th]. This is an antiunitary operator that maps $\mathrm{Ran}\,P^0_{-}$ onto $\mathrm{Ran}(1-P^0_{-})$. In our convention [@Th] it is defined by the formula: $$\label{di_chargeconj} \forall\,\psi\in L^2({\ensuremath{\mathbb{R}^3}}),\ {\ensuremath{\mathrm{C}}}\psi(x)=i\beta\alpha_2\overline{\psi}(x),$$ where $\overline{\psi}$ denotes the usual complex conjugation. More precisely: $$\label{di_chargeconjprec} {\ensuremath{\mathrm{C}}}\cdot \begin{pmatrix}\psi_1\\ \psi_2\\ \psi_2\\\psi_4\end{pmatrix}=\begin{pmatrix}\overline{\psi}_4\\ -\overline{\psi}_3\\ -\overline{\psi}_2\\\overline{\psi}_1\end{pmatrix}.$$ In our convention it is also an *involution*: ${\ensuremath{\mathrm{C}}}^2=\text{id}$. An important property is the following: $$\label{di_denspsi} \forall\,\psi\in\,L^2,\forall\,x\in\mathbb{R}^3,\ |{\ensuremath{\mathrm{C}}}\psi(x)|^2=|\psi(x)|^2.$$ The Dirac operator anti-commutes with $D_0$, or equivalently there holds $$-{\ensuremath{\mathrm{C}}}D_0 {\ensuremath{\mathrm{C}}}^{-1}=-{\ensuremath{\mathrm{C}}}D_0{\ensuremath{\mathrm{C}}}=D_0.$$ – There exists another simple symmetry. We define $$\label{di_Isym} {\ensuremath{\mathrm{I}_{\mathrm{s}}}}:=\begin{pmatrix}0 & -\mathrm{Id}_{\mathbb{C}^2}\\ \mathrm{Id}_{\mathbb{C}^2}& 0 \end{pmatrix}\in\mathbb{C}^{4\times 4}.$$ This operator is $-i$ the *time reversal operator* $\text{L}_T$ [@Th 2.5.7] in $\mathfrak{H}$, interpreted as a unitary reprsentation of the Poincar[é]{} group. It acts on the spinor by simple multiplication, furthermore we have ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}^2=-\mathrm{Id}$ and $${\ensuremath{\mathrm{I}_{\mathrm{s}}}}:\begin{array}{rcl} \mathrm{Ran}\,P^0_-&\overset{\simeq}{\longrightarrow}& \mathrm{Ran}\,(1-P^0_-)\\ \psi(x)&\mapsto& {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi(x) \end{array}$$ Similarly we have $ -{\ensuremath{\mathrm{I}_{\mathrm{s}}}}D_0 {\ensuremath{\mathrm{I}_{\mathrm{s}}}}^{-1}={\ensuremath{\mathrm{I}_{\mathrm{s}}}}D_0 {\ensuremath{\mathrm{I}_{\mathrm{s}}}}= D_0.$ – To end this part we recall that $\mathbf{SU}(2)$ acts on $\mathfrak{H}$ [@Th]. Writing $\boldsymbol{\alpha}:=(\alpha_j)_{j=1}^3$ and $$\label{di_L,S} \mathbf{p}:=-i\hbar\nabla,\ {\ensuremath{\mathbf{L}}}:=\mathbf{x}\wedge \mathbf{p},\ {\ensuremath{\mathbf{S}}}:=-\frac{i}{4}\boldsymbol{\alpha}\wedge \boldsymbol{\alpha}=\frac{1}{2}\begin{pmatrix}\boldsymbol{\sigma}&0\\ 0&\boldsymbol{\sigma} \end{pmatrix},$$ we define $$\label{di_J_moment} {\ensuremath{\mathbf{J}}}:={\ensuremath{\mathbf{L}}}+{\ensuremath{\mathbf{S}}}.$$ The operator $\mathbf{L}$ is the angular momentum operator and $\mathbf{J}$ is the total angular momentum. From a geometrical point of view, $-i\mathbf{J}$ gives rise to a unitary representation of $\mathbf{SU}(2)$ in $\mathfrak{H}$ by the following formula: $$\left\{\begin{array}{l}e^{-i\theta\mathbf{J}\cdot{\ensuremath{\omega}}}\psi(x)=e^{-i\mathbf{S}\cdot{\ensuremath{\omega}}}\psi\big( \mathbf{R}^{-1}_{{\ensuremath{\omega}},\theta}\big),\\ \forall\theta\in[0,4\pi),\forall\psi\in\mathfrak{H},\forall{\ensuremath{\omega}}\in\mathbb{S}^2, \end{array}\right.$$where $\mathbf{R}_{{\ensuremath{\omega}},\theta}\in\mathrm{SO}(3)$ is the rotation with axis ${\ensuremath{\omega}}$ and angle $\theta$. As each $\mathrm{S}_j$ is diagonal by block, it is clear that this group representation can be decomposed in two representations, the first acting on the upper spinors $\phi\in L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2)$ and the second on the lower spinors $\chi\in L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2)$: $$\psi=:\begin{pmatrix}\phi\\\chi\end{pmatrix}.$$ In [@Th pp. 122-129] it is proved that $D_0$ commutes with the action of $\mathbf{SU}(2)$, thus the representation can also be decomposed with respect to $\mathrm{Ran}\,P^0_-$ and $\mathrm{Ran}\,(1-P^0_-)$. From an algebraic point of view, there exists a group morphism ${\ensuremath{\Phi_{\mathrm{SU}}}}:\mathbf{SU}(2)\to \mathbf{U}({\ensuremath{\mathfrak{H}_\Lambda}})$ where $\mathbf{U}(\mathfrak{H})$ is the set of unitary operator of $\mathfrak{H}$. We write $$\mathbf{S}:={\ensuremath{\Phi_{\mathrm{SU}}}}\big(\mathbf{SU}(2) \big).$$ The irreducible representations of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ are known and are expressed in terms of eigenspaces of ${\ensuremath{\mathbf{J}}}^2,{\ensuremath{\mathbf{S}}}$. The proofs of the following can be found in [@Th pp. 122-129]. The operators ${\ensuremath{\mathbf{J}}}^2,\mathrm{J}_3,{\ensuremath{\mathbf{K}}}$ all commute with each other, and ${\ensuremath{\mathbf{J}}}^2,{\ensuremath{\mathbf{K}}}$ with $D_0$. Moreover ${\ensuremath{\mathbf{K}}}$ commutes with the action ${\ensuremath{\Phi_{\mathrm{SU}}}}$. We have ${\ensuremath{\mathfrak{H}_\Lambda}}\subset L^2(\mathbb{R}^3)\simeq L^2((0,\infty),dr)\otimes L^2(\mathbb{S}^2)^4$, and ${\ensuremath{\mathbf{J}}}$, ${\ensuremath{\mathbf{L}}}$ only act on the part $L^2(\mathbb{S}^2)^4$. Restricted to $L^2(\mathbb{S}^2)^4$, we have $$\sigma\,({\ensuremath{\mathbf{J}}}^2)=\big\{j(j+1),\ j\in\frac{1}{2}+\mathbb{Z}_+\big\},$$ and for each eigenvalue $j(j+1)\in \sigma\,{\ensuremath{\mathbf{J}}}^2$, the eigenspace $\mathrm{Ker}\big({\ensuremath{\mathbf{J}}}^2-j(j+1)\big)$ may be decomposed with respect to the eigenspaces of $\mathrm{J}_3$ and ${\ensuremath{\mathbf{S}}}$. The corresponding eigenvalues are 1. $m_j=-j,-j+1,\cdots,j-1,j$ for $\mathrm{J}_3$, 2. $\kappa_j=\pm\big(j+\frac{1}{2}\big)$ for ${\ensuremath{\mathbf{S}}}$. The eigenspace $\mathfrak{k}_{m_j,\kappa_j}$ of a triplet $(j,m_j,\kappa_j)$ has dimension $2$ and is spanned by $\Phi^+_{m_j,\kappa_j}\perp\Phi^-_{m_j,\kappa_j}$, which have respectively a zero lower spinor and zero upper spinor. \[di\_irreduc\] For each irreducible subrepresentation $\Phi'_{\mathrm{SU}}$ of ${\ensuremath{\Phi_{\mathrm{SU}}}}$, there exists $$(j,{\ensuremath{\varepsilon}},\mathbf{z}=[z_1:z_2], a_1(r),a_2(r))\in \big(\frac{1}{2}+\mathbb{Z}_+\big)\times\{+,-\}\times\mathbb{C}P^1\times \big(\mathbb{S}L^2((0,\infty),dr)\big)^2,$$ such that the representation $\Phi'_{\mathrm{SU}}$ is spanned by $\psi(x)$ defined as follows: $$\forall x=r{\ensuremath{\omega}}\in{\ensuremath{\mathbb{R}^3}}, \psi(x):=z_1 ra_1(r)\Phi^+_{j, {\ensuremath{\varepsilon}}(j+\tfrac{1}{2})}({\ensuremath{\omega}})+z_2 ra_2(r)\Phi^-_{j,{\ensuremath{\varepsilon}}(j+\tfrac{1}{2})}.$$ \[di\_def\_mathbb\_s\] We recall that for any Hilbert space $\mathfrak{h}$ and any subspace $V\subset \mathfrak{h}$, we define $\mathbb{S}V$ as the unitary vector in $V$: $$\mathbb{S}V:=\{x\in V,\ \lVert x\rVert_{\mathfrak{h}}=1\}.$$ We will use this notation throughout this paper. We prove this Lemma in Section \[di\_proofmanif\]. \[di\_rep\_type\] An irreducible subrepresentation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ is characterized by the two numbers $(j,\kappa_j)$. Indeed, the irreducible representations of $\mathbf{SU}(2)$ are known: they can be described by homogeneous polynomials, and for any $n\in \mathbb{Z}_+$, there is but one irreducible representation of dimension $n+1$, up to isomorphism. In the case of ${\ensuremath{\Phi_{\mathrm{SU}}}}$, the two cases $\kappa_j=\pm(j+\tfrac{1}{2})$ are different but *isomorphic*. An irreducible subrepresentation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ spanned by an eigenvector of ${\ensuremath{\mathbf{J}}}^2$ and ${\ensuremath{\mathbf{K}}}$ with respective eigenvalues $j(j+1)$ and ${\ensuremath{\varepsilon}}(j+\tfrac{1}{2})$ will be refered as beeing of type $(j,{\ensuremath{\varepsilon}})$ (where ${\ensuremath{\varepsilon}}\in\{+,-\}$). Throughout this paper we write $\text{Proj}\,E$ to mean the orthonormal projection onto the vector space $E$. The BDF model ------------- This model is a no-photon approximation of quantum electrodynamics (QED) which was introduced by Chaix and Iracane in 1989 [@CI], and studied in many papers [@stab; @ptf; @Sc; @mf; @at; @gs; @sok]. It allows to take into account the Dirac vacuum together an electronic system in the presence of an external field. This is a Hartree-Fock type approximation in which a state of the system “vacuum plus real electrons” is given by an infinite Slater determinant $\psi_1\wedge\psi_2\wedge \cdots$. Such a state is represented by the projector onto the space spanned by the $\psi_j$’s: its so-called one-body density matrix. For instance $P^0_-$ represents the free Dirac vacuum. We do not recall the derivation of the BDF model from QED: we refer the reader to [@CI; @ptf; @mf] for full details. To simplify the notations, we choose relativistic units in which, the mass of the electron $m_e$, the speed of light $c$ and $\hbar$ are set to $1$. Let us say that there is an external density $\nu$, *e.g.* that of some nucleus. We write $\alpha>0$ the so-called *fine structure constant* (physically $e^2/(4\pi{\ensuremath{\varepsilon}}_0\hbar c)$, where $e$ is the elementary charge and ${\ensuremath{\varepsilon}}_0$ the permittivity of free space). The relative energy of a Hartree-Fock state represented by its 1pdm $P$ with respect to a state of reference ($P^0_-$ in [@CI; @ptf]) turns out to be a function of $Q=P-P^0_-$, the so-called reduced one-body density matrix. A projector $P$ is the one-body density matrix of a Hartree-Fock state in $\mathcal{F}_{\text{elec}}$ *iff* $P-P^0_-$ is Hilbert-Schmidt, that is compact such that its singular values form a sequence in $\ell^2$ [@ptf Appendix]. An ultraviolet cut-off $\Lambda>0$ is needed: we only consider electronic states in $${\ensuremath{\mathfrak{H}_\Lambda}}:=\big\{ f\in\mathfrak{H},\ \text{supp}\,{\ensuremath{\widehat{f}}}\subset B(0,{\ensuremath{\Lambda}})\big\},$$ where ${\ensuremath{\widehat{f}}}$ is the Fourier transform of $f$. This procedure gives the BDF energy introduced in [@CI] and studied in [@ptf; @Sc]. Our convention for the Fourier transform $\mathscr{F}$ is the following $$\forall\,f\in L^1({\ensuremath{\mathbb{R}^3}}),\ {\ensuremath{\widehat{f}}}(p):=\frac{1}{(2\pi)^{3/2}}{\ensuremath{\displaystyle\int}}f(x)e^{-ixp}dx.$$ Let us notice that ${\ensuremath{\mathfrak{H}_\Lambda}}$ is invariant under $D_0$ and so under $P^0_-$. We write $\Pi_{\ensuremath{\Lambda}}$ for the orthogonal projection onto ${\ensuremath{\mathfrak{H}_\Lambda}}$: $\Pi_{\ensuremath{\Lambda}}$ is the Fourier multiplier $\mathscr{F}^{-1}\chi_{B(0,\Lambda)}\mathscr{F}$. By means of a thermodynamical limit, Hainzl *et al.* showed that the formal minimizer and hence the reference state should not be given by $\Pi_{\ensuremath{\Lambda}}P^0_-$ but by another projector ${\ensuremath{\mathcal{P}^0_-}}$ in ${\ensuremath{\mathfrak{H}_\Lambda}}$ that satisfies the self-consistent equation [@mf]: $$\label{di_PP_self} \left\{ \begin{array}{ccl} {\ensuremath{\mathcal{P}^0_-}}-\tfrac{1}{2}&=&-\text{sign}\big({\ensuremath{\mathcal{D}^0}}\big),\\ {\ensuremath{\mathcal{D}^0}}&=&D_0\Pi_{\ensuremath{\Lambda}}-\dfrac{\alpha}{2}\dfrac{({\ensuremath{\mathcal{P}^0_-}}-\tfrac{1}{2})(x-y)}{|x-y|} \end{array} \right.$$ We have ${\ensuremath{\mathcal{P}^0_-}}=\chi_{(-\infty,0)}({\ensuremath{\mathcal{D}^0}})$. This operator ${\ensuremath{\mathcal{D}^0}}$ was previously introduced by Lieb *et al.* in [@ls]. In $\mathfrak{H}$, the operator ${\ensuremath{\mathcal{D}^0}}$ coincides with a bounded, matrix-valued Fourier multiplier whose kernel is ${\ensuremath{\mathfrak{H}_\Lambda}}^{\perp}\subset \mathfrak{H}$. Throughout this paper we write $$m=\inf \sigma \big(|{\ensuremath{\mathcal{D}^0}}|\big)\ge 1,$$ and $${\ensuremath{\mathcal{P}^0_+}}:=\Pi_{\ensuremath{\Lambda}}-{\ensuremath{\mathcal{P}^0_-}}=\chi_{(0,+\infty)}({\ensuremath{\mathcal{D}^0}}).$$ The resulting BDF energy $\mathcal{E}^\nu_{\text{BDF}}$ is defined on Hartree-Fock states represented by their one-body density matrix $P$: $$\mathscr{N}:=\big\{P\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\ P^*=P^2=P,\ P-{\ensuremath{\mathcal{P}^0_-}}\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})\big\}.$$ We recall that $\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}})$ is the set of bounded operators and that for $p\ge1$, $\mathfrak{S}_p({\ensuremath{\mathfrak{H}_\Lambda}})$ is the set of compact operators $A$ such that ${\ensuremath{\mathrm{Tr}}}\big(|A|^p\big)<+\infty$ [@ReedSim; @Sim]. In particular $\mathfrak{S}_{\infty}({\ensuremath{\mathfrak{H}_\Lambda}})$ is the set $\text{Comp}({\ensuremath{\mathfrak{H}_\Lambda}})$ of compact operators. This energy depends on three parameters: the fine structure constant $\alpha>0$, the cut-off ${\ensuremath{\Lambda}}>0$ and the external density $\nu$. We assume that $\nu$ has finite *Coulomb energy*, that is $${\ensuremath{\widehat{\nu}}}\ \text{measurable\ and\ }D(\nu,\nu):=4\pi\underset{{\ensuremath{\mathbb{R}^3}}}{{\ensuremath{\displaystyle\int}}}\frac{|{\ensuremath{\widehat{\nu}}}(k)|^2}{|k|^2}dk<+\infty.$$ The above integral coincides with $\underset{{\ensuremath{\mathbb{R}^3}}\times{\ensuremath{\mathbb{R}^3}}}{\iint}\frac{\nu(x)^*\nu(y)}{|x-y|}dxdy$ whenever this last one is well-defined. The same symmetries holds for ${\ensuremath{\mathcal{P}^0_-}}$ and ${\ensuremath{\mathcal{P}^0_+}}$: the charge conjugation ${\ensuremath{\mathrm{C}}}$ and the operator ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$ maps $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_-}}$ onto $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$. Moreover thanks to [@Th pp. 122-129] we can easily check that ${\ensuremath{\mathcal{D}^0}}$ also commutes with the action of $\mathbf{SU}(2)$ and with the operators ${\ensuremath{\mathbf{J}}}^2$ and ${\ensuremath{\mathbf{K}}}$. Minimizers and critical points ------------------------------ For $P\in\mathscr{N}$, we have the identity $$\label{di_eqq} (P-{\ensuremath{\mathcal{P}^0_-}})^2={\ensuremath{\mathcal{P}^0_+}}(P-{\ensuremath{\mathcal{P}^0_-}}){\ensuremath{\mathcal{P}^0_+}}-{\ensuremath{\mathcal{P}^0_-}}(P-{\ensuremath{\mathcal{P}^0_-}}){\ensuremath{\mathcal{P}^0_-}}\in \mathfrak{S}_1.$$ The charge of a state $P$ is given by the ${\ensuremath{\mathcal{P}^0_-}}$-trace of $P-{\ensuremath{\mathcal{P}^0_-}}$, defined by the formula: $$\begin{aligned} {\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big(P-{\ensuremath{\mathcal{P}^0_-}}\big)&:={\ensuremath{\mathrm{Tr}}}\big({\ensuremath{\mathcal{P}^0_-}}(P-{\ensuremath{\mathcal{P}^0_-}}){\ensuremath{\mathcal{P}^0_-}}+{\ensuremath{\mathcal{P}^0_+}}(P-{\ensuremath{\mathcal{P}^0_-}}){\ensuremath{\mathcal{P}^0_+}}\big),\\ &=\text{Dim}\mathrm{Ran}({\ensuremath{\mathcal{P}^0_+}})\cap \mathrm{Ran} (P)-\text{Dim}\mathrm{Ran}({\ensuremath{\mathcal{P}^0_-}})\cap \mathrm{Ran} (1-P).\end{aligned}$$ A minimizer over states with charge $N\in\mathbb{N}$ is interpreted as a ground state of a system with $N$ electrons, in the presence of an external density $\nu$ The existence problem was studied in several papers [@at; @sok; @sokd]: by [@at Theorem 1], it is sufficient to check binding inequalities. The following results hold under technical assumptions on $\alpha$ and ${\ensuremath{\Lambda}}$ (different for each result). In [@at], Hainzl *et al.* proved existence of minimizers for the system of $N$ electrons with $\nu\ge 0$, provided that $N-1<\int \nu$ . In [@sok], we proved the existence of a ground state for $N=1$ and $\nu=0$: an electron can bind alone in the vacuum. This surprising result holds due to the vacuum polarization. In [@sokd], we studied the charge screening effect: due to vacuum polarization, the observed charge of a minimizer $P\neq {\ensuremath{\mathcal{P}^0_-}}$ is different from its real charge ${\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}(P-{\ensuremath{\mathcal{P}^0_-}})$. We also proved it is possible to keep track of this effect in the non-relativistic limit $\alpha\to 0$: the resulting limit is an altered Hartree-Fock energy. Here we are looking for states with an equal number of electrons and positrons, that is we study $\mathcal{E}^0_{\text{BDF}}$ on $$\mathscr{M}:=\Big\{P\in\mathscr{N},\ {\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big(P-{\ensuremath{\mathcal{P}^0_-}}\big)=0\Big\}.$$ From a geometrical point of view $\mathscr{M}$ is a Hilbert manifold and $\mathcal{E}^0_{\text{BDF}}$ is a differentiable map on $\mathscr{M}$ (Propositions \[di\_manim\] and \[di\_gragra\]). We thus seek a critical point on $\mathscr{M}$, that is some $P\in\mathscr{M},\ P\neq{\ensuremath{\mathcal{P}^0_-}}$ such that $\nabla \mathcal{E}^0_{\text{BDF}}(P)=0$. In [@pos_sok], we have found the ortho-positronium by studying the BDF energy restricted to states with the ${\ensuremath{\mathrm{C}}}$-symmetry: $$\label{di_mm_cc} P\in\mathscr{M}\text{\ s.t.\ }P+{\ensuremath{\mathrm{C}}}P{\ensuremath{\mathrm{C}}}=\mathrm{Id}_{{\ensuremath{\mathfrak{H}_\Lambda}}}.$$ We write $\mathscr{M}_{\mathscr{C}}$ the set of such states. We will seek the para-positronium in the set $\mathscr{M}_{\mathscr{I}}$ of states having the ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$-symmetry. $$\label{di_ii_ss} \mathscr{M}_{\mathscr{I}}:=\{P\in\mathscr{M}\text{\ s.t.\ }P+{\ensuremath{\mathrm{I}_{\mathrm{s}}}}P{\ensuremath{\mathrm{I}_{\mathrm{s}}}}^{-1}=P-{\ensuremath{\mathrm{I}_{\mathrm{s}}}}P{\ensuremath{\mathrm{I}_{\mathrm{s}}}}=\mathrm{Id}_{{\ensuremath{\mathfrak{H}_\Lambda}}}\}.$$ Equivalently $P\in\mathscr{M}_{\mathscr{I}}$ if and only if $Q:=P-{\ensuremath{\mathcal{P}^0_-}}$ is Hilbert-Schmidt and satisfies $$-{\ensuremath{\mathrm{I}_{\mathrm{s}}}}Q {\ensuremath{\mathrm{I}_{\mathrm{s}}}}^{-1}={\ensuremath{\mathrm{I}_{\mathrm{s}}}}Q {\ensuremath{\mathrm{I}_{\mathrm{s}}}}=Q.$$ We seek a projector $P$ “close” to a state $P_0$ that can be written as:$$\label{di_imagine} P_0={\ensuremath{\mathcal{P}^0_-}}+{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_-\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_-|}\xspace}-{\ensuremath{|\psi_-\rangle}\xspace}{\ensuremath{\langle \psi_-|}\xspace},\ {\ensuremath{\mathcal{P}^0_+}}\psi_-=0.$$ To deal with the dipositronium, we impose an additional symmetry: we define $\mathscr{W}\subset \mathscr{M}_{\mathscr{C}}$ as follows. $$\label{di_def_w} \mathscr{W}:=\big\{P\in \mathscr{M}_{\mathscr{C}},\ \forall U\in\mathbf{S},\ UP U^{-1}=P \big\}.$$ Equivalently $$P\in\mathscr{W}\,\iff\,Q:=P-{\ensuremath{\mathcal{P}^0_-}}\mathrm{\ satisfies\ }-{\ensuremath{\mathrm{C}}}Q{\ensuremath{\mathrm{C}}}=Q\mathrm{\ and\ }UQU^{-1}=Q,\ \forall\,U\in \mathbf{S}.$$ Those sets $\mathscr{M}_{\mathscr{C}},\mathscr{M}_{\mathscr{I}},\mathscr{W}$ have fine properties: they are all submanifolds of $\mathscr{M}$, invariant under the gradient flow of $\mathcal{E}^0_{\text{BDF}}$ (Proposition \[di\_mani\_ci\_sym\]). However while $\mathscr{M}_{\mathscr{C}}$ has two connected components, $\mathscr{M}_{\mathscr{I}}$ has only one connected component and $\mathscr{W}$ has countable connected components. So we may find critical points by searching a minimizer of the BDF energy over the different connected components of $\mathscr{W}$. For the para-positronium, a critical point is found by an argument of mountain pass. \[di\_conn\_comp\] There is a one-to-one correspondence between the connected components of $\mathscr{W}$ and the set $\mathbb{Z}_2^2[X]$ of polynomials with coefficients in the ring $\mathbb{Z}_2\times \mathbb{Z}_2$. Let $P$ be in $\mathscr{W}$. The vector space $E_1:=\mathrm{Ran}\,P\cap \mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$ has finite dimension and is invariant under ${\ensuremath{\Phi_{\mathrm{SU}}}}$. We decompose it into irreducible representations. The projector is associated to $\sum_{\ell=1}^{\ell_0}t_\ell X^\ell$ with $t_\ell=(t_{\ell,1};t_{\ell,-1})$ if and only if for any $j\in\tfrac{1}{2}+\mathbb{Z}_+$: 1. The number $b_{j-\tfrac{1}{2},1}$ of irreducible representations of $E_1$ of type $(j,+)$ satisfies $b_{j-\tfrac{1}{2},1}\equiv t_{j-\tfrac{1}{2},1}[2]$. 2. The number $b_{j-\tfrac{1}{2},-1}$ of irreducible representations of $E_1$ of type $(j,-)$ satisfies $b_{j-\tfrac{1}{2},-1}\equiv t_{j-\tfrac{1}{2},-1}[2]$. \[di\_a\_c\_i\] The symbols $\mathscr{Y}$ and ${\ensuremath{\mathrm{Y}}}$ denotes respectively $\mathscr{C}$ and ${\ensuremath{\mathrm{C}}}$ or $\mathscr{I}$ and ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$. Furthermore the different connected components of $\mathscr{W}$ are written $\mathscr{W}_{p(X)}$ with $p(X)\in\mathbb{Z}_2^2[X]$. To state our main Theorems, we need to introduce the mean-field operator. An operator $Q\in \mathscr{V}$ is Hilbert-Schmidt and we write $Q(x,y)$ its integral kernel. Its density $\rho_Q$ is defined by the formula $$\label{di_dens_def} \forall x\in{\ensuremath{\mathbb{R}^3}},\ \rho_Q(x):={\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathbb{C}^4}}}\big(Q(x,x)\big),$$ we prove in the next Section that it is well-defined. The mean-field operator $D^{({\ensuremath{\Lambda}})}_{Q}$ associated to $Q$ *in the vacuum* is : $$\label{di_mean_field} D^{({\ensuremath{\Lambda}})}_{Q}:=\Pi_{\ensuremath{\Lambda}}\Big({\ensuremath{\mathcal{D}^0}}+\alpha \big(\rho_Q*\frac{1}{|\cdot|}-\frac{Q(x,y)}{|x-y|}\big)\Big).$$ \[di\_main\] There exist $\alpha_0,L_0,{\ensuremath{\Lambda}}_0>0$ such that if $$\alpha\le \alpha_0;\ \alpha{\ensuremath{\log(\Lambda)}}:=L\le L_0\text{\ and\ }{\ensuremath{\Lambda}}^{-1}\le {\ensuremath{\Lambda}}_0^{-1},$$ then there exists a critical point ${\ensuremath{\overline{P}}}={\ensuremath{\overline{Q}}}+{\ensuremath{\mathcal{P}^0_-}}$ of $\mathcal{E}^0_{\text{BDF}}$ in $\mathscr{M}_{\mathscr{I}}$ that satisfies the following equation. $$\exists 0<\mu<m,\ \exists \psi_a\in \mathrm{Ker}\big(D_{{\ensuremath{\overline{Q}}}}^{({\ensuremath{\Lambda}})}-\mu\big),\ {\ensuremath{\overline{P}}}=\chi_{(-\infty,0)}\big(D_{{\ensuremath{\overline{Q}}}}^{({\ensuremath{\Lambda}})}\big)+{\ensuremath{|\psi_a\rangle}\xspace}{\ensuremath{\langle \psi_a|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a|}\xspace}.$$ As $\alpha$ tends to $0$, the upper spinor of $U_{{\ensuremath{\lambda}}}\psi_a:={\ensuremath{\lambda}}^{3/2}\psi_a({\ensuremath{\lambda}}(\cdot))$ with ${\ensuremath{\lambda}}:=\tfrac{g'_1(0)^2}{\alpha m}$ tends to a Pekar minimizer. – *We recall that the Pekar energy is defined as follows* $$\forall\,\psi\in H^1,\ \mathcal{E}_{\text{PT}}(\psi):={\ensuremath{\lVert\nabla\psi\rVert_{L^{2}}}}^2-D\big(|\psi|^2,|\psi|^2\big).$$ *The infimum over $\mathbb{S}L^2\cap H^1$ is written $E_{\text{PT}}(1)$.* \[di\_main\_1\] There exist $L_0,{\ensuremath{\Lambda}}_0>0$, and for any $j\in\tfrac{1}{2}+\mathbb{Z}_+$, there exists $\alpha_j$ such that if $$\alpha\le \alpha_j;\ \alpha{\ensuremath{\log(\Lambda)}}:=L\le L_0\text{\ and\ }{\ensuremath{\Lambda}}^{-1}\le {\ensuremath{\Lambda}}_0^{-1},$$ then there exists a minimizer $P_{\mathbf{t}X^{\ell_0}}=Q+{\ensuremath{\mathcal{P}^0_-}}$ of $\mathcal{E}^0_{\text{BDF}}$ over the connected component of $\mathscr{W}_{\mathbf{t}X^{\ell_0}}$ with $\mathbf{t}\in\{(1,0),(0,1)\}$. Moreover there exists $0<\mu_{\ell_0,\mathbf{t}}<1$ and $\psi\in \mathrm{Ker}\big(D_Q^{({\ensuremath{\Lambda}})}-\mu_{\ell_0,\mathbf{t}}\big)$ such that $$P_{\mathbf{t}X^{\ell_0}}=\chi_{(-\infty,0)}(D_Q^{({\ensuremath{\Lambda}})})+\mathrm{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)-\mathrm{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi).$$ Any upper spinor ${\ensuremath{\widetilde{{\ensuremath{\varphi}}}}}$ of ${\ensuremath{\widetilde{\psi}}}\in {\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)$ can be written as $$\forall\,x=r{\ensuremath{\omega}}_x\in{\ensuremath{\mathbb{R}^3}},\ {\ensuremath{\widetilde{{\ensuremath{\varphi}}}}}=:ra(r)\sum_{m=-j}^j c_m({\ensuremath{\widetilde{{\ensuremath{\varphi}}}}})\Phi^+_{m,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})},\ c_m({\ensuremath{\widetilde{{\ensuremath{\varphi}}}}})\in\mathbb{C}.$$ Furthermore, as $\alpha$ tends to $0$, the function $ U_{{\ensuremath{\lambda}}} a(r)={\ensuremath{\lambda}}^{3/2}a({\ensuremath{\lambda}}r)$ tends to a minimizer of the energy $\mathcal{E}_{\mathbf{t}X^{\ell_0}}$ over $\mathbb{S}L^2(\mathbb{R}_+,r^2dr)\cap H^1(\mathbb{R}_+,r^2dr):$ $$\label{di_non_rel_w} \mathcal{E}_{\mathbf{t}X^{\ell_0}}\big(f(r)\big):= {\ensuremath{\mathrm{Tr}}}\big(-\Delta\, \mathrm{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(rf(r)\Phi^+_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}) \big)-{\ensuremath{\lVert\mathrm{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,(rf(r)\Phi^+_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})})\rVert_{\text{Ex}}}}^2.$$ In particular, the dipositronium corresponds to the case $\ell_0=j_0-\tfrac{1}{2}=0$. The minimum is written $E_{\mathbf{t}X^{\ell_0}}^{nr}$ for the non-relativistic energy and $E_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}$ for the BDF energy over $\mathscr{W}_{\mathbf{t}X^{j_0-1/2}}$. \[di\_eps\_t\] For $\mathbf{t}X^{\ell_0}\in \mathbb{Z}_2^2[X]$ as in Theorem \[di\_main\_1\], ${\ensuremath{\varepsilon}}(\mathbf{t})\in\{+,-\}$ denotes $+$ if $\mathbf{t}=(1,0)$ or $-$ if $\mathbf{t}=(0,1)$. We expect the existence of minimizers over any connected components of $\mathscr{W}$ (associated to $p(X)\in \mathbb{Z}_2^2[X]$), provided that $\alpha$ is smaller than some $\alpha_{p(X)}$. \[di\_prec\_non\_rel\] The non-relativistic energy can be computed: $$\left\{\begin{array}{rcl} \mathcal{E}_{\mathbf{t}X^{\ell_0}}\big(f(r)\big)&:=&(2j_0+1)\underset{0}{\overset{+\infty}{{\ensuremath{\displaystyle\int}}}} \Big[r^2|f'(r)|^2+(j_0+{\ensuremath{\varepsilon}}\tfrac{1}{2})(j_0+1+{\ensuremath{\varepsilon}}\tfrac{1}{2})|f(r)|^2\Big]dr\\ && \ \ \ -\underset{\mathbb{R}_+^2}{{\ensuremath{\displaystyle\iint}}}r_1^2r_2^2|f(r_1)|^2|f(r_2)|^2w_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}(r_1,r_2),\\ w_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}(r_1,r_2)&:=&\underset{(\mathbb{S}^2)^2}{{\ensuremath{\displaystyle\iint}}}\frac{dn_1 dn_2}{|r_1n_1-r_2n_2|}\Big({\ensuremath{\displaystyle\sum}}_{m_1,m_2}((\Phi^+_{m_1,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})})^*\Phi^+_{m_1,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})})(n_1) \Big)\\ &&\ \ \ \times\Big({\ensuremath{\displaystyle\sum}}_{m_1,m_2}((\Phi^+_{m_1,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})})^*\Phi^+_{m_1,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})})(n_2) \Big). \end{array}\right.$$ It corresponds to the energy $$\mathcal{E}_{nr}\big({\ensuremath{\Gamma}}\big):= {\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\Gamma}}\big)-{\ensuremath{\lVert{\ensuremath{\Gamma}}\rVert_{\text{Ex}}}}^2,\ 0\le {\ensuremath{\Gamma}}\le 1,\ {\ensuremath{\Gamma}}\in\mathfrak{S}_1(H^1({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2))$$ restricted to the subspace $$\mathscr{S}_{(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))}:=\big\{{\ensuremath{\Gamma}},\ {\ensuremath{\Gamma}}^*={\ensuremath{\Gamma}}^2={\ensuremath{\Gamma}},\ \mathrm{Ran}\,({\ensuremath{\Phi_{\mathrm{SU}}}})_{\big|_{{\ensuremath{\Gamma}}}}\ \text{irreducible\ of\ type\ }(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))\big\}. $$ This subspace is invariant under the action of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ and it is easy to see that it is a submanifold of $ \big\{ {\ensuremath{\Gamma}},\ {\ensuremath{\Gamma}}^*={\ensuremath{\Gamma}}^2={\ensuremath{\Gamma}},\ {\ensuremath{\mathrm{Tr}}}\,{\ensuremath{\Gamma}}=2j_0+1\big\}$. The subspace $\mathscr{S}_{(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))}$ is invariant under the flow of $\mathcal{E}_{nr}$. The energies can be estimated. \[di\_est\] In the same regime as in Theorem \[di\_main\], the following holds. The critical point ${\ensuremath{\overline{P}}}$ of the BDF functional over $\mathscr{M}_{\mathscr{I}}$ satisfies $$\label{di_en_para} \mathcal{E}^0_{\text{BDF}}({\ensuremath{\overline{P}}})=2m +\frac{\alpha^2m}{g'_1(0)^2}E_{\text{PT}}(1)+\mathcal{O}(\alpha^3).$$ Furthermore the minimizer ${\ensuremath{\overline{P}}}_{\ell_0}$ over $\mathscr{W}_{\mathbf{t}X^{\ell_0}}$ satisfies: $$\label{di_est_mult} \mathcal{E}^0_{\text{BDF}}({\ensuremath{\overline{P}}}_{\ell_0})=2(2j_0+1)+\frac{\alpha^2 m}{g'_1(0)^2} E_{\mathbf{t}X^{\ell_0}}^{nr}+\mathcal{O}(\alpha^3K(j_0)).$$ The Pekar model describes an electron trapped in its own hole in a polarizable medium. Thus it is not surprising to find it here. We recall that there is a unique minimizer of the Pekar energy up to translation and a phase in $\mathbb{S}^7$ (in $\mathbb{C}^4$). The asymptotic expansion coincides with that of the ortho-positronium [@pos_sok]. In fact, it can be proved that the first difference between the energies occurs at order $\alpha^4$. Throughout this paper we write $K$ to mean a constant independent of $\alpha,{\ensuremath{\Lambda}}$. Its value may differ from one line to the other. When we write $K(a)$, we mean a constant that depends solely on $a$. We also use the symbol $\apprle$: $0\le a\apprle b$ means there exists $K>0$ such that $a\le Kb$. We also recall the reader our use of the notation $\mathbb{S}V$ for any subspace $V$ of some Hilbert space that denotes the set of unitary vector in $V$. Remarks and notations about ${\ensuremath{\mathcal{D}^0}}$ ---------------------------------------------------------- ${\ensuremath{\mathcal{D}^0}}$ has the following form [@mf]: $$\label{di_D_form} {\ensuremath{\mathcal{D}^0}}=g_0(-i\nabla)\beta -i\boldsymbol{\alpha}\cdot \frac{\nabla}{|\nabla|}g_1(-i\nabla)$$ where $g_0$ and $g_1$ are smooth radial functions on $B(0,{\ensuremath{\Lambda}})$. Moreover we have: $$\forall\,p\in B(0,{\ensuremath{\Lambda}}),\ 1\le g_0(p),\text{\ and\ }|p|\le g_1(p)\le |p|g_0(p).$$ For $\alpha{\ensuremath{\log(\Lambda)}}$ sufficiently small, we have $m=g_0(0)$ [@LL; @sok]. The smallness of $\alpha$ is needed to get estimates that hold close to the non-relativistic limit. The smallness of $\alpha{\ensuremath{\log(\Lambda)}}$ is needed to get estimates of ${\ensuremath{\mathcal{D}^0}}$: in this case ${\ensuremath{\mathcal{D}^0}}$ can be obtained by a fixed point scheme [@mf; @LL], and we have [@sok Appendix A]: $$\label{di_estim_g} \begin{array}{c} g'_0(0)=0,\ \text{and}\ {\ensuremath{\lVertg'_0\rVert_{L^{\infty}}}},{\ensuremath{\lVertg_0''\rVert_{L^{\infty}}}}\le K\alpha\\{\ensuremath{\lVertg'_1-1\rVert_{L^{\infty}}}}\le K\alpha{\ensuremath{\log(\Lambda)}}\le \tfrac{1}{2}\ \text{and}\ {\ensuremath{\lVertg_1''\rVert_{L^{\infty}}}}\apprle 1. \end{array}$$   Description of the model ======================== The BDF energy -------------- For any ${\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}}'\in\{+,-\}$ and $A\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}})$, we write $$A^{{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}}'}:=\mathcal{P}^0_{{\ensuremath{\varepsilon}}}A\mathcal{P}^0_{{\ensuremath{\varepsilon}}'}.$$ For an operator $Q\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})$, we write $R_Q$ the operator given by the integral kernel: $$R_Q(x,y):=\frac{Q(x,y)}{|x-y|}.$$ Let $\alpha>0,{\ensuremath{\Lambda}}>0$ and $\nu\in\mathcal{S}'({\ensuremath{\mathbb{R}^3}})$ a generalized function with $D(\nu,\nu)<+\infty$. For $P\in\mathscr{N}$ we write $Q:=P-{\ensuremath{\mathcal{P}^0_-}}$ and $$\label{di_formule_bdf} \left\{\begin{array}{l} \mathcal{E}^0_{\text{BDF}}(Q)={\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big({\ensuremath{\mathcal{D}^0}}Q \big)-\alpha D(\rho_Q,\nu)+\dfrac{\alpha}{2}\Big(D(\rho_Q,\rho_Q)-{\ensuremath{\lVertQ\rVert_{\text{Ex}}}}^2\Big),\\ \forall\,x,y\in{\ensuremath{\mathbb{R}^3}},\ \rho_Q(x):={\ensuremath{\mathrm{Tr}}}_{\mathbb{C}^4}\big(Q(x,x)\big),\ {\ensuremath{\lVertQ\rVert_{\text{Ex}}}}^2:={\ensuremath{\displaystyle\iint}}\frac{|Q(x,y)|^2}{|x-y|}dxdy, \end{array}\right.$$ where $Q(x,y)$ is the integral kernel of $Q$. The term ${\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big({\ensuremath{\mathcal{D}^0}}Q \big)$ is the kinetic energy, $-\alpha D(\rho_Q,\nu)$ is the interaction energy with $\nu$. The term $\dfrac{\alpha}{2}D(\rho_Q,\rho_Q)$ is the so-called *diract term* and $-\dfrac{\alpha}{2}{\ensuremath{\lVertQ\rVert_{\text{Ex}}}}^2$ is the *exchange term*. Let us see that formula is well-defined whenever $Q$ is ${\ensuremath{\mathcal{P}^0_-}}$-trace-class [@ptf; @at]. #### $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ and the variational set $\mathcal{K}$ The set $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ of ${\ensuremath{\mathcal{P}^0_-}}$-trace class operator is the following Banach space: $$\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}=\big\{Q\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}}),\ Q^{++},Q^{--}\in\mathfrak{S}_1({\ensuremath{\mathfrak{H}_\Lambda}})\big\},$$ with the norm $$\lVert Q\rVert_{\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}}:={\ensuremath{\lVertQ^{+-}\rVert_{\mathfrak{S}_{2}}}}+{\ensuremath{\lVertQ^{-+}\rVert_{\mathfrak{S}_{2}}}}+{\ensuremath{\lVertQ^{++}\rVert_{\mathfrak{S}_{1}}}}+{\ensuremath{\lVertQ^{--}\rVert_{\mathfrak{S}_{1}}}}.$$ We have $\mathscr{N}\subset {\ensuremath{\mathcal{P}^0_-}}+\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ thanks to . The closed convex hull of $\mathscr{N}-{\ensuremath{\mathcal{P}^0_-}}$ under $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ is $$\mathcal{K}:=\big\{Q\in\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}}),\ Q^*=Q,\ -{\ensuremath{\mathcal{P}^0_-}}\le Q\le {\ensuremath{\mathcal{P}^0_+}}\big\}$$ and we have [@ptf; @Sc] $$\forall\,Q\in \mathcal{K},\ Q^2\le Q^{++}-Q^{--}.$$ #### The BDF energy for $Q\in \mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ We have $${\ensuremath{\mathcal{P}^0_-}}({\ensuremath{\mathcal{D}^0}}Q){\ensuremath{\mathcal{P}^0_-}}=-|{\ensuremath{\mathcal{D}^0}}|Q^{--}\in\,\mathfrak{S}_1({\ensuremath{\mathfrak{H}_\Lambda}}),\ \text{because}\, |{\ensuremath{\mathcal{D}^0}}|\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),$$ this proves that the kinetic energy is defined. By the Kato-Seiler-Simon (KSS) inequality [@Sim], $Q$ is locally trace-class: $$\forall\,\phi\in \mathbf{C}^\infty_0({\ensuremath{\mathbb{R}^3}}),\ \phi \Pi_{\ensuremath{\Lambda}}\in\mathfrak{S}_2\text{\ so\ }\phi Q \phi=\phi\Pi_{\ensuremath{\Lambda}}Q\phi\in\mathfrak{S}_1(L^2({\ensuremath{\mathbb{R}^3}})).$$ We recall this inequality states that for all $2\le p\le \infty$ and $d\in\mathbb{N}$, we have $$\forall\,f,g\in L^p(\mathbb{R}^d),\ f(x)g(-i\nabla)\in\mathfrak{S}_{p}({\ensuremath{\mathfrak{H}_\Lambda}})\text{\ and\ }{\ensuremath{\lVertf(x)g(-i\nabla)\rVert_{\mathfrak{S}_{p}}}}\le (2\pi)^{-d/p}{\ensuremath{\lVertf\rVert_{L^{p}}}}{\ensuremath{\lVertg\rVert_{L^{p}}}}.$$ It follows that the *density* $\rho_Q$ of $Q$, defined in is well-defined. By the KSS inequality, we can also prove that ${\ensuremath{\lVert\rho_Q\rVert_{\mathcal{C}}}}\apprle K({\ensuremath{\Lambda}})\lVert Q \rVert_{\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}}$ [@gs Proposition 2]. By Kato’s inequality: $$\label{di_kato} \dfrac{1}{|\cdot|}\le \dfrac{\pi}{2}|\nabla|,$$ the exchange term is well-defined. Moreover the following holds: if $\alpha < \tfrac{4}{\pi}$, then the BDF energy is bounded from below on $\mathcal{K}$ [@stab; @Sc; @at]. We have $$\label{di_below} \forall\,Q_0\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}}),\ \mathcal{E}^0_{\text{BDF}}(Q_0)\ge \big(1-\alpha\frac{\pi}{4}\big){\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}||Q_0|^2\big).$$ Here we assume it is the case. This result will be often used throughout this paper. #### Minimizers For $Q\in\mathcal{K}$, its charge is its ${\ensuremath{\mathcal{P}^0_-}}$-trace: $q={\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}(Q)$. We define the Charge sector sets: $$\forall\,q\in{\ensuremath{\mathbb{R}^3}},\ \mathcal{K}^q:=\big\{Q\in\mathcal{K},\ {\ensuremath{\mathrm{Tr}}}(Q)=q\big\}.$$ A minimizer of $\mathcal{E}^\nu_{\text{BDF}}$ over $\mathcal{K}$ is interpreted as the polarized vacuum in the presence of $\nu$ while a minimizer over charge sector $N\in\mathbb{N}$ is interpreted as the ground state of $N$ electrons in the presence of $\nu$, by Lieb’s principle [@at Proposition 3], such a minimizer is in $\mathscr{N}-{\ensuremath{\mathcal{P}^0_-}}$. We define the energy functional $E^\nu_{\text{BDF}}$: $$\forall\,q\in{\ensuremath{\mathbb{R}^3}},\ E^\nu_{\text{BDF}}(q):=\inf\big\{\mathcal{E}^\nu_{\text{BDF}}(Q),\ Q\in\mathcal{K}^q\big\}.$$ We also write: $$\label{di_koci} \mathcal{K}^0_{\mathscr{Y}}:=\{ Q\in\mathcal{K},\ \text{Tr}_{{\ensuremath{\mathcal{P}^0_-}}}(Q)=0,\ -{\ensuremath{\mathrm{Y}}}Q {\ensuremath{\mathrm{Y}}}^{-1} =Q\}.$$ Proposition \[di\_weakclosed\] states that this set is sequentially weakly-$*$ closed in $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}})$. Structure of manifold --------------------- We consider $$\mathscr{V}=\big\{P-{\ensuremath{\mathcal{P}^0_-}},\ P^*=P^2=P\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\ {\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big( P-{\ensuremath{\mathcal{P}^0_-}}\big)=0\big\}\subset \mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}}).$$ and write: $\mathscr{M}:={\ensuremath{\mathcal{P}^0_-}}+\mathscr{V}=\big\{P,\ P^*=P^2=P,\ {\ensuremath{\mathrm{Tr}}}_{{\ensuremath{\mathcal{P}^0_-}}}\big( P-{\ensuremath{\mathcal{P}^0_-}}\big)=0\big\}.$ We recall the following proposition, proved in [@pos_sok]. \[di\_manim\] The set $\mathscr{M}$ is a Hilbert manifold and for all $P\in\mathscr{M}$, $$\mathrm{T}_P \mathscr{M}=\{ [A,P],\,A\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\ A^*=-A\text{\ and\ }PA(1-P)\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})\}.$$ Writing $$\mathfrak{m}_P:=\{ A\in\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\ A^*=-A,\ PAP=(1-P)A(1-P)=0\text{\ and\ }PA(1-P)\in\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})\},$$ any $P_1\in\mathscr{M}$ can be written as $P_1=e^A P e^{-A}$ where $A\in\mathfrak{m}_P$. The BDF energy $\mathcal{E}_{\text{BDF}}^\nu$ is a differentiable function in $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}})$ with: $$\label{di_eqdebdf} \left\{ \begin{array}{l} \forall\, Q,\delta Q\in\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}}),\ \text{d}\mathcal{E}_{\text{BDF}}^\nu(Q)\cdot \delta Q=\text{Tr}_{{\ensuremath{\mathcal{P}^0_-}}}\big(D_{Q,\nu}\delta Q\big).\\ D_{Q,\nu}:={\ensuremath{\mathcal{D}^0}}+\alpha \big((\rho_Q-\nu)*\frac{1}{|\cdot|}-R_Q\big). \end{array}\right.$$ We may rewrite as follows: $$\forall\, Q,\delta Q\in\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}}),\ \text{d}\mathcal{E}_{\text{BDF}}^\nu(Q)\cdot \delta Q=\text{Tr}_{{\ensuremath{\mathcal{P}^0_-}}}\big(\Pi_\Lambda D_{Q,\nu}\Pi_\Lambda \delta Q\big)$$ We recall the mean-field operator $ D_Q^{({\ensuremath{\Lambda}})}$ is defined in Notation \[di\_mean\_field\]. \[di\_gragra\] Let $(P,v)$ be in the tangent bundle $\mathrm{T}\mathscr{M}$ and $Q=P-{\ensuremath{\mathcal{P}^0_-}}$. Then we have $[[\Pi_\Lambda D_Q \Pi_\Lambda,P],P]\in\mathrm{T}_P\mathscr{M}$ and: $$\label{di_difftan} \mathrm{d}\mathcal{E}_{\text{BDF}}^0(P)\cdot v=\text{Tr}\Big(\big[\big[ D_{Q}^{({\ensuremath{\Lambda}})} ,P\big],P\big]v\Big).$$ In other words: $$\label{di_defgradient} \forall\,P\in\mathscr{M},\ \nabla \mathcal{E}_{\text{BDF}}^0(P)=\big[\big[\Pi_{\ensuremath{\Lambda}}D_Q \Pi_{\ensuremath{\Lambda}},P\big],P\big].$$ The operator $[[\Pi_{\ensuremath{\Lambda}}D_Q\Pi_{\ensuremath{\Lambda}},P],P]$ is the “projection” of $\Pi_{\ensuremath{\Lambda}}D_Q \Pi_{\ensuremath{\Lambda}}$ onto $\text{T}_P\mathscr{M}$. In [@pos_sok], we proved that $\mathscr{M}_{\mathscr{C}}$ is a submanifold of $\mathscr{M}$. We recall that the notations $\mathscr{Y}$, ${\ensuremath{\mathrm{Y}}}$ are specified in Notation \[di\_a\_c\_i\]. \[di\_mani\_ci\_sym\] The sets $\mathscr{M}_{\mathscr{I}}$ and $\mathscr{W}$ are *submanifolds* of $\mathscr{M}$, which are *invariant* under the flow of $\mathcal{E}_{\text{BDF}}^0$. The following holds: for any $P\in \mathscr{M}_{\mathscr{Y}}$, writing $$\mathfrak{m}^{\mathscr{Y}}_P=\{a\in \mathfrak{m}_P,\ {\ensuremath{\mathrm{Y}}}a {\ensuremath{\mathrm{Y}}}^{-1}=a\},$$ we have $$\label{di_tangentc} \mathrm{T}_P \mathscr{M}_{\mathscr{Y}}=\{[a,P],\ a\in \mathfrak{m}_P^{\mathscr{Y}}\}=\{v\in\mathrm{T}_P \mathscr{M},\ -{\ensuremath{\mathrm{Y}}}v {\ensuremath{\mathrm{Y}}}^{-1}=v\}.$$ Furthermore, for any $P\in\mathscr{M}_{\mathscr{Y}}$ we have $\rho_{P-{\ensuremath{\mathcal{P}^0_-}}}=0.$ For $P\in\mathscr{W}$, the same holds with $$\left\{ \begin{array}{rcl} \mathfrak{m}^{\mathscr{W}}_{P}&:=&\big\{a\in \mathfrak{m}^{\mathscr{C}}_{P},\ \forall\,U\in\mathbf{S},\ U a U^{-1}=a\big\},\\ \mathrm{T}_P\mathscr{W}&:=&\big\{[a,P],\ a\in\mathfrak{m}_P^{\mathscr{W}}\big\}. \end{array} \right.$$ \[Lagrangians\] The operator ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$ induced a symplectic structure on the *real* Hilbert space $({\ensuremath{\mathfrak{H}_\Lambda}},\mathfrak{Re}{\ensuremath{\langle \cdot\,,\,\cdot\rangle}\xspace}_{\mathfrak{H}})$: $$\forall\,f,g\in{\ensuremath{\mathfrak{H}_\Lambda}},\ {\ensuremath{\omega}}_{\mathrm{I}}(f,g):=\mathfrak{Re}{\ensuremath{\langle f\,,\,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}g\rangle}\xspace}.$$ The manifold $\mathscr{M}_{\mathscr{I}}$ is constituted by *Lagrangians* of ${\ensuremath{\omega}}_{\mathrm{I}}$ that are in $\mathscr{M}$. We end this section by stating technical results. Form of trial states -------------------- The following Theorem is stated in [@at Appendix] and proved in [@pos_sok]. \[di\_structure\] Let $P_1,P_0$ be in $\mathscr{N}$ and $Q=P_1-P_0$. Then there exist $M_+,M_-\in\mathbb{Z}_+$ such that there exist two orthonormal families $$\begin{array}{ll} (a_1,\ldots,a_{M_+})\cup(e_i)_{i\in\mathbb{N}}& \mathrm{in}\ \mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}},\\ (a_{-1},\ldots,a_{-M_+})\cup(e_{-i})_{i\in\mathbb{N}}&\mathrm{in}\ \mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_-}}, \end{array}$$ and a nonincreasing sequence $({\ensuremath{\lambda}}_i)_{i\in\mathbb{N}}\in\ell^2$ satisfying the following properties: 1. The $a_i$’s are eigenvectors for $Q$ with eigenvalue $1$ (resp. $-1$) if $i>0$ (resp. $i<0$). 2. For each $i\in\mathbb{N}$ the plane $\Pi_i:=\text{Span}(e_i,e_{-i})$ is spanned by two eigenvectors $f_i$ and $f_{-i}$ for $Q$ with eigenvalues ${\ensuremath{\lambda}}_i$ and $-{\ensuremath{\lambda}}_i$. 3. The plane $\Pi_i$ is also spanned by two orthogonal vectors $v_i$ in $\mathrm{Ran}(1-P)$ and $v_{-i}$ in $\mathrm{Ran}(P)$. Moreover ${\ensuremath{\lambda}}_i=\sin(\theta_i)$ where $\theta_i\in (0,\tfrac{\pi}{2})$ is the angle between the two lines $\mathbb{C}v_i$ and $\mathbb{C}e_i$. 4. There holds: $$Q={\ensuremath{\displaystyle\sum}}_i^{M_+}{\ensuremath{|a_i\rangle}\xspace}{\ensuremath{\langle a_i|}\xspace}-{\ensuremath{\displaystyle\sum}}_i^{M_-}{\ensuremath{|a_{-i}\rangle}\xspace}{\ensuremath{\langle a_{-i}|}\xspace}+{\ensuremath{\displaystyle\sum}}_{j\in \mathbb{N}}{\ensuremath{\lambda}}_j({\ensuremath{|f_j\rangle}\xspace}{\ensuremath{\langle f_j|}\xspace}-{\ensuremath{|f_{-j}\rangle}\xspace}{\ensuremath{\langle f_{-j}|}\xspace}).$$ We have $$\label{di_++--} \begin{array}{l} Q^{++}={\ensuremath{\displaystyle\sum}}_i^{M_+}{\ensuremath{|a_i\rangle}\xspace}{\ensuremath{\langle a_i|}\xspace}+{\ensuremath{\displaystyle\sum}}_{j\in\mathbb{N}}\sin(\theta_j)^2{\ensuremath{|e_j\rangle}\xspace}{\ensuremath{\langle e_j|}\xspace},\\ Q^{--}=-{\ensuremath{\displaystyle\sum}}_i^{M_-}{\ensuremath{|a_{-i}\rangle}\xspace}{\ensuremath{\langle a_{-i}|}\xspace}-{\ensuremath{\displaystyle\sum}}_{j\in\mathbb{N}}\sin(\theta_j)^2{\ensuremath{|e_{-j}\rangle}\xspace}{\ensuremath{\langle e_{-j}|}\xspace}. \end{array}$$ Thanks to Theorem \[di\_structure\], it is possible to characterize states in $\mathscr{M}_{\mathscr{Y}}$ and $\mathscr{W}$. We restate a proposition of [@pos_sok] and add the case of ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$. \[di\_chasym\] Let ${\ensuremath{\gamma}}=P-{\ensuremath{\mathcal{P}^0_-}}$ be in $\mathscr{M}_{\mathscr{Y}}$. For $-1\le \mu\le 1$ and $X\in\{{\ensuremath{\gamma}},{\ensuremath{\gamma}}^2\}$, we write $$E^X_\mu=\mathrm{Ker}(X-\mu).$$ Then for any $\mu\in\sigma({\ensuremath{\gamma}})$, ${\ensuremath{\mathrm{Y}}}E^{\ensuremath{\gamma}}_\mu=E^{\ensuremath{\gamma}}_{-\mu}$. Moreover for $|\mu|<1$ if we decompose $E^{\ensuremath{\gamma}}_{\mu}\oplus E^{\ensuremath{\gamma}}_{-\mu}$ into a sum of planes $\Pi$ as in Theorem \[di\_structure\], then 1. If ${\ensuremath{\mathrm{Y}}}={\ensuremath{\mathrm{I}_{\mathrm{s}}}}$, then we can choose the $\Pi$’s to be ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$-invariant. 2. If ${\ensuremath{\mathrm{Y}}}={\ensuremath{\mathrm{C}}}$, then each $\Pi$ is *not* ${\ensuremath{\mathrm{C}}}$-invariant and $\mathrm{Dim}\,E^{\ensuremath{\gamma}}_{\mu}$ is even. Equivalently $\text{Dim}\,E^{{\ensuremath{\gamma}}^2}_{\mu^2}$ is divisible by $4$. Moreover there exists a decomposition $$E^{{\ensuremath{\gamma}}^2}_{\mu^2}=\underset{1\le j\le \tfrac{N}{2}}{\overset{\perp}{\oplus}}V_{\mu,j}\text{\ and\ }V_{\mu,j}=\Pi^a_{\mu,j}\overset{\perp}{\oplus}{\ensuremath{\mathrm{C}}}\Pi^a_{\mu,j}$$ where the $\Pi^a_{\mu,j}$’s and ${\ensuremath{\mathrm{C}}}\Pi^a_{\mu,j}$’s are spectral planes described in Theorem \[di\_structure\]. The Cauchy expansion -------------------- In this part, we introduce a useful trick in the model. The Cauchy expansion is an application of functional calculus: we refer the reader to [@ptf; @sok] for further details. We assume $Q_0\in \mathfrak{S}_2$ with $$\label{di_supp} \alpha {\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}Q_0\rVert_{\mathfrak{S}_{2}}}}\ll 1.$$ We recall the following inequality, proved in [@sok] $$\label{di_cauchy_est0} \forall\,Q_0\in\mathfrak{S}_2,\ {\ensuremath{\lVertR_{Q_0}\tfrac{1}{|\nabla|^{1/2}}\rVert_{\mathfrak{S}_{2}}}}^2\apprle {\ensuremath{\lVertQ\rVert_{\text{Ex}}}}^2\apprle {\ensuremath{\displaystyle\iint}}|p+q||{\ensuremath{\widehat{Q}}}(p,q)|^2dpdq,$$ From now on, we only deal with $Q_0$ whose density vanishes: $\rho_{Q_0}=0$. The mean-field operator $D_{Q_0}^{({\ensuremath{\Lambda}})}$ is away from $0$ thanks to . Indeed, there holds $$\begin{aligned} |\Pi_{\ensuremath{\Lambda}}R_{Q_0}\Pi_{\ensuremath{\Lambda}}|^2&\le |\nabla|^{1/2}\,\frac{\Pi_{\ensuremath{\Lambda}}}{|\nabla|^{1/2}}R_{Q_0}^* R_{Q_0}\frac{\Pi_{\ensuremath{\Lambda}}}{|\nabla|^{1/2}}\,|\nabla|^{1/2}\\ &\le \Pi_{\ensuremath{\Lambda}}|\nabla|{\ensuremath{\lVert\tfrac{1}{|\nabla|^{1/2}} R_{Q_0}\rVert_{\mathcal{B}}}}^2\\ &\apprle \Pi_{\ensuremath{\Lambda}}|\nabla|{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}\apprle |{\ensuremath{\mathcal{D}^0}}|^2{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}^2,\end{aligned}$$ thus $$|D_{Q_0}^{({\ensuremath{\Lambda}})}|\apprge |{\ensuremath{\mathcal{D}^0}}|\big(1-\alpha K{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}\big).$$ The Cauchy expansion gives an expression of $${\ensuremath{\gamma}}_0:=\chi_{(-\infty,0)}\big(D_{Q_0}^{({\ensuremath{\Lambda}})}\big)-{\ensuremath{\mathcal{P}^0_-}}:={\ensuremath{\overline{\boldsymbol{\pi}}}}_0.$$ We have [@ptf] $$\label{di_cauchy0} \chi_{(-\infty,0)}\big(D_{Q_0}^{({\ensuremath{\Lambda}})}\big)-{\ensuremath{\mathcal{P}^0_-}}=\frac{1}{2\pi}{\ensuremath{\displaystyle\int}}_{-\infty}^{+\infty}\frac{d {\ensuremath{\omega}}}{{\ensuremath{\mathcal{D}^0}}+i\omega}\big(\alpha\Pi_{\ensuremath{\Lambda}}R_{Q_n}\Pi_{\ensuremath{\Lambda}}\big)\dfrac{1}{D_{Q_0}+i{\ensuremath{\omega}}}\Pi_{\ensuremath{\Lambda}}.$$ We also expand in power of $Y[Q_0]:=-\alpha \Pi_{\ensuremath{\Lambda}}R_{Q_0}\Pi_{\ensuremath{\Lambda}}$: $$\label{di_cauchy20} \left\{ \begin{array}{rcl} {\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}&=&{\ensuremath{\displaystyle\sum}}_{j\ge 1}\alpha^j M_j[Y[Q_0]],\\ M_j[Y_n]&=&-\dfrac{1}{2\pi}{\ensuremath{\displaystyle\int}}_{-\infty}^{+\infty}\frac{ d{\ensuremath{\omega}}}{{\ensuremath{\mathcal{D}^0}}+i{\ensuremath{\omega}}}\Big(Y_n\frac{1}{{\ensuremath{\mathcal{D}^0}}+i{\ensuremath{\omega}}} \Big)^{j}. \end{array} \right.$$ Each $M_j[Y[Q_0]]$ is polynomial in $\Pi_{\ensuremath{\Lambda}}R_{Q_0} \Pi_{\ensuremath{\Lambda}}$ of degree $j$. By using , the decomposition is well-defined in several Banach space, provided that $\alpha {\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}$ is small enough. – First, integrating the norm of bounded operator in , we obtain $${\ensuremath{\lVert{\ensuremath{\overline{\boldsymbol{\pi}}}}_0-{\ensuremath{\mathcal{P}^0_-}}\rVert_{\mathcal{B}}}}\apprle \alpha {\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}<1.$$ – We take the Hilbert-Schmidt norm [@ptf; @sok]: we get $$\label{di_estim_gn} {\ensuremath{\lVert{\ensuremath{\gamma}}_{0}\rVert_{\mathfrak{S}_{2}}}}\apprle \alpha {\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}.$$ – We take the norm ${\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}(\cdot)\rVert_{\mathfrak{S}_{2}}}}$ we get the rough estimate $$\label{di_estim_kin} {\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}{\ensuremath{\gamma}}_{0}\rVert_{\mathfrak{S}_{2}}}}\apprle \min(\sqrt{L\alpha}{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}},\alpha {\ensuremath{\lVertR_{Q_0}\rVert_{\mathfrak{S}_{2}}}}\big)+\alpha^2{\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}^2.$$ \[di\_diff\_ch\] The same estimates holds for the differential of $Q_0\mapsto {\ensuremath{\gamma}}_0$, for sufficiently small $\alpha$. As shown in [@sok], the upper bound of each norm is a power series of kind $$\lVert{\ensuremath{\gamma}}_0\rVert\le \alpha \lVert M_1[Y[Q_0]]\rVert+{\ensuremath{\displaystyle\sum}}_{j=1}^{+\infty}\sqrt{j}\alpha^j\big(K{\ensuremath{\lVert Q_0\rVert_{\text{Ex}}}} \big)^j.$$ In the case of the differential, we get an upper bound of kind $$\lVert \text{d}{\ensuremath{\gamma}}_0\rVert\le \alpha \lVert M_1[Y[Q_0]]\rVert+{\ensuremath{\displaystyle\sum}}_{j=1}^{+\infty}j^{3/2}\alpha^j\big(K{\ensuremath{\lVert Q_0\rVert_{\text{Ex}}}} \big)^j.$$ The power series converge for sufficiently small $\alpha {\ensuremath{\lVertQ_0\rVert_{\text{Ex}}}}$. – It is also possible to consider other norms, using from the fact that a (scalar) Fourier multiplier $F(\mathbf{p}-\mathbf{q})=F(-i\nabla_x+i\nabla_y)$ commutes with the operator $R[\cdot]:Q(x,y)\mapsto \tfrac{Q(x,y)}{|x-y|}$. We can also consider the norm $$\lVert Q_0\rVert_{w}^2:={\ensuremath{\displaystyle\iint}}w(p-q)({\ensuremath{\widetilde{E}\left(p\right)}}+{\ensuremath{\widetilde{E}\left(q\right)}})|{\ensuremath{\widehat{Q}}}_0(p,q)|^2dpdq,$$ where $w(\cdot)\ge 0$ is any weight satisfying a subadditive condition [@sok]: $$\forall\,p,q\in{\ensuremath{\mathbb{R}^3}},\ \sqrt{w(p+q)}\le K(w)\big(\sqrt{w(p)}+\sqrt{w(q)}\big).$$ Proof of Theorems \[di\_main\] and \[di\_main\_1\] ================================================== Strategy and tools of the proof: the dipositronium -------------------------------------------------- ### Topologies The existence of a minimizer over $\mathscr{W}_{\mathbf{t}X^\ell}$ (with $\mathbf{t}\in\mathbb{Z}_2^2$) is proved with the same method used in [@pos_sok]. We use a lemma of Borwein and Preiss [@borw; @at], a smooth generalization of Ekeland’s Lemma [@ek]: we study the behaviour of a specific minimizing sequence $(P_n)_n$ or equivalently $(P_n-{\ensuremath{\mathcal{P}^0_-}}=:Q_n)_n$. This sequence satisfies an equation close to the one satisfied by a real minimizer and we show this equation remains in some weak limit. \[di\_topo\] We recall different topologies over bounded operator, besides the norm topology ${\ensuremath{\lVert\cdot\rVert_{\mathcal{B}}}}$ [@ReedSim]. 1. The so-called *strong topology*, the weakest topology $\mathcal{T}_s$ such that for any $f\in{\ensuremath{\mathfrak{H}_\Lambda}}$, the map $$\begin{array}{rcl} \mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}})&\longrightarrow&{\ensuremath{\mathfrak{H}_\Lambda}}\\ A&\mapsto& Af \end{array}$$ is continuous. 2. The so-called *weak operator topology*, the weakest topology $\mathcal{T}_{w.o.}$ such that for any $f,g\in{\ensuremath{\mathfrak{H}_\Lambda}}$, the map $$\begin{array}{rcl} \mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}})&\longrightarrow&\mathbb{C}\\ A&\mapsto& {\ensuremath{\langle A f\,,\,g\rangle}\xspace} \end{array}$$ is continuous. We can also endow $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$ with its weak-$*$ topology, the weakest topology such that the following maps are continuous: $$\begin{array}{|l} \begin{array}{rcl} \mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}&\longrightarrow&\mathbb{C}\\ Q&\mapsto& {\ensuremath{\mathrm{Tr}}}\big(A_0(Q^{++}+Q^{--})+A_2(Q^{+-}+Q^{-+})\big) \end{array}\\ \forall\,(A_0,A_2)\in\mathrm{Comp}({\ensuremath{\mathfrak{H}_\Lambda}})\times \mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}}). \end{array}$$ \[di\_weakclosed\] The set $\mathcal{K}^0_{\mathscr{Y}}$, defined in , is weakly-$*$ sequentially closed in $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}({\ensuremath{\mathfrak{H}_\Lambda}})$. This Lemma was stated for $\mathscr{Y}=\mathscr{C}$ in [@pos_sok]. For $\mathscr{Y}=\mathscr{I}$ the proof is the same and we refer the reader to this paper. ### The Borwein and Preiss Lemma We recall this Theorem as stated in [@at]: \[di\_bp\_lemma\] Let $\mathcal{M}$ be a closed subset of a Hilbert space $\mathcal{H}$, and $F:\mathcal{M}\to (-\infty,+\infty]$ be a lower semi-continuous function that is bounded from below and not identical to $+\infty$. For all ${\ensuremath{\varepsilon}}>0$ and all $u\in \mathcal{M}$ such that $F(u)<\inf_{\mathcal{M}}+{\ensuremath{\varepsilon}}^2$, there exist $v\in\mathcal{M}$ and $w\in{\ensuremath{\overline{\mathrm{Conv}(\mathcal{M})}}}$ such that 1. $F(v)< \inf_{\mathcal{M}}+{\ensuremath{\varepsilon}}^2$, 2. $\lVert u-v\rVert_{\mathcal{H}}<\sqrt{{\ensuremath{\varepsilon}}}$ and $\lVert v-w\rVert_{\mathcal{H}}<\sqrt{{\ensuremath{\varepsilon}}}$, 3. $F(v)+{\ensuremath{\varepsilon}}\lVert v-w\rVert_{\mathcal{H}}^2=\min\big\{F(z)+{\ensuremath{\varepsilon}}\lVert z-w\rVert_{\mathcal{H}}^2,\ z\in\mathcal{M}\big\}.$ – Here we apply this Theorem with $\mathcal{H}=\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})$, $\mathcal{M}=\mathscr{W}_{p(X)}-{\ensuremath{\mathcal{P}^0_-}}$ and $F=\mathcal{E}^0_{\mathrm{BDF}}$. The BDF energy is continuous in the $\mathfrak{S}_1^{{\ensuremath{\mathcal{P}^0_-}}}$-norm topology, thus its restriction over $\mathscr{V}$ is continuous in the $\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})$-norm topology. This subspace $\mathcal{H}$ is closed in the Hilbert-Schmidt norm topology because $\mathscr{V}=\mathscr{M}_{\mathscr{C}}$ is closed in $\mathfrak{S}_2({\ensuremath{\mathfrak{H}_\Lambda}})$ and $\mathscr{E}_{-1}-{\ensuremath{\mathcal{P}^0_-}}$ is closed in $\mathscr{V}$. Moreover, we have $${\ensuremath{\overline{\text{Conv}(\mathscr{W}_{p(X)}-{\ensuremath{\mathcal{P}^0_-}})}}}^{\mathfrak{S}_2}\subset \mathcal{K}_{\mathscr{C}}^0.$$ – For every $\eta>0$, we get a projector $P_\eta\in\mathscr{W}_{p(X)}$ and $A_\eta\in \mathcal{K}_{\mathscr{C}}^0$ such that $P$ that minimizes the functional $F_\eta: P\in\mathscr{E}_{-1}\mapsto \mathcal{E}_{\text{BDF}}^0(P-{\ensuremath{\mathcal{P}^0_-}})+{\ensuremath{\varepsilon}}{\ensuremath{\lVertP-{\ensuremath{\mathcal{P}^0_-}}-A_\eta\rVert_{\mathfrak{S}_{2}}}}^2.$ We write $$\label{di_almost} Q_\eta:= P_\eta -{\ensuremath{\mathcal{P}^0_-}},\ {\ensuremath{\Gamma}}_\eta:=Q_\eta -A_\eta,\ {\ensuremath{\widetilde{D}}}_{Q_\eta}:=\Pi_{\ensuremath{\Lambda}}\big({\ensuremath{\mathcal{D}^0}}-\alpha R_{Q_\eta}+2\eta {\ensuremath{\Gamma}}_\eta\big)\Pi_{\ensuremath{\Lambda}}.$$ Studying its differential on $\text{T}_{P_\eta} \mathscr{W}$, we get: $$\label{di_eq_almost} \big[{\ensuremath{\widetilde{D}}}_{Q_\eta}, P_\eta\big]=0.$$ In particular, by functional calculus, we have: $$\label{di_pimoins} \big[\boldsymbol{\pi}_-^{\eta},P_\eta\big]=0,\ \boldsymbol{\pi}_{\eta}^-:=\chi_{(-\infty,0)}({\ensuremath{\widetilde{D}}}_{Q_\eta}).$$ We also write $$\label{di_piplus} \boldsymbol{\pi}_{\eta}^+:=\chi_{(0,+\infty)}({\ensuremath{\widetilde{D}}}_{Q_\eta})=\Pi_{\ensuremath{\Lambda}}-\boldsymbol{\pi}_{\eta}^-.$$ We decompose ${\ensuremath{\mathfrak{H}_\Lambda}}$ as follows (here R means $\mathrm{Ran}$): $$\label{di_decomp_hl} {\ensuremath{\mathfrak{H}_\Lambda}}=\text{R}(P_\eta)\cap \text{R}({\ensuremath{\boldsymbol{\pi}}}_{\eta}^-)\overset{\perp}{\oplus}\text{R}(P_\eta)\cap \text{R}({\ensuremath{\boldsymbol{\pi}}}_{\eta}^+)\overset{\perp}{\oplus}\text{R}(\Pi_{\ensuremath{\Lambda}}-P_\eta)\cap \text{R}({\ensuremath{\boldsymbol{\pi}}}_{\eta}^-)\overset{\perp}{\oplus}\text{R}(\Pi_{\ensuremath{\Lambda}}-P_\eta)\cap \text{R}({\ensuremath{\boldsymbol{\pi}}}_{\eta}^+).$$ We will prove 1. $\mathrm{Ran}\,P\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}_{\eta}^+$ has dimension $2j+1$ and is invariant under ${\ensuremath{\Phi_{\mathrm{SU}}}}$, spanned by a unitary $\psi_\eta\in{\ensuremath{\mathfrak{H}_\Lambda}}$. 2. As $\eta$ tends to $0$, up to translation and a subsequence, $\psi_\eta\rightharpoonup \psi_a\neq 0$, $Q_\eta\rightharpoonup {\ensuremath{\overline{Q}}}$. There holds ${\ensuremath{\overline{P}}}_{j_0}={\ensuremath{\overline{Q}}}+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{W}_{p(X)}$, $\psi_a$ is a unitary eigenvector of $ D_{{\ensuremath{\overline{Q}}}}^{({\ensuremath{\Lambda}})} $ and $$\label{di_eq_min} {\ensuremath{\overline{Q}}}+{\ensuremath{\mathcal{P}^0_-}}=\chi_{(-\infty,0)}\big( D_{{\ensuremath{\overline{Q}}}}^{({\ensuremath{\Lambda}})} \big)+\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi_a)-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi_a),$$ where $\text{Proj}\,E$ means the orthonormal projection onto the vector space $E$. In the following part we write the spectral decomposition of trial states and prove Lemma \[di\_weakclosed\]. ### Spectral decomposition Let $(Q_n)_n$ be any minimizing sequence for $E_{\mathbf{t}X^{(j_0-1/2)}}^{nr}$ for $j_0\in \tfrac{1}{2}+\mathbb{Z}_+$. Thanks to the upper bound, $\text{Dim}\,\mathrm{Ker}(Q_n-1)=1$, as shown in Subsection \[di\_subscritic\]. There exist a *non-increasing* sequence $({\ensuremath{\lambda}}_{j;n})_{j\in\mathbb{N}}\in\ell^2$ of eigenvalues and an orthonormal family $\mathbf{B}_n$ of $\mathrm{Ran}\, Q_n$: $$\label{di_basen} \mathbf{B}_n:=(\psi_n,{\ensuremath{\mathrm{C}}}\psi_n)\cup (e_{j;n}^a,e_{j;n}^b,{\ensuremath{\mathrm{C}}}e_{j;n}^a,{\ensuremath{\mathrm{C}}}e_{j;n}^b),\ {\ensuremath{\mathcal{P}^0_-}}\psi_n={\ensuremath{\mathcal{P}^0_-}}e_{j;n}^{\star}=0,\ \star\in\{a,b\},$$ such that the following holds. We omit the index $n$. 1\. For any $j$, the vector spaces $V_{j;n}^\star:={\ensuremath{\Phi_{\mathrm{SU}}}}(e_{j;n}^\star)$ are irreducible, and so is $V_{0;n}:={\ensuremath{\Phi_{\mathrm{SU}}}}(\psi_n)$. 2\. That last one is of type $(\ell_0,{\ensuremath{\varepsilon}}(\mathbf{t}))$ (see Notation \[di\_eps\_t\]). 3\. Moreover for any $j\in \mathbb{N}$ we write: \[di\_formtrial\] $$\label{di_formtrial1} \begin{array}{| l} e_{-j}^a:=-{\ensuremath{\mathrm{C}}}e_{j}^b\text{\ and\ } e_{-j}^b:={\ensuremath{\mathrm{C}}}e_j^a,\\ V_{-j}^a:={\ensuremath{\Phi_{\mathrm{SU}}}}\,e_{-j}^a\text{\ and\ }V_{-j}^b:={\ensuremath{\Phi_{\mathrm{SU}}}}\,e_{-j}^b. \end{array}$$ $$\label{di_formtrial2} \begin{array}{rll} f_{j}^\star&:=& \sqrt{\tfrac{1-{\ensuremath{\lambda}}_j}{2}} e_{-j}^\star+\sqrt{\tfrac{1+{\ensuremath{\lambda}}_j}{2}}e_{j}^\star,\\ f_{-j}^\star&:=& -\sqrt{\tfrac{1+{\ensuremath{\lambda}}_j}{2}}e_{-j}^\star+\sqrt{\tfrac{1+{\ensuremath{\lambda}}_j}{2}} e_{j}^\star, \end{array}$$ and $$\label{di_formtrial22} \forall\,j\in \mathbb{Z}^*,\ F_j^\star:={\ensuremath{\Phi_{\mathrm{SU}}}}(f_j^\star).$$ The trial state $Q_n$ has the following form. $$\label{di_formtrial3} \left\{\begin{array}{rll} Q_n&=&\text{Proj}\,V_{0,n}-\text{Proj}\,{\ensuremath{\mathrm{C}}}V_{0,n}+{\ensuremath{\displaystyle\sum}}_{j\ge 1}{\ensuremath{\lambda}}_jq_{j;n} \\ q_{j;n}&=&\text{Proj}\,F_{j}^a-\text{Proj}\,F_{-j}^a+\text{Proj}\,F_{j}^b-\text{Proj}\,F_{-j}^b. \end{array} \right.$$ \[di\_diag\_extrac\] Thanks to the cut-off the sequences $(\psi_n)_n$ and $(e_{j;n})_n$ are $H^1$-bounded. Up to translation and extraction ($(n_k)_k\in\mathbb{N}^{\mathbb{N}}$ and $(x_{n_k})_k\in(\mathbb{R}^3)^{\mathbb{N}}$), we can assume that the weak limit of $(\psi_n)_n$ is non-zero (if it were then there would hold $E_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}=2m(2j_0+1)$). We can consider the weak limit of each $(e_n)$: by means of a diagonal extraction, we assume that all the $(e_{j,n_{k}}(\cdot -x_{n_k}))_k$ and $(\psi_{j,n_k}(\cdot-x_{n_k}))_k$, converge along the same subsequence $(n_k)_k$. We also assume that $$\label{di_spec_conv} \forall\,j\in\mathbb{N},\ {\ensuremath{\lambda}}_{j,n_k}\to\mu_j,\ (\mu_j)_j\in\ell^2,\ (\mu_j)_j\text{\ non-increasing},$$ and that the above convergences also hold in $L^2_{\text{loc}}$ and almost everywhere. Upper bound and rough lower bound of $E_{j_0,\pm}$ {#di_subscritic} -------------------------------------------------- We aim to prove the upper bound of Proposition \[di\_est\]. The method will also give a rough lower bound of $E_{j_0,\pm}$. We write: $$C(j_0):=j_0^2\underset{-j_0\le m\le j_0}{\sup}{\ensuremath{\lVert\Psi_{m,j_0\pm \tfrac{1}{2}}\rVert_{L^{\infty}}}}^4,$$ where the functions $\Psi_{m,j_0\pm\tfrac{1}{2}}$ are defined in [@Th p. 125]: they are the upper or lower spinors of the $\Phi^{\pm}_{m,\kappa_{j_0}}$’s. For $E_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t}})$, we only consider $\mathbf{t}\in\{(1,0);(0,1)\}$ and ${\ensuremath{\varepsilon}}(\mathbf{t})$ is defined in Notation \[di\_eps\_t\]. – We consider trial state of the following form: $$Q=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi),$$ where ${\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)$ is of type $(\ell_0+\tfrac{1}{2},{\ensuremath{\varepsilon}}(\mathbf{t}))$ and ${\ensuremath{\mathcal{P}^0_-}}\psi=0$. For short, we write $$N_\psi:=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi)\text{\ and\ }N_{{\ensuremath{\mathrm{C}}}\psi}:=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi).$$ The set of these states is written $\mathscr{W}_{\mathbf{t}X^{\ell_0}}^0$. We will prove that the energy of a particular $Q$ gives the upper bound. The BDF energy of $Q\in \mathscr{W}_{\mathbf{t}X^{\ell_0}}^0$ is: $$\label{di_form_no_pol} 2{\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|N_{\psi}\big)-\alpha{\ensuremath{\lVertN_\psi\rVert_{\text{Ex}}}}^2-\alpha \mathfrak{Re}\,{\ensuremath{\mathrm{Tr}}}\big(N_\psi R[N_{{\ensuremath{\mathrm{C}}}\psi}]\big).$$ – We will study the non-relativistic limit $\alpha\to 0$. – To get an upper bound, we choose a specific trial state in $\mathscr{W}_{\mathbf{t}X^{\ell_0}}$, the idea is the same as in [@sok; @pos_sok]: the trial state is written in . Before that, we precise the structure of elements in $\mathscr{W}_{\mathbf{t}X^{\ell_0}}^0$. #### Minimizer for $E_{\mathbf{t}X^{\ell_0}}^{nr}$ By an easy scaling argument, there exists a minimizer for the non-relativistic energy $E_{\mathbf{t}X^{\ell_0}}^{nr}$ . The scaling argument enables us to say that this energy is negative. Then it is clear that a minimizing sequence converges to a minimizer ${\ensuremath{\overline{{\ensuremath{\Gamma}}}}}$, up to extraction. Writing $$H_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}:=-\Delta-R_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}},$$ this minimizer satisfies the self-consistent equation $$\big[H_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}, {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big]=0.$$ This comes from Remark \[di\_prec\_non\_rel\]. In particular, $H_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}$ restricted to $\mathrm{Ran}\,{\ensuremath{\Gamma}}$ is a homothety by some $-e^2<0$, so $$\forall\,\psi\in\mathrm{Ran}\,{\ensuremath{\overline{{\ensuremath{\Gamma}}}}},\ {\ensuremath{\lVert\psi\rVert_{L^{2}}}}=1,\ {\ensuremath{\lVert\Delta \psi\rVert_{L^{2}}}}\le {\ensuremath{\lVertR_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}\psi\rVert_{L^{2}}}}\apprle {\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}{\ensuremath{\lVert\,|\nabla|^{1/2}\psi\rVert_{L^{2}}}},$$ and we get $${\ensuremath{\lVert\Delta \psi\rVert_{L^{2}}}}^{3/4}\apprle {\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}\ i.e.\ {\ensuremath{\lVert\Delta \psi\rVert_{L^{2}}}}\apprle {\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}^{4/3}\apprle (2j_0+1)^{2/3}.$$ The last estimate comes from a simple study of a minimizer for $E_{\mathbf{t}X^{\ell_0}}^{nr}$: we have $${\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)-\frac{\pi}{2}{\ensuremath{\mathrm{Tr}}}\big(|\nabla|{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\le \mathcal{E}_{nr}({\ensuremath{\overline{{\ensuremath{\Gamma}}}}})<0,$$ thus ${\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\apprle j_0^2$ and $ {\ensuremath{\mathrm{Tr}}}\big((-\Delta)^2{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\apprle j_0^{5/2}.$ We end this bootstrap argument at ${\ensuremath{\lVert|\nabla|^{3}\psi\rVert_{L^{2}}}}$ for $\psi\in\mathrm{Ran}\,\psi$: we have $$\begin{aligned} |\nabla|^3\psi&=\frac{-\Delta}{e^2-\Delta}\Big([|\nabla|,R_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}]\psi+R_{{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}}\psi\Big),\\ {\ensuremath{\lVert\,|\nabla|^3\psi\rVert_{L^{2}}}}&\apprle {\ensuremath{\lVert\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\mathfrak{S}_{2}}}}+{\ensuremath{\lVert\nabla {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\mathfrak{S}_{2}}}}\apprle j_0^{5/2}.\end{aligned}$$ #### Trial state We take the following trial state. First, let ${\ensuremath{\overline{{\ensuremath{\Gamma}}}}}=\text{Proj}\,ra_0(r)\Psi_{j_0,j_0+{\ensuremath{\varepsilon}}(\mathbf{t})\tfrac{1}{2}}$ be a minimizer for $E_{\mathbf{t}X^{\ell_0}}^{nr}$. We form $$\label{di_trial_non_rel_1} {\ensuremath{\overline{N}}}_+:= \text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\, {\ensuremath{\mathcal{P}^0_+}}U_{{\ensuremath{\lambda}}^{-1}}(ra_0(r)\Phi^{+}_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})(j_0+\tfrac{1}{2})})$$ where we recall that $${\ensuremath{\lambda}}:=\frac{g'_1(0)^2}{\alpha m}\text{\ and\ }U_a \phi(x):=a^{3/2} \phi(ax),\ a>0.$$ This corresponds to dilating ${\ensuremath{\overline{{\ensuremath{\Gamma}}}}}$ by ${\ensuremath{\lambda}}^{-1}$ and projecting the range of the dilation onto $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$. Of course $ {\ensuremath{\Gamma}}\in \mathfrak{S}_1(L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2))$ is embedded in $\mathfrak{S}_1(L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2\times\mathbb{C}^2))$ as follows: $${\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\mapsto \begin{pmatrix} {\ensuremath{\overline{{\ensuremath{\Gamma}}}}} & 0 \\ 0 & 0\end{pmatrix}\in \mathfrak{S}_1(L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C}^2\times\mathbb{C}^2)).$$ Then we define $$\label{di_trial_non_rel_2} {\ensuremath{\overline{N}}}_-:={\ensuremath{\mathrm{C}}}{\ensuremath{\overline{N}}}_-{\ensuremath{\mathrm{C}}}^{-1}={\ensuremath{\mathrm{C}}}{\ensuremath{\overline{N}}}_-{\ensuremath{\mathrm{C}}}.$$ Our trial state is $$\label{di_trial_non_rel_end} {\ensuremath{\overline{N}}}:={\ensuremath{\overline{N}}}_+-{\ensuremath{\overline{N}}}_-.$$ #### Upper bound for $E_{j_0,\pm}$ We compute $\mathcal{E}^0_{\text{BDF}}({\ensuremath{\overline{N}}})$. Before that, we study a general projector $\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi$ where ${\ensuremath{\mathcal{P}^0_-}}\psi=0$ and ${\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi$ irreducible of type $(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))$. As an element of $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$, the wave function $\psi$ can be written $$\psi={\ensuremath{\mathcal{P}^0_+}}\begin{pmatrix}{\ensuremath{\varphi}}\\ 0\end{pmatrix}.$$ As it spans an irreducible representation of type $(j_0,{\ensuremath{\varepsilon}}(\mathbf{t}))$, we can choose $$\forall\, x=r{\ensuremath{\omega}}_x\in{\ensuremath{\mathbb{R}^3}},\ {\ensuremath{\varphi}}(x):= ia(r)\Psi_{j_0+{\ensuremath{\varepsilon}}(\mathbf{t})\tfrac{1}{2}}^{j_0}({\ensuremath{\omega}}_x),\ a(r)\in L^2\big((0,\infty),r^2dr\big),$$ where we used notations of [@Th p. 126]. This corresponds to taking $$\psi:={\ensuremath{\mathcal{P}^0_+}}ra(r)\Phi^+_{j_0,{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})},\ {\ensuremath{\varepsilon}}={\ensuremath{\varepsilon}}(\mathbf{t}).$$ We recall the following formulae of [@Th pp. 125-127] (with $\boldsymbol{{\ensuremath{\omega}}}:x\mapsto\tfrac{x}{|x|}$) $$\label{di_form_th_125} \begin{array}{l} -i\boldsymbol{\alpha}\cdot \nabla=-i(\boldsymbol{\alpha}\cdot \boldsymbol{{\ensuremath{\omega}}})\partial_r+\frac{i}{r}(\boldsymbol{\alpha}\cdot \boldsymbol{{\ensuremath{\omega}}})(2{\ensuremath{\mathbf{S}}}\cdot {\ensuremath{\mathbf{L}}}),\\ \big\{{\ensuremath{\mathbf{S}}}\cdot {\ensuremath{\mathbf{L}}},\boldsymbol{\alpha}\cdot \boldsymbol{{\ensuremath{\omega}}}\big\}=-\boldsymbol{\alpha}\cdot \boldsymbol{{\ensuremath{\omega}}}\text{\ and\ }i\boldsymbol{\sigma}\cdot \boldsymbol{{\ensuremath{\omega}}}\Psi^{m_j}_{j\pm \tfrac{1}{2}}=\Psi^{m_j}_{j\mp\tfrac{1}{2}}. \end{array}$$ This gives $$\label{di_form_en_trial1} \begin{array}{rcl} {\ensuremath{\mathcal{P}^0_+}}a(r)\Phi^+_{m,{\ensuremath{\varepsilon}}(\mathbf{t})(j_0+\tfrac{1}{2})}&=&\dfrac{1}{2}\begin{pmatrix}i\big(1+\frac{g_0(|\nabla|)}{|{\ensuremath{\mathcal{D}^0}}|} \big)a(r) \Psi^m_{j_0+{\ensuremath{\varepsilon}}\tfrac{1}{2}}\\ \frac{g_1(|\nabla|)}{|{\ensuremath{\mathcal{D}^0}}||\nabla|}\big(\partial_r (a(r))+{\ensuremath{\varepsilon}}(j_0+\tfrac{1}{2})\tfrac{a(r)}{r}\big)\Psi^m_{j_0-{\ensuremath{\varepsilon}}\tfrac{1}{2}}\end{pmatrix},\\ &=:&\begin{pmatrix} ia_{\uparrow}(r)\Psi^m_{j_0+{\ensuremath{\varepsilon}}\tfrac{1}{2}}\\ a_{\downarrow}({\ensuremath{\varepsilon}},j_0;r)\Psi^m_{j_0-{\ensuremath{\varepsilon}}\tfrac{1}{2}}\end{pmatrix}. \end{array}$$ We write $ \mathrm{Op}:=\frac{g_1(|\nabla|)}{|{\ensuremath{\mathcal{D}^0}}||\nabla|}:$ the following holds. $$\label{di_form_en_trial2} \begin{array}{rcl} \Big| {\ensuremath{\mathrm{Tr}}}\big(N_\psi R[N_{{\ensuremath{\mathrm{C}}}\psi}]\big)\Big|&\apprle &j_0^2\sup_{m}{\ensuremath{\lVert\Psi^m_{j_0\pm\tfrac{1}{2}}\rVert_{L^{\infty}}}}^2{\ensuremath{\lVert\,|a_{\uparrow}a_{\downarrow}({\ensuremath{\varepsilon}},j_0,\cdot)|\rVert_{\mathcal{C}}}}^2\\ &\apprle& C(j_0)D\Big(|a_{\uparrow}|^2; |\mathrm{Op}\cdot\partial_r (a(r))|^2+j_0^2|\mathrm{Op}\cdot r^{-1}a(r)|^2\Big),\\ &\apprle& C(j_0) {\ensuremath{\langle |\nabla|\psi\,,\,\psi\rangle}\xspace}{\ensuremath{\lVert\nabla\psi\rVert_{L^{2}}}}^2=: \mathcal{R}em_0(j_0,\psi). \end{array}$$ In fact, we have ${\ensuremath{\mathrm{Tr}}}\big(N_\psi R[N_{{\ensuremath{\mathrm{C}}}\psi}] \big)\ge 0$ by direct computation. Let us deal with ${\ensuremath{\lVertN_\psi\rVert_{\text{Ex}}}}^2$. We write ${\ensuremath{P_{\uparrow}}}$ the projection onto the upper part of $\mathbb{C}^2\times\mathbb{C}^2$ and ${\ensuremath{P_{\downarrow}}}$ the projection onto the lower part. That is: ${\ensuremath{P_{\uparrow}}}\psi$ has no lower spinor and the same upper spinor as $\psi$. Similarly, $$\begin{aligned} {\ensuremath{\lVertN_\psi\rVert_{\text{Ex}}}}^2-{\ensuremath{\lVert{\ensuremath{P_{\uparrow}}}N_\psi {\ensuremath{P_{\uparrow}}}\rVert_{\text{Ex}}}}^2&={\ensuremath{\mathrm{Tr}}}\big({\ensuremath{P_{\uparrow}}}N_\psi {\ensuremath{P_{\downarrow}}}R_{N_\psi}\big)+{\ensuremath{\mathrm{Tr}}}\big({\ensuremath{P_{\downarrow}}}N_\psi {\ensuremath{P_{\uparrow}}}R_{N_\psi}\big)\\ &\ \ +{\ensuremath{\lVert{\ensuremath{P_{\downarrow}}}N_\psi {\ensuremath{P_{\downarrow}}}\rVert_{\text{Ex}}}},\\ &\apprle \mathcal{R}em(j_0,\psi)+C(j_0){\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2{\ensuremath{\lVert\tfrac{|\nabla|^{3/2}}{|D_0|} \psi\rVert_{L^{2}}}}^2,\\ &=:\mathcal{R}em_1(j_0,\psi).\end{aligned}$$ For the trial state , this gives: $$\begin{aligned} {\ensuremath{\lVert{\ensuremath{\overline{N}}}_+\rVert_{\text{Ex}}}}^2&={\ensuremath{\lVert{\ensuremath{P_{\uparrow}}}N_\psi {\ensuremath{P_{\uparrow}}}\rVert_{\text{Ex}}}}^2+\mathcal{O}\Big( C(j_0)\big( \alpha^3j_0+\alpha^5j_0^{5/3}\big)\Big)\\ &=\frac{\alpha m}{g'_1(0)^2}{\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}^2(1+\mathcal{O}({\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2))\\ &\ \ \ +\mathcal{O}\big( {\ensuremath{\lVert\tfrac{\Delta}{1-\Delta} \psi\rVert_{L^{2}}}}^2({\ensuremath{\lVert\tfrac{|\nabla|^{5/2}}{1-\Delta} \psi\rVert_{L^{2}}}}+{\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2)\big),\\ &=\frac{\alpha m}{g'_1(0)^2}{\ensuremath{\lVert{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\rVert_{\text{Ex}}}}^2\\ &+\mathcal{O}\Big[C(j_0)\Big(\alpha^3j_0^{5/3}+ \underset{0\le s\le 1}{\inf}(\alpha^{4s}j_0^{4s/3})\big(\alpha^2j_0^{2/3}+\underset{2^{-1}\le s\le 1}{\inf}(\alpha^{4s}j_0^{4s/3})\big) \Big)\Big].\end{aligned}$$ We compute the kinetic energy as in [@sok; @pos_sok]: we get $$\begin{aligned} {\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}| {\ensuremath{\overline{N}}}_+\big)&=\frac{\alpha^2m}{g'_1(0)^2}{\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\big(1+K\alpha\big)+\mathcal{O}\big(\alpha^4{\ensuremath{\mathrm{Tr}}}\big((\Delta)^2{\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)\big),\\ &=\frac{\alpha^2m}{g'_1(0)^2}{\ensuremath{\mathrm{Tr}}}\big(-\Delta {\ensuremath{\overline{{\ensuremath{\Gamma}}}}}\big)+\mathcal{O}\big(\alpha^3 j_0+\alpha^4j_0^{5/2}\big).\end{aligned}$$ This proves $$\begin{array}{|l} E_{j_0,{\ensuremath{\varepsilon}}(\mathbf{t})}\le 2m(2j_0+1)+\frac{\alpha^2m}{g'_1(0)^2}E_{\mathbf{t}X^{\ell_0}}^{nr}+\mathcal{O}\big(\varrho(\alpha,j_0)\big)\\ \varrho(\alpha,j_0):=\alpha^3 j_0+\alpha^4j_0^{5/2}+C(j_0)\Big(\alpha^3j_0^{5/3}+ \underset{0\le s\le 1}{\inf}(\alpha^{4s}j_0^{4s/3})\big(\alpha^2j_0^{2/3}+\underset{2^{-1}\le s\le 1}{\inf}(\alpha^{4s}j_0^{4s/3})\big) \Big). \end{array}$$ First, by Kato’s inequality , we have $${\ensuremath{\lVertN_\psi-N_{{\ensuremath{\mathrm{C}}}\psi}\rVert_{\text{Ex}}}}^2\le \frac{\pi}{2}{\ensuremath{\mathrm{Tr}}}\big(|\nabla| (N_\psi+N_{{\ensuremath{\mathrm{C}}}\psi})\big)=\pi{\ensuremath{\mathrm{Tr}}}\big(|\nabla|N_{\psi}\big).$$ So $$\mathcal{E}^0_{\text{BDF}}(Q)\ge 2\Big({\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|N_{\psi}\big)-\alpha\frac{\pi}{2}{\ensuremath{\mathrm{Tr}}}\big(|\nabla|N_{\psi}\big)\Big)=:2\big((2j_0+1)m+\mathcal{F}(N_\psi)\big).$$ As $\alpha$ tends to $0$, a minimizer over $\mathscr{W}_{\mathbf{t}X^{\ell_0}}^0$ should be localized in Fourier space around $0$. Indeed, for $\alpha,L$ sufficiently small, we have $$\forall\,p\in B(0,{\ensuremath{\Lambda}}),\ {\ensuremath{\widetilde{E}\left(p\right)}}-m=\frac{g_0(p)^2-m^2+g_1(p)^2}{{\ensuremath{\widetilde{E}\left(p\right)}}+m}\ge \frac{p^2}{2{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}|D_0|},$$ and for any $0<s\le 2$: $$\frac{p^2}{2{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}|D_0|}\ge s\frac{\alpha\pi}{2}|p|\iff |p|\ge \frac{\alpha s\pi {\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}}{\sqrt{1-(\alpha s\pi{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}})^2}}=:\vartheta_{s}.$$ We get $$2\mathcal{F}\big( \Pi_{\vartheta_1} N_{\psi} \Pi_{\vartheta_1}\big)\le \mathcal{E}^0_{\text{BDF}}(Q)-2(2j_0+1)m.$$ By Cauchy-Schwartz inequality, we get a rough lower bound $${\ensuremath{\mathrm{Tr}}}\big(-\Delta \Pi_{\vartheta_1} N_{\psi} \Pi_{\vartheta_1} \big)\apprle \alpha^2(2j_0+1)\text{\ and\ }\mathcal{E}^0_{\text{BDF}}(Q)-2(2j_0+1)m\apprge -\alpha^2(2j_0+1).$$ For an almost minimizer $Q$, the same argument shows that $$\label{di_alm_min} {\ensuremath{\mathrm{Tr}}}\big(\frac{-\Delta}{|{\ensuremath{\mathcal{D}^0}}|}Q^2\big)\apprle \alpha^2 (2j_0+1).$$ A precise lower bound is obtained once we know that there exists a minimizer ${\ensuremath{\overline{P}}}_{j_0}$. This state satisfies the self-consistent equation : see Subsection \[di\_low\_bound\]. The same method can be used to get an upper bound of $E_{p(X)}^{nr}$ for any $p(X)=\sum_{\ell=0}^{\ell_0}\mathbf{t}_{\ell}X^\ell$. By scaling we have $E_{p(X)}^{nr}<0.$ Strategy of the proof: the para-positronium ------------------------------------------- The method is more subtle because $\mathscr{M}_{\mathscr{I}}$ has only one connected component. We first consider the subset $\mathscr{M}_{\mathscr{I}}^{1}$ defined by: $$\label{di_trial_isym} \mathscr{M}_{\mathscr{I}}^{1}=\big\{P_\psi:={\ensuremath{\mathcal{P}^0_-}}+{\ensuremath{|\psi\rangle}\xspace}{\ensuremath{\langle \psi|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi|}\xspace},\ \psi\in\mathbb{S}\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}\big\}.$$ \[di\_infimum\_1\] Let $F_{\mathscr{I}}$ be the infimum of the BDF energy over $\mathscr{M}_{\mathscr{I}}^{1}$. Then we have $$F_{\mathscr{I}}\ge 2m-\alpha^2\frac{E_{\mathrm{PT}}(1) m}{g'_1(0)^2}+\mathcal{O}(\alpha^3).$$ We will prove the existence of a critical point in the neighbourhood of $\mathscr{M}_{\mathscr{I}}^{1}$ *via* a mountain pass argument. In this part, we aim to prove the following Proposition. \[di\_para\_method\] 1. In the regime of Theorem \[di\_main\], there exists a bounded sequence in $\mathscr{M}_{\mathscr{I}}-{\ensuremath{\mathcal{P}^0_-}}$ of almost critical points: $(Q_n=P_n-{\ensuremath{\mathcal{P}^0_-}})_n$ such that $$\underset{n\to+\infty}{\lim}{\ensuremath{\lVert\nabla \mathcal{E}^0_{\text{BDF}}(P_n)\rVert_{\mathfrak{S}_{2}}}}=0\mathrm{\ with\ }\mathcal{E}^0_{\text{BDF}}(Q_n)= 2m-\frac{\alpha^2 m}{g'_1(0)^2}E_{\text{PT}}(1)+\mathcal{O}(\alpha^3).$$ Furthermore, for sufficiently big $n$, there exists $\psi_{a;n}$ such that $$\mathbb{C}\psi_{a;n}=\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,\chi_{(0,+\infty)}\big(D_{Q_n}^{({\ensuremath{\Lambda}})}-\nabla \mathcal{E}^0_{\text{BDF}}(P_n)\big)$$ and $P_n=\chi_{(-\infty,0)}\big(D_{Q_n}^{({\ensuremath{\Lambda}})}-\nabla \mathcal{E}^0_{\text{BDF}}(P_n)\big)+{\ensuremath{|\psi_{a;n}\rangle}\xspace}{\ensuremath{\langle \psi_{a;n}|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_{a;n}\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_{a;n}|}\xspace}.$ 2\. Up to a subsequence and up to translation the sequence tends to a critical point $Q_{\infty}$ of $\mathcal{E}^0_{\text{BDF}}$ in $\mathscr{M}_{\mathscr{I}}-{\ensuremath{\mathcal{P}^0_-}}$. Moreover, writing ${\ensuremath{\overline{P}}}=Q_\infty+{\ensuremath{\mathcal{P}^0_-}}$, there exists $0<\mu<m$ and $\psi_a\in \mathbb{S}\,{\ensuremath{\mathfrak{H}_\Lambda}}$ such that $$\left\{\begin{array}{ccl} {\ensuremath{\overline{P}}}&=&\chi_{(-\infty,0)}(D_{Q_\infty}^{({\ensuremath{\Lambda}})})+{\ensuremath{|\psi_a\rangle}\xspace}{\ensuremath{\langle \psi_a|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a|}\xspace},\\ \mathbb{C}\psi_a&=&\mathrm{Ker}\big(D_{Q_\infty}^{({\ensuremath{\Lambda}})}-\mu\big),\\ \inf\sigma(|D_{Q_\infty}^{({\ensuremath{\Lambda}})}|)&=&\mu. \end{array} \right.$$ #### Proof of Proposition \[di\_para\_method\]: first part For any $\psi\in \mathbb{S}\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$, we define: $$\label{di_def_c(t)} c_{\psi}:\begin{array}{rcl} [0,1] &\longrightarrow& \mathscr{M}_{\mathscr{I}}-{\ensuremath{\mathcal{P}^0_-}}\\ s&\mapsto& {\ensuremath{|\sin(\pi s)\psi+\cos(\pi s){\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\rangle}\xspace}{\ensuremath{\langle \sin(\pi s)\psi+\cos(\pi s){\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi|}\xspace}. \end{array}$$ \[di\_remark\_cross\] The loop $c_\psi+{\ensuremath{\mathcal{P}^0_-}}$ crosses $\mathscr{M}_{\mathscr{I}}^{1}$ at $t_0=\tfrac{1}{2}$ where the BDF energy is maximal: $$\underset{s\in[0,1]}{\sup}\mathcal{E}^0_{\text{BDF}}(c(s)).$$ Indeed, there holds $$\mathcal{E}^0_{\text{BDF}}(c(s))=2\sin(\pi s)^2{\ensuremath{\langle |{\ensuremath{\mathcal{D}^0}}|\psi\,,\,\psi\rangle}\xspace}-\alpha\sin(\pi s)^2\big[D\big(|\psi|^2,|\psi|^2\big)+\cos(2\pi s)D\big(\psi^*{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi, \psi^*{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\big)\big],$$ and the derivative with respect to $s$ is: $$\begin{array}{l} \frac{d}{d s} \mathcal{E}^0_{\text{BDF}}(c(s_0))=2\pi\sin(2\pi s_0)\Big({\ensuremath{\langle |{\ensuremath{\mathcal{D}^0}}|\psi\,,\,\psi\rangle}\xspace}-\frac{\alpha}{2}\big[D\big(|\psi|^2,|\psi|^2\big)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +(\sin(\pi s_0)^2-\tfrac{1}{2}\cos(2\pi s_0)) \alpha D\big(\psi^*{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi, \psi^*{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\big)\big]\Big). \end{array}$$ For sufficiently small $\alpha$, this quantity vanishes only at $2\pi s_0\equiv 0[\pi]$. What happens when we apply the gradient flow $\Phi_{\text{BDF},t}$ of the BDF energy ? The loop $c_{\psi}$ is transformed into $c_{t}:=\Phi_{\text{BDF},t}(c_\psi)$ and we still have $$c_t(s=0)=c_t(s=1)=0.$$ This follows from the fact that ${\ensuremath{\mathcal{P}^0_-}}$ is the global minimizer of $\mathcal{E}^0_{\text{BDF}}$. We recall that for all $s\in[0,1]$, the function $c_t(s)$ satisfies the equation $$\forall\,t_0\in\mathbb{R}_+,\ \frac{d}{dt}(c_{t_0}(s))=-\nabla \mathcal{E}^0_{\text{BDF}}(c_{t_0}(s))\in \text{T}_{c_{t_0}(s)+{\ensuremath{\mathcal{P}^0_-}}}\mathscr{M}_{\mathscr{I}}.$$ The non-trivial result holds. \[di\_non\_triv\] Let $P_\psi\in\mathscr{M}_{\mathscr{I}}^1$ be a state whose energy is close to the infimum $F_{\mathscr{I}}$: $$\mathcal{E}^0_{\text{BDF}}\big(P_\psi\big)<F_{\mathscr{I}}+\alpha^3.$$ Let $c_\psi$ be the loop associated to $\psi$ (see ) and $c_t:=\Phi_{\text{BDF},t}(c_\psi)$. Then for all $t\in\mathbb{R}_+$, the loop $c_t$ crosses the set $\mathscr{M}_{\mathscr{I}}^{1}$ at some ${\ensuremath{\widetilde{s}}}(t)\in(0,1)$. \[di\_ex\_critic\] Let $(c_t)_{t\ge 0}$ be the family of loops defined in Lemma \[di\_non\_triv\] and let $(s(t))_{t\ge 0}$ be a family of reals in $(0,1)$ such that $$\forall\,t\ge 0,\ \mathcal{E}^0_{\text{BDF}}\big(c_t(s(t))\big)=\underset{s\in[0,1]}{\sup}\mathcal{E}^0_{\text{BDF}}(c_t(s)).$$ Then there exists an increasing sequence $(t_n)_{n\in\mathbb{N}}$ the sequence $(c_{t_n}(s(t_n)))_{n\ge 0}$ satisfies the first point of Proposition \[di\_para\_method\] We prove Lemmas \[di\_infimum\_1\] and \[di\_non\_triv\] in Subsection \[di\_fait\_ch\]. We assume they are true to prove Lemma \[di\_ex\_critic\] and Proposition \[di\_para\_method\]. The proof of Lemma \[di\_non\_triv\] uses an index argument. We kept it elementary but it is possible to rephrase it in terms of the Maslov index [@LagGrass] once we notice that ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$ induces a symplectic structure and that the projectors in $\mathscr{M}_{\mathscr{I}}$ are Lagrangians (see Remark \[Lagrangians\]). ##### Spectral decomposition of $P_n$ We define $$F_1:=\liminf_{t\to +\infty}\mathcal{E}^0_{\text{BDF}}(c_t(s(t)))=\liminf_{t\to+\infty}\underset{s\in[0,1]}{\sup}\mathcal{E}^0_{\text{BDF}}(c_t(s)).$$ We assume $(t_n)_{n\ge 0}$ is a minimizing sequence for $F_1$. We may assume that $\lim_{n\to+\infty}t_n=+\infty$. – First we prove that along the path $c_t$ the gradient $\nabla \mathcal{E}^0_{\text{BDF}}$ (see ) is bounded in $\mathfrak{S}_2$. Indeed, for all $P=Q+{\ensuremath{\mathcal{P}^0_-}}\in\mathscr{M}$, we write $${\ensuremath{\widetilde{Q}}}:=P-\chi_{(-\infty,0)}\big(\Pi_{\ensuremath{\Lambda}}D_Q\Pi_{\ensuremath{\Lambda}}\big),$$ We recall that $D_Q^{({\ensuremath{\Lambda}})}:=\Pi_{\ensuremath{\Lambda}}D_Q \Pi_{\ensuremath{\Lambda}}$: $$\label{di_form_grad} \begin{array}{rcl} \nabla \mathcal{E}^0_{\text{BDF}}(P)&=&\big[\big[D_Q^{{\ensuremath{\Lambda}}},P\big],P\big]=\big\{|D_Q^{({\ensuremath{\Lambda}})}|;{\ensuremath{\widetilde{Q}}}\big\}-2{\ensuremath{\widetilde{Q}}} D_{Q}^{({\ensuremath{\Lambda}})} {\ensuremath{\widetilde{Q}}},\\ \lVert \nabla \mathcal{E}^0_{\text{BDF}}(P)\rVert_{\mathfrak{S}_2}&\apprle&{\ensuremath{\lVert{\ensuremath{\widetilde{Q}}}\rVert_{\mathfrak{S}_{2}}}}{\ensuremath{\widetilde{E}\left({\ensuremath{\Lambda}}\right)}}\Big[(1+{\ensuremath{\lVertQ\rVert_{\mathfrak{S}_{2}}}})(1+{\ensuremath{\lVert{\ensuremath{\widetilde{Q}}}\rVert_{\mathfrak{S}_{2}}}})\Big]\\ &\apprle& K({\ensuremath{\Lambda}},F_1+\alpha^3). \end{array}$$ We have used the Cauchy expansion to get an expression $$\chi_{(-\infty,0)}\big(D_Q^{({\ensuremath{\Lambda}})}\big)-{\ensuremath{\mathcal{P}^0_-}}={\ensuremath{\displaystyle\sum}}_{k=1}^{+\infty}\alpha^k M_k[Y[Q]]$$ where $M_k[Y[Q]]$ is a polynomial function of $\pi_{\ensuremath{\Lambda}}R_{Q}\Pi_{\ensuremath{\Lambda}}$ of degree $k$. We refer the reader to these papers or to - above for more details. From formula and Remark \[di\_diff\_ch\] we see that the gradient, as a function of $Q$ is *locally Lipschitz*, at least in some ball $\{Q_0:\,{\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2} Q_0\rVert_{\mathfrak{S}_{2}}}}\le C_0\}$ in which there holds $$\inf \sigma\big(|D_{Q_0}^{({\ensuremath{\Lambda}})}|\big)\ge K(C_0),$$ where $C_0$ is some constant. The Lipschitz constant depends on the constant $C_0$ and in the present case, we can take $C_0\apprle 1$. Let us prove that $$\label{di_grad_zero} \lim_{n\to+\infty}{\ensuremath{\lVert\nabla \mathcal{E}^0_{\text{BDF}}(c_{t_n}(s(t_n))) \rVert_{\mathfrak{S}_{2}}}}=0.$$ If not, the $\limsup$ is bigger than some $\eta>0$ and then we get a contradiction when we consider $n_0$ large enough such that $$|F_1-\mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}} (s(t_{n_0})))|\ll \eta\text{\ and\ }{\ensuremath{\lVert\nabla \mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}}(s(t_{n_0})))\rVert_{\mathfrak{S}_{2}}}}\ge \frac{\eta}{2},$$ because $$\forall\,\tau>0,\ \mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}+\tau}(s(t_{n_0})))-\mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}}(s(t_{n_0})))=-{\ensuremath{\displaystyle\int}}_0^{\tau}{\ensuremath{\lVert\nabla \mathcal{E}^0_{\text{BDF}}(c_{t_{n_0}+u})(s_{t_{n_0}}) \rVert_{\mathfrak{S}_{2}}}}^2du.$$ – We recall that the gradient at $P\in\mathscr{M}$ is the “projection” of the mean-field operator onto the tangent plane $\text{T}_{P}\mathscr{M}$, in the sens that $$\begin{array}{l} \forall\,v\in \text{T}_{P}\mathscr{M},P D_Q (1-P)\in\mathfrak{S}_1\text{\ and\ }\\ \ \ \ \ \ \ \ \ \ \ \ \ {\ensuremath{\mathrm{Tr}}}\big(P D_Q (1-P) v+(1-P)D_Q P v\big)={\ensuremath{\mathrm{Tr}}}\big( \nabla \mathcal{E}^0_{\text{BDF}} \big) \end{array}$$ For short, we write $$Q_n:=c_{t_n}\big(s(t_n)\big)\text{\ and\ }P_n:=Q_n\text{\ and\ }v_n:=\nabla \mathcal{E}^0_{\text{BDF}}(Q_n).$$ Moreover, we write $${\ensuremath{\widetilde{D}}}_{Q_n}:=D_{Q_n}-v_n\text{\ and\ } {\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}:=\chi_{(-\infty,0)}\big(D_{Q_n}^{({\ensuremath{\Lambda}})}-v_n\big).$$ We have shown that $\lim_{n\to+\infty}{\ensuremath{\lVertv_n\rVert_{\mathfrak{S}_{2}}}}=0.$ But as $v_n$ is an element of the tangent plane $\text{T}_{P_n}\mathscr{M}$, we have $$\big[ \big[v_n,P_n\big] ,P_n\big]=P_n v_n(1-P_n)+(1-P_n)v_nP_n=v_n$$ thus $$\big[ \big[ D_{Q_n}^{({\ensuremath{\Lambda}})}-v_n,P_n\big] ,P_n\big]=0.$$ Equivalently, we have $$\label{di_comm_alm} \big[{\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})},P_n \big]=(1-P_n){\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}P_n-P_n{\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}(1-P_n)=0.$$ Thus the projector $P_n$ commutes with the distorted mean-field operator ${\ensuremath{\widetilde{D}}}_{Q_n}$. We recall that $$\lim_n {\ensuremath{\lVert{\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}-D_{Q_n}^{({\ensuremath{\Lambda}})}\rVert_{\mathfrak{S}_{2}}}}=0,$$ and thus up to taking $n$ big enough, we can neglect the distortion $v_n$: all its Sobolev norms tend to zero as $n$ tends to infinity *thanks to the cut-off*. – Thanks to Lemma \[di\_infimum\_1\] we have the following energy condition: $$2m+\mathcal{O}(\alpha^2)\le F_1\le \mathcal{E}^0_{\text{BDF}}(Q_n)\le F_1+\alpha^3=2m+\mathcal{O}(\alpha^2).$$ Using the Cauchy expansion -, we have $${\ensuremath{\lVert\,|{\ensuremath{\mathcal{D}^0}}|^{1/2}({\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}-{\ensuremath{\mathcal{P}^0_-}})\rVert_{\mathfrak{S}_{2}}}}\apprle \sqrt{L\alpha}{\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}\apprle \sqrt{L\alpha}.$$ Thus we get $$\big|{\ensuremath{\lVertQ_n\rVert_{\mathfrak{S}_{2}}}}-{\ensuremath{\lVertP_n-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}\rVert_{\mathfrak{S}_{2}}}}\big|\le {\ensuremath{\lVert{\ensuremath{\mathcal{P}^0_-}}-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}\rVert_{\mathfrak{S}_{2}}}}\apprle \sqrt{L\alpha}.$$ As ${\ensuremath{\widetilde{D}}}_{Q_n}$ and $P_n$ commutes, then necessarily ${\ensuremath{\lVertP_n-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}\rVert_{\mathfrak{S}_{2}}}}^2$ is an integer equal to twice the dimension of $\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,(1-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n})$. But we know that $$m{\ensuremath{\lVertQ_n\rVert_{\mathfrak{S}_{2}}}}^2\le {\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|Q_n^2\big)\le \frac{1}{1-\alpha \tfrac{\pi}{4}}\mathcal{E}^0_{\text{BDF}}(Q_n)\le \frac{2m}{1-\alpha\frac{\pi}{4}}=2m+\mathcal{O}(\alpha).$$ Then the above dimension is lesser than $1$ and it cannot be $0$ because of the energy condition $$\mathcal{E}^0_{\text{BDF}}(Q_n)\ge F_{\mathscr{I}}\ge 2m-K\alpha^2\gg \sqrt{L\alpha}.$$ This proves the first part of Proposition \[di\_para\_method\]. We have $\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,(1-{\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n})=\mathbb{C}\psi_{a;n}$ where $\psi_{a;n}$ is unitary. It is an eigenvector for ${\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}$ with eigenvalue $\mu_n$. From the equality: $$\mathcal{E}^0_{\text{BDF}}(Q_n)=\mathcal{E}^0_{\text{BDF}}({\ensuremath{\widetilde{\boldsymbol{\pi}}}}_{-;n}-{\ensuremath{\mathcal{P}^0_-}})+2\mu_n-\frac{\alpha}{2}{\ensuremath{\displaystyle\iint}}\frac{|\psi_{a;n}\wedge {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_{a;n}(x,y)|^2}{|x-y|}dxdy,$$ we get $0<\mu_n<m$. We end the proof as follows. #### Proof of Proposition \[di\_para\_method\]: second part We follow the method of [@pos_sok]. We recall the main steps and refer the reader to this paper for further details. – The idea is simple: we must ensure that there exists a non-vanishing weak-limit and that this weak-limit is in fact a critical point. Let us say that $\psi_{a;n}$ is associated to the eigenvalue $\mu_n$. – The condition of the energy ensures that the sequence $(\psi_{a;n})_n$ does not vanish in the sense that we *do not* have the following: $$\forall\,A>0,\ \limsup_n \sup_{x\in{\ensuremath{\mathbb{R}^3}}}{\ensuremath{\displaystyle\int}}_{B(x,A)}|\psi_{a;n}|^2=0.$$ Up to translation and extraction of a subsequence, we may suppose that $(Q_n)$ (resp. $(\psi_{a;n})$) converges in the weak topology of $H^1$ to $Q_\infty\neq 0$ (resp. $\psi_{a}\neq 0$). In particular these sequences also converge in $L^2_{loc}$ and *a.e.* We recall that thanks to the cut-off and Kato’s inequality , we have $Q_n\in H^1({\ensuremath{\mathbb{R}^3}}\times {\ensuremath{\mathbb{R}^3}})$ with $${\ensuremath{\lVert|D_0|Q_n\rVert_{\mathfrak{S}_{2}}}}^2\le {\ensuremath{\widetilde{E}\left({\ensuremath{\Lambda}}\right)}}{\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}Q_n\rVert_{\mathfrak{S}_{2}}}}^2\le \frac{{\ensuremath{\widetilde{E}\left({\ensuremath{\Lambda}}\right)}}}{1-\alpha \pi/4}\sup_n \mathcal{E}^0_{\text{BDF}}(Q_n).$$ A similar estimate hold for $(\psi_{a;n})$. We also suppose that $\lim_n\mu_n=\mu_{\infty}$. – As shown in [@pos_sok], the operator $R_{Q_n}$ converges in the strong operator topology to $R_{Q_\infty}$. Thanks to the Cauchy expansion , we also have $$\text{s}.\,\lim_n\Big[\chi_{(-\infty,0)}\big(D_{Q_n}^{({\ensuremath{\Lambda}})}-\nabla \mathcal{E}^0_{\text{BDF}}(P_n)\big)-{\ensuremath{\mathcal{P}^0_-}}\Big]=\chi_{(-\infty,0)}\big(D_{Q_\infty}^{({\ensuremath{\Lambda}})}\big)-{\ensuremath{\mathcal{P}^0_-}}.$$ By that strong convergence, we also have the weak-convergence of ${\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}\psi_{a;n}$ to $D_{Q_\infty}^{({\ensuremath{\Lambda}})}\psi_a$ in $L^2$ and it follows that: $$D_{Q_\infty}^{({\ensuremath{\Lambda}})}\psi_a=\mu_{\infty}\psi_a\neq 0.$$ – The condition of the energy ensures that for $\alpha$ sufficiently small, the $\psi_{a;n}$’s are close to a scaled Pekar minimizer: for any $n$, there exists a Pekar minimizer ${\ensuremath{\widetilde{\phi}}}_n$ such that $$\lVert\psi_{a;n}-{\ensuremath{\lambda}}^{-3/2}{\ensuremath{\widetilde{\phi}}}_n({\ensuremath{\lambda}}^{-1}(\cdot))\rVert_{H^1}^2\le \alpha K\text{\ where\ }{\ensuremath{\lambda}}:=\frac{g'_1(0)^2}{\alpha m}.$$ The constant $K$ depends on the energy estimate of Proposition \[di\_para\_method\]. – Thanks to that, for all $n$, $\mu_n$ is an isolated eigenvalue of ${\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}$, uniformly in $n$: we have $$\mathbb{C}\psi_{a;n}=\mathrm{Ker}\big({\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}-\mu_n\big),$$ and $$\text{dist}\Big(\mu_n;\sigma\big({\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}\big)\backslash \{ \mu_n\}\Big)>K\alpha^2.$$ By functional calculus, we finally get the norm convergence of $(\psi_{a;n})_n$ to $\psi_a$ in $L^2$. – This proves that $$\text{s}.\,\lim_n P_n=\chi_{(-\infty,0)}\big(D_{Q_\infty}^{({\ensuremath{\Lambda}})}\big)+{\ensuremath{|\psi_a\rangle}\xspace}{\ensuremath{\langle \psi_a|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_a|}\xspace}\in \mathscr{M}_{\mathscr{I}},$$ and ends the proof. Existence of a minimizer for $E_{j_0,\pm}$ ------------------------------------------ We consider a family of almost minimizers $(P_{\eta_n})_n$ of type where $(\eta_n)_n$ is any decreasing sequence. We also consider the spectral decomposition of any $Q_n:=P_{\eta_n}-{\ensuremath{\mathcal{P}^0_-}}$. For short we write $P_n:=P_{\eta_n}$ and we replace the subscript $\eta_n$ by $n$ (for instance $\psi_n:=\psi_{\eta_n}$). Moreover, we will often write ${\ensuremath{\varepsilon}}$ instead of ${\ensuremath{\varepsilon}}(\mathbf{t})$. We study weak limits of $(Q_{n})_n$. We recall that $Q_n$ can be written as follows:$$\label{di_spec_no} \left\{ \begin{array}{l} N_{+;n}={\ensuremath{\mathcal{P}^0_+}}N_{+;n}=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{\eta_n}\text{\ and\ }N_{-;n}={\ensuremath{\mathrm{C}}}N_{+;n}{\ensuremath{\mathrm{C}}},\\ Q_n=N_{+;n}-N_{-;n}+{\ensuremath{\gamma}}_n,\ \mathrm{Ran}\,N_{\pm;n}\cap\mathrm{Ker}\,{\ensuremath{\gamma}}_n=\{ 0\}. \end{array} \right.$$ We can suppose $$\psi_n={\ensuremath{\mathcal{P}^0_+}}a_n(r)\Phi^+_{j_0,{\ensuremath{\varepsilon}}{\mathbf{t}}},\ a_n(r)\in \mathbb{S}L^2(\mathbb{R}_+,r^2dr).$$ \[di\_newton\_rem\] The functions $\psi\in\mathrm{Ran}\,N_{\pm;n}$ are “almost” radial. We recall , giving $$\label{di_radial} \begin{array}{| l} \forall\,x=r{\ensuremath{\omega}}_x\in{\ensuremath{\mathbb{R}^3}},\ |\psi(x)|\le {\ensuremath{\lVert\psi\rVert_{L^{2}}}}|s_n(r)|{\ensuremath{\lVert\Phi^{\pm}_{j_0,\pm(j_0+\tfrac{1}{2})}\rVert_{L^{\infty}}}},\\ 4|s_n(r_0)|^2:=\big|(1+\frac{g_0(|\nabla|)}{|{\ensuremath{\mathcal{D}^0}}|})a_n\big|(r_0)^2+\big|\frac{g_1(|\nabla|)}{|\nabla||{\ensuremath{\mathcal{D}^0}}|}(\partial_r a_n+{\ensuremath{\varepsilon}}\tfrac{a_n}{r})\big|(r_0)^2. \end{array}$$ In particular by Newton’s Theorem for radial function we have: $$\label{di_newton} \forall\,\psi\in \mathrm{Ran}\,N_{\pm;n},\ |\psi|^2*\frac{1}{|\cdot|}(x_0)\le K(j_0)\frac{{\ensuremath{\lVert\psi\rVert_{L^{2}}}}^2}{|x_0|}.$$ – We first prove that there is no vanishing, that is $$\exists A>0,\ \limsup_n \sup_{z\in{\ensuremath{\mathbb{R}^3}}}\underset{B(z,A)}{{\ensuremath{\displaystyle\int}}}|\psi_{n}(x)|^2dx>0.$$ Indeed, let assume this is false. Then using , it is clear that $${\ensuremath{\lVertN_{\pm;n}\rVert_{\text{Ex}}}}^2\to 0,$$ and we get $\liminf \mathcal{E}^0_{\text{BDF}}\ge 2(2j_0+1)m+\liminf \mathcal{E}^0_{\text{BDF}}({\ensuremath{\gamma}}_n)\ge 2(2j_0+1)m,$ an inequality that is false as shown in the previous section. **Thus, we have: $Q_n\rightharpoonup Q_{\infty}\neq 0$.** – As the BDF energy is sequential weakly lower continuous [@Sc], we have $$E_{j_0,{\ensuremath{\varepsilon}}}\ge \mathcal{E}_{\text{BDF}}^0(Q_{\infty}).$$ Our aim is to prove that $Q_{\infty}+{\ensuremath{\mathcal{P}^0_-}}\in\mathscr{W}_{\mathbf{t}X^{\ell_0}}$: in other words that $Q_{\infty}$ is a minimizer for $E_{j_0,{\ensuremath{\varepsilon}}}$. The spectral decomposition is not the relevant one: let us prove we can describe $P_n$ in function of the spectral spaces of the “mean-field operator” ${\ensuremath{\widetilde{D}}}_{Q_n}$: the first step is to prove below. We recall that $Q_n$ satisfies Eq. , that we have the decomposition . Using , we have for all $\psi$ in $\mathbb{S}\mathrm{Ran}\,N_{+;n}$: $$\begin{aligned} {\ensuremath{\langle {\ensuremath{\widetilde{D}}}_{Q_n} \psi\,,\,\psi\rangle}\xspace}-m&={\ensuremath{\langle (|{\ensuremath{\mathcal{D}^0}}|-m)\psi\,,\,\psi\rangle}\xspace}- {\ensuremath{\langle (\alpha R_{Q_n}+2\eta_n {\ensuremath{\Gamma}}_n )\psi\,,\,\psi\rangle}\xspace},\\ &\apprge -\alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}{\ensuremath{\lVert\,|\nabla|^{1/2}\psi\rVert_{L^{2}}}}-\\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\apprge -\alpha^2(2j_0+1).\end{aligned}$$ Thus $\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}^{n}_+\neq \{0\}.$ – Let us prove this subspace has dimension $2j_0+1$: we use the minimizing property of $Q_n$. The condition on the first derivative gives . The estimation of the energy (from above and below) obtained in the previous section gives this result. Indeed, using the Cauchy expansion and the method of [@sok], we have $$\label{di_kin_no_proof} \begin{array}{|l} \sqrt{{\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|{\ensuremath{\gamma}}_{vac;n}^2\big)}\apprle \alpha({\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}})\apprle \sqrt{L\alpha}\sqrt{\alpha j_0} ,\\ {\ensuremath{\gamma}}_{vac;n}:=\chi_{(-\infty,0)}\big({\ensuremath{\widetilde{D}}}_{Q_n}\big)-{\ensuremath{\mathcal{P}^0_-}}. \end{array}$$ The Cauchy expansion is explained in - below, we assume the above estimate for the moment (see ). We write $Q_n=N_n+{\ensuremath{\overline{{\ensuremath{\gamma}}}}}_{n}$: there holds $$\big|{\ensuremath{\lVertN_n\rVert_{\mathfrak{S}_{2}}}}^2-{\ensuremath{\lVertQ_n\rVert_{\mathfrak{S}_{2}}}}^2\big|\apprle L^{1/2}\alpha(2j_0+1).$$ As $2(2j_0+1)\le {\ensuremath{\lVertQ_n\rVert_{\mathfrak{S}_{2}}}}^2\le 2(2j_0+1)\big(1-\alpha \pi/4\big)^{-1}$, then necessarily $$\label{di_arg_en} \big|{\ensuremath{\lVertN_n\rVert_{\mathfrak{S}_{2}}}}^2-2(2j_0+1)\big|\apprle \alpha (2j_0+1),$$ and for $\alpha$ sufficiently small, the upper bound is smaller than $4$. This proves $$\text{Dim}\,\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}^{n}_+=2j_0+1.$$ \[di\_form\_nn\] There exists a unitary $\psi_{a;n}$ such that $${\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{a;n}=\mathrm{Ran}\,P_n\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}^{n}_+.$$ We can assume that $\psi_{a;n}\in\mathrm{Ker}\big( \mathrm{J}_3-j_0\big)$. Then we have $$N_n:=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{a;n}-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,{\ensuremath{\mathrm{C}}}\psi_{a;n}.$$ Equivalently writing $\psi_{w;n}:={\ensuremath{\mathrm{C}}}\psi_{a;n}$ there holds ${\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{w;n}=\mathrm{Ran}\,(1-P_n)\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}^{n}_-$. – We have: $$\label{di_spec_yes} P_n=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{a;n}-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{w;n}+{\ensuremath{\boldsymbol{\pi}}}_-^n.$$ We thus write $$Q_n=N_n+{\ensuremath{\gamma}}_{vac;n}.$$ As $ \mathrm{Ran}\,P_n$ is ${\ensuremath{\widetilde{D}}}_{Q_n}$ invariant and that ${\ensuremath{\widetilde{D}}}_{Q_n}$ is bounded (with a bound that depends on ${\ensuremath{\Lambda}}$), necessarily $${\ensuremath{\widetilde{D}}}_{Q_n}\psi_{a;n}=\mu_n\psi_{a;n},\ \mu_n\in \mathbb{R}_+.$$ As in [@pos_sok], studying the Hessian we have$$m-\mu_n+2\eta_n\ge 0.$$ – As for $\psi_n$, there is no vanishing for $(\psi_{a,n})_n$ for $\alpha$ sufficiently small: decomposing $\psi_+\in\mathrm{Ran}\,P_n$: $$\psi_+=a\psi_{a;n}+\phi,\ \phi\in \mathrm{Ran}\,P_n\cap \mathrm{Ran}\,{\ensuremath{\boldsymbol{\pi}}}_-^n,$$ we have $$|a|^2\ge\frac{1}{\mu}\big(m+{\ensuremath{\langle |{\ensuremath{\widetilde{D}}}_{Q_n}|\phi\,,\,\phi\rangle}\xspace}-K(\alpha^2j_0+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}) \big).$$ Provided that $\mu_n$ is close to $m$, the absence of vanishing for $\psi_n$ implies that of $\psi_{a;n}$. By Kato’s inequality : $$\begin{aligned} {\ensuremath{\widetilde{D}}}_{Q_n}^2&\ge |{\ensuremath{\mathcal{D}^0}}|\big(1-2\alpha{\ensuremath{\lVertR_{Q_n}|{\ensuremath{\mathcal{D}^0}}|^{-1}\rVert_{\mathcal{B}}}}-4\eta_n {\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathcal{B}}}}\big)|{\ensuremath{\mathcal{D}^0}}|\\ &\ge |{\ensuremath{\mathcal{D}^0}}|^2\big(1-\alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}-4\eta_n {\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\big)\end{aligned}$$ Thus $$\big| {\ensuremath{\widetilde{D}}}_{Q_n}\big|\ge |{\ensuremath{\mathcal{D}^0}}|\big(1-\alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}-2\eta_n {\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\big)\text{\ and\ }\mu_n\ge 1-K(\alpha^2 j_0 +\eta_n {\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}).$$ In the same way we can prove that $$|\mu_n-m|\apprle \alpha^2j_0+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}$$ So $$\psi_{a,n}\rightharpoonup\psi_{a}\neq 0.$$ – We decompose ${\ensuremath{\gamma}}_{vac;n}={\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}\in\mathscr{W}_{0}-{\ensuremath{\mathcal{P}^0_-}}$ as in : using Cauchy’s expansion -, we have $$\label{di_cauchy} {\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}=\frac{1}{2\pi}{\ensuremath{\displaystyle\int}}_{-\infty}^{+\infty}\frac{d {\ensuremath{\omega}}}{{\ensuremath{\mathcal{D}^0}}+i\omega}\big(2\eta_n {\ensuremath{\Gamma}}_n-\alpha\Pi_{\ensuremath{\Lambda}}R_{Q_n}\Pi_{\ensuremath{\Lambda}}+2\eta_n {\ensuremath{\Gamma}}_n \big)\dfrac{1}{{\ensuremath{\widetilde{D}}}_{Q_n}+i{\ensuremath{\omega}}}\Pi_{\ensuremath{\Lambda}}.$$ To justify this equality, we remark that $|{\ensuremath{\widetilde{D}}}_{Q_n}|$ is uniformly bounded from below, it follows that the r.h.s. of is well-defined provided that $\alpha\le \alpha_{j_0}$: $$\begin{aligned} \Pi_{\ensuremath{\Lambda}}R_{Q_n}\Pi_{\ensuremath{\Lambda}}^2&\apprle |\nabla|{\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}^2\apprle \alpha(2j_0+1)|\nabla|\le \alpha(2j_0+1)|{\ensuremath{\mathcal{D}^0}}|^2.\end{aligned}$$ We must ensure that $\alpha \sqrt{\alpha(2j_0+1)}$ is sufficiently small. Integrating the norm of bounded operator in , we obtain $${\ensuremath{\lVert{\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}\rVert_{\mathcal{B}}}}\apprle \alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}<1.$$ We also expand in power of $Y_n:=-\alpha \Pi_{\ensuremath{\Lambda}}R_{Q_n}\Pi_{\ensuremath{\Lambda}}+2\eta_n {\ensuremath{\Gamma}}_n$ as in $$\label{di_cauchy2} \begin{array}{rcl} {\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}&=&{\ensuremath{\displaystyle\sum}}_{j\ge 1}\alpha^j M_j[Y_n]. \end{array}$$ We have $$\label{di_estim_gn1} {\ensuremath{\lVert{\ensuremath{\gamma}}_{vac;n}\rVert_{\mathfrak{S}_{2}}}}\apprle \alpha {\ensuremath{\lVertQ_n\rVert_{\text{Ex}}}}+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\apprle \alpha^2.$$ We take the norm ${\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}(\cdot)\rVert_{\mathfrak{S}_{2}}}}$: $$\label{di_estim_kin1} {\ensuremath{\lVert|{\ensuremath{\mathcal{D}^0}}|^{1/2}{\ensuremath{\gamma}}_{vac;n}\rVert_{\mathfrak{S}_{2}}}}\apprle \sqrt{L\alpha}{\ensuremath{\lVertQ_N\rVert_{\text{Ex}}}}+\eta_n{\ensuremath{\lVert{\ensuremath{\Gamma}}_n\rVert_{\mathfrak{S}_{2}}}}\apprle L^{1/2}\alpha j_0.$$ – We thus write $$\begin{array}{rcl} {\ensuremath{\gamma}}_{vac;n}&=&{\ensuremath{\displaystyle\sum}}_{j\ge 1}{\ensuremath{\lambda}}_{j;n}q_{j;n}, \end{array}$$ where $q_{j;n}$ has the same form as the one in . Up to a subsequence, we may assume all weak convergence as in Remark : the sequence of eigenvalues $({\ensuremath{\lambda}}_{j;n})_n$ tends to $(\mu_j)_j\in\ell^2$ and each $(e_{j;n}^\star)_n$ (with $\star\in\{a,b\}$) tends to $e_{j;\infty}^\star$, $(\psi_{e;n})_n$ tends to $\psi_{e}$. We can also assume that the sequence $(\mu_n)_n$ tends to $\mu$ with $0\le \mu\le m$. For shot we write $\psi_v:={\ensuremath{\mathrm{C}}}\psi_e$. Furthermore, we write ${\ensuremath{\overline{P}}}:=Q_{\infty}+{\ensuremath{\mathcal{P}^0_-}}$ and ${\ensuremath{\overline{\boldsymbol{\pi}}}}:=\chi_{(-\infty,0)}(D_{Q_\infty}^{({\ensuremath{\Lambda}})})$. – We will prove that 1. $\big[D^{({\ensuremath{\Lambda}})}_{Q_\infty} ,{\ensuremath{\overline{P}}}\big]=0$, 2. $D_{Q_\infty}^{({\ensuremath{\Lambda}})}\psi_{a}=\mu\psi_a$ and so ${\ensuremath{\overline{\boldsymbol{\pi}}}}\psi_a=0$. Moreover $D_{Q_\infty}^{({\ensuremath{\Lambda}})}{\ensuremath{\mathrm{C}}}\psi_{a}=-\mu{\ensuremath{\mathrm{C}}}\psi_a$ and ${\ensuremath{\langle {\ensuremath{\mathrm{C}}}\psi_a\,,\,\psi_a\rangle}\xspace}=0$. 3. $$\label{di_marre_form} {\ensuremath{\overline{\boldsymbol{\pi}}}}={\ensuremath{\overline{P}}}-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}(\psi_a)+\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}({\ensuremath{\mathrm{C}}}\psi_a)=:{\ensuremath{\overline{P}}}-N.$$ These results follow from the strong convergence $$\label{di_strong} \text{s}.\,\lim_n R_{Q_n}=R_{Q_\infty}.$$ This fact enables us to show $$\begin{array}{|l} \lim_n R_{Q_n}\psi_{a;n}=R_{Q_\infty}\psi_a\text{\ in\ }L^2,\\ \text{s.\,op.}\ \lim_n\big({\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}\big)={\ensuremath{\overline{\boldsymbol{\pi}}}}-{\ensuremath{\mathcal{P}^0_-}}\text{\ in\ }\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\\ \text{w.\,op.}\ \lim_n P_n={\ensuremath{\overline{\boldsymbol{\pi}}}}-{\ensuremath{\mathcal{P}^0_-}}+\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_a-\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\psi_w\text{\ in\ }\mathcal{B}({\ensuremath{\mathfrak{H}_\Lambda}}),\\ \lim_n\psi_{a;n}=\psi_a\text{\ in\ }L^2. \end{array}$$ \[di\_assume\] We only write in this paper the proof of $$R_{Q_n}\psi_{a;n}\underset{n\to+\infty}{\overset{L^2}{\longrightarrow}}R_{Q_\infty}\psi_a\text{\ and\ }\psi_{a;n}\underset{n\to+\infty}{\overset{L^2}{\longrightarrow}}\psi_a.$$ The convergence in the weak-topology can be proved using the same method as in [@pos_sok]. For the first limit this follows from the convergence of $R_{Q_n}$ in the strong topology. For the proof of this fact and of the strong convergence of ${\ensuremath{\gamma}}_{vac;n}={\ensuremath{\boldsymbol{\pi}}}_-^n-{\ensuremath{\mathcal{P}^0_-}}$, we refer the reader to [@pos_sok]. For $R_{Q_n}$, it suffices to remark that $Q_n(x,y)$ converges in $L^2_{loc}$ and $a.e.$. To estimate the mass at infinity, we simply use the term $\tfrac{1}{|x-y|}$ in $\tfrac{Q_n(x,y)}{|x-y|}$. The strong convergence of ${\ensuremath{\gamma}}_{vac;n}$ follows from that of $R_{Q_n}$ and the Cauchy expansion . Then, assuming all these convergences, the convergence of $Q_n$ resp. $\big[ {\ensuremath{\widetilde{D}}}_{Q_n}^{({\ensuremath{\Lambda}})}; P_n\big]$ in the weak operator topology to $Q_\infty$ resp. $\big[ D_{Q_\infty}^{({\ensuremath{\Lambda}})},{\ensuremath{\overline{P}}}\big]$ are straightforward. Similarly, using , it is clear that $${\ensuremath{\widetilde{D}}}_{Q_n}\psi_{a;n}\underset{n\to+\infty}{\rightharpoonup}D_{Q_\infty}\psi_a,$$ and that $$D_{Q_\infty}^{({\ensuremath{\Lambda}})}\psi_a=\mu\psi_a.$$ To get the existence of minimizer, it suffices to prove that ${\ensuremath{\lVert\psi_a\rVert_{L^{2}}}}=1$ or equivalently $\lim_n\psi_{a;n}=\psi_a$ in $L^2$. – To prove the norm convergence of $\psi_{a;n}$ to $\psi_a$, we need a uniform upper bound of $\mu_n$, or precisely, we need the following: $$\label{di_need} \limsup_n(m-\mu_n)>0.$$ Indeed, we then get $$\label{di_lim_l2} ({\ensuremath{\mathcal{D}^0}}-\mu_n)\psi_{a;n}=\alpha R_{Q_n}\psi_{a;n}-2\eta_n {\ensuremath{\Gamma}}_n\psi_{a;n}\text{\ and\ }\psi_{a;n}=\frac{\alpha}{{\ensuremath{\mathcal{D}^0}}-\mu_n}\big(R_{Q_n}\psi_{a;n}-2\eta_n {\ensuremath{\Gamma}}_n\psi_{a;n}\big).$$ Provided that holds and that we have norm convergence of $R_{Q_n}\psi_{a;n}$ we obtain the norm convergence of $\psi_{a;n}$. – To prove the norm convergence of $R_{Q_n}\psi_{a;n}$ to $R_{Q_\infty}\psi_a$, we use the fact that the element of ${\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_{a;n}$ are “almost radial” (see in Remark \[di\_newton\_rem\]). We recall holds. In the following, we write $\delta Q_n:=Q_n-Q_\infty$ and $\delta \psi_n:=\psi_{a;n}-\psi_a$ and use Cauchy-Schwartz inequality: for any $A>0$ there hold $$\begin{aligned} {\ensuremath{\displaystyle\int}}_{|x|\ge A}\Big|{\ensuremath{\displaystyle\int}}\frac{\delta Q_n(x,y)}{|x-y|}\psi_{a;n}(y)dy\Big|^2dx&\le {\ensuremath{\lVert\delta Q_n\rVert_{\text{Ex}}}}^2 \frac{K(j_0)}{A},\\ {\ensuremath{\displaystyle\int}}_{|x|\le A}\Big|{\ensuremath{\displaystyle\int}}\frac{\delta Q_n(x,y)}{|x-y|}\psi_{a;n}(y)dy\Big|^2dx&\le \frac{2\pi}{2}{\ensuremath{\langle |\nabla| \psi_{a;n}\,,\,\psi_{a;n}\rangle}\xspace}\underset{B(0,A)\times B(0,2A)}{{\ensuremath{\displaystyle\iint}}}\frac{|\delta Q_n(x,y)|^2}{|x-y|}dxdy\\ &\ \ \ +\frac{2}{A^2}{\ensuremath{\lVert\delta Q_n\rVert_{\mathfrak{S}_{2}}}}^2{\ensuremath{\lVert\psi_{a;n}\rVert_{L^{2}}}}^2.\end{aligned}$$ Thus $$\limsup_n {\ensuremath{\lVertR[Q_n-Q_\infty]\psi_{a;n}\rVert_{L^{2}}}}=0.$$ Similarly $$\begin{aligned} {\ensuremath{\displaystyle\int}}_{|x|\ge A}\Big|\frac{Q_\infty(x,y)}{|x-y|}\delta \psi_n(y)dy\Big|^2dx&\le \frac{2}{A-\tfrac{A}{2}}{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}^2{\ensuremath{\lVert\delta\psi_n\rVert_{L^{2}}}}^2+2{\ensuremath{\lVert\delta \psi_n\rVert_{L^{2}}}}^2\frac{2}{A}{\ensuremath{\lVertQ_\infty\rVert_{\text{Ex}}}}^2,\\ {\ensuremath{\displaystyle\int}}_{|x|\le A}\Big|\frac{Q_\infty(x,y)}{|x-y|}\delta \psi_n(y)dy\Big|^2dx&\le\frac{2\pi}{2}{\ensuremath{\langle |\nabla|\delta \psi_n\,,\,\delta\psi_n\rangle}\xspace}\underset{B(0,A)\times B(0,2A)}{{\ensuremath{\displaystyle\iint}}}\frac{|\delta Q_n(x,y)|^2}{|x-y|}dxdy\\ &\ \ \ +\frac{2}{A^2}{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}^2{\ensuremath{\lVert\delta\psi_n\rVert_{L^{2}}}}^2,\end{aligned}$$ and $$\limsup_n{\ensuremath{\lVertR_{Q_\infty}(\psi_{a;n}-\psi_a)\rVert_{L^{2}}}}=0.$$ This proves that $$\lim_{n\to+\infty}{\ensuremath{\lVertR_{Q_n}\psi_{a;n}-R_{Q_\infty}\psi_a\rVert_{L^{2}}}}=0.$$ – Let us prove . We have: $$\label{di_need_to_proof} \begin{array}{rcl} 2\mu_n(2j_0+1)&=&{\ensuremath{\mathrm{Tr}}}\Big({\ensuremath{\widetilde{D}}}_{Q_n}N_n\Big),\\ &=&{\ensuremath{\mathrm{Tr}}}\Big({\ensuremath{\widetilde{D}}}_{{\ensuremath{\gamma}}_{vac;n}}N_n\Big)-\alpha {\ensuremath{\lVertN_n\rVert_{\text{Ex}}}}^2,\\ &=&\mathcal{E}^0_{\text{BDF}}(Q_n)-\mathcal{E}^0_{\text{BDF}}({\ensuremath{\gamma}}_{vac;n})-\frac{\alpha}{2}{\ensuremath{\lVertN_n\rVert_{\text{Ex}}}}^2,\\ &<&2m(2j_0+1)-K(j_0)\alpha^2. \end{array}$$ This upper bound holds provided that $\alpha\le \alpha_{j_0}$ thanks to the upper bound of $E_{j_0,{\ensuremath{\varepsilon}}}$ obtained in the previous section. Lower bound of $E_{j_0,\pm}$ {#di_low_bound} ---------------------------- Our aim is to prove the estimate of Proposition \[di\_est\]. We consider the minimizer $Q_{\infty}=N+{\ensuremath{\gamma}}_{vac}$ found in the previous subsection. It satisfies Eq. where $$\label{di_marre_form_re} {\ensuremath{\overline{P}}}={\ensuremath{\mathcal{P}^0_-}}+Q_\infty\text{\ and\ }{\ensuremath{\gamma}}_{vac}=\chi_{(-\infty,0)}(D_{Q_infty}^{({\ensuremath{\Lambda}})})-{\ensuremath{\mathcal{P}^0_-}}.$$ – The proof is the same as that in [@sok; @pos_sok] and relies on estimates on the Sobolev norms ${\ensuremath{\lVert\,|\nabla|^s N_+\rVert_{\mathfrak{S}_{2}}}}$ where we write $$\label{di_eq_recall} N_+:=\text{Proj}\,{\ensuremath{\Phi_{\mathrm{SU}}}}\,\psi_a=\mathrm{Ker}\,(D_{Q_\infty}^{({\ensuremath{\Lambda}})}-\mu).$$ Using , we get $$\begin{aligned} {\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|^2N_+\big)&= 2(2j_0+1)\mu^2+2\alpha \mu{\ensuremath{\mathrm{Tr}}}\big(R_{Q_\infty}N_+\big)+\alpha^2{\ensuremath{\mathrm{Tr}}}\big(R_{Q_\infty}^2N_+\big),\\ &\le 2(2j_0+1)\mu^2+4\alpha \mu{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}{\ensuremath{\lVert\nabla N_+\rVert_{\mathfrak{S}_{2}}}}+4\alpha^2{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}^2{\ensuremath{\lVert\nabla N_+\rVert_{\mathfrak{S}_{2}}}}^2\end{aligned}$$ and provided that $\alpha\le \alpha_{j_0}$, we get $${\ensuremath{\mathrm{Tr}}}\big((-\Delta)N_+\big)\apprle \frac{\alpha^2(2j_0+1)}{1-4\alpha^2(2j_0+1)-2{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}{\ensuremath{\lVertg_0''\rVert_{L^{\infty}}}}}.$$ We have used Hardy’s inequality: $$\label{di_hardy} \dfrac{1}{4|\cdot|^2}\le -\Delta\text{\ in\ }{\ensuremath{\mathbb{R}^3}}.$$ We recall that $$0\le {\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}-1\apprle \alpha{\ensuremath{\log(\Lambda)}}\text{\ and\ } {\ensuremath{\lVertg_0''\rVert_{L^{\infty}}}}\apprle \alpha.$$ See (or [@sok Appendix A] for more details). Thus for sufficiently small $\alpha$, we have $$\label{di_wf_to_scale} \forall\,\psi\in\mathbb{S}\mathrm{Ran}\,N_+,\ {\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2\apprle \frac{\alpha^2}{1-4\alpha^2(2j_0+1)-2{\ensuremath{\lVertg_0\rVert_{L^{\infty}}}}{\ensuremath{\lVertg_0''\rVert_{L^{\infty}}}}}\apprle \alpha^2.$$ – By *bootstrap argument*, we can estimate ${\ensuremath{\lVert\,\Delta N_+\rVert_{\mathfrak{S}_{2}}}}$. We have: $$\label{di_marre_boot} \forall\,\psi\in\mathbb{S}\mathrm{Ran}\,N_+,\ {\ensuremath{\lVert\,|\nabla|^{3/2}\psi\rVert_{L^{2}}}}^2\apprle \alpha^{3}\sqrt{2j_0+1}\text{\ and\ }{\ensuremath{\lVert\Delta \psi\rVert_{L^{2}}}}\apprle \alpha^4(2j_0+1)^{3/2}.$$ We prove this result below. Furthermore, using the Cauchy expansion and , we get $$\begin{array}{| rcl} {\ensuremath{\lVert\,|{\ensuremath{\mathcal{D}^0}}|^{1/2}{\ensuremath{\gamma}}_{vac}\rVert_{\mathfrak{S}_{2}}}}&\apprle&\alpha {\ensuremath{\lVert\nabla N\rVert_{\mathfrak{S}_{2}}}}+\sqrt{L\alpha}{\ensuremath{\lVert{\ensuremath{\gamma}}_{vac}\rVert_{\text{Ex}}}}+\alpha^2{\ensuremath{\lVertQ_\infty\rVert_{\text{Ex}}}}^2\big({\ensuremath{\lVert\nabla N\rVert_{\mathfrak{S}_{2}}}}+{\ensuremath{\lVert{\ensuremath{\gamma}}_{vac}\rVert_{\text{Ex}}}} \big), \end{array}$$ hence $$\label{di_marre_gvac} {\ensuremath{\lVert\,|{\ensuremath{\mathcal{D}^0}}|^{1/2}{\ensuremath{\gamma}}_{vac}\rVert_{\mathfrak{S}_{2}}}}\apprle \alpha^2\sqrt{2j_0+1}.$$ Now, if we assume -, then we get $$\text{For\ } \alpha\le \alpha_{j_0},\ \mathcal{E}^0_{\text{BDF}}\big(Q_\infty\big)=2m(2j_0+1)+\frac{\alpha^2m}{g'_1(0)^2}E_{\mathbf{t}X^{\ell_0}}^{nr}+\mathcal{O}\big(\alpha^3 K(j_0)\big).$$ We do not prove this fact: the method is the same as in [@sok; @pos_sok] (in the proof of the lower bound of $E^0_{\text{BDF}}(1)$ resp. $E_{1,1}$). We just recall how we get . ##### Proof of We scale the wave functions of by ${\ensuremath{\lambda}}:=\frac{g'_1(0)^2}{\alpha m}$: $$\forall\,x\in {\ensuremath{\mathbb{R}^3}},\ U_{{\ensuremath{\lambda}}}\psi(x)={\ensuremath{\underline{\psi}}}(x):={\ensuremath{\lambda}}^{3/2}\psi({\ensuremath{\lambda}}x),$$ and we split $\psi$ (resp. ${\ensuremath{\underline{\psi}}}$) into the upper spinor ${\ensuremath{\varphi}}$ (resp. ${\ensuremath{\underline{{\ensuremath{\varphi}}}}}$) and the lower spinor $\chi$ (resp. ${\ensuremath{\underline{\chi}}}$). Thanks to , we have $$\alpha^{-2}(m-\mu)=:\alpha^{-2}\delta m\ge K(j_0)>0$$ provided that $\alpha$ is sufficiently small $(\alpha\le \alpha_{j_0})$. We write $$\forall\,Q_0\in\mathfrak{S}_2,\ {\ensuremath{\underline{Q_0}}}:=U_{{\ensuremath{\lambda}}} Q_{0}U_{{\ensuremath{\lambda}}}^{-1}=U_{{\ensuremath{\lambda}}} Q_{0}U_{{\ensuremath{\lambda}}^{-1}}.$$ For all $\psi$ in $\mathbb{S}\mathrm{Ran}{\ensuremath{\underline{N_+}}}$ we have $$\label{di_eq_scale} \left\{\begin{array}{rcl} {\ensuremath{\lambda}}^2\delta m{\ensuremath{\underline{{\ensuremath{\varphi}}}}}&=&i{\ensuremath{\lambda}}\boldsymbol{\sigma}\cdot \nabla {\ensuremath{\underline{\chi}}}+\alpha {\ensuremath{\lambda}}\big(R_{{\ensuremath{\underline{Q}}}_\infty} {\ensuremath{\underline{\psi}}}\big)_{\uparrow},\\ {\ensuremath{\underline{\chi}}}&=&\frac{-i{\ensuremath{\lambda}}\boldsymbol{\sigma}\cdot \nabla {\ensuremath{\underline{{\ensuremath{\varphi}}}}}}{{\ensuremath{\lambda}}(m+\mu)}-\tfrac{\alpha}{{\ensuremath{\lambda}}} \big(R_{{\ensuremath{\underline{Q}}}_\infty} {\ensuremath{\underline{\psi}}}\big)_{\downarrow}. \end{array} \right.$$ – We recall $$\label{di_marre_Rtrois} \forall\,Q_0\in\mathfrak{S}_2,\ \lVert \big[\nabla,R_{Q_0}\big]\tfrac{1}{|\nabla|^{1/2}} \rVert_{\mathcal{B}}^2\apprle {\ensuremath{\displaystyle\iint}}|p-q|^2|p+q||{\ensuremath{\widehat{Q_0}}}(p,q)|^2dpdq.$$ This result was previously proved in [@pos_sok] and follows from the fact that a (scalar) Fourier multiplier $F(\mathbf{p}-\mathbf{q})=F(-i\nabla_x+i\nabla_y)$ commutes with the operator $R[\cdot]:Q(x,y)\mapsto \tfrac{Q(x,y)}{|x-y|}$. Then it suffices to use Hardy’s inequality : $${\ensuremath{\lVert\big[\nabla,R_{{\ensuremath{\underline{Q_\infty}}}}\big]{\ensuremath{\underline{\psi}}}\rVert_{L^{2}}}}^2\apprle {\ensuremath{\lambda}}^2{\ensuremath{\displaystyle\iint}}|p-q|^2|{\ensuremath{\widehat{Q}}}_{\infty}(p,q)|^2dpdq\times {\ensuremath{\lVert\nabla {\ensuremath{\underline{\psi}}}\rVert_{L^{2}}}}^2.$$ By Hardy’s inequality and , the following holds: $$\label{di_est_un_scale} \begin{array}{| rcl} {\ensuremath{\lVert{\ensuremath{\underline{\chi}}}\rVert_{\mathfrak{S}_{2}}}}^2&\le& \frac{2}{4{\ensuremath{\lambda}}^2m^2}{\ensuremath{\lVert\nabla {\ensuremath{\underline{{\ensuremath{\varphi}}}}}\rVert_{\mathfrak{S}_{2}}}}^2+2\alpha^2{\ensuremath{\lVertR_{{\ensuremath{\underline{Q_\infty}}}}{\ensuremath{\underline{\psi}}}\rVert_{\mathfrak{S}_{2}}}}^2\apprle \alpha^2,\\ {\ensuremath{\lVert\nabla {\ensuremath{\underline{\chi}}}\rVert_{\mathfrak{S}_{2}}}}^2&\le&2({\ensuremath{\lambda}}\delta m)^2+2\alpha^2{\ensuremath{\lVertR_{{\ensuremath{\underline{Q_\infty}}}} {\ensuremath{\underline{\psi}}}\rVert_{\mathfrak{S}_{2}}}}^2\apprle \frac{(\delta m)^2}{\alpha^2}+\alpha^2(2j_0+1),\\ {\ensuremath{\lVert\Delta {\ensuremath{\underline{{\ensuremath{\varphi}}}}}\rVert_{\mathfrak{S}_{2}}}}^2&\le&2{\ensuremath{\lambda}}^2 m{\ensuremath{\lVert\nabla {\ensuremath{\underline{\chi}}}\rVert_{L^{2}}}}^2+2\alpha^2({\ensuremath{\lVert\big[\nabla,R_{{\ensuremath{\underline{Q_\infty}}}}\big]{\ensuremath{\underline{\psi}}}\rVert_{L^{2}}}}+{\ensuremath{\lVertR_{Q_\infty}\rVert_{L^{2}}}}\nabla {\ensuremath{\underline{\psi}}})^2 \\ &\apprle& \frac{(\delta m)^2}{\alpha^4}+(2j_0+1)+\alpha^2(2j_0+1)^{3/2}, \\ {\ensuremath{\lVert\Delta {\ensuremath{\underline{\chi}}}\rVert_{\mathfrak{S}_{2}}}}^2&\le& 2{\ensuremath{\lambda}}^2(\delta m)^2{\ensuremath{\lVert\nabla {\ensuremath{\underline{{\ensuremath{\varphi}}}}}\rVert_{L^{2}}}}+2\alpha^2({\ensuremath{\lVert\big[\nabla,R_{{\ensuremath{\underline{Q_\infty}}}}\big]{\ensuremath{\underline{\psi}}}\rVert_{L^{2}}}}+{\ensuremath{\lVertR_{Q_\infty}\rVert_{L^{2}}}}\nabla {\ensuremath{\underline{\psi}}})^2 \\ &\apprle& \frac{(\delta m)^2}{\alpha^2}+(2j_0+1)+\alpha^2(2j_0+1)^{3/2}. \end{array}$$ – There remains to estimate $${\ensuremath{\displaystyle\iint}}|p-q|^2|{\ensuremath{\widehat{Q_0}}}(p,q)|^2dpdq,\ \text{for\ }Q_0=N\text{\ and\ }{\ensuremath{\gamma}}_{vac}.$$ For $Q_0=N$, we just have to estimate ${\ensuremath{\mathrm{Tr}}}\big(|\nabla|^2 N_+\big)$. The case $Q_0={\ensuremath{\gamma}}_{vac}$ is dealt with as in [@sok; @sokd]: by a *fixed-point* argument (valid for $\alpha\le \alpha_{j_0}$), we prove that $$\left\{{\ensuremath{\displaystyle\iint}}|p-q|^2|{\ensuremath{\widehat{{\ensuremath{\gamma}}_{vac}}}}(p,q)|^2dpdq\right\}^{1/2}\apprle \alpha\min\big({\ensuremath{\lVert\Delta N\rVert_{\mathfrak{S}_{2}}}},{\ensuremath{\lVert\,|\nabla|^{3/2}N\rVert_{\mathfrak{S}_{2}}}}\big).$$ Now, we can prove that $${\ensuremath{\mathrm{Tr}}}\big(|\nabla|^{3}N_+\big)\apprle \alpha^{5/2}(2j_0+1)^{3/2}.$$ For a unitary $\psi$ in $\mathrm{Ran}\,N_+$, there holds $$\label{di_sobtrois} \begin{array}{rcl} {\ensuremath{\lVert\,|\nabla|^{1/2}{\ensuremath{\mathcal{D}^0}}\psi\rVert_{L^{2}}}}^2&\le& \mu^2{\ensuremath{\langle |\nabla|\psi\,,\,\psi\rangle}\xspace}+\alpha K {\ensuremath{\lVert\,|\nabla|^{1/2}\psi\rVert_{L^{2}}}}{\ensuremath{\lVertR_{Q_\infty}\psi\rVert_{L^{2}}}}\\ &&\ \ \ +\alpha^2\big({\ensuremath{\lVert[R_{Q_\infty},|\nabla|^{1/2}] \psi\rVert_{L^{2}}}}+2{\ensuremath{\lVertQ_\infty\rVert_{\mathfrak{S}_{2}}}}{\ensuremath{\lVert\,|\nabla|^{3/2}\rVert_{L^{2}}}}\big)^2. \end{array}$$ Similarly, in Fourier space we have: $$\Big|\mathscr{F}\big([R_{Q_\infty},|\nabla|^{1/2}];p,q\big) \Big|\apprle |p-q|^{1/2}|{\ensuremath{\widehat{R}}}_{Q_\infty}(p,q)|,$$ and by Hardy’s inequality $${\ensuremath{\lVert[R_{Q_\infty},|\nabla|^{1/2}] \psi\rVert_{L^{2}}}}^2\apprle {\ensuremath{\displaystyle\iint}}|p-q| |{\ensuremath{\widehat{Q_\infty}}}(p,q)|^2dpdq{\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2\apprle {\ensuremath{\mathrm{Tr}}}\big(|\nabla| Q_\infty^2\big){\ensuremath{\lVert\nabla \psi\rVert_{L^{2}}}}^2.$$ Substituting in , we get $${\ensuremath{\langle |\nabla|^3\psi\,,\,\psi\rangle}\xspace}\apprle \alpha^{5/2}\sqrt{2j_0+1},\text{\ hence\ }{\ensuremath{\mathrm{Tr}}}\big(|\nabla|^3N_+ \big)\apprle \alpha^{5/2}(2j_0+1)^{3/2}.$$ Proof of Lemmas \[di\_infimum\_1\] and \[di\_non\_triv\] {#di_fait_ch} -------------------------------------------------------- ### Proof of Lemma \[di\_infimum\_1\] We consider a trial state $P_\psi\in\mathscr{M}_{\mathscr{I}}^1$: $$Q_\psi:=P_\psi-{\ensuremath{\mathcal{P}^0_-}}={\ensuremath{|\psi\rangle}\xspace}{\ensuremath{\langle \psi|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi|}\xspace},\ {\ensuremath{\mathcal{P}^0_+}}\psi=\psi\in\mathbb{S}\,{\ensuremath{\mathfrak{H}_\Lambda}}.$$ Its BDF energy is $$\begin{aligned} \mathcal{E}^0_{\text{BDF}}(Q_\psi)&=2{\ensuremath{\langle |{\ensuremath{\mathcal{D}^0}}|\psi\,,\,\psi\rangle}\xspace}-\frac{\alpha}{2}{\ensuremath{\displaystyle\iint}}\frac{|\psi\wedge {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi(x,y)|^2}{|x-y|}dxdy\\ &\ge 2m+2{\ensuremath{\langle \big(|{\ensuremath{\mathcal{D}^0}}|-m\big)\psi\,,\,\psi\rangle}\xspace}-\alpha D\big(|\psi|^2,\psi^2\big)=:2m+\mathcal{G}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}(\psi).\end{aligned}$$ We recall the following $$\begin{aligned} |{\ensuremath{\mathcal{D}^0}}|-m&=\frac{1}{|{\ensuremath{\mathcal{D}^0}}|+m}\big((g_0(-i\nabla)-m)(g_0(-i\nabla)+m)+g_1(-i\nabla)^2\big).\end{aligned}$$ Thanks to Estimates and Kato’s inequality , we have$$\mathcal{G}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}(\psi)\le (1-K\alpha){\ensuremath{\langle \tfrac{-\Delta}{2|{\ensuremath{\mathcal{D}^0}}|} \psi\,,\,\psi\rangle}\xspace}-\alpha\frac{\pi}{4}{\ensuremath{\langle |\nabla|\psi\,,\,\psi\rangle}\xspace}$$ We split $\psi$ into two with respect to the frequency cut-off $\Pi_{\alpha K_0}$: we get $$\psi=\Pi_{\alpha K_0}\psi+\psi_2=\psi_1+\psi_2.$$ The constant $K_0$ is chosen such that $$\frac{\alpha^2 K_0^2}{2{\ensuremath{\widetilde{E}\left(\alpha K_0\right)}}}\apprge \alpha \pi \alpha K_0.$$ Then we have $$\begin{aligned} D\big(|\psi|^2,|\psi|^2\big)&=D\big(|\psi_1|^2,|\psi_1|^2\big)+\mathcal{O}\big({\ensuremath{\langle |\nabla|\psi_2\,,\,\psi_2\rangle}\xspace}+{\ensuremath{\lVert|\psi_1|^2\rVert_{\mathcal{C}}}}{\ensuremath{\lVert\,|\nabla|^{1/2}\psi_2\rVert_{L^{2}}}}\big)\\ &=D\big(|\psi_1|^2,|\psi_1|^2\big)+\mathcal{O}\big({\ensuremath{\langle |\nabla|\psi_2\,,\,\psi_2\rangle}\xspace}+\sqrt{\alpha}{\ensuremath{\lVert\,|\nabla|^{1/2}\psi_2\rVert_{L^{2}}}}\big),\end{aligned}$$ where we recall that ${\ensuremath{\lVert\rho\rVert_{\mathcal{C}}}}^2=D(\rho,\rho)$. This gives $$\begin{array}{rcl} \tfrac{1}{2}\mathcal{G}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}(\psi)&=&{\ensuremath{\langle \frac{g_1(-i\nabla)^2}{|{\ensuremath{\mathcal{D}^0}}|+m}\psi_1\,,\,\psi_1\rangle}\xspace}-\alpha\frac{\pi}{2}D\big(|\psi_1|^2,|\psi_1|^2\big)\\ &&\ \ \ +K{\ensuremath{\langle \frac{g_1^2(-i\nabla)}{|{\ensuremath{\mathcal{D}^0}}|} \psi_2\,,\,\psi_2\rangle}\xspace}+\mathcal{O}(\alpha^3),\\ &\ge&\frac{\alpha^2 g'_1(0)^2}{2m}{\ensuremath{\lVert\nabla\psi_1\rVert_{L^{2}}}}^2-\frac{\alpha}{2}D\big(|\psi_1|^2,|\psi_1|^2\big)+\mathcal{O}(\alpha^3),\\ &\ge&\frac{\alpha^2 m}{2g'_1(0)^2}E_{\text{PT}}(1)+\mathcal{O}(\alpha^3). \end{array}$$ We have obtained a lower bound. Let us prove that it is attained up to an error $\mathcal{O}(\alpha^3)$. That is let us prove there exists a unitary $\psi_0\in\mathrm{Ran}{\ensuremath{\mathcal{P}^0_+}}$ such that $$\label{di_di_test} \begin{array}{rcl} \mathcal{E}^0_{\text{BDF}}(Q_{\psi_0})-2m&=&\mathcal{G}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}(\psi_0)+\mathcal{O}(\alpha^3)\\ &=& \frac{\alpha^2 m}{g'_1(0)^2}E_{\text{PT}}(1)+\mathcal{O}(\alpha^3). \end{array}$$ As in [@pos_sok], we consider the unique positive radially symetric Pekar minimizer $\phi_{\text{PT}}$ in $L^2({\ensuremath{\mathbb{R}^3}},\mathbb{C})$. We form $$\label{di_test0} \phi_1:=\begin{pmatrix}\phi_{\text{PT}}\\ 0\\0\\0 \end{pmatrix}\in L^2({\ensuremath{\mathbb{R}^3}},{\ensuremath{\mathbb{C}^4}}),$$ which is a Pekar minimizer in the space of spinors. We scale this wave function by ${\ensuremath{\lambda}}^{-1}:=\frac{\alpha m}{g'_1(0)^2}$: $$\label{di_test1} \forall\,x\in{\ensuremath{\mathbb{R}^3}},\ \phi_{{\ensuremath{\lambda}}^{-1}}(x):={\ensuremath{\lambda}}^{-3/2}\phi_1({\ensuremath{\lambda}}^{-1}x).$$ To get a proper $\psi_0\in\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$, we form $$\label{di_test2} \psi_0:=\frac{1}{{\ensuremath{\lVert{\ensuremath{\mathcal{P}^0_+}}\phi_{{\ensuremath{\lambda}}^{-1}}\rVert_{L^{2}}}}}{\ensuremath{\mathcal{P}^0_+}}\phi_{{\ensuremath{\lambda}}^{-1}}.$$ Our trial state is: $$\label{di_test3} Q_0:={\ensuremath{|\psi_0\rangle}\xspace}{\ensuremath{\langle \psi_0|}\xspace}-{\ensuremath{|{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_0\rangle}\xspace}{\ensuremath{\langle {\ensuremath{\mathrm{I}_{\mathrm{s}}}}\psi_0|}\xspace}.$$ We do not compute its energy: the method is as in [@pos_sok] (except that instead of ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$, the operator ${\ensuremath{\mathrm{C}}}$ is considered in [@pos_sok], but that does not change anything). Eventually we refer the reader to the proof of the upper bound of $E_{\mathbf{t}X^{\ell_0}}$ above in Section \[di\_subscritic\] for the ideas. ### Proof of Lemma \[di\_non\_triv\] We remark the following fact. \[di\_lem\_orientation\] Let $\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}\subset {\ensuremath{\mathfrak{H}_\Lambda}}$ be the set $$\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}=\big\{ f\in {\ensuremath{\mathfrak{H}_\Lambda}},\ {\ensuremath{\lVertf\rVert_{L^{2}}}}=1,\ {\ensuremath{\langle f\,,\,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\rangle}\xspace}=0\big\}=\big\{ f\in {\ensuremath{\mathfrak{H}_\Lambda}},\ {\ensuremath{\lVertf\rVert_{L^{2}}}}=1,\ \mathfrak{Im}{\ensuremath{\langle {\ensuremath{\mathcal{P}^0_-}}f\,,\,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}{\ensuremath{\mathcal{P}^0_+}}f\rangle}\xspace}=0\big\}.$$ There exists a smooth angle operator $\mathcal{A}:\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}\to \mathbb{R}/ \pi \mathbb{Z}$. For two $\mathbb{C}$-colinear wave functions $f_1,f_2$ in $\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ we have $\mathcal{A}(f_1)=\mathcal{A}(f_2)$. Furthermore we have $\mathcal{A}^{-1}(0)=\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_-}}$ and $\mathcal{A}^{-1}(\tfrac{\pi}{2})=\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$. #### Proof: Let $f$ be in $\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$: the space $\text{Span}_{\mathbb{C}}(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f)$ is spanned by the eigenvectors $g_{-}:=\tfrac{f+i{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f}{{\ensuremath{\lVertf+i{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\rVert_{L^{2}}}}}$ and $g_+:=\tfrac{f-i{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f}{{\ensuremath{\lVertf-i{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\rVert_{L^{2}}}}}$. We have $$\text{Span}_{\mathbb{C}}(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f)=\text{Span}({\ensuremath{\mathcal{P}^0_-}}g_{\pm},{\ensuremath{\mathcal{P}^0_+}}g_{\pm}).$$ It follows that $\mathcal{P}^0_{\pm} f\parallel \mathcal{P}^0_{\pm} g_+$ and ${\ensuremath{\mathcal{P}^0_-}}f \parallel {\ensuremath{\mathrm{I}_{\mathrm{s}}}}{\ensuremath{\mathcal{P}^0_+}}f$. As $f\in \mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$, for ${\ensuremath{\varepsilon}}\in\{ +,-\}$ with $\mathcal{P}^0_{{\ensuremath{\varepsilon}}} f\neq 0$, we have $$\mathcal{P}^0_{-{\ensuremath{\varepsilon}}} f\in \text{Span}_{\mathbb{R}}(\mathcal{P}^0_{{\ensuremath{\varepsilon}}} f).$$ Thus we have with $$\label{di_di_cond} \mathrm{Span}_{\mathbb{R}}(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f)=\mathrm{Span}_{\mathbb{R}}(e_-,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-),\ e_-\in\text{Ran}\,{\ensuremath{\mathcal{P}^0_-}}\text{\ and\ }{\ensuremath{\lVerte_-\rVert_{L^{2}}}}=1.$$ Indeed if ${\ensuremath{\mathcal{P}^0_-}}f\neq 0$ we can choose $e_-:=\tfrac{{\ensuremath{\mathcal{P}^0_-}}f}{{\ensuremath{\lVert{\ensuremath{\mathcal{P}^0_-}}f\rVert_{L^{2}}}}}$, else we can choose $e_-:={\ensuremath{\mathrm{I}_{\mathrm{s}}}}\tfrac{{\ensuremath{\mathcal{P}^0_+}}f}{{\ensuremath{\lVert{\ensuremath{\mathcal{P}^0_+}}f\rVert_{L^{2}}}}}$. Then we decompose $f$ w.r.t. the basis $(e_-,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-)$ and there exists $\theta\in \mathbb{R}/(2\pi \mathbb{Z})$ with $f=\cos(\theta) e_-+\sin(\theta) {\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-$. In fact the function $f\mapsto (e_-,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-)$ that maps $f$ to a basis is bi-valued: if $(e_-,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-)$ is a possibility, then $(-e_-,-{\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-)$ is another possibility. It follows that the angle $\theta$ is defined up to $\pi$: we thus obtain a function $$\mathcal{A}:\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}\to \mathbb{R}/ \pi \mathbb{Z}.$$ The smoothness of $\mathcal{A}$ is straightforward. The end of the proof is also clear. We use the angle operator to get a mountain pass argument: see Lemma \[di\_mount\] below. We use and Theorem \[di\_structure\] and Proposition \[di\_chasym\]. Let $\mathscr{U}\subset \mathscr{M}_{\mathscr{I}}$ be the *open* subset $$\mathscr{U}\subset \mathscr{M}_{\mathscr{I}}:=\Big\{P=Q+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{M}_{\mathscr{I}},\ \text{dim}\,\mathrm{Ker}(Q-{\ensuremath{\lVertQ\rVert_{\mathcal{B}}}})=1 \Big\}.$$ For all $P=Q+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{U}$, the eigenspace $\mathrm{Ker}(Q-{\ensuremath{\lVertQ\rVert_{\mathcal{B}}}})$ is spanned by a unitary vector $f_0$. By ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$-symmetry, we have $${\ensuremath{\mathrm{I}_{\mathrm{s}}}}\mathrm{Ker}(Q-{\ensuremath{\lVertQ\rVert_{\mathcal{B}}}})=\mathrm{Ker}(Q+{\ensuremath{\lVertQ\rVert_{\mathcal{B}}}}),$$ and we have ${\ensuremath{\langle f_0\,,\,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f_0\rangle}\xspace}=0.$ By Proposition \[di\_chasym\], the plane $\text{Span}_{\mathbb{C}}\,(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f)$ is spanned by $f_-\in\text{Ran}\,P$ and $f_+\in\text{Ran}\,(1-P)$. By ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$-symmetry, we have ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}f_-\in\mathbb{R}f_+$. In other words: **the wave function $f_-$ is in $\mathbb{S}_{{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$**. Let $Q+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{U}\subset \mathscr{M}_{\mathscr{I}}$ and $f_-$ as above. We define the smooth function $\mathcal{A}_U$ as follows: $$\mathcal{A}_U:Q+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{U}\subset \mathscr{M}_{\mathscr{I}}\mapsto \mathcal{A}(f_-).$$ It is clear it does not depend on the choice of $f_-$ but is a function of $\mathbb{C}f_-$. Furthermore, we have $$\forall\,P\in \mathscr{U},\ \nabla \mathcal{A}_U(P)\neq 0$$ The following Lemma is an application of classical results in geometry. \[di\_mount\] Let $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ be the subset $$\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}:=\big\{Q+{\ensuremath{\mathcal{P}^0_-}}\in\mathscr{U},\ {\ensuremath{\lVertQ\rVert_{\mathcal{B}}}}=1\big\}=\mathcal{A}_U^{-1}\big(\big\{\frac{\pi}{2}\big\}\big),$$ in other words the set of projectors in $\mathscr{U}$ whose range intersects nontrivially $\mathrm{Ran}\,{\ensuremath{\mathcal{P}^0_+}}$. For any differentiable function $c:(-{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}})\to \mathscr{M}_{\mathscr{I}}$ such that ${\ensuremath{\varepsilon}}>0$, $c(0)\in \mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ and $${\ensuremath{\mathrm{Tr}}}\big(\nabla \mathcal{A}_U(c(0))^* \frac{d}{ds}c(0)\big)\neq 0,$$ the following holds: any sufficiently small smooth perturbation $$c+\delta c:(-{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}})\to \mathscr{M}_{\mathscr{I}},$$ in the norm $$\lVert \widetilde{c}\rVert:=\sup_{s\in (-{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}})}{\ensuremath{\lVert\widetilde{c}(s)-{\ensuremath{\mathcal{P}^0_-}}\rVert_{\mathfrak{S}_{2}}}}+\sup_{s\in (-{\ensuremath{\varepsilon}},{\ensuremath{\varepsilon}})}{\ensuremath{\lVert\tfrac{d}{ds}\widetilde{c}(s)\rVert_{\mathfrak{S}_{2}}}}$$ still intersects $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ at some $s(\delta c)$. – Let us now prove Lemma \[di\_non\_triv\]. We recall that we have defined a loop $c_\psi=c_0$ that crosses $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ at $s=\tfrac{1}{2}$ and we can easily check that ${\ensuremath{\mathrm{Tr}}}\big(\mathcal{A}_U(c(2^{-1}))^*\frac{d}{ds}c(2^{-1})\big)=1\neq 0.$ Furthermore we have defined the family $(c_t)_{t\ge 0}$ by $c_t:=\Phi_{\text{BDF};t}(c_\psi)$ where $\Phi_{\text{BDF};t}$ is the gradient flow of the BDF energy. – By Lemma \[di\_mount\], the loop $c_t$ still intersects $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ for sufficiently small $t$. We must ensure that this fact holds for all $t\ge 0$ to end the proof. We use a continuation principle and set $$t_{\infty}:=\sup\Big\{t\ge 0,\ \forall\,0\le \tau\le t,\exists s_0\in[0,1] c_\tau\text{\ crosses\ }\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}\text{\ at\ }s=s_0\Big\}.$$ We also define for all $0\le \tau<t_{\infty}$: $$\begin{array}{| rcl} s_{-}(\tau)&=&\sup\{ s\in[0,1],\ \forall\,s'\le s,\ {\ensuremath{\lVertc_\tau(s')\rVert_{\mathcal{B}}}}<1\}>0,\\ s_{+}(\tau)&=&\inf\{ s\in[0,1],\ \forall\,s'\ge s,\ {\ensuremath{\lVertc_\tau(s')\rVert_{\mathcal{B}}}}<1\}<1. \end{array}$$ – We assume that $t_\infty<+\infty$ and prove this implies a contradiction. The initial loop $c_0$ induces $$\mathcal{L}_0:s\in [0,1]\mapsto \mathcal{A}_U(c_0(s))=\pi s\in \mathbb{T},$$ and we notice that $\mathcal{L}_0$ has a non-trivial homotopy. Thus, at least for $\tau$ close to $0$, the following holds. 1. There exist $0<\eta_\tau,\eta_\tau'\ll 1$ such that $$\label{di_cont_0} \mathcal{A}_U\big[c_\tau\big((s_-(\tau)-\eta_\tau,s_-(\tau)) \big)\big]\cap (\tfrac{\pi}{2},\tfrac{\pi}{2}+\eta_\tau')=\varnothing.$$ 2. There exist $0<\eta_\tau,\eta_\tau'\ll 1$ such that $$\label{di_cont_1} \mathcal{A}_U\big[c_\tau\big((s_+(\tau),s_+(\tau)+\eta_\tau) \big)\big]\cap (\tfrac{\pi}{2}-\eta_\tau',\tfrac{\pi}{2})=\varnothing.$$ The functions $\tau\ge 0\mapsto s_{\pm}(\tau)$ are well-defined and continuous in a neighbourhood of $0$ with $ s_-(0)=s_+(0)=\tfrac{1}{2}.$ – We prove that by continuity in $\tau$ we have $$\label{di_contz} \forall\,s\in[0,1],\ {\ensuremath{\lVertc_{\tau}(s)\rVert_{\mathcal{B}}}}=1\Rightarrow c_{\tau}(s)+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$$ and in particular $$\label{di_cont} c_\tau(s_\pm(\tau))\in \mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}-{\ensuremath{\mathcal{P}^0_-}}.$$ If not, this implies that as $\tau$ increases, the second highest eigenvalue of $c_\tau(s_0)$ also increases to reach $1$ where becomes false, at some $(\tau_0,s_0)$. This cannot occurs because of the energy condition: if this was true, we would have by Kato’s inequality $$\mathcal{E}^0_{\text{BDF}}\big(c_{\tau_0}(s_0)\big)\ge (1-\alpha \tfrac{\pi}{4}){\ensuremath{\mathrm{Tr}}}\big(|{\ensuremath{\mathcal{D}^0}}|c_{\tau_0}(s_0)^2 \big)\ge 4m(1-\alpha \tfrac{\pi}{4})>2m.$$ Thus - hold for all $0\le \tau<t_\infty$. – Thanks to this fact, by continuity for all $0\le \tau<t_\infty$, - hold: if we follow the point $s_{\pm}(\tau)$ from $\tau=0$, we see that there cannot exist $\tau_0$ such that or becomes false, because the set $\{t\ge 0,\ \forall\,0\le \tau< t,$ (resp. ) holds for $\tau \}$ is non-empty and open. – Up to an isomorphism of $[0,1]$, we can suppose that for all $0\le \tau\le t_\infty$, $$\forall\, s\in[0,1],\ {\ensuremath{\lVert\partial_s c_\tau(s_0)\rVert_{\mathfrak{S}_{2}}}}\apprle 1.$$ In $\mathfrak{S}_2$, the function $\partial_sc_t(s_0)$ satsifies the following equation: $$\frac{d}{dt}\partial_sc_t(s_0)=\partial_s \nabla \mathcal{E}^0_{\text{BDF}}(c_t(s_0))\in\mathfrak{S}_2.$$ These new loops are written ${\ensuremath{\widetilde{c}}}_\tau$ and have the same range as the $c_\tau$’s and define the same arc length. Studying the limit of ${\ensuremath{\widetilde{c}}}_\tau$ as $\tau$ tends to $t_\infty$, we get that at $t=t_\infty$, - still holds for the loop ${\ensuremath{\widetilde{c}}}_{t_\infty}$ at some $0< s_{-}(t_\infty)\le s_+(t_\infty)< 1$. Then necessarily, the loop ${\ensuremath{\widetilde{c}}}_{t_\infty}$ crosses $\mathscr{M}_{U,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}}$ at some $s\in [s_-(t_\infty),s_+(t_\infty)]$. Going back to $c_{t_\infty}$, this proves that the same holds for $c_{t_\infty}$, which contradicts the definition of $t_\infty$. Proofs on results on the variational set {#di_proofmanif} ======================================== Proof of Lemma \[di\_irreduc\] {#di_irr_proof} ------------------------------ Let $${\ensuremath{\Phi_{\mathrm{SU}}}}':\mathbf{SU}(2)\to \mathbf{U}(E),\ E\subset{\ensuremath{\mathfrak{H}_\Lambda}}$$ be an irreducible representation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$. As ${\ensuremath{\mathbf{J}}}^2$ and ${\ensuremath{\mathbf{S}}}$ commutes with the action of $\mathbf{SU}(2)$, then necessarily $E$ is an eigenspace for ${\ensuremath{\mathbf{J}}}^2$ and ${\ensuremath{\mathbf{S}}}$, associated to $j(j+1)$ and $\kappa_j={\ensuremath{\varepsilon}}(j+\tfrac{1}{2})$ where $j\in \tfrac{1}{2}+\mathbb{Z}_+$ and ${\ensuremath{\varepsilon}}=\pm$. The eigenspaces are known [@Th p. 126]: they are spanned by wave functions of type $$\label{di_irr_0} \forall\,x=r{\ensuremath{\omega}}_x\in{\ensuremath{\mathbb{R}^3}},\ \psi(x):=a(r)\Phi^{\pm}_{m,\kappa_j},\ m=-j,-j+1,\ldots,j,$$ where \[di\_courage\] $$\label{di_irr_1} a(r)\in L^2(\mathbb{R}_+,r^2dr),$$ $$\label{di_irr_2} \Phi^+_{m,\pm(j+\tfrac{1}{2})}:=\begin{pmatrix}i\Psi^{m}_{j\pm\tfrac{1}{2}}\\ 0\end{pmatrix}\text{\ and\ }\Phi^-_{m,\pm(j+\tfrac{1}{2})}:=\begin{pmatrix}0 \\ \Psi^{m}_{j\mp \tfrac{1}{2}} \end{pmatrix}$$ $$\Psi^{m}_{j-\tfrac{1}{2}}=\frac{1}{\sqrt{2j}}\begin{pmatrix}\sqrt{j+m}Y^{m-\tfrac{1}{2}}_{j-\tfrac{1}{2}}\\ \sqrt{j-m}Y^{m+\tfrac{1}{2}}_{j-\tfrac{1}{2}}\end{pmatrix}\text{\ and\ }\Psi^{m}_{j+\tfrac{1}{2}}=\frac{1}{\sqrt{2j+2}}\begin{pmatrix} \sqrt{j+1-m}Y^{m-\tfrac{1}{2}}_{j+\tfrac{1}{2}}\\ -\sqrt{j+1+m}Y^{m+\tfrac{1}{2}}_{j+\tfrac{1}{2}}\end{pmatrix}.$$ We recall that the $Y^m_{\ell}$ are the spherical harmonics (eigenvectors of ${\ensuremath{\mathbf{L}}}^2$). Hence $E$ is spanned by a wave function which is a linear combination of that of type . We recall that for any integer $n\ge 1$ there is but one irreducible representation of $\mathbf{SU}(2)$ of dimension $n$ up to isomorphism. They can be found by the number of eigenvalues of $J_3'$, the infinitesimal “rotation” around the $z$ axis which induces a representation of $\mathbf{SO}(3)$.. Here $J'_3$ corresponds to $J_3$. Thus we get that for ${\ensuremath{\varepsilon}}\in\{+,-\}$ $$E_{{\ensuremath{\varepsilon}}}:={\ensuremath{\Phi_{\mathrm{SU}}}}\,a(r)\Phi^{\ensuremath{\varepsilon}}_{j,\kappa_j}$$ is irreducible with respect to ${\ensuremath{\Phi_{\mathrm{SU}}}}$. By unicity of the irreducible representation of dimension $2j+1$, there exists an isomorphism from $E_-$ to $E_+$. As there must be a correspondence between the eigenspace of $J_3(E_-)$ and that of $J_3(E_+)$, necessarily $\mathbb{C}a\Phi^-_{m,\kappa_j}$ is sent to $\mathbb{C}a\Phi^+_{m,\kappa_j}$. In particular as ${\ensuremath{P_{\uparrow}}}E$ and ${\ensuremath{P_{\downarrow}}}E$ are also representation of $\mathbf{SU}(2)$ with same eigenvalues of ${\ensuremath{\mathbf{J}}}^2,{\ensuremath{\mathbf{S}}}$ (or $=\{0\}$). If one of them is zero then $E$ is of type $E_\pm$. If both are non-zero, then there exists $a_\uparrow(r),a_{\downarrow}(r)$ such that $${\ensuremath{P_{\uparrow}}}E={\ensuremath{\Phi_{\mathrm{SU}}}}a_\uparrow(r)\Phi^{+}_{j,\kappa_j}\text{\ and\ }{\ensuremath{P_{\downarrow}}}E={\ensuremath{\Phi_{\mathrm{SU}}}}a_{\downarrow}(r)\Phi^{-}_{j,\kappa_j}.$$ Both ${\ensuremath{P_{\uparrow}}}E$ and ${\ensuremath{P_{\downarrow}}}E$ are irreducible. We can suppose that there exists $f\in E$ with $${\ensuremath{P_{\uparrow}}}f= a_\uparrow(r)\Phi^{+}_{j,\kappa_j}\text{\ and\ }{\ensuremath{P_{\downarrow}}}f=a_{\downarrow}(r)\Phi^{-}_{j,\kappa_j}.$$ The isomorphism between the two representations implies that $$E={\ensuremath{\Phi_{\mathrm{SU}}}}\big( a_\uparrow(r)\Phi^{+}_{j,\kappa_j}+a_{\downarrow}(r)\Phi^{-}_{j,\kappa_j}\big).$$ Proof of Proposition \[di\_mani\_ci\_sym\] ------------------------------------------ We have to prove that $\mathscr{M}_{\mathscr{I}}$ and $\mathscr{W}$ are submanifold of $\mathscr{M}$. The method is similar to the one used in [@pos_sok] to prove that $\mathscr{M}_{\mathscr{C}}$ is a submanifold of $\mathscr{M}$. Let $P_0=Q_0+{\ensuremath{\mathcal{P}^0_-}}\in \mathscr{M}$. We will prove that in a neighbourhood of $P_0$ in ${\ensuremath{\mathcal{P}^0_-}}+\mathfrak{S}_2$, the projectors $P_1$ in $\mathscr{M}_{\mathscr{I}}$ (resp. $\mathscr{W}$) can be written as $$P_1=e^{A}P_0e^{-A},$$ where $A\in\mathfrak{m}^{\mathscr{I}}_{P_0}$ (resp. $\mathfrak{m}^{\mathscr{W}}_{P_0}$). – If we assume this point, then it is clear that the two sets are submanifolds of $\mathscr{M}$. Indeed $e^A$ is a global linear isometry of ${\ensuremath{\mathfrak{H}_\Lambda}}$, whose restriction to the $\mathfrak{m}_P^{\cdot}$’s maps $\mathfrak{m}_{P_0}^{\cdot}$ onto $\mathfrak{m}_{P_1}^{\cdot}$. Equivalently it maps the first tangent plane onto the other: $$\{ [a,P_0],\ a\in\mathfrak{m}_{P_0}^{\cdot}\}\underset{\simeq}{\to} \{ [a,P_1],\ a\in\mathfrak{m}_{P_1}^{\cdot}\}.$$ – We use Theorem \[di\_structure\] to write $$\label{di_di_chiant} Q_0={\ensuremath{\displaystyle\sum}}_{j=1}^{+\infty}{\ensuremath{\lambda}}_j\big({\ensuremath{|f_j\rangle}\xspace}{\ensuremath{\langle f_j|}\xspace}-{\ensuremath{|f_{-j}\rangle}\xspace}{\ensuremath{\langle f_{-j}|}\xspace}\big)$$ where $({\ensuremath{\lambda}}_i)_i\in\ell^2$ is non-increasing and the $f_i$’s form an orthonormal basis of $\mathrm{Ran}\,Q$. Provided that $${\ensuremath{\lVertP_1-P_0\rVert_{\mathfrak{S}_{2}}}}<1,$$ then ${\ensuremath{\lambda}}_1<1$ and there is no $j$ such that $f_j$ or $f_{-j}$ is in the range of ${\ensuremath{\mathcal{P}^0_+}}$ or ${\ensuremath{\mathcal{P}^0_-}}$. We decompose with respect with the eigenvalues $\mu_1>\mu_2>\cdots>0$ as follows: $$Q_0={\ensuremath{\displaystyle\sum}}_{k=1}^{+\infty}\mu_k\big(\text{Proj}\ \mathrm{Ker}(Q_0-\mu_k)-\text{Proj}\ \mathrm{Ker}(Q_0+\mu_k)\big).$$ For short we write $\mu_{-k}:=-\mu_{k}$, and $$M_k:=\text{Proj}\ \mathrm{Ker}(Q_0-\mu_k)\text{\ and\ }E_{\mu_k}^{Q_0}:=\mathrm{Ker}(Q_0-\mu_k).$$ As any ${\ensuremath{\mathrm{Y}}}\in\{{\ensuremath{\mathrm{C}}},{\ensuremath{\mathrm{I}_{\mathrm{s}}}}\}$ is an isometry (linear or antilinear) and as the eigenvalues are the sine of the angles between vectors in $P_0$ and ${\ensuremath{\mathcal{P}^0_-}}$, for any $k$ we have $$\label{di_invar_y} {\ensuremath{\mathrm{Y}}}E_{\mu_k}^{Q_0}=E_{-\mu_k}^{Q_0}$$ and the eigenspaces $E_{\mu_k}^{Q_0}\oplus E_{-\mu_k}^{Q_0}=\mathrm{Ker}(Q_0^2-\mu_k^2)$ are invariant under ${\ensuremath{\mathrm{Y}}}$. #### Case of $\mathscr{W}$ – In the case ${\ensuremath{\mathrm{Y}}}={\ensuremath{\mathrm{C}}}$ and $P_0\in \mathscr{W}$, each eigenspace $\mathrm{Ker}(Q_0^2-\mu_k^2)$ is also invariant under the action of ${\ensuremath{\Phi_{\mathrm{SU}}}}$. In other words, $\mathrm{Ker}(Q_0^2-\mu_k^2)$ is a finite dimensional representation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$, and we can decompose it into irreducible representations $E_{\mu_k}^{(\ell)}$, where $0\le \ell\le \ell_k$. By ${\ensuremath{\mathrm{C}}}$-symmetry, we have $${\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell_1)}=E_{-\mu_k}^{(\ell_1')},$$ there is a one-to-one correspondence between irreducible representations of type $E_{\mu_k}^{(\ell)}$ and that of type $E_{-\mu_k}^{(\ell)}$. Up to changing indices $\ell'_j$, we can suppose that $${\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell)}=E_{-\mu_k}^{(\ell)},\ 0\le \ell\le \ell_k.$$ Decomposing $E_{\mu_k}^{(\ell)}$ with respect with ${\ensuremath{\mathcal{P}^0_-}}$ and ${\ensuremath{\mathcal{P}^0_+}}$, we see that $$\mathcal{P}^0_{\pm} E_{\mu_k}^{(\ell)}\ \text{is\ irreducible},$$ and from the spectral decomposition of $Q_0$ $${\ensuremath{\mathcal{P}^0_-}}E_{\mu_k}^{(\ell)}\oplus {\ensuremath{\mathcal{P}^0_+}}E_{\mu_k}^{(\ell)}=E_{\mu_k}^{(\ell)}\oplus F_{-\mu_k},$$ where $F_{-\mu_k}$ is an irreducible subset of $\mathrm{Ker}(Q_0+\mu_k)$. – Let us show that $$\label{di_inters} F_{-\mu_k}\cap {\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell)}=\{ 0\}.$$ Indeed, from Lemma \[di\_irreduc\] and the expression of the $\Phi^{\pm}_{m,\kappa}$, we see that $${\ensuremath{\mathrm{C}}}\mathrm{Ker}\big(J_3-m\big)=\mathrm{Ker}\big(J_3+m\big).$$ Thus if the intersection is non-zero, then we have by ${\ensuremath{\mathrm{C}}}$-symmetry and ${\ensuremath{\Phi_{\mathrm{SU}}}}$-symmetry: $$F_{-\mu_k}={\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell)}.$$ But as shown in [@pos_sok], this cannot happen: let us say that $E_{\mu_k}^{(\ell)}$ is associated to the eigenvalues $j_0(j_0+1),\kappa$ of ${\ensuremath{\mathbf{J}}}^2$ resp. ${\ensuremath{\mathbf{S}}}$. We consider: $$\mathrm{Ker}(J_3-m)\cap \mathcal{P}^0{\pm} E_{\mu_k}^{(\ell)}=\mathbb{C}e_{\pm;m},\ -j_0\le m\le j_0,\ {\ensuremath{\lVerte_{\pm;m}\rVert_{L^{2}}}}=1.$$ We would have $${\ensuremath{\mathrm{C}}}e_{\pm;m}=\text{exp}{i\theta(\pm;m)}e_{\mp;-m}.$$ The constant $\theta(\pm;m)$ does not depend on $m$ by ${\ensuremath{\Phi_{\mathrm{SU}}}}$-symmetry. Moreover, if $$\mathrm{Ker}(J_3-m)\cap E_{\mu_k}^{(\ell)}=\mathbb{C}f_{m},$$ then $$\mathcal{P}^0_{\pm} f_m\parallel e_{\pm;m}.$$ As in [@pos_sok] for $\mathscr{M}_{\mathscr{C}}$, the condition ${\ensuremath{\mathrm{C}}}^2=1$ implies $\theta_+-\theta_-\equiv 0[2\pi]$ while $$-{\ensuremath{\mathrm{C}}}Q_0{\ensuremath{\mathrm{C}}}=Q_0$$ implies $\theta_+-\theta_-\equiv \pi[2\pi]$, which cannot occur. Similarly, we can prove that holds and that in fact $F_{-\mu_k}$ is orthogonal to $ {\ensuremath{\mathrm{C}}}E_{\mu_k}^{(\ell)}$. As a consequence, the number of $E_{\mu_k}^{(\ell)}$’s is even, or equivalently, the number of ${\ensuremath{\mathcal{P}^0_-}}E_{\mu_k}^{(\ell)}$ is even. – The fact that $$\label{di_check} P_1=e^A P_0 e^{-A},\ \text{with}\ {\ensuremath{\Phi_{\mathrm{SU}}}}A=A,\ {\ensuremath{\mathrm{C}}}A {\ensuremath{\mathrm{C}}}=A,\ {\ensuremath{\lVertA\rVert_{\mathfrak{S}_{2}}}}<+\infty,$$ follows from Theorem \[di\_structure\] and the different symmetries. The $f_j$’s in can be written as (${\ensuremath{\lambda}}_j=\sin(\theta_j)$) $$f_j=\sqrt{\frac{1-{\ensuremath{\lambda}}_j}{2}}e_{-;j}+\sqrt{\frac{1+{\ensuremath{\lambda}}_j}{2}}e_{+;j},\ \mathcal{P}^0_{\pm}e_{\pm;j}=e_{\pm;j}.$$ We also have $$f_{-j}=-\sqrt{\frac{1+{\ensuremath{\lambda}}_j}{2}}e_{-;j}+\sqrt{\frac{1-{\ensuremath{\lambda}}_j}{2}}e_{+;j}.$$ Then we define $$\label{di_aaa} A={\ensuremath{\displaystyle\sum}}_{j=1}^{+\infty}\theta_j\big({\ensuremath{|e_{+;j}\rangle}\xspace}{\ensuremath{\langle e_{-;j}|}\xspace}-{\ensuremath{|e_{-;j}\rangle}\xspace}{\ensuremath{\langle e_{+;j}|}\xspace}\big).$$ It is easy to check that $A$ satisfies . In fact, we can assume that $f_j$ spans an irreducible representation of $\mathbf{SU}(2)$, and in this case the same holds for $e_{+;j}$ and $e_{-;j}$. As in Section \[di\_irr\_proof\], the correspondence $e_{-;j}\mapsto e_{+;j}$ induces an isomorphism between ${\ensuremath{\Phi_{\mathrm{SU}}}}e_{-;j}$ and ${\ensuremath{\Phi_{\mathrm{SU}}}}e_{+;j}$. This fact together with the ${\ensuremath{\Phi_{\mathrm{SU}}}}$-symmetry implies that $$\forall\,U\in\mathrm{Ran}\,{\ensuremath{\Phi_{\mathrm{SU}}}},\ UAU^{-1}=A.$$ The fact that ${\ensuremath{\mathrm{C}}}A{\ensuremath{\mathrm{C}}}=A$ was proved in [@pos_sok] in the case $P_0,P_1\in\mathscr{M}_{\mathscr{C}}$. Here this remains true because $$\mathscr{W}\subset \mathscr{M}_{\mathscr{C}}.$$ – We can now determine the connected component of $\mathscr{W}$. Let $P_0,P_1$ be in $\mathscr{W}$ and let $Q=P_1-P_0$. We consider $$E_1^{Q}:=\mathrm{Ker}(Q-1).$$ If $E_1^Q=\{0\}$, then we can write $P_1=e^A P_0 e^{-A}$ as in . And we see that the path in $\ell^2$: $$t\in[0,1]\mapsto (t\theta_j)_j\in \ell^2$$ induces a path connecting $P_0$ and $P_1$. If $E_1^Q\neq \{ 0\}$, we count the number of irreducible representation in $E_1^Q$: let $b_{j,\kappa_j}$ be the number of irr. rep. in $$\mathrm{Ker}\big({\ensuremath{\mathbf{J}}}^2-j(j+1)\big)\cap\mathrm{Ker}\big({\ensuremath{\mathbf{S}}}-\kappa_j\big).$$ If all the $b_{j,\kappa_j}$’s are even, we can still write $P_1$ as $P_1=e^A P_0 e^{-A}$ with $A$ as in with the first $\theta_j$ equal to $\tfrac{\pi}{2}$. In particular the two projectors can be connected by a path in $\mathscr{W}$. Let us say that $b_{j_0,\kappa_{0}}\equiv 1[2]$ for some $j_0,\kappa_0$. We have shown that for $P\in\mathscr{W}$ with ${\ensuremath{\lVertP-P_0\rVert_{\mathcal{B}}}}<1$, the number of planes $\Pi_j$’s in the decomposition of Theorem \[di\_structure\] is even. Precisely, due to the ${\ensuremath{\mathrm{C}}}$-symmetry, there exists a sequence $(\ell_\mu(j,\kappa))_j$ in $\mathbb{N}$, with $$\begin{array}{l} \mathrm{Ker}\big((P-P_0)-\mu\big)\cap\mathrm{Ker}\big({\ensuremath{\mathbf{J}}}^2-j(j+1)\big)\cap \mathrm{Ker}\big({\ensuremath{\mathbf{S}}}-\kappa\big)\\ \ \ \ =\underset{1\le \ell\le \ell_\mu(j,\kappa)}{\bigoplus}E^{(\ell)}_{\mu}, \end{array}$$ where each $E^{(\ell)}_{\mu}$ is irreducible as a representation of ${\ensuremath{\Phi_{\mathrm{SU}}}}$ and $\ell_\mu(j,\kappa)$ is *even*. We show that there cannot exist a continuous path linking $P_0$ and $P_1$ by a contradiction argument. Let us say that ${\ensuremath{\gamma}}:t\in [0,1]\to \mathscr{W}$ is a continuous path with ${\ensuremath{\gamma}}(0)=P_0$ and ${\ensuremath{\lVert{\ensuremath{\gamma}}(1)-P_0\rVert_{\mathcal{B}}}}=1$. Then by the previous remarks, we have by continuity: $$\begin{array}{l} \forall t\in[0,1], \forall\,j\in\frac{1}{2}+\mathbb{Z}_+,\ \forall\kappa\in\big\{\pm\big(j+\frac{1}{2}\big)\big\},\\ \ \ \ \ \ell_1(Q_t={\ensuremath{\gamma}}(t)-P_0;j,\kappa)\equiv 0[2]. \end{array}$$ In particular it is not possible to have ${\ensuremath{\gamma}}(1)=P_1$. #### Case of $\mathscr{M}_{\mathscr{I}}$ For ${\ensuremath{\mathrm{Y}}}={\ensuremath{\mathrm{I}_{\mathrm{s}}}}$ and $P_0\in \mathscr{M}_{\mathscr{I}}$, we use . For each $f\in E_\mu^Q$, we have ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}\in E_{-\mu}^Q$ where $\mu\in \sigma(Q)$. We may assume that $\mu>0$. Thus the plane $$\Pi:=\text{Span}\big(f,{\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\big)$$ is invariant under $Q$ and ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}$. We decompose $f$ and ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}f$ with respect to $P_0$ and $1-P_0$. By a dimension argument: 1. either $\mu=1$, $P_0 f=0$ and $(1-P_0){\ensuremath{\mathrm{I}_{\mathrm{s}}}}f=0$, 2. or $0<\mu<1$ and $$\mathbb{C}P_0 f=\mathbb{C} P_0 {\ensuremath{\mathrm{I}_{\mathrm{s}}}}f\text{\ and\ }\mathbb{C}(1-P_0) f=\mathbb{C} (1-P_0) {\ensuremath{\mathrm{I}_{\mathrm{s}}}}f.$$ In each case, we write $e_{-}$ a unitary vector in $\mathrm{Ran}\,P_0\cap \Pi$ and $e_+={\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_-$. If we consider the sequence $(\mu_i)_i$ of positive eigenvalues of $Q$ (counted with multiplicities), we get the correspondent sequences $(e_{-;j})_j$ and $(e_{+;j})$. Moreover by Theorem \[di\_structure\], we know that $\mu_j=\sin(\theta_j)$ where $\theta_j\in[0,\tfrac{\pi}{2}]$ is the angle between the two lines $\mathbb{C}e_{-;j}$ and $\mathbb{C}f_j$. Provided that we take $-\theta_j$ instead of $\theta_j$ and up to a phase, we can suppose that $$f_j=\cos(\theta_j)e_{-;j}+\sin(\theta_j){\ensuremath{\mathrm{I}_{\mathrm{s}}}}e_{-;j}.$$ In particular we have $$P_1=e^A P_0 e^{-A},$$ with $$A={\ensuremath{\displaystyle\sum}}\theta_j\big({\ensuremath{|e_{+;j}\rangle}\xspace}{\ensuremath{\langle e_{-;j}|}\xspace}-{\ensuremath{|e_{-;j}\rangle}\xspace}{\ensuremath{\langle e_{+;j}|}\xspace}\big).$$ It is straightforward to check that ${\ensuremath{\mathrm{I}_{\mathrm{s}}}}A {\ensuremath{\mathrm{I}_{\mathrm{s}}}}^{-1}=A$. *Acknowledgment* The author wishes to thank Éric séré for useful discussions and helpful comments. This work was partially supported by the Grant ANR-10-BLAN 0101 of the French Ministry of research.

ArXiv

--- abstract: | We review the current status of the study of rotation curve (RC) of the Milky Way, and present a unified RC from the Galactic Center to the galacto-centric distance of about 100 kpc. The RC is used to directly calculate the distribution of the surface mass density (SMD). We then propose a method to derive the distribution of dark matter (DM) density in the in the Milky Way using the SMD distribution. The best-fit dark halo profile yielded a local DM density of We also review the estimations of the local DM density in the last decade, and show that the value is converging to a value at $\rho_\odot=0.39\pm 0.09$ .\ [**Key words**]{} galaxies: DM—galaxies: individual (Milky Way)—galaxies: rotation curve\ ([*Invited review accepted for Galaxies to appear in special issue on “Debate on the Physics of Galactic Rotation and the Existence of Dark Matter”*]{}) author: - | Yoshiaki Sofue\ Institute of Astronomy, The University of Tokyo, Mitaka, Tokyo 181-0015, Japan\ E-mail: [email protected] title: Rotation Curve of the Milky Way and the Dark Matter Density --- 0[ V\_0 ]{} Introduction ============ The rotation curve (RC) of the Milky Way was obtained by observations of galactic objects in the non-MOND (MOdified Newtonian Dynamics)frame work. The existence of the dark halo (DH) has been confirmed by the analysis of the observed RCs, assuming that  Newtonian dynamics applies evenly to the result of the observations. In this article, current works of RC observations are briefly reviewed, and a new estimation of the local dark matter (DM) density is presented in the framework of Newtonian dynamics. An RC is defined as the mean circular velocity $\Vrot$ around the nucleus plotted as a function of the galacto-centric radius $R$. Non-circular streaming motion due to the triaxial mass distribution in a bar is crucial for kinematics in the innermost region, though it does not affect the mass determination much in the disk and halo. Spiral arms are another cause for local streaming, which affect the mass determination by several percent, while they do not influence the mass determination of the dark halo much. There are several reviews on RCs and mass determination of galaxies \[[@SofueRubin2001; @Sofue2017; @Salucci2019]\]. In this review, we revisit recent RC studies and determination of the local DM density in our Milky Way. In Section \[Section2\], we briefly review the current status of the RC determinations along with the methods. We adopt the galactic constants: $(\rzero,\vzero)$=(8.0 kpc, 238 ) \[[@Honma+2012; @Honma+2015]\], where $\rzero$ is the distance of the Sun from the galactic center (GC) and $\vzero$ is the circular velocity of the local standard of rest (LSR) at the Sun \[[@Fich+1991]\]. Rotation Curve of the Milky Way {#Section2} =============================== Progress in the Last Decades ---------------------------- The galactic RC is dependent on the galactic constants. Accordingly, the uncertainty and error in the RC include uncertainties of the constants. Currently recommended, determined, or measured values are summarized in Table \[tabGal\], where they appear to be converging to around $\sim 8.0-8.3$ and . In this paper, we adopt $R_0= 8.0$ kpc and $V_0 =238$ from the recent measurements with VERA (VLBI Experiments for Radio Astrometry) \[[@Honma+2012; @Honma+2015]\]. **Authors (Year)** **(kpc)** **()** ---------------------------------------------------------------- ----------------- ------------- IAU recommended (1982) 8.2 220 Review before 1993 (Reid 1993) \[[@Reid1993]\] $8.0 \pm 0.5$ Olling and Dehnen 2003 \[[@Olling+2003]\] $7.1\pm 0.4$ $184\pm 8$ VLBI Sgr A$^*$ (Ghez et al. 2008) \[[@Ghez+2008]\] $8.4 \pm 0.4$ ibid (Gillessen et al. 2009) \[[@Gillessen+2009]\] $8.33 \pm 0.35$ Maser astrometry (Reid et al. 2009) \[[@Reid+2009]\] $8.4\pm 0.6$ $254\pm 16$ Cepheids (Matsunaga et al. 2009) \[[@Matsunaga+2009]\] $8.24 \pm 0.42$ VERA (Honma et al. 2012, 2015) \[[@Honma+2012; @Honma+2015]\]. $8.05\pm 0.45$ $238\pm 14$ Adopted in this paper 8.0 238 : Galactic constants ($R_0,V_0$). []{data-label="tabGal"} The RC of the galaxy has been obtained by various methods as described in the next subsection, and many authors presented their results based on different galactic constants (Table \[tabrcmw\]). **Authors (Year)** **Radii (kpc)** **Method** ----------------------------------------------------------------------------------------------------------- ----------------- ------------------------- Burton and Gordon (1978)\[[@Burton+1978]\] 0–8 HI tangent Blitz et al. (1979) \[[@Blitz+1979]\] 8–18 OB-CO assoc. Clemens (1985)\[[@Clemens1985]\] 0 -18 CO/compil. Dehnen and  Binney (1998)\[[@Dehnen+1998]\] 8–20 compil. + model Genzel et al. (1994–), Ghez et al. (1998–)\[[@Genzel+2010; @Ghez+2008]\] 0–0.0001 GC IR spectr. Battinelli, et al. (2013)\[[@Battinelli+2013]\] 9–24 C stars Bhattacharjee et al.(2014)\[[@Bhattacharjee+2014]\] 0–200 Non-disk objects Lopez-Corredoira (2014)\[[@Lopez2014]\] 5–16 Red-clump giants $\mu$ Boby et al. (2012)\[[@Bovy+2012b]\] 4-14 NIR spectroscopy Bobylev (2013); — & Bajkova (2015)\[[@Bobylev2013; @Bobylev+2015]\] 5–12 Masers/OB stars Reid et al. (2014)\[[@Reid+2014]\] 4-16 Masers SF regions, VLBI Honma et al. (2012, 2015)\[[@Honma+2012; @Honma+2015]\] 3–20 Masers,VLBI Iocco et al. (2015, 2016); Pato & Iocco (2017a,b)\[[@Iocco+2015; @Iocco+2016; @Pato+2017a; @Pato+2017b]\] 1–25 kpc CO/HI/opt/maser/compil. Huang et al. (2016)\[[@Huang+2016]\] 4.5–100 HI/opt/red giants Kre[ł]{}owski et al (2018)\[[@Krelowski+2018]\] 8–12 GAIA Lin and Li (2019)\[[@Lin+2019]\] 4–100 compil. Eilers et al (2019)\[[@Eilers+2019]\] 5–25 Wise, 2Mass, GAIA Mróz et al. (2019)\[[@Mroz+2019]\] 4–20 Classical cepheids Sofue et al. (2009); Sofue (2013, 2015, this work)\[[@Sofue+2009; @Sofue2013; @Sofue2015]\] 0.01–1000 CO/HI/maser/opt/compil. : Rotation curves (RCs) of the Milky Way galaxy.[]{data-label="tabrcmw"} In the 1970–1980s, the inner RC was extensively measured using the terminal-velocities of HI (neutral hydrogen) and CO (carbon monoxide) gases \[[@Burton+1978; @Clemens1985; @Fich+1989]\]. In the late 1980s to the 2000s, outer rotation velocities were measured by combining optical distances of OB \[[@Blitz+1979; @Demers+2007]\]. The HI thickness method was also useful to measure rotation of the entire disk \[[@Merrifield1992; @Honma+1997]\]. The innermost mass distributions inside the GC have been obtained extensively since the 1990s using the motion of infrared stellar objects \[[@Genzel+2010; @Ghez+2008; @Lindqvist+1992; @Gillessen+2009]\]. Trigonometric determinations of both the 3D positions and velocities have provided the strongest tool to date for measurement of the galactic rotation \[[@Honma+2007; @Honma+2012; @Honma+2015; @Sakai+2015; @Nakanishi+2015]\]. A number of optical parallax measurements of stars such with GAIA have been obtained for RC determination \[[@Lopez2014; @Krelowski+2018]\]. The total mass of the galaxy, including the extended dark halo, has been measured by analyzing the outermost RC and motions of satellite galaxies orbiting the galaxy, and the mass up to kpc has been estimated to be $\sim 3 \times 10^{11}\Msun$ \[[@Sofue2015; @Callingham+2019]\]. Methods to Determine the Galactic RC ------------------------------------ The particular location of the Sun inside the Milky Way makes it difficult to measure the rotation velocity of the galactic objects. Sophisticated methods have been developed to solve this problem, as briefly described below. ### Tangent-Velocity Method Inside the solar circle ($-90^\circ \le l \le 90^\circ$), the galactic gas disk has tangential points, at which the rotation velocity is parallel to the line of sight and attains the maximum radial velocity $ {\vr}_{\rm ~max}$ (terminal or tangent-point velocity). The rotation velocity $V(R)$ at galacto-centric distance $R=\Rsun \sin~ l$ is calculated simply correcting for the solar motion. ### Radial-Velocity + Distance Method If the distance $r$ of the object is measured by spectroscopic and/or trigonometric observations, the rotation velocity is obtained by geometric conversion of the radial velocity, distance, and the longitude. The distance has to be measured independently, often using spectroscopic distances of OB stars, and the distances are assumed to be the same as those of associated molecular clouds and HII (ionized hydrogen) regions, whose radial velocities are observed by radio lines. Since the photometric distances have often large errors, obtained RC plots show large scatter. ### Trigonometric Method If the proper motion and radial velocity along with the distance are measured at the same time, or from different observations, the 3D velocity vector, and therefore the rotation velocity, of any source is uniquely determined without being biased by assumption of circular motion as well as the galactic constants. VLBI (very long baseline interferometer) measurements of maser sources \[[@Honma+2007; @Honma+2012; @Honma+2015; @Nakanishi+2015]\] and optical/IR trigonometry of stars \[[@Roeser+2010; @Lopez2014]\] have given the most accurate RC. ### Disk-Thickness Method The errors in the above methods are mainly caused by the uncertainty of the distance measurements. This disadvantage is eased by the HI-disk thickness method \[[@Merrifield1992; @Honma+1997]\]. The angular thickness of the HI disk along an annulus ring is related to can be used to determine the rotation velocity by combining with radial velocity distribution along the longitude. ### Pseudo-RC from Non-Disk Objects Beyond or outside the galactic disk, globular clusters and satellite galaxies are used to estimate the pseudo-circular velocity from their radial velocities based on the Virial theorem, assuming that their motions are at random, or the  rotation velocity is calculated by $\Vrot \sim \sqrt{2} v_g$, where $v_g$ is the galacto-centric radial velocity. [On the other hand, Huang et al. (2016) \[[@Huang+2016]\] have recently employed more sophisticated, probably more reliable, method to solve the Jeans equations for the non-disk stars and clusters.]{} Unified RC ---------- A RC covering a wide region of the galaxy has been obtained by compiling the existing data by re-scaling the distances and velocities to the common galactic constants \[[@Sofue+2009]\], and later to (8.0 kpc, 238 ) \[[@Sofue2013; @Sofue2017]\]. In these works, the central RC inside the GC has been obtained from analyses of the kinematics of the molecular gas and infrared stellar motions as well as the supposed Keplerian motion representing the central massive black hole. Outer RC beyond $R\sim 30$ kpc has been determined from the radial motions of satellite galaxies and globular clusters. The RC determination has been improved recently by compiling a large amount of data from a variety of spectroscopic as well as trigonometric measurements from radio to optical wavelengths. An extensive compilation of the data of rotation velocities of the galactic disk has been published recently, and is available as an internet data base \[[@Iocco+2015; @Iocco+2016; @Pato+2015a; @Pato+2015b; @Pato+2017a; @Pato+2017b]\]. Figure \[mwrc\]a shows the presently obtained unified RC using the curves from\[[@Sofue2015; @Sofue2017]\]and RC by Huang et al. (2016)\[[@Huang+2016]\] between $R=4.6$ and $\sim 100$ kpc. Although Huang et al. employed the galactic constants of (8.34 pc, 240 ), we did not apply rescaling to (8.0, 238), because the galacto-centric distances of off-plane objects are less dependent on the solar position compared to the disk objects as used for our RC at $<\sim 20$ kpc where the rotation velocity is rather flat, and also because their $V_0=240$ is close to our 238 . ![ (**a**) Unified RC of the Milky Way used in this paper for the mass distribution obtained by averaging the RCs from references \[[@Sofue2015; @Sofue2017; @Huang+2016]\]. The bars are standard deviations within each Gaussian-averaging bin. The plotted values are listed in the tables in Appendix \[AppendixA\]. (**b**) Logarithmic RC of the Milky Way from \[[@Sofue2015; @Sofue2017]\] (circles), compared with those from the recent literature: Green circles with error bars are from the compilation by \[[@Pato+2017a; @Pato+2017b]\]and blue triangles are their running averages. Red triangles stand for data from \[[@Krelowski+2018]\]based on GAIA data. [These two data are re-scaled to ($R_0, V_0$)=(8.0 kpc, 238 ).]{} Pink rectangles are the RC by \[[@Huang+2016]\].[without re-scaling]{}. (**c**) Same, but in linear scale. (**d**) Same, but close up in the solar vicinity.[]{data-label="mwrc"}](fig1.eps){width="8.5cm"} The unified RC was obtained by taking Gaussian running averages of rotation velocities from the used RCs in each of newly settled radius bins, where the  statistical weight of each input point was given by the inverse of the squared error. In Figure \[mwrc\]b,c we compare the unified RC with the recent measurements by \[[@Pato+2017a; @Pato+2017b; @Krelowski+2018]\]re-scaled to the galactic constants of (8.0 kpc, 238 ) following the method described in \[[@Sofue+2009]\].Although individual data points are largely scattered, their averages well coincide with the unified RC. In the figures we also compare the data with the RC by \[[@Huang+2016]\]up to $\sim 100$ kpc without rescaling, which also coincides with the other data within the scatter. We here comment on the property of the unified RC built by averaging the published data. It must be remembered that the averaging procedure does not satisfy the condition of statistics in the strict meaning, because the data are compiled from different authors using a variety of instruments and analysis methods, which makes it difficult to evaluate common statistical weights for the used data points. So, remembering such a property, in view that the unified RC well approximates the original curves as well as for its convenience for the determination of the mass distribution by the least-squares and/or $\chi^2$ fitting, we shall employ it in our present analysis. Mass Components --------------- The rotation velocity is related to the gravitational potential, hence to the mass distribution, as V(R)==, where $\Phi_i$ is the gravitational potential of the $i$-th component and $V_i$ is the corresponding circular velocity. The rotation velocity is often represented by superposition of the central black hole (BH), bulge, disk, and the dark halo as V(R)= . Here, the subscript BH represents black hole, b stands for bulge, d for disk, and h for the dark halo. The contribution from the black hole can be neglected in sufficiently high accuracy, when the dark halo is concerned. The mass components are usually assumed to have the following functional forms. ### Massive Black Hole The GC of the Milky Way is known to nest a massive black hole of mass of $M_{\rm BH}\sim 4 \times 10^6\Msun$ \[[@Genzel+2010; @Ghez+2008; @Gillessen+2009]\].The RC is assumed to be expressed by a curve following the Newtonian potential of a point mass at the nucleus. ### De Vaucouleurs Bulge The commonly used SMD profile to represent the central bulge, which is assumed to be proportional to the empirical optical profile of the surface brightness, is the  law \[[@deV1958]\], \_[b]{}(R) = \_[be]{} [exp]{} , \[eq-smdb\] where $ \Sigma_{\rm be} $ is the value at radius $R_{\rm b}$ enclosing a half of the integrated surface mass \[[@Sofue2017]\].Note that the surface profile, also the exponential disk, has a finite value at the center. The volume mass density $\rho(r)$ at radius $r$ for a spherical bulge is calculated using the SMD by (r) = \_r\^ dx, \[eq-rhob\] and the mass inside $R$ is M(R) =4\_0\^R r\^2(r)dr. The circular velocity is thus obtained by V\_[b]{}(R) = . More general form $e^{-(R/r_e)^n}$ called the law is discussed in relation to its dynamical relation to the galactic structure based on the more general profile \[[@Ciotti1991; @Trujillo2002]\]. ### Exponential Disk The galactic disk is generally represented by an exponential disk \[[@Freeman1970]\],where the SMD is expressed as \_[d]{} (R)=\_[d]{} [exp]{}(-R/R\_[d]{}). \[eq-smdd\] Here, $\Sigma_{\rm d}$ is the central value, $R_d$ is the scale radius. The total mass of the exponential disk is given by $M_{\rm disk}= 2 \pi \Sigma_{dc} R_{\rm d}^2$. The RC for a thin exponential disk is expressed by \[[@Binney+1987]\]V\_[d]{}(R)=, where $y=R/ (2R_{\rm d}) $, and $I_i$ and $K_i$ are the modified Bessel functions. The dark halo is described in the next section Dark Halo {#Section3} ========= The existence of dark halos in spiral galaxies has been firmly evidenced from the well established difference between the galaxy mass predicted by the luminosity and the mass predicted by the rotation velocities \[[@SofueRubin2001; @Sofue2017; @Salucci2019]\]. In the Milky Way, extensive analyses of RC and motions of non-disk objects such as globular clusters and dwarf galaxies in the Local Group have shown flat rotation up to $\sim 30$ kpc, beyond which the RC declines smoothly up to $\sim 300$ kpc \[[@Sofue2013; @Sofue2015]\]. Further analyses of non-disk tracer objects have also shown that the outer RC declines in a similar manner \[[@Bhattacharjee+2014; @Huang+2016; @Li+2017]\]. The fact that the rotation velocity beyond $R\sim 30$ kpc declines monotonically indicates that the isothermal model can be ruled out in representing the Milky Way’s halo. Dark Halo Models ---------------- There have been various proposed DH models, which may be categorized into two types: The cored halo models \[[@Burkert1995; @Salucci+2000; @Brownstein+2006]\] are a modification of the isothermal model with a steeper decrease of density at large radii. The central cusp models \[[@Navarro+1995; @Navarro+1997; @Moore+1999; @Fukushige+2004]\] are based on extensive $N$-body numerical simulations of the structural evolution in the cold dark matter scenario in the expanding universe, which predict an infinitely increasing central peak. In either type, all the DH models predict decreasing DM density beyond $h$ as $\rho \propto R^{-3}$, or declining rotation velocity as $\Vrot \propto \sqrt{{\rm ln} \ R/R}$. The cored halo models exhibit a central plateau of finite density with scale radius, or the core radius, $h$, and are often represented by the following functions, where $x=R/h$. [**Isothermal halo**]{}: \_[Iso]{} (x)=, \[eq-iso\] [**Beta model with $\beta=1$** ]{} \[[@Navarro+1995]\] :\_(x)= . [**Burkert model**]{} \[[@Burkert1995; @Salucci+2000]\] : \_[Bur]{} (x)=, \[eq-bur\] [**Brownstein model**]{} \[[@Brownstein+2006]\] : \_[Bro]{} (x)=. \[eq-bro\] On the other hand, the central cusp models are often represented by the following functions. [**NFW model**]{} \[[@Navarro+1996; @Navarro+1997]\] : \_[NFW]{} (x)=, \[eq-nfw\] [**Moore model**]{} \[[@Moore+1999; @Fukushige+2004]\] with $\alpha=1.5$: \_[Moo]{} (x) = =. \[eq-nfw\] Figure \[rhoModels\] shows schematic density profiles for various DH models with $h=10$ kpc combined with the bulge and exponential disk, where the halo density is normalized at $R=20$ kpc.. ![ (**Top**) Schematic density profiles of [NFW]{} (Navarro, Frenk, White) (thick solid), Moore (upper long dash), Burkert (long dash), Brownstein (dot), $\beta$ (dash), and isothermal (thin solid) models with $h=10$ kpc normalized at 20 kpc, compared with the disk (straight line) and bulge (inner thick dash). Uppermost thin lines are the sum of bulge, disk and halo. (**Bottom**) Same, but in log–log plot. The NFW cusp and cored halos do not much contribute to the mass density in the GC, whereas the Moore cusp somehow resembles the bulge profile. []{data-label="rhoModels"}](fig2.eps){width="10cm"} Cusp vs Cored Halo ------------------ The density profiles for the NFW (Navarro, Frenk and White), Moore, Burkert, $\beta$, and Brownstein models are almost identical beyond the core radius $h$, where they tend to $\propto R^{-3}$. Differences among the models appear within the Solar circle. The cusp models (NFW and Moore models) predict steep increase of density toward the center with a singularity. The cored halo models predict a mild and low density plateau in the center with the peak densities not much differing from each other within a factor of two. However, the Burkert model has a singularity with the density gradient being not continuous across the nucleus. Most of the DH models predict lower density in the innermost galaxy by two to several orders of magnitudes than the bulge’s density. This implies that the DH does not much influence the kinematics in the inner galaxy. Namely, it is practically impossible to detect the DM cusp by analyzing the RC. Only the Moore model predicts cusp density exceeding the bulge’s density in the very center at $R<\sim 0.1$ pc, whereas the applicability of the model to such small sized region is not obvious \[[@Fukushige+2004]\]. Central DM Density ------------------ If we assume that the functional form of the NFW model is valid in the very central region, the SMD at $R\sim 100$ pc could be estimated to be about $\Sigma \sim 2.2\times 10^3\Msun {\rm pc}^{-2}$. This yields an approximate volume density on the order of $\rho \sim \Sigma/R\sim 11\Msun {\rm pc}^{-3} \sim 840$ GeV cm$^{-3}$ for a detector of $\sim 1.4\deg$ resolution. Such estimations could be a key to the indirect detection experiments of DM in the GC . However, it is stressed that the DM density in the GC is two to several orders of magnitudes smaller than the bulge’s density on the order of $10^4-10^5$ GeV cm$^{-3}$, making the kinematical detection of DM difficult. Interestingly, the column density of DM, hence brightness (flux/steradian) of self-annihilation emission ($\gamma$-ray) stays almost constant against the radius and is therefore constant regardless the resolution of the detector. On the other hand, the emission measure $\sim \rho^2 R$ varies as $\propto R^{-1}$, hence, the brightness of collision-origin emission ($\gamma$ or microwave haze) increases toward the center \[e.g.,[@Finkbeiner2004]\], so that the detection rate will increase with the detector’s resolution. Another concern about the DM cusp is the kinetic energy of individual particles. In order for the cusp to be stationary, the particles must be bound to the gravitational potential, so that the particle’s speed must be lower than the escaping velocity $v\sim \sqrt{2}\Vrot \sim 300$ . This will give a constraint on the cross section $\sigma_A$ of the DM annihilation, if the collision rate $\sigma_A v$ is fixed by the detection of DM-origin emissions. The cored halo models (isothermal, Burkert, Brownstein, and the $\beta$ models) predict a mild and finite-density plateau with scale radius of $h$ ($\sim 10$ kpc). Their central densities are also several orders of magnitude less than the bulge’s density, hence do not contribute to the kinematics of the gas and stars in the GC. DM Density from Direct SMD {#Section4} ========================== SMD from RC ----------- In the decomposition method of the RC, the resulting mass distribution depend on the assumed functional forms of the model profiles. In order to avoid this inconvenience, the RC can be used to directly calculate the surface mass distribution without employing any functional form. Only an assumption has to be made, either if the galaxy’s shape is a sphere or a flat disk. On the assumption of spherical distribution, the mass inside radius $R$ is given by $$M(R)=\frac{R {V(R)}^{2}}{G}. \label{masssphere}$$ Then the surface-mass density (SMD) ${\Sigma}_{S}(R)$ at $R$ is calculated by \_[S]{}(R) = 2 \_0\^ (r) dz, \[smdsphere\] where $$\rho(r) =\frac{1}{4 \pi r^2} \frac{dM(r)}{dr}. \label{rhosphere}$$ If the galaxy is assumed to be a flat thin disk, the SMD ${\Sigma}_{\rm d}(R)$ is calculated by solving Poisson’s equation (Freeman 1970; Binney and  Tremaine 1987) by $${\Sigma}_{\rm d}(R) =\frac{1}{{\pi}^2 G} \left[ \frac{1}{R} \int\limits_0^R {\left(\frac{dV^2}{dr} \right)}_x K \left(\frac{x}{R}\right)dx + \int\limits_R^{\infty} {\left(\frac{dV^2}{dr} \right)}_x K \left (\frac{R}{x}\right) \frac{dx}{x} \right]. \label{smdflat}$$ Here, $K$ is the complete elliptic integral, which becomes very large when $x\simeq R$. The SMD distributions in the galaxy for the sphere and flat-disk cases have been calculated for the recent RCs \[[@Sofue2017]\].In this paper we apply the same method to the here obtained unified RC (Figure \[mwrc\]). Since we aim at studying the dark halo, which is postulated to be rather spherical than a flat disk, we assume spherical mass distribution. The calculated SMD distribution is shown in Figure \[smd\_fit\]. ![(**Top**) Direct surface-mass density (SMD) calculated for the unified RC in figure \[mwrc\] in spherical symmetry assumption (dots with error bars) in semi-logarithmic representation. The solid line is the $\chi^2$ fit, and red, blue, and dashed lines represent the NFW halo, disk, and bulge, respectively. (**Bottom**) Same, but in log–log plots. The semi-logarithmic plot makes it easier to discriminate the dark halo from exponential disk, which appears as a straight line. The plotted values are listed in the tables in Appendix \[AppendixA\]. []{data-label="smd_fit"}](fig3.eps){width="10cm"} The SMD is strongly concentrated toward the center, reaching a value as high as $\sim 10^5 \Msun~{\rm pc}^{-2}$ within $R\sim 10$ pc, representing the core of the central bulge with the extent of several hundred pc. It is followed by a straightly declining profile from $R\sim 2$ to 8 kpc in the semi-logarithmic plot, representing the exponential nature of the galactic disk. In the outer galaxy beyond $\sim 8$ kpc, the SMD profile tends to be displaced from the straight disk profile, and is followed by an extended outskirt with a slowly declining profile, representing a massive halo extending to the end of the RC measurement at . Fitting by Bulge, Disk, and Dark Halo ------------------------------------- In order to separate the dark halo from the disk and bulge components, the well established RC decomposition method has been extensively applied to the RCs \[[@Sofue2017; @Salucci2019]\].Besides this traditional method, we here propose to use the SMD distribution. For this, we assume three mass components of bulge, exponential disk, and dark halo. In order to represent the, we employ the NFW profile as a ’tool’ for its popularity and for the dynamics background based on the extensive numerical simulations. We employ the least $\chi^2$ fitting method, where $\chi^2$ is defined by \^2=\_i\[(SMD\_i\^[direct]{} - SMD\_i\^[calc]{})/\_i\]\^2, with $i$ denoting the value at the $i$-th data point, and $\sigma_i$ is the standard deviation around each data point in the running averaging procedure of the SMD distribution. Fitting parameters are the scale radius $a_d$ and central SMD $\Sigma_d^0$ for the disk, and the scale (core) radius $h$ and representative DM density $\rho_{\rm model}^0$ for the halo. The bulge SMD is fixed to an assumed profile, which is negligible in the present fitting range at $R\ge 1$ kpc. The fitting was obtained between $R=1$ and 100 kpc. The fitting result for the NFW halo model is shown in Figure \[smd\_fit\]. The solid line is the $\chi^2$ fit to SMD, and red, blue, and dashed lines represent the halo, disk, and bulge components, respectively. Note that the semi-logarithmic plot makes it visually easier to recognize the dark halo significantly displaced from the exponential disk, which appears as a straight line. Local DM Density ---------------- We thus obtained the NFW DM halo parameters to be $h=10.94\pm 1.05$ kpc, $\rho_{\rm NFW}^0=0.787\pm 0.037$ , which yields the local DM density $\rho_\odot=0.359\pm 0.017$ . The best-fit parameters for the disk are determined to be $a_d=4.38\pm 0.35 $ kpc and $\Sigma_0=(1.28\pm 0.09)\times 10^3 \Msun {\rm pc}^{-2}$. Table \[tab\_fit\] lists the fitted result along with the minimized $\chi^2$ value. **Component** **Parameter** **Fitted Value** ------------------------- -------------------- ------------------------------------------------- -------- Expo. disk $a_d$ $4.38\pm 0.35 $ kpc $\Sigma_0$ $(1.28\pm 0.09)\times 10^3 \Msun {\rm pc}^{-2}$ NFW dark halo $h$ $10.94\pm 1.05$ kpc $\rho_{\rm NFW}^0$ $0.787\pm 0.037$ $\rho_\odot$ $0.359\pm 0.017$ 11.9 Burkert$^\dagger$ $\rho_\odot$ $\sim 0.30\pm 0.02$ $17.3$ Brownstein$^\dagger$ $\rho_\odot$ $\sim 0.40\pm 0.02$ $17.9$ $\beta$ model$^\dagger$ $\rho_\odot$ $\sim 0.31\pm 0.02$ $17.3$ : Best-fit parameters of the direct SMD by NFW halo and exponential disk.[]{data-label="tab_fit"} $^\dagger$ Rough fitting, not conclusive. We also obtained $\chi^2$ fitting using the Burkert, Brownstein, and $\beta$ profiles, and listed the local DM density and minimized $\chi^2$ in Table \[tab\_fit\]. In these three models, the $\chi^2 \sim 17-18$ were found to be systematically greater than that for the NFW model ($\chi^2=11.9$). The reason for the difference is due to the systematic difference in the functional behavior between NFW and the other three models: NSF has a cusp steeply increasing toward the center with sharpening scale radius, which results in the possibility of finer fitting to the slightly curved SMD profile at $R<\sim 10$ kpc in the semi-log plot. On the contrary, the other three models predict almost negligible SMD there, so that halo parameters contribute less intensively to the fitting in the innermost region, or the fitting must be done only by the disk’s two parameters there, resulting in worse fitting. ----------------------------------------------------------------------- ------------------- ----------- -------- -- [**Reference**]{} **(kpc)** **()** Weber and de Boer (2010)\[[@Weber+2010]\] 0.2 - 0.4 Catena and Ulio (2010)\[[@Catena+2010]\] $0.389 \pm 0.025$ Bovy and Tremaine (2012) \[[@Bovy+2012]\] $0.3\pm 0.1$ Piffl et al. (2014) \[[@Piffl+2014]\] 0.58 Pato et al (2015), Pato & Iocco (2015) \[[@Pato+2015a; @Pato+2015b]\] $0.42\pm 0.25$ 230 Huang et al. (2016)\[[@Huang+2016]\] $0.32 \pm 0.02$ 8.34 240 McMillan (2017)\[[@McMillan2017]\] $0.38\pm 0.04$ 8.21 233.1 Lin and Li (2019)\[[@Lin+2019]\] $0.51 \pm 0.09$ 8.1 240 Salucci et al. (2010, 2019) \[[@Salucci+2010; @Salucci2019]\] $0.43\pm 0.06$ 8.29 239 Eilers et al (2019) \[[@Eilers+2019]\] $0.3\pm 0.03$ 8.1 229 de Salas et al. (2019) \[[@deSalas+2019]\] $0.3 - 0.4$ Cautun et al (2019) \[[@Cautun+2019]\] $0.34\pm 0.02$ 8 229 Karukes et al (2019) \[[@Karukes+2019]\] $0.43 \pm 0.02$ 8.34 240 Sofue (2013) \[[@Sofue2013]\] $0.40 \pm 0.04$ 8.0 238 —– (2020 this paper) $0.36\pm 0.02$ 8.0 238 Average$^\ddagger$ $0.387 \pm 0.080$ ----------------------------------------------------------------------- ------------------- ----------- -------- -- : Current determinations of the local DM density and the literature.[]{data-label="tab_localdm"} $^\dagger$ =38.2 $\Msun\ {\rm pc}^{-3}$. $^\ddagger$ Simple average of the listed values with equal weighting.\ The local DM density is a key quantity in laboratory experiments by the direct detection of DM, and has been estimated by a number of authors with a variety of methods. In Table \[tab\_localdm\] we list the local DM densities from the literature along with the present value for NFW profile. They are also plotted in Figure \[localDMauthors\] against publication years. The $\rho_\odot$ values seem to be nearly constant in the decade. Averaging all the listed values with an equal weighting yields $\rho_\odot=0.39 \pm 0.09$ , which may be taken as a ’canonical’ value. ![Local dark matter (DM) density from the literature (Table \[tab\_localdm\]) plotted against publication year. The dashed line indicates a simple mean of the plots at $\rho_\odot=0.39 \pm 0.09$ .[]{data-label="localDMauthors"}](fig4.eps){width="9cm"} Dependence on the Galactic Constants ------------------------------------ We have re-scaled the adopted RC to $(R_0,V_0)=(8.0, 238)$ (kpc, ), which may vary within several %. The resulting local DM density will vary accordingly, depending on the constants. The local mass density of the spherical component is dependent on the constants as $\rho_0 \propto R_0 V_0^2/R_0^3 \sim V_0^2 R_0^{-2}$. For small corrections $\delta V_0$ and $\delta R_0$, the DM density will change as $\delta \rho_0/\rho_0 \sim 2(\delta V_0/ V_0-\delta R_0/R_0)$. For example, for $\delta V_0 \sim \pm $ 10 , the estimated local density varies by $\delta \rho_0/\rho_0 \sim \pm 0.08$, or for $\delta R_0 \sim \pm 0.1$ kpc, $\delta \rho_0/\rho_0 \sim \mp 0.025$. Summary ======= We reviewed the current status of determination of the RC of the Milky Way, and presented a unified RC from the GC to outer halo at $R\sim 100$ kpc. The RC was used to directly calculate the SMD without assuming any functional form. The disk appears as a straight line on the semi-logarithmic plot of SMD against $R$, and is visually well discriminated from the DH having an extended outskirt. The SMD distribution was fitted by a bulge, disk, and NFW dark halo using the $\chi^2$ method. The best-fit DH profile yielded the local DM density of $0.359 \pm 0.017$ . We also reviewed the current estimations from the literature in the last decade, which appear to be converging to a mean value of $\rho_\odot=0.39 \pm 0.09$ . [[**Acknowledgments**]{}]{} [The data analysis was performed at the Center of Astronomical Data Analysis of the National Astronomical Observatory of Japan. The author is grateful to Professor A. Hofmeister for inviting him to this special issue. ]{} Tables concerning the RC and SMD of the Milky Way {#AppendixA} ================================================= Tables \[tabrcA\] and \[tabrcB\] list the running-averaged RC of the Milky Way using the data from \[[@Sofue2015; @Sofue2017; @Huang+2016]\],which is used to calculate the SMD in Figure \[smd\_fit\]. Tables \[tab\_smdsA\] and \[tab\_smdsB\] lists the directly calculated SMD from the RC \[0.7\] ------------ ------- ------------------- **Radius** **Standard Dev**. (kpc) () () 0.100 144.9 3.7 0.110 147.4 4.2 0.121 150.4 4.8 0.133 153.8 6.1 0.146 158.9 10.3 0.161 167.4 16.1 0.177 180.1 22.4 0.195 196.6 27.1 0.214 213.6 26.9 0.236 227.8 22.7 0.259 237.9 17.0 0.285 244.4 11.8 0.314 248.2 7.6 0.345 250.2 4.7 0.380 251.0 2.9 0.418 250.7 2.1 0.459 249.7 2.3 0.505 248.0 2.9 0.556 245.9 3.7 0.612 243.2 4.6 0.673 239.8 5.7 0.740 235.8 6.4 0.814 231.7 6.5 0.895 227.8 6.0 0.985 224.5 5.2 1.083 221.7 4.5 1.192 219.1 4.0 1.311 216.8 3.7 1.442 214.7 3.4 1.586 212.7 3.1 1.745 210.9 2.8 1.919 209.5 2.3 2.111 208.5 1.8 2.323 208.2 1.6 ------------ ------- ------------------- : Rotation curve of the Milky Way used in Figure \[mwrc\].[]{data-label="tabrcA"} \[0.7\] ------------ ------- ------------------- **Radius** **Standard Dev**. (kpc) () () 2.555 208.9 2.2 2.810 210.7 3.6 3.091 213.4 4.8 3.400 217.2 5.9 3.740 222.0 6.6 4.114 226.6 5.7 4.526 229.5 4.4 4.979 231.6 4.3 5.476 234.1 5.3 6.024 237.2 5.7 6.626 239.5 5.0 7.289 240.1 4.1 8.018 239.0 4.4 8.820 236.7 5.4 9.702 234.5 6.0 10.672 234.2 7.1 11.739 237.1 9.8 12.913 242.8 12.4 14.204 248.5 13.3 15.625 249.7 14.8 17.187 246.2 17.4 18.906 243.3 18.3 20.797 243.9 17.5 22.876 245.6 15.6 25.164 243.7 15.2 27.680 237.3 16.1 30.448 229.6 15.5 33.493 222.5 14.1 36.842 215.0 14.0 40.527 207.1 13.8 44.579 200.3 12.7 49.037 194.7 11.9 53.941 189.8 11.3 59.335 186.2 10.4 65.268 184.7 9.6 71.795 183.9 9.3 78.975 181.4 11.0 86.872 175.5 14.6 95.560 167.7 16.3 ------------ ------- ------------------- : Continued from Table \[tabrcA\].[]{data-label="tabrcB"} \[0.7\] ------------ --------- ------------------- **Radius** **Standard Dev.** **(kpc)** 0.100 29933.0 861.3 0.110 29054.0 654.8 0.121 28384.0 666.0 0.133 28160.0 570.8 0.146 28319.0 637.2 0.161 28203.0 1406.6 0.177 27368.0 2481.1 0.195 25014.0 3514.2 0.214 21548.0 4357.7 0.236 17908.0 4806.8 0.259 14804.0 4733.1 0.285 12369.0 4231.5 0.314 10489.0 3549.3 0.345 8978.9 2929.7 0.380 7736.5 2384.1 0.418 6700.8 1959.1 0.459 5830.8 1636.9 0.505 5090.6 1374.2 0.556 4452.0 1158.1 0.612 3899.9 973.5 0.673 3464.9 803.4 0.740 3145.2 644.7 0.814 2904.3 510.8 0.895 2701.7 415.1 0.985 2510.1 354.4 1.083 2320.9 319.0 1.192 2144.8 291.9 1.311 1985.0 266.7 1.442 1843.6 240.7 1.586 1718.4 214.5 1.745 1611.4 188.2 1.919 1519.3 164.5 2.111 1440.6 144.4 2.323 1368.3 130.9 ------------ --------- ------------------- : Directly calculated SMD by spherical assumption of the mass distribution.[]{data-label="tab_smdsA"} \[0.7\] ------------ ------------------------- ------------------------- **Radius** Standard Dev. (kpc) $\Msun \ {\rm pc}^{-2}$ $\Msun \ {\rm pc}^{-2}$ 2.555 1296.4 125.1 2.810 1220.5 126.3 3.091 1139.8 134.1 3.400 1055.2 146.1 3.740 944.3 157.4 4.114 824.6 161.5 4.526 734.9 150.5 4.979 668.3 133.7 5.476 600.8 123.8 6.024 523.5 119.3 6.626 446.8 113.6 7.289 383.7 101.6 8.018 339.1 83.4 8.820 314.1 62.2 9.702 303.4 43.9 10.672 293.2 39.0 11.739 272.0 48.2 12.913 229.5 58.5 14.204 170.5 65.2 15.625 127.5 62.4 17.187 114.7 46.4 18.906 110.0 35.2 20.797 91.8 33.5 22.876 61.2 34.5 25.164 37.2 33.2 27.680 28.0 25.4 30.448 24.7 17.1 33.493 20.3 11.6 36.842 17.3 7.8 40.527 17.1 4.7 44.579 16.7 3.0 49.037 15.2 2.3 53.941 14.4 2.3 59.335 13.4 3.3 65.268 10.6 4.3 71.795 6.2 4.9 ------------ ------------------------- ------------------------- : Continued from Table \[tab\_smdsA\].[]{data-label="tab_smdsB"} [**References**]{} [999]{} 1\. Sofue, Y.; Rubin, V.  Rotation Curves of Spiral Galaxies. *Annu. Rev. Astron. Astrophys.* **2001**, *39*, 137. 2\. 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ArXiv

--- abstract: 'A Hermitian symplectic manifold is a complex manifold endowed with a symplectic form $\omega$, for which the bilinear form $\omega(I\cdot,\cdot)$ is positive definite. In this work we prove ${dd^c}$-lemma for 1- and (1,1)-forms for compact Hermitian symplectic manifolds of dimension 3. This shows that Albanese map for such manifolds is well-defined and allows one to prove Kählerness if the dimension of the Albanese image of a manifold is maximal.' author: - Grigory Papayanov date: 2015 title: Cohomological properties of Hermitian sympletic threefolds --- [**Cohomological properties of Hermitian\ symplectic threefolds** ]{}\ Grigory Papayanov\ Introduction {#introduction .unnumbered} ============ A Hermitian symplectic manifold is a complex manifold $(M,I)$ together with a symplectic form $\omega$, for which the bilinear form $\omega(I\cdot,\cdot)$ is positive definite (that is, $\omega(IX,X)>0$ for any vector field $X$ on $M$). Any Kähler manifold is obviously Hermitian symplectic, and it is an open problem whether there exist other examples of Hermitian symplectic manifolds. Hermitian symplectic manifolds were studied by Streets and Tian in [@Streets_Tian:pluriclosed] and [@Streets_Tian:flow]; they constructed an appropriate Ricci flow on Hermitian symplectic manifolds, and studied its convergency properties. Since then, many people searched for non-trivial examples of Hermitian symplectic manifolds. The search for non-Kähler examples of Hermitian symplectic manifolds was vigorous, but ultimately unsuccessful. All common sources of examples of non-Kähler manifolds were tapped at some point. For complex dimension 2, Hermitian symplectic structures are all Kähler. This was shown by Streets and Tian in [@Streets_Tian:pluriclosed]. Another proof could be obtained from the Lamari ([@Lamari]) result about existence of positive, exact $(1,1)$-current on any non-Kähler complex surface. In [@Peternell], it was shown that any non-Kähler Moishezon manifold admits an exact, positive $(n-1,n-1)$-current; therefore, Moishezon manifolds which are Hermitian symplectic are also Kähler. In [@Enrietti_Fino_Vezzoni] it was shown that no complex nilmanifold can admit a Hermitian symplectic structure, and in [@Fino_Kasuya_Vezzoni] this result was extended to all complex solvmanifolds and Oeljeklaus-Toma manifolds. Existence of Kähler metric implies some restrictions on the cohomology of a manifold: for example the Frölicher spectral sequence of Kähler manifold always degenerates at the first page. Results of Cavalcanti ([@Cavalcanti:SKT]) show that the Frölicher spectral sequence for Hermitian symplectic manifolds degenerates at the first page. In this work we define some Laplacian-like operators, kernels of which conjecturally isomorphic to the spaces of cohomology, and, with the help of these operators, prove ${dd^c}$-lemma for (1,1)-forms on Hermitian symplectic threefolds. Argument of Gauduchon ([@Gauduchon]) shows that ${dd^c}$-lemma for (1,1)-forms is equivalent to the equality $b^1=2h^{0,1}$. It follows that the Albanese map is well-defined and, if its image is not a point, the generic fiber of ${\operatorname{Alb}}$ is Kähler. The question of existence of special (e.g. Kähler or balanced) metrics on total spaces of maps with Kähler base and fibers is studied, for example, in [@HL] and [@Michelsohn]. Using the Albanese map, we are able to prove that if a Hermitian symplectic threefold $M$ has ${\operatorname{dim}}{\operatorname{Alb}}(M)=3$, then it admits a Kähler metric, and if ${\operatorname{dim}}{\operatorname{Alb}}(M)=1$, $M$ is balanced. If $dd^c$-lemma holds for $(2,2)$-forms, then by [@HL] ${\operatorname{dim}}{\operatorname{Alb}}(M)=2$ would imply that $M$ is Kähler, but, unfortunately, we have not proven ${dd^c}$-lemma in full generality yet. [**Acknowledgements.**]{} The author would like to thank M.Verbitsky for many extremely helpful discussions. Work on sections 1–3 was supported by RSCF, grant number 14-21-00053, within the Laboratory of Algebraic Geometry. Work on section 4 was supported by RFBR 15-01-09242. Preliminaries ============= [ ]{}Let $M$ be a smooth manifold of dimension 2n, $I:TM {{\:\longrightarrow\:}}TM$ an integrable complex structure, ${\mathcal{A}}^{p,q}$ the corresponding Hodge decomposition on the bundle of differential forms: ${\mathcal{A}}^n\otimes {{\Bbb C}}=\bigoplus_{n=p+q}{\mathcal{A}}^{p,q}$, ${\omega^{1,1}}$ a form in ${\mathcal{A}}^{1,1}$. We will say that ${\omega^{1,1}}$ is [*Hermitian*]{} if the tensor $h(\cdot,\cdot):={\omega^{1,1}}(\cdot,I\cdot)$ is a Riemannian metric on $M$, and we will say that ${\omega^{1,1}}$ is [*Hermitian symplectic*]{} if there exists a symplectic form $\omega$ such that ${\omega^{1,1}}$ is the (1,1)-component in the Hodge decomposition of $\omega$. If $M$ is endowed with such ${\mathcal{I}}$ and ${\omega^{1,1}}$, we will call it a Hermitian symplectic manifold. For a Hermitian symplectic manifold $(M,I,\omega)$, let $d: {\mathcal{A}}^\bullet{{\:\longrightarrow\:}}{\mathcal{A}}^{\bullet+1}$ be the usual de Rham differential acting on forms, $d^c:=IdI^{-1}: {\mathcal{A}}^{\bullet}{{\:\longrightarrow\:}}{\mathcal{A}}^{\bullet+1}$ the twisted differential, $L: A^\bullet{{\:\longrightarrow\:}}A^{\bullet+2}$ the operator of (left) multiplication by $\omega$, $L(\eta):= \omega\wedge \eta$, $\Lambda: {\mathcal{A}}^{\bullet}{{\:\longrightarrow\:}}{\mathcal{A}}^{\bullet-2}$ the adjoint operator ([@Yau_Tseng]). In the local Darboux coordinates $p_i, q_i$ where $\omega=\sum dp_i\wedge dq_i$, operator $\Lambda$ looks like $\sum i_{\!\frac{{\partial}}{{\partial}p_i}}i_{\!\frac{{\partial}}{{\partial}q_i}}$. We will denote by $L^{1,1}$ the operator of multiplication by the hermitian form ${\omega^{1,1}}$, and by $\Lambda^{1,1}$ the adjoint operator to $L^{1,1}$. [ ]{}\[SKT\] The form ${\omega^{1,1}}$ is the SKT form, that is, ${\partial}{\overline}{\partial}{\omega^{1,1}}=0$. [**Proof:**]{} Let $\omega={\omega^{1,1}}+\alpha$, where $\alpha$ lies in ${\mathcal{A}}^{2,0}\oplus{\mathcal{A}}^{0,2}$. Since $d\omega=0$, ${\partial}{\omega^{1,1}}=-{\overline}{\partial}\alpha$ and ${\partial}{\overline}{\partial}{\omega^{1,1}}={\overline}{\partial}^2\alpha=0$. [ ]{}Let $\alpha$ be a differential form on $M$. We will say that $\alpha$ is [*primitive with respect to $\omega$*]{} if $\Lambda\alpha=0$, and that $\alpha$ is primitive with respect to ${\omega^{1,1}}$ if $\Lambda^{1,1}\alpha=0$. [ ]{}(The Weil identities). Let $B^{p,q}$ be a primitive with respect to ${\omega^{1,1}}$ $(p,q)$-form, $p+q=r$. Then the following formula holds ([@Voisin Proposition 6.29]): $$*B^{p,q}=(-1)^{\frac{r(r+1)}{2}}(\sqrt{-1})^{p-q}\frac{1}{(n-r)!}({\omega^{1,1}})^{n-k}\wedge B^{p,q}.$$ [ ]{}An operator $\Delta$ defined as double graded commutator, $\Delta:=\{d,\{d^c,\Lambda^{1,1}\}\}$ is called [*the Hermitian symplectic*]{} Laplacian. [ ]{}$\Delta$ is not a Laplacian associated to the Riemannian metric $h$. Nevertheless they differ by a differential operator of first order (see e.g. [@Liu_Yang] for the exact formula), therefore they have equal symbols, so $\Delta$ is elliptic. Recall the graded Jacobi identity for the graded commutator: $$\{a,\{b,c\}\}=\{\{a,b\},c\}+(-1)^{deg(a)deg(b)}\{b,\{a,c\}\}.$$ [ ]{}\[commutators\] $\Delta=\{d^c,\{d,\Lambda^{1,1}\}\}$. Therefore $\Delta$ commutes with $d$ and with $d^c$. [**Proof:**]{} Follows simply from the Jacobi identity. [ ]{}\[spectral\] (Spectral theorem). Let $(M, I,\omega)$ be a compact Hermitian symplectic manifold. Then the space of differential forms decomposes as a topological direct sum of generalized eigenspaces of $\Delta$: ${\mathcal{A}}^\bullet(M)=\bigoplus_{\lambda_i}{\mathcal{A}}^\bullet_{\lambda_i}(M)$, each component of this decomposition is finite-dimensional and preserved by $d$, $d^c$ and $\delta$. [**Proof:**]{} Decomposition is in fact proven in [@BGV Proposition 2.36] ($\Delta$ is a generalized laplacian in their terminology); one has to apply spectral theorem for compact operators: compact operator on Hilbert space has a canonical Jordan form with finite-dimensional generalized eigenvalues ([@Conway]). By \[commutators\], $\Delta$ commutes with $d$ and $d^c$, so all generalized eigenspaces are in fact subcomplexes. [ ]{}Let $\alpha$ be a closed form in $\bigoplus_{\lambda_i\ne 0}{\mathcal{A}}^\bullet_{\lambda_i}(M)$. Then $\alpha$ is exact. [**Proof:**]{} When restricted to $\bigoplus_{\lambda_i\ne 0}{\mathcal{A}}^\bullet_{\lambda_i}(M)$, Laplacian $\Delta$ has an inverse, $\Delta^{-1}$. So $$\alpha=\Delta\Delta^{-1}\alpha=(\pm dd^c\Lambda \pm d\Lambda d^c)\Delta^{-1}\alpha.$$ Forms on a Hermitian symplectic manifold ======================================== In this section $M$ is assumed to be compact. [ ]{}\[ddc1\] ($dd^c$-lemma for 1-forms). Let $\alpha$ be a $d$-exact, $d^c$-closed (or $d^c$-exact and $d$-closed) 1-form. Then $\alpha=0$. [**Proof:**]{} Suppose $\alpha$ is $d$-exact, $\alpha=df$. Then $dd^cf=0$. By Hopf maximum principle ([@_Gilbarg_Trudinger_]), $f$ is constant, hence $\alpha=df=0$. We will now investigate whether holomorphic forms on $M$ are closed. [ ]{}Let the $n$ be the complex dimension of $M$. Then every holomorphic $n-2$-form is closed. [**Proof:**]{} Let $\alpha$ be a holomorphic $n-2$-form, $\alpha \in {\mathcal{A}}^{n-2,0}$, ${\overline}{\partial}\alpha=0$. Then $d\alpha={\partial}\alpha$ is primitive with respec to ${\omega^{1,1}}$, by dimension reasons. So, by Weil identities, $$||d\alpha||^2=\int d\alpha\wedge d{\overline}\alpha \wedge {\omega^{1,1}}=\int {\partial}\alpha\wedge {\overline}{\partial}{\overline}\alpha \wedge {\omega^{1,1}}= \alpha \wedge {\overline}\alpha \wedge {\partial}{\overline}{\partial}{\omega^{1,1}}=0.$$ Hence $\alpha$ is closed. [ ]{}\[holoforms\] Obviously, on any compact complex manifold of complex dimension $n$, every holomorphic function and every holomorphic $n$-form is closed. Every holomorphic $n-1$-form is also closed, as the simple argument with the integration shows. So, any holomorphic form on a Hermitian symplectic threefold is closed. $dd^c$-lemma for (1,1)-forms ============================ Recall that by \[spectral\] every differential form $\alpha$ decomposes by generalized eigenspaces of $\Delta$: $\alpha=\alpha_0 + \alpha_{\ne 0}$, where $\Delta^N(\alpha_0)=0$ for some $N$, and $\alpha_{\ne 0}=\Delta\Delta^{-1}\alpha_{\ne 0}$. Suppose that $\alpha$ is $d$-exact and $d^c$-closed. Then $\alpha_0$ and $\alpha_{\ne 0}$ are also $d$-exact and $d^c$-closed. [ ]{}In notations as above, $\alpha_{\ne 0}$ is $dd^c$-exact. [**Proof:**]{} by \[commutators\], $\Delta^{-1}$ commutes with $d$ and $d^c$, so $\Delta\Delta^{-1}\alpha_{\ne 0}=dd^c\Lambda^{1,1}\Delta^{-1}\alpha_{\ne 0}=\alpha$. [ ]{}\[primitivness\] Suppose exact (1,1)-form $\eta=d\gamma$ lies in the kernel of $\Delta^{1,1}$. Then $\eta$ is primitive (with respect both to $\omega$ and to $\omega^{1,1}$). [**Proof:**]{} $\Delta\eta=dd^c\Lambda^{1,1}\eta=0$, so, by Hopf maximum principle [@_Gilbarg_Trudinger_] $\Lambda^{1,1}\eta=c$, where $c$ is some constant. It means that $\Lambda\eta$ also equals $c$. If $\Lambda\eta=c,$ then $\eta=c\omega+B$, where $B$ is a primitive form. Since $\eta=d\gamma$, the cohomology classes of $c\omega$ and $B$ are equal, but the cohomology class of a symplectic form cannot be represented by a primitive form. Indeed, $\omega\wedge\omega^{n-1}$ is a volume form, hence nonzero in cohomology, but $B\wedge\omega^{n-1}=0$. So $c=0$ and $\eta$ is primitive. [ ]{}\[vanish\] Suppose ${\operatorname{dim}}(M)=3$, $\eta=dd^cf$ is $dd^c$-exact primitive $(1,1)$-form. Then $\eta=0$. [**Proof:**]{} Note first that, since $\eta$ is primitive with respect to $\omega$, it is primitive with respect to $\omega^{1,1}$, so, by Weil identities, $*\eta=\eta\wedge(\omega^{1,1})^{\wedge n-2}$, where $*$ is the Hodge star operator associated with the Hermitian metric $h$ with corresponding 2-form equal to $\omega^{1,1}$ ([@Griffiths_Harris]). Then $h(\eta,\eta)=$ $$=\int \eta\wedge*\eta=\int \eta\wedge\eta\wedge(\omega^{1,1})^{\wedge n-2}=\int f\eta\wedge dd^c(\omega^{1,1})^{\wedge n-2}.$$ But $dd^c\omega^{1,1}=0$ on a Hermitian symplectic manifold, so the integral vanishes. Since $h$ is a hermitian metric, $\eta$ also equals to zero. [ ]{}Let ${\operatorname{dim}}(M)=3$. Suppose that an exact (1,1)-form $\eta=d\gamma$ lies in the kernel of $(\Delta)^n,$ $n>1$. Then $\eta$ lies in the kernel of $(\Delta)^{n-1}.$ [**Proof:**]{} $(\Delta)^{n-1}\eta$ is an exact (1,1)-form lying in the kernel of $\Delta$, so, by \[primitivness\] it is primitive. Since $d\eta=d^c\eta=0, (\Delta)^{n-1}\eta=(dd^c\Lambda)^{n-1}\eta$, it is $dd^c$-exact, therefore, by \[vanish\], it vanishes. In order to complete the proof of ${dd^c}$-lemma for (1,1)-forms on Hermitian symplectic manifolds, we have to prove that an exact, primitive (1,1)-form vanishes. [ ]{} Let $M$ be a Hermitian symplectic manifold of dimension 3, $\eta$ be an exact, primitive (1,1)-form on $M$. Then $\eta=0$. [**Proof:**]{} Square of Hermitian norm of $\eta$ is equal to $\int \eta\wedge\eta\wedge\omega^{1,1}$, but in dimension 3 we have the equality $\eta\wedge\eta\wedge\omega^{1,1}=\eta\wedge\eta\wedge\omega$; the latter form is exact, therefore $\eta=0$. [ ]{}\[ddc11\] Let $M$ be a compact Hermitian symplectic threefold, $\alpha$ is a $d$-closed, $d^c$-exact $(1,1)$-form. Then $\alpha=dd^cf$ for some function $f$. Applications ============ [ ]{}(Gauduchon, [@Gauduchon]). For a complex manifold $M$, $dd^c$-lemma for $(1,1)$-forms is equivalent to the equality $b^1=2h^{1,0}$. [**Proof:**]{} Consider the cohomology sequence associated to the short exact sequence of sheaves of the form $0 {{\:\longrightarrow\:}}\sqrt{-1}{{\Bbb R}}{{\:\longrightarrow\:}}\mathcal{O} \stackrel{Re}{{{\:\longrightarrow\:}}} {{\mathcal{H}}}{{\:\longrightarrow\:}}0$, where ${{\mathcal{H}}}$ is the sheaf of pluriharmonic functions. The relevant piece looks like $$... \stackrel{0}{{{\:\longrightarrow\:}}} H^1(M,\sqrt{-1}{{\Bbb R}}) {{\:\longrightarrow\:}}H^1(M,\mathcal{O}) {{\:\longrightarrow\:}}H^1(M,{{\mathcal{H}}}) {{\:\longrightarrow\:}}H^2(M,\sqrt{-1}{{\Bbb R}}) {{\:\longrightarrow\:}}...$$ It is well-known that $$H^1(M,{{\mathcal{H}}})=\frac{{\operatorname{Ker}}d:{\mathcal{A}}^{1,1} {{\:\longrightarrow\:}}{\mathcal{A}}^3}{{\operatorname{Im}}dd^c: {\mathcal{A}}^0 {{\:\longrightarrow\:}}{\mathcal{A}}^{1,1}}.$$ So $dd^c$-lemma for $(1,1)$-forms holds if and only if the third arrow is an isomorphism, and, by exactness, if and only if the first arrow is an isomorphism. So, the equality $b^1=2h^{1,0}$ holds on compact Hermitian symplectic threefolds. It follows that we have the Hodge decomposition on the first cohomology of $M$: $H^1(M,{{\Bbb C}})=H^{0,1}(M) \oplus H^{1,0}(M)$, and ${\operatorname{dim}}H^{0,1}={\operatorname{dim}}H^{1,0}$. So the rank of the abelian group $H^1(M,{{\Bbb Z}})$ is equal to the dimension of the real vector space $H^{0,1}(M)$. It follows that the Albanese torus is defined correctly and we have the Albanese map ${\operatorname{Alb}}: M {{\:\longrightarrow\:}}H^{0,1}(M)^*/H_1(M,{{\Bbb Z}})$. Its image ${\operatorname{Alb}}(M)$ is a subvariety (possibly singular) of a torus. [ ]{}Suppose ${\operatorname{dim}}{\operatorname{Alb}}(M)=3$. Then $M$ is Kähler. [**Proof:**]{} If ${\operatorname{Alb}}(M)$ is smooth, then ${\operatorname{Alb}}$ is an immersion, and pullback of the Kähler form ${\operatorname{Alb}}^*\omega$ is the Kähler form on $M$. Otherwise, we can desingularize the morphism ${\operatorname{Alb}}$ to obtain the Kähler metric on some manifold $\tilde{M}$ bimeromorphic to $M$ ($M$ is then a manifold in the Fujiki class C). On the other hand, $M$ admits an SKT structure (\[SKT\]). From the theorem of Chiose ([@Chiose]) it follows that $M$ is Kähler. [ ]{}It would be interesting to know what one can extract from the Albanese map if ${\operatorname{dim}}{\operatorname{Alb}}(M)=1$ or $2$. For example, if ${\operatorname{Alb}}(M)$ is a smooth curve $C$, fibers of ${\operatorname{Alb}}(M)$ are Hermitian symplectic (and therefore Kähler) surfaces, and the pullback of the volume form ${\operatorname{Alb}}^* {\operatorname{Vol}}_C$ is a closed, non-exact $(1,1)$-form on $M$. By $dd^c$-lemma for $(1,1)$-forms and \[holoforms\], it could not be cohomologous to a form of type $(2,0)+(0,2)$. By a theorem of Michelsohn ([@Michelsohn]), in that situation there exists a [*balanced*]{} metric on $M$, that is, a Hermitian form $\omega$ such that $d\omega^{{\operatorname{dim}}M - 1}=0$. Actually, the smoothness of $C$ is not necessary, because a manifold bimeromorphic to a balanced manifold is balanced itself ([@AB]). [100]{} L. Alessandrini, G. Bassanelli, [*Modifcations of compact balanced manifolds*]{}, In: C.R. Acad. Sci. Paris Math. [**320**]{} (1995), 1517–1522 Nicole Berline, Ezra Getzler, and Michèle Vergne, [*Heat Kernels and Dirac Operators*]{}, Grundlehren Math. Wiss., vol. 298, Springer-Verlag, New York, 1992. Brylinski, Jean-Luc; [*A differential complex for Poisson manifolds*]{}, In: J. Differential Geom. Volume 28, Number 1 (1988), 93–114. Gil R. Cavalcanti, [*Hodge theory and deformations of SKT manifolds*]{}, arXiv:1203.0493 Chiose, I.; [*Obstructions to the existence of Kähler structures on compact complex manifolds*]{}, In: Proc. of the Amer. Math. Soc., [**142**]{} (2014), no. 10, 3561–3568. John B. Conway, [*A course in Functional Analisys*]{}, Graduate Texts in Mathematics 96, Springer 1990. Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis; [*Real homotopy theory of Kähler manifolds*]{}, In: Invent. Math. [**29**]{} (1975), no. 3, 245–274. Enrietti, Nicola; Fino, Anna; Vezzoni, Luigi; [*Tamed symplectic forms and SKT metrics,*]{} arxiv:1002.3099. Fino, Anna; Kasuya, Hisashi; Vezzoni, Luigi; [*Tamed complex structures on solvmanifolds*]{}, arxiv:1302.5569. Gauduchon, Paul; [*La 1-forme de torsion d’une variété hermitienne compacte*]{}, Math. Ann. 267, 495-518 (1984). D. Gilbarg,N. S. Trudinger, [*Elliptic Partial Differential Equations of Second Order,*]{} Springer-Verlag, 1983 Griffiths, Phillip; Harris, Joe; [*Principles of Algebraic Geometry*]{}, Wiley-Interscience, New York, 1978. Harvey, Reese; Lawson, H. Blaine; [*An intrinsic characterization of Kähler manifolds*]{}, Invent. math. [**74**]{}, 169–198 (1983). A. Lamari. *Courants kählériens et surfaces compactes* (French) \[Kähler currents and compact surfaces\]. Ann. Inst. Fourier (Grenoble) **49** (1999), no. 1, 263–285. Kefeng Liu, Xiaokui Yang; *Geometry of Hermitian manifolds,* arXiv:1011.0207 Michelsohn, M. L., [*On the existence of special metrics in complex geometry.*]{} Acta Math. 149 (1982), no. 3-4, 261–295. Peternell, Th., [*Algebraicity criteria for compact complex manifolds.*]{} Math. Ann. 275 (1986), no. 4, 653-672. Streets, Jeffrey; Tian, Gang; [*Symplectic curvature flow*]{}, arXiv:1012.2104. Streets, Jeffrey; Tian, Gang [*A parabolic flow of pluriclosed metrics*]{}, Int. Math. Res. Not. IMRN 2010, no. 16, 3101-3133. Voisim, Claire; [*Hodge Theory and Complex Algebraic Geometry I: Volume 1*]{}, Cambridge Studies in Advanced Mathematics. Tseng, Li-Sheng; Yau, Shing-Tung; [*Cohomology and Hodge Theory on Symplectic Manifolds: I*]{}, arXiv:0909.5418. [: [Laboratory of Algebraic Geometry,\ National Research University HSE,\ Department of Mathematics, 7 Vavilova Str. Moscow, Russia,]{}\ [email protected]]{}.

ArXiv

--- abstract: 'Auger recombination is a non-radiative process, where the recombination energy of an electron-hole pair is transferred to a third charge carrier. It is a common effect in colloidal quantum dots that quenches the radiative emission with an Auger recombination time below nanoseconds. In self-assembled QDs, the Auger recombination has been observed with a much longer recombination time in the order of microseconds. Here, we use two-color laser excitation on the exciton and trion transition in resonance fluorescence on a single self-assembled quantum dot to monitor in real-time every quantum event of the Auger process. Full counting statistics on the random telegraph signal give access to the cumulants and demonstrate the tunability of the Fano factor from a Poissonian to a sub-Poissonian distribution by Auger-mediated electron emission from the dot. Therefore, the Auger process can be used to tune optically the charge carrier occupation of the dot by the incident laser intensity; independently from the electron tunneling from the reservoir by the gate voltage. Our findings are not only highly relevant for the understanding of the Auger process, it also demonstrates the perspective of the Auger effect for controlling precisely the charge state in a quantum system by optical means.' author: - 'P. Lochner' - 'A. Kurzmann' - 'J. Kerski' - 'P. Stegmann' - 'J. König' - 'A. D. Wieck' - 'A. Ludwig' - 'A. Lorke' - 'M. Geller' title: 'Real-time detection of every Auger recombination in a self-assembled quantum dot' --- Keywords: Quantum dots, Resonance fluorescence, Auger recombination, Full counting statistics, Random telegraph signal\ The excitonic transitions in self-assembled quantum dots (QDs) [@Bimberg1999; @Petroff2001] realize perfectly a two-level system in a solid-state environment. These transitions can be used to generate single photon sources [@Michler2000; @Yuan2001] with high photon indistinguishability[@Santori2002; @Matthiesen2013], an important prerequisite to use quantum dots as building blocks in (optical) quantum information and communication technologies[@Kimble2008; @Ladd2010]. Moreover, self-assembled QDs are still one of the best model systems to study in an artificial atom the carrier dynamics[@Kurzmann2016b; @Geller2019], the spin- and angular-momentum properties[@Bayer2000; @Vamivakas2009] and charge carrier interactions[@Labud2014]. One important effect of carrier interactions is the Auger process: An electron-hole pair recombines and instead of emitting a photon, the recombination energy is transferred to a third charge carrier, which is then energetically ejected from the QD[@Kharchenko1996; @Efros1997; @Fisher2005; @Jha2009]. This is a common effect, mostly studied in colloidal QDs, where it quenches the radiative emission with recombination times in the order of picoseconds to nanoseconds[@Vaxenburg2015; @Klimov2000; @Park2014]. This limits the efficiency of optical devices containing QDs like LEDs[@Caruge2008; @Cho2009] or single photon sources[@Brokmann2004; @Michler2000a; @Lounis2000]. In self-assembled QDs, Auger recombination was speculated to be absent, and only recently, it was directly observed in optical measurements on a single self-assembled QD coupled to a charge reservoir with recombination times in the order of microseconds[@Kurzmann2016]. As a single Auger process is a quantum event, it is unpredictable and only the statistical evaluation of many processes gives access to the physical information of the recombination process[@Levitov1996; @Blanter2000]. The most in-depth evaluation - the so-called full counting statistics - becomes possible when each single quantum event in a time trace is recorded. Such real-time detection in optical experiments on a single self-assembled QD have until now only been shown for the statistical process of electron tunneling between the QD and a charge reservoir, where tunneling and spin-flip rates could be tuned by the applied electric and magnetic field[@Kurzmann2019]. Here, Auger recombination in a single self-assembled QD is investigated by optical real-time measurements of the random telegraph signal. With the technique of two-laser excitation, we are able to detect every single quantum event of the Auger recombination. These events take place in the single QD, leaving the quantum dot empty until single-electron tunneling into the QD from the charge reservoir takes place again. This reservoir is coupled to the QD with a small tunneling rate in the order of ms$^{-1}$. The laser intensity, exciting the trion transition, precisely controls the electron emission by the Auger recombination and, hence, the average occupation with an electron. It also tunes the Fano factor from a Poissonian to a sub-Poissonian distribution, which we observe in analyzing the random telegraph signal by methods of full counting statistics. The investigated sample was grown by molecular beam epitaxy (MBE) with a single layer of self-assembled In(Ga)As QDs embedded in a p-i-n diode (see Supporting Information for details). A highly n-doped GaAs layer acts as charge reservoir, which is coupled to the QDs via a tunneling barrier, while a highly p-doped GaAs layer defines an epitaxial gate[@Ludwig2017]. An applied gate voltage $V_\text{G}$ shifts energetically the QD states with respect to the Fermi energy in the electron reservoir and controls the charge state of the dots by electron tunneling through the tunneling barrier. The sample is integrated into a confocal microscope setup within a bath cryostat at 4.2K for resonant fluorescence (RF) measurements (see Methods). {width="1\columnwidth"} Figure \[1\] shows the RF of the neutral exciton (X^0^) and the negatively charged exciton, called trion (X^-^). A RF measurement as function of gate voltage in Figure \[1\]**b** shows the fine-structure split exciton[@Hoegele2004] with an average linewidth of about 1.8$\upmu$eV at low excitation intensity ($1.6\cdot10^{-3}\,\upmu$W/$\upmu$m$^2$). Please note, that this measurement was recorded at a laser energy where the exciton gets into resonance at negative gate voltages because here, the measurement conditions were the best. The quantum-confined Stark effect shifts the exciton resonance X^0^ for higher gate voltages to higher frequencies up to 325.760THz, seen in Figure \[1\]**a**. This quadratic Stark shift of the two exciton transitions[@Li2000] is indicated by two white lines. At a voltage of about 0.375V (dashed vertical line in Fig. \[1\]**a**), the electron ground state in the dot is in resonance with the Fermi energy in the charge reservoir. An electron tunnels into the QD and the exciton transition vanishes while the trion transition can be excited at lower frequencies from 324.5095THz to 324.5115THz. The spectrum of the exciton (blue dots) and the trion transition (red dots) under two-laser excitation is shown in Figure \[1\]**c**. The trion transition is measured at a laser frequency of 324.511THz (corresponding to the red line, “Laser 1” in Fig. \[1\]**a**) and a laser excitation intensity of $8\cdot10^{-6}\,\upmu$W/$\upmu$m$^2$ at a gate voltage of 0.515V. The exciton spectrum in Figure \[1\]**c** was obtained simultaneously by a second laser 2 on the exciton transition (blue line in Fig. \[1\]**a** at 325.7622THz) with a laser excitation intensity of $1.6\cdot10^{-3}\,\upmu$W/$\upmu$m$^2$, as the Auger recombination with rate $\gamma_\text{a}$ leads to an empty QD until an electron tunnels into the dot from the reservoir with rate $\gamma_\text{In}=\gamma_\text{In}^0+\gamma_\text{In}^\text{X}$. This rate comprises the tunneling into the empty dot $\gamma_\text{In}^0$ and the tunneling into the dot charged with an exciton $\gamma_\text{In}^\text{X}$[@Seidl2005] (see Fig. \[1\]**d** for a schematic representation). This has been explained previously in Kurzmann et al.[@Kurzmann2016] with the important conclusion that the intensity ratio between trion/exciton intensity in equilibrium measurements is given by the ratio between Auger/tunneling rate $\gamma_\text{a}/\gamma_\text{In}$. As the tunneling rate $\gamma_\text{In}$ in the sample used here is in the range of ms$^{-1}$, the Auger rate $\gamma_\text{a}$ exceeds the tunneling rate by more than two orders of magnitude (see below). As a consequence, the intensity of the trion transition in equilibrium is by more than two orders of magnitude smaller than the exciton transition. {width="0.5\columnwidth"} The interplay between electron tunneling and optical-driven Auger recombination can be studied in more detail by a real-time random telegraph signal of the resonance fluorescence. In these measurements, the time stamp of every detected RF photon is recorded, see Figure \[2\], enabling the evaluation by full counting statistics. As the intensity of the trion is very weak, the random telegraph signal has been investigated in a two-color excitation scheme. The bright exciton transition with count rates exceeding 10MCounts/s (see Supporting Information) is used as an optical detector for the telegraph signal of the Auger recombination. In this two-color laser excitation scheme, the “exciton off” signal corresponds to the “trion on” signal and vice versa[@Kurzmann2016]. Hence, the trion statistics can directly be determined from the “inverse” exciton signal. The intensity of the exciton excitation laser 2 is held constant at $1.6\cdot 10^{-3}\,\upmu$W/$\upmu$m$^2$. This intensity is far below the saturation of the RF signal of the exciton (see Supporting Information) and avoids the photon-induced electron capture at high excitation intensities[@Kurzmann2016a]. However, this laser intensity yields count rates above 200kcounts/s (see Fig. \[1\]**b**), sufficiently-high for recording single quantum events in a real-time measurement[@Kurzmann2019]. While the intensity of the exciton detection laser 2 is kept constant, the laser intensity of the trion excitation laser 1 is increased from $1.6\cdot10^{-7}\,\upmu$W/$\upmu$m$^2$ up to $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$. For every trion laser intensity, the time-resolved RF signal is recorded for 15 minutes using a fast (350ps) avalanche photo diode and a bin time of 100$\upmu$s. Figure \[2\] shows parts of three different time traces at three different trion laser 1 intensities. As the exciton laser 2 intensity always exceeds the trion laser 1 intensity by at least nearly two orders of magnitude, the small amount of RF counts from the trion can be neglected. As a consequence, the detected RF signal of the exciton is directly related to the Auger recombination: An Auger recombination empties the dot and the exciton transition detects an empty dot (no trion transition possible) with a count rate of about 25 counts per bin time (100$\upmu$s). After a time $\tau_\text{On}$, an electron tunnels into the QD in Figure \[2\] and the exciton RF signal quenches for a charged dot (trion transition possible) until, after a time $\tau_\text{Off}$, another Auger recombination happens. Increasing the trion laser intensity from $8\cdot10^{-7}$ up to $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$ in Figure \[2\] increases the probability of an electron emission with rate $\gamma_\text{E}= n \gamma_\text{a}$ by an Auger process, as the probability for occupation of the dot with a trion $n$ increases with increasing laser 1 intensity. Therefore, the exciton transition is observed most frequently for the highest trion laser intensity. This can be observed in Figure \[2\], where the optical random telegraph signal is compared for three different trion excitation intensities. A threshold between exciton “on” and “off” is set for the following statistical evaluation[@Gustavsson2009; @Gustavsson2006]. All exciton RF intensities smaller than this threshold (dashed red line at 7counts/0.1ms in Fig. \[2\]) are counted as “exciton off” (white areas), all intensities above the threshold are counted as “exciton on” (blue areas). {width="\columnwidth"} From these time-resolved RF data sets, the Auger and tunneling rates can be determined by analysing the probability distributions of the “off”-times $\tau_\text{Off}$ and the “on”-times $\tau_\text{On}$ for every 15 minutes long data set[@Gustavsson2009]. A representative distribution at a trion laser intensity of $8\cdot10^{-7}\,\upmu$W/$\upmu$m$^2$ can be seen in Figure \[3\]**a**. An exponential fit to the “on”-times (blue line in Fig. \[3\]**a**) yields the tunneling rate $\gamma_\text{In}$ into the QD, while an exponential fit to the “off”-times (red line in Fig. \[3\]**a**) yields the emission rate $\gamma_\text{E}=n\gamma_\text{a}$ for this specific trion laser 1 intensity. In the example in Figure \[3\]**a**, we find $\gamma_\text{In}=0.80\,$ms$^{-1}$ and $\gamma_\text{E}=0.074\,$ms$^{-1}$. As discussed above, the probability for emitting an electron by an Auger recombination process increases with the occupation probability of the QD with a trion $n$. The occupation probability with a trion $n$ depends on the laser 1 excitation intensity and has been determined from a pulsed measurement of the trion RF intensity, where the highest trion intensity corresponds to an occupation probability of $n=$ 0.5[@Loudon2000] (see Supporting Information for more details). Figure \[3\]**b** shows the expected linear dependence of the electron emission rate $\gamma_\text{E}=n \gamma_\text{a}$ on the occupation probability of the QD with a trion $n$; tuning the emission rate $\gamma_\text{E}$ from almost zero to more than $\gamma_\text{E}= 2$ ms$^{-1}$. The Auger rate is the proportional factor $\gamma_\text{a}$=1.7$\upmu\text{s}^{-1}$ (red data points) and in good agreement with the value obtained before for a different QD with slightly different size[@Kurzmann2016]. The tunneling rate $\gamma_\text{In}$ remains approximately constant at a mean value of 0.74ms$^{-1}$ (blue data points in Fig. \[3\]**b**). This is in agreement with the probability for an electron to tunnel into the empty QD at a constant gate voltage: it is independent on the trion laser intensity. That means, we are able to use the Auger recombination to tune optically the electron emission rate independently from the gate voltage, influencing the emission rate without changing the rate for capturing an electron into the QD (here by the tunneling rate $\gamma_\text{In}$). An independent tuning of electron emission and capture rate is usually not possible for a QD that is tunnel-coupled to one charge reservoir. Changing the coupling strength or Fermi energy by a gate voltage always changes both rates for tunneling into and out of the dot simultaneously. Using the standard methods of full counting statistics[@Gustavsson2009; @Flindt2009] in the following, first of all the asymmetry $a=\frac{\gamma_\text{In}-\gamma_\text{E}}{\gamma_\text{In}+\gamma_\text{E}}$ between the tunneling $\gamma_\text{In}$ and emission rate $\gamma_\text{E}$ has been evaluated. The asymmetry in Figure \[3\]**c** can be tuned by the trion excitation laser intensity from -1 up to 0.55 at a maximum laser intensity of $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$. Important to mention here: At high trion laser intensities above $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$, the electron emission by Auger recombination after an tunneling event from the reservoir happens much faster than the bin time of 0.1ms. Therefore, the RF intensity within the bin time is not falling below the threshold and these events are not detected, i.e. the maximum bandwidth of 10kHz (given by the bin time) of the optical detection scheme distorts the statistical analysis at trion laser intensities above $1.6\cdot10^{-5}\,\upmu$W/$\upmu$m$^2$. Below this laser intensity, every single Auger recombination event is detected in the real-time telegraph signal. ![**Probability distribution and cumulants of the time-resolved RF random telegraph signal.** Panel **a** and **b** show the probability $P(N)$ for a number $N$ of Auger events in a bin time of 200ms (blue bars) and the Poissonian distribution related to the mean value of the probability $P(N)$ (red curve). At a trion excitation intensity (laser 1) of $3\cdot10^{-7}\,\upmu$W/$\upmu$m$^2$ (panel **a**), which corresponds to an asymmetry close to -1, the probability $P(N)$ is close to a Poissonian distribution. At a trion excitation intensity of $6\cdot10^{-6}\,\upmu$W/$\upmu$m$^2$ (panel **b**), which corresponds to an asymmetry close to 0, the probability $P(N)$ is sub-Poissonian. Panel **c** shows the second (blue) and third (red) normalized cumulant as a function of the asymmetry. Symbols are measured values, lines are calculated curves for a two-state system[@Gustavsson2006].[]{data-label="4"}](Figure4.pdf){width="\columnwidth"} Finally, full counting statistics[@Gustavsson2009; @Fricke2007; @Gorman2017] is performed on the telegraph signal: Every 15-min long telegraph signal is divided into sections with length $t_0$. The number $N$ of Auger events within the time interval $t_0$ is counted. Figure \[4\]**a** and **b** show two examples for the corresponding probability distributions $P(N)$ in the limit of large $t_0$ (0.2s). At an asymmetry close to -1 (a trion laser intensity of $3\cdot 10^{-7}\,\upmu$W/$\upmu$m$^2$, Fig. \[4\]**a**), the probability is close to a Poissonian distribution. At an asymmetry of about 0 (laser intensity of $6\cdot 10^{-6}\,\upmu$W/$\upmu$m$^2$, Fig. \[4\]**b**), the probability distribution is sub-Poissonian, indicating a relation between Auger recombination and electron tunneling. The Auger recombination emits an electron after an electron has tunneled from the reservoir into the dot. Vice versa, the electron can only tunnel after the Auger recombination has emptied the QD. From the probability distributions, the cumulants $C_m(t_0)=\partial_z^m \ln \mathcal{M}(z,t_0)|_{z=0}$ can be derived with the generating function $\mathcal{M}(z,t_0)=\sum_{N}e^{zN}P(N)$[@Gustavsson2009]. The first cumulant $C_1$ corresponds to the mean value, the second cumulant $C_2$ is the variance and the third one describes the skewness of the distribution. The second and third normalized cumulant in the limit of large $t_0$ (20ms and 5ms, respectively) can be seen as data points in Figure \[4\]**c**. For a two-state system, theory predicts for these normalized cumulants in the long-time limit $C_2/C_1=(1+a^2)/2$ (also called “Fano factor”) and $C_3/C_1=(1+3a^4)/4$[@Gustavsson2006], shown as lines in Figure \[4\]**c**. The data for the second and third normalized cumulant coincide perfectly with the calculated curves. We can conclude from the statistical analysis that the QD behaves like a two-state system, where one state is the QD charged with one electron (or a trion after optical excitation) and the other state is the empty dot (or charged with an exciton). The QD charged with one electron cannot be distinguished from the dot containing a trion (same for empty dot and exciton) as the optical transition times in the order of nanoseconds are orders of magnitude faster than the tunneling and emission time by the Auger recombination [@Zrenner2002]. The statistical analysis demonstrates the influence of the Auger recombination on the cumulants, especially on the Fano factor, which can be tune from $F=1$ to $F=0.5$ by increasing the incident laser intensity on the trion transition. In summary, we performed real-time RF random telegraph measurements and studied full counting statistics of the Auger effect in a single self-assembled QD. With this technique, we were able to measure every single Auger recombination as a quantum jump from a charged to an uncharged QD; followed by single-electron tunneling. The full counting statistics gives access to the normalized cumulants and demonstrates the tunability of the Fano factor from Possonian to sub-Poissonian distribution by the incident laser intensity on the trion transition. Comparison with theoretical prediction shows that the empty and charged QD with the Auger recombination and tunneling follows a dynamical two-state system. For future quantum state preparation, the Auger process can be used to control optically the charge state in a quantum system by optical means. Methods ======= As the same measurement technique is used, this methods section follows the Supplemental information of Kurzmann et al.[@Kurzmann2019]. Optical measurements -------------------- Resonant optical excitation and collection of the fluorescence light is used to detect the optical response of the single self-assembled QD, where the resonance condition is achieved by applying a specific gate voltage between the gate electrode and the Ohmic back contact. The QD sample is mounted on a piezo-controlled stage under an objective lens with a numerical aperture of $NA=0.65$, giving a focal spot size of about 1$\upmu$m diameter. All experiments are carried out in a liquid He confocal dark-field microscope at 4.2K with a tunable diode laser for excitation and an avalanche photodiode (APD) for fluorescence detection. The resonant laser excitation and fluorescence detection is aligned along the same path with a microscope head that contains a 90:10 beam splitter and two polarizers. Cross-polarization enables a suppression of the spurious laser scattering into the detection path by a factor of more than $10^7$. The counts of the APD (dead time of 21.5ns) were binned by a QuTau time-to-digital converter with a temporal resolution of 81ps. This work was supported by the German Research Foundation (DFG) within the Collaborative Research Centre (SFB) 1242, Project No. 278162697 (TP A01), and the individual research grant No. GE2141/5-1. 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ArXiv

--- author: - 'K. Sandstrom' - 'O. Krause' - 'H. Linz' - 'E. Schinnerer' - 'G. Dumas' - 'S. Meidt' - 'H.-W. Rix' - 'M. Sauvage' - 'F. Walter' - 'R. C. Kennicutt' - 'D. Calzetti' - 'P. Appleton' - 'L. Armus' - 'P. Beirão' - 'A. Bolatto' - 'B. Brandl' - 'A. Crocker' - 'K. Croxall' - 'D. Dale' - 'B. T. Draine' - 'C. Engelbracht' - 'A. Gil de Paz' - 'K. Gordon' - 'B. Groves' - 'C.-N. Hao' - 'G. Helou' - 'J. Hinz' - 'L. Hunt' - 'B. D. Johnson' - 'J. Koda' - 'A. Leroy' - 'E. J. Murphy' - 'N. Rahman' - 'H. Roussel' - 'R. Skibba' - 'J.-D. Smith' - 'S. Srinivasan' - 'L. Vigroux' - 'B. E. Warren' - 'C. D. Wilson' - 'M. Wolfire' - 'S. Zibetti' title: 'Mapping far-IR emission from the central kiloparsec of [NGC$\,1097$]{}[^1]' --- Introduction ============ The central regions of galaxies host some of the most intense star-formation that we can observe in the local Universe in circumnuclear starburst rings. Starburst rings are believed to be the consequence of the pile-up of inflowing gas and dust, driven by a non-axisymmetric potential from a stellar bar, on orbits located near the Inner Lindblad Resonance of the bar [@combes85; @athanassoula92]. The high surface densities that exist in the ring lead to high star-formation rates. Indeed starburst rings are one of few regions in non-interacting galaxies where the formation of “super star clusters” commonly occurs [@maoz96]. The stars formed in the ring can be numerous enough to drive the structural evolution of the galaxy [@norman96; @kormendy04] and can be the dominant power source for the galaxy’s infrared (IR) emission. Star-formation in circumnuclear rings occurs under conditions not normally found in the disks of galaxies: in addition to their high gas surface densities, these regions have dynamical timescales that are comparable to the lifetimes of massive stars. Understanding star formation in circumnuclear rings has been a long-standing problem [@combes96]. There are two main models: the “popcorn” model [@elmegreen94], where star-formation is driven by stochastic gravitational fragmentation along the ring, and the “pearls on a string” model, where gas flowing into the ring is compressed near the contact points (i.e. locations where the dust lanes intersect the ring) and then forms stars a short distance downstream [e.g., @boker08]. The “pearls on a string” model predicts a gradient in the ages of young stellar clusters as one moves away from the contact points. This has been observed in a number of starburst rings [e.g., @mazzuca08; @boker08]. Conversely, many well-studied rings show no evidence for an age gradient [@maoz01]. It is not obvious, however, that a single mode of star-formation must occur in all rings or even at all times in a given ring [@vandeven09]. KINGFISH (Key Insights into Nearby Galaxies: A Far-Infrared Survey with [*Herschel*]{}, PI R. Kennicutt) is an Open-Time Key Program to study the interstellar medium (ISM) of nearby galaxies with far-IR/sub-mm photometry and spectroscopy. Among the unique aspects of the KINGFISH science program is the ability to observe thermal dust emission at unprecedented spatial resolution ($\sim$ 5.6, 6.8 and 11.3at 70, 100 and 160 [$\mu$m]{}) using PACS (Photodetector Array Camera and Spectrometer) imaging. High spatial resolution is crucial for observing processes occurring in the central regions of galaxies. These regions represent our best opportunity to study in detail the interplay between dynamics, star-formation and feedback that regulate the fueling of nuclear activity, be it a starburst or an active galactic nucleus (AGN). Below we present PACS imaging of the galaxy NGC 1097, one of the first KINGFISH targets observed during the [*Herschel*]{} Science Demonstration Program (SDP) (for PACS spectroscopy of NGC 1097 see Beirão et al. 2010 and for [*SPIRE*]{} observations see Engelbracht et al. 2010). The source NGC 1097 is a barred spiral galaxy located at a distance of 19.1 Mpc [@willick97 1$\approx 92$ pc]. In its central kpc it hosts an intensely star-forming [$\sim 5$  yr$^{-1}$; @hummel87] ring with a radius of $\sim 900$ pc. The ring’s rotation speed of $\sim 300$ km s$^{-1}$ [corrected for inclination, @storchi-bergmann96], corresponds to a rotation period of $\sim 18$ Myr. The galaxy’s nucleus is classified as a LINER from optical emission line diagnostics [@phillips84], but is shown to be a Seyfert 1 by its double-peaked H$\alpha$ profile [@storchi-bergmann93]. UV spectroscopy has revealed a few Myr old burst of star-formation in the central 9 pc of the galaxy [@storchi-bergmann05]. With the high spatial resolution of [*Herschel*]{} PACS, we can resolve the starburst ring and inner 600 pc of NGC 1097 for the first time at wavelengths near the peak of the dust spectral energy distribution (SED). Observations and data reduction =============================== The galaxy NGC 1097 was observed with the PACS instrument (Poglitsch et al. 2010) on the [*Herschel*]{} Space Observatory (Pilbratt et al. 2010) on 2009 December 20 during the SDP. We obtained 15 long scan-maps in two orthogonal directions at the medium scan speed (20s$^{-1}$). The scan position angles (45$^\circ$ relative to the scan direction) provide homogeneous coverage over the mapped region. The total on-source times per pixel were approximately 150, 150, and 300 seconds for 70, 100 and 160 [$\mu$m]{}, respectively. The raw data were reduced with HIPE (Ott 2010), version 3.0, build 455. Besides the standard steps leading to level-1 calibrated data, second-level deglitching and correction for offsets in the detector sub-matrices were performed. The data were then highpass-filtered using a median window of 5 to remove the effects of bolometer sensitivity drifts and 1/f noise. We masked out emission structures (visible in a first iteration of the map-making) with a 5-wide mask before computing this running median to prevent subtraction of source emission. Although the filtering may remove some extended flux from the galaxy, because we are primarily interested in the very bright central 1 of NGC 1097 this effect will be negligible. Finally, the data were projected onto a coordinate grid with 1 pixels. After pipeline processing we applied flux correction factors from the PACS team to adjust the calibration. The current calibration has uncertainties of $\sim 10$, 10, and 20% for the 70, 100 and 160 [$\mu$m]{} bands, respectively (Poglitsch et al. 2010). Because we aim to compare our PACS observations with ancillary data at other wavelengths, we adjusted the relative astrometry of the PACS observations to match that of the [*Spitzer*]{} 24 [$\mu$m]{} from SINGS [Spitzer Infrared Nearby Galaxies Survey: @kennicutt03]. This was done by measuring the positions of background point-sources in the MIPS 24 [$\mu$m]{} (Multi-Band Imaging Photometer) and PACS 100 [$\mu$m]{} images, adjusting the PACS 100 [$\mu$m]{} astrometry, assuming the relative astrometry for the PACS bands is well-calibrated and transferring the solution to the other bands. The offset between the PACS and MIPS astrometry was $\sim 2$. The one-sigma surface brightness sensitivities per pixel of the final maps are 5.9, 6.2 and 3.3 MJy sr$^{-1}$. In Fig \[fig:rgb\] we show the three PACS images with a logarithmic stretch to highlight the spiral arms. Note that below we extract photometry from the images at their native resolution using apertures larger than the beam size of the lowest resolution map. [*Herschel* PACS observations of the circumnuclear ring in NGC 1097]{} ====================================================================== The most prominent far-IR structure in NGC 1097 is its circumnuclear starburst ring, shown in Fig \[fig:ring\] at a variety of wavelengths. The PACS angular resolution allows us to clearly separate the contribution of the ring and nucleus from the galaxy’s emission for the first time at wavelengths that probe the peak of the dust SED. Summing the emission within a radius of 20(1.8 kpc) of the center and comparing it with the total flux from the galaxy, we find that the ring and nucleus emit 75, 60 and 55% of the total flux of NGC 1097 at 70, 100 and 160 [$\mu$m]{}, respectively (there is some galactic emission within 1.8 kpc that is not associated with the ring or nucleus, but this component is negligible). These measurements imply that the SED of the more extended galactic emission peaks at longer wavelengths than the SED of the ring. Indeed, by fitting a modified blackbody to the MIPS and [*SPIRE*]{} photometry of the galaxy, Engelbracht et al. (2010) find that the central region of NGC 1097 is 22% warmer than the disk. The mid- and far-IR images of the ring in Fig \[fig:ring\] show similar structures. The ring is continuous (i.e. no obvious gaps) with a series of bright knots. The same knots are visible in each PACS image, although at 160 [$\mu$m]{} they are not well-resolved. At 70 and 100 [$\mu$m]{}, the surface brightness of the ring varies by less than $\pm 15$% about the mean on 600 pc scales. The variations at 24 [$\mu$m]{} on the same spatial scales are $\pm 25$%. The similarities from mid- to far-IR suggest that dust temperatures are not varying substantially in the ring, which we quantify below. It is interesting to note that the same pattern of bright knots is not observed in carbon monoxide (CO) (shown in panel h of Fig \[fig:ring\]) or other dense gas tracers at comparable resolution [@kohno03; @hsieh08]. Instead the CO intensity peaks near the contact points and is much fainter over the rest of the ring. The differences between the far-IR and CO emission may be due to different CO excitation mechanisms in the shocked gas near the contact points or by the consumption and/or dissociation of molecular gas by star-formation events shortly downstream from the contact points. Three of the bright knots are also prominent in 3.5 cm radio continuum (as shown in panel f of Fig \[fig:ring\]). @beck05 showed that the radio knots have a flatter radio spectral index than the rest of the ring, most likely due to either a contribution from free-free emission from H II regions or synchrotron emission from young supernova remnants (SNRs), which has an intrinsically flatter spectrum. Because young SNRs will heat only a small fraction of the dust, an enhancement of thermal radio continuum and dust heating in and around H II regions may be the best explanation for the origin of the coincident bright radio and far-IR knots. In Fig \[fig:randaz\] we show the mid- and far-IR band ratios as a function of azimuthal angle. The surface brightness was measured in 9 azimuthal bins with inner and outer radii of 5 and 15 to adequately sample the PSF out to 100 [$\mu$m]{}. The largest variations are in the 24/70 ratio, which peaks shortly downstream from the northernmost contact point, and varies by $\pm 15$%. The 70/100 ratio varies by less than $\pm 5$% around the ring. If there is a well-defined age gradient along the ring as predicted by the “pearls on a string” model, one might expect a gradient in dust temperatures moving away from the contact points. In the youngest star-forming regions, the radiation field will be more intense due to the presence of the most massive and short-lived stars and the regions will be more compact. Both of these effects lead to hotter dust temperatures. For instance, @groves08 modeled the spectra of the H II regions plus surrounding photo-dissociation region for star clusters with ages between 0.1$-$10 Myr. They find that for a cluster mass of $\sim 10^5$  and an ISM pressure of P/k $\sim 10^6$ K cm$^{-3}$ [approximately what has been deduced for the ISM in the circumnuclear ring by @hsieh08], the 24/70 and 70/100 band ratios decrease by 90 and 70% (factors of $\sim 7$ and 3, respectively) as the cluster ages from 1 to 10 Myr. The @groves08 models represent an upper bound to the band ratio variation we could expect if the ring was comprised solely of a well-defined sequence of aging clusters between 1$-$10 Myr old. That we do not see large mid- and far-IR band ratio gradients does not rule out the existence of “pearls on a string” in favor of “popcorn” in NGC 1097, however. It may be the case that dust in the ring is predominantly heated by the radiation field from older stars (e.g. B stars with lifetimes of 10-100 Myr), which are uniformly distributed around the ring after a number of revolutions. Stellar population studies of the central kpc of NGC 1097 have shown that intermediate age stars make a considerable contribution to the UV radiation field in the vicinity of the ring [e.g. @bonatto98]. In this situation, the variation due to an age gradient would be diluted depending on the relative contribution of young clusters to the total dust heating. In addition, one might expect that given the fast dynamical time in the ring that local enhancements of dust heating are quickly wiped out. Stars and interstellar matter in NGC 1097’s ring traverses the distance between the two contact points in $\sim 9$ Myr. Even assuming that the cluster formation happens instantaneously after entering the ring, there will still be an abundance of massive stars by the time the cluster crosses half the ring. Limits on the nuclear flux in the far-IR ======================================== High resolution imaging at mid- and near-IR wavelengths of the nucleus of NGC 1097 shows an unresolved central point source [@prieto05; @mason07] which contains a nuclear starburst [@storchi-bergmann05] and the AGN. No observations can yet resolve the AGN or central starburst, but it is still possible to distinguish the contributions of the different sources to some degree. @mason07, for instance, found that the 12 and 18 [$\mu$m]{}emission arises primarily from dust heated by the nuclear starburst rather than the AGN torus. In NGC 1097, previous far-IR flux limits for the nucleus were dominated by the starburst ring and provide limited information about the nuclear starburst or the AGN. With our [*Herschel*]{} observations, we can place limits on the flux arising in the central $\sim 600$ pc. We use the PACS PSF observations of Vesta with a 20s$^{-1}$ scan speed scaled to match the peak intensity of the point source we see in the center. Our scaled-PSF photometry is possible at 70 and 100 [$\mu$m]{}, but not at 160 [$\mu$m]{}where the central source is not well-resolved. The best scaled PSF has a total flux of 3.5 and 7.3 Jy at 70 and 100 [$\mu$m]{}. Without more detailed modeling of the nuclear region these measurements should only be considered upper limits. However, they improve constraints on the nuclear flux by more than an order of magnitude as shown in Fig \[fig:sed\], which presents the SED of the nucleus from a compilation by @prieto10. Conclusions =========== We have presented [*Herschel*]{} PACS observations from KINGFISH of the inner kpc of the barred spiral galaxy NGC 1097. These are the first observations to resolve a starburst ring at wavelengths probing the peak of the dust SED. We show a comparison of the ring in a variety of tracers and find similar bright knots in the mid- and far-IR and radio continuum. These knots do not correspond to the same knots traced by CO. We find modest variation azimuthally in the mid- and far-IR band ratios suggesting that either there is no azimuthal age gradient, as would be predicted by the “pearls on a string” mode of star-formation, that dust heating is dominated by an older stellar population and/or that the dust heating variations get quickly erased over the short ring orbital period ($\sim 18$ Myr). Finally, we place an order-of-magnitude tighter constraint on the far-IR emission originating in the central $\sim 600$ pc of the galaxy. The authors thank R. Beck for the radio continuum data and P.-Y. Hsieh for the CO data. K.S. would like to thank G. van de Ven and L. Burtscher for useful discussions. PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF- IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain). [1]{} , E. 1992, , 259, 345 , R., [Fletcher]{}, A., [Shukurov]{}, A., [et al.]{} 2005, , 444, 739 Beirão et al. 2010 , T., [Falc[ó]{}n-Barroso]{}, J., [Schinnerer]{}, E., [Knapen]{}, J. H., & [Ryder]{}, S. 2008, , 135, 479 , C., [Pastoriza]{}, M. G., [Alloin]{}, D., & [Bica]{}, E. 1998, , 334, 439 , F. 1996, in ASP Conf. Ser., Vol. 91, IAU Colloq. 157: Barred Galaxies, ed. 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[Y. Mizumoto, K.-I. Morita, & M. Ohishi]{}, in press , M. M., [Pagel]{}, B. E. J., [Edmunds]{}, M. G., & [Diaz]{}, A. 1984, , 210, 701 Pilbratt et al. 2010 Poglitsch et al. 2010 , M. A., [Maciejewski]{}, W., & [Reunanen]{}, J. 2005, , 130, 1472 , M. A., [Reunanen]{}, J., [Tristram]{}, K. R. W., [et al.]{} 2010, , 402, 724 , T., [Baldwin]{}, J. A., & [Wilson]{}, A. S. 1993, , 410, L11 , T., [Nemmen]{}, R. S., [Spinelli]{}, P. F., [et al.]{} 2005, , 624, L13 , T., [Wilson]{}, A. S., & [Baldwin]{}, J. A. 1996, , 460, 252 , G. & [Chang]{}, P. 2009, , 697, 619 , J. A., [Courteau]{}, S., [Faber]{}, S. M., [et al.]{} 1997, , 109, 333 [^1]: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.

ArXiv

--- abstract: 'We performed Self-Consistent Greens Function (SCGF) calculations for symmetric nuclear matter using realistic nucleon-nucleon (NN) interactions and effective low-momentum interactions ($V_{low-k}$), which are derived from such realistic NN interactions. We compare the spectral distributions resulting from such calculations. We also introduce a density-dependent effective low-momentum interaction which accounts for the dispersive effects in the single-particle propagator in the medium.' author: - 'P. Bożek[^1]' - 'D. J. Dean[^2]' - 'H. Müther[^3]' title: Correlations and effective interactions in nuclear matter --- Introduction ============ The description of bulk properties of nuclear systems starting from realistic nucleon-nucleon (NN) interactions is a long-standing and unsolved problem. Various models for the NN interaction have been developed, which describe the experimental NN phase shifts up to the threshold for pion production with high accuracy[@cdbonn; @arv18; @nijmw; @n3lo]. A general feature of all these interaction models are strong short-range and tensor components, which lead to corresponding correlations in the nuclear many-body wave-function. Hartree-Fock mean-field theory, which represents the the lowest-order many-body calculations one can perform with such realistic NN interactions, fails to produce bound nuclei [@reviewartur; @localint] precisely because Hartree-Fock does not fully incorporate many-body correlation effects. That correlations beyond the mean field are important is supported by experiments exploring the spectral distribution of the single-particle strength. One experimental fact found in all nuclei is the global depletion of the Fermi sea. A recent experiment from NIKHEF puts this depletion of the proton Fermi sea in ${}^{208}$Pb at a little less than 20% [@bat01] in accordance with earlier nuclear matter calculations [@vond1]. Another consequence of the presence of short-range and tensor correlations is the appearance of high-momentum components in the ground state wave-function to compensate for the depleted strength of the mean field. Recent JLab experiments [@rohe:04] indicate that the amount and location of this strength is consistent with earlier predictions for finite nuclei [@mudi:94] and calculations of infinite matter [@frmu:03]. These data and their analysis, however, are not sufficient to allow for a detailed comparison with the predictions derived from the various interaction models at high momenta. In this paper, we want to investigate a possibility to separate the predictions for correlations at low and medium momenta, which are constrained by the NN scattering matrix below pion threshold, from the high momentum components, which may strongly depend on the underlying model for the NN interaction. For that purpose we will perform nuclear many-body calculations within a model space that allows for the explicit evaluation of low-momentum correlations. The effective Hamiltonian for this model space will be constructed from a realistic interaction to account for for correlations outside the model space. This concept of a model space and effective operators appropriately renormalized for this model space has a long history in approaches to the nuclear many-body physics. As an example we mention the effort to evaluate effective operators to be used in Hamiltonian diagonalization calculations of finite nuclei. For a review on this topic see e.g. [@morten:04]. The concept of a model space for the study of infinite nuclear matter was used e.g. by Kuo et al.[@kumod1; @kumod2; @kumod3]. Also the Brueckner-Hartree-Fock (BHF) approximation can be considered as a model space approach. In this case one restricts the model space to just one Slater-determinant and determines the effective interaction through a calculation of the G-matrix, the solution of the Bethe-Goldstone equation. The effective hamiltonians for such model space calculations have frequently been evaluated within the Rayleigh-Schrödinger perturbation theory, leading to a non-hermitian and energy-dependent result. The energy-dependence can be removed by considering the so-called folded-diagrams as has been discussed e.g. by Brandow[@brandow:67] and Kuo[@kuo:71]. We note that the folded-diagram expansion yields effective interaction terms between three and more particle, even if one considers a realistic interaction with two-body terms only[@polls:83; @polls:85]. During the last years the folded-diagram technique has been applied to derive an effective low-momentum potential $V_{low-k}$[@bogner:03] from a realistic NN interaction. By construction, $V_{low-k}$ potentials reproduce the deuteron binding-energy, the low-energy phase shifts and the half-on-shell $T$ matrix calculated from the underlying realistic NN interaction up to the chosen cut-off parameter. The resulting $V_{low-k}$ turns out to be rather independent on the original NN interaction if this cut-off parameter for the relative momenta is below the value of the pion-production threshold in NN scattering. The off-shell characteristics of the $V_{low-k}$ effective interaction are not constrained by experimental data and can influence the many-body character of the interaction. For finite nuclei we find that one does indeed obtain different binding energies for $^{16}$O depending on the underlying NN interaction from which one derives the $V_{low-k}$ interaction. For example, using coupled-cluster techniques at the singles and doubles level (CCSD) [@dean04] we find binding energies for $^{16}$O at a lab-momentum cutoff of $\Lambda=2.0$ fm$^{-1}$ to be $-143.4\pm 0.4$ MeV and $-153.3\pm 0.4$ MeV for the N$^3$LO [@n3lo] and CD-Bonn two-body interactions, respectively. The CCSD calculations were carried out at up to 7 major oscillator shells (with extrapolations to an infinite model space) using the intrinsic Hamiltonian defined as $H=T-T_{cm}+V_{low-k}$ where $T_{cm}$ is the center of mass kinetic energy. Attractive energies are obtained if such a $V_{low-k}$ interaction is used in a Hartree-Fock calculation of nuclear matter or finite nuclei[@corag:03; @kuck:03]. High-momentum correlations, which are required to obtain bound nuclear systems from a realistic NN interaction (see above) are taken into account in the renormalization procedure which leads to $V_{low-k}$. Supplementing these Hartree-Fock calculations with corrections up to third order in the Goldstone perturbation theory leads to results for the ground-state properties of $^{16}O$ and $^{40}Ca$, which are in fair agreement with the empirical data[@corag:03]. (One should note that $T_{cm}$ was not included in these calculations.) Calculations in infinite matter demonstrate that $V_{low-k}$ seems to be quite a good approximation for the evaluation of low-energy spectroscopic data. The results for the pairing derived from the bare interaction are reproduced[@kuck:03]. The prediction of pairing properties also agree with results obtained phenomenological interactions like the Gogny force[@gogny; @sedrak:03]. The $V_{low-k}$ interaction also yields a good approximation for the calculated binding energy of nuclear matter at low densities. At high densities, however, BHF calculations using $V_{low-k}$ yield too much binding energy and do not reproduce the saturation feature of nuclear matter[@kuck:03]. This is due to the fact that $V_{low-k}$ does not account for the effects of the dispersive quenching of the two-particle propagator, as it is done e.g. in the Brueckner $G$-matrix derived from a realistic NN interaction. The saturation can be obtained if a three-body nucleon is added to the hamiltonian[@bogner:05]. An alternative technique to determine an effective hamiltonian for a model space calculation is based on a unitary transformation of the hamiltonian. It has been developed by Suzuki[@suzuki:82] and leads to an energy-independent, hermitian effective interaction. The unitary-model-operator approach (UMOA) has also been used to evaluate the ground-state properties of finite nuclei[@suz13; @suz15; @fuji:04; @roth:05]. In the present study we are going to employ the unitary transformation technique to determine an effective interaction, which corresponds to the $V_{low-k}$ discussed above. This effective interaction will then be used in self-consistent Green’s function (SCGF) calculation of infinite nuclear matter. Various groups have recently developed techniques to solve the corresponding equations and determine the energy- and momentum-distribution of the single-particle strength in a consistent way[@frmu:03; @bozek0; @bozek1; @bozek2; @dewulf:03; @rd; @frmu:05]. Therefore we can study the correlation effects originating from $V_{low-k}$ inside the model space and compare it to the correlations derived from the bare interaction. Furthermore we use the unitary transformation technique to determine an effective interaction which accounts for dispersive effects missing in the original $V_{low-k}$ (see discussion above). After this introduction we will present the method for evaluating the effective interaction in section 2 and briefly review the basic features of the SCGF approach in section 3. The results of our investigations are presented in section 4, which is followed up by the conclusions. Effective interaction ===================== For the definition and evaluation of an effective interaction to be used in a nuclear structure calculation, which is restricted to a subspace of the Hilbert space, the so-called model space, we follow the usual notation and define a projection operator $P$, which projects onto this model space. The operator projecting on the complement of this subspace is identified by $Q$ and these operators satisfy the usual relations like $P+Q=1$, $P^2=P$, $Q^2=Q$, and $PQ=0=QP$. It is the aim of the Unitary Model Operator Approach (UMOA) to define a unitary transformation $U$ in such a way, that the transformed Hamiltonian does not couple the $P$ and $Q$ space, i.e. $QU^{-1}HUP=0$. For a many-body system the resulting Hamiltonian can be evaluated in a cluster expansion, which leads to many-body terms. This is very similar to the folded diagram expansion, which has been discussed above. In UMOA studies of finite nuclei terms up to three-body clusters have been evaluated[@suz13; @suz15] indicating a convergence of the expansion up to this order. In the present study we would like to determine an effective two-body interaction and therefore consider two-body systems only. We define the effective interaction as $$V_{eff} = U^{-1}\left( h_0 + v_{12}\right) U - h_0\,,\label{eq:veff1}$$ with $v_{12}$ representing the bare NN interaction. The operator $h_0$ denotes the one-body part of the two-body system and contains the kinetic energy of the interacting particles. This formulation will lead to an effective interaction corresponding to $V_{low-k}$. Since, however, we want to determine an effective interaction of two nucleons in the medium of nuclear matter, we will also consider the possibility to add a single-particle potential to $h_0$. Note that in any case $h_0$ commutes with the projection operators $P$ and $Q$. The operator for the unitary transformation $U$ can be expressed as[@suz24] $$U=(1+\omega-\omega ^{\dagger})(1+\omega \omega ^{\dagger} +\omega ^{\dagger}\omega )^{-1/2}\,,\label{eq:veff2}$$ with an operator $\omega$ satisfying $\omega=Q\omega P$ such that $\omega^2 = \omega^{\dagger 2} = 0$. In the following we will describe how to determine the matrix elements of this operator $\omega$. As a first step we solve the two-body eigenvalue equation $$\left( h_0 + v_{12}\right)\vert \Phi _{k}\rangle =E_{k}\vert \Phi _{k}\rangle\,. \label{eq:veff2a}$$ This can be done separately for each partial wave of the two-nucleon problem. Partial waves are identified by total angular momentum $J$, spin $S$ and isospin $T$. The relative momenta are appropriately discretized such that we can reduce the eigenvalue problem to a matrix diagonalization problem. Momenta below the cut-off momentum $\Lambda$ define the $P$ space and will subsequently be denoted by $\vert p\rangle$ and $\vert p'\rangle$. Momenta representing the $Q$ space will be labeled by $\vert q\rangle$ and $\vert q'\rangle$, while states $\vert i\rangle$, $\vert j\rangle$, $\vert k \rangle$ and $\vert l \rangle$ refer to basis states of the total $P+Q$ space. From the eigenstates $\vert \Phi _{k}\rangle$ we determine those $N_P$ ($N_P$ denoting the dimension of the $P$ space) eigenstates $\vert \Phi _{p}\rangle$, which have the largest overlap with the $P$ space and determine $$\label{eq:veff3} \langle q\vert\omega\vert p'\rangle =\sum_{p=1}^{N_P}\langle q\vert Q\vert \Phi _{p}\rangle \langle \tilde{\varphi}_{p}\vert p'\rangle,$$ with $\vert \varphi_{p}\rangle = P\vert \Phi_{p}\rangle$ and $\langle \tilde{\varphi}_{p}\vert$ denoting the biorthogonal state, satisfying $$\sum_{p}\langle \tilde{\varphi} _{k}|p\rangle \langle p|\varphi _{k'}\rangle \quad \mbox{and} \quad \sum_{k}\langle p'|\tilde{\varphi} _{k}\rangle \langle \varphi _{k}|p\rangle =\delta _{p,p'}\,.\label{eq:veff4}$$ In the next step we solve the eigenvalue problem in the $P$ space $$\omega ^{\dagger}\omega\vert\chi_{p}\rangle =\mu _{p}^{2}|\chi_{p} \rangle\, ,\label{eq:veff5}$$ and use the results to define $$\vert\nu _{p}\rangle =\frac{1}{\mu _{p}}\omega \vert\chi _{p}\rangle ,\label{eq:veff5a}$$ which due to the fact that $\omega=Q\omega P$, can be written as $$\label{eq:veff5b} \langle q|\nu _{p}\rangle =\frac{1}{\mu _{p}} \sum_{p'}\langle q|\omega |p'\rangle \langle p'|\chi_{p}\rangle\, .$$ Using Eqs. (\[eq:veff5\]) - (\[eq:veff5b\]) and the representation of $U$ in Eq. (\[eq:veff2\]), the matrix elements of the unitary transformation operator $U$ can be written $$\begin{aligned} \label{eq:Up'p} \langle p''|U|p'\rangle &=&\langle p''|(1+\omega^{\dagger}\omega )^{-1/2}|p'\rangle \nonumber \\ &=&\sum_{p=1}^{N_P}(1+\mu_{p}^{2})^{-1/2} \langle p''|\chi_{p}\rangle \langle \chi_{p}|p'\rangle \,,\end{aligned}$$ $$\begin{aligned} \label{eq:Uqp} \langle q|U|p'\rangle &=&\langle q|\omega (1+\omega^{\dagger}\omega )^{-1/2}|p'\rangle \nonumber \\ &=&\sum_{p=1}^{N_P}(1+\mu_{p}^{2})^{-1/2}\mu _{p} \langle q|\nu _{p}\rangle \langle \chi _{p}|p'\rangle\, ,\end{aligned}$$ $$\begin{aligned} \label{eq:Upq} \langle p'|U|q\rangle &=&-\langle p'|\omega ^{\dagger}(1+\omega \omega ^{\dagger})^{-1/2} |q\rangle \nonumber \\ &=&-\sum_{p=1}^{N_P}(1+\mu_{p}^{2})^{-1/2}\mu_{p} \langle p'\vert\chi_{p}\rangle \langle \nu_{p}\vert q\rangle \,, \end{aligned}$$ $$\begin{aligned} \label{eq:Uq'q} \langle q'|U|q\rangle &=&\langle q'|(1+\omega \omega ^{\dagger})^{-1/2}\vert q\rangle \nonumber \\ &=&\sum_{p=1}^{N_P}\{(1+\mu_{p}^{2})^{-1/2}-1\} \langle q'|\nu _{p}\rangle \langle \nu _{p}|q\rangle + \delta _{q,q'}\,.\end{aligned}$$ These matrix elements of $U$ can then be used to determine the matrix elements of the effective interaction $V_{eff}$ according to Eq.(\[eq:veff1\]). They might also be used to define matrix elements of other effective operators. Self-consistent Green’s function approach ========================================= One of the key quantities within the Self-consistent Green’s Function (SCGF) approach is the retarded single-particle (sp) Green’s function or sp propagator $G(k,\omega)$ (see e.g.[@diva:05]). Its imaginary part can be used to determine the spectral function $$\label{spec_g2} A(k,\omega)=-2\,{\mathrm{Im}}\,G(k,\omega+{\mathrm{i}}\eta)\,.$$ The spectral function provides the information about the energy- and momentum-distribution of the single-particle strength, i.e. the probability for adding or removing a particle with momentum $k$ and leaving the residual system at an excitation energy related to $\omega$. In the limit of the mean-field or quasi-particle approximation the spectral function is represented by a $\delta$-function and takes the simple form $$A(k,\omega)=2\pi\delta(\omega -\varepsilon_k) \,,\label{eq:specqp}$$ with the quasi-particle energy $\varepsilon_k$ for a particle with momentum $k$. The sp Green’s function can be obtained from the solution of the Dyson equation, which reduces for the system of homogeneous infinite matter to a a simple algebraic equation $$\left[\omega -\frac{k^2}{2m}-\Sigma(k,\omega)\right] G(k,\omega) = 1\,,\label{eq:dyson}$$ where $\Sigma(k,\omega)$ denotes the complex self-energy. The self-energy can be decomposed into a generalized Hartree-Fock part plus a dispersive contribution $$\label{spec_Sigma} \Sigma(k,\omega)=\Sigma^{HF}(k)-\frac{1}{\pi}\int_{-\infty}^{+\infty} {\mathrm{d}}\omega^{\prime} \, \frac{{\mathrm{Im}}\Sigma(k,\omega^{\prime}+ {\mathrm{i}}\eta)} {\omega-\omega^{\prime}}.$$ The next step is to obtain the self energy in terms of the in-medium two-body scattering $T$ matrix. It is possible to express ${\mathrm{Im}}\Sigma(k,\omega+{\mathrm{i}}\eta)$ in terms of the retarded $T$ matrix [@frmu:03; @bozek3; @kadanoff] (for clarity, spin- and isospin quantum number are suppressed) $$\begin{aligned} \label{im_sigma} {\mathrm{Im}}\Sigma(k,\omega+{\mathrm{i}}\eta)&=& \frac{1}{2}\int \frac{{\mathrm{d}}^3k^{\prime}}{(2\pi)^3} \int_{-\infty}^{+\infty} \frac{{\mathrm{d}}\omega^{\prime}}{2\pi} \left<{\mathbf{kk}}^{\prime}| {\mathrm{Im}}T(\omega+\omega^{\prime}+{\mathrm{i}}\eta)| {\mathbf{kk}}^{\prime}\right> \nonumber \\ && \qquad \times [f(\omega^{\prime})+b(\omega+\omega^{\prime})] A(k^{\prime},\omega^{\prime}).\end{aligned}$$ Here and in the following $f(\omega)$ and $b(\omega)$ denote the Fermi and Bose distribution functions, respectively. These functions depend on the chemical potential $\mu$ and the inverse temperature $\beta$ of the system. The in-medium scattering matrix $T$ is to be determined as a solution of the integral equation $$\begin{aligned} \left<{\mathbf{kk}}^{\prime}|T(\Omega+{\mathrm{i}}\eta)| {\mathbf{pp}}^{\prime}\right> & = &\left<{\mathbf{kk}}^{\prime}|V| {\mathbf{pp}}^{\prime}\right> + \int \frac{d^3q\,d^3q^\prime}{\left(2\pi\right)^6} \left<{\mathbf{kk}}^{\prime}|V| {\mathbf{qq}}^{\prime}\right>G^0_{\mathrm{II}}(\mathbf{qq}^\prime,\Omega+i\eta) \nonumber \\ &&\quad\quad\times \left<{\mathbf{qq}}^{\prime}|T(\Omega+{\mathrm{i}}\eta)| {\mathbf{pp}}^{\prime}\right>\,,\label{eq:tscat0}\end{aligned}$$ where $$\label{two_pp} G^0_{\mathrm{II}}(k_1,k_2,\Omega+i\eta)= \int_{-\infty}^{+\infty}\frac{{\mathrm{d}}\omega}{2\pi} \int_{-\infty}^{+\infty}\frac{{\mathrm{d}}\omega^{\prime}}{2\pi} A(k_1,\omega)A(k_2,\omega^{\prime}) \frac{1-f(\omega)-f(\omega^{\prime})} {\Omega-\omega-\omega^{\prime}+i\eta}\,.$$ stands for the two-particle Green’s function of two non-interacting but dressed nucleons. The matrix elements of the two-body interaction $V$ represent either the bare NN interaction $v_{12}$ or the effective interaction $V_{eff}$, in which case the integrals are cut at the cut-off parameter $\Lambda$. The in-medium scattering equation (\[eq:tscat0\]) can be reduced to a set of one-dimensional integral equations if the two-particle Green’s function in (\[two\_pp\]) is written as a function of the total and relative momenta of the interacting pair of nucleons and the usual angle-average approximation is employed (see *e.g.* [@angleav] for the accuracy of this approximation). This leads to integral equations in the usual partial waves, which can be solved very efficiently if the two-body interaction is represented in terms of separable interaction terms of a sufficient rank[@bozek1]. Finally, we consider the generalized Hartree-Fock contribution to the self-energy in (\[spec\_Sigma\]), which takes the form $$\label{hf_sigma} \Sigma^{HF}(k) = \frac{1}{2} \int \frac{{\mathrm{d}}^3k^{\prime}}{(2\pi)^3} \left<{\mathbf{k}},{\mathbf{k}}^{\prime}\right| V \left|{\mathbf{k}},{\mathbf{k}}^{\prime}\right> n(k^{\prime}),$$ where $n(k)$ is the correlated momentum distribution, which is to be calculated from the spectral function by $$\label{occupation} n(k)= \int_{-\infty}^{+\infty} \frac{{\mathrm{d}}\omega}{2\pi} f(\omega) A(k,\omega).$$ Also the energy per particle, $E/A$, can be calculated from the spectral function using Koltun’s sum rule $$\label{eda} \frac{E}{A}=\frac{1}{\rho} \int \frac{{\mathrm{d}}^3k}{(2\pi)^3} \int_{-\infty}^{+\infty} \frac{{\mathrm{d}}\omega}{2\pi} \frac{1}{2}\left(\frac{k^2}{2m}+\omega\right)A(k,\omega)f(\omega)\,.$$ Eqs.(\[spec\_g2\])-(\[occupation\]) define the so-called $T$-matrix approach to the SCGF equations. They form a symmetry conserving approach in the sense of [@kadanoff], which means that thermodynamical relations like the Hughenholtz-Van Hove theorem[@hugenholtz; @bozek1] are obeyed. The Brueckner-Hartree-Fock (BHF) approximation, which is very popular in nuclear physics, can be regarded as a simple approximation to this $T$-matrix approach. In the BHF approximation one reduces the spectral function $A(k,\omega)$ to the quasiparticle approximation (\[eq:specqp\]). Furthermore one ignores the hole-hole scattering terms in the scattering Eq.(\[eq:tscat0\]), which means that one replaces $$\left(1-f(\omega)-f(\omega')\right) \quad\rightarrow \quad \left(1-f(\omega) \right) \left(1-f(\omega')\right)\,, \label{eq:pauliop}$$ which is the usual Pauli operator (at finite temperature). This reduces the in-medium scattering equation to the Bethe-Goldstone equation. The removal of the hole-hole scattering terms leads to real self-energies $\Sigma(k,\omega)$ at energies $\omega$ below the chemical potential, i.e. for the hole states. Results and discussion ====================== In the following we discuss results for symmetric nuclear matter obtained from Self-Consistent Greens Function (SCGF) calculations. These calculations are either performed in the complete Hilbert space using the bare CD-Bonn [@cdbonn] interaction or in the model space, which is defined by a cut-off parameter $\Lambda$ = 2 fm$^{-1}$ in the two-body scattering equation, employing the corresponding effective interaction $V_{low-k}$, which is derived from the CD-Bonn interaction using the techniques described in Sect II. We note that using this unitary model operator technique we were able to reproduce the results of the BHF calculations presented in [@kuck:03], which used tabulated matrix elements of [@bogner:03], with good accuracy. The NN interaction has been restricted to partial waves with total angular momentum $J$ less than 6. Results for the calculated energy per nucleon are displayed in Fig. \[fig:becd1\] for various densities, which are labeled by the corresponding Fermi momentum $k_F$. The effective interaction $V_{low-k}$ accounts for a considerable fraction of the short-range NN correlations, which are induced by realistic interactions like the CD-Bonn interactions. Therefore, already the Hartree-Fock approximation using this $V_{low-k}$ yields reasonable results for the energies as can be seen from the dotted line of Fig. \[fig:becd1\]. Hartree-Fock calculations using the bare CD-Bonn interaction yield positive energies ranging between 2 MeV per nucleon and 15 MeV per nucleon for the densities considered in this figure. Note that the CD-Bonn interaction should be considered as a soft realistic interaction. Interaction models, which are based on local potentials, like the Argonne interaction [@arv18], yield more repulsive Hartree-Fock energies [@localint]. ![(Color online) Binding energy per nucleon for symmetric nuclear matter as function of the Fermi momentum: Results of self-consistent $T$-matrix calculations for the CD-Bonn potential (dashed line), are compared to results of calculations using $V_{low-k}$ with $\Lambda=2$fm$^{-1}$ in the Hartree-Fock approximation (dotted line), the self-consistent second order approximation (dashed-dotted line) and for the self-consistent $T-$matrix approximation (solid line) within the model space. []{data-label="fig:becd1"}](becd1.eps){width="10.5cm"} The inclusion of correlations within the model space yields a substantial decrease of the energy. The self-consistent $T$-matrix approach provides additional attraction ranging between 6 MeV per nucleon at a density of 0.4 $\rho_0$ (with $\rho_0$ the empirical saturation density) and 3 MeV per nucleon at 2 $\rho_0$. The fixed cut-off parameter $\Lambda$ seems to reduce the phase-space available for correlations beyond the mean-field approach at higher densities. Therefore the energy calculated in the self-consistent $T$-matrix approach reduces to the Hartree-Fock result at large densities. Fig. \[fig:becd1\] also displays the energies resulting from a SCGF calculation within the model space, in which the $T$-matrix has been approximated by the corresponding scattering matrix including only terms up to second order in the NN interaction $V$. The results of such second-order calculations in $V_{low-k}$ are represented by the dashed-dotted line and show a very good agreement with the model-space calculations including the full $T$-matrix. This confirms the validity of approaches, which consider correlation effects within the model-space in a perturbative way. All these model space calculations using $V_{low-k}$, however, fail to reproduce the results of the SCGF calculations, which are obtained in the complete space using the bare NN interaction, which are labeled by CD Bonn T-matrix in Fig. \[fig:becd1\]. In particular, the model space calculations yield to attractive energies at high densities and therefore do not exhibit a minimum for the energy as a function of density. This confirms the results of the BHF calculations of [@kuck:03]. It has been argued [@kuck:03] that this overestimate of the binding energy at high densities is due to the fact that $V_{low-k}$ does not account for the quenching of correlation effects, which is due to the Pauli principle and the dispersive effects in the single-particle propagator getting more important with increasing density. Therefore we try to account for the dispersive quenching effects by adopting the following two-step procedure. In a vein similar to the use of a G-matrix within a self-consistent BHF calculation, as a first step we perform BHF calculations using $V_{low-k}$. The resulting single-particle spectrum is approximated by an effective mass parameterization. This parameterization of the mean field is employed to define the single-particle operator $h_0$, used in Eq. (\[eq:veff1\]) and the following equations of Sect. II (see also [@fuji:04]). The resulting effective interaction is used again for a BHF calculation within the model space, leading to an update of the mean field parameterization. The procedure is repeated until a self-consistent result is obtained. Since the mean field parameterization depends on the density, this method yields an effective density-dependent interaction, which in the limit of the density $\rho\to 0$ coincides with $V_{low-k}$. Therefore we call this effective interaction the density dependent $V_{low-k}$ or in short $V_{low-k}(\rho)$. Such a procedure amounts to summing up certain higher order terms in the full many-body problem. ![(Color online) Same as Fig. \[fig:becd1\] but for $V_{low-k}(\rho)$ calculated at each density[]{data-label="fig:becd2"}](becd2.eps){width="10.5cm"} In a second step this $V_{low-k}(\rho)$ is used in SCGF calculations at the corresponding density. Energies resulting from such model space calculations using $V_{low-k}(\rho)$ are presented in Fig. \[fig:becd2\]. The comparison of the various calculations within the model space exhibits the same features as discussed above for the original $V_{low-k}$. The correlation within the model space provide a substantial reduction of the energy as can be seen from the comparison of the self-consistent $T$-matrix approach with the Hartree-Fock results. The approach treating correlations up to second order in $V_{low-k}(\rho)$ yields energies which are very close to the complete $T$-matrix approach. The density dependence of the effective interaction $V_{low-k}(\rho)$ yields a significant improvement for the comparison between the model space calculations and the SCGF calculation using the bare CD-Bonn interaction. Note that the energy scale has been adjusted going from Fig. \[fig:becd1\] to Fig. \[fig:becd2\]. The discrepancy remaining at densities above $\rho_0$ might be due to the effects of the Pauli quenching, which are not included in $V_{low-k}(\rho)$. These deviations could also originate from the simple parameterization of the dispersive quenching in $V_{low-k}(\rho)$. Our investigations also provide the possibility to explore the effects of correlations evaluated within the model space using the effective interaction $V_{low-k}$. We can furthermore compare these correlation effects with the corresponding effects determined by the bare interaction in the unrestricted space. As a first example, we discuss the imaginary part of the self-energy calculates at the empirical saturation density $\rho_0$ for various nucleon momenta $p$ as displayed in Fig. \[fig:im10\]. The calculations within the model space reproduce the results of the unrestricted calculations with a good accuracy in the energy interval for $\omega$ ranging between 50 MeV below and 50 MeV above the chemical potential $\mu$. The remaining differences around the Fermi energy can be attributed to the difference in the effective masses obtained using the $V_{low-k}$ and the bare potential [@wi98]. The agreement between the $T$-matrix results around $\omega=\mu$ using the two potentials is improved if one rescales by the ratio of the effective masses. The imaginary part calculated with $V_{low-k}$, however, is much smaller than the corresponding result obtained for the bare interaction at energies $\omega -\mu$ above 100 MeV. Furthermore the model space calculation do not reproduce the imaginary part for energies below the chemical potential at momenta $k$ above 400 MeV/c. ![(Color online) Imaginary part of the self-energy as a function of the energy $\omega$ for various momenta $p$ as indicated in the panels (see Eq. ( )). The results have been determined for the empirical saturation density $\rho_0$; using $V_{low-k}$ in the $T$-matrix approximation (solid line), using $V_{low-k}$ in the second order approximation (dashed-dotted line), and employing CD-Bonn interaction in the $T$-matrix approximation (dotted line). The dashed line in the first panel denotes the results of the $T$-matrix calculation with the CD-Bonn potential rescaled by the ratio of the effective masses at the Fermi momentum obtained with the $V_{low-k}$ and the bare CD-Bonn potential. []{data-label="fig:im10"}](im10.eps){width="10.5cm"} The imaginary part of the self-energy is a very important ingredient for the evaluation of the spectral function $A(k,\omega )$ and therefore also for the calculation of the occupation probability $n(k)$ (see Eq. (\[occupation\])). The small values for the imaginary part of the self-energy at high momenta $k$ and negative energies $\omega -\mu$ leads to occupation probabilities at these momenta, which are much smaller than the corresponding predictions derived from bare realistic NN interactions as can be seen from Fig. \[fig:nofk\]. This missing strength in the prediction of $V_{low-k}$ at high momenta is accompanied by larger occupation probabilities at low momenta. The self-consistent $T$-matrix approximation using CD-Bonn yields an occupation probability at $k=0$ of 0.897, while the corresponding number using $V_{low-k}$ is 0.920. At this density, the calculation including only terms up to second order in $V_{low-k}$ yields a rather good approximation to the self-consistent $T$-matrix approximation within the model space. ![(Color online) Momentum distribution $n(k)$ (see Eq. ()) calculated for nuclear matter at the empirical saturation density $\rho_0$. Results of the $T$-matrix approximation within the model space (solid line) are compared to results of the second order approximation (dashed-dotted line) and the $T$-matrix approximation (dotted line) in the unrestricted space.[]{data-label="fig:nofk"}](nofk.eps){width="10.5cm"} As a second example we consider the imaginary part of the self-energy calculated at a lower density $\rho=0.4\times \rho_0$. The results displayed in Fig. \[fig:im04\] refer to nucleons with momentum $k=0$. Also at this density we find that the imaginary part evaluated with $V_{low-k}$ drops to zero at large positive energies much faster than the predictions derived from the bare interaction (see upper panel on the left in Fig. \[fig:im04\]). It is worth noting, that at this low density the second order approximation is not such a good approximation to the full $T$-matrix approach as it is for the higher densities. Characteristic differences between the dashed-dotted and the solid line show up at energies $\omega$ close to the chemical potential. In order to trace the origin of these differences we display in Fig. \[fig:im04\] the contributions of various partial waves of NN interaction channels to this imaginary part. It turns out that the differences are largest in the $^3S_1-^3D_1$ and the $^1S_0$ channels. This means that the perturbative approach is not very successful in those two channels which tend to form quasi-bound states. In these channels all particle-particle hole-hole ladders have to be summed up to obtain the pairing solution. Note, that the pairing solutions are suppressed at higher densities, if the effects of short-range correlations are properly taken into account[@bozek4; @muwi05]. Furthermore we would like to point out that a different scale is used in the two lower panels of Fig. \[fig:im04\]. Taking this into account it is evident from this figure that the main contribution to the imaginary part of the self-energy, and that means the main contribution to the character of the deviation of the spectral function from the mean-field approach originates from the NN interaction in the $^3S_1-^3D_1$ channel. ![Imaginary part of the self-energy as a function of the energy $\omega$ for nucleons with momentum $k=0$ calculated at the density $\rho=0.4\times \rho_0$. Results of the $T$-matrix approach (solid line) and the second order approximation (dashed-dotted line) within the model space are compared to results obtained in the unrestricted calculation (dotted line).[]{data-label="fig:im04"}](im04all.eps){width="10.5cm"} Conclusions =========== During the last few years it has become very popular to perform nuclear structure calculations using effective low-momentum NN interactions. These $V_{low-k}$ interactions are based on a realistic model of the NN interaction. They are constructed to be different from zero only within a model space defined by a cut-off $\Lambda$ in the relative momenta of the interacting nucleons. Within this model space they reproduce the NN data of the underlying bare interaction, although the many-body solutions may show differences with different starting NN interactions. For this study we performed Self-Consistent Greens Functions (SCGF) calculations of symmetric nuclear matter employing $V_{low-k}$ effective interactions as well as the bare CD Bonn interaction they are based on. Special attention was paid to the correlations which can be described within this model space as compared to correlations predicted by the underlying interaction within the unrestricted space. Using a cut-off $\Lambda$ = 2 fm$^{-1}$ we find that the spectral distribution of the single-particle strength in an energy window of plus minus 50 MeV around the Fermi energy is rather well reproduced by the calculation using $V_{low-k}$. The effective interaction $V_{low-k}$ is softer than typical realistic NN interactions. Therefore for many observables it is sufficient to approximate the full in-medium scattering matrix $T$ by the approximation including terms up to second order in $V_{low-k}$. This justifies the use of the resummed effective interaction in many-body approximations that do not the include ladder-diagram resummation. Special attention must be paid to nuclear systems at smaller densities: the possible formation of quasi-bound states may require the non-perturbative treatment of the NN scattering in the medium. This also has implications for the use of $V_{low-k}$ in studies of weakly bound nuclear systems. The model space approach cannot reproduce correlation effects, which lead to spectral strength at high energies and high momenta. For nuclear matter at the empirical saturation density $\rho_0$ the momentum distribution is reliably predicted up to a momentum of 400 MeV/c. The $V_{low-k}$ approach overestimates the binding energy per nucleon at high densities. Therefore we introduced a density-dependent effective interaction $V_{low-k}(\rho)$ which we constructed along the same line as the original $V_{low-k}$. 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ArXiv

--- abstract: 'We evaluate quantum corrections to conductivity in an electrically gated thin film of a three-dimensional (3D) topological insulator (TI). We derive approximate analytical expressions for the low-field magnetoresistance as a function of bulk doping and bulk-surface tunneling rate. Our results reveal parameter regimes for both weak localization and weak antilocalization, and include diffusive Weyl semimetals as a special case.' author: - Ion Garate and Leonid Glazman title: 'Weak Localization and Antilocalization in Topological Insulator Thin Films with Coherent Bulk-Surface Coupling' --- Introduction and Overview ========================= The theoretical discovery[@ti] of 3D topological insulators (TIs) in 2006 precipitated an avalanche of experiments aimed at detecting the signature behavior of these unconventional solids. Since then, angle-resolved photoemission spectra[@hasan2011] have given evidence for the Dirac-like dispersion and the momentum-dependent spin texture of TI surface states, whereas local STM probes have indicated a characteristic suppression of backscattering off surface imperfections.[@stm] However, the most desired observation of a hallmark dc conduction confined to the surface layer of a 3D TI remains elusive.[@dimi2011] The main problem is conduction through the bulk: 3D TIs are narrow-gap semiconductors, rich in bulk carriers that are either thermally activated and/or donated by crystalline lattice imperfections. Along with attempts to reduce bulk charge carriers, experimentalists are developing techniques which allow to register a separate conduction channel on the surface of a 3D TI.[@kim2012] Chief among these are measurements of low-field magnetoresistance combined with electrostatic gating of thin-film samples.[@wang2011; @chen2010; @checkelsky2011; @he2011; @chen2011; @steinberg2011; @hong2012] Low-field magnetoresistance measurements unveil the interference correction $\delta\sigma$ to the Drude conductivity $\sigma_D$.[@inter] At low temperatures, $\sigma_D$ is defined by independent acts of scattering of electrons off the crystal’s imperfections, and is proportional to the classical diffusion constant $D$. When the phase relaxation length $l_\phi$ is parametrically longer than the scattering mean free path, quantum interference affects the conductivity to a measurable extent. The sign of the interference correction depends on the strength of spin-orbit interactions. For weak spin-orbit interactions ($l_{\rm so}\gg l_\phi$, where $l_{\rm so}$ is the spin-orbit scattering length), it follows that $\delta\sigma<0$. This is called weak localization (WL). In contrast, strong spin-orbit interaction ($l_{\rm so}\ll l_\phi$) leads to suppression of backscattering and thus $\delta\sigma>0$. This is called weak antilocalization (WAL). Being interference effects, WL and WAL are degraded by a magnetic field $H$ when $H\gtrsim H_\phi\equiv \Phi_0/(8\pi l_\phi^2)$, where $\Phi_0=h/e$ is the flux quantum. Yet, $\sigma_D$ is nearly immune to $H$ at such low fields. Therefore, the low-field magnetoconductivity reads $\Delta\sigma(H)\equiv\sigma(H)-\sigma(0)\simeq\delta\sigma(H)-\delta\sigma(0)$. All experiments to date report WAL in 3D TI thin films,[@kapitulnik2012] and ascribe it to the strong spin-orbit interaction in the electronic bands of these materials. For film thickness less than $l_\phi$, the measured $\Delta\sigma(H)$ agrees well with the functional form provided by 2D WAL theory, namely $$\label{eq:hikami} \Delta\sigma(H)\simeq \alpha\,(e^2/2\pi^2 \hbar) f(H_\phi/H),$$ where $f(z)\equiv \ln z-\psi(1/2+z)$, with $\psi$ and $\alpha$ being the digamma function and a number,[@hikami1980] respectively. In a system with a single conduction channel, $\alpha$ is universal and equals $1/2$. The WAL contributions add for systems which are isolated from each other. For example, having two independent parallel conduction channels yields $\alpha=1$, irrespective of the ratio of Drude conductivities of the two subsystems. The relation between $\alpha$ and the number of parallel channels is at the heart of recent magnetoresistance experiments in 3D TIs.[@checkelsky2011; @chen2011; @steinberg2011] Overall, the coefficient $\alpha$ is found to depend on the gate voltage. For some devices,[@checkelsky2011; @chen2011; @steinberg2011] it changes from $\alpha=1/2$ all the way to $\alpha=1$. A plausible interpretation for this variation is presented in Ref. \[\]. At zero or positive bias applied to the top gate, electrons from the $n$-doped bulk reach the surface states easily: the entire film acts as a single electron system, and $\alpha=1/2$. At negative bias, electrons are repeled from the top surface and, for strong enough bias, a depletion layer is formed adjacent to it. This depletion region separates the film into two subsystems: bulk carriers (combined with surface carriers from the bottom surface) on one side, and top-surface carriers on the other side. For a wide enough depletion layer, $\alpha=1$. In spite of the ongoing scrutiny on the experimental front, quantum corrections to conductivity in 3D TIs have stimulated relatively little theoretical activity. Even though the WAL contribution from TI surface states has been calculated explicitly,[@lu2011a; @tkachov2011] there are no calculations that incorporate conducting 3D bulk states. The main reason for this omission may be the prevailing view that quantum corrections originating from bulk TI states ought to be conceptually identical to those in ordinary strongly spin-orbit coupled systems, i.e. of WAL type. Recently, an objection to this viewpoint has been raised,[@lu2011] declaring that quantum well states in ultrathin TI films may contribute via WL rather than WAL. Although suggestive, the calculation of Ref. \[\] is limited to quasi-2D films and disregards the coupling between bulk and surface states, which leaves out several experiments of interest. Besides, its extrapolation to 3D bulk states has not been carried out properly. In this paper we evaluate $\Delta\sigma$ for gated 3D thin films, as a function of the bulk carrier concentration and accounting for the coupling between surface and bulk states. Our calculation applies to TI films that are thicker than the bulk mean free path, thinner than $l_\phi$, and not highly doped. In these films, bulk carriers are three-dimensional and are concentrated around the $\Gamma$ point of the electronic band structure. The resulting approximate analytical expressions for $\Delta\sigma$ (Eqs. (\[eq:magres\_bulk\]), (\[eq:res\_tot\]) and (\[eq:res\_tot5\])) are aimed at improving the interpretation of magnetoresistance measurements in TIs, in Weyl semimetals,[@burkov2011] and in some class of topologically trivial materials. Although a few of our observations resemble those developed for graphene[@mccann2006] and 2D TIs,[@tkachov2011] there are qualitative differences originating from the 3D Dirac nature of bulk carriers in 3D TIs. Altogether, the results reported here paint a richer picture than previously anticipated. On one hand, we confirm the conventional crossover between $\alpha=1/2$ and $\alpha=1$ as a function of the gate voltage: the former corresponds to the case of coherently-coupled bulk and surface electron states, while the latter indicates a single decoupled Dirac cone on the top surface along with generic WAL from the rest of the film (containing coupled bulk and bottom surface). On the other hand, less conventional results arise when the Fermi energy is close to the bulk band edge or when the Fermi energy is much larger than the bulk bandgap: in the former regime the bulk exhibits WL with $\alpha=-1$, whereas in the latter regime the bulk exhibits an anomalous WAL with $\alpha=1$. These two “unusual” bulk regimes, combined with the surface contributions, may result in a range of $\alpha$ including $\alpha<0$ and $\alpha>1$. The rest of this work is organized as follows. In Section II we evaluate quantum corrections to [*bulk*]{} conductivity. Readers not interested in technical details should read subsection IIA and quickly scan through IIB and IIC in order to get acquainted with the nomenclature; the main results of the section are collected in Section IID. The well-known message from IIA is that at low energies bulk electrons of TI films behave as massive 3D Dirac fermions with spin and valley (or orbital) degrees of freedom. The direction of spin is locked with that of momentum, and valleys are coupled to one another by the mass of the Dirac fermions. The special case in which the Dirac mass vanishes is a time- and inversion-symmetric Weyl semimetal. In Section IID we identify and count the number of “soft” Cooperon modes, which determine the magnitude and sign of $\Delta\sigma$ in the bulk. Each soft Cooperon obeys a classical difussion equation and is thus associated with a conserved physical quantity. Since charge is conserved, there is at least one soft Cooperon in (non-magnetic) bulk TIs. We find that additional soft Cooperons can emerge depending on the bulk doping concentration as well as the bulk bandgap. This realization leads to the most important results in IID, Eqs. (\[eq:res\_bulk\])-(\[eq:magres\_bulk\]), which indicate that for bulk states $\alpha$ may acquire three different universal values. On one hand, WL with $\alpha=-1$ is possible when the bulk Fermi surface is “small” (as defined in the text), because in this case the spin-momentum locking of bulk states becomes weak and the spin of electrons is nearly conserved. In contrast, WAL with $\alpha=1$ can arise for bulk TIs with particularly small bandgaps, because in such case bulk electrons can be described by a 3D analogue of graphene with two nearly decoupled valleys, each contributing $1/2$ to $\alpha$. For a more generic case, in which neither valley nor spin are approximately conserved, the quantum interference is similar to that of an ordinary film with strong spin-orbit coupling and therefore $\alpha=1/2$. Magnetic fields perpendicular to the TI film can be used to induce crossovers between different universal regimes of $\alpha$. The accessible values of $\alpha$ and the corresponding crossover fields depend on the bulk electron density. In Section III we evaluate the [*full*]{} $\Delta\sigma$ in 3D TI thin films, which comprises coupled bulk and surface contributions. Sections IIIA and IIIB cover preliminary material that is needed to derive the main results in IIIC. Section IIIA reviews the well-established fact that, in absence of magnetic order, isolated TI surface states exhibit WAL with $\alpha=1/2$ (in this paper we assume one Dirac cone per surface). Section IIIB develops a diagrammatic framework for evaluating quantum corrections to conductivity in ordinary tunnel-coupled layers. Readers who are not interested in technicalities can disregard the diagrams in the figures and concentrate on the outcome of the calculation (Eqs. (\[eq:q1\])-(\[eq:Dii\])), as well as on the subsequent discussion. One qualitative point made therein is that the crossover from weak to strong coupling (which is accompanied by a change in $\alpha$ from $1$ to $1/2$) occurs when the interlayer resistance for a square of area $l_\phi^2$ becomes smaller than the sum of the classical intralayer resistances. Section IIIC combines results from IID, IIIA and IIIB in order to figure out quantum corrections to conductivity in experimentally realized TI films. The most important results in IIIC are Eqs. (\[eq:res\_tot\]) and  (\[eq:res\_tot5\]), which describe how $\Delta\sigma$ depends on the bulk doping concentration, on the phase relaxation rate, and on the bulk-surface tunneling rate. Some special cases of these results are highlighted in Appendix \[sec:special\]. A salient conclusion is that the WL regime of isolated bulk states is generally eliminated when either one of the film surfaces is strongly coupled to bulk states, in which case the film displays $1/2\leq\alpha\leq 1$. However, WL can still be present if the TI surfaces have short phase relaxation lengths. Finally, Section IIID characterizes the electrostatics of the depletion layer and estimates the bulk-surface tunneling rate in TI films. This estimate confirms experimental indications showing that both weak and strong bulk-surface coupling are accessible by mediation of a gate voltage. Quantum Corrections to Bulk Conductivity ======================================== This section is devoted to evaluating $\delta\sigma$ for the bulk states of a 3D TI. As a byproduct, we derive $\delta\sigma$ for a time-reversal symmetric Weyl semimetal. The contribution from TI surface states will be discarded until the next section. Model ----- The bulk band structure of a 3D TI near the $\Gamma$ point can be approximated by the following ${\bf k}\cdot{\bf p}$ Hamiltonian: [@zhang2009] $$\begin{aligned} \label{eq:model_b} &{\cal H}=\sum_{\bf k}\Psi^\dagger_{\bf k} h({\bf k}) \Psi_{\bf k}\nonumber\\ & h({\bf k}) \simeq \epsilon({\bf k}){\bf 1}_4+M({\bf k}){\bf 1}_2\,\tau^z+\hbar\left(v_z k_z\sigma^z+v_\perp {\bf k}_\perp\cdot{\boldsymbol \sigma}^\perp\right)\tau^x, \end{aligned}$$ where ${\boldsymbol \tau}$ is an orbital pseudospin ($\tau^z=T,B$), ${\boldsymbol \sigma}$ is the real spin ($\sigma^z=\uparrow,\downarrow$), ${\bf k}=({\bf k}_\perp,k_z)$ is the momentum measured from the $\Gamma$ point of the Brillouin zone, ${\bf 1}_N$ is an $N\times N$ identity matrix, $\Psi=(\Psi_{T\uparrow},\Psi_{T\downarrow},\Psi_{B\uparrow},\Psi_{B\downarrow})$ is a 4-spinor, $\epsilon({\bf k})=\epsilon(-{\bf k})$ is the part of the Hamiltonian that is independent of spin/pseudospin indices, $v_z$ and $v_\perp$ are the Fermi velocities, and $M({\bf k})=M_0-M_1 k_\perp^2-M_2 k_z^2$ is the mass term (independent of spin). $M_0$, $M_1$ and $M_2$ are constants. Equation (\[eq:model\_b\]) captures the bottom of the conduction band and the top of the valence band in the vicinity of the $\Gamma$ point ($k\equiv 0$), where the bandgap is smallest. It models 3D Dirac fermions with a Dirac mass that equals half the energy gap. For the purposes of this paper we ignore $\epsilon({\bf k})$, and assume $M({\bf k})=M={\rm const}>0$ as well as spherical symmetry ($v_z=v_\perp=v$). These assumptions simplify calculations without incurring in qualitative loss of generality. For instance, the XXZ anisotropy can be modeled by promoting the diffusion constant from a scalar to a matrix. Also, the $k^2$ terms in $M({\bf k})$ can be incorporated into our final results by $M\to |M({\bf k}_F)|$, where ${\bf k}_F$ is the Fermi wave vector. Note that in absence of spherical symmetry the Fermi surface does not have a constant mass; this complication will be disregarded in the present paper. Finally, $\epsilon({\bf k})$ can be absorbed into the definition of the Fermi energy. ![Bulk energy bands of an $n$-doped 3D TI near the $\Gamma$ point, in the spherical approximation. The momentum $k$ is measured from the $\Gamma$ point. The energies $\epsilon_F$ and $M$ are measured with respect to midgap.[]{data-label="fig:bands"}](./bands.eps) The energy eigenvalues for $h({\bf k})$ in the spherical approximation are $E_{{\bf k}\pm}=\pm\sqrt{\hbar^2v^2 k^2+M^2}$, each doubly degenerate (Fig. \[fig:bands\]). The corresponding Bloch states can be written as $$\label{eq:eigen0} |\Psi_{{\bf k}\alpha}\rangle=(1/\sqrt{V})\exp(i {\bf k}\cdot{\bf r})|\alpha {\bf k}\rangle,$$ where $V$ is the volume of the TI and $\alpha\in\{1, 2, 3, 4\}$ is a band index ($1$ and $2$ denote conduction bands, while $3$ and $4$ denote valence bands). This $\alpha$ is obviously unrelated to that of Eq. (\[eq:hikami\]); from here on it will be clear from the context which one we are referring to. For concreteness we set the chemical potential in the bulk conduction band, although all results obtained below will be directly applicable to $p$-doped bulk TIs as well. The density of conduction band electrons is then $$\label{eq:n} n\simeq \frac{\left(\epsilon_F^2-M^2\right)^{3/2}}{\pi^2\hbar^3 v^3},$$ where $\epsilon_F$ is the Fermi energy measured from the middle of the bulk energy gap. Adopting the basis $\{|T\uparrow\rangle, |T\downarrow\rangle,|B\uparrow\rangle,|B\downarrow\rangle\}$, the two eigenspinors corresponding to the conduction bands near the $\Gamma$ point are $$\begin{aligned} \label{eq:eigenstates} |1 {\bf k}\rangle&=&\sqrt{\frac{E_k+M}{2 E_k}}\left(1,0,\frac{\hbar v k_z}{E_k+M},\frac{\hbar v k_+}{E_k+M}\right)\nonumber\\ |2 {\bf k}\rangle&=&\sqrt{\frac{E_k+M}{2 E_k}}\left(0,1,\frac{\hbar v k_-}{E_k+M},\frac{-\hbar v k_z}{E_k+M}\right),\end{aligned}$$ where $k_\pm=k_x\pm i k_y$ and $E_k=E_{{\bf k},+}$. Since all non-Hall dc transport properties of good conductors involve states close to the Fermi energy, we hereafter ignore valence bands. Unlike in the ${\bf k}\cdot{\bf p}$ Hamiltonians for graphene and 2D (or quasi-2D) TIs, Eq. (\[eq:model\_b\]) cannot be decomposed into two $2\times 2$ block diagonal matrices (due to $M\neq 0$). In addition, the $k_z$ band dispersion absent in 2D cannot be neglected in our case. These two features make the calculations and results of this section quite different from those of Refs. \[\]. Equation (\[eq:model\_b\]) becomes inaccurate when the chemical potential moves up in the conduction band and electron pockets away from the $\Gamma$ point begin to be populated. These additional pockets contribute to quantum interference, and the total $\delta\sigma$ depends on the scattering rate between different electron pockets. Although a realistic study of the full band structure is beyond the scope of this paper, we expect calculations based on Eq. (\[eq:model\_b\]) to provide a generic understanding of quantum corrections to conductivity in 3D Dirac materials at low-to-moderate doping concentrations. Formalism --------- In order to quantify the conductivity of a bulk TI, we begin by characterizing the simplest possible disorder potential: $V_{\rm dis}({\bf r})=V({\bf r}){\bf 1}_4$, which is time-independent (elastic) and independent of spin as well as orbital degrees of freedom. For simplicity we assume $V({\bf r})$ to be slowly-varying at the atomic scale, yet short-ranged compared to the mean free path: $V({\bf r})=V_0\delta({\bf r})$. It is due to its slow spatial variation on atomic lenghtscales that $V_{\rm dis}$ becomes an identity operator in orbital space. With such disorder realization, the Fermi-surface lifetime $\tau_0$ for the $\alpha=1,2$ eigenstates in Eq. (\[eq:eigen0\]) obeys $$\begin{aligned} \label{eq:lifetime} \frac{1}{\tau_0} &=\frac{2\pi u_0}{\hbar}\int_{{\bf k}'}\sum_{\alpha'}|\langle\alpha {\bf k}_F|\alpha' {\bf k}_F'\rangle|^2 \delta(\epsilon_F-E_{{\bf k}'\alpha'})\nonumber\\ &\simeq\frac{\pi u_0 \nu}{\hbar} \left(1+\frac{M^2}{\epsilon_F^2}\right),\end{aligned}$$ where $\int_{\bf k}\equiv\int d^3 k/(2\pi)^3$, $u_0\equiv n_i V_0^2$, $n_i$ is the density of impurities, and $\nu$ is the density of states per band and per unit volume at $\epsilon_F$. A related quantity is the transport scattering rate $\tau^{-1}$, $$\begin{aligned} \frac{1}{\tau} &\equiv \frac{2\pi u_0}{\hbar}\int_{{\bf k}'}\sum_{\alpha'} (1-\hat{{\bf k}}_F\cdot\hat{{\bf k}}_F') |\langle\alpha {\bf k}_F|\alpha' {\bf k}_F'\rangle|^2 \delta(\epsilon_F-E_{{\bf k}'\alpha'})\nonumber\\ &= \frac{2}{3\tau_0}\frac{\epsilon_F^2+2 M^2}{\epsilon_F^2+M^2}.\end{aligned}$$ The momentum-dependence of $|\alpha {\bf k}\rangle$ makes $\tau_0\neq\tau$ even for $\delta$-function impurity potentials. Throughout this work we impose $(\epsilon_F-M)\tau\gg \hbar$ or equivalently $k_F l\gg 1$, where $l=(D\tau)^{1/2}$ is the elastic mean free path, $$k_F=(\epsilon_F^2-M^2)^{1/2}/(\hbar v)$$ is the Fermi wave vector and $$D=v_F^2\tau/3=v^2\tau(1-M^2/\epsilon_F^2)/3$$ is the classical diffusion constant. Next, we consider a TI with spatial dimensions $L\times L$ in the $xy$ plane and a thickness $W$ along the $z$ direction. We take a thin film geometry with $L\gg l_\phi\gg l$ and $l_\phi\gg W\gg l$, where $l_\phi=(D\tau_\phi)^{1/2}$ is the coherence length and $\tau_\phi$ is the phase relaxation time. The conductivity of this film is $$\sigma=\sigma_D+\delta\sigma,$$ where $\sigma_D$ is the classical (Drude) part and $\delta\sigma$ is the part coming from quantum interference. On one hand, the Drude conductivity can be approximated as $$\label{eq:sd} \sigma_D\simeq\frac{e^2\hbar}{2\pi}\sum_{\alpha,\beta}\int_{\bf k} v^x_{\alpha\beta}({\bf k})\tilde{v}^x_{\beta\alpha}({\bf k}) G^R_\alpha({\bf k}) G^A_\beta({\bf k}),$$ where we have assumed a spatially uniform dc electric field and $\alpha,\beta\in\{1,2\}$. $v^x_{\alpha\beta}=\langle\alpha{\bf k}|{\bf v}\cdot\hat{x}|\beta {\bf k}\rangle$ is a matrix element for the $x$-component of the bare velocity operator ${\bf v}=v\tau^x{\boldsymbol\sigma}$, which obeys $$\label{eq:vbare} {\bf v}_{\alpha\beta}({\bf k})=\hbar v^2 ({\bf k}/E_k) \delta_{\alpha\beta}\,\,\,\mbox{(for $\alpha,\beta\in\{1,2\}$)}.$$ Disorder vertex corrections renormalize Eq. (\[eq:vbare\]) to $$\tilde{{\bf v}}_{\alpha\beta}={\bf v}_{\alpha\beta}(\tau/\tau_0),$$ see Appendix \[sec:ren\]. In addition, $$\label{eq:ds} G_\alpha^{R(A)}({\bf k})= \left[\epsilon_F-E_{{\bf k}\alpha}+(-) \frac{i\hbar}{2\tau_0}\right]^{-1}$$ is the zero-frequency retarded (advanced) Green’s function in the band eigenstate basis. Using $G^{R(A)}_1({\bf k})=G^{R(A)}_2({\bf k})\equiv G^{R(A)}({\bf k})$, Eq. (\[eq:sd\]) yields $$\sigma_D=2 e^2 \nu D.$$ ![(a) Feynman diagram for $\delta\sigma_1$, defined in the text. Filled squares denote velocity operators (including disorder vertex corrections), $C$ is the Cooperon. (b) Diagrammatic representation of the Bethe-Salpeter equation for the Cooperon. Crosses correspond to impurity scattering centers. Solid lines with arrows are disorder-averaged Green’s functions. (c) Additional Feynman diagrams that contribute to conductivity of 3D TIs even when impurity scattering is isotropic.[]{data-label="fig:cofig"}](./diag0.eps) On the other hand, the quantum correction $\delta\sigma$ can be written as $\delta\sigma\simeq\delta\sigma_1+\delta\sigma_2$, represented pictorially in Fig. \[fig:cofig\]. Following standard approximations, the expression for $\delta\sigma_1$ is $$\begin{aligned} \label{eq:ds} \delta\sigma_1 &\simeq \frac{e^2\hbar}{2\pi}\sum_{\alpha,\alpha',\beta,\beta'}\int_{\bf k} \tilde{v}^x_{\alpha\beta}({\bf k}) \tilde{v}^x_{\beta'\alpha'}(-{\bf k}) G^R_\alpha({\bf k}) G^R_{\alpha'}(-{\bf k})\nonumber\\ &~~~\times G^A_\beta({\bf k}) G^A_{\beta'}(-{\bf k})\frac{1}{W}\int \frac{d^2 Q}{(2\pi)^2} C^{\beta\beta'}_{\alpha'\alpha}({\bf k},{\bf k},{\bf Q}).\end{aligned}$$ In the second line of Eq. (\[eq:ds\]) we have exploited the condition $W\ll l_\phi$, which allows to set $Q_z=0$ everywhere. $C^{\beta\beta'}_{\alpha'\alpha}({\bf k}_1,{\bf k}_2,{\bf Q})$ are the matrix elements of the Cooperon matrix $\hat{C}$ in the band eigenstate basis. ${\bf Q}=(Q_x,Q_y)$ is the momentum of the Cooperon, whose magnitude ranges from $0$ to $\simeq (D\tau)^{-1/2}$. $\hat{C}$ obbeys the Bethe-Salpeter equation (Fig. \[fig:cofig\]b): $$\begin{aligned} \label{eq:bs} &C^{\beta\beta'}_{\alpha'\alpha}({\bf k}_1,{\bf k}_2,{\bf Q})=\Gamma^{\beta\beta'}_{\alpha'\alpha}({\bf k}_1,{\bf k}_2,{\bf Q})+ \int_{{\bf k}_3}\Gamma^{\beta\beta''}_{\alpha'\alpha''}({\bf k}_1,{\bf k}_3,{\bf Q})\nonumber\\ &~~~~~~~\times G^A_{\beta''}({\bf k}_3) G^R_{\alpha''}(-{\bf k}_3+{\bf Q}) C^{\beta''\beta'}_{\alpha''\alpha}({\bf k}_3,{\bf k}_2,{\bf Q}),\end{aligned}$$ where a sum over repeated indices is implied and $$\Gamma^{\beta\beta'}_{\alpha'\alpha}({\bf k}_1,{\bf k}_2,{\bf Q})\equiv u_0\langle\beta {\bf k}_1|\beta' -{\bf k}_2+{\bf Q}\rangle\langle\alpha' -{\bf k}_1+{\bf Q}|\alpha {\bf k}_2\rangle\nonumber$$ is the bare disorder vertex (first term on the right hand side of Fig. \[fig:cofig\]b). Equation (\[eq:bs\]) is a complicated integral equation because $C^{\beta\beta'}_{\alpha'\alpha}$ is a function of three momenta. This is unlike in simplest examples, where the Cooperon depends only on ${\bf Q}$. The difficulty originates from the momentum-dependence of $|\alpha {\bf k}\rangle$, which cannot be overlooked as it crucially determines both the magnitude and the sign of $\delta\sigma$. One procedure[@garate2009] to solve Eq. (\[eq:bs\]) starts by writing the Cooperon in the two-particle spin/orbit basis $\{|m,m'\rangle\}$, where $m\in\{T\uparrow,T\downarrow,B\uparrow,B\downarrow\}$: $$\begin{aligned} \label{eq:trans} &C^{\beta \beta'}_{\alpha' \alpha}({\bf k}_1, {\bf k}_2, {\bf Q})=\sum_{m,m',n,n'}\langle\alpha',-{\bf k}_1+{\bf Q}|m'\rangle\langle\beta {\bf k}_1|m\rangle\nonumber\\ &\times\langle n|\beta',-{\bf k}_2+{\bf Q}\rangle\langle n'|\alpha {\bf k}_2\rangle C^{m n }_{m' n'}({\bf Q}).\end{aligned}$$ We then make the ansatz that $C^{m n}_{m' n'}$ depends on ${\bf Q}$ but not on ${\bf k}_1$ and ${\bf k_2}$; the entire ${\bf k}_1$- and ${\bf k}_2$-dependence of $C^{\beta \beta'}_{\alpha' \alpha}({\bf k}_1,{\bf k}_2,{\bf Q})$ originates from the overlap matrix elements of Eq. (\[eq:trans\]). The internal consistency of this ansatz can be demonstrated by substituting Eq. (\[eq:trans\]) in Eq. (\[eq:bs\]), which produces an algebraic equation for $C^{m n}_{m' n'}$ that is more tractable than the original integral equation:$$\label{eq:C_mat} C^{m n}_{m' n'}({\bf Q})= u_0 \delta_{m n}\delta_{m' n'}+\sum_{l,l'}U^{m l}_{m' l'}({\bf Q}) C^{l n}_{l' n'}({\bf Q}),$$ where $$\label{eq:U} U^{m l}_{m' l'}({\bf Q})=u_0\int \frac{d^3 k}{(2\pi)^3} G_{m l}^A({\bf k})G^R_{m' l'}(-{\bf k}+{\bf Q})$$ and $$G^{R (A)}_{m l}({\bf k})=\sum_\alpha \langle m|\alpha {\bf k}\rangle G_\alpha^{R (A)}({\bf k}) \langle\alpha {\bf k}|l\rangle.$$ In matrix language, Eq. (\[eq:C\_mat\]) can be rewritten as $$\label{eq:C_mat2} \hat{C}=({\bf 1}_{16}-\hat{U})^{-1} \hat{C}^{(0)},$$ where $\hat{C}^{(0)}=u_0 {\bf 1}_{16}$. Once we obtain $C^{m n}_{m' n'}$, we use Eq. (\[eq:trans\]) in order to recover $C^{\beta\beta'}_{\alpha'\alpha}$. During this operation we neglect ${\bf Q}$ in the overlap matrix elements, which is a good approximation because $\delta\sigma$ is dominated by elements of $C^{m n}_{m' n'}({\bf Q})$ that are strongly peaked at $Q\simeq 0$. For $\epsilon_F$ in the conduction band, we once again limit ourselves to $\alpha,\beta,\alpha',\beta'\in\{1,2\}$ in Eq. (\[eq:ds\]). Then we can approximate ${\bf k}\simeq {\bf k}_F$ inside the Cooperon, and an integration over $|{\bf k}|$ yields $$\label{eq:ds2bis} \delta\sigma_1 \simeq -6 \frac{e^2}{\hbar^2} \nu D \tau\tau_0\frac{1}{W}\int\frac{d^2 Q}{(2\pi)^2}\overline{C}({\bf Q}),$$ where $$\label{eq:overline_c} \overline{C}({\bf Q}) \equiv \int \frac{d\Omega_{\bf k}}{4\pi}\hat{\bf k}_x^2\sum_{\alpha,\alpha'=1,2}C^{\alpha \alpha'}_{\alpha' \alpha}({\bf k}_F,{\bf k}_F,{\bf Q})$$ and $d\Omega_{\bf k}$ is the differential solid angle subtended by $\hat{{\bf k}}$. Note that $\delta\sigma_1$ depends on the lifetime $\tau_0$ of Bloch states as well as on the transport relaxation time $\tau$. As mentioned above, the difference between $\tau$ and $\tau_0$ comes from the momentum dependence of $|\alpha {\bf k}\rangle$ states. At any rate, the full correction $\delta\sigma$ depends only on $\tau$ due to the additional contribution from $\delta\sigma_2$ (see Fig. \[fig:cofig\]c and Eq. (\[eq:nasty\])). Equation (\[eq:nasty\]) can be evaluated using the same procedure as for $\delta\sigma_1$. For instance, in Appendix \[sec:ds2\] we derive $$\label{eq:ds2} \delta\sigma_2\simeq\left\{\begin{array}{ccc} 0 & {\rm if } & (\epsilon_F-M)/M\ll 1\\ -(1/3)\delta\sigma_1 & {\rm if }& (\epsilon_F-M)/M\gg 1.\\ \end{array}\right.$$ The full form of the quantum correction, $\delta\sigma_1+\delta\sigma_2$, depends only on the transport mean free path $\tau$ and has (in appropriate limits) a universal magnitude, see Eqs. (\[eq:res\_bulk\]) and  (\[eq:magres\_bulk\]). Calculations ------------ The road map to $\delta\sigma$ starts from a calculation of $\hat{U}$ in Eq. (\[eq:U\]). In Appendix \[sec:u\] we derive $$\begin{aligned} \label{eq:coeffs} U^{m l}_{m' l'} &= a\,\delta_{m l}\delta_{m' l'}+ \sum_\mu b_\mu\,\Lambda^\mu_{m' l'}\delta_{m l}\nonumber\\ &+\sum_\mu c_\mu\,\Lambda^\mu_{m l}\delta_{m' l'}+\sum_{\mu,\nu} d_{\mu\nu}\,\Lambda^\mu_{m l}\Lambda^\nu_{m' l'},\end{aligned}$$ where $\mu,\nu\in\{1,2,3,4\}$, $\Lambda^i=\sigma^i\tau^x$ for $i\in\{1,2,3\}$ and $\Lambda^4={\bf 1}_{2}\,\tau^z$. In addition, $a$, $b_\mu$, $c_\nu$ and $d_{\mu\nu}$ are ${\bf Q}$-dependent coefficients whose explicit expressions are shown in Appendix \[sec:u\]. With those, $\hat{U}$ is fully determined. The next task is to get $C^{m n}_{m' n'}({\bf Q})$ from Eq. (\[eq:C\_mat2\]). While $({\bf 1}_{16}-\hat{U}({\bf Q}))$ can be inverted numerically, it is not possible to do so analytically for $Q\neq 0$. Since we are interested in analytical expressions, we follow an approximate three-step route. First, we diagonalize $({\bf 1}_{16}-\hat{U})$ for $Q=0$, analytically. All eigenvalues can be written in the form $\Delta_g\tau_0$, where $\Delta_g$ is the “intrinsic” Cooperon gap or mass. We find that one of the eigenvalues has $\Delta_g=0$ for any $\epsilon_F$ and $M$, which is a reflection of combined time-reversal symmetry and charge conservation. As we elaborate in the next subsection, there may be additional eigenvalues with $\Delta_g\simeq 0$ when $(\epsilon_F-M)/M\ll 1$ and $(\epsilon_F-M)/M\gg 1$. Hereafter we refer to eigenvectors of $\Delta_g\simeq 0$ eigenvalues as gapless (or massless, or “soft”) modes. Because $\Delta_g\simeq 0$ eigenvalues make $\hat{C}$ large, $\delta\sigma$ is determined mainly by soft modes. Second, we extrapolate the $Q=0$ case to $Q\neq 0$ perturbatively, with the objective of finding how the eigenvalues of the gapless modes depend on $Q$. To that end $\delta \hat{U} ({\bf Q})\equiv \hat{U}({\bf 0})-\hat{U}({\bf Q})$ is written in the basis that diagonalizes $\hat{U}({\bf 0})$. The shift of $Q=0$ eigenvalues under $\delta\hat{U}({\bf Q})$ is then evaluated through standard second order perturbation theory. The need to go to second order in $\delta \hat{U}$ originates from the fact that several matrix elements of $U^{m n}_{m' n'}({\bf Q})$ are linear in $Q$ (see Appendix \[sec:u\]). When $(\epsilon_F-M)/M\ll 1$ and $(\epsilon_F-M)/M\gg 1$, perturbation theory leads to eigenvalues $(D Q^2+\Delta_g)\tau_0$. The fact that $D$ contains the transport time $\tau$ rather than the scattering time $\tau_0$ is generally crucial in order to arrive at the correct result for $\delta\sigma$. Third, we invert the diagonalized matrix, and transform its outcome to the $|m,m'\rangle$ basis by using the $Q=0$ eigenvector matrix (the change of unperturbed eigenvectors under $\delta\hat{U}({\bf Q})$ is deemed unimportant.) This yields $C^{m n}_{m' n'}({\bf Q})$. Once we have $C^{m n}_{m' n'}({\bf Q})$, we use Eq. (\[eq:trans\]) in order to extract $C^{\alpha\beta}_{\beta\alpha}({\bf k},{\bf k}',{\bf Q})$. This is then plugged in Eqs. (\[eq:ds2bis\]) and  (\[eq:nasty\]). Results ------- The diagonalization of Eq. (\[eq:C\_mat2\]) at $Q=0$ shows one genuinely gapless Cooperon mode ($\Delta_g=0$, c.f. Sec. IIC), with a spin-singlet and orbital-triplet eigenvector: $$\label{eq:g1} \left[\frac{\epsilon_F+M}{2\sqrt{\epsilon_F^2+M^2}}|T T\rangle+ \frac{\epsilon_F-M}{2\sqrt{\epsilon_F^2+M^2}}|B B\rangle\right]\left(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle\right).$$ The fact that Eq. (\[eq:g1\]) remains gapless for any $\epsilon_F/M$ is a physical manifestation of charge conservation. This situation differs qualitatively from 2D TIs in HgTe quantum wells,[@tkachov2011] where a nonzero mass term gaps all Cooperons. The reason for the difference is that in 2D TIs the mass term acts somewhat like a Zeeman field in a 2D electron gas with Rashba spin-orbit interaction. Importantly, the diagonalization of Eq. (\[eq:C\_mat2\]) reveals two qualitatively distinct regimes of quantum interference, $(\epsilon_F-M)/M\ll 1$ and $(\epsilon_F-M)/M\gg 1$, which potentially host additional gapless Cooperon modes. As we discuss below, these additional gapless modes can change and even reverse the contribution to $\delta\sigma$ coming from Eq. (\[eq:g1\]). When $(\epsilon_F-M)/M\gg 1$, we identify a slightly gapped (soft) Cooperon mode with $$\label{eq:tau_s} \Delta_g=2 (M^2/\epsilon_F^2)\tau_0^{-1}\equiv\tau_v^{-1}\ll\tau_0^{-1},$$ whose eigenvector is a spin-singlet and an orbital-triplet: $$\label{eq:g2} \frac{1}{2}\left(|T B\rangle+|B T\rangle\right)\left(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle\right).$$ Physically, $\tau_v^{-1}$ is the rate of “intervalley” transitions ($|T\rangle+|B\rangle \to |T\rangle-|B\rangle$) induced by the “mass term” ($M\tau^z$) in Eq. (\[eq:model\_b\]). Because both Eq. (\[eq:g1\]) and Eq. (\[eq:g2\]) are spin-singlets, their contributions to $\delta\sigma$ are of WAL type (this is proven below). Incidentally, $M=0$ is the physically relevant regime for Weyl semimetals, which have two degenerate Dirac points with linear energy dispersion along the three momenta axes. Unlike in graphene,[@mccann2006] where there are $4$ gapless Cooperon modes (in absence of atomically sharp defects and hexagonal warping), in a Weyl semimetal we obtain only $2$ gapless Cooperon modes. This difference stems from the fact that the SU(2) “valley symmetry” of graphene[@mccann2006] gets reduced to a U(1) symmetry in Weyl semimetals, due to the band dispersion along $z$. Acting somewhat like a Zeeman field would in a free electron gas, the $k_z$ dispersion generates a mass for orbital-singlet modes, which is why the nearly-gapless Cooperons in Eq. (\[eq:g1\]) and  (\[eq:g2\]) are orbital-triplets. When $(\epsilon_F-M)/M\ll 1$, there are three soft modes with gap $$\label{eq:tau_v} \Delta_g=(2/9)(1-M/\epsilon_F)^2\tau_0^{-1}\equiv\tau_s^{-1}\ll\tau_0^{-1}.$$ Physically, $\tau_s^{-1}$ is the rate of spin-flip transitions induced by the “spin-orbit term” ($v {\bf k}\cdot{\boldsymbol\sigma} \tau^x$) in Eq. (\[eq:model\_b\]). The eigenvectors for the three slightly gapped modes are $$\begin{aligned} \label{eq:trip} &(\lambda_1|T T\rangle+\lambda_2|B B\rangle)|\downarrow\downarrow\rangle\nonumber\\ & (\lambda_1|T T\rangle+\lambda_2|B B\rangle)|\uparrow\uparrow\rangle\nonumber\\ & (\lambda_3|T T\rangle +\lambda_4|B B\rangle)\left(|\uparrow\downarrow\rangle+\downarrow\uparrow\rangle\right),\end{aligned}$$ where $\lambda_1,...,\lambda_4$ are coefficients that depend only on $\epsilon_F/M$, such that $\lambda_1\simeq\lambda_3\simeq 1+O[(\epsilon_F/M-1)^2]$ and $\lambda_2\simeq\lambda_4\simeq O[(\epsilon_F/M-1)]$. Therefore, the three soft modes in Eq. (\[eq:trip\]) are all spin and orbital triplets. As will be demonstrated momentarily, their contribution to $\delta\sigma$ is of WL type. Next we determine $\overline{C}$ (c.f. Eq. (\[eq:overline\_c\])) by diagonalizing Eq. (\[eq:C\_mat\]) at $Q\neq 0$ and doing the angular integration in Eq. (\[eq:overline\_c\]). For $(\epsilon_F-M)/M\ll 1$ we obtain $$\label{eq:cav1} \overline{C}\simeq \frac{\hbar}{6\pi\nu\tau^2}\left[-\frac{1}{D Q^2+\tau_\phi^{-1}}+\frac{3}{D Q^2+\tau_\phi^{-1}+\tau_s^{-1}}\right].$$ For $(\epsilon_F-M)/M\gg 1$, we instead get $$\label{eq:cav2} \overline{C}\simeq \frac{3 \hbar}{8\pi\nu\tau^2}\left[-\frac{1}{D Q^2+\tau_\phi^{-1}}-\frac{1}{D Q^2+\tau_\phi^{-1}+\tau_v^{-1}}\right].$$ In the derivation of Eqs. (\[eq:cav1\]) and  (\[eq:cav2\]) we have included the phase relaxation time $\tau_\phi$ and exploited $D Q^2\tau_0\ll 1$. The first term in the square brackets of Eqs. (\[eq:cav1\]) and  (\[eq:cav2\]) is large at $Q\to 0$ irrespective of $\epsilon_F/M$, and originates from the spin-singlet Cooperon mode in Eq. (\[eq:g1\]). Its negative sign means that it makes a contribution towards WAL. Equation (\[eq:cav1\]) displays a competition between WL and WAL, which is no different from that found in an ordinary metal with spin-orbit interactions. WL terms originate from the three spin triplet modes of Eq. (\[eq:trip\]). WL prevails if $\tau_\phi^{-1}\gg\tau_s^{-1}$, whereas WAL rules if $\tau_{\phi}^{-1}\ll\tau_s^{-1}$. Equation (\[eq:cav2\]) unveils two different regimes of WAL. On one hand, if $\tau_\phi^{-1}\gg\tau_v^{-1}$, the spin-singlet Cooperon mode of Eq. (\[eq:g2\]) makes a contribution to $\delta\sigma$ that equals that of Eq. (\[eq:g1\]). In this limit, quantum interference can be interpreted as coming from two identical and nearly-decoupled Dirac valleys. On the other hand, if $\tau_\phi^{-1}\ll\tau_v^{-1}$, the contribution from Eq. (\[eq:g2\]) becomes relatively unimportant and the magnitude of WAL is halved. In other words, when the intervalley transition rate induced by the mass term $M\tau^z$ is fast compared to the phase relaxation rate, the two valleys contribute as one. This is quite different from graphene, where strong intervalley scattering changes WAL into WL.[@mccann2006] The underlying reason for such a qualitative difference is that in graphene a gapless valley-singlet mode is responsible for producing WL, whereas in a Weyl semimetal the valley-singlet Cooperons are strongly gapped by the $k_z$ band dispersion. Substituting Eqs. (\[eq:cav1\]) and  (\[eq:cav2\]) in Eq. (\[eq:ds2\]) and doing the $Q$-integral, we arrive at $$\begin{aligned} \label{eq:res_bulk} &\delta G\simeq\alpha\, G_q\ln(\tau_\phi/\tau)\nonumber\\ &\alpha=\left\{\begin{array}{ccc} -1 & {\rm if } & \tau_\phi\ll\tau_s\\ 1/2 & {\rm if }& \tau_\phi\gg(\tau_v,\tau_s)\\ 1 & {\rm if } & \tau_\phi\ll\tau_v, \end{array}\right.\end{aligned}$$ where $\delta G\equiv W\delta\sigma$ is the quantum interference correction to [*conductance*]{} and $$G_q\equiv e^2/(2\pi^2 \hbar)$$ is a universal conductance unit. In the derivation of Eq. (\[eq:res\_bulk\]) we have used Eq. (\[eq:ds2\]). The reason why $\alpha=1/2$ when $\tau_\phi\gg(\tau_v,\tau_s)$ is that in such regime there is only one gapless Cooperon mode (hence $|\alpha|=1/2$), which is a spin-singlet (hence $\alpha=|\alpha|$). While Eq. (\[eq:res\_bulk\]) is valid in absence of external magnetic fields, the magnetoconductance $\Delta G(H)\equiv G(H)-G(0)\simeq\delta G(H)-\delta G(0)$ can be easily obtained from Eq. (\[eq:res\_bulk\]) for $H$ perpendicular to the TI thin film. The replacement of $\int d^2 Q$ by an appropriate sum over Landau levels[@hikami1980] results in $$\begin{aligned} \label{eq:magres_bulk} &\Delta G\simeq\alpha\, G_q f(H_\phi/H)\nonumber\\ &\alpha=\left\{\begin{array}{ccc} -1 & {\rm if } & \tau_H\ll\tau_s\\ 1/2 & {\rm if } & \tau_H\gg(\tau_v,\tau_s)\\ 1 & {\rm if } & \tau_H\ll\tau_v, \end{array}\right.\end{aligned}$$ where $f(z)\equiv\ln z-\psi(1/2+z)$ with asymptotes $f(z)\propto z^{-2}$ for $z\gg 1$ and $f(z)\propto\ln(1/z)$ for $z\ll 1$, $\psi$ is the digamma function, $$\tau_H^{-1}\equiv \tau_\phi^{-1}+2 e D H/\hbar\,\,\mbox{ and }\,\, H_\phi\equiv\hbar/(4 e D \tau_\phi).$$ Three conclusions of experimental relevance can be extracted from Eqs. (\[eq:res\_bulk\]) and  (\[eq:magres\_bulk\]), which apply when highest occupied electronic states are all located near the $\Gamma$ point. First, bulk TI bands can display $\alpha=-1$ (WL) as long as the chemical potential is sufficiently close to the bottom of the bulk conduction band. Second, bulk TI bands can produce $\alpha=1$ when $\epsilon_F/M$ is sufficiently large. Third, when $(\epsilon_F-M)/M$ is neither large nor small, $\alpha=1/2$ ensues; this is the conventional result expected for ordinary conducting thin films with strong spin-orbit coupling, and is the one that has been often presumed in experiments on TI films.[@chen2010; @checkelsky2011; @wang2011; @he2011; @chen2011; @steinberg2011] At $\tau_H\simeq\tau_s$ there is a crossover between $\alpha=-1$ and $\alpha=1/2$; likewise, at $\tau_H\simeq\tau_v$ there is a crossover between $\alpha=1/2$ and $\alpha=1$. The particular expressions for $\tau_s$ and $\tau_v$ in Eqs. (\[eq:tau\_s\]) and (\[eq:tau\_v\]) rely on our assumption of $V_{\rm dis}\propto {\bf 1}_4$. Spin-orbit coupled impurities and/or atomically sharp disorder potentials would induce additional spin- and valley-flip processes, whose rates $\tau_{sf}^{-1}$ and $\tau_{vf}^{-1}$ would have to be incorporated via $\tau_s^{-1}\to\tau_s^{-1}+\tau_{sf}^{-1}$ and $\tau_v^{-1}\to\tau_v^{-1}+\tau_{vf}^{-1}$. If $\tau_{vf}^{-1}$ and $\tau_{sf}^{-1}$ are strong enough and insensitive to the value of $\epsilon_F/M$, then the only surviving regime of interference corrections is the conventional $\alpha=1/2$. The conventional $\alpha=1/2$ can be found in a wide range of parameter space at low temperatures, whereas the unconventional $\alpha=-1$ and $\alpha=1$ emerge in the relatively narrow regimes $\tau\ll\tau_H\ll\tau_s$ and $\tau\ll\tau_H\ll\tau_v$ (respectively). How accessible are these unconventional regimes? Suppose $M\simeq 150 {\rm meV}$, $v\simeq 5\times 10^5 {\rm m/s}$ and a bulk carrier density of $n\simeq 3\times 10^{18} {\rm cm}^{-3}$. This situation corresponds to having a small bulk Fermi surface. Then, it follows that $\alpha\simeq -1$ for a fairly wide range of magnetic fields ($l_H/(12 l)\ll 1$, where $l_H\equiv(D \tau_H)^{1/2}$). The limit $\alpha\to 1$ is not accesible in this regime. Instead, $\alpha\simeq 1$ should be accessible in (i) Weyl semimetals or in TIs with very narrow bandgaps, (ii) in TIs with large bandgap but $M({\bf k}_F)\simeq 0$. For the latter case it must be kept in mind that in the absence of spherical symmetry $M({\bf k}_F)$ is not constant on the Fermi surface. Suppose $M\simeq 5 {\rm meV}$ and a bulk carrier density of $\simeq 2\times 10^{18} {\rm cm}^{-3}$. Then, $\alpha\simeq 1$ in the range of fields corresponding to $l_H/(10 l)\ll 1$. For typical thin films, the requirements for $\alpha=\pm 1$ are compatible with $k_F l\gg 1$. Materials like BiTl(S$_{1-\delta}$Se$_\delta$)$_2$, where controlled changes of $\delta$ can tune $M$ from 0 to large values,[@xu2011] appear to be good candidates to observe crossovers between different regimes of magnetoresistance in Eq. (\[eq:magres\_bulk\]). Our analysis has thus far neglected surface states of the TI, which can also contribute to the measured magnetoresistance. It can be argued that surface states are unimportant and Eq. (\[eq:magres\_bulk\]) suffices in trivial insulators described by Eq. (\[eq:model\_b\]), as well as in time-reversal-invariant Weyl semimetals and in TIs with very small bulk bandgaps ($\lesssim\hbar/\tau_0$). In contrast, when the surface states of the TI are robust, Eq. (\[eq:magres\_bulk\]) is incomplete and must be generalized. Such generalization is the subject for the rest of this paper. Quantum Corrections to Conductivity from Coupled Bulk and Surface States ======================================================================== In this section we consider the combined bulk-surface contribution to $\delta\sigma$ in 3D TIs with relatively large bandgaps. We concentrate on a particular setup that consists of a TI thin film gated on one surface. The gate voltage can produce a depletion layer that spatially separates bulk and surface carriers (Fig. \[fig:dep\]), and carriers tunnel back and forth across the depletion layer. We assume the bulk-surface tunneling rate to be much smaller than the elastic scattering time on either side of the depletion layer, so that electrons scatter many times within the bulk (surface) before tunneling to the surface (bulk). This assumption is experimentally realistic, and it simplifies the microscopic theory of this section considerably. Single isolated TI surface -------------------------- As a preliminary step, we recall the expression for $\delta\sigma$ on a single TI surface that is decoupled from the bulk. Taking $\epsilon_{Fs} \tau\gg 1$, where $\epsilon_{Fs}$ is the Fermi energy measured from the Dirac point of the surface states, one arrives[@tkachov2011; @lu2011a] at $$\label{eq:res_s} \Delta G/G_q=(1/2) f(H_\phi/H)$$ for any $\tau_H$. The prefactor $1/2$ is consistent with having a gapless spin-singlet Cooperon (the spin-triplet Cooperons have large gaps due to the strong spin-momentum coupling on the surface). Two coupled 2D layers without spin-orbit coupling ------------------------------------------------- As another preliminary step, here we compute $\delta\sigma$ for two ordinary metallic 2D layers separated by a tunnel barrier. In a double layer system, the current flowing in layer $i$ can be written as ${\bf j}_i=\sum_j \sigma_{i j} {\bf E}_j$, where ${\bf E}_j$ is the electric field in layer $j$. For concreteness we take ${\bf E}_1={\bf E}_2\equiv {\bf E}$, so that the measured current is ${\bf j}={\bf j}_1+{\bf j}_2=\sigma{\bf E}$ with $\sigma=\sum_{i j}\sigma_{i j}$. Consequently, the quantum corrections to conductivity are $\delta\sigma=\sum_{i j}\delta\sigma_{i j}$. The goal of this section is to compute $\delta\sigma$ from microscopic theory. ![Diagrammatic representation for $\delta\sigma_{i j}$, where $i$ and $j$ are layer indices. For 2D layers without spin-orbit coupling, the Cooperon matrix elements are fully characterized by layer indices. The velocity operator is diagonal in the layer index; therefore, the Cooperons $C^{1 1}_{2 2}$ and $C^{2 2}_{1 1}$ do not enter in the expression for $\delta\sigma_{i j}$.[]{data-label="fig:dsij"}](./diag6.eps) ![Typical microscopic process that gives rise to $\delta\sigma_{1 2}$. It can be neglected when the intralayer disorder potentials in the two layers are uncorrelated.[]{data-label="fig:ds12"}](./diag3.eps) The interference correction $\delta\sigma_{i j}$ has the diagrammatic representation shown in Fig. \[fig:dsij\]. Because the velocity operator is diagonal in the layer index, the only Cooperons that enter in the conductivity are $C^{i j}_{j i}$, with $i,j\in\{1,2\}$. In particular $\delta\sigma_{i i}$ involves intralayer Cooperons $C^{i i}_{i i}$, whereas $\delta\sigma_{1 2}$ and $\delta\sigma_{2 1}$ involve interlayer Cooperons $C^{1 2}_{2 1}$ and $C^{2 1}_{1 2}$ (Fig. \[fig:ds12\]). Assuming that disorder potentials in the two layers are uncorrelated, $C^{i j}_{j i}=0$ for $i\neq j$. This is a reasonable assumption when electrons in the two layers scatter off different sets of impurities. Hence, we are left with $\delta\sigma=\sum_i\delta\sigma_{i i}$. From here on we simplify the notation via $C^{i i}_{i i}\equiv C_i$. When evaluating $\delta\sigma_{i i}$ we will neglect spin-orbit interactions; however, the main lessons learned in this subsection will be transferrable to the spin-orbit coupled case studied in the next subsection. In absence of interlayer coupling, a standard calculation yields $$\label{eq:dnotu} \delta\sigma_{i i}^{(0)}\simeq -4 \frac{e^2}{\hbar^2} \nu_i D_i \tau_{d i}^2\int_{\bf Q} C_i^{(0)}({\bf Q}),$$ where $\int_{\bf Q}\equiv \int d^2 Q/(2\pi)^2$, an extra factor of $2$ is due to spin degeneracy, $\tau_{d i}$ is the scattering time in layer $i$ due to elastic impurities (we assume purely s-wave scattering, so that there is no difference between the transport scattering time and the quantum lifetime), $\nu_i$ is the density of states per unit area in layer $i$ and $$C_i^{(0)} ({\bf Q})=\frac{\hbar}{2\pi \nu_i \tau_{d i}^2} \frac{1}{D_i Q^2+\tau_{\phi i}^{-1}}$$ is the Cooperon for an isolated layer. In presence of interlayer tunneling, $C_i^{(0)}$ in Eq. (\[eq:dnotu\]) is replaced by $C_i$: $$\label{eq:dd} \delta\sigma_{i i}\simeq -4 \frac{e^2}{\hbar^2} \nu_i D_i \tau_{d i}^2\int_{\bf Q}C_i({\bf Q}),$$ in whose prefactor we have neglected terms containing the ratio between the tunneling rate and the elastic scattering rate. ![Single-particle Green’s functions. (a) Dressing of Bloch states due to intralayer impurity scattering. (b) Dressing of disorder-averaged Green’s functions due to interlayer tunneling. The tunneling amplitude is regarded as a random variable.[]{data-label="fig:gfig"}](./diag.eps) In order to compute $C_i$, we recognize that the influence of interlayer coupling occurs at two different levels. On one hand, it modifies the single-particle Green’s function for each layer (Fig. \[fig:gfig\]). Because the thickness of the depletion layer typically shows microscopic variations within the same film as well as from sample to sample, the interlayer tunneling amplitude can be regarded as a random variable. Consequently, the change in the ensemble-averaged Green’s function due to tunneling can be captured via $\tau_{d i}^{-1}\to \tau_{d i}^{-1}+\tau_{t i}^{-1}$, where $$\tau_{t i}^{-1}=(2\pi/\hbar) \langle|t|^2\rangle S\, \nu_j$$ is the tunneling rate from layer $i$ onto layer $j\neq i$, $\langle|t|^2\rangle$ is the averaged square of the tunneling matrix element and $S$ is the layer area. Note that $\langle|t|^2\rangle$ scales like $S^{-1}$, so that $\tau_{t i}^{-1}$ is independent of the layer area. ![(a) Cooperon $C_i^{(0)}$ without interlayer tunneling. (b) Partially dressed Cooperon $\tilde{C}_i^{(0)}$, where tunneling is included solely in the single-particle Green’s functions. $\tilde{C}_i^{(0)}$ can be directly obtained from $C_i^{(0)}$ via $\tau_{\phi i}\to\tilde{\tau}_{\phi i}$. (c) Fully dressed Cooperon $C_i$, where tunneling is incorporated both in the single-particle Green’s function and in the particle-particle correlations.[]{data-label="fig:cfig"}](./diag5.eps) ![Typical processes not included in Fig. \[fig:cfig\], as they are subdominant for $\tau_{t i}\gg\tau_{d i}$.[]{data-label="fig:nfig"}](./diag4b.eps) On the other hand, interlayer tunneling modifies particle-particle correlations that build up Cooperons. An approximate diagrammatic expression for these correlations is shown in Fig. \[fig:cfig\]. The equation of Fig. \[fig:cfig\]c can be solved in momentum space and it yields $$\label{eq:cii} C_i =\frac{\hbar}{2\pi\nu_i\tau_{d i}^2}\frac{D_j Q^2+\tilde{\tau}_{\phi j}^{-1}}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1})-\tau_{t 1}^{-1}\tau_{t 2}^{-1}}$$ for $j\neq i$. In the derivation of Eq. (\[eq:cii\]) we have introduced $$\label{eq:tau_tilde} \tilde{\tau}_{\phi i}^{-1}\equiv\tau_{\phi i}^{-1}+\tau_{t i}^{-1}$$ as an effective phase relaxation rate that incorporates tunneling, and have used $$\int_{\bf k} G_i^R ({\bf k}) G^A_i(-{\bf k}+{\bf Q})\simeq \frac{2\pi \nu_i\tau_{d i} }{\hbar}\left(1-\frac{\tau_{d i}}{\tilde{\tau}_{\phi i}}-D_i Q^2 \tau_{d i}\right).\nonumber$$ Microscopic processes depicted in Fig. \[fig:cfig\] leave out those in which two consecutive tunneling events occur without any intralayer scattering in between. Likewise, they ignore electron trajectories in which a tunneling event precedes any intralayer scattering (Fig. \[fig:nfig\]). These processes are relatively unimportant if $\tau_{t i}\gg \tau_{d i}$. Not surprisingly, Eq. (\[eq:cii\]) arises in the coupled equations for the classical diffusive conductivity as well (see Appendix \[sec:cond\]). It is convenient to rewrite $C_i$ in Eq. (\[eq:cii\]) as $$\label{eq:q1} C_i=\frac{\hbar}{2\pi\nu_i D_i \tau_{d i}^2}\left[\frac{A_i}{Q^2+q_a^2}+\frac{B_i}{Q^2+q_b^2}\right],$$ where $$\label{eq:ab1} 2 q_{a (b)}^2=\frac{1}{\tilde{l}_{\phi 1}^{2}}+\frac{1}{\tilde{l}_{\phi 2}^{2}}\pm \sqrt{\left(\frac{1}{\tilde{l}_{\phi 1}^{2}}-\frac{1}{\tilde{l}_{\phi 2}^{2}}\right)^2+ \frac{4}{l_{t 1}^2 l_{t 2}^2}}$$ and $$\label{eq:ab2} A_i = 1-B_i= (\tilde{l}_{\phi j}^{-2}-q_a^2)/(q_b^2-q_a^2)\,\,\mbox{ for $j\neq i$.}$$ In Eq. (\[eq:ab1\]) we have defined $\tilde{l}_{\phi i}\equiv(D_i\tilde{\tau}_{\phi i})^{1/2}$ as an effective coherence length and $l_{t i}\equiv(D_i\tau_{t i})^{1/2}$ as the interlayer leakage length. Besides, $q_a^2 (q_b^2)$ gets the positive (negative) sign in front of the square root. Combining Eq. (\[eq:dd\]) with Eq. (\[eq:q1\]) and using $A_1+A_2=B_1+B_2=1$, we get $$\delta\sigma=\sum_i\delta\sigma_{i i}=-2 \frac{e^2}{\pi\hbar} \int_{\bf Q} \left[\frac{1}{Q^2+q_a^2}+\frac{1}{Q^2+q_b^2}\right].$$ Therefore, the low-field magnetoconductance reads $$\label{eq:Dii} \Delta\sigma=\sum_i\Delta\sigma_{i i}=-G_q\left[ f\left(\frac{H_a}{H}\right)+ f\left(\frac{H_b}{H}\right) \right],$$ where $$H_{a (b)}\equiv \hbar\, q_{a (b)}^2/(4 e).$$ In the limit of very strong tunneling ($\tau_{t i}/\tau_{\phi i}\to 0$), Eq. (\[eq:Dii\]) becomes $\Delta\sigma\simeq -G_q f(H_b/H)$, as though there was a single layer. In the limit of very weak tunneling ($\tau_{t i}/\tau_{\phi i}\to\infty$), $\Delta\sigma$ is the sum of contributions from two independent films. It is helpful to understand the weak and strong coupling regimes in terms of measurable quantities like the interlayer conductance per square, $$g_t=(2\pi e^2/\hbar)\langle|t|^2\rangle S \nu_1\nu_2=\sigma_{D i}/l_{t i}^2,$$ where $\sigma_{D i}$ is the Drude conductivity in layer $i$. For simplicity suppose that $\tau_{\phi 1}\simeq\tau_{\phi 2}\equiv\tau_\phi$. In this case the crossover from weak to strong tunneling occurs when $$\label{eq:cross} \frac{1}{g_t l_\phi^2}\lesssim \frac{1}{\sigma_{D 1}}+\frac{1}{\sigma_{D 2}}\,\,\,\mbox{(crossover condition)},$$ namely when the tunneling resistance for a square of area $l_\phi^2$ becomes smaller than the sum of the classical intralayer resistivities. Let us define $$g_c^{-1}\equiv (\sigma_{D 1}^{-1}+\sigma_{D 2}^{-1}) l_\phi^2.$$ If $g_t\ll g_c$, then $\Delta\sigma/G_q\simeq -2\ln(H/H_\phi)$ for $H\gg H_\phi$. If $g_t\gg g_c$, then $\Delta\sigma/G_q\simeq -\ln(H/H_\phi)$ for $H_\phi\ll H\ll H_\phi (g_t/g_c)$. Thus changing the interlayer conductance results in a factor-of-two change for the magnitude of the WL correction. ![Example of an interlayer scattering process that is allowed in multilayer systems. Its analog in multivalley semiconductors of Ref. \[\] is forbidden.[]{data-label="fig:fuku"}](./diag7.eps) Limits reminiscent of the above were first discussed in inversion layers of multivalley semiconductors like Si,[@fukuyama1980] where the role of the layers is played by different electron pockets in the Brillouin zone. Similarities notwithstanding, there are clear differences between our microscopic theory and that of multivalley semiconductors. On one hand, the separation in momentum between valleys of Si prevents scattering processes such as the one in Fig. \[fig:fuku\]. These processes are not only allowed in our case, but also lead to the Cooperon dressing shown in Fig \[fig:cfig\]c. On the other hand, in our case the interlayer Cooperon vanishes due to uncorrelated disorder potentials in the two spatially separated layers. That is not the case in multivalley semiconductors, where both valleys scatter off the same set of real-space impurities and intervalley Cooperons contribute crucially to $\delta\sigma$. Finally, it should be mentioned that Eqs. (\[eq:ab1\]), (\[eq:ab2\]) and (\[eq:Dii\]) coincide with those derived by G. Bergmann,[@bergmann1989] who invoked macroscopic arguments based on the diffusion equation. The microscopic theory of this subsection supports Bergmann’s results, insofar as $\tau_{t i}\gg \tau_{d i}$ and the disorder potentials in the two layers are uncorrelated. Incidentally, yet another way to arrive at the same results is unveiled in Appendix \[sec:coupled\_cooper\]; this later method will prove convenient in the upcoming subsection. 3D TI film with bulk-surface coupling ------------------------------------- We now consider a 3D TI film (Fig. \[fig:dep\]) with a gate electrode placed near its top surface. At the moment we neglect the bottom surface of the TI, which will be incorporated below. For ease of notation we use subscript “1” to refer to “bulk”, and subscript “2” to refer to “top surface”. Like in the preceding subsection we assume bulk-surface disorder correlations to be negligible, so that the quantum corrections to conductance can be written as $\delta G=\delta G_{1 1}+\delta G_{2 2}=W \delta\sigma_{1 1}+\delta\sigma_{2 2}$. $\delta G$ is approximately independent of the film thickness $W$ as long as $W\ll\tilde{l}_{\phi 1}$ , where $\tilde{l}_{\phi 1}$ was defined below Eq. (\[eq:ab2\]). The goal of this subsection is to calculate $\delta G$ from microscopic theory. Unlike in the previous subsection, here both “layers” are spin-orbit coupled. We assume that tunneling events, albeit being time-reversal invariant, conserve neither spin nor orbital degrees of freedom. Indeed, in a TI spin is not conserved for non-momentum-conserving tunneling. Similarly, the orbital degree of freedom is not conserved due to broken inversion symmetry near the surface. Let us begin with no tunneling. On one hand, there are four surface Cooperon modes: one gapless spin-singlet mode and three spin-triplet modes with large ($\sim\tau_{d 2}^{-1}$) gaps. On the other hand, there are sixteen bulk Cooperons, of which a spin-singlet mode (Eq. (\[eq:g1\])) is always gapless. In addition, four of the bulk modes (the spin-singlet of Eq. (\[eq:g2\]) and three spin-triplets of Eq. (\[eq:trip\])) can be “soft” depending on $\epsilon_F/M$. The rest of the bulk Cooperon modes have large gaps of order $\tau_{d 1}^{-1}$. Let us now turn on tunneling. Since $\tau_{t i}\gg \tau_{d i}$, we can limit ourselves to analyzing the effects of tunneling within the low-energy subspace formed by the soft Cooperons. If there are no magnetic impurities in the depletion layer, the total spin of the Cooperon is a good quantum number even in presence of tunneling. Accordingly tunneling does not mix spin-singlet modes with spin-triplet modes, and the full (dressed) Cooperons can also be classified into spin-singlets and a spin-triplet. In the regimes $\tau_{\phi 1}\ll\tau_s$ and $\tau_{\phi 1}\gg (\tau_s,\tau_v)$, tunneling dresses one soft spin-singlet Cooperon in the bulk with another soft spin-singlet Cooperon on the surface. This dressing is completed as explained in Section IIIB: first by renormalizing the phase relaxation time $\tau_{\phi i}\to \tilde{\tau}_{\phi i}$, and afterwards proceeding with the series expansion of Fig. \[fig:cfig\]c. All “blocks” appearing in this series expansion are spin-singlets. When $\tau_{\phi 1}\ll\tau_s$, the soft spin-triplet Cooperons from the bulk are dressed simply through $\tau_{\phi 1}\to\tilde{\tau}_{\phi 1}$: they do not get appreciably admixed with the spin-triplet Cooperon on the surface because the latter has a large gap. In the regime $\tau_{\phi 1}\ll\tau_v$, there are two gapless singlet Cooperons in the bulk, each of which can hybridize with the singlet gapless Cooperon on the surface. For this situation, Fig. \[fig:cfig\]c does not capture all possible processes and the calculation from the previous subsection must be generalized; this generalization is carried out in Appendix \[sec:coupled\_cooper\]. With the above considerations in mind, we combine Eqs. (\[eq:magres\_bulk\]) and  (\[eq:res\_s\]) in order to obtain the total contribution to low-field magnetoconductance: $$\begin{aligned} \label{eq:res_tot} & \frac{\Delta G}{G_q}\simeq\frac{1}{2}\left\{\begin{array}{ccc} f\left(\frac{H_a}{H}\right)+f\left(\frac{H_b}{H}\right)-3 f\left(\frac{\tilde{H}_{\phi 1}}{H}\right) &{\rm if } & \tilde{\tau}_H\ll\tau_s\\ f\left(\frac{H_a}{H}\right)+f\left(\frac{H_b}{H}\right) &{\rm if } & \tilde{\tau}_H\gg(\tau_s,\tau_v)\\ f\left(\frac{H_c}{H}\right)+f\left(\frac{H_d}{H}\right)+f\left(\frac{\tilde{H}_{\phi 1}}{H}\right) &{\rm if } &\tilde{\tau}_H\ll\tau_v, \end{array}\right.\end{aligned}$$ where $H_l=\hbar\, q_l^2/(4 e)$ for $l\in\{a,b,c,d\}$, $$\tilde{H}_{\phi 1}\equiv \hbar/(4 e D_1 \tilde{\tau}_{\phi 1}),\,\,\,{\rm and}\,\,\, \tilde{\tau}_H^{-1}\equiv \tilde{\tau}_{\phi 1}^{-1}+ 2 e D_1 H/\hbar.$$ Note that the effective phase relaxation rate increases linearly with the bulk-to-surface tunneling rate (c.f. Eq. (\[eq:tau\_tilde\])). The characteristic momenta $q_{a (b)}$ have been introduced earlier in Eq. (\[eq:ab1\]). The additional momenta $q_{c(d)}$ are identical to $q_{a (b)}$, except for $\tau_{t 2}^{-1}\to 2\,\tau_{t 2}^{-1}$. The reason for this difference is that the surface Cooperon can decay into two gapless bulk Cooperons when $\tau_{\phi 1}\ll\tau_v$. The first line of Eq. (\[eq:res\_tot\]) displays a competition between WL and WAL, and suggests that it is possible to induce a WAL-to-WL transition with a varying gate voltage. In the weak tunneling regime WL prevails, whereas in the strong tunneling regime WAL takes over. Similarly, a gate voltage can induce transitions between three different WAL coefficients: $\alpha\in(1/2,1)$ in the second line, and $\alpha\in(1/2,3/2)$ in the third line. The second line of Eq. (\[eq:res\_tot\]) describes quantum corrections as if they originated from two independent thin films with mixed bulk-surface character; indeed, universal results expected for the simplectic symmetry class are recovered when the effective phase relaxation times become the longest timescales of the problem. Some simple limiting cases of Eq. (\[eq:res\_tot\]) are discussed in Appendix \[sec:special\]. Thus far we have considered the coupling between the bulk and [*one*]{} (the top) surface of the TI film. As a consequence, Eq. (\[eq:res\_tot\]) applies to a TI film only if the phase relaxation time of the bottom surface (adjacent to the substrate) is short compared to other phase relaxation and tunneling times in the problem. This condition is likely not met in some recent experiments,[@checkelsky2011; @chen2011] which report on independent contributions from both surfaces. Partly motivated by these experiments, we now generalize Eq. (\[eq:res\_tot\]) so as to capture two surfaces, each coupled to bulk states. We consider the scenario depicted in Fig. \[fig:dep\], where the bottom surface contains bulk carriers. Since there is no depletion layer near $z=W$, we assume that the bulk-surface tunneling rate therein is strong compared to the phase relaxation rate, yet weak compared to disorder scattering rate. Hence we describe the hybrid of bottom surface and bulk states via Eq. (\[eq:res\_tot2\]), and thereafter couple this hybrid to the top surface along the lines of Eq. (\[eq:res\_tot\]). The resulting expression for $\Delta G$ can be approximated as $$\label{eq:res_tot5} \frac{\Delta G}{G_q}\simeq\frac{1}{2} f\left(\frac{H'_a}{H}\right)+\frac{1}{2}f\left(\frac{H'_b}{H}\right),$$ where $H'_{a(b)}\equiv\hbar (q'_{a(b)})^2/(4 e)$. The characteristic momenta $q'_a$ and $q'_b$ obey Eq. (\[eq:ab1\]), where “1” labels the top surface and “2” labels a hybrid between the bottom surface and the bulk. Notably, Eq. (\[eq:res\_tot5\]) implies that WL is no longer possible once the bottom surface is strongly coupled to bulk states. Instead, conventional WAL ensues with $\alpha\in(1/2,1)$. This observation not only sheds light on why current experiments see no indication for WL, but it also gives insight as to how WL could be observed in TI films. A possible strategy is to degrade the surfaces, e.g. by depositing magnetic impurities on them, and decoupling them from the bulk by double-sided gating. One may expect WL even if only the top surface is decoupled, while the (degraded) bottom surface is in contact with the bulk. In this case, Eq. (\[eq:res\_tot\]) reduces to Eq. (\[eq:magres\_bulk\]) derived for the sole bulk conduction, with the replacement $\tau_{\phi 1}^{-1}\to\tau_{\phi 1}^{-1}+\tau_{t 3}^{-1}$, where $\tau_{t 3}$ is the tunneling rate of electrons from bulk to the bottom surface. If the film is thick enough, then $\tau_{t 3}^{-1}$ may become sufficiently small to provide some dynamic range for observing WL behavior. This same strategy can also facilitate the observation of WAL with $\alpha>1$. Estimates for the bulk-surface coupling --------------------------------------- ![Schematic energy band profile for a gated 3D TI thin film. $z=0$ corresponds to the top surface of the device, immediately under the gate. $z=W$ corresponds to the bottom (ungated) surface. The vertical (blue) solid lines at $z=0, W$ are surface states. The curved solid (red) line is the bulk conduction band, and the dot-dashed (brown) curve is the bulk valence band. The chemical potential is depicted by a horizontal dashed line. $z_d$ is the thickness of the depletion layer, where neither bulk nor surface carriers are present. $\epsilon_{F s}$ is the Fermi energy of the surface states measured from the Dirac point ($\epsilon_{Fs}<0$ in this figure). $\epsilon_F$ is the Fermi energy of the bulk states, measured with respect to the midgap point. []{data-label="fig:dep"}](./dep.eps) This subsection is devoted to an approximate electrostatic and quantum mechanical analysis of the depletion layer in a TI film, which will result in quantitative estimates for the bulk-surface coupling. For a TI with an $n$-doped bulk, a negative charge per unit area $(-Q_g)$ placed at the gate repels electrons from bulk bands at $z=0$ as well as from the surface states at $z=0$. This leaves a positive net charge on the top surface, which is equivalent to a downward shift in the local chemical potential at $z=0$: $\Delta\mu_s=\epsilon_F-\epsilon_{F s}$. Recall that $\epsilon_{Fs}$ is the Fermi energy of the surface states measured from the Dirac point (for simplicity the Dirac point is assumed to be in the middle of the bandgap at $z=0$) and $\epsilon_F$ is the Fermi energy of the bulk states measured from the middle of the bandgap. Since the chemical potential deep inside the bulk must be unaffected by the gate, $\Delta\mu_s\neq 0$ implies a band bending of magnitude $\phi_s=\Delta\mu_s$ near the gated surface (Fig. \[fig:dep\]). When $\Delta\mu_s>(\epsilon_F-M)$ there are no bulk carriers left at $z=0$ and a depletion layer appears at $z\in(0,z_d)$, where $z_d$ is determined below. For each value of $Q_g$, $\Delta\mu_s$ (or equivalently $\epsilon_{F s})$ can be uniquely determined from the overall neutrality condition $Q_s+Q_d=Q_g$, where $Q_s$ is the positive net charge induced on the surface, and $Q_d$ is the positive net charge in the depletion layer. In the depletion approximation[@sze2002] one has $Q_d\simeq n z_d$, where $n$ (c.f. Eq. (\[eq:n\])) is equal to the density of charged donors in the depleted region. The electrostatic energy profile in the depleted region then obeys $$\label{eq:fi} \phi(z)=\phi_b-\frac{1}{2}\frac{e^2 n}{\kappa}(z-z_d)^2,$$ where $\phi_b\equiv\phi_s-(\epsilon_F-M)=M-\epsilon_{F s}$, $\kappa$ is the static dielectric constant and $$\label{eq:depl} z_d=\sqrt{\frac{2\kappa \phi_b}{e^2 n}}.$$ In the derivation of Eq. (\[eq:fi\]) we have assumed that the electric field vanishes at $z=z_d$, which is accurate within a screening radius. As the gate voltage is made more negative, the maximum width of the depletion layer ($z_d^{\rm max}$) is achieved when $\phi_b\simeq 2 M$. For $\phi_b>2 M$, the bulk bands get inverted at $z=0$ and $z_d$ saturates. We estimate $z_d^{\rm max}\simeq 20 {\rm nm}$ for some typical parameter values ($M=150 {\rm meV}$, $n\simeq 4\times 10^{18} {\rm cm}^{-3}$, $\kappa=50$). Once the electrostatic profile of the TI film is characterized, we can analyze the quantum mechanical tunneling of electrons across the depletion layer. The tunneling conductance per unit area is roughly $$\label{eq:kappa} g_t\sim (e^2/h) \lambda_F^{-2} \exp(-2\chi),$$ where $\lambda_F$ is the smallest between bulk and surface Fermi wavelengths, and $$\label{eq:kappa2} \chi\simeq \int_0^{z_d} dz\frac{\phi_b-\phi(z)}{\hbar v}\simeq\frac{1}{6}\frac{e^2 n z_d^3}{\kappa\,\hbar v}.$$ In Eq. (\[eq:kappa2\]) we have ignored effective mass and Fermi velocity mismatches across the depletion layer. The WKB exponent $\chi$ can be tuned by a gate voltage: as $z_d$ varies from $0$ to $z_d^{\rm max}$, $\chi$ goes from $0$ to $\simeq 6$. Drawing from the previous subsection (c.f. Eq. (\[eq:cross\])), the crossover from weak to strong bulk-surface coupling occurs when $$\label{eq:cross1} \frac{1}{g_t l_\phi^2}\lesssim\frac{1}{\sigma_{D 1} W}+\frac{1}{\sigma_{D 2}}\simeq \frac{1}{\sigma_{D 2}},$$ where in the second equality we have assumed that $\sigma_{D 1} W\gg\sigma_{D 2}$. This is a good assumption provided that (i) the bulk mean free path is of the same order as the surface mean free path, and (ii) $k_F W\gg 1$. Plugging Eq. (\[eq:kappa\]) in Eq. (\[eq:cross1\]), the latter becomes $$\label{eq:cross2} \frac{l_\phi}{\lambda_F}\gtrsim (k_{Fs} l_2)^{1/2} \exp(\chi),$$ where $k_{Fs}=|\epsilon_{F s}|/{\hbar v}$ is the Fermi wave vector for the surface states and we have used $\sigma_{D 2}\sim (e^2/h) k_{F s} l_2$. When $z_d=z_d^{\rm max}$, the right hand side of Eq. (\[eq:cross2\]) reaches $\simeq 1000$, which exceeds the typical $l_\phi/\lambda_F$ in TI thin films by at least an order of magnitude. Therefore, when the depletion layer has its maximum width, the top surface and the bulk of the TI film can be regarded as weakly coupled. This state of affairs changes rapidly when the depletion layer is made thinner by a gate voltage. For instance, when $z_d=z_d^{\rm max}/\sqrt2$, the right hand side of Eq. (\[eq:cross2\]) equals $\simeq 30$, which is comparable to the typical $l_\phi/\lambda_F$. Further slight reductions in $z_d$ can subsequently drive the film into a regime of strong bulk-surface coupling. These estimates justify the interpretation of experimental data given in e.g. Ref. \[\]. Summary and conclusions ======================= We have completed a theoretical study of low-field magnetoresistance in electrostatically gated 3D TI films. The concise analytical expressions presented here \[Eqs. (\[eq:magres\_bulk\]), (\[eq:res\_tot\]) and (\[eq:res\_tot5\])\] may shed light on the quantum magnetoresistance of TIs, Weyl semimetals, as well as some topologically trivial materials. Only magnetic fields that are perpendicular to the TI thin film have been considered in this work; for in-plane fields and small bulk bandgaps, quantum interference contributions might be masked by classical magnetoresistance anomalies.[@son2012] A number of predictions from this work have not been articulated in previous studies and await experimental confirmation. For instance, we find that TI thin films with low bulk doping may exhibit weak localization (WL) or negative magnetoresistance, instead of the often presumed weak antilocalization (WAL) or positive magnetoresistance. Admittedly, the parameter space for WL is relatively narrow, and vanishes when either surface of the TI film is strongly coupled to bulk states. However, WL may be experimentally accessible in thicker films, or in thin films where the surfaces have short phase relaxation times. Under these conditions, a gate can induce a crossover between WL and WAL. On a separate note, we find that the “universal” prefactor for WAL varies depending on the bandgap of the TI, on the bulk doping concentration, on the phase relaxation times, and on the applied gate voltage. The results from this work are applicable to conducting yet lighly doped TIs, with thicknesses ranging between the bulk transport mean free path and the bulk phase relaxation length. It may be useful to find out how the results derived here change in highly doped TIs containing additional electrons pockets away from the $\Gamma$ point. Likewise, it may be helpful to extend our results to thinner films. Other potentially interesting tasks involve investigating universal conductance fluctuations and determining the influence of electron-electron interactions in the magnetoresistance of doped TI films. This research has been financially supported by a fellowship from Yale University (I.G.), and by NSF DMR Grant No. 0906498 (L.G.). L.G. thanks Pablo Jarillo-Herrero for a discussion that initiated the present work, I.G. thanks Ewelina Hankiewicz for an informative conversation, and both authors thank Aharon Kapitulnik for bringing Ref. \[\] to their attention. Renormalized velocity operator {#sec:ren} ============================== The velocity operators appearing in the expressions for $\sigma_D$ and $\delta\sigma$ (c.f. Sec. IIB) must be renormalized with ladder diagrams containing impurity scattering. The Dyson equation for the renormalized velocity operator is (Fig. \[fig:vertex\]) $$\label{eq:vd} \tilde{{\bf v}}_{\alpha\beta}({\bf k})= {\bf v}_{\alpha\beta}({\bf k})+u_0\sum_{\alpha,\beta\in\{1,2\}}\int_{{\bf k}'}\langle\alpha {\bf k}|\alpha' {\bf k}'\rangle\langle\beta'{\bf k}'|\beta{\bf k}\rangle G^A({\bf k}') G^R({\bf k}') \tilde{{\bf v}}_{\alpha'\beta'}({\bf k}'),$$ where ${\bf v}_{\alpha\beta}({\bf k})=\delta_{\alpha\beta} \hbar v^2 {\bf k}/E_k$ is a matrix element for the bare velocity operator. We solve Eq. (\[eq:vd\]) by guessing a solution of the form $$\label{eq:guess} \tilde{{\bf v}}_{\alpha\beta}({\bf k})=\gamma_k {\bf k}\delta_{\alpha\beta},$$ where $\gamma_k$ is a scalar that depends on $|{\bf k}|$ but not $\hat{\bf k}$. Although it is [*a priori*]{} not obvious that the renormalized velocity operator should be diagonal in the band indices, substituting Eq. (\[eq:guess\]) in Eq. (\[eq:vd\]) and using Eq. (\[eq:eigenstates\]) we find that $\tilde{{\bf v}}_{\alpha\beta}({\bf k})\propto\delta_{\alpha\beta}$ is indeed appropriate provided that $$\gamma_k=\frac{\hbar v^2}{E_k}\frac{\tau}{\tau_0}.$$ Here $$\frac{\hbar}{\tau}=2\pi\nu u_0\int\frac{d\Omega_{{\bf k}'}}{4\pi}\sum_{\alpha'}|\langle\alpha {\bf k}_F|\alpha' {\bf k}'_F\rangle|^2 (1-\hat{\bf k}_F\cdot\hat{\bf k}'_F)$$ is the transport scattering time. Therefore, the final result for the renormalized velocity is $\tilde{{\bf v}}_{\alpha\beta}({\bf k})={\bf v}_{\alpha\beta}({\bf k}) (\tau/\tau_0)$. ![Impurity vertex corrections for the velocity operator []{data-label="fig:vertex"}](./vertex.eps) Evaluation of $\delta\sigma_2$ in some simple cases {#sec:ds2} =================================================== The expression for $\delta\sigma_2$ (depicted in Fig. \[fig:cofig\]c) reads $$\label{eq:nasty} \delta\sigma_2\simeq -2\frac{e^2\hbar}{2\pi}\int_{{\bf k}, {\bf k}'} \tilde{v}^x({\bf k}) \tilde{v}^x({\bf k}') G^A({\bf k}) G^A({\bf k}') G^A(-{\bf k}) G^A(-{\bf k}') G^R(-{\bf k}') G^R({\bf k})\sum_{\alpha\beta\alpha'\beta'}\Gamma^{\alpha\alpha'}_{\beta'\beta}({\bf k},-{\bf k}',0) \frac{1}{W}\int\frac{d^2 Q}{(2\pi)^2} C^{\beta\alpha}_{\alpha'\beta'}({\bf k},{\bf k}',{\bf Q}),$$ where the overall factor of two stems from the fact that the two diagrams in Fig. \[fig:cofig\]c give identical contribution, and the band indices $\alpha,\beta$ etc. are summed over $1,2$. For generic $(\epsilon_F-M)/M$, the calculation of $\delta\sigma_2$ is cumbersome. Here we focus on two simple limits that are of interest: $(\epsilon_F-M)/M\ll 1$ and $(\epsilon_F-M)/M\gg 1$. When $(\epsilon_F-M)/M\ll 1$, the momentum dependence of $|\alpha {\bf k}_F\rangle$ is negligible. Consequently, $C^{\beta\alpha}_{\alpha'\beta'}({\bf k}_F,{\bf k}_F',{\bf Q})$ and $\Gamma^{\alpha\alpha'}_{\beta'\beta}({\bf k}_F,-{\bf k}_F',0)$ become independent of ${\bf k}_F$ and ${\bf k}_F'$. Since the matrix elements of the velocity operator are odd under ${\bf k}\to -{\bf k}$ and ${\bf k}'\to -{\bf k}'$, it is clear that $$\delta\sigma_2\simeq 0.$$ The limit of $(\epsilon_F-M)/M\gg 1$ is less trivial. In this regime the Hamiltonian is approximately block diagonal both in absence and in presence of disorder, because the disorder potential we take is spin- and orbital-indpendent. Therefore we may focus on a $2\times 2$ Hamiltonian (describing a Weyl node of positive chirality), $$\label{eq:h_simple} h'({\bf k})=\hbar v {\bf k}\cdot{\boldsymbol \sigma}+V_0({\bf r}){\bf 1}_{2\times 2},$$ where ${\bf k}=k(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. The result for $\delta\sigma$ obtained from such Hamiltonian needs to be multiplied by two at the end, as each block makes an equal contribution. The eigenstates for $h'({\bf k})$ are $|+,{\bf k}\rangle=(\cos(\theta/2),\exp(i\phi)\sin(\theta/2))^T$ and $|-,{\bf k}\rangle=(\sin(\theta/2),-\exp(i\phi)\cos(\theta/2))^T$. One significant simplification from Eq. (\[eq:h\_simple\]) is that there is only one band at the Fermi energy. This allows us to rewrite Eq. (\[eq:nasty\]) as $$\begin{aligned} \label{eq:less_ugly} \delta\sigma_2=& -4\frac{e^2\hbar^3}{2\pi} u_0\frac{\tau^2}{\tau_0^2}\left[\int\frac{dk k^2}{(2\pi)^2}\frac{k v^2}{E_k} (G^A)^2 G^R\right]^2\frac{1}{W}\int\frac{d^2 Q}{(2\pi)^2}\nonumber\\ &\times\int\frac{d\Omega_{{\bf k}}}{4\pi}\int\frac{d\Omega_{{\bf k}'}}{4\pi} \sin\theta\cos\phi\sin\theta'\cos\phi'\langle +,{\bf k}_F|+,{\bf k}_F'\rangle\langle +,-{\bf k}_F|+,-{\bf k}_F'\rangle C^{+ +}_{+ +}({\bf k}_F,{\bf k}_F',{\bf Q}),\end{aligned}$$ where the aforementioned extra factor of two has been accounted for. It is illustrative to compare Eq. (\[eq:less\_ugly\]) with its counterpart in $\delta\sigma_1$: $$\label{eq:pretty} \delta\sigma_1=-2\frac{e^2 \hbar^3}{2\pi}\frac{\tau^2}{\tau_0^2}\int\frac{dk k^2}{(2\pi)^2}\frac{k^2 v^4}{E_k^2} (G^R)^2 (G^A)^2 \frac{1}{W}\int\frac{d^2 Q}{(2\pi)^2} \sin^2\theta\cos^2\phi\, C^{+ +}_{+ +}({\bf k}_F,{\bf k}_F,{\bf Q}).$$ In Section II we detailed the steps to follow for the evaluation of Eq. (\[eq:pretty\]). Applying those same steps to Eq. (\[eq:less\_ugly\]) and using $$\int\frac{dk k^2}{2\pi^2} \frac{k^2}{E_k^2} (G^R)^2 (G^A)^2 \simeq \frac{4\pi\nu\tau_0^3}{\hbar^5 v^2}\,\,\,\mbox{ and }\,\,\,\left[\int\frac{dk k^2}{(2\pi)^2}\frac{k}{E_k} (G^A)^2 G^R\right]^2 u_0 \simeq -\frac{4\pi \nu \tau_0^3}{\hbar^5 v^2},$$ we arrive at $$\delta\sigma_2=-\frac{1}{3}\delta\sigma_1=-\frac{1}{3} G_q \ln\left(\frac{\tau_\phi}{\tau}\right).$$ Evaluation of matrix elements for $\hat{U}$ {#sec:u} =========================================== In this Appendix we calculate the coefficients entering in Eq. (\[eq:coeffs\]). These coefficients generally depend on the frequency $\Omega$ and wave vector ${\bf Q}$ of the external perturbation. Even though only $\Omega=0$ is needed for our evaluation of $\delta\sigma$, for completeness here we allow for $\Omega\neq 0$ as well. The calculation is facilitated by rewriting Eq. (\[eq:model\_b\]) as $$h({\bf k})=\sum_\mu\eta_\mu({\bf k}) \Lambda^\mu,$$ where $\mu\in\{1,2,3,4\}$, $\eta_i({\bf k})= \hbar v k_i$ and $\Lambda^i=\sigma^i\tau^x$ for $i\in\{1,2,3\}$, $\eta_4({\bf k})=M$ and $\Lambda^4={\bf 1}_{2}\,\tau^z$. Then, the finite-frequency retarded and advanced Green’s functions read $$\label{eq:gf} G_{m n}^{R(A)}({\bf k},\Omega)=\frac{\epsilon^{R(A)}\delta_{m n}^0+\sum_\mu\eta_\mu \Lambda_{m n}^\mu}{[\epsilon^{R(A)}]^2-E_k^2},$$ where $\epsilon^R\equiv\epsilon_F+i\gamma$ and $\epsilon^A\equiv\epsilon_F+\hbar\Omega-i\gamma$, with $\gamma\equiv\hbar/(2\tau_0)$ (c.f. Eq. (\[eq:lifetime\])). Substituting Eq. (\[eq:gf\]) in Eq. (\[eq:U\]), we get $$U^{m l}_{m' l'} = a\,\delta_{m l}\delta_{m' l'}+ \sum_\mu b_\mu\,\Lambda^\mu_{m' l'}\delta_{m l}+\sum_\mu c_\mu\,\Lambda^\mu_{m l}\delta_{m' l'}+\sum_{\mu\nu} d_{\mu\nu}\,\Lambda^\mu_{m l}\Lambda^\nu_{m' l'},$$ where $$\begin{aligned} \label{eq:ints} a&= u_0\int \frac{d^3 k}{(2\pi)^3}\frac{\epsilon^R(\epsilon^A+\hbar \Omega)}{[(\epsilon^R)^2-E_{-{\bf k}}^2][(\epsilon^A+\hbar \Omega)^2-E_{{\bf k}+{\bf Q}}^2]} \,\,\,\mbox{ ; }\,\,\,b_\mu=u_0\int \frac{d^3 k}{(2\pi)^3}\frac{\epsilon^R d_\mu({\bf k}+{\bf Q})}{[(\epsilon^R)^2-E_{-{\bf k}}^2][(\epsilon^A+\hbar \Omega)^2-E_{{\bf k}+{\bf Q}}^2]}\nonumber\\ c_\mu&=u_0\int \frac{d^3 k}{(2\pi)^3}\frac{(\epsilon^A+\hbar \Omega) d_\mu(-{\bf k})}{[(\epsilon^R)^2-E_{-{\bf k}}^2][(\epsilon^A+\hbar \Omega)^2-E_{{\bf k}+{\bf Q}}^2]}\,\,\,\mbox{ ; }\,\,\,d_{\mu\nu}=u_0\int \frac{d^3 k}{(2\pi)^3}\frac{d_\mu(-{\bf k}) d_\nu({\bf k}+{\bf Q})}{[(\epsilon^R)^2-E_{-{\bf k}}^2][(\epsilon^A+\hbar \Omega)^2-E_{{\bf k}+{\bf Q}}^2]},\nonumber\\\end{aligned}$$ and $\mu,\nu\in\{1,2,3,4\}$. In the diffusive transport regime, namely $(\epsilon_F-M)\gg\gamma\gg (\hbar v Q,\hbar \Omega)$, the integrals in Eq. (\[eq:ints\]) can be analytically performed and the outcome is $$\begin{aligned} \label{eq:coeffs2} a&\simeq a^{(0)}\left[1-\frac{1}{12}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar^2 v^2 Q^2}{\gamma^2}-\frac{i\hbar \Omega}{2\gamma}\right]\nonumber\\ b_1 &=-c_1\simeq \frac{i}{6} a^{(0)}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar v Q_x}{\gamma}\,\,\,\mbox{ ; }\,\,\,b_2=-c_2\simeq \frac{i}{6} a^{(0)}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar v Q_y}{\gamma}\,\,\,\mbox{ ; }\,\,\,b_4=c_4=\frac{M}{\epsilon_F} a\nonumber\\ d_{1 1}&\simeq -\frac{1}{3}\left(1-\frac{M^2}{\epsilon_F^2}\right) a^{(0)}\left[1-\frac{1}{20}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{(3 Q_x^2+Q_y^2) \hbar^2 v^2}{\gamma^2}-\frac{i\hbar \Omega}{2\gamma}\right]\nonumber\\ d_{2 2}&\simeq -\frac{1}{3}\left(1-\frac{M^2}{\epsilon_F^2}\right) a^{(0)}\left[1-\frac{1}{20}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{(3 Q_y^2+Q_x^2) \hbar^2 v^2}{\gamma^2}-\frac{i\hbar \Omega}{2\gamma}\right]\nonumber\\ d_{3 3}&\simeq -\frac{1}{3}\left(1-\frac{M^2}{\epsilon_F^2}\right) a^{(0)}\left[1-\frac{1}{20}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar^2 v^2 Q^2}{\gamma^2}-\frac{i\hbar \Omega}{2\gamma}\right]\nonumber\\ d_{4 4}&\simeq\frac{M^2}{\epsilon_F^2} a\nonumber\\ d_{1 2} &=d_{2 1}\simeq a^{(0)}\frac{1}{30}\left(1-\frac{M^2}{\epsilon_F^2}\right)^2\frac{\hbar^2 v^2 Q_x Q_y}{\gamma^2}\nonumber\\ d_{1 4}&=-d_{4 1}\simeq -i a^{(0)}\frac{M}{\epsilon_F}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar v Q_x}{\gamma} \,\,\,\mbox{ ; }\,\,\,d_{2 4}=-d_{4 2}\simeq -i a^{(0)}\frac{M}{\epsilon_F}\left(1-\frac{M^2}{\epsilon_F^2}\right)\frac{\hbar v Q_y}{\gamma},\end{aligned}$$ where $a^{(0)}\equiv[2 (1+M^2/\epsilon_F^2)]^{-1}$, and the elements omitted above are zero. It is worth noting that Eq. (\[eq:coeffs2\]) can be used to investigate the dynamical spin-charge coupling in doped TIs. Since this task is not directly related to the theme of this paper, it will be pursued elsewhere. Classical conductivity of two coupled layers {#sec:cond} ============================================ In this Appendix we analyze the classical conductivity of two coupled layers. The current in layer $i$ is given by ${\bf j}_i=\sum_j \sigma_{i j} {\bf E}_j$. It is illustrative to write $\sigma_{i j}$ in terms of the diffusive density-density response, using the continuity equation $$\frac{\partial\rho_i}{\partial t}+\nabla\cdot{\bf j}+\lambda\sum_j (\rho_j-\rho_i)=0$$ along with the constitutive equation ${\bf j}_i=-D_i {\boldsymbol\nabla}\rho_i-e^2\nu_i D_i {\bf E}_i$. $\lambda$ is the interlayer tunneling rate. Thus it follows that $$\sigma_{i j}({\bf q},\omega)=-\frac{i\omega}{q^2} \chi_{i j}+\frac{\lambda}{q^2}\sum_k (\chi_{i j}-\chi_{k j}),$$ where $\chi_{i j} ({\bf q},\omega)=e^2 \nu_j D_j q^2 p_{i j}({\bf q},\omega)$ is the density-density response function and $$p_{i j}({\bf q},\omega) = \left\{\begin{array}{ccc} \tilde{p}_i^{(0)}/(1-\lambda^2 p_1^{(0)} p_2^{(0)}) & {\rm if } & i=j\\ \lambda \tilde{p}_1^{(0)} \tilde{p}_2^{(0)}/(1-\lambda^2 p_1^{(0)} p_2^{(0)}) & {\rm if } & i\neq j\end{array}\right.,$$ with $\tilde{p}_i^{(0)} \equiv(D_i q^2-i\omega+\lambda)^{-1}$. The dressed diffusion probability $p_{i i}$, derived here from the continuity equation, has identical form as Eq. (\[eq:cii\]), which was derived microscopically in Section IIIB. Here $\omega$ and ${\bf q}$ are the frequency and momentum associated with the applied electric field. A straightforward calculation shows that $\sigma_{1 2}=\sigma_{2 1}=0$ when ${\bf E}_i$ is spatially uniform (${\bf q}=0$). Equations for coupled Cooperons {#sec:coupled_cooper} =============================== In the first part of this Appendix we present an alternative derivation for the results of Section IIIB. In the second part of the Appendix we generalize the derivation to make it suitable for TI thin films with $\tau_{\phi 1}\ll\tau_v$, which contain two gapless singlet Cooperons in the bulk and one gapless singlet Cooperon on the surface. The outcome of such generalization is the third line of Eq. (\[eq:res\_tot\]). Two 2D layers without spin-orbit coupling ----------------------------------------- In this subsection we use “1” and “2” to label the two layers. The relevant Cooperon modes are then $C_{1 1}$, $C_{1 2}$, $C_{2 1}$ and $C_{2 2}$. Recognizing that Cooperons must obey a diffusion equation in absence of phase relaxation, we posit the following coupled equations: $$\begin{aligned} \label{eq:1} (D_1 Q^2 +\tau_{\phi 1}^{-1}) C_{1 1}+\lambda (C_{1 1}-C_{2 1}) &=\hbar/(2\pi\nu_1\tau_{d 1}^2)\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 1}+\lambda (C_{2 1}-C_{1 1}) &= 0\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 2}+\lambda(C_{2 2}-C_{1 2}) &=\hbar/(2\pi\nu_2\tau_{d 2}^2)\nonumber\\ (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{1 2}+\lambda (C_{1 2}-C_{2 2})&=0,\end{aligned}$$ where $\lambda$ is the interlayer tunneling rate. Note that the source term appears only for the diagonal terms of the $2\times 2$ Cooperon matrix. The solution of Eq. (\[eq:1\]) reads $$\begin{aligned} C_{1 1}&=\frac{\hbar}{2\pi\nu_1\tau_1^2}\frac{D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1})-\lambda^2}\nonumber\\ C_{2 2}&=\frac{\hbar}{2\pi\nu_2\tau_2^2}\frac{D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1}}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1})-\lambda^2}\nonumber\\ C_{1 2}&=C_{2 1}=\frac{\lambda}{D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}} C_{1 1},\end{aligned}$$ where $\tilde{\tau}_{\phi i}^{-1}\equiv \tau_{\phi i}^{-1}+\lambda$. The expressions for $C_{1 1}$ and $C_{2 2}$ agree with Eq. (\[eq:cii\]). In addition, $C_{1 2}$ and $C_{2 1}$ agree with the expressions for $p_{1 2}$ and $p_{2 1}$ derived in Appendix \[sec:cond\] (where we discussed the classical diffusive conductivity). $C_{i i}$ of Eq. (\[eq:1\]) is equivalent to $C^{i i}_{i i}$ of Fig. \[fig:dsij\]. Likewise, $C_{1 2}$ and $C_{2 1}$ of Eq. (\[eq:1\]) correspond to $C^{1 1}_{2 2}$ and $C^{2 2}_{1 1}$ of Fig. \[fig:dsij\]. Although $C_{1 2}$ and $C_{2 1}$ are nonzero, they do not contribute to $\delta\sigma$ because the velocity operator is diagonal in the layer index. Therefore, we reproduce the expression of Section IIIB for $\delta\sigma$. TI film with two gapless bulk Cooperons and one gapless surface Cooperon ------------------------------------------------------------------------ In this subsection we use “1” and “3” to label the two bulk Cooperons, and “2” to label the surface Cooperon. The generalization of Eq. (\[eq:1\]) is $$\begin{aligned} \label{eq:2} (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{1 1}+\lambda(C_{1 1}-C_{2 1})&=\hbar/(2\pi\nu_1\tau_{d 1}^2)\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 1}+\lambda(2 C_{2 1}-C_{1 1}-C_{3 1})&=0\nonumber\\ (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{3 1}+\lambda(C_{3 1}-C_{2 1})&=0,\end{aligned}$$ $$\begin{aligned} \label{eq:3} (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{1 2}+\lambda (C_{1 2}-C_{2 2}) &= 0\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 2}+\lambda( 2 C_{2 2}-C_{1 2}-C_{3 2})&=\hbar/(2\pi\nu_2\tau_{d 2}^2)\nonumber\\ (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{3 2}+\lambda(C_{3 2}-C_{2 2})&=0\end{aligned}$$ and $$\begin{aligned} \label{eq:4} (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{1 3}+\lambda(C_{1 3}-C_{2 3})&=0\nonumber\\ (D_2 Q^2+\tau_{\phi 2}^{-1}) C_{2 3}+\lambda(2 C_{2 3}-C_{1 3}-C_{3 3})&=0\nonumber\\ (D_1 Q^2+\tau_{\phi 1}^{-1}) C_{3 3}+\lambda(C_{3 3}-C_{2 3})&=\hbar/(2\pi\nu_1\tau_{d 1}^2),\end{aligned}$$ Once again in Eqs. (\[eq:2\])-(\[eq:4\]) the source term appears for the diagonal components of the $3\times 3$ Cooperon matrix. In addition, a factor of $2$ has been multiplied in front of some tunneling rates associated to surface Cooperons. The rationale behind this is that the Cooperon on the surface can decay into two bulk modes, i.e. the effective decay rate becomes $\tau_{\phi 2}^{-1}+2\lambda$. Aside from this, we have assumed a unique tunneling rate $\lambda$ between all pairs of Cooperons. The quantum correction to conductance can be written as $$\label{eq:ds_app} \delta G=2\frac{e^2}{\hbar^2}\nu_1 D_1 \tau_{d 1}^2\int_{\bf Q} (C_{1 1}+C_{3 3})+ 2\frac{e^2}{\hbar^2}\nu_2 D_2 \tau_{d 2}^2\int_{\bf Q} C_{2 2}.$$ Solving Eqs. (\[eq:2\])-(\[eq:4\]) requires some algebra. The results for the Cooperons of interest are $$\begin{aligned} \label{eq:cii_app} C_{1 1}&=C_{3 3}=\frac{\hbar}{2\pi\nu_1\tau_{d 1}^2}\frac{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)-\lambda^2}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})\left[(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)-2 \lambda^2\right]}\nonumber\\ C_{2 2} &=\frac{\hbar}{2\pi\nu_2\tau_{d 2}^2}\frac{D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1}}{(D_1 Q^2+\tilde{\tau}_{\phi 1}^{-1})(D_2 Q^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)-2 \lambda^2},\end{aligned}$$ which are not illuminating expressions. It is better to rewrite them as $$\begin{aligned} \label{eq:ciib_app} C_{1 1} &=C_{3 3}=\frac{\hbar}{2\pi\nu_1\tau_{d 1}^2}\frac{1}{D_1}\left[\frac{X}{Q^2+q_x^2}+\frac{Y}{Q^2+q_y^2}+\frac{Z}{Q^2+q_z^2}\right]\nonumber\\ C_{2 2} &=\frac{\hbar}{2\pi\nu_2\tau_{d 2}^2} \frac{1}{D_2}\left[\frac{A}{Q^2+q_a^2}+\frac{B}{Q^2+q_b}\right],\end{aligned}$$ so that Eq. (\[eq:ds\_app\]) transforms into $$\label{eq:ds2_app} \delta G=\frac{e^2}{\pi\hbar}\int_{\bf Q} \left[2\frac{X}{Q^2+q_x^2}+2\frac{Y}{Q^2+q_y^2}+2\frac{Z}{Q^2+q_z^2}+\frac{A}{Q^2+q_a^2}+\frac{B}{Q^2+q_b^2}\right].$$ Comparing Eqs. (\[eq:cii\_app\]) and  (\[eq:ciib\_app\]), we arrive at $$\begin{aligned} \label{eq:A_app} A &=\frac{\frac{1}{D_1\tilde{\tau}_{\phi 1}}-q_a^2}{q_b^2-q_a^2}\,\,\,\mbox{ ; }\,\,\,B = 1-A\nonumber\\ X &= \frac{(D_1 q_x^2-\tilde{\tau}_{\phi 1}^{-1})(D_2 q_x^2-\tilde{\tau}_{\phi 2}^{-1}-\lambda)-\lambda^2}{D_1 D_2 (q_x^2-q_y^2)(q_x^2-q_z^2)}\nonumber\\ Y &= \frac{ D_2 q_y^2 \tilde{\tau}_{\phi 1}^{-1}-\tilde{\tau}_{\phi 1}^{-1}(\tilde{\tau}_{\phi 2}^{-1}+\lambda)+D_1 q_y^2 (-D_2 q_y^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)+\lambda^2}{D_1 D_2 (q_x^2-q_y^2)(q_y^2-q_z^2)}\nonumber\\ Z &= \frac{ D_2 q_z^2 \tilde{\tau}_{\phi 1}^{-1}-\tilde{\tau}_{\phi 1}^{-1}(\tilde{\tau}_{\phi 2}^{-1}+\lambda)+D_1 q_z^2 (-D_2 q_z^2+\tilde{\tau}_{\phi 2}^{-1}+\lambda)+\lambda^2}{D_1 D_2 (q_x^2-q_z^2)(q_z^2-q_y^2)}\end{aligned}$$ and $$\begin{aligned} \label{eq:qa_app} 2 q_{a(b)}^2 &=\frac{1}{D_1\tilde{\tau}_{\phi 1}}+\frac{1}{D_2\tilde{\tau}_{\phi 2}}+\frac{\lambda}{D_2}\pm\sqrt{\left(\frac{1}{D_1\tilde{\tau}_{\phi 1}}-\frac{1}{D_2 \tilde{\tau}_{\phi 2}}-\frac{\lambda}{D_2}\right)^2+\frac{8\lambda^2}{D_1 D_2}}\nonumber\\ q_{x (y)}^2 &=q_{a (b)}^2\,\,\,\mbox{ ; }\,\,\,q_z^2 = 1/(D_1\tilde{\tau}_{\phi 1}).\end{aligned}$$ Note that $q_{a (b)}=q_{x (y)}$, which will be important below. Also note that the expressions for $A$, $B$ and $q_{a(b)}$ are identical to the ones in Section IIIB, except for the following difference: the effective inelastic scattering rate for layer $2$ is now $\tau_{\phi 2}^{-1}+2\lambda$ instead of $\tau_{\phi 2}^{-1}+\lambda$, for the reason explained above. Although Eqs.  (\[eq:A\_app\]) and  (\[eq:qa\_app\]) look cumbersome, after substituting Eq. (\[eq:qa\_app\]) back in Eq. (\[eq:A\_app\]) we find some remarkable simplifications. In particular $$Z=1/2\,\,\,\mbox{ , }\,\,\,2 X+ A=1 \,\,\,\mbox{ and }\,\,\, 2 Y+ B =1.$$ Replacing these in Eq. (\[eq:ds2\_app\]) immediately leads to $$\delta G=\frac{e^2}{\pi\hbar}\int_{\bf Q} \left[\frac{1}{Q^2+q_a^2}+\frac{1}{Q^2+q_b^2}+\frac{1}{Q^2+q_z^2}\right].$$ In consequence, we recover the third line of Eq. (\[eq:res\_tot\]) for the low-field magnetoconductance: $$\label{eq:dg} \frac{\Delta G}{G_q}=\frac{1}{2}\left[f\left(\frac{H_a}{H}\right)+f\left(\frac{H_b}{H}\right)+f\left(\frac{H_z}{H}\right)\right],$$ where $H_a=\hbar q_a^2/(4 e)$, etc. As a reality check, let us take some simple limits. First, consider the case of no bulk-surface coupling, $\lambda\to 0$. In this case $H_a=H_z=\hbar/(4 e D_1 \tau_{\phi 1})$ and $H_b=\hbar/(4 e D_2 \tau_{\phi 2})$, which produces $$\frac{\Delta G}{G_q}=\frac{1}{2}\left[2 f\left(\frac{H_a}{H}\right)+f\left(\frac{H_b}{H}\right)\right].$$ This is indeed the result one would have expected when bulk and surface are decoupled. Second, suppose both $\tau_{\phi 1}$ and $\tau_{\phi 2}$ are infinitey large, for arbitrary tunneling rate. Then it follows that $H_b=0$, $$H_a=\frac{\hbar}{4 e}\lambda\left(\frac{1}{D_1}+\frac{2}{D_2}\right)\,\,\,\mbox{ and }\,\,\, H_z=\frac{\hbar}{4 e}\frac{\lambda}{D_1}$$ Then, $$\frac{\Delta G}{G_q}=\frac{1}{2}\left[f\left(\frac{H_a}{H}\right)+f\left(\frac{H_z}{H}\right)\right].$$ The fact that $H_b=0$ means that we recover the conventional WAL case (as we should when the phase relaxation times are infinitely long). Finally, consider the case of very strong tunneling between bulk and surface states. In this case $H_a$ and $H_z$ become very large ($\propto\lambda$), whereas $H_b$ becomes independent of $\lambda$. Consequently $$\frac{\Delta G}{G_q}=\frac{1}{2} f\left(\frac{H_b}{H}\right),$$ as if we had a single channel contributing to WAL. This seems to make sense too, because when tunneling is strong, $C_{i i}$ are strongly coupled to one another ($i=1,2,3$). Some special cases of Eq. (\[eq:res\_tot\]) {#sec:special} =========================================== In this Appendix we analyze some simple limiting cases of Eq. (\[eq:res\_tot\]), which considers a single TI surface coupled to bulk states. First, suppose that surface-bulk tunneling is strong, so that $\tau_{t i}\ll \tau_{\phi i}$ for $i=1,2$. In this case $(H_a, H_c, \tilde{H}_1)\gg (H_b,H_d)$ and thus Eq. (\[eq:res\_tot\]) turns into $$\begin{aligned} \label{eq:res_tot2} & \frac{\Delta G}{G_q}=\frac{1}{2}\left\{\begin{array}{ccc} f(H_b/H) &{\rm if } & \tilde{\tau}_H\ll\tau_s\\ f(H_b/H) &{\rm if } & \tilde{\tau}_H\gg(\tau_v,\tau_s)\\ f(H_d/H) &{\rm if } & \tilde{\tau}_H\ll\tau_v, \end{array}\right.\end{aligned}$$ where $H_b \simeq \hbar/(4 e)(1/\tau_{\phi 1}+1/\tau_{\phi 2})/(D_1+D_2)$ and $H_d\simeq \hbar/(4 e)(2/\tau_{\phi 1}+1/\tau_{\phi 2})/(2 D_1+D_2)$. For simplicity we have taken $\tau_{t 1}=\tau_{t 2}$, but this assumption can be easily relaxed. In sum, WL is [*not*]{} possible when the bulk-surface coupling is strong, and the film exhibits conventional WAL ($\alpha=1/2$) regardless of the bulk carrier concentration. Next, we consider a weak surface-bulk tunneling, so that $\tau_{t i}\gg \tau_{\phi i}$ for $i=1,2$. In this case the outcome depends on whether $D_1\tau_{\phi 1}>D_2\tau_{\phi 2}$ or $D_1\tau_{\phi 1}<D_2\tau_{\phi 2}$. Without loss of generality suppose that $D_1\tau_{\phi 1}>D_2\tau_{\phi 2}$. Then Eq. (\[eq:res\_tot\]) yields $$\begin{aligned} \label{eq:res_tot3} & \frac{\Delta G}{G_q}\simeq\frac{1}{2}\left\{\begin{array}{ccc} f(H_{\phi 2}/H)-2 f(H_{\phi 1}/H) &{\rm if } & \tilde{\tau}_H\ll\tau_s\\ f(H_{\phi 2}/H)+f(H_{\phi 1}/H) &{\rm if } & \tilde{\tau}_H\gg(\tau_v,\tau_s)\\ f(H_{\phi 2}/H)+2 f(H_{\phi 1}/H) &{\rm if } & \tilde{\tau}_H\ll\tau_v, \end{array}\right.\end{aligned}$$ where $H_{\phi i}= \hbar/(4 e D_i \tau_{\phi i})$ for $i=1,2$. When $H_{\phi 1}$ and $H_{\phi 2}$ are of the same order, the first line of Eq. (\[eq:res\_tot3\]) displays WL with $\alpha=-1/2$ and the third line exhibits WAL with $\alpha=3/2$. If instead $H_{\phi 1}\ll H_{\phi 2}$, $\Delta G$ is the same as if there were no surface states. This latter regime can be experimentally accessible by e.g. depositing magnetic impurities on the surface of the TI. Last, we consider the case $\tau_{t 1}\gg\tau_{\phi i}\gg\tau_{t 2}$ for $i=1,2$. This situation may be relevant for some thicker TI films where $\tau_{t 1}/\tau_{t 2}=W\nu_1/\nu_2\gg 1$ (for thicker films, surface states have more bulk states to decay onto). 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ArXiv

--- abstract: 'We aim to design adaptive online learning algorithms that take advantage of any special structure that might be present in the learning task at hand, with as little manual tuning by the user as possible. A fundamental obstacle that comes up in the design of such adaptive algorithms is to calibrate a so-called step-size or learning rate hyperparameter depending on variance, gradient norms, etc. A recent technique promises to overcome this difficulty by maintaining multiple learning rates in parallel. This technique has been applied in the MetaGrad algorithm for online convex optimization and the Squint algorithm for prediction with expert advice. However, in both cases the user still has to provide in advance a Lipschitz hyperparameter that bounds the norm of the gradients. Although this hyperparameter is typically not available in advance, tuning it correctly is crucial: if it is set too small, the methods may fail completely; but if it is taken too large, performance deteriorates significantly. In the present work we remove this Lipschitz hyperparameter by designing new versions of MetaGrad and Squint that adapt to its optimal value automatically. We achieve this by dynamically updating the set of active learning rates. For MetaGrad, we further improve the computational efficiency of handling constraints on the domain of prediction, and we remove the need to specify the number of rounds in advance.' bibliography: - 'biblio.bib' title: Lipschitz Adaptivity with Multiple Learning Rates in Online Learning --- Introduction ============ We consider *online convex optimization* (OCO) of a sequence of convex functions $f_1,\ldots,f_T$ over a given bounded convex domain, which become available one by one over the course of $T$ rounds [@ShalevShwartz2011; @HazanOCOBook2016]. Typically $f_t(\w) = {\textsc{loss}}(\w,\x_t,y_t)$ represents the *loss* of predicting with parameters $\w$ on the $t$-th data point $(\x_t,y_t)$ in a machine learning task. At the start of each round $t$, a learner has to predict the best parameters $\w_t$ for the function $f_t$ before finding out what $f_t$ is, and the goal is to minimize the *regret*, which is the difference in the sum of function values between the learner’s predictions $\w_1,\ldots,\w_T$ and the best fixed oracle parameters $\u$ that could have been chosen if all the functions had been given in advance. A special case of OCO is prediction with expert advice [@cesa06], where the functions $f_t(\w) = \w^\top \vloss_t$ are convex combinations of the losses $\vloss_t = (\loss_{t,1},\ldots,\loss_{t,K})^\top$ of $K$ expert predictors and the domain is the probability simplex. Central results in these settings show that it is possible to control the regret with almost no prior knowledge at all about the functions. For instance, knowing only an upper bound $G$ on the $\ell_2$-norms of the gradients ${\operatorname{grad}_}t = \nabla f_t(\w_t)$, the online gradient descent (OGD) algorithm guarantees $O(G \sqrt{T})$ regret by tuning its learning rate hyperparameter $\eta_t$ proportional to $1/(G\sqrt{t})$ [@Zinkevich2003], and in the case of prediction with expert advice the Hedge algorithm achieves regret $O(L\sqrt{T\ln K})$ knowing only an upper bound $L$ on the range $\max_k \ell_{t,k} - \min_k \ell_{t,k}$ of the expert losses [@FreundSchapire1997]. Here $G$ is the $\ell_2$-Lipschitz constant of the learning task[^1], and $L/2$ is the $\ell_1$-Lipschitz constant over the probability simplex. The above guarantees are tight if we make no further assumptions about the functions $f_t$ [@HazanOCOBook2016; @CesaBianchiEtAl1997], but they can be significantly improved if the functions have additional special structure that makes the learning task easier. The literature on online learning explores multiple orthogonal dimensions in which tasks may be significantly easier in practice (see ‘related work’ below). Here we focus on the following regret guarantees that are known to exploit multiple types of easiness at the same time: $$\begin{aligned} \text{OCO:}& &O\left(\sqrt{V_T^\u d \log T}\right) \text{ for all $\u$,} \quad \text{with $V_T^\u = \sum_{t=1}^T ((\w_t - \u)^\top {\operatorname{grad}_}t)^2$,}\label{eqn:ourmetagradbound}\\ \text{Experts:}& &O\left(\sqrt{\operatorname{\mathbb{E}}_{\rho(k)}[V_T^k] \operatorname{KL}(\rho\|\pi)}\right) \text{ for all $\rho$,} \quad \text{with $V_T^k = \sum_{t=1}^T ((\w_t - \e_k)^\top \vloss_t)^2$,}\label{eqn:oursquintbound}\end{aligned}$$ where $d$ is the number of parameters and $\operatorname{KL}(\rho\|\pi) = \sum_{k=1}^K \rho(k) \ln \rho(k)/\pi(k)$ is the Kullback-Leibler divergence of a data-dependent distribution $\rho$ over experts from a fixed prior distribution $\pi$. The OCO guarantee is achieved by the MetaGrad algorithm [@Erven2016], and implies regret that grows at most logarithmic in $T$ both in case the losses are curved (exp-concave, strongly convex) and in the stochastic case whenever the losses are independent, identically distributed samples with variance controlled by the Bernstein condition [@Erven2016; @koolen2016]. The guarantee for the expert case is achieved by the Squint algorithm [@koolen2015; @squintPAC]. It also exploits special structure along two dimensions simultaneously, because the $V_T^k$ term is much smaller than $L^2 T$ in many cases [@GaillardStoltzVanErven2014; @koolen2016] and the so-called *quantile bound* $\operatorname{KL}(\rho\|\pi)$ is much smaller than the worst case $\ln K$ when multiple experts make good predictions [@ChaudhuriFreundHsu2009; @ChernovVovk2010]. Squint and MetaGrad are both based on the same technique of tracking the empirical performance of *multiple learning rates* in parallel over a quadratic approximation of the original loss. A computational difference though is that Squint is able to do this by a continuous integral that can be evaluated in closed form, whereas MetaGrad uses a discrete grid of learning rates. Unfortunately, to achieve and , both MetaGrad and Squint need knowledge of the Lipschitz constant ($G$ or $L$, respectively). Overestimating $G$ or $L$ by a factor of $c > 1$ has the effect of reducing the effective amount of available data by the same factor $c$, but underestimating the Lipschitz constant is even worse because it can make the methods fail completely. In fact, the ability to adapt to $G$ has been credited [@WardWuBottou2018] as one of the main reasons for the practical success of the AdaGrad algorithm [@DuchiHazanSinger2011; @McMahanStreeter2010]. Thus getting the Lipschitz constant right makes the difference between having practical algorithms and having promising theoretical results. For OCO, an important first step towards combining Lipschitz adaptivity to $G$ with regret bounds of the form was taken by @cutkosky2017, who aimed for but had to settle for a weaker result with $G \sum_{t=1}^T \|{\operatorname{grad}_}t\|_2 \|\w_t - \u\|_2^2$ instead of $V_T^\u$. Although not sufficient to adapt to the Bernstein condition, they do provide a series of stochastic examples where their bound already leads to fast $O(\ln^4 T)$ rates. For the expert setting, @Wintenberger2017 has made significant progress towards a version of without the quantile bound improvement, but he is left with having to specify an initial guess $L_\text{guess}$ for $L$ that enters as $O(\ln \ln (L/L_\text{guess}))$ in his bound, which may yet be arbitrarily large when the initial guess is on the wrong scale. #### Main Contributions Our main contributions are that we complete the process began by @cutkosky2017 and @Wintenberger2017 by showing that it is indeed possible to achieve and without prior knowledge of $G$ or $L$. In fact, for the expert setting we are able to adapt to the tighter quantity $B \geq \max_k |(\w_t - \e_k)^\top \vloss_t|$. We achieve these results by dynamically updating the set of active learning rates in MetaGrad and Squint depending on the observed Lipschitz constants. In both cases we encounter a similar tuning issue as @Wintenberger2017, but we avoid the need to specify any initial guess using a new restarting scheme, which restarts the algorithm when the observed Lipschitz constant increases too much. In addition to these main results, we remove the need to specify the number of rounds $T$ in advance for MetaGrad by adding learning rates as $T$ gets larger, and we improve the computational efficiency of how it handles constraints on the domain of prediction: by a minor extension of the black-box reduction for projections of @cutkosky2018, we incur only the computational cost of projecting on the domain of interest in *Euclidean* distance. This should be contrasted with the usual projections in time-varying Mahalanobis distance for second-order methods like MetaGrad. #### Related Work If adapting to the Lipschitz constant were our only goal, a well-known way to achieve it for OCO would be to change the learning rate in OGD to $\eta_t \propto 1/\sqrt{\sum_{s\leq t} \|{\operatorname{grad}_}s\|_2^2}$, which leads to $O(\sqrt{\sum_{t\leq T} \|{\operatorname{grad}_}t\|_2^2}) = O(G \sqrt{T})$ regret. This is the approach taken by AdaGrad (for each dimension separately) [@DuchiHazanSinger2011; @McMahanStreeter2010]. In prediction with expert advice, Lipschitz adaptive methods are sometimes called and have previously been obtained by @cbms07 [@rooij14] with generalizations to OCO by @OrabonaPal2015. In addition, the first two of these works obtain a data-dependent variance term that is different from $V_T^k$ in , but no quantile bounds are known for the former. Results for the latter have previously been obtained by @GaillardStoltzVanErven2014 [@Wintenberger2014Arxiv] without quantile bounds, and with a slightly weaker notion of variance by @AdaNormalHedge. Quantile bounds without variance adaptivity were introduced by @ChaudhuriFreundHsu2009 [@ChernovVovk2010]. These may be interpreted as measures of the complexity of the comparator $\rho$. The corresponding notion in OCO is to adapt to the norm of $\u$, which has been achieved in various different ways, see for instance [@McMahanAbernethy2013; @cutkosky2018]. For curved functions, existing results achieve fast rates assuming that the degree of curvature is known [@HazanAgarwalKale2007], measured online [@BartlettHazanRakhlin2007; @Do2009] or entirely unknown [@Erven2016; @cutkosky2018]. Fast rates are also possible for slowly-varying linear functions and, more generally, optimistically predictable gradient sequences [@hazan2010extracting; @GradualVariationInCosts2012; @RakhlinSridharan2013]. We view our results as a step towards developing algorithms that automatically adapt to multiple relevant measures of difficulty at the same time. It is not a given that such combinations are always possible. For example, @CutkoskyBoahen2017Impossible show that Lipschitz adaptivity and adapting to the comparator complexity in OCO, although both achievable independently, cannot both be realized at the same time (at least not without further assumptions). A general framework to study which notions of task difficulty do combine into achievable bounds is provided by @FosterRakhlinSridharan2015. @FosterRakhlinSridharan2017 characterize the achievability of general data-dependent regret bounds for domains that are balls in general Banach spaces. #### Outline We add Lipschitz adaptivity to Squint for the expert setting in Section \[Squint2\]. Then, in Section \[MetaC\], we do the same for MetaGrad in the OCO setting. The developments are analogous at a high level but differ in the details for computational reasons. We highlight the differences along the way. Section \[MetaC\] further describes how to avoid specifying $T$ in advance for MetaGrad. Then, in Section \[four\], we add efficient projections for MetaGrad, and finally Section \[sec:conclusion\] concludes with a discussion of directions for future work. section[Introduction]{} Any source on prediction with expert advice will start with the celebrated minimax regret bound of the following form $$R_T^k ~\le~ \sqrt{\frac{T}{2} \ln K} \qquad \text{for each expert $k \in \{1,\ldots,K\}$} ,$$ and follow up with the remark that its multiplicative constant is optimal in the limit of large $T,K$ [@cesa06 Theorem 3.7]. Despite the mathematical strength and elegance, matching minimax algorithms are found to underwhelm in practice, whereas simple heuristics shine. This observation spurred multiple lines of research into adaptive algorithms with individual-sequence regret bounds with refined dependencies on the data and the comparator that hold under possibly relaxed assumptions. Properties of bounds that have been identified as desirable are - **quantile** bounds improve when multiple experts are good, a necessity when dealing with continuous expert spaces. These bounds also typically allow guarantees that are non-uniform across experts (adapting e.g. to available prior knowledge) - **first-order** bounds improve when the loss $L^*$ of the comparator is small. - **second-order** bounds improve when some measure of variance is small. The literature distinguishes two flavours of variance, namely the variance of the loss [@hazan10] and the variance of the excess loss [@Gaillard2014]. The latter is particularly interesting because it can be shown that algorithms with guarantees for squared excess losses have constant regret in many statistical cases [@koolen2016]. - **scale-free** bounds are for settings where no a-priori range of the losses can be assumed. Scale-free algorithms are unaffected by scaling the losses, while scale-free bounds scale along with any scaling of the losses imposed. @cbms07 call algorithms/bounds that are also invariant under translations of the losses **fundamental**. - **timeless** algorithms/bounds, as advocated by @rooij14, are unaffected when rounds are inserted with all-identical losses. Timelessness was proposed as a sanity-check to “protest” crudely measuring the complexity of the problem by its number of rounds. An algorithm can always be made timeless by simply ignoring any all-identical-loss round, but this is clunky and discontinuous, calling for prediction rules that are naturally smoothly timeless. Taking stock (see Table \[tab:stock\]), we see that no algorithm currently has all desirable features. The closest candidates are <span style="font-variant:small-caps;">AdaHedge</span> by @rooij14, which is fundamental second-order timeless but not quantile, and the later <span style="font-variant:small-caps;">Squint</span> by @koolen2015 which is second-order quantile timeless but not scale-free.[^2] Moreover, second-order bounds are by nature fundamental and timeless. This strongly suggests that it is possible to obtain everything. However, this seems to require a new idea in terms of algorithm design. That is where we come in. ----------------------- -------------------------------------------------------- ------------------------------------------------------ ------------------------------------------------------- <span style="font-variant:small-caps;">AdaHedge</span> <span style="font-variant:small-caps;">Squint</span> <span style="font-variant:small-caps;">Squint2</span> @rooij14 @koolen2015 this paper Quantile Second-order Scale-free Translation-invariant Timeless ----------------------- -------------------------------------------------------- ------------------------------------------------------ ------------------------------------------------------- : State of algorithms on the above two dimensions.[]{data-label="tab:stock"} --------------------------------------------------------------------- -------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------- <span style="font-variant:small-caps;">MetaGrad</span> <span style="font-variant:small-caps;">FreeRex-</span><span style="font-variant:small-caps;">Momentum</span> [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} @Erven2016 @cutkosky2017 this paper No prior knowledge of the gradient range No prior knowledge of the horizon Log. regret under exp-concavity Log. regret under the Bernstein condition [^3] Worst-case time complexity per round in the constrained OCO setting $O(C_{\mathcal{U}} \ln T$) $O(C_{\mathcal{U}})$[^4] $O(C_{\mathcal{U}} + d^2 \ln T)$ --------------------------------------------------------------------- -------------------------------------------------------- -------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------- : State of algorithms on the above two dimensions. $C_{\mathcal{U}}$ denotes the worst-case cost of performing a projection into the set $\mathcal{U}\subset \mathbb{R}^d$. []{data-label="tab:metagrad"} Related Work ------------ ### Adaptive Expert Algorithms - First-order adaptivity: Auer et al: $L^*$ - Second-order: Adahedge/CBMS, modified prod (Gaillard,Stoltz,Van Erven) - Quantile bounds: NormalHedge, Chernov-Vovk, (precursor: Poland&Hutter for non-uniform prior) - Second-order+quantile: AdaNormalHedge, Squint - Optimistic (predictible/slowly varying/...) ### Adaptive OCO Algorithms - Lipschitz-adaptivity: GD with $\eta_t = \frac{D}{\sqrt{\epsilon + \sum_{s=1}^t \|\bm{g}_s\|^2}}$ \[Don’t know reference for this; @McMahan2017 discusses it for $d=1$\], diagonal version of AdaGrad [@DuchiHazanSinger2011; @McMahanStreeter2010] uses this per dimension. @McMahan2017 reviews data-dependent regularizers and has a good discussion of AdaGrad (explicitly interpreted as per-coordinate learning rates); he claims that you need an initial guess $\epsilon = G$ for the FTRL-version of AdaGrad, but you can get away with $\epsilon = 0$ for the Mirror-Descent version of AdaGrad; otherwise not much novelty in [@McMahan2017], so don’t cite too much. @WardWuBottou2018 (who call this AdaGrad even though it is not) prove Lipschitz-adaptivity for this method in non-convex stochastic optimization with a bound $c_G^2$ on the *expected* gradient norm squared. They actually get an extra term $\log \frac{c_G^2}{B_0}$ in their convergence rate. @cutkosky2018 cite [@SrebroEtAl2010] for the claim that a regret bound in terms of $\|\bm{g}_t\|^2$ implies fast rates for smooth losses. This is not precise: @SrebroEtAl2010 do not directly say anything about regret in this case, but their Lemma 3.1 says that for smooth, *non-negative* loss the dual norm of the gradients can be bounded in terms of the square root of the function value. This might be small if the algorithm is converging to the minimum function value and this minimum value is 0, but showing that takes some work that is not done in [@SrebroEtAl2010]. - Adapt to $G$ and $D$ simultaneously: not possible [@CutkoskyBoahen2017Impossible] (actually, Manfred and Wojtek have a paper submitted to ICML where they show you can get around this impossibility result if you know $\bm{x}_t$ before making a prediction) @OrabonaPal2015 do have an “adaptive” bound $O(D^2\sqrt{\sum_{t=1}^T \|\bm{g}_t\|^2})$, which they achieve by generalizing AdaHedge to FTRL, but it has suboptimal dependence on $D$ (no square root) because they just omit it when tuning the learning rate. They also try to hide this suboptimality in their discussion, which is not good. They do give a table with overview of ‘scale-free’ algorithms: notably, AdaGrad is only scale-free in its MD version and not in its FTRL version. - [@cutkosky2017] adapt to $G$ and a class of stochastic settings. They say they would like to combine adaptivity to $G$ with the MetaGrad regret bound, but instead they settle for $R_T = O\left(\sqrt{G_T \sum_{t=1}^T \|\bm{g}_t\| \|\bm{w}_t - \bm{w}^*\|^2}\right)$, which implies “logarithmic regret” $O(\log^4(T))$ for “$\alpha$-acutely-convex” functions. $\alpha$-acutely convex is a stochastic condition, which implies the $(B,\beta)$-Bernstein condition with $B=G_T/\alpha$ and $\beta=1$, so even the constants are what we would expect (@cutkosky2017 have a factor $G_T$ in their regret bound, which we incur via the Bernstein constant). - Adapt to curvature: @BartlettHazanRakhlin2007 adapt to strong convexity (need to observe strong convexity per round); @cutkosky2018 adapt to strong convexity (in Banach spaces) without needing to observe strong convexity per round, but they lose logarithmic factors; MetaGrad adapts to exp-concavity and strong convexity without needing to observe the strong convexity per round, but loses a factor $d$ for strong convex losses \[this should be fixable, but we have not done it\] - Other well-known second-order methods in the mistake-bound model: AROW and the second-order perceptron - (Offset-) Rademacher complexity and its empirical versions. Notes: if you need a reference showing that $O(DG \sqrt{T})$ is the optimal rate in some sense, then @cutkosky2017 refers to Jacob Abernethy, Peter L Bartlett, Alexander Rakhlin, and Ambuj Tewari, COLT 2008. I could check this out. Problem Setting and Notation ============================ In OCO, a learner repeatedly chooses actions $\w_t$ from a closed convex set ${\mathcal{U}}\subseteq {\mathbb{R}}^d$ during rounds $t=1,\ldots,T$, and suffers losses $f_t(\w_t)$, where $f_t: {\mathcal{U}}\to {\mathbb{R}}$ is a convex function. The learner’s goal is to achieve small *regret* $R_T^\u = \sum_{t=1}^T f_t(\w_t) - \sum_{t=1}^T f_t(\u)$ with respect to any comparator action $\u \in {\mathcal{U}}$, which measures the difference between the cumulative loss of the learner and the cumulative loss they could have achieved by playing the oracle action $\u$ from the start. A special case of OCO is prediction with expert advice, where $f_t(\w) = \w^\top \vloss_t$ for $\vloss_t \in {\mathbb{R}}^K$ and the domain ${\mathcal{U}}$ is the probability simplex ${\triangle}_K = \{(w_1,\ldots,w_K) : w_i \geq 0, \sum_i w_i = 1\}$. In this context we will further write $\p$ instead of $\w$ for the parameters to emphasize that they represent a probability distribution. We further define $[K] = \{1,\ldots,K\}$. Adaptive Second-order Quantile Method for Experts {#Squint2} ================================================= In this section, we present an extension of the <span style="font-variant:small-caps;">Squint</span> algorithm that adapts automatically to the loss range in the setting of prediction with expert advice. Throughout this section, we denote $r^k_t \coloneqq {\langle{\widehat{\bm{p}}}_t- \bm{e}_k,\bm{\ell}_t\rangle}$ and $v^k_t \coloneqq (r^k_t)^2$, where ${\widehat{\bm{p}}}_t \in \triangle_K$ is the weight vector played by the algorithm at round $t$ and $\bm{\ell}_t$ is the observed loss vector. The cumulative regret with respect to expert $k$ is given by ${R}^k_t\coloneqq \sum_{s=1}^t r^k_s$. We use ${V}^k_t \coloneqq \sum_{s=1}^t v^k_s$ to denote the cumulative squared excess loss (which can be regarded as a measure of variance) of expert $k$ at round $t$. In the next subsection, we review the <span style="font-variant:small-caps;">Squint</span> algorithm. The <span style="font-variant:small-caps;">Squint</span> Algorithm {#AdaptiveSquint} ------------------------------------------------------------------ We first describe the original <span style="font-variant:small-caps;">Squint</span> algorithm, as introduced by [@koolen2015]. Let $\pi$ and $\gamma$ be prior distributions with supports on $[K]$ and $\left]0, \frac{1}{2}\right]$, respectively. Then <span style="font-variant:small-caps;">Squint</span> outputs predictions $$\begin{gathered} \label{Squintforcaster} \p_{t+1} \propto \underset{\pi(k)\gamma(\eta)}{\mathbb{E}}\left[ \eta e^{- \sum_{s=1}^t f_s(k,\eta)} \bm{e}_k \right], \shortintertext{where $f_t(k,\eta)$ are quadratic \emph{surrogate losses} defined by} \label{surrogatesquint0} f_t(k,\eta) \coloneqq - \eta {\langle{\widehat{\bm{p}}}_t-\bm{e}_k,\bm{\ell}_t\rangle} + \eta^2 {\langle{\widehat{\bm{p}}}_t-\bm{e}_k,\bm{\ell}_t\rangle}^2.\end{gathered}$$ [@koolen2015] propose to use the *improper* prior $\gamma(\eta) = \frac{1}{\eta}$ which does not integrate to a finite value over its domain, but because of the weighting by $\eta$ in the predictions $\p_{t+1}$ are still well-defined. The benefit of the improper prior is that it allows calculating $\p_{t+1}$ in closed form [@koolen2015]. For any distribution $\rho \in {\triangle}_K$, <span style="font-variant:small-caps;">Squint</span> achieves the following bound: $$\begin{aligned} {R}^{\rho}_T = O\left(\sqrt{{V}^{\rho}_T\left( \operatorname{KL}(\rho || \pi ) + \ln \ln T\right)}\right), \label{Squintbound}\end{aligned}$$ where $R_T^{\rho} = \mathbb{E}_{\rho(k)}\left[R_T^{k} \right]$ and $V_T^{\rho} = \mathbb{E}_{\rho(k)}\left[V_T^{k} \right]$. This version of Squint assumes the loss range $\max_k \ell_{t,k} - \min_k \ell_{t,k}$ is at most $1$, and can fail otherwise. In the next subsection, we present an extension of <span style="font-variant:small-caps;">Squint</span> which does not need to know the Lipschitz constant. Lipschitz Adaptive Squint ------------------------- |We first design a version of <span style="font-variant:small-caps;">Squint</span>, called [<span style="font-variant:small-caps;">Squint+C</span>]{}, that still requires an initial estimate $B > 0$ of the Lipschitz constant. The next section will be devoted to setting this parameter online. For now, we consider it fixed. In addition to this, the algorithm takes a prior distribution $\pi \in \triangle_K$. In a sequence of rounds $t = 1, 2, \ldots$ the algorithm predicts with $\hat\p_t \in \triangle_K$, and receives a loss vector $\vloss_t^k \in \mathbb R^K$. We denote the *instantaneous regret of expert $k$ in round $t$* by $r_t^k \df \tuple{\hat \p_t - \e_k, \vloss_t}$. We denote the observed Lipschitz constant in round $t$ at point $\hat \p_t$ by $ b_t \df \max_k \abs{r_t^k}$, and we denote its running maximum by $B_t \df B \lub \max_{s \le t} b_s$, and we use the convention that $B_0=B$. In addition, we will also require a clipped version of the loss vector $\scale{\vloss_t} = \frac{B_{t-1}}{B_t} \vloss_t$, and we denote by $\scale{r}_t^k = \tuple{\hat \p_t - \e_k, \scale \vloss_t}$ the rescaled instantaneous regret. We will use that $\abs{\scale r_t^k} \le B_{t-1}$. It suffices to control the regret for the clipped loss, because the cumulative difference is a negligible lower-order constant[^5]: $$\label{eq:ashok} R_T^k - \scale R_T^k ~\df~ \sum_{t=1}^T \del*{r_t^k - \scale r_t^k} ~=~ \sum_{t=1}^T \del*{B_t - B_{t-1}} \frac{r_t^k}{B_t} ~\le~ B_T - B_0 .$$ This means we can focus on regret for $\scale \vloss_t$, for which the range bound $\abs{\scale r_t^k} \le B_{t-1}$ is available *ahead* of each round $t$. To motivate [<span style="font-variant:small-caps;">Squint+C</span>]{}, we define the potential function after $T$ rounds by $$\label{eq:sq.pot} \Phi_T \df \sum_k \pi_k \int_0^\frac{1}{2 B_{T-1}} \frac{e^{\eta \scale{R}_T^k - \eta^2 \scale{V}_T^k} -1}{\eta} \dif \eta \quad \text{where} \quad \scale R_T^k \df \sum_{t=1}^T \scale r_t^k ~~ \text{and} ~~ \scale V_T^k \df \sum_{t=1}^T (\scale r_t^k)^2 .$$ We also define $\Phi_0 = 0$ (due to the integrand being zero), even though it involves the meaningless $B_{-1}$ in the upper limit. The algorithm is now derived from the desire to keep this potential under control. As we will see in the analysis, this requirement uniquely forces the choice of weights $$\label{eq:sq.weights} \hat p_{T+1}^k ~\propto~ \pi_k \int_0^\frac{1}{2 B_T} e^{\eta \scale{R}_T^k - \eta^2 \scale{V}_T^k} \dif \eta .$$ Like the original <span style="font-variant:small-caps;">Squint</span>, we see that the weights $\hat \p_{t+1}$ can be evaluated in closed form using Gaussian CDFs. The regret analysis consists of two parts. First, we show that the algorithm keeps the potential small. \[lem:pot.is.small\] Given parameter $B\geq0$, [<span style="font-variant:small-caps;">Squint+C</span>]{} ensures $\Phi_T \le \ln \frac{B_{T-1}}{B}$. The next step of the argument is to show that small potential is useful. The argument here follows [@koolen2015], specifically the version by [@squintPAC]. We have \[lem:small.is.good\] Definition implies that for any comparator distribution $\rho \in \triangle_K$ the regret is at most $$\begin{gathered} \scale R_T^\rho ~\le~ \sqrt{2 \scale V_T^\rho} \del*{ 1+ \sqrt{2 C_T^{\rho}} } + 5 B_{T-1} \del*{C_T^{\rho}+ \ln 2}, \quad \text{where,} \\ C^{\rho}_T ~\df~ \operatorname{KL}\delcc*{\rho}{\pi} + \ln \del*{ \Phi_T + \frac{1}{2} + \ln \left(2+ \sum_{t=1}^{T-1} \frac{b_t}{B_t} \right) } \end{gathered}$$ Keeping only the dominant terms, this reads $$\scale R_T^\rho ~=~ O\del*{ \sqrt{\scale V_T^\rho \del*{\operatorname{KL}\delcc*{\rho}{\pi} + \ln \Phi_T + \ln \ln T}} } .$$ The significance of , Lemmas \[lem:pot.is.small\] and  \[lem:small.is.good\] is that we obtain a bound of the form $$R_T^\rho ~=~ O \del*{ \sqrt{V_T^\rho \del*{\operatorname{KL}\delcc*{\rho}{\pi} + \ln \ln \frac{TB_{T-1}}{B}}} + 5 B_T \del*{\operatorname{KL}\delcc*{\rho}{\pi} + \ln \ln \frac{TB_{T-1}}{B} } } .$$ However, there does not seem to be any safe a-priori way to tune $B=B_0$. If we set it too small, the factor $\ln \ln \frac{B_{T-1}}{B}$ explodes. If we set it too large, the lower-order contribution $B_{T-1} \ge B$ blows up. It does not appear possible to bypass this tuning dilemma within the current construction. Fortunately, we are able to resolve it using restarts. Algorithm \[bb1alg\], which applies to both [<span style="font-variant:small-caps;">Squint+C</span>]{} and [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} (presented in the next section), monitors a condition of the sequences $(b_t)$ and $(B_t)$ to trigger restarts. is either [[<span style="font-variant:small-caps;">Squint+C</span>]{}]{} or [[<span style="font-variant:small-caps;">MetaGrad+C</span>]{}]{}, taking as input parameter an initial scale $B$ Play $\w_1$ until the first time $t=\tau_1$ that $b_t \neq 0$. \[line:runmetagrad\] Run [[<span style="font-variant:small-caps;">Alg</span>]{}]{} with input $B = B_{\tau_1}$ until the first time $t=\tau_2$ that $\displaystyle \frac{B_t}{B_{\tau_1}} > \sum_{s=1}^t \frac{b_s}{B_s}$.\ Set $\tau_1 = \tau_2$ and goto line \[line:runmetagrad\]. \[blackboxreduction0\] Let [<span style="font-variant:small-caps;">Squint+L</span>]{} be the result of applying Algorithm \[bb1alg\] to [<span style="font-variant:small-caps;">Squint+C</span>]{} (as <span style="font-variant:small-caps;">Alg</span>). [<span style="font-variant:small-caps;">Squint+L</span>]{} guarantees, for any comparator $\rho\in \triangle_K$, $$\begin{aligned} R_T^\rho ~\le~ 2\sqrt{ V_T^\rho} \del*{ 1+ \sqrt{2 \Gamma_T^{\rho}} } + 10 B_{T} \del*{ \Gamma_T^{\rho} + \ln 2} + 4 B_T,\end{aligned}$$ where $ \Gamma_T^{\rho} ~\df~\operatorname{KL}\delcc*{\rho}{\pi}+ \ln \del*{\ln \left(\sum_{t=1}^{T-1} \frac{b_{t}}{B_t}\right) + \frac{1}{2}+ \ln \left(2+\sum_{t=1}^{T-1} \frac{b_{t}}{B_t}\right)}$. Theorem \[blackboxreduction0\] shows that the bound on the regret of [<span style="font-variant:small-caps;">Squint+L</span>]{} has a term of order $O(\ln \ln \sum_{t=1}^{T-1} \frac{b_{t}}{B_t})=O(\ln \ln T)$, which does not depend on the initial guess $B_0$ anymore. Adaptive Method for Online Convex Optimization {#MetaC} ============================================== We consider the Online Convex Optimization (OCO) setting where at each round $t$, the learner predicts by playing ${\widehat{\bm{u}}}_t$ in a closed convex set $\mathcal{U} \subset \mathbb{R}^d$, then the environment announces a convex function $\ell_t : \mathcal{U}\rightarrow [0,+\infty[$ and the learner suffers loss $\ell_t({\widehat{\bm{u}}}_t)$. The goal of the learner is to minimize the regret with respect to the single best action $\bm{u}\in \mathcal{U}$ in hindsight (after $T$ rounds); that is, minimizing ${R}^{\bm{u}}_T \coloneqq \sum_{t=1}^T \ell_t({\widehat{\bm{u}}}_t) - \sum_{t=1}^T \ell_t(\bm{u})$ for the worst case $\bm{u}\in \mathcal{U}$. Since the losses are convex, it suffices to bound the sum of linearized losses $\tilde{R}^{\bm{u}}_T \df \sum_{t=1}^T {\langle{\widehat{\bm{u}}}_t - \bm{u},\bm{g}_t\rangle}$, where $\bm{g}_t \coloneqq \nabla \ell_t({\widehat{\bm{u}}}_t)$. We will assume that the set $\mathcal{U}$ is bounded and let $D\in ]0,+\infty[$ be its diameter $$\begin{aligned} \label{rad}D \coloneqq \sup_{\bm{u}, \bm{v}\in \mathcal{U}} {\left\lVert\bm{u} - \bm{v}\right\rVert}_2.\end{aligned}$$ Our main contribution in this section is to devise a simple modification of <span style="font-variant:small-caps;">MetaGrad</span> — [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} — which, without prior knowledge of the maximum value of the gradient range $G \coloneqq \max_{t \le T} {\left\lVert\nabla \ell_t({\widehat{\bm u}}_t)\right\rVert}$, guarantees the following regret bound $$\begin{aligned} \forall \bm{u}\in \mathcal{U}, \quad R_T^\u \leq \tilde{R}^{\bm{u}}_T ~=~ O \del*{\sqrt{{V}^{\bm{u}}_T d\ln T } + B_T d \ln T}, \label{mresult} \end{aligned}$$ where ${V}^{\bm{u}}_T \coloneqq \sum_{t=1}^T {\langle{\widehat{\bm{u}}}_t - \bm{u},\bm{g}_t\rangle}^2$. Consequently, this algorithm inherits the fast convergence results of standard <span style="font-variant:small-caps;">MetaGrad</span> [@Erven2016]. In particular, it was shown that due to the form of the bound in , <span style="font-variant:small-caps;">MetaGrad</span> achieves a logarithmic regret when the sequence of losses are exp-concave [@Erven2016]. Furthermore, when the sequence of gradient functions $(\nabla \ell_t)$ are i.i.d distributed with common distribution $\mathbb{P}$ and satisfy the ($B, \beta$)-Bernstein condition for $B > 0$ and $\beta \in [0, 1]$ with respect to the risk minimizer $\bm{u}^* =\operatorname*{argmin}_{\bm{u}\in \mathcal{U}}\mathbb{E}_{f \sim \mathbb{P}}[f(\bm{u})]$, then <span style="font-variant:small-caps;">MetaGrad</span> (and thus [<span style="font-variant:small-caps;">MetaGrad+C</span>]{}) achieves the expected regret $$\begin{aligned} \mathbb{E}\left[{R}^{\bm{u}^*}_T\right] ~=~ O \del*{ (d \ln T)^{\frac{1}{2-\beta}} T^{\frac{1-\beta}{2-\beta}} + d \ln T}.\end{aligned}$$ See [@koolen2016] for more detail. The <span style="font-variant:small-caps;">MetaGrad</span> Algorithm -------------------------------------------------------------------- The <span style="font-variant:small-caps;">MetaGrad</span> algorithm runs several sub-algorithms at each round; namely, a set of slave algorithms, which learn the best action in $\mathcal{U}$ given a learning rate $\eta$ in some predefined grid $\mathcal{G}$, and the master algorithm, which learns the best learning rate. The goal of <span style="font-variant:small-caps;">MetaGrad</span> is to maximize the sum of payoff functions $\sum_{t=1}^T f_t(\bm{u},\eta)$ over all $\eta \in \mathcal{G}$ and $\bm{u}\in \mathcal{U}$ simultaneously, where $$\begin{aligned} \label{surrmeta} f_t(\bm{u},\eta) \coloneqq - \eta {\langle{\widehat{\bm{u}}}_t - \bm{u},\bm{g}_t\rangle} + \eta^2 {\langle{\widehat{\bm{u}}}_t - \bm{u},\bm{g}_t\rangle}^2,\quad t\in [T],\end{aligned}$$ and ${\widehat{\bm{u}}}_t$ is the master prediction at round $t\geq 1$. Each slave algorithm takes as input a learning rate from a finite, exponentially-spaced grid $\mathcal{G}$ (with ${\lceil{\log_2 \sqrt{T}}\rceil}$ points) within the interval $\left[\frac{1}{5DG\sqrt{T}}, \frac{1}{5DG}\right]$, where $G$ is an upper bound on the norms of the gradients. In this case, the bound $G$ must be known in advance. In what follows, we let $\mathbf{M}_t \coloneqq \sum_{s=1}^t \bm{g}_s\bm{g}_s^{\operatorname{\intercal}}$, for $ t \geq 0$. #### Slave predictions. Every slave $\eta \in \mathcal G$ starts with ${\widehat{\bm{u}}}_1^\eta = \bm{0}$. At the end of round $t \ge 1$, it receives the master prediction ${\widehat{\bm{u}}}_t$ and updates the prediction in two steps $$\begin{gathered} {\bm{u}}^{\eta}_{{t+1}} \coloneqq {\widehat{\bm{u}}}^{\eta}_t - \eta \mathbf{\Sigma}^{\eta}_{{t+1}} \bm{g}_t \left(1 + 2 \eta \left( {\widehat{\bm{u}}}^{\eta}_t -{\widehat{\bm{u}}}_t \right)^{\operatorname{\intercal}}\bm{g}_t\right) , \text{ where }\ \mathbf{\Sigma}^{\eta}_{{t+1}} \coloneqq \left( \tfrac{\mathbf{I}}{D^2} +2\eta^2\mathbf{{M}}_t \right)^{-1}, \label{quadprog0}\\ \label{gaussian} \text{ and } \ \ {\widehat{\bm{u}}}^{\eta}_{{t+1}} = \operatorname*{argmin}_{\bm{u}\in \mathcal{U}} \left(\bm{u}_{{t+1}}^{\eta}- \bm{u} \right)^{\operatorname{\intercal}}\left( \mathbf{\Sigma}^{\eta}_{{t+1}}\right)^{-1} \left(\bm{u}_{{t+1}}^{\eta}- \bm{u} \right) ,\end{gathered}$$ #### Master predictions After receiving the slaves predictions, $\left({\widehat{\bm{u}}}^{\eta}_t\right)_{\eta \in \mathcal{G}}$, the master algorithm aggregates them and outputs ${\widehat{\bm{u}}}_t\in \mathcal{U}$ according to: $$\begin{aligned} {\widehat{\bm{u}}}_{t} \coloneqq \frac{\sum_{\eta\in \mathcal{G}} \eta w^{\eta}_t {\widehat{\bm{u}}}^{\eta}_{t} }{\sum_{\eta \in \mathcal{G}}\eta w^{\eta}_t };\quad w^{\eta}_t \coloneqq e^{- \sum_{s=1}^{t-1} f_s({\widehat{\bm{u}}}^{\eta}_s,\eta)}, \label{masterpred}\end{aligned}$$ As mentioned earlier, the <span style="font-variant:small-caps;">MetaGrad</span> algorithm requires the knowledge of the maximum value of the gradient range $G$ and the horizon $T$ in advance. These are needed to define the grid of the slave algorithms. In the analysis of <span style="font-variant:small-caps;">MetaGrad</span>, it is crucial for the $\eta$’s to be in the right interval in order to invoke a Gaussian exp-concavity result for the surrogate losses in (see e.g. [@Erven2016 Lemma 10]). In the next subsection, we explore a natural extension of <span style="font-variant:small-caps;">MetaGrad</span> which does not require the knowledge of the gradient range or the horizon $T$. An Extension of <span style="font-variant:small-caps;">MetaGrad</span> for Unknown Gradient Range and Horizon ------------------------------------------------------------------------------------------------------------- We present a natural extension of <span style="font-variant:small-caps;">MetaGrad</span>, called [<span style="font-variant:small-caps;">MetaGrad+C</span>]{}, which does not assume any knowledge on the gradient range or the horizon. Contrary to the original <span style="font-variant:small-caps;">MetaGrad</span> which requires knowledge of the horizon $T$ to define the grid for the slaves, [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} circumvents this by defining an infinite grid $\mathcal{G}$, in which, at any given round $t\geq1$, only a finite number of slaves (up to $\log_2 t$ many) output a prediction (see Remark \[numslaves\]). Each slave $\eta$ in this grid receives a prior weight $\pi(\eta) \in[0,1]$, where $\sum_{\eta\in \mathcal{G}} \pi(\eta) =1$. The expressions of $\mathcal{G}$ and $\pi$ are given by $$\begin{aligned} \mathcal{G} \coloneqq \left\{ \eta_i \coloneqq \tfrac{2^{-i}}{5 B}: i \in \mathbb{N} \right\} \label{Ggrid} ;\quad \pi(\eta_i) \coloneqq \tfrac{1}{(i+1)(i+2)}, \ i\in \mathbb{N}.\end{aligned}$$ where $B>0$ is the input to [<span style="font-variant:small-caps;">MetaGrad+C</span>]{}. ### Algorithm Description #### Preliminaries. As in the previous subsection, we let ${\widehat{\bm{u}}}_t$ and ${\widehat{\bm{u}}}^{\eta}_{t}$ be the predictions of the master and slave $\eta$, respectively, at round $t\geq1$ (we give their explicit expressions further below). Let $(b_t)$ and $(B_t)$ be the sequences in $\mathbb{R}_{\geq0}$ defined by $$\begin{aligned} b_t \coloneqq D {\left\lVert\bm{g}_t\right\rVert}_2; \quad \quad \quad B_t \coloneqq B \vee \max_{s\in [t]} b_s, \quad t\in[T], \label{littleb}\end{aligned}$$ where $B$ is the input of [<span style="font-variant:small-caps;">MetaGrad+C</span>]{}, and we use the convention that $B_0 = B$. Using the sequence $(B_t)$, we define the clipped gradients $\bar{\bm{g}}_t \coloneqq \frac{B_{t-1}}{B_t} \bm{g}_t$, and $\forall \bm{u}\in\mathcal{U},\forall t\geq 1, \forall \eta >0$, we let $$\begin{aligned} \bar{r}^{\bm{u}}_t \coloneqq {\langle{\widehat{\bm{u}}}_t-\bm{u},\bar{\bm{g}}_t\rangle},\quad \quad \bar{f}_t(\bm{u},\eta)\coloneqq - \eta \bar{r}^{\bm{u}}_t + \left(\eta \bar{r}^{\bm{u}}_t\right)^2,\quad \quad \bar{\mathbf{M}}_t\coloneqq \sum_{s=1}^t \bar{\bm{g}}_s \bar{\bm{g}}_s^{\operatorname{\intercal}}. \label{clippedstuff}\end{aligned}$$ For each slave $\eta\in \mathcal{G}$, we define the time $s_\eta$ to be $$\begin{aligned} \label{threshold} s_\eta ~\df~ \min\setc*{t \ge 0}{ \eta \geq \frac{1}{D \sum_{s=1}^t {\left\lVert\bar{\bm{g}}_s\right\rVert}_2 + B_t} },\end{aligned}$$ and define the set $\mathcal{A}_t$ of “active” slaves by $$\begin{aligned} \mathcal{A}_t \coloneqq \{ \eta \in \mathcal{G}_t : s_\eta < t \}, \quad \text{where} \quad \mathcal{G}_t \coloneqq \mathcal{G} \cap \left[0, \tfrac{1}{5B_{t-1}}\right] , \quad t\geq 1.\end{aligned}$$ #### Slaves’ predictions. A slave $\eta \in \mathcal{A}_t$ issues its first prediction ${\widehat{\bm{u}}}_t^\eta = \bm{0}$ in round $t=s_\eta+1$. From then on, it receives the master prediction ${\widehat{\bm{u}}}_t$ as input and updates in two steps as $$\begin{gathered} \bm{u}^{\eta}_{{t+1}} \coloneqq {\widehat{\bm{u}}}^{\eta}_t - \eta \mathbf{\Sigma}^{\eta}_{{t+1}} \bar{\bm{g}}_t \left(1 + 2 \eta \left( {\widehat{\bm{u}}}^{\eta}_t -{\widehat{\bm{u}}}_t \right)^{\operatorname{\intercal}}\bar{\bm{g}}_t\right), \text{ where }\ \mathbf{\Sigma}^{\eta}_{{t+1}} \coloneqq \left( \tfrac{\mathbf{I}}{D^2} +2\eta^2\left(\bar{\mathbf{M}}_t -\bar{\mathbf{M}}_{s_{\eta}}\right) \right)^{-1}, \nonumber \\ \text{ and }\ \ {\widehat{\bm{u}}}^{\eta}_{{t+1}} = \operatorname*{argmin}_{\bm{u}\in \mathcal{U}} \left(\bm{u}_{{t+1}}^{\eta}- \bm{u} \right)^{\operatorname{\intercal}}\left( \mathbf{\Sigma}^{\eta}_{{t+1}}\right)^{-1} \left(\bm{u}_{{t+1}}^{\eta}- \bm{u} \right). \label{quadprog}\end{gathered}$$ Slaves that are outside the set $\mathcal{A}_t$ at round $t$ are irrelevant to the algorithm[^6]. Note that restricting the slaves to the set $\mathcal{G}_t$ is similar to clipping the upper integral range in the [<span style="font-variant:small-caps;">Squint+C</span>]{} case. #### Master predictions. At each round $t\geq1$, the master algorithm receives the slaves predictions $({\widehat{\bm{u}}}_t^{\eta})_{t\in \mathcal{A}_{t}}$ and outputs the $\widehat{\bm{u}}_t$: $$\begin{aligned} \label{newmaster} {\widehat{\bm{u}}}_t = \frac{\sum_{\eta \in \mathcal{A}_{t}}\eta w^{\eta}_t {\widehat{\bm{u}}}_t^{\eta}}{\sum_{\eta \in \mathcal{A}_{t}} \eta w^{\eta}_t }; \quad w^{\eta}_t \coloneqq \pi(\eta) e^{- \sum_{s=s_{\eta}+1}^{t-1} \bar{f}_s({\widehat{\bm{u}}}^{\eta}_s,\eta)}, \quad t\geq 1.\end{aligned}$$ \[numslaves\] At any round $t\geq 1$, the number of active slaves is at most ${\left\lfloor{\log_2 t}\right\rfloor}$. In fact, if $\eta \in \mathcal{A}_t$, then by definition $\eta \geq 1/(D\sum_{s=1}^{s_{\eta}}{\left\lVert\bm{g}_s\right\rVert}_2 + B_{s_{\eta}}) \geq 1/(t B_{t-1})$ (since $s_{\eta}\leq t-1$), and thus $\mathcal{A}_t \subset [1/(tB_{t-1}), 1/(5B_{t-1})]$. Since $\mathcal{A}_t$ is an exponentially-spaced grid with base $2$, there are at most ${\left\lfloor{\log_2 t}\right\rfloor}$ slaves in $\mathcal{A}_t$. ### Analysis To analyse the performance of [<span style="font-variant:small-caps;">MetaGrad+C</span>]{}, we consider the potential function $$\begin{aligned} \label{masterpot} \Phi_t \coloneqq \pi(\mathcal{G}_t\setminus \mathcal{A}_t) + \sum_{\eta\in \mathcal{A}_t} \pi(\eta) e^{-\sum_{s=s_{\eta}+1}^t \bar{f}_s({\widehat{\bm{u}}}^{\eta}_s,\eta)}, \quad t\geq 0.\end{aligned}$$ For $\bm{u}\in \mathcal{U}$, we define the pseudo-regret $\tilde{R}^{\bm{u}}_T \coloneqq \sum^T_{t=1} {\langle{\widehat{\bm{u}}}_t - \bm{u},{\bm{g}}_t\rangle}$ and its clipped version ${\bar{R}}^{\bm{u}}_T \coloneqq \sum^T_{t=1} {\langle{\widehat{\bm{u}}}_t - \bm{u},\bar{\bm{g}}_t\rangle}$. The following analogue to relates these two regrets. \[relatingtheregret\] Let $(b_t)$ and $(B_t)$ be as in , respectively, then for all $\bm{u}\in \mathcal{U}$, $$\begin{aligned} \label{clippedrel} \tilde{R}^{\bm{u}}_{T} \leq {\bar{R}}^{\bm{u}}_{T} +B_T.\end{aligned}$$ Similarly to the <span style="font-variant:small-caps;">Squint</span> case, one can use the prod-bound to control the growth of this potential function as shown in the proof of the following lemma (see Appendix \[MetaGrad2proofs\]): \[lemmameta\] [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} guarantees that $\Phi_T \leq \dots \leq \Phi_0 = 1$, for all $T \in \mathbb{N}$. We now give a bound on the clipped regret ${\bar{R}}^{\u}_T$ in terms of the clipped variance ${\bar{V}}^{\u}_T \coloneqq \sum_{t=1}^T (\bar{r}^{\u}_t)^2$: \[naivemeta\] Given input $B>0$, the clipped pseudo-regret for [[<span style="font-variant:small-caps;">MetaGrad+C</span>]{}]{} is bounded by $${\bar{R}}_T^\u\leq 3\sqrt{{\bar{V}}_T^\u C_T} + 15 B_T C_T \quad \text{for any $\u \in {\mathcal{U}}$,} \label{naivebound}$$ where $C_T \coloneqq d\ln\left(1 + \frac{2 \sum_{t=1}^{T-1} b_t^2 + 2 B^2_{T-1}}{25 d B^2_{T-1}}\right) + 2 \ln \left( \log^+_2 \frac{\sqrt{\sum_{t=1}^Tb^2_t }}{B} +3 \right) + 2$ and $\log_2^+ = 0 \vee \log_2 $. \[truebound\] We can relate the clipped pseudo-regret to the ordinary regret via $R_T^\u \leq {\tilde{R}}_T^\u \leq {\bar{R}}_T^\u + B_T$ (see ) and on the right-hand side we can also use that ${\bar{V}}_T^\u \leq V_T^\u$. An important thing to note from the result of Theorem \[naivemeta\] is that the ratio $\sqrt{\sum_{t=1}^Tb^2_t}/B$, could in principle be arbitrarily large if the input $B$ is too small compared to the actual regret range. To resolve this issue, one can use the same restart trick as in the <span style="font-variant:small-caps;">Squint</span> case: \[blackboxreduction1\] Let [[<span style="font-variant:small-caps;">MetaGrad+L</span>]{}]{} be the result of applying Algorithm \[bb1alg\] to [[<span style="font-variant:small-caps;">MetaGrad+C</span>]{}]{}. Then the regret for [[<span style="font-variant:small-caps;">MetaGrad+L</span>]{}]{} is bounded by $$\begin{aligned} {\tilde{R}}_T^\u \leq 3\sqrt{V_T^\u \Gamma_T} + 15 B_T \Gamma_T + 4 B_T \quad \text{for all $\u \in {\mathcal{U}}$,}\label{bbbound}\end{aligned}$$ where $\Gamma_T \coloneqq 2 d\ln\left(\frac{27}{25} + \frac{2}{25d} \sum_{t=1}^{T} \frac{b_t^2}{B^2_{t}}\right) + 4 \ln \left( \log^+_2 \sqrt{\sum_{t=1}^T (\sum_{s=1}^t \frac{b_s}{B_s})^2} +3 \right)+ 4 = O(d \ln T)$. In Theorem \[blackboxreduction1\], we have replaced the ratio $\sqrt{\sum_{t=1}^Tb^2_t} /B$ appearing in the (clipped) pseudo-regret bound of [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} by the term $\sigma_T \coloneqq \sqrt{\sum_{t=1}^T (\sum_{s=1}^t \frac{b_s}{B_s})^2}$ which is always smaller than $T^{\frac{3}{2}}$, but this is acceptable since $\sigma_T$ appears inside a $\ln \ln$. From the bound of Theorem \[blackboxreduction1\] on can easily recover an ordinary regret bound, i.e. a bound on $R^{\bm{u}}_t, \bm{u}\in \mathcal{U}$ (see Remark \[truebound\]).\ Efficient Implementation Through a Reduction to the Sphere {#four} ========================================================== Using [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} (or <span style="font-variant:small-caps;">MetaGrad</span>), the computation of each vector ${\widehat{\bm{u}}}^{\eta}_t$ requires a (Mahalanobis) projection step onto an arbitrary convex set $\mathcal{U}$. Numerically, this typically requires $O(d^p)$ floating point operations (flops), for some $p \in \mathbb{N}$ which depends on the topology of the set $\mathcal{U}$. Since $p$ can be large in many applications, evaluating ${\widehat{\bm{u}}}^{\eta}_{t}$ at each grid point $\eta$ can become computationally prohibitive, especially when the number of grid points grows with $T$ — in the case or [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} there can be at most ${\left\lfloor{\log_2 T}\right\rfloor}$ slaves at round $T\geq1$ (see Remark \[numslaves\]). An efficient implementation of [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} on the ball {#ballefficient} ------------------------------------------------------------------------------------------------------- In this subsection, we assume that $\mathcal{U}$ is the ball of diameter $D$, i.e. $\mathcal{U}=\mathcal{B}_{D} \coloneqq \left\{\bm{u} \in \mathbb{R}^d \colon {\left\lVert\bm{u}\right\rVert}_2 \leq D/2 \right\}$. In order to compute the slave prediction ${\widehat{\bm{u}}}^{\eta}_{t+1}$, for $t\geq 1$ and $\eta \in \mathcal{A}_t$, the following quadratic program needs to be solved: $$\begin{aligned} {\widehat{\bm{u}}}^{\eta}_{t+1} = \operatorname*{argmin}_{\bm{u}\in \mathcal{U}} \left(\bm{u}_{t+1}^{\eta}- \bm{u} \right)^{\operatorname{\intercal}}\left( \mathbf{\Sigma}^{\eta}_{t+1}\right)^{-1} \left(\bm{u}_{t+1}^{\eta}- \bm{u} \right), \label{quadprog2}\end{aligned}$$ where $\bm{u}^{\eta}_{t+1}$ (the unprojected prediction) and $\mathbf{\Sigma}^{\eta}_{t+1}$ (the co-variance matrix) are defined in . Since $\mathcal{U}$ is a ball, can be solved efficiently using the result of following lemma: \[redquad\] Let $t\geq 1$, $\eta \in \mathcal{A}_t$, and $\bm{v}^{\eta}_{t+1}\coloneqq \left(\tfrac{\mathbf{I}}{D^2} +2\eta^2\left(\bar{\mathbf{M}}_t -\bar{\mathbf{M}}_{s_{\eta}}\right)\right) \bm{u}^{\eta}_{t+1}$. Let $\mathbf{Q}_t$ be an orthogonal matrix which diagonalizes $\bar{\mathbf{M}}_{t}$, and $\mathbf{\Lambda}_t \coloneqq \left[\lambda^i_t\right]_{i=1}^t$ the diagonal matrix which satisfies $\mathbf{Q}_t \bar{\mathbf{M}}_t \mathbf{Q}^{\operatorname{\intercal}}_t = \mathbf{\Lambda}_t$. The solution of is given by ${\widehat{\bm{u}}}^{\eta}_{t+1}=\bm{u}^{\eta}_{t+1}$, if $\bm{u}^{\eta}_{t+1} \in \mathcal{U}$; and otherwise, ${\widehat{\bm{u}}}^{\eta}_{t+1} = \mathbf{Q}_t^{\operatorname{\intercal}} (x_t^{\eta}\mathbf{I} +2 \eta^2 (\mathbf{\Lambda}_t- \mathbf{\Lambda}_{s_{\eta}} ))^{-1} \mathbf{Q}_t \bm{v}^{\eta}_{t+1}$, where $x_{t}^{\eta}$ is the unique solution of $$\begin{aligned} \rho_t^{\eta}(x) \coloneqq \sum_{i=1}^{d} \frac{{\langle\bm{e}_i,\mathbf{Q}_t \bm{v}^{\eta}_{t+1}\rangle}^2}{(x+2\eta^2 (\lambda^i_t -\lambda^i_{s_{\eta}}))^2} =\frac{D^2}{4}, \label{proxyfun}\end{aligned}$$ The proof of the lemma is in Appendix \[fourproof\]. Note that since the matrix $\bar{\mathbf{M}}_t$ is symmetric for all $t\geq 1$, the existence of the matrices $\mathbf{Q}_t$ and $\mathbf{\Lambda}_t$ in Lemma \[redquad\] is always guaranteed. Since $\rho_t^{\eta}$ in is strictly convex and decreasing, one can use the Newton method to efficiently solve $\rho_t^{\eta}(x)=D^2/4$. Thus, since the computation of $\mathbf{Q}_t \bm{v}^{\eta}_{t+1}$ only involves matrix-vector products, Lemma \[redquad\] gives an efficient way of solving . : A bounded convex set $\mathcal{U}\in \mathbb{R}^d$ with diameter $D$, and a fast [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} implementation on the ball $\mathcal{B}_{D}$, [<span style="font-variant:small-caps;">MetaGrad+C</span>]{}, taking input $B$. Get ${\widehat{\bm{u}}}_t$ from [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} Play ${\widehat{\bm{w}}}_t = \Pi_{\mathcal{U}}({\widehat{\bm{u}}}_t)$, receive $\mathring{\bm{g}}_t = \nabla \ell_t({\widehat{\bm{w}}}_t)$ Set $\bm{g}_t \in \tfrac{1}{2} \left( \mathring{\bm{g}}_t +{\left\lVert\mathring{\bm{g}}_t\right\rVert} \partial {\operatorname{d}}_{\mathcal{U}}({\widehat{\bm{u}}}_t) \right)$ Send $\bm{g}_t$ to [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} #### Implementation on the ball. At round $t\geq 1$, the implementation of [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} on the ball $\mathcal{B}_{D}$ keeps in memory the orthogonal matrix $\mathbf{Q}_{t-1}$ which diagonalizes $\bar{\mathbf{M}}_{t-1}$. In this case, since $\bar{\mathbf{M}}_t = \bar{\mathbf{M}}_{t-1}+ \bar{\bm{g}}_t \bar{\bm{g}}_t^{\operatorname{\intercal}}$ it is possible to compute the new matrices $\mathbf{Q}_t$ and $\mathbf{\Lambda}_t$ in $O(d^2)$ flops [@stor2015]. Note that this operation only needs to be performed once — the diagonalization does not depend on $\eta$. Therefore, computing $\mathbf{Q}_t \bm{v}^{\eta}_{t+1}$ (and thus ${\widehat{\bm{u}}}^{\eta}_{t+1}$) can be performed in only $O(d^2)$ flops. Thus, aside from the matrix-vector products, the time complexity involved in computing ${\widehat{\bm{u}}}_{t+1}^{\eta}$ for a given $\eta\in \mathcal{A}_t$ is of the same order as that involved in solving $\rho_t^{\eta}(x)=D^2/4$. A Reduction to the ball ----------------------- In this subsection, we make use of a recent technique by [@cutkosky2018] that reduces constrained optimization problems to unconstrained ones, to reduce any OCO problem on an arbitrary bounded convex set $\mathcal{U}\subset\mathbb{R}^d$ to an OCO problem on a ball, where we can apply [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} efficiently. Let $D$ be the diameter of $\mathcal{U}\in \mathbb{R}^d$ as in , so that the ball $\mathcal{B}_D$ of radius $D/2$ encloses $\mathcal{U}$. For $\bm{u}\in \mathcal{U}$, we denote ${\operatorname{d}}_{\mathcal{U}}(\bm{u}) = \min_{\bm{w} \in \mathcal U} {\left\lVert\bm{u}-\bm{w}\right\rVert}_2$ the *distance function* from the set $\mathcal{U}$, and we define $\Pi_{\mathcal{U}}(\u)\coloneqq \{\w\in \mathcal{U}: {\left\lVert\bm{w}-\u\right\rVert}_2 = {\operatorname{d}}_{\mathcal{U}}(\u) \}$. Algorithm \[OCOGeneral\] reduces the OCO problem on the set $\mathcal{U}$ to one on the ball $\mathcal{B}_{D}$, where [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} is used as a black-box to solve it efficiently. As a result, Algorithm \[OCOGeneral\] (including its [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} subroutine) only performs a single Euclidean projection (as opposed to the projection in Mahalanobis distance as in ) onto the set $\mathcal{U}$, which is applied to the output of [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} — the [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} subroutine only performs projections onto the ball $\mathcal{B}_D$, which can be done efficiently as described in the previous subsection. Let $\mathring{{R}}^{\bm{u}}_T \coloneqq \sum_{t=1}^T {\langle{\widehat{\bm{w}}}_t - \bm{u},\mathring{{\operatorname{grad}}}_t\rangle}$ and $\mathring{{V}}^{\bm{u}}_T \coloneqq \sum_{t=1}^T {\langle{\widehat{\bm{w}}}_t - \bm{u},\mathring{{\operatorname{grad}}}_t\rangle}^2$ be the pseudo-regret and variance of Algorithm \[OCOGeneral\]. The following theorem, whose proof is in Appendix \[fourproof\], shows how the regret guarantee of [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} readily transfers to Algorithm \[OCOGeneral\]: \[reductionbound\] Algorithm \[OCOGeneral\], which uses [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} as a black-box, guarantees: $$\begin{aligned} \sum_{t=1}^T \left(\ell_t({\widehat{\bm{w}}}_t) - \ell_t(\bm{u})\right) \leq \mathring{{R}}^{\bm{u}}_T \leq 3 \sqrt{\mathring{{V}}^{\bm{u}}_T {\Gamma}_T} + 24 B_T {\Gamma}_T +B_T, \ \text{ for }\u\in \mathcal{U}, \label{alg1bound}\end{aligned}$$ where $ {\Gamma}_T \coloneqq d\ln\left(\frac{27}{25} + \frac{2 \sum_{t=1}^{T-1} {b}_t^2}{25 d{B}^2_{T-1}}\right) + 2 \ln \left( \log^+_2 \frac{\sqrt{\sum_{t=1}^T {b}_t^2}}{B} +3 \right) + 2= O(d \ln T)$, and $$\begin{aligned} {b}_t \coloneqq D {\left\lVert{\bm{g}}_t\right\rVert}_2; \quad \quad \quad {B}_t \coloneqq B \vee \max_{s\in [t]} {b}_s, \quad t\in[T]. \label{littlebbre}\end{aligned}$$ Note that Algorithm \[OCOGeneral\], guarantees the same type of regret as [<span style="font-variant:small-caps;">MetaGrad+C</span>]{}, and thus can also adapt to exp-concavity of the losses $(\ell_t)$ and the Bernstein condition. Conclusion {#sec:conclusion} ========== We present algorithms that adapt to the Lipschitz constant of the loss for OCO and experts. Stepping back, we see that an interesting combination of problem complexity dimensions can be adapted to, with hardly any overhead in either regret or computation. The main question for future work is to obtain a better understanding of the landscape of interactions between measures of problem complexity and their algorithmic reflection. One surprising conclusion from our work, which provides a curious contrast with incompatibility of Lipschitz adaptivity with comparator complexity adaptivity in general OCO [@CutkoskyBoahen2017Impossible], is the following observation. Our results for the expert setting, which we phrased for a finite set of $K$ experts, in fact generalise unmodified to priors with infinite support. Considering a countable set of experts, we find a scenario where the comparator complexity $\operatorname{KL}(\rho\|\pi)$ is unbounded, yet our Squint strategy adapts to the Lipschitz constant of the loss without inflating the regret compared to an a-priori known complexity by more than a constant. A final very interesting question is when it is possible to exploit scenarios with large ranges that occur only very infrequently. A example of this is found in statistical learning with heavy-tailed loss distributions. Martingale methods for such scenarios that are related to our potential functions suggest that it may be necessary to replace the “surrogate” negative quadratic term $f_t(\u,\eta)$ that our algorithms include in the exponent by another function appropriate for the specific distribution [@linecrossing Table 3]. It is not currently clear what individual sequence analogues can be obtained. \[3\][\#3]{} \[3\][\#2]{} Proofs of Section \[Squint2\] {#proofssquint} ============================= [**of Lemma \[lem:pot.is.small\]**]{} We proceed by induction on $T$. By definition $\Phi_0 = 0$. For $T \ge 0$, the definition gives $$\Phi_{T+1} ~=~ \underbrace{ \sum_k \pi_k \int_0^\frac{1}{2 B_T} \frac{ e^{\eta \scale{R}_{T}^k - \eta^2 \scale{V}_{T}^k} \del*{ e^{\eta \scale{r}_{T+1}^k - \eta^2 (\scale{r}_{T+1}^k)^2} - 1}}{\eta} \dif \eta }_{\fd Q_1} + \underbrace{ \sum_k \pi_k \int_0^\frac{1}{2 B_T} \frac{e^{\eta \scale{R}_{T}^k - \eta^2 \scale{V}_{T}^k} -1}{\eta} \dif \eta }_{\fd Q_2} .$$ To control the first term $Q_1$, we apply the “prod bound” $e^{x-x^2} \le 1+x$ for $x \ge -1/2$ to $x = \eta \scale r_{T+1}^k$, which we may do as $\eta \scale r_{T+1} \ge - \frac{1}{2 B_T} B_T$. Linearity and the definition of the weights further yield $$Q_1 ~\le~ \sum_k \pi_k \int_0^\frac{1}{2 B_T} \frac{ e^{\eta \scale{R}_{T}^k - \eta^2 \scale{V}_{T}^k} \eta \scale{r}_{T+1}^k }{\eta} \dif \eta ~=~ \tuple*{ \sum_k \pi_k \int_0^\frac{1}{2 B_T} e^{\eta \scale{R}_{T}^k - \eta^2 \scale{V}_{T}^k} \del*{\hat \p_{T+1} - \e_k} \dif \eta , \scale \vloss_{T+1}} ~=~ 0 .$$ To control the second term $Q_2$, we extend the range of the integral to find $$Q_2 ~\le~ \sum_k \pi_k \int_0^\frac{1}{2 B_{T-1}} \frac{e^{\eta \scale{R}_{T}^k - \eta^2 \scale{V}_{T}^k} -1}{\eta} \dif \eta + \ln \frac{B_T}{B_{T-1}} ~=~ \Phi_T + \ln \frac{B_T}{B_{T-1}} .$$ [**of Lemma \[lem:small.is.good\]**]{} For any $\epsilon \in [0, \frac{1}{2 B_{T-1}}]$, we may split the potential as follows $$\Phi_T ~=~ \underbrace{ \sum_k \pi_k \int_0^\epsilon \frac{e^{\eta \scale{R}_T^k - \eta^2 \scale{V}_T^k} -1}{\eta} \dif \eta }_{\fd Q_1} + \underbrace{ \sum_k \pi_k \int_\epsilon^\frac{1}{2 B_{T-1}} \frac{e^{\eta \scale{R}_T^k - \eta^2 \scale{V}_T^k} -1}{\eta} \dif \eta }_{\fd Q_2} .$$ For convenience, let us introduce $\scale b_t \df \max_k \abs{\scale r_t^k} = \frac{B_{t-1}}{B_t} b_t$ and abbreviate $\scale S_T \df \sum_{t=1}^T \scale b_t$. To bound the left term $Q_1$ from below, we use $e^x -1 \ge x$. Then combined with $\scale R_T^k \ge - \scale S_T$ and $\scale V_T^k \le \sum_{t=1}^{T-1} \scale b_t^2 \le B_{T-1} \scale S_T$ we find $$Q_1 ~\ge~ \sum_k \pi_k \int_0^\epsilon \scale{R}_T^k - \eta \scale{V}_T^k \dif \eta ~\ge~ - \del*{\epsilon + \frac{\epsilon^2}{2} B_{T-1}} \scale S_T .$$ For the right term $Q_2$, we use KL duality to find $$\begin{aligned} Q_2 &~=~ \sum_k \pi_k \int_\epsilon^\frac{1}{2 B_{T-1}} \frac{e^{\eta \scale{R}_T^k - \eta^2 \scale{V}_T^k}}{\eta} \dif \eta + \ln \del*{2 B_{T-1} \epsilon} \\ &~\ge~ e^{-\operatorname{KL}\delcc*{\rho}{\pi}} \int_\epsilon^\frac{1}{2 B_{T-1}} \frac{e^{\eta \scale{R}_T^\rho - \eta^2 \scale{V}_T^\rho}}{\eta} \dif \eta + \ln \del*{2 B_{T-1} \epsilon} \end{aligned}$$ Way pick the admissible $\epsilon = \frac{1}{2(\scale S_T + B_{T-1})}$ for which $\del*{\epsilon + \frac{\epsilon^2}{2} B_{T-1}} \scale S_T \le \frac{1}{2}$, and find $$\Phi_T ~\ge~ e^{-\operatorname{KL}\delcc*{\rho}{\pi}} \int_\epsilon^\frac{1}{2 B_{T-1}} \frac{e^{\eta \scale{R}_T^\rho - \eta^2 \scale{V}_T^\rho}}{\eta} \dif \eta - \frac{1}{2} - \ln \del*{1+ \frac{\scale S_T}{B_{T-1}}}$$ which we may reorganise to $$Q_3 \df \ln \int_\frac{1}{2(\scale S_T + B_{T-1})}^\frac{1}{2 B_{T-1}} \frac{e^{\eta \scale{R}_T^\rho - \eta^2 \scale{V}_T^\rho}}{\eta} \dif \eta ~\le~ \operatorname{KL}\delcc*{\rho}{\pi} + \ln \del*{ \Phi_T + \frac{1}{2} + \ln \del*{1+ \frac{\scale S_T}{B_{T-1}}} } .$$ The argument to bound the integral in $Q_3$ splits in 3 cases. Let us abbreviate $R \equiv \scale R_T^\rho$ and $V \equiv \scale V_T^\rho$. Let $\hat \eta = \frac{R}{2 V}$ be the maximiser of $\eta \to \eta R - \eta^2 V$. 1. First the important case, where $\intcc{\hat \eta - \frac{1}{\sqrt{2 V}}, \hat \eta} \subseteq \intcc{ \frac{1}{2(\scale S_T + B_{T+1})}, \frac{1}{2 B_{T-1}}}$. Then $$\begin{aligned} Q_3 &~\ge~ \ln \int_{\hat \eta - \frac{1}{\sqrt{2 V}}}^{\hat \eta} \frac{e^{\eta R - \eta^2 V}}{\eta} \dif \eta ~\ge~ \ln \int_{\hat \eta - \frac{1}{\sqrt{2 V}}}^{\hat \eta} \frac{e^{(\hat \eta - \frac{1}{\sqrt{2 V}}) R - (\hat \eta - \frac{1}{\sqrt{2 V}})^2 V}}{\eta} \dif \eta \\ &~=~ (\hat \eta - \frac{1}{\sqrt{2 V}}) R - (\hat \eta - \frac{1}{\sqrt{2 V}})^2 V + \ln \ln \frac{\hat \eta}{\hat \eta - \frac{1}{\sqrt{2 V}}} \\ &~=~ \frac{R^2}{4 V} - \frac{1}{2} + \ln \ln \frac{1}{1 - \frac{\sqrt{2 V}}{R}} ~\ge~ \frac{1}{2} \del*{\frac{R}{\sqrt{2 V}} -1}^2 \end{aligned}$$ where the last inequality uses $\ln \ln \frac{1}{1 - \frac{1}{x}} \ge 1 - x$ for $x \ge 1$, which can be easily verified by a one-dimensional plot. We conclude $$R ~\le~ \sqrt{2 V} \del*{ 1+ \sqrt{ 2 \operatorname{KL}\delcc*{\rho}{\pi} + 2 \ln \del*{ \Phi_T + \frac{1}{2} + \ln\del*{1+ \frac {\scale S_T}{B_{T-1}}} } } } .$$ 2. Then in the case where $\hat \eta - \frac{1}{\sqrt{2 V}} < \frac{1}{\scale S_T}$, we have $$R ~<~ \sqrt{2 V} + \frac{2 V}{\scale S_T} ~\le~ \sqrt{2 V} + 2 B_{T-1}$$ and we are done again. 3. We come to the final case where $\hat \eta > \frac{1}{2 B_{T-1}}$, meaning that $R > \frac{V}{B_{T-1}}$. Here we use that for any $u \in \intcc{\frac{1}{2(\scale S_T + B_{T-1})}, \frac{1}{2 B_{T-1}}}$ $$Q_3 ~\ge~ \ln \int_u^\frac{1}{2 B_{T-1}} \frac{e^{u R - u^2 V}}{\eta} \dif \eta ~\ge~ u R (1 - u B_{T-1}) + \ln \ln \frac{1}{2 u B_{T-1}}$$ and hence $$R \le \frac{ Q_3 - \ln \ln \frac{1}{2 u B_{T-1}} }{ u (1 - u B_{T-1}) } .$$ Picking the feasible $u = \frac{5 - \sqrt{5}}{10 B_{T-1}}$ and using $ - \ln \ln \frac{5}{5 - \sqrt{5}} \le \ln 2$ results in $$R ~\le~ 5 B_{T-1} \del*{ \operatorname{KL}\delcc*{\rho}{\pi} + \ln \del*{ \Phi_T + \frac{1}{2} + \ln \del*{1+ \frac{\scale S_T}{B_{T-1}}} } + \ln 2 } .$$ Finally, using the fact that $$\frac{\scale S_T}{B_{T-1}}= \frac{1}{B_{T-1}} \sum_{t=1}^{T} \frac{B_{t-1}}{B_t} b_t \leq 1+ \sum_{t=1}^{T-1} \frac{b_t}{B_t}$$ concludes the proof. iin [1,...,10]{} [ (ni) at (i,0) ; ]{} (begin) at (-1,0); (dots) at (0,0)[$\dots$]{}; ; ; ; ; (6.5,-1) – (6.5,+1) node\[above\] [final restart]{}; (3.5,-1) – (3.5,+1) node\[above\] [penultimate restart]{}; (6.5+.05,-1) – (10-.05,-1) node \[pos=0.5,anchor=north,yshift=-2em\] [$\sqrt{V}$ bound]{}; (3.5+.05,-1) – (6.5-.05,-1) node \[pos=0.5,anchor=north,yshift=-2em\] [$\sqrt{V}$ bound]{}; (-1+.05,-1) – (3.5-.05,-1) node \[pos=0.5,anchor=north,yshift=-2em\] [tiny]{}; (6.5,-1.5) to \[bend left=80\] (1.25,-2.5); at (3.5,-3) [implies]{}; [**of Theorem \[blackboxreduction0\]**]{} The idea of the proof is to analyse the rounds in three parts, as shown in Figure \[fig:bd.strategy\]. For comparator $\rho \in \triangle_K$, $B>0$ and $\tau_1,\tau_2 \in \mathbb{N}$ such that $\tau_1 < \tau_2$, we define the regret $R^{\rho}_{(\tau_1,\tau_2]}$ and variance $V^{\rho}_{(\tau_1,\tau_2]}$ of [<span style="font-variant:small-caps;">Squint+C</span>]{} started at round $\tau_1+1$ (with input $B_{\tau_1}$) and terminated after round $\tau_2$ by $$\begin{gathered} R^{\rho}_{(\tau_1,\tau_2]} \coloneqq \sum_{t=\tau_1+1}^{\tau_2} \mathbb{E}_{\rho(k) } \left[r^{k}_t\right], \quad V^{\rho}_{(\tau_1,\tau_2]} \coloneqq \sum_{t=\tau_1+1}^{\tau_2} \mathbb{E}_{\rho(k) } \left[(r^{k}_t)^2\right]. \end{gathered}$$ We also define $$\begin{aligned} \Gamma^{\rho}_{(\tau_1,\tau_2]} ~\df~ \operatorname{KL}\delcc*{\rho}{\pi} + \ln \del*{ \ln \sum_{t=1}^{\tau_2-1} \frac{b_t}{B_t} + \frac{1}{2} + \ln \left(2+ \sum_{t=\tau_1+1}^{\tau_2-1} \frac{b_{t}}{B_t} \right) }. \end{aligned}$$ If we assume that $\frac{B_{\tau_2-1}}{B_{\tau_1}} \leq \sum_{t=1}^{\tau_2-1} \frac{b_t}{B_t}$ (this corresponds to the case when the restart condition in line \[line:runmetagrad\] of Algorithm \[bb1alg\] is not triggered at round $\tau_2-1$), then from Lemma \[lem:pot.is.small\] the potential function $\Phi_{\tau_2}$, can be bounded by $$\begin{aligned} \label{boundedpotent} \Phi_{\tau_2} \leq \ln \frac{B_{\tau_2-1}}{B_{\tau_1}}\leq \sum_{t=1}^{\tau_2-1} \frac{b_t}{B_t}.\end{aligned}$$ Using this together with Lemma \[lem:small.is.good\] and , we get the following bound on the regret for [<span style="font-variant:small-caps;">Squint+L</span>]{} : $$\begin{gathered} R^{\rho}_{(\tau_1,\tau_2]} \leq \sqrt{2 V_{(\tau_1,\tau_2]}^\rho} \del*{ 1+ \sqrt{2 \Gamma_{(\tau_1,\tau_2]}^{\rho}} } + 5 B_{\tau_2} \del*{\Gamma_{(\tau_1,\tau_2]}^{\rho}+ \ln 2} +B_{\tau_2}. \end{gathered}$$ Now assume without loss of generality that $b_1 \neq 0$. Then the regret of [<span style="font-variant:small-caps;">Squint+L</span>]{} in round one is bounded by $B_1 \leq B_T$, and [<span style="font-variant:small-caps;">Squint+C</span>]{} is started for the first time in round $t=2$ with input $B=B_1$. Now suppose first that the restart condition in line \[line:runmetagrad\] of Algorithm \[bb1alg\] is never triggered, which means that $\frac{B_t}{B_1}\leq \sum_{s=1}^t \frac{b_s}{B_s}$ for all rounds $t=2,\ldots,T$. Then for any comparator distribution $\rho \in \triangle_K$, the result follows from Lemma \[lem:small.is.good\] and the fact that $V^{\rho}_{(1:T]} \leq V_T^{\rho}$ and the fact that $\Gamma^{\rho}_{(1:T]} \leq \Gamma_T^{\rho}$. Alternatively, suppose there is at least one restart. Then let $1 \leq \tau_1 < \tau_2< T$ be such that $(\tau_1,\tau_2]$ and $(\tau_2,T]$ are the two intervals over which the last two runs of [<span style="font-variant:small-caps;">Squint+C</span>]{} occurred. We invoke Theorem \[lem:small.is.good\] separately for both these intervals to bound $$\begin{aligned} R_{(\tau_1,T]}^{\rho} &\leq \sqrt{2 V_{(\tau_1,\tau_2]}^\rho} \del*{ 1+ \sqrt{2 \Gamma_{(\tau_1,\tau_2]}^{\rho}} } + 5 B_{\tau_2} \del*{\Gamma_{(\tau_1,\tau_2]}^{\rho}+ \ln 2} +B_{\tau_2}. \nonumber\\ &\quad+ \sqrt{2 V_{(\tau_2,T]}^\rho} \del*{ 1+ \sqrt{2 \Gamma_{(\tau_2,T]}^{\rho}} } + 5 B_{T} \del*{\Gamma_{(\tau_2,T]}^{\rho}+ \ln 2} +B_{T}. \nonumber\\ &\leq 2 \sqrt{ V_{(\tau_1,T]}^\rho} \del*{ 1+ \sqrt{2 \Gamma_{(\tau_1,T]}^{\rho}} } + 10 B_{T} \del*{\Gamma_{(\tau_1,T]}^{\rho}+ \ln 2} +2B_T, \label{seclast} \\ & \leq 2 \sqrt{ V_T^\rho} \del*{ 1+ \sqrt{2 \Gamma_T^{\rho}} } + 10 B_{T} \del*{\Gamma_T^{\rho}+ \ln 2} +2B_T.\nonumber\end{aligned}$$ where in we used the fact that $\sqrt{x} +\sqrt{y}\leq \sqrt{2x+2y}$. If there is exactly one restart, then this implies the desired result. If there are multiple restarts, then the proof is completed by bounding the contribution to the regret of all rounds $2,\ldots,\tau_1$ by $$\begin{aligned} R_{(1,\tau_1]}^\u \leq \sum_{t=2}^{\tau_1} b_t \leq B_{\tau_1}\sum_{t=1}^{\tau_1} \frac{b_t}{B_t} \leq B_{\tau_1}\sum_{t=1}^{\tau_2} \frac{b_t}{B_t} < B_{\tau_2} \leq B_T,\end{aligned}$$ where the second to last inequality holds because there was a restart at $t=\tau_2$. Finally, by bounding the instantaneous regret from the first round by $B_T$, we obtain the desired result. Proofs of Section \[MetaC\] {#MetaGrad2proofs} =========================== [**of Lemma \[relatingtheregret\]**]{} Let $\bm{u} \in \mathcal{U}$, and denote $r^{\bm{u}}_t \coloneqq {\langle{\widehat{\bm{u}}}_t-\bm{u},{\operatorname{grad}_}t\rangle}$ and $\bar{r}^{\bm{u}}_t \coloneqq {\langle{\widehat{\bm{u}}}_t-\bm{u},\bar{\bm{g}}_t\rangle}$. We have $$\begin{aligned} \label{eq:ashoknew} \tilde{R}_T^{\bm{u}} - \scale R_T^{\bm{u}} ~\df~ \sum_{t=1}^T \del*{r_t^{\bm{u}}- \scale r_t^{\bm{u}}} ~=~ \sum_{t=1}^T \del*{B_t - B_{t-1}} \frac{r_t^{\bm{u}}}{B_t} ~\le~ B_T - B_0 . \end{aligned}$$ where to get to the last inequality we used Cauchy Schwarz inequality and the fact that $\mathcal{U}$ has diameter $D$, which imply that $|r^{\bm{u}}_t| \leq B_t$. [**of Lemma \[lemmameta\]**]{} Let $t\geq 1$. To simplify notation, we denote $\bar{r}_s^{\eta} \coloneqq {\langle{\widehat{\bm{u}}}_s - {\widehat{\bm{u}}}^{\eta}_s,\bar{\bm{g}}_s\rangle}$, for $\bm{u}\in \mathcal{U}$ and $s\in \mathbb{N}$. By appealing to the prod-bound, we have $$\begin{aligned} \Phi_{t+1} &~=~ \pi(\mathcal{G}_{t+1}\setminus \mathcal{A}_{t+1}) + \sum_{\eta\in \mathcal{A}_{t+1}} w^{\eta}_{t+1} \del*{e^{\eta \bar{r}^{\eta}_{t+1} - \eta (\bar{r}^{\eta}_{t+1})^2}-1} + \sum_{\eta\in \mathcal{A}_{t+1}} w^{\eta}_{t+1} \\ &~\le~ \pi(\mathcal{G}_{t+1}\setminus \mathcal{A}_{t+1}) + \sum_{\eta\in \mathcal{A}_{t+1}} w^{\eta}_{t+1} \eta \bar{r}^{\eta}_{t+1} + \sum_{\eta\in \mathcal{A}_{t+1}} w^{\eta}_{t+1}\end{aligned}$$ Now by $$\sum_{\eta\in \mathcal{A}_{t+1}} w^{\eta}_{t+1} \eta \bar{r}^{\eta}_{t+1} ~=~ \sum_{\eta\in \mathcal{A}_{t+1}} \eta w^{\eta}_{t+1} ({\widehat{\bm{u}}}_{t+1} - {\widehat{\bm{u}}}^{\eta}_{t+1})^{\operatorname{\intercal}} \bar{\bm{g}}_t ~=~ 0.$$ Moreover, by definition of $\mathcal G_t$ and $\mathcal A_t$, $$\begin{aligned} & \pi(\mathcal{G}_{t+1}\setminus \mathcal{A}_{t+1}) + \sum_{\eta\in \mathcal{A}_{t+1}} w^{\eta}_{t+1} ~=~ \pi(\set*{\eta \in \mathcal{G}_{t+1} : s_\eta > t}) + \sum_{\eta \in \mathcal G_{t+1} : s_\eta \le t} w^{\eta}_{t+1} \\ &~\le~ \pi(\set*{\eta \in \mathcal{G}_{t} : s_\eta > t}) + \sum_{\eta \in \mathcal G_{t} : s_\eta \le t} w^{\eta}_{t+1} ~=~ \pi(\set*{\eta \in \mathcal{G}_{t} : s_\eta \ge t}) + \sum_{\eta \in \mathcal G_{t} : s_\eta < t} w^{\eta}_{t+1} \\ &~=~ \pi(\mathcal{G}_{t} \setminus \mathcal A_t) + \sum_{\eta \in \mathcal A_{t}} w^{\eta}_{t+1} ~=~ \Phi_t .\end{aligned}$$ Where we used that $w_{s_\eta+1}^\eta = \pi(\eta)$. Finally, as $\mathcal A_0 = \emptyset$ and $\mathcal G_0 = \mathcal G$, we find $\Phi_0 = \pi(\mathcal{G}) = 1$. [**of Theorem \[naivemeta\]**]{} Throughout this proof we will deal with slaves $\eta \in \mathcal G_T \setminus \mathcal A_T$ that are provisioned but not active yet by time $T$, and we will interpret their $s_\eta = T$ for uniform treatment, even though technically all we know from is that $s_\eta \ge T$. First due to Lemma \[lemmameta\], we have $\Phi_T\leq 1$, where $\Phi_T$ is the potential defined in . Taking logarithms and rearranging, we find $$\begin{aligned} \forall \eta \in \mathcal{G}_T,\quad -\sum_{t=s_{\eta}+1}^T \bar{f}_t({\widehat{\bm{u}}}^{\eta}_t,\eta) \leq - \ln \pi(\eta). \label{boundmaster}\end{aligned}$$ On the other hand, every slave $\eta\in \mathcal{G}_T$ guarantees the following regret for the rounds $t=s_\eta+1,\dots,T$ (see @Erven2016 [Lemma 5]): $$\begin{aligned} \sum_{t=s_\eta+1}^T \left( \bar{f}_t({\widehat{\bm{u}}}^{\eta}_t,\eta) - \bar{f}_t(\bm{u},\eta) \right) & \leq \ln \det\left(\mathbf{I} + 2 \eta^2 D^2 (\bar{\mathbf{M}}_T -\bar{\mathbf{M}}_{s_\eta}) \right) +\tfrac{{\left\lVert\bm{u}\right\rVert}^2}{2 D^2}, \nonumber \\ &\leq d\ln\left( 1 +\tfrac{2 D^2}{25 d B^2_{T-1}}{\operatorname{tr}}{\bar{\mathbf{M}}_T} \right)+\tfrac{{\left\lVert\bm{u}\right\rVert}^2}{2 D^2},\label{boundslave}\end{aligned}$$ where in we used concavity of $\ln\det$, $\bar{\mathbf{M}}_{s_\eta} \succeq \bm{0}$ and the fact that $\eta \in \mathcal{G}_T \subset \left[0,\frac{1}{5 B_{T-1}}\right]$. We then invert the wakeup condition at time $s_\eta-1$ to infer $$\begin{aligned} -\sum_{t=1}^{s_\eta} \bar{f}_t(\bm{u},\eta) &\le \eta \sum_{t=1}^{s_\eta} \bar r_t^\u \le \frac{ \sum_{t=1}^{s_\eta-1} \bar r_t^\u + \bar r_{s_\eta}^\u }{D \sum_{t=1}^{s_\eta-1} {\left\lVert\bar{\bm{g}}_t\right\rVert}_2 + B_{s_\eta-1}} \leq 1. \label{eq:lowbound} \end{aligned}$$ Combining the bounds , , and , then dividing through by $\eta$, gives: $$\begin{aligned} \forall \eta\in \mathcal{G}_T, \quad {\bar{R}}^{\bm{u}}_T \leq \eta {\bar{V}}^{\u}_T +\tfrac{1}{\eta} C_T(\eta),\label{gridpointregret}\end{aligned}$$ where $C_T(\eta) \coloneqq d\ln\left( 1 +\tfrac{2D^2}{25 d B^2_{T-1}}{\operatorname{tr}}{\bar{\mathbf{M}}_T} \right) - \ln \pi(\eta) + 2$. Let $C_T$ be as in the theorem statement and let $\eta_*$ be the estimator defined by $\eta_*\coloneqq \sqrt{C_T/{\bar{V}}^{\u}_T}$. Suppose that $\eta_*\leq \frac{1}{5B_{T-1}}$. By construction of the grid, there exists $\hat{\eta} \in \mathcal{G}_T$ such that $\hat{\eta} \in \left[\eta_*/2, \eta_*\right]$. On the other hand, the estimator $\eta_*$ can be lower bounded by $1/\sqrt{{{\bar{V}}}^{\bm{u}}_T}$, since $C_T \geq 1$. From this lower bound on $\eta_*$ and the fact that there exists $ i\in \mathbb{N}$ such that $2^{- i}/(5B_0) = \hat{\eta} \in \left[\eta_*/2, \eta_*\right]$, we have $2^{-i}/(5B_0) \geq \frac{1}{2\sqrt{{{\bar{V}}}^{\bm{u}}_T}}$. This implies that the prior weight on $\hat{\eta}$ satisfies $$\begin{aligned} \frac{1}{\pi(\hat{\eta})} = ( i +1)( i +2) \leq \left( \log_2 \tfrac{2 \sqrt{ {{\bar{V}}}^{\bm{u}}_T}}{5B_0}+1 \right) \left( \log_2 \tfrac{2 \sqrt{ {{\bar{V}}}^{\bm{u}}_T}}{5B_0}+2\right) \leq \left(\log_2 \tfrac{ \sqrt{{{\bar{V}}}^{\bm{u}}_T}}{B_0}+3\right)^2. \label{priorbound} \end{aligned}$$ Note also that from the fact that $1/\sqrt{{{\bar{V}}}^{\bm{u}}_T} \leq \eta_* \leq 1/(5B_{T-1})\leq 1/(5B_0)$, we have $\sqrt{{{\bar{V}}}^{\bm{u}}_T} /B_0 \geq 2$. This fact combined with , implies that $C_T(\hat{\eta})\leq C_T$, where $C_T$ is as in the statement of the theorem. Plugging $\hat{\eta}$ into and using the fact that $ \hat{\eta} \in\left[\eta_*/2, \eta_*\right]$, gives $$\begin{aligned} \label{firstbound} {\bar{R}}^{\bm{u}}_{T} \leq \hat{\eta} {\bar{V}}^{\bm{u}}_T + \tfrac{1}{\hat{\eta}} C_T(\hat{\eta}) \leq \eta_* {\bar{V}}^{\bm{u}}_T + \tfrac{2}{\eta_*} C_T =3 \sqrt{{\bar{V}}^{\bm{u}}_T C_T}.\end{aligned}$$ Now suppose that $\eta_*>\frac{1}{5B_{T-1}}$. Let $\hat{\eta} \coloneqq \max \mathcal{G}_T \geq \frac{1}{10B_{T-1}}$, where the last inequality follows by construction of $\mathcal{G}_T$. Note that in this case $\frac{1}{\pi(\hat{\eta})} \leq (\log_2 \frac{2B_{T-1}}{B_0}+1)(\log_2 \frac{2B_{T-1}}{B_0}+2)$, and thus, we still have $C_T(\hat{\eta})\leq C_T$. Plugging $\hat{\eta}$ into and using the assumption on $\eta_*$, we obtain $$\begin{aligned} \label{secondbound} {\bar{R}}^{\bm{u}}_{T} \leq \hat{\eta} {\bar{V}}_T^{\bm{u}} + \tfrac{1}{\hat{\eta}} C_T(\hat{\eta}) \leq \hat{\eta} {\bar{V}}_T^{\bm{u}} + \tfrac{1}{\hat{\eta}} C_T \leq 15 B_TC_T.\end{aligned}$$ By combining and , we get the desired result. [**of Theorem \[blackboxreduction1\]**]{} \[naivemetarestartproof\] Assume without loss of generality that $b_1 \neq 0$. Then the regret of [[<span style="font-variant:small-caps;">MetaGrad+L</span>]{}]{} in round one is bounded by $B_1 \leq B_T$, and [[<span style="font-variant:small-caps;">MetaGrad+C</span>]{}]{} is started for the first time in round $t=2$ with parameter $B=B_1$. Let $V_{(1:T]}^\u$ and $C_{(1:T]}$ represent the quantities denoted by $V^{\bm{u}}_T$ and $C_T$ in Theorem \[naivemeta\] but measured on rounds $2,\ldots,T$. Now suppose first that the restart condition in line \[line:runmetagrad\] of Algorithm \[bb1alg\] is never triggered, which means that $$\begin{aligned} \frac{B_t}{B_1}\leq \sum_{s=1}^t \frac{b_s}{B_s}, \quad \text{ for all rounds $t=2,\dots,T$} \label{norestart} \end{aligned}$$ Then the result follows from Theorem \[naivemeta\] and $$\begin{aligned} V_{(1:T]}^\u &\leq V_T^\u,\\ C_{(1:T]}, &= d\ln\left(\frac{27}{25} + \frac{2}{25d} \frac{\sum_{t=2}^{T-1} b_t^2 }{B^2_{T-1}} \right) + 2 \ln \left( \log^+_2 \frac{\sqrt{\sum_{t=2}^T b_t^2}}{B_1} +3 \right) + 2,\nonumber \\ &\leq d\ln\left( \frac{27}{25} + \frac{2}{25d} \frac{\sum_{t=2}^{T-1} b_t^2}{B^2_{T-1}}\right) + 2 \ln \left( \log^+_2 \sqrt{\sum_{t=2}^T \left(\sum_{s=1}^t \frac{b_s}{B_s}\right)^2} +3 \right) + 2, \label{usethis}\\ & \leq \Gamma_T,\nonumber\end{aligned}$$ where in , we used . Alternatively, suppose there is at least one restart. Then let $1 \leq \tau_1 < \tau_2< T$ be such that $(\tau_1,\tau_2]$ and $(\tau_2,T]$ are the two intervals over which the last two runs of [[<span style="font-variant:small-caps;">MetaGrad+C</span>]{}]{} occurred. We invoke Theorem \[naivemeta\] separately for both these intervals to bound $$\begin{aligned} R_{(\tau_1,T]}^\u &\leq 3\sqrt{V_{(\tau_1,\tau_2]}^\u C_{(\tau_1,\tau_2]}} + 15 B_T C_{(\tau_1,\tau_2]} + B_{\tau_2}\\ &\quad+ 3\sqrt{V_{(\tau_2,T]}^\u C_{(\tau_2,T]}} + 15 B_T C_{(\tau_2,T]} + B_T\\ &\leq 3\sqrt{V_{(\tau_1,\tau_2]}^\u \Gamma_T/2} + 3\sqrt{V_{(\tau_2,T]}^\u \Gamma_T/2} + 15 B_T \Gamma_T + 2 B_T\\ &\leq 3\sqrt{V_{(\tau_1,T]}^\u \Gamma_T} + 15 B_T \Gamma_T + 2 B_T,\end{aligned}$$ where a subscript $(\tau_1,\tau_2]$ indicates a quantity measured only on rounds $\tau_1+1,\ldots,\tau_2$ and the last inequality uses $\sqrt{x} + \sqrt{y} \leq \sqrt{2x + 2y}$. If there is exactly one restart, then this implies the desired result. If there are multiple restarts, then the proof is completed by bounding the contribution to the regret of all rounds $2,\ldots,\tau_1$ by $$\begin{aligned} R_{(1,\tau_1]}^\u \leq \sum_{t=2}^{\tau_1} b_t \leq B_{\tau_1}\sum_{t=1}^{\tau_1} \frac{b_t}{B_t} \leq B_{\tau_1}\sum_{t=1}^{\tau_2} \frac{b_t}{B_t} < B_{\tau_2} \leq B_T,\end{aligned}$$ where the second to last inequality holds because there was a restart at $t=\tau_2$. Finally, by bounding the instantaneous regret from the first round by $B_T$, we obtain the desired result. Proofs of Section \[four\] {#fourproof} ========================== [**of Lemma \[redquad\]**]{} We use the Lagrangian multiplier to solve . To this end let $$\begin{aligned} \mathcal{L}(\bm{u},\mu) \coloneqq \left(\bm{u}_{t+1}^{\eta}- \bm{u} \right)^{\operatorname{\intercal}}\left( \mathbf{\Sigma}^{\eta}_{t+1}\right)^{-1} \left(\bm{u}_{t+1}^{\eta}- \bm{u} \right) +\mu (\bm{u}^{\operatorname{\intercal}} \bm{u} -D^2). \end{aligned}$$ Setting $\frac{\partial \mathcal{L}}{\partial \bm{u}} =0$ implies that $2 \left( \mathbf{\Sigma}^{\eta}_{t+1}\right)^{-1} \left( \bm{u}-\bm{u}_{t+1}^{\eta} \right) +2\mu \bm{u}=0$. After rearranging, this becomes $$\begin{aligned} \bm{u} &= \left( \left(\mu + \tfrac{1}{D^2} \right) \mathbf{I} + 2 \eta^2 (\bar{\mathbf{M}}_t - \bar{\mathbf{M}}_{s_{\eta}}) \right)^{-1} \left(\mathbf{\Sigma}^{\eta}_{t+1}\right)^{-1} \bm{u}^{\eta}_t, \label{solution} \\ & = \mathbf{Q}^{\operatorname{\intercal}}_t \left(x \mathbf{I} +2 \eta^2 (\mathbf{\Lambda}_t-\mathbf{\Lambda}_{s_{\eta}}) \right)^{-1} \mathbf{Q}_t \bm{v}^{\eta}_{t+1}, \nonumber\end{aligned}$$ where we set $x\coloneqq \mu +\frac{1}{D^2}$. The result follows by observing that $\bm{u}^{\operatorname{\intercal}} \bm{u} = D^2/4 \iff \rho_t^{\eta}(x)=D^2/4$. [**of Theorem \[reductionbound\]**]{} Let $\mathring{R}^{\bm{u}}_T \coloneqq \sum_{t=1}^T {\langle{\widehat{\bm{u}}}_t - \bm{u},\mathring{\bm{g}}_t\rangle}$ and $\mathring{V}^{\bm{u}}_T \coloneqq \sum_{t=1}^T {\langle{\widehat{\bm{u}}}_t - \bm{u},\mathring{\bm{g}}_t\rangle}^2$ be the pseudo-regret and variance of Algorithm \[OCOGeneral\]. From Theorem \[naivemeta\], the bound on the pseudo-regret $\tilde{R}^{\bm{u}}_T =\sum_{i=1}^T {\langle{\widehat{\bm{u}}}_t -\bm{u},{\bm{g}}_t\rangle}$ with respect to ${V}_T^{\bm{u}} =\sum_{t=1}^T {\langle{\widehat{\bm{u}}}_t -\bm{u},{\bm{g}}_t\rangle}^2$ for the [<span style="font-variant:small-caps;">MetaGrad+C</span>]{} subroutine in Algorithm \[OCOGeneral\], can be written as $$\begin{aligned} \forall \bm{u}\in \mathcal{U}, \forall \eta >0, \quad \eta {\tilde{R}}^{\bm{u}}_T - \eta^2 {V}^{\bm{u}}_T \leq \tfrac{9}{4}{\Gamma}_T + 15 \eta B_T \left( {\Gamma}_T + \tfrac{1}{15} \right), \label{compbound}\end{aligned}$$ where ${\Gamma}_T^\u \coloneqq d\ln\left(\frac{27}{25} + \frac{2 \sum_{t=1}^{T-1} {b}_t^2}{25d {B}^2_{T-1}}\right) + 2 \ln \left( \log^+_2 \frac{\sqrt{\sum_{t=1}^T {b}^2_t}}{B} +3 \right) + 2$. As in the proof of [@cutkosky2018 Proposition 1], we have $$\begin{aligned} \label{cutkow} {\langle{\widehat{\bm{w}}}_t - \bm{u},\mathring{\bm{g}}_t\rangle} \leq 2 \mathring{\ell}_t({\widehat{\bm{u}}}_t) - 2\mathring{\ell}_t(\bm{u}),\end{aligned}$$ where ${\widehat{\bm{w}}}_t = \Pi_{\mathcal{U}}({\widehat{\bm{u}}}_t)$ is the prediction of Algorithm \[OCOGeneral\] at round $t$ and $\mathring{\ell}_t$ is the function defined by $\mathring{\ell}_t(\bm{u}) \coloneqq \frac{1}{2}\left( {\langle\mathring{\bm{g}}_t,\bm{u}\rangle} + {\left\lVert\mathring{\bm{g}}_t\right\rVert} {\operatorname{d}}_{\mathcal{U}}(\bm{u})\right)$. By the convexity of $\mathring{\ell}_t$ and the fact that ${\bm{g}}_t \in \partial \mathring{\ell}_t({\widehat{\bm{u}}}_t)$, we have $$\begin{aligned} {\langle{\widehat{\bm{u}}}_t-\bm{u},{\bm{g}}_t\rangle} \geq \mathring{\ell}_t({\widehat{\bm{u}}}_t) - \mathring{\ell}_t(\bm{u})\geq \tfrac{1}{2}{\langle{\widehat{\bm{w}}}_t - \bm{u},\mathring{\bm{g}}_t\rangle},\label{useful} \end{aligned}$$ where the right-most inequality follows from . Since the function $x\mapsto x - x^2$ is strictly increasing on $]-\infty, \frac{1}{2}[$, implies $$\begin{aligned} \tfrac{\eta}{2}{\langle{\widehat{\bm{w}}}_t - \bm{u},\mathring{\bm{g}}_t\rangle} - \tfrac{\eta^2}{4}{\langle{\widehat{\bm{w}}}_t - \bm{u},\mathring{\bm{g}}_t\rangle} ^2 \leq \eta {\langle{\widehat{\bm{u}}}_t - \bm{u},{\bm{g}}_t\rangle} - \eta^2 {\langle{\widehat{\bm{u}}}_t - \bm{u},{\bm{g}}_t\rangle}, \nonumber \end{aligned}$$ for all $\eta \in \left]0, \frac{1}{2B_{T}} \right]$. Summing over $t=1..T$ and using the bound , we get $$\begin{aligned} \forall \eta \in \left]0, \tfrac{1}{2B_T} \right],\quad \mathring{R}^{\bm{u}}_T-\tfrac{\eta}{2} \mathring{V}^{\bm{u}}_T \leq \tilde{R}^{\bm{u}}_T- {V}^{\bm{u}}_T \leq \tfrac{9}{4\eta}{\Gamma}_T + 15 B_T \left( {\Gamma}_T + \tfrac{1}{15} \right),\label{compbound11}\end{aligned}$$ which leads to, for all $\eta \in \left]0, \tfrac{1}{2B_{T}} \right]$, $$\begin{aligned} \mathring{R}^{\bm{u}}_T \leq \tfrac{\eta}{2} \mathring{V}^{\bm{u}}_T+ \tfrac{9}{4\eta}{\Gamma}_T + 15 B_T \left( {\Gamma}_T + \tfrac{1}{15} \right),\label{compbound2}\end{aligned}$$ The $\eta$ which minimizes the right-hand side of is given by $\eta_*\coloneqq \sqrt{\frac{9{\Gamma}_T}{2\mathring{V}^{\bm{u}}_T}}$. We consider two cases; suppose first that $\hat{\eta}\leq \frac{1}{2B_T}$. Then, by setting $\eta =\eta_*$, we have $$\begin{aligned} \label{firstbound2} \tfrac{\eta}{2} \mathring{V}_T^{\bm{u}} + \tfrac{9}{4\eta} {\Gamma}_T =3 \sqrt{\mathring{V}^{\bm{u}}_T {\Gamma}_T}.\end{aligned}$$ Now suppose that $\eta_*>\frac{1}{2B_T}$. Then for $\eta = \frac{1}{2B_T}$ we have $$\begin{aligned} \tfrac{\eta}{2} \mathring{V}_T^{\bm{u}} + \tfrac{9}{4\eta} {\Gamma}_T \leq 9 B_T {\Gamma}_T. \label{secondbound2}\end{aligned}$$ Combining - yields the desired result. [^1]: We slightly abuse terminology here, because the standard definition of a Lipschitz constant requires an upper bound on the gradient norms for any parameters $\w$, not just for $\w = \w_t$, and may therefore be larger. [^2]: Note that <span style="font-variant:small-caps;">AdaHedge</span> is timeless, and so are its refined bounds. Similarly, <span style="font-variant:small-caps;">Squint</span> is timeless (regardless of the prior on the learning rate $\eta$), and so is its CV bound but not its improper bound (which features a $\ln \ln T$). [^3]: @cutkosky2017 show that their algorithm still achieves a regret of order $O(\ln^4 T)$ under a condition they name $\alpha$-acute convexity. The link of the latter to the more common Bernstein condition is unclear. [^4]: While the original algorithm is designed for unbounded OCO, their algorithm can still be used for bounded $\mathcal{U}$ via a simple reduction proposed by [@cutkosky2018] [^5]: We learned this technique from Ashok Cutkosky [^6]: The predictions of the slaves outside $\mathcal{A}_t$ do not appear anywhere in the description or analysis of the algorithm. Alternatively, we may think of each slave $\eta$ as operating with $\eta_t=0$ in the first $s_\eta$ rounds and with $\eta_t=\eta$ afterwards. The presence of the factor $\eta$ in renders the master oblivious to inactive slaves.

ArXiv

--- abstract: | We present 21-cm [H[I]{}]{} line and 13-cm continuum observations, obtained with the Australian Long Baseline Array, of the Seyfert 2 galaxy IC 5063. This object appears to be one of the best examples of Seyfert galaxies where shocks produced by the radio plasma jet influence both the radio as well as the near-infrared emission. The picture resulting from the new observations of IC 5063 confirms and completes the one derived from previous Australia Telescope Compact Array (ATCA) lower resolution observations. A strong interaction between the radio plasma ejected from the nucleus and a molecular cloud of the ISM is occurring at the position of the western hot spot, about 0.6 kpc from the active nucleus. Because of this interaction, the gas is swept up forming, around the radio lobe, a cocoon-like structure where the gas is moving at high speed. Due to this, part of the molecular gas is dissociated and becomes neutral or even ionised if the UV continuum produced by the shocks is hard and powerful enough. In the 21-cm [H[I]{}]{} line new data, we detect only part of the strong blue-shifted [H[I]{}]{} absorption that was previously observed with the ATCA at lower resolution. In particular, the main component detected in the VLBI absorption profile corresponds to the most blue-shifted component in the ATCA data, with a central velocity of 2786 [km s$^{-1}$]{} and therefore blue-shifted $\sim$614 [km s$^{-1}$]{} with respect to the systemic velocity. Its peak optical depth is 5.4%. The corresponding column density of the detected absorption, for a spin temperature of 100 K, is $N_{\rm HI} \sim 2 \times 10^{21} $atoms cm$^{-2}$. Most of the remaining blue-shifted components detected in the ATCA [H[I]{}]{} absorption profile are now undetected, presumably because this absorption occurs against continuum emission that is resolved out in these high-resolution observations. The [H[I]{}]{} absorption properties observed in IC 5063 appear different from those observed in other Seyfert galaxies, where the [H[I]{}]{} absorption detected is attributed to undisturbed foreground gas associated with the large-scale galaxy disk. In the case of IC 5063, only a small fraction of the absorption can perhaps be due to this. The reason for this can be that the western jet in IC 5063 passes through a particularly rich ISM. Alternatively, because of the relatively strong radio flux produced by this strong interaction, and the high spectral dynamic range of our observations, broad absorption lines of low optical depth as detected in IC 5063 may have remained undetected in other Seyferts that are typically much weaker radio emitters or for which existing data is of poorer quality. author: - 'T.A. Oosterloo, R. Morganti,' - 'A. Tzioumis, J. Reynolds, E. King,' - 'P. McCulloch,' - 'Z. Tsvetanov' title: 'A strong jet/cloud interaction in the Seyfert galaxy IC 5063: VLBI observations' --- Introduction ============ The study of the effects of interactions between the radio plasma ejected from an active nucleus and the interstellar medium (ISM) of the hosting galaxy is presently attracting a lot of interest. In particular, Seyfert and high-redshift radio galaxies appear to be the kind of objects where the effects of such interactions can be very important. They can range from shaping the morphology of the gas in the ISM (with the radio plasma sweeping up material as it advances in the ISM), to the ionisation of the gas itself. While there is little doubt on the presence of such interactions in objects like Seyferts or high-$z$ radio galaxies, the actual importance of these effects in determining the overall characteristics of these sources is still a matter of debate. In some Seyfert galaxies the morphological association between the radio plasma and the optical line-emitting clouds, as well as the presence of disturbed kinematics in these clouds, is striking. In particular, the narrow-line regions (NLR) in Seyfert galaxies (i.e. regions of highly ionised, kinematically complicated gas emission that occupy the central area – up to $\sim 1$ kpc from the nucleus) often appears to form a ‘cocoon’ around the radio continuum emission (see e.g. Wilson 1997 for a review; Capetti et al. 1996; Falcke, Wilson & Simpson (1998) and references therein). Moreover, outflow phenomena are observed in the warm gas of several Seyfert galaxies (see Aoki et al. 1996 for a summary). Thus, the NLRs represent some of the best examples of regions where interaction between the local ISM and the radio plasma takes place and can be studied in detail. The situation appears to be different for the atomic hydrogen. Observations of the [H[I]{}]{} 21-cm line, in absorption, can trace the distribution of this gas in front of the brightest radio components, that are usually observed in the central region of Seyferts (of kpc or sub-kpc size, i.e. [*co-spatial with the NLRs*]{}). Thus, the study of the distribution and kinematics of the [*cold*]{} component of the circumnuclear ISM can nicely complement the optical data. Although [H[I]{}]{} absorption has been detected in a number of Seyfert galaxies (e.g. NGC 4151 Pedlar et al. 1992; NGC 5929, Cole et al. 1998; Mkn 6, Gallimore et al. 1998 ; see also Brinks & Mundell 1996 and Gallimore et al. 1999 and references therein), most of the investigated objects show single localised [H[I]{}]{} absorption components that can be explained as rotating, inclined disks or rings aligned with the outer galaxy disk (Gallimore et al.1999) and only very seldom with gas in a parsec-scale circumnuclear torus (NGC 4151, Mundell et al. 1995). These components are therefore originated by gas that is not in interaction with the radio plasma. However, more complex [Hi]{} absorption profiles that cannot be explained by the above mechanism have been observed in at least one Seyfert galaxy, IC 5063. Australia Telescope Compact Array (ATCA) observations of this galaxy (Morganti, Oosterloo & Tsvetanov 1998, hereafter M98) have revealed a very interesting absorption system with velocities up to $\sim 700$ [km s$^{-1}$]{} blue-shifted with respect to the systemic velocity. In this object, unlike in other Seyfert galaxies, at least some of the observed [H[I]{}]{} absorption is originating from regions of interaction between the radio plasma and the ISM, producing an outflow of the neutral gas. This object, therefore, poses a number of interesting questions as: where is the interaction occurring, what are the physical conditions, why such interaction is not seen more often in neutral gas in other Seyfert galaxies? Previous [H[I]{}]{} observations were limited by low spatial resolution. In this paper we present the results from new VLBI observations aimed at investigating in more detail its nuclear radio structure and locating where the complex [H[I]{}]{} absorption observed with ATCA is really occurring. Throughout the paper we adopt a Hubble constant of $H_\circ = 50$ [km s$^{-1}$]{}, so that 1 arcsec corresponds to 0.32 kpc at the redshift of IC 5063. Summary of the properties of IC 5063 ==================================== IC 5063 is a nearby ($z = 0.0110$) early-type galaxy that hosts a Seyfert 2 nucleus that emits particularly strong at radio wavelengths ($P_{\rm 1.4\,GHz} = 6.3\times 10^{23}$ W Hz$^{-1}$). This object has been recently studied, both in radio continuum at 8 GHz and in the 21-cm line of [Hi]{}, using the ATCA (M98). In the continuum, on the arcsecond scale, we find a linear triple structure (see Fig. 1) of about 4 arcsec size ($\sim 1.3$ kpc), that shows a close spatial correlation with the optical ionised gas, very similar in nature to what is observed in several other Seyfert galaxies (see e.g. Wilson 1997) and indicating that the radio plasma is important in shaping the NLR. In the [H[I]{}]{} 21-cm line, apart from detecting the emission from the large-scale disk of IC 5063, very broad ($\sim 700$ kms$^{-1}$), mainly blue-shifted absorption was detected against the central continuum source. These line observations could only be obtained with $\sim 7$ arcsec resolution, the highest resolution achievable with the ATCA at this wavelength. This resolution is too low to resolve the linear continuum structure detected in the 8-GHz continuum image. However, and what makes this absorption particularly interesting is that we were able to conclude (by a careful analysis of the data, see M98 for the detailed discussion) that at least the most blue-shifted absorption is likely to originate against the western (and brighter) radio knot and not against the central radio feature seen at 8 GHz. The large, blue-shifted velocities observed in the absorption profile make it very unlikely that these motions have a gravitational origin (the most blue-shifted [H[I]{}]{} emission associated with the large-scale [H[I]{}]{} disk occurs at roughly $300$ [km s$^{-1}$]{}with respect to the systemic velocity), and are more likely to be connected to a fast outflow of the ISM caused by an interaction with the radio plasma. The identification of the central radio feature as the core, and hence that the absorption is occurring against the western lobe, is an important element in interpreting the nature of the absorption detected in IC 5063. In the literature, the core of this galaxy has sometimes been identified with the bright western knot (Bransford et al. 1998), however in our opinion there is compelling evidence that the identification of M98 is correct. The superposition of the 8-GHz radio image with an optical [*WFPC2*]{} image available from the [*HST*]{} public archive and with a ground based narrow-band  image suggests that the nucleus coincides with the central radio knot (see Figs 3 and 4 from M98). Although, as usual, there is some freedom in aligning the [*WFPC2*]{} image with the 8-GHz radio image, aligning the western radio knot with the nucleus would require too large a shift. Given that the [*WFPC2*]{} image was taken through the F606W filter, it contains the bright emission lines of $\lambda 5007$, H$\alpha$ and $\lambda\lambda 6548, 6584$, and it gives a good idea of the morphology of the ionised gas. By aligning the nucleus with the central radio knot, a good overall correspondence between the radio morphology and the bright region of optical emission lines is obtained, both in the [*WFPC2*]{} image and the ground-based  image, similar in nature to what is observed in many other Seyfert galaxies. Choosing this alignment, the western radio knot falls right on top of a very bright, unresolved, spike in the [*WFPC2*]{} image, i.e. the western radio knot would also have a counterpart in the [*WFPC2*]{} image. The filamentary morphology of the ionised gas of the region just around this spike is suggestive of an interaction between the radio plasma and the ISM and the identification of the western radio lobe with this feature seems natural. Using optical spectroscopy, Wagner & Appenzeller (1989) found off-centre blue-shifted broad emission lines with similar widths as the detected [H[I]{}]{}absorption at a position 1-2 arcsec west of the nucleus, i.e. coincident with the spike seen in the [*WFPC2*]{} image. This also suggests that at this position a violent interaction is occurring. The identification of the core with the central radio knot has been recently confirmed by Kulkarni et al. (1998) from NICMOS images. Three well resolved knots were detected in the emission lines of \[Fe[II]{}\], Pa$\alpha$ and H$_2$. This emission-line structure shows a direct correspondence with the radio continuum structure. In broad band near IR images they detected a very red point source coincident with the central source seen in the emission lines, consistent with previous suggestions of a dust-obscured active nucleus. The strong \[Fe[II]{}\] and H$_2$ emission are usually interpreted as evidence for fast shocks and the direct correspondence between these regions and the radio emission suggest that shocks associated with the radio jet play a role in the excitation of the emission-line knot. By the same authors, an asymmetry in the H$_2$ distribution was found, with the eastern lobe showing a much weaker emission than the western lobe. This asymmetry can be explained, e.g, if an excess of molecular gas is present on the western side (for example, if the radio jet has struck a molecular cloud). In the optical, IC5063 shows a very high-excitation emission line spectrum (including \[Fe[VII]{}\]$\lambda\lambda$5721, 6087; Colina, Spark & Macchetto 1991). The high-excitation lines are detected within 1 - 1.5 arcsec on both sides of the nucleus, about the distance between the radio core and both the lobes. These lines indicate the presence of a powerful and hard ionising continuum in the general area of the nucleus and the radio knots in IC 5063. We have estimated (M98) the energy flux in the radio plasma to be an order of magnitude smaller than the energy flux emitted in emission lines. The shocks associated with the jet-ISM interaction are, therefore, unlikely to account for the overall ionisation and the NLR must be, at least partly, photoionised by the nucleus, unless the lobe plasma contains a significant thermal component (Bicknell et al. 1998). VLBI observations ================= IC 5063 was observed with the Australian Long Baseline Array (LBA) initially in continuum at 13 cm (2.3 GHz), followed by spectral-line observations at the frequency corresponding to the redshifted [H[I]{}]{}. The 13-cm observations in June 1996 comprised five stations; Parkes (64 m), Mopra (22 m), the Australia Telescope Compact Array (5$\times$22-m dishes as tied-array), the Mount Pleasant 26-m antenna of the University of Tasmania and the Tidbinbilla 70-m antenna of the Canberra Deep Space Communications Complex (CDSCC) near Canberra. The observations used the S2 recording system to record a single 16 MHz band in right-circular polarisation and were correlated at the LBA S2 VLBI correlator of the Australia Telescope National Facility at Marsfield, Sydney. The 13-cm data were edited and calibrated using the AIPS processing system. After this, the data were exported to DIFMAP (Sheperd 1997) for model fitting and imaging. The final image is presented in Fig. 2 and was made with uniform weighting. Although the observations were not phase-referenced, absolute position calibration for the 13-cm LBA image was extracted from the delay and rate data, allowing the radio image to be fixed at the $\sim$0.1 arcsec level in each coordinate, adequate for registration with other images. The 21-cm observations were made in September 1997 at the redshifted [H[I]{}]{}frequency of 1407 MHz, recording 16 MHz bandwidths in each circular polarisation. The same array was used, except for the Tidbinbilla 70-m antenna, which has no 21 cm capability. Correlation was in spectral-line mode with 256 spectral channels on each baseline and polarisation. The editing and part of the calibration of the 21-cm line data was done in AIPS and then the data were transfered to MIRIAD (Sault, Teuben & Wright 1995) for the bandpass calibration. The calibration of the bandpass was done using additional observations of the strong calibrators PKS 1921–293 and PKS 0537–441. Problems were encountered at Mopra which limited the usefulness of those data. It proved not possible to image the source from the final dataset and instead a simpler analysis using the time-averaged baseline spectra was employed. The sub-kpc structure ===================== The radio continuum morphology ------------------------------ The final 13-cm image, shown in Fig. 2, has a beam of $\sim56\times 15\,$mas in position angle (p.a.) $-40^\circ$. The r.m.s. noise is $\sim 0.7$ mJy beam$^{-1}$. The total flux is 210 mJy. Because of the high accuracy of the astrometry of this VLBI image, we know that the observed structure corresponds (as expected) to the brighter, western, lobe observed in the 8-GHz ATCA image (see Fig. 2). It is therefore situated at about 0.6 kpc from the nucleus. The image shows that the lobe appears to have a relatively bright peak ($77$ mJy beam$^{-1}$) and some extended emission to the north-east in p.a. $\sim 40^\circ$ of total size of about 50 mas (or $\sim 16$ pc). The p.a. is quite different from the p.a. of the arcsecond sized structure seen in the ATCA 8-GHz data (p.a. $\sim 295^\circ$), so there appears to be structure perpendicular to the main radio structure. These kind of distortions are often seen in the radio structure of Seyfert galaxies (e.g. Falcke et al. 1998) and could perhaps result from the interaction of the radio plasma with the environment. From our data, a brightness temperature of $\rm{T_{B}}\sim 10^{7}$K can be inferred for the VLBI source. This brightness temperature is several orders of magnitude less than the typical values seen in milliarcsecond AGN cores or inner (pc-scale) jets that typically have brightness temperatures between $10^{9}$ and $10^{11}$K. However, this temperature is quite commonly found for radio knots detected in Seyfert galaxies (e.g. knot C in NGC 1068, Roy et al. 1998). Unfortunately, we do not have a spectral index of this region on the VLBI scale. The overall spectral index inferred from the ATCA 8.6 and 1.4-GHz images is steep, $\alpha\sim -1$, and indeed consistent with a radio lobe or jet. However, unless a detailed multi-frequency spectral index study can be carried out, it is difficult to derive conclusions from this result alone given the complexity often observed in the spectral index of the central regions of Seyfert galaxies. In summary, we can conclude that the radio morphology, the spectrum and the brightness temperature of the VLBI source are consistent with what expected in a radio lobe. The [H[I]{}]{} absorption ------------------------- As mentioned above, because from the 21-cm line observations useful data could only be obtained on the Parkes-ATCA baseline, we will present only an time-integrated spectral profile of the [H[I]{}]{} on this baseline. These data correspond to a spatial scale of about 0.1 arcsec. Fig. 3 shows the continuum-weighted [H[I]{}]{} absorption profile. Heliocentric, optical velocities are used. For comparison, the spectrum obtained from the previous ATCA observations (with much lower spatial resolution) is superimposed (dashed line). In Fig. 3 we have also indicated the range of the velocities observed as measured for the [H[I]{}]{} emission of the large-scale disk of IC 5063, as well as the systemic velocity of the galaxy of $3400$ km s$^{-1}$ as derived from the kinematics of the [H[I]{}]{} emission. The r.m.s. limit to the optical depth is $\sim 0.3$%. Fig. 3 shows that a strong absorption signal is detected against the VLBI source. Since from the 13-cm data it followed that the VLBI source corresponds to the western radio lobe, these data now confirm what was believed to be the case from the ATCA data, namely that the absorption is occurring against the western radio lobe. Fig. 4 shows the same data as in Fig. 3, except that both profiles have been normalised to the same optical depth for the most blue-shifted component. Figs 3 and 4 show quite clearly that the shape of the absorption profile obtained at the high resolution of the VLBI data is quite different in character than that obtained with the ATCA. While in the ATCA data the absorption is relatively uniform in velocity, in the VLBI spectrum the most blue-shifted component is clearly the dominant one. This shows that the most blue-shifted absorption is occurring against a compact radio source, while the absorption at lower velocities is against a more diffuse source. Component ($A$) has a central velocity of 2786 [km s$^{-1}$]{}, over 600 [km s$^{-1}$]{} blue-shifted with respect to the systemic velocity (3400 [km s$^{-1}$]{}), with its bluest wing extending to about 2650 [km s$^{-1}$]{}, or –750 [km s$^{-1}$]{}relative to the systemic velocity. Component $A$ corresponds to the most blue-shifted component found in the ATCA profile, as is illustrated in Fig. 4. At slightly less blue-shifted velocities, but still outside the range of velocities observed in emission, the VLBI data show a second component ($B$). The absorption with velocities within the range of the [H[I]{}]{} emission, as detected in the ATCA profile, is only partly detected in the VLBI spectrum with component $C$. No absorption is detect in the velocity range 3000-3200 [km s$^{-1}$]{}. Hence, the absorption seen in the ATCA data at velocities above 3000 [km s$^{-1}$]{} has become much less prominent compared to the more blue-shifted absorption. Note that this effect is probably even stronger than the data shows, since the low resolution of the ATCA will have caused some filling of the absorption with emission of the [H[I]{}]{} disk and the ‘true’ absorption is likely to be stronger at these velocities. The ATCA spectrum also showed a faint red-shifted absorption component that is perhaps also detected in the VLBI spectrum. The column density $N_{\rm HI}$ of the obscuring neutral hydrogen is given by $N_{\rm HI} = 1.823\times 10^{18} T_s \int \tau dv$ cm $^{-2}$ where $T_{\rm s}$ is the spin temperature of the electron. Assuming a spin temperature of 100 K we derive a column density of $\sim 1.7 \times 10^{21}$ atoms cm$^{-2}$ for the components $A$ and $B$ and a column density of $\sim 2.5 \times 10^{20}$ atoms cm$^{-2}$ for the component $C$. The main source of uncertainty for the derived column density comes from the assumption in the value of the spin temperature. The presence of a strong continuum source near the [H[I]{}]{} gas can make the radiative excitation of the [H[I]{}]{} hyperfine state to dominate over the, usually more important, collisional excitation (see e.g.Bahcall & Ekers 1969). Gallimore et al. (1999) argue that the [H[I]{}]{} causing the absorption against Seyferts jets is in general at too high densities ($\sim$$10^5$ cm$^{-3}$) for these effects to be relevant. However, the argument used by Gallimore et al. applies to [H[I]{}]{} in pressure equilibrium with the NLR, while the absorbing gas in Seyferts in general is [*not*]{} co-spatial with the NLR, but is at larger radii. Because of this the density of the absorbing gas is lower, but the regions are also further removed from the central engine and the spin temperature approaches the kinetic temperature at much lower densities. In our model for IC 5063, the [H[I]{}]{} causing the most blue-shifted absorption is the skin of a molecular cloud that is being stripped off by the jet (see also §5). Given that typical densities in molecular clouds are in the range $10^4$ - $10^6$ cm$^{-3}$, this is an upper limit to the density of the absorbing gas. But given the large velocities involved, the actual density of the blue-shifted gas could be substantially lower. The effects on the excitation of the fine-structure line by the local radiation field were already discussed by M98 for the case of IC 5063, where it was concluded that these effects are perhaps important. The column density derived by Kulkarni et al.  from the NICMOS observation ($N_{\rm HI} \sim 5 \times 10^{21}$ atoms cm$^{-2}$) is slightly higher than our estimate based on a $T_{\rm spin}$ of 100 K, also suggesting that perhaps the spin temperature is somewhat higher than 100 K. For $T_{\rm spin} = 100$ K the derived column density is much lower than the value of $\sim10^{23}$atoms cm$^{-2}$ found from X-ray data (Koyama et al. 1992). H[ **I**]{}, H$_2$ and radio plasma: a possible scenario for the interaction ============================================================================ Summarising, the main result from our new observations is that with the improved spatial resolution, the absorption at velocities outside the range allowed by the rotational kinematics of the large-scale [H[I]{}]{} disk has become much stronger, while the absorption in the range of velocities of the [H[I]{}]{}disk has become much less prominent. From the 13-cm radio continuum VLBI data we have been able to image only the western part of the source observed by the ATCA, while the remaining structure is resolved out. Thus, all this confirms and completes the picture we derived from the ATCA data, namely that [*a strong interaction between the radio plasma and the ISM is occurring at the position of the western radio lobe*]{}. Fig. 5 gives a schematic diagram of what we believe is happening in the western lobe of IC 5063. Following the results from NICMOS (Kulkarni et al.  1998), it is likely that the asymmetry observed in the brightness of the H$_2$ (western side brighter than the eastern side) may be explained by an excess of molecular gas on the western side. Thus, the radio plasma ejected from the nucleus appears to interact directly with such a molecular cloud. Because of this interaction, the jet is drilling a hole in the dense ISM, sweeping up the gas and forming a cocoon-like structure around the radio lobe where the gas is moving at high speed and an outflow of gas is created. The increased ultraviolet radiation due to the presence of shocks generated from the interaction, can dissociate part of the molecular gas. This creates neutral hydrogen or even ionised gas if the UV continuum produced by the shocks is hard and powerful enough. The region of ionised gas would correspond to the part of the cocoon closer to where the interaction is occurring, possibly corresponding to both the shocked gas and to the precursor. The complex kinematics of the emission lines in this region observed from the optical emission lines (Wagner & Appenzeller 1989) is consistent with this. As for the neutral gas, we will observe only the component in front of the radio continuum and therefore, as effect of the outflow produced by the interaction, we will observe only the blue-shifted component. The most blue-shifted component will be seen against the hot spot where the interaction is most intense. Somewhat away from this location, the [H[I]{}]{} will driven out by the expanding cocoon, but since this is away from the hot spot, this will occur at lower velocities. Moreover, the radio continuum emission from this region is also more extended compared to the small-sized hot spot. Hence the VLBI observation do not detect the absorption at lower velocities, but only the highest velocities against the hot spot (as illustrated in Fig.5). The origin of the H$_2$ emission can be related to UV or shocks (Draine, Roberge & Dalgano 1983, Sternberg & Dalgano 1989). Although we are not able to distinguish between these two mechanisms, this scenario suggest that there should be in IC 5063 a strong shock component. The H$_2$ emission observed by NICMOS would therefore come from the very dense region (again due to the compression of the gas associated with the interaction) of the molecular cloud. As we noticed above, not all the components observed in the ATCA data are also visible in the VLBI data. A possible explanation for this is that the components that are missing from the VLBI [H[I]{}]{} absorption are against continuum emission that is resolved out in our VLBI data, indicating that the cocoon of shocked gas is quite extended and cover at least all the western radio lobe. Alternatively, part of the absorption undetected in the VLBI spectrum can be due to the large-scale disk associated with the dust-lane and also seen in [H[I]{}]{} in emission, although the continuity of the ATCA absorption profile does not suggest this. By looking at the velocity field derived for this disk from the [H[I]{}]{} emission observations (see Fig. 5 in M98) we can see as the western side is the approaching side, therefore showing a blue-shifted velocity relative to the systemic. This means that the large-scale disk being in front of the hot spot could be responsible for component $C$, but that it cannot explain the weak red-shifted component unless non circular motions are present in the foreground gas associated with the dust lane. In M98 we hypothesised that the red-shifted component could be associated with a nuclear torus/disk. This was motivated by the fact that the width of the red-shifted component appeared to be similar to the width of the CO profile as observed by Wiklind et al. (1995). However, there seems to be no indication of detection of the nuclear component from the visibility of the continuum associated with the 21-cm data (and by extrapolating the arcsec data, the core flux is probably too weak to be detected and, even more, to produce an absorption) so this hypothesis has to be ruled out. A final possibility is that, apart from the bulk outflow, turbulent motions produced in the shocked region can give rise to clouds with red-shifted velocity. Comparison with other Seyfert galaxies ====================================== How does IC 5063 compare with other Seyferts galaxies? The results on IC 5063 confirm the more general results obtained by Gallimore et al. (1999) on a sample of Seyfert galaxies, that the [H[I]{}]{} absorption is not occurring against the core and that the absorbing gas in Seyferts does not trace (except for NGC 4151) the pc-scale gas. In the galaxies studied by Gallimore et al., the absorption is occurring at a few hundred parsec from the core and is caused by the inner regions of, or gas associated with, the large-scale [H[I]{}]{} disks. This is also happening in IC 5063. The important difference is that in IC 5063 the jet is physically strongly interacting with this [H[I]{}]{} disk, causing the fast outflow observed for the absorbing material. This makes IC 5063 unique. Only component $C$ would exactly fit the scenario proposed by Gallimore et al. It is quite likely a gas cloud at large radius (given its column density), unrelated to the interaction, projected in front of the radio hot spot. One obvious question is why such kind of absorption (i.e. broad blue-shifted absorption) is so rare. Are the physical conditions in IC 5063 rare, or is there an observational bias? Some arguments suggest that IC 5063 is a special case. IC 5063 is a very strong radio emitter compared to other Seyfert galaxies. Most of the strong radio flux of IC 5063 is produced in the western radio knot, indicating that the interaction is particularly strong. Also the fact that the western radio knot is much brighter than the eastern one indicates that the conditions near the western lobe are special. It has been noted that IC 5063 belongs to a group of “radio-excess infrared galaxies” (Roy & Norris 1997), objects that could represent active galactic nuclei hosted in an unusual environment or perhaps dust-enshrouded quasars or their progenitors. It appears that the jet-cloud interaction in IC 5063 is particularly strong. This would make IC 5063 a very suitable object for further detailed studies of jet-cloud interactions in Seyfert galaxies. One factor is of course that in order to create the strong interaction and the very broad absorption, the jet has to lie more or less in the plane of the [H[I]{}]{} disk. Only then can the jet have a strong interaction with the ISM. The orientation of the AGN in Seyferts is not correlated with that of the large-scale disk, so the effects seen in IC 5063 should then only occur in a minority of cases. On the other hand, interactions between the radio plasma and the ISM are common in Seyferts, given that in many Seyfert galaxies, very large velocity widths of the optical emission lines are observed in the NLR (e.g. Aoki et al. 1996 and references therein). Perhaps the high sensitivity in $\tau$ of our observations also plays a role. IC 5063 is a strong radio source compared to other Seyfert galaxies that are between 10 and 100 times weaker at radio wavelengths. Therefore only [H[I]{}]{} absorption with much higher optical depth can be observed against those objects. For example, the Seyfert 2 galaxy NGC 5929 shows a striking morphological similarity (both in the optical and radio) with IC 5063. However, the peak of the radio emission in NGC 5929 is only 24 mJy beam$^{-1}$, so in this object absorption of a few percent would not be detectable with the noise level of the current observations (Cole et al. 1998). For almost all the galaxies in the sample studied by Gallimore et al. (1999) is the sensitivity not enough to have been able to detect faint, broad absorption like in IC 5063. Moreover, in order to detect broad profiles of the level as in IC 5063 even in strong sources, good spectral dynamic range is required, which is not always easy to obtain (e.g. NGC 1068; Gallimore et al. 1999). It is quite well possible that more cases like IC 5063 will be found if more sensitive observations are performed. Conclusions =========== Using the Australian Long Baseline Array, we have detected a compact radio source of about of 50 mas (or $\sim 16$ pc) in size (at 13 cm) in the Seyfert galaxy IC 5063. Because of the high positional accuracy of these measurements, we can unambiguously identify this radio knot with the western radio lobe. The hot spot is extended in a direction almost perpendicular to the radio jet. In 21-cm line observations, we detect absorption very much blue-shifted ($\sim$700 km s$^{-1}$) with respect to the systemic velocity. Together with the 13-cm observations, this confirms that the [H[I]{}]{} absorption is not taking place against the core, but that it is against the western radio knot. At the position of the western radio knot a very strong interaction must be occurring between the radio jet and the ISM. Various arguments suggest that this interaction is particularly strong compared to other Seyfert galaxies. This makes IC 5063 a good candidate for studying the physics of jet-cloud interactions in Seyfert galaxies. The HI absorption characteristics of IC 5063 are only partially consistent with other absorption studies of Seyfert galaxies. The major absorption component is occuring against the bright radio knot offset a few hundred parsecs from the core. 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ArXiv

--- abstract: 'We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to the Taylor series expansion approach used by Bramble, Dupont and Tomée [@BrDuTh72] in the context of Nitsche’s method and, in the case of *inf–sup* stable multiplier methods, prove a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.' author: - Erik Burman - Peter Hansbo - 'Mats G. Larson' date: - - title: Dirichlet Boundary Value Correction using Lagrange Multipliers --- Introduction ============ In this contribution we develop a modified Lagrange multiplier method based on the idea of boundary value correction originally proposed for standard finite element methods on an approximate domain in [@BrDuTh72] and further developed in [@Du74]. More recently boundary value correction have been developed for cut and immersed finite element methods [@BuHaLa18; @BuHaLa18b; @BBCL18; @MaSc18; @MaSc18b]. Using the closest point mapping to the exact boundary, or an approximation thereof, the boundary condition on the exact boundary may be weakly enforced using multipliers on the boundary of the approximate domain. Of particular practical importance in this context is the fact that we may use a piecewise linear approximation of the boundary, which is very convenient from a computational point of view since the geometric computations are simple in this case and a piecewise linear distance function may be used to construct the discrete domain. We prove optimal order a priori error estimates, in the energy and $L^2$ norms, in terms of the error in the boundary approximation and the meshsize. The proof utilizes the a priori error estimates derived in [@BuHaLa18] for the cut boundary value corrected Nitsche method together with a bound, which shows that the solution to the boundary value corrected Lagrange method is close to the corresponding Nitsche solution for which optimal bounds are available. We obtain optimal order convergence for polynomial approximation up to order 3 of the solution. Note that without boundary correction one typically requires $O(h^{p+1})$ accuracy in the $L^\infty$ norm for the approximation of the domain, which leads to significantly more involved computations on the cut elements for higher order elements, see [@JoLa13]. We present numerical results illustrating our theoretical findings. The outline of the paper is as follows: In Section 2 we formulate the model problem and our method, in Section 3 we present our theoretical analysis, in Section 4 we discuss the choice of finite element spaces in cut finite element methods, in Section 5 we present the numerical results, and finally in Section 6 we include some concluding remarks. Model problem and method ======================== The domain ---------- Let $\Omega$ be a domain in $\mathbb{R}^d$ with smooth boundary $\partial \Omega$ and exterior unit normal ${\boldsymbol n}$. We let $\varrho$ be the signed distance function, negative on the inside and positive on the outside, to $\partial \Omega$ and we let $U_\delta(\partial \Omega)$ be the tubular neighborhood $\{{\boldsymbol x}\in {\mathbb{R}}^d : |\varrho({\boldsymbol x})| < \delta\}$ of $\partial \Omega$. Then there is a constant $\delta_0>0$ such that the closest point mapping ${\boldsymbol p}({\boldsymbol x}):U_{\delta_0}(\partial \Omega) \rightarrow \partial \Omega$ is well defined and we have the identity ${\boldsymbol p}({\boldsymbol x}) = {\boldsymbol x}- \varrho({\boldsymbol x}){\boldsymbol n}({\boldsymbol p}({\boldsymbol x}))$. We assume that $\delta_0$ is chosen small enough that ${\boldsymbol p}({\boldsymbol x})$ is well defined. See [@GilTru01], Section 14.6 for further details on distance functions. The model problem ----------------- We consider the problem: find $u:\Omega \rightarrow {\mathbb{R}}$ such that $$\begin{aligned} {2}\label{eq:poissoninterior_strong} -\Delta u &= f \qquad && \text{in $\Omega$} \\ \label{eq:poissonbc_strong} u &= g \qquad && \text{on $\partial\Omega$}\end{aligned}$$ where $f\in H^{-1}(\Omega)$ and $g\in H^{1/2}(\partial \Omega)$ are given data. It follows from the Lax-Milgram Lemma that there exists a unique solution to this problem and we also have the elliptic regularity estimate $$\label{eq:ellipticregularity} \|u\|_{H^{s+2}(\Omega)} \lesssim \|f\|_{H^s(\Omega)}, \qquad s \geq -1.$$ Here and below we use the notation $\lesssim$ to denote less or equal up to a constant. Using a Lagrange multiplier to enforce the boundary condition we can write the weak form of – as: find $(u,\lambda) \in H^1(\Omega) \times H^{-1/2}(\partial\Omega)$ such that $$\begin{aligned} {2}\label{eq:poissoninterior} \int_{\Omega}\nabla u \cdot\nabla v \,\text{d}\Omega +\int_{\partial\Omega}\lambda\, v\, \text{d}s &= \int_{\Omega}f v\, \text{d}\Omega\qquad \forall v\in H^1(\Omega) \\ \label{eq:poissonbc} \int_{\partial\Omega}u\, \mu\, \text{d}s &= \int_{\partial\Omega}g\, \mu\, \text{d}s\qquad \forall \mu\in H^{-1/2}(\partial\Omega)\end{aligned}$$ The mesh and the discrete domain -------------------------------- Let ${\mathcal{K}}_{h}, h \in (0,h_0]$, be a family of quasiuniform partitions, with mesh parameter $h$, of $\Omega$ into shape regular triangles or tetrahedra $K$. The partitions induce discrete polygonal approximations $\Omega_h = \cup_{K \in {\mathcal{K}}_h}K$, $h \in (0,h_0]$, of $\Omega$. We assume neither $\Omega_h \subset \Omega$ nor $\Omega \subset \Omega_h$, instead the accuracy with which $\Omega_h$ approximates $\Omega$ will be crucial. To each $ \Omega_h$ is associated a discrete unit normal ${\boldsymbol n}_h$ and a discrete signed distance $\varrho_h:\partial \Omega_h \rightarrow \mathbb{R}$, such that if ${\boldsymbol p}_h({\boldsymbol x},\varsigma):={\boldsymbol x}+ \varsigma {\boldsymbol n}_h({\boldsymbol x})$ then ${\boldsymbol p}_h({\boldsymbol x},\varrho_h({\boldsymbol x})) \in \partial \Omega$ for all ${\boldsymbol x}\in \partial \Omega_h$. We will also assume that ${\boldsymbol p}_h({\boldsymbol x},\varsigma) \in U_{\delta_0}(\Omega):=U_{\delta_0}(\partial\Omega)\cup\Omega$ for all ${\boldsymbol x}\in \partial \Omega_h$ and all $\varsigma$ between $0$ and $\varrho_h({\boldsymbol x})$. For conciseness we will drop the second argument of ${\boldsymbol p}_h$ below whenever it takes the value $\varrho_h({\boldsymbol x})$, and thus we have the map $\partial \Omega_h \ni {\boldsymbol x}\mapsto {\boldsymbol p}_h({\boldsymbol x}) \in \partial \Omega$. We assume that the following assumptions are satisfied $$\label{eq:geomassum-a} \delta_h := \| \varrho_h \|_{L^\infty(\partial \Omega_h)} = o(h), \qquad h \in (0,h_0]$$ and $$\label{eq:geomassum-c} \| {\boldsymbol n}_h - {\boldsymbol n}\circ {\boldsymbol p}_h \|_{L^\infty(\partial \Omega_h)} = o(1), \qquad h \in (0,h_0]$$ where $o(\cdot)$ denotes the little ordo. We also assume that $h_0$ is small enough to guarantee that $$\label{eq:geomassum-b} \partial \Omega_h \subset U_{\delta_0}(\partial \Omega), \qquad h\in(0,h_0]$$ and that there exists $M>0$ such for any ${\boldsymbol y}\in U_{\delta_0}(\partial \Omega)$ the equation, find ${\boldsymbol x}\in \partial \Omega_h$ and $ |\varsigma| \leq \delta_h$ such that $$\label{eq:assump_olap} {{\boldsymbol p}}_h({\boldsymbol x},\varsigma) = {\boldsymbol y}$$ has a solution set $\mathcal{P}_h$ with $$\label{eq:card_hyp} \mbox{card}(\mathcal{P}_h) \leq M$$ uniformly in $h$. The rationale of this assumption is to ensure that the image of ${\boldsymbol p}_h$ can not degenerate for vanishing $h$; for more information cf. [@BuHaLa18]. We note that it follows from (\[eq:geomassum-a\]) that $$\label{eq:geomassum-exact-normal} \|\varrho \|_{L^\infty(\partial \Omega_h)} \lesssim \|\varrho_h \|_{L^\infty(\partial \Omega_h)} = o(h)$$ since $|\varrho_h({\boldsymbol x})| \geq |\varrho({\boldsymbol x})|$, ${\boldsymbol x}\in U_{\delta_0}(\partial \Omega)$. We also assume the additional regularity $$\label{eq:residualregularity} f+ \Delta u \in H^{l\textcolor{black}{+\frac12+\epsilon}}(U_{\delta_0}(\Omega))$$ The finite element method ------------------------- ### Boundary value correction The basic idea of the boundary value correction of [@BrDuTh72] is to use a Taylor series at ${\boldsymbol x}\in {\partial\Omega_h}$ in the direction ${\boldsymbol n}_h$, and let this series represent $u_h \vert_{\partial\Omega}$. In the present work we will restrict ourselves to $$\label{def:Taylor} u_h\circ {\boldsymbol p}_h({\boldsymbol x}) \approx u_h({\boldsymbol x}) + \varrho_h({\boldsymbol x}){\boldsymbol n}_h({\boldsymbol x})\cdot\nabla u_h({\boldsymbol x})$$ which is the case of most practical interest. Choosing appropriate discrete spaces $V_h$ and $\Lambda_h$ for the approximation of $u$ and $\lambda$, respectively (particular choices are considered in Section \[sec:numex\]), we thus seek $(u_h,\lambda_h)\in V_h\times\Lambda_h$ such that $$\begin{aligned} {2}\label{eq:poissoninterior_FEM} \int_{\Omega_h}\nabla u_h \cdot\nabla v \,\text{d}\Omega_h +\int_{\partial\Omega_h}\lambda_h\, v\, \text{d}s &= \int_{\Omega_h}f v\, \text{d}\Omega_h\qquad \forall v\in V_h \\ \label{eq:poissonbc_FEM} \int_{\partial\Omega_h}(u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla u_h)\, \mu\, \text{d}s &= \int_{\partial\Omega_h}\tilde{g}\, \mu\, \text{d}s\qquad \forall \mu\in \Lambda_h\end{aligned}$$ where we introduced the notation $\tilde{g}:= g\circ {\boldsymbol p}_h$ for the pullback of $g$ from $\partial \Omega$ to $\partial \Omega_h$. Using Green’s formula we note that the first equation implies that $\lambda_h = -{\boldsymbol n}_h\cdot\nabla u_h$, and therefore we now propose the following modified method: Find $(u_h,\lambda_h)\in V_h\times\Lambda_h$ such that $$\begin{aligned} {2}\label{eq:multinterior} \int_{\Omega_h}\nabla u_h \cdot\nabla v \,\text{d}\Omega_h +\int_{\partial\Omega_h}\lambda_h\, v\, \text{d}s &= \int_{\Omega_h}f v\, \text{d}\Omega_h\qquad \forall v\in V_h \\ \label{eq:multbc} \int_{\partial\Omega_h}u_h\, \mu\, \text{d}s-\int_{\partial\Omega_h} \varrho_h\lambda_h\,\mu\, \text{d}s &= \int_{\partial\Omega}\tilde{g}\, \mu\, \text{d}s\qquad \forall \mu\in \Lambda_h\end{aligned}$$ or $$A(u_h,\lambda_h;v,\mu) = (f,v)_{\Omega_h} + (\tilde{g},\mu)_{\partial\Omega_h}\quad \forall (u_h,\lambda_h)\in V_h\times\Lambda_h\label{eq:mainproblem}$$ where $(\cdot,\cdot)_{M}$ denotes the $L_2$ scalar product over $M$, with $\| \cdot\|_{M}$ the corresponding $L_2$ norm, and $$A(u,\lambda;v,\mu) := (\nabla u ,\nabla v )_{\Omega_h} +(\lambda , v)_{\partial\Omega_h} +( u,\mu)_{\partial\Omega_h} -(\varrho_h\lambda ,\mu)_{\partial\Omega_h}.$$ Relation to Nitsche’s method with boundary value correction ----------------------------------------------------------- Problem (\[eq:mainproblem\]) can equivalently be formulated as finding the stationary points of the Lagrangian $$\mathcal{L}(u,\lambda) := \frac12\|\nabla u\|^2_{\Omega_h} + (\lambda,u)_{\partial\Omega_h}-\|\varrho^{1/2}_h\lambda\|^2_{\partial\Omega_h} -(f,u)_{\Omega_h} - (\tilde{g},\lambda)_{\partial\Omega_h}$$ We now follow [@BuHa17] and add a consistent penalty term and seek stationary points of the augmented Lagrangian $$\mathcal{L}_\text{aug}(u,\lambda) := \mathcal{L}(u,\lambda) + \frac{1}{2} \|\gamma^{1/2}(u-\varrho_h\lambda-\tilde{g})\|^2_{\partial\Omega_h}$$ where $\gamma > 0$ remains to be chosen. The corresponding optimality system is $$\begin{aligned} \nonumber (f,v)_{\Omega_h} + (\tilde{g},\mu)_{\partial\Omega_h} = {}& A(u_h,\lambda_h;v,\mu) +(\gamma (u_h-\varrho_h\lambda_h-\tilde{g}),v)_{\partial\Omega_h}\\ & -(\gamma\varrho_h (u_h-\varrho_h\lambda_h-\tilde{g}),\mu)_{\partial\Omega_h} \end{aligned}$$ Now, formally replacing $\lambda_h$ by $-{\boldsymbol n}_h\cdot\nabla u_h$ and $\mu$ by $-{\boldsymbol n}_h\cdot\nabla v$ we obtain $$\begin{aligned} \nonumber (f,v)_{\Omega_h} - (\tilde{g},{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h} = {}& (\nabla u_h ,\nabla v )_{\Omega_h} -({\boldsymbol n}_h\cdot\nabla u_h,v)_{\partial\Omega_h} \\ \nonumber & -(u_h,{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h} -(\varrho_h {\boldsymbol n}_h\cdot\nabla u_h,{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h}\\ & +(\gamma (u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla u_h-\tilde{g}),v+\varrho_h{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h}\end{aligned}$$ Setting now $\gamma = \gamma_0/h$, with $\gamma_0$ sufficiently large to ensure coercivity, we obtain the symmetrized version of the boundary value corrected Nitsche method proposed in [@BrDuTh72] with optimal convergence up to order $p=3$ assuming $\varrho_h\ge - C h$, for some sufficiently small constant. This means that $\partial \Omega_h$ either has to be a good approximation of $\partial \Omega$, or where it approximates poorly, $\Omega_h$ must approximation $\Omega$ from the inside. For future reference we define this method as: Find $u_h \in V_h$ such that $$\label{eq:Nitform} A_{Nit}(u_h,v_h) = (f,v_h)_{\partial \Omega_h} + (\tilde g, {\boldsymbol n}_h \cdot \nabla v_h)_{\partial \Omega_h}+ (\gamma \tilde g, v_h + \varrho_h {\boldsymbol n}_h \cdot \nabla v_h)_{\partial \Omega_h}$$ for all $v_h \in V_h$. Here the bilinear form is defined by $$\begin{aligned} \nonumber A_{Nit}(w_h,v_h) &:= (\nabla w_h ,\nabla v_h )_{\Omega_h} -({\boldsymbol n}_h\cdot\nabla w_h,v_h+\varrho_h {\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} -(w_h+\varrho_h {\boldsymbol n}_h\cdot\nabla w_h,{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h} \\ &\qquad +(\varrho_h {\boldsymbol n}_h\cdot\nabla w_h,{\boldsymbol n}_h\cdot\nabla v)_{\partial\Omega_h} + (\gamma (w_h+\varrho_h{\boldsymbol n}_h\cdot\nabla w_h,v+\varrho_h{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h}.\end{aligned}$$ Elements of analysis ==================== In this section we will prove some basic results on the stability and the accuracy of the method (\[eq:mainproblem\]). We will restrict ourselves to a discussion of the case $- C h \leq \varrho_h$, for some $C$ small enough. We assume that $\Lambda_h$ is the space of piecewise polynomial functions of order $k-1$ and $V_h$ is the space of continuous piecewise polynomial functions of order $k$, that we will denote $V_h^k$, enriched with higher order bubbles on the faces in $\partial \Omega_h$ so that inf-sup stability holds. The precise condition is given in equation (\[eq:infsup\]) below. For details on stable choices of the multiplier space we refer to [@BM97; @BD98; @KLPV01]. We introduce the triple norm defined on $ H^1(\Omega_h) \times L^2(\partial \Omega_h)$: $${|\mspace{-1mu}|\mspace{-1mu}|}(v,\mu) {|\mspace{-1mu}|\mspace{-1mu}|}:= \|\nabla v\|_{\Omega_h} + \|h^{-\frac12} v\|_{\partial \Omega_h} + \|h^{\frac12} \mu\|_{\partial \Omega_h}.$$ We let $\pi_h:L^2(\partial \Omega_h) \to \Lambda_h$ denote the $L^2$-orthogonal projection and we assume that the bound $$\|v - \pi_h v\|_{\partial \Omega_h} \lesssim h \|\nabla_\partial v\|_{\partial \Omega_h}$$ for all $v \in H^1(\partial \Omega_h)$ and where $\nabla_\partial$ denotes the gradient on the boundary. The formulation (\[eq:mainproblem\]) satisfies the following stability result \[prop:infsup\] Assume that $\varrho_h \ge - C_{\partial \Omega} h$ and that $V_h \times \Lambda_h$ satisfies the inf-sup condition. Then for $C_{\partial \Omega}$ sufficiently small, for all $(y_h,\eta_h) \in V_h \times \Lambda_h$, there exists $(v_h,\mu_h) \in V_h \times \Lambda_h$ such that $${|\mspace{-1mu}|\mspace{-1mu}|}(y_h,\eta_h) {|\mspace{-1mu}|\mspace{-1mu}|}\lesssim A(y_h,\eta_h;v_h,\mu_h)$$ and $${|\mspace{-1mu}|\mspace{-1mu}|}(v_h,\mu_h) {|\mspace{-1mu}|\mspace{-1mu}|}\lesssim {|\mspace{-1mu}|\mspace{-1mu}|}(y_h,\eta_h){|\mspace{-1mu}|\mspace{-1mu}|}.$$ First observe that $$A(y_h,\eta_h;y_h,-\eta_h) = \|\nabla y_h\|^2_{\Omega_h} + (\varrho_h \eta_h, \eta_h)_{\partial \Omega_h}.$$ Then recall that since the space satisfies the inf-sup condition there exists $v_\eta \in V_h$ such that $$\label{eq:infsup} \|h^{\frac12} \eta_h\|_{\partial \Omega_h} \lesssim (\eta_h,v_\eta)_{\partial \Omega_h} \quad \mbox{and} \quad \|\nabla v_\eta\|_{\Omega_h}+\|h^{-\frac12} v_\eta\|_{\partial \Omega_h} \lesssim \|h^{\frac12} \eta_h\|_{\partial \Omega_h}.$$ It follows that for some $c_\eta,\, C_{\partial \Omega_h}$ sufficiently small $$\begin{aligned} \|\nabla y_h\|^2_{\Omega_h}+\|h^{\frac12} \eta_h\|^2_{\partial \Omega} &\lesssim \|\nabla y_h\|^2_{\Omega_h} + (\varrho_h \eta_h, \eta_h)_{\partial \Omega_h} + \|h^{\frac12} \eta_h\|^2_{\partial \Omega} \\ &\lesssim A(y_h,\eta_h;y_h + c_\eta v_\eta,-\eta_h ).\end{aligned}$$ Here we used equation , $$(\varrho_h \eta_h, \eta_h)_{\partial \Omega_h} + \|h^{\frac12} \eta_h\|^2_{\partial \Omega} \ge (1 - C_{\partial \Omega_h}) \|h^{\frac12} \eta_h\|^2_{\partial \Omega}$$ and $$(\nabla y_h, y_h + c_\eta v_\eta)_{\Omega_h} \ge -\frac12 \|\nabla y_h\|^2_{\Omega_h} - 2 c_\eta^2 \|v_\eta\|^2_{\partial \Omega}.$$ Finally let $\mu_y =\pi_h y_h$ and observe that $$\begin{aligned} \nonumber &(y_h, h^{-1} \mu_y)_{\partial \Omega} - (\rho_h \eta_h, h^{-1} \mu_y) _{\partial \Omega} \\ &\qquad \ge \|h^{-\frac12} y_h\|_{\partial \Omega_h}^2 - \|h^{-\frac12} (y_h - \mu_y)\|_{\partial \Omega_h}^2 - \frac12 C_{\partial \Omega_h}^2 \|h^{\frac12} \eta_h\|_{\partial \Omega_h}^2 - \frac12 \|h^{-\frac12} \mu_y\|_{\partial \Omega_h}^2 \\ &\qquad \ge \frac12 \|h^{-\frac12} y_h\|_{\partial \Omega_h}^2 - C^2_t \|\nabla y_h\|^2_{\Omega_h} - \frac12 C_{\partial \Omega}^2 \|h^{\frac12} \eta_h\|_{\partial \Omega_h}^2.\end{aligned}$$ Where we used the approximation property of $\pi_h$ and a trace inequality $$\label{eq:trace} h_K^{\frac12}\|v_h\|_{\partial K} + h_K \|\nabla v_h\|_{K} \lesssim \|v_h\|_K.$$ for all elements $K \in {{\mathcal{K}}_h}$ and polynomials $v_h \in \mathbb{P}(K)$, to show that $$\|h^{-\frac12} (y_h - \mu_y)\|_{\partial \Omega_h} \leq C_t \|\nabla y_h\|_{\Omega_h}.$$ The first claim follows by taking $v_h = y_h + c_ \eta v_\eta$ and $\mu_h = - \eta_h + c_y h^{-1} \mu_y$ with $c_\eta$ and $c_y$ both $O(1)$, sufficiently small and assuming that $C_{\partial \Omega_h}$ is small enough. To conclude the proof we need to show that $${|\mspace{-1mu}|\mspace{-1mu}|}(v_h,\mu_h) {|\mspace{-1mu}|\mspace{-1mu}|}\lesssim {|\mspace{-1mu}|\mspace{-1mu}|}(y_h,\eta_h) {|\mspace{-1mu}|\mspace{-1mu}|}.$$ By the triangle inequality we have $${|\mspace{-1mu}|\mspace{-1mu}|}(v_h,\mu_h) {|\mspace{-1mu}|\mspace{-1mu}|}\leq {|\mspace{-1mu}|\mspace{-1mu}|}(y_h,\eta_h) {|\mspace{-1mu}|\mspace{-1mu}|}+{|\mspace{-1mu}|\mspace{-1mu}|}( c_ \eta v_\eta,c_y h^{-1} \mu_y) {|\mspace{-1mu}|\mspace{-1mu}|}.$$ By definition $${|\mspace{-1mu}|\mspace{-1mu}|}( c_ \eta v_\eta,c_y h^{-1} \mu_y) {|\mspace{-1mu}|\mspace{-1mu}|}=c_ \eta \|\nabla v_\eta\|_{\Omega_h} + c_ \eta\|h^{-\frac12} v_\eta\|_{\partial \Omega_h} + c_y \|h^{-\frac12} \mu_y\|_{\partial \Omega_h}$$ and the proof follows from (\[eq:infsup\]) together with the stability of $\pi_h$ in $L^2$. We will now use this stability result to prove an error estimate. For simplicity we here assume that $\varrho_h>0$, i.e. $\Omega_h \subset \Omega$. Let $u \in H^{k+1}(\Omega)$ denote the solution to (\[eq:poissoninterior\])–(\[eq:poissonbc\]). Let $u_h, \lambda_h \in V_h \times \Lambda_h$ denote the solution of (\[eq:mainproblem\]). Assume that the polynomial order of $V_h$ is $k \in \{1,2,3\}$, with enrichment on the boundary and $\Lambda_h \equiv X_h^{k-1}$. Assume that $V_h \times \Lambda_h$ satisfies (\[eq:infsup\]). Then there holds, with $\tilde \lambda = {\boldsymbol n}_h \cdot \nabla u\vert_{\partial \Omega_h}$, $$\begin{aligned} \label{eq:errorestenergy} {|\mspace{-1mu}|\mspace{-1mu}|}(u - u_h, \tilde \lambda- \lambda_h) {|\mspace{-1mu}|\mspace{-1mu}|}&\lesssim h^{k} \|u \|_{H^{k+1}(\Omega)} + h^{-1/2} \delta_h^{2} \sup_{0\leq t \leq \delta_0} \| D^{k+1} u\|_{L^2(\partial \Omega_t)} \\ \nonumber &\qquad + \textcolor{black}{h^{1/2}} \delta_h^{l+1} \sup_{-\delta_0\leq t < 0} \| D_n^{l} (f + \Delta u)\|_{L^2(\partial \Omega_t)}.\end{aligned}$$ Let $\tilde u_h \in V_h$ denote the solution to (\[eq:Nitform\]). We recall from [@BrDuTh72; @BuHaLa18] that the following error bound holds $$\begin{aligned} \nonumber {|\mspace{-1mu}|\mspace{-1mu}|}(u - \tilde u_h,0) {|\mspace{-1mu}|\mspace{-1mu}|}&+\|h^{\frac12} {\boldsymbol n}_h \cdot \nabla (u - \tilde u_h)\|_{\partial \Omega_h}+ \|h^{-\frac12} (\tilde u_h + \varrho {\boldsymbol n}_h \cdot \nabla \tilde u_h - \tilde g)\|_{\partial \Omega_h} \\ \label{eq:errorNit} &\lesssim h^{k} \|u \|_{H^{k+1}(\Omega)} + h^{-1/2} \varrho_h^{2} \sup_{0\leq t \leq \delta_0} \| D^{k+1} u\|_{L^2(\partial \Omega_t)} \\ \nonumber &\qquad + \textcolor{black}{h^{1/2}} \varrho_h^{l+1} \sup_{-\delta_0\leq t < 0} \| D_n^{l} (f + \Delta u)\|_{L^2(\partial \Omega_t)}.\end{aligned}$$ Let $i_h u$ denote the nodal interpolant of $u$. We then form the discrete errors $e_h = u_h - \tilde u_h$ and $\varsigma_h = \lambda_h - \zeta_h$ for some $\zeta_h \in \Lambda_h$. Using the triangle inequality and we have $${|\mspace{-1mu}|\mspace{-1mu}|}(u - u_h, \tilde \lambda- \lambda_h) {|\mspace{-1mu}|\mspace{-1mu}|}\leq {|\mspace{-1mu}|\mspace{-1mu}|}(u - \tilde u_h, \tilde \lambda- \zeta_h) {|\mspace{-1mu}|\mspace{-1mu}|}+ {|\mspace{-1mu}|\mspace{-1mu}|}(e_h, \varsigma_h) {|\mspace{-1mu}|\mspace{-1mu}|}.$$ Since the first term on the left hand side is bounded by standard interpolation and (\[eq:errorNit\]). We only need to consider the second term. By the stability estimate of Proposition \[prop:infsup\] we have $${|\mspace{-1mu}|\mspace{-1mu}|}(e_h, \varsigma_h) {|\mspace{-1mu}|\mspace{-1mu}|}\lesssim A(e_h,\varsigma_h; v_h,\mu_h).$$ Using the method (\[eq:mainproblem\]) and the definition of $\tilde u_h$ we find that $$\begin{aligned} \label{eq:gal_ortho} A(e_h,\varsigma_h; v_h,\mu_h) & = (f,v_h)_{\Omega_h} + (\tilde g, \mu_h)_{\partial \Omega_h} \\ \nonumber &\qquad - (\nabla \tilde u_h,\nabla v_h) _{\Omega_h}+(\zeta_h,v_h)_{\partial \Omega_h} \\ \nonumber &\qquad -(\tilde u_h, \mu_h)_{\partial \Omega_h} - (\varrho_h\zeta_h, \mu_h) _{\partial \Omega_h}.\end{aligned}$$ The definition of Nitsche’s method (\[eq:Nitform\]) implies the equality $$\begin{aligned} (f,v_h)_{\Omega_h} - (\nabla \tilde u_h ,\nabla v_h )_{\Omega_h} = {}& (\tilde{g},{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} -({\boldsymbol n}_h\cdot\nabla \tilde u_h,v_h)_{\partial\Omega_h} \\ \nonumber & -(\tilde u_h,{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} -(\varrho_h {\boldsymbol n}_h\cdot\nabla \tilde u_h,{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} \\ \nonumber & +(\gamma (\tilde u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla \tilde u_h-\tilde{g}),v_h+\varrho_h{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} \\ = {} & -({\boldsymbol n}_h\cdot\nabla \tilde u_h,v_h)_{\partial\Omega_h} + (\tilde{g} - \tilde u_h - \varrho_h {\boldsymbol n}_h\cdot\nabla \tilde u_h,{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} \\ \nonumber & +(\gamma (\tilde u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla \tilde u_h-\tilde{g}),v_h+\varrho_h{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h}.\end{aligned}$$ Combining then (\[eq:gal\_ortho\]) with (\[eq:Nitform\]) we have $$\begin{aligned} A(e_h,\varsigma_h; v_h,\mu_h) = &(\tilde g-\tilde u_h-\varrho_h \zeta_h, \mu_h)_{\partial \Omega_h} \\ \nonumber &+(\zeta_h-{\boldsymbol n}_h \cdot \nabla \tilde u_h,v_h)_{\partial \Omega_h}\\ \nonumber & + (\tilde{g} - \tilde u_h - \varrho_h {\boldsymbol n}_h\cdot\nabla \tilde u_h,{\boldsymbol n}_h\cdot\nabla v_h)_{\partial\Omega_h} \\ \nonumber & +(\gamma (\tilde u_h+\varrho_h{\boldsymbol n}_h\cdot\nabla \tilde u_h-\tilde{g}),v_h+\varrho_h{\boldsymbol n}_h\cdot\nabla v_h)_{\partial \Omega_h}\\ = & I+II+III+IV.\end{aligned}$$ We will now bound the terms $I-IV$. First note that, $$\begin{aligned} I+III+IV \leq & (\|h^{-\frac12}(\tilde g-\tilde u_h-\varrho_h {\boldsymbol n}_h \nabla\cdot \tilde u_h)\|_{\partial \Omega_h}+\|h^{-\frac12}\varrho_h (\zeta_h- {\boldsymbol n}_h \cdot \nabla \tilde u_h) \|_{\partial \Omega_h}) \\ \nonumber & \times (\| \nabla v_h\|_{\Omega_h} + \|h^{-\frac12} v_h\|_{\partial \Omega_h} + \|h^{\frac12} \mu_h\|_{\partial \Omega_h}).\end{aligned}$$ For term $II$ there holds using Cauchy-Schwarz inequality $$\begin{aligned} II=(\zeta_h-{\boldsymbol n}_h \cdot \nabla \tilde u_h,v_h)_{\partial \Omega_h} \lesssim \|h^{\frac12}(\zeta_h - {\boldsymbol n}_h \cdot\nabla \tilde u_h)\|_{\partial \Omega_h} \|h^{-\frac12} v_h\|_{\partial \Omega_h}.\end{aligned}$$ Summing up we have using the assumption that $\|\rho_h\|_{L^\infty(\partial \Omega_h)} \leq O(h)$, $$\begin{aligned} I+II+III+IV &\leq (\|h^{-\frac12}(\tilde g-\tilde u_h-\varrho_h {\boldsymbol n}_h \cdot \nabla \tilde u_h)\|_{\partial \Omega_h} \\ \nonumber &\qquad +\|h^{\frac12} (\zeta_h- {\boldsymbol n}_h \cdot \nabla \tilde u_h) \|_{\partial \Omega_h}) {|\mspace{-1mu}|\mspace{-1mu}|}(v_h,\mu_h){|\mspace{-1mu}|\mspace{-1mu}|}.\end{aligned}$$ For the term $\|h^{-\frac12}(\tilde g-\tilde u_h-\varrho_h {\boldsymbol n}_h \cdot \nabla \tilde u_h)\|_{\partial \Omega_h}$ we may use the bound (\[eq:errorNit\]). It only remains to bound $\|h^{\frac12} (\zeta_h- {\boldsymbol n}_h \nabla \tilde u_h) \|_{\partial \Omega_h} $. To this end consider $\pi_{k-1} \nabla \tilde u_h \in [X_h]^d$ and let $\zeta_h = {\boldsymbol n}_h \cdot \pi_{k-1} \nabla \tilde u_h\vert_{\partial \Omega_h}$. For this choice we have using a trace inequality $$\begin{aligned} \|h^{\frac12} (\zeta_h- {\boldsymbol n}_h \nabla \tilde u_h) \|_{\partial \Omega_h} = {}&\|h^{\frac12} {\boldsymbol n}_h \cdot (\pi_{k-1} \nabla \tilde u_h- \nabla \tilde u_h) \|_{\partial \Omega_h} \\ \leq & \|\pi_{k-1} \nabla \tilde u_h- \nabla \tilde u_h\|_{\Omega_h}.\end{aligned}$$ To bound the term in the right hand side we add and subtract $\nabla u - \pi_{k-1} \nabla u$ and use the triangle inequality and the stability of the $L^2$-projection $\pi_{k-1}$ to obtain $$\begin{aligned} \|\pi_{k-1} \nabla \tilde u_h- \nabla \tilde u_h\|_{\Omega_h} &\leq \|\pi_{k-1} (\nabla \tilde u_h- \nabla u)\|_{\Omega_h}+ \|\pi_{k-1} \nabla u- \nabla u\|_{\Omega_h}+\|\nabla u- \nabla \tilde u_h\|_{\Omega_h} \\ & \leq \|\pi_{k-1} \nabla u- \nabla u\|_{\Omega_h}+2 \|\nabla u- \nabla \tilde u_h\|_{\Omega_h}.\end{aligned}$$ For the first term in the right hand side we have the approximation bound $$\|\pi_{k-1} \nabla u- \nabla u\|_{\Omega_h} \lesssim h^k \|D^{k+1} u\|_{\Omega_h}.$$ The second term is bounded by (\[eq:errorNit\]). We conclude by applying the second inequality of Proposition \[prop:infsup\]. Remarks on cut finite element methods ===================================== In the context of cut finite element methods the discontinuous multiplier spaces used above can no longer be expected to be stable. It is possible to stabilise the multiplier using Barbosa-Hughes stabilisation. However, fluctuation based multipliers are unlikely to be suitable in this context since the weak consistency of the fluctuations of the multiplier between elements depends on the geometry approximation through the interface normal. Since the method is of interest when the geometry approximation is of relatively low order, this limits the possibility to use fluctuation based stabilisation. For closed smooth boundaries, one may prove inf-sup stability and optimal convergence, without stabilisation, when using continuous approximation of polynomial order less than or equal to $2$, for both the bulk variable and the multiplier provided $\rho_h = O(h^2)$. The approximation order of the interface normal, which is $O(h)$ prohibits higher order convergence if the interface approximation is piecewise affine. For instance, piecewise cubic continuous approximation will not necessarily achieve higher order convergence that the piecewise quadratic approximation. Numerical examples {#sec:numex} ================== We show examples of higher order triangular elements with linearly interpolated boundary and low order rectangular elements with staircase boundary, using discontinuous multiplier spaces. In all examples we define the meshsize $h=1/\sqrt{\text{NNO}}$, where NNO corresponds to the number of nodes of the lowest order FEM on the mesh in question (bilinear or affine). Triangular elements ------------------- We first consider the case of affine triangulations of a ring $1/4\leq r\leq 3/4$, $r=\sqrt{x^2+y^2}$. We use the manufactured solution $u=(r-1/4)(3/4-r)$ and compute the corresponding right–hand side analytically. An elevation of the a typical discrete solution is given in Fig.  \[fig:trisol\]. We use continuous piecewise $P^k$ polynomials, $k=2,3$ for the approximation of $u$, and for the approximation of $\lambda$ we use piecewise $P^{k-1}$ polynomials, discontinous on each element edge on $\Gamma_h$. To ensure [*inf–sup*]{} stability, we add hierarchical $P^{k+1}$ bubbles on each edge in the approximation of $u$. #### Second order elements. In Fig.  \[fig:errtri\] we show the convergence in $L_2(\Omega_h)$ and $H^1(\Omega_h)$ with and without boundary modification. In Fig.  \[fig:errlam\] we show the error in multiplier computed as $\| (-{\boldsymbol n}\cdot\nabla u)\vert_{\partial\Omega_h} - \lambda_h\|_{\partial\Omega_h}$. Optimal order convergence is observed for the modified method, convergence $O(h^3)$ in $L_2(\Omega_h)$ and $O(h^2)$ in $H^1(\Omega_h)$; the multiplier error is approximately $O(h^2)$. #### Third order elements. Next we use continuous piecewise third order polynomials for the approximation of $u$, and for the approximation of $\lambda$ we use piecewise quadratic polynomials, discontinous on each element edge on $\Gamma_h$. In Fig. \[fig:errtri2\] we show the convergence in $L_2(\Omega_h)$ and $H^1(\Omega_h)$ with and without boundary modification. In Fig.  \[fig:errlam2\] we show the error in multiplier computed as above. Optimal order convergence is again observed for the modified method, convergence $O(h^4)$ in $L_2(\Omega_h)$ and $O(h^3)$ in $H^1(\Omega_h)$; the multiplier error is approximately $O(h^3)$. Note that no improvement over $P^2$ approximations can be seen in the unmodified method due to the geometry error being dominant. #### An unstable pairing of spaces. We finally make the observation that our modification has a stabilising influence on the approximation. We try continuous $P^2$ approximations of $u$ and discontinuous $P^2$ approximations of $\lambda$. In this case we get no convergence without the modification due to the violation of the [*inf–sup*]{} condition, whereas with modification we obtain the optimal convergence pattern in $u$ and a stable multiplier convergence given in Fig \[fig:conunstab\]. Rectangular elements -------------------- This example shows that it is possible to achieve optimal convergence even on a staircase boundary. We use a continuous piecewise $Q_1$ approximation on the (affine) rectangles, again enhanced for [*inf–sup*]{}, now by hierarchical $P^2$ bubble function on the boundary edges, together with edgewise constant multipliers on $\Gamma_h$. We use the manufactured solution $u=\sin(x^3)\cos(8y^3)$ on the domain inside the ellipse $x^2/4+y^2 = 1$. Our computational grids consist of elements completely inside this ellipse; a typical coarse grid is shown if Fig. \[fig:coarse\] where we note the staircase boundary. In Fig. \[fig:elevq\] we show elevations of the numerical solutions on a finer grid without and with boundary correction. In Fig. \[fig:errquad\] we show the errors of the unmodified and modified methods. Again we observe optimal order convergence for the modified method, $O(h^2)$ in $L_2(\Omega_h)$ and $O(h)$ in $H^1(\Omega_h)$. Concluding remarks ================== We have introduced a symmetric modification of the Lagrange multiplier approach to satisfying Dirichlet boundary conditions for Poisson’s equation. This novel approach allows for affine approximations of the boundary, and thus affine elements, up to polynomial approximation order 3 without loss of convergence rate as compared to higher order boundary fitted meshes. The modification is easy to implement and only requires that the distance to the exact boundary in the direction of the discrete normal can be easily computed. In fact, the modification stabilises the multiplier method so that unstable pairs of spaces can be used, as long as there is a uniform distance to the boundary. #### Acknowledgement. This research was supported in part by EPSRC, UK, Grant No. EP/P01576X/1, the Swedish Foundation for Strategic Research Grant No. AM13-0029, the Swedish Research Council Grants No. 2013-4708, 2017-03911, 2018-05262, and Swedish strategic research programme eSSENCE. [10]{} F. Ben Belgacem and Y. Maday. The mortar element method for three-dimensional finite elements. , 31(2):289–302, 1997. T. Boiveau, E. Burman, S. Claus, and M. Larson. Fictitious domain method with boundary value correction using penalty-free [N]{}itsche method. , 26(2):77–95, 2018. D. Braess and W. Dahmen. Stability estimates of the mortar finite element method for [$3$]{}-dimensional problems. , 6(4):249–263, 1998. J. H. Bramble, T. Dupont, and V. Thom[é]{}e. Projection methods for [D]{}irichlet’s problem in approximating polygonal domains with boundary-value corrections. , 26:869–879, 1972. E. Burman and P. Hansbo. Deriving robust unfitted finite element methods from augmented [L]{}agrangian formulations. In S. P. A. Bordas, E. Burman, M. G. Larson, and M. A. Olshanskii, editors, [*Geometrically Unfitted Finite Element Methods and Applications*]{}, pages 1–24. Springer, Cham, 2017. E. Burman, P. Hansbo, and M. G. Larson. A cut finite element method with boundary value correction. , 87(310):633–657, 2018. E. Burman, P. Hansbo, and M. G. Larson. A cut finite element method with boundary value correction for the incompressible [S]{}tokes’ equations. In F. A. Radu, K. Kumar, I. Berre, J. M. Nordbotten, and I. S. Pop, editors, [*Numerical Mathematics and Advanced Applications ENUMATH 2017*]{}, pages 183–192. Springer, Cham, 2019. T. Dupont. ${L}_2$ error estimates for projection methods for parabolic equations in approximating domains. In C. de Boor, editor, [*Mathematical Aspects of Finite Elements in Partial Differential Equations*]{}, pages 313–352. Academic Press, New York, 1974. D. Gilbarg and N. S. Trudinger. . Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. A. Johansson and M. G. Larson. A high order discontinuous [G]{}alerkin [N]{}itsche method for elliptic problems with fictitious boundary. , 123(4):607–628, 2013. C. Kim, R. D. Lazarov, J. E. Pasciak, and P. S. Vassilevski. Multiplier spaces for the mortar finite element method in three dimensions. , 39(2):519–538, 2001. A. Main and G. Scovazzi. The shifted boundary method for embedded domain computations. [Part I: Poisson and Stokes]{} problems. , 372:972–995, 2018. A. Main and G. Scovazzi. The shifted boundary method for embedded domain computations. [Part II: Linear advection–diffusion and incompressible Navier–Stokes equations]{}. , 372:996–1026, 2018. ![Elevation of the discrete solution on triangles.[]{data-label="fig:trisol"}](elevation.eps) ![Errors with and without boundary modification, $P^2$ case.[]{data-label="fig:errtri"}](convp2.eps) ![Errors in the multiplier with and without boundary modification, $P^2$ case.[]{data-label="fig:errlam"}](convp2lam.eps) ![Errors with and without boundary modification, $P^3$ case.[]{data-label="fig:errtri2"}](convp3.eps) ![Errors in the multiplier with and without boundary modification, $P^3$ case.[]{data-label="fig:errlam2"}](convp3lam.eps) ![Error plots for the unstable triangular element example.[]{data-label="fig:conunstab"}](conunstab.eps) ![A coarse mesh inside the elliptical domain.[]{data-label="fig:coarse"}](ellipse.eps) ![Elevation of the discrete solution on rectangles for the unmodified (top) and for the modified (bottom) schemes.[]{data-label="fig:elevq"}](elevationunstab.eps) ![Elevation of the discrete solution on rectangles for the unmodified (top) and for the modified (bottom) schemes.[]{data-label="fig:elevq"}](elevationstab.eps) ![Error plots for the rectangular element example.[]{data-label="fig:errquad"}](errorplotquad.eps)

ArXiv

=1 Introduction ============ The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. Our original motivation was to understand the possible obstructions to the third Lie theorem for algebraic Lie pseudogroups. This article is concerned with the simply transitive case. These obstructions should appear in the Galois group of certain connection associated to a Lie algebroid. However, we have written the article in the language of regular and rational parallelisms of algebraic varieties and their symmetries. A theorem of P. Deligne says that any Lie algebra can be realized as a parallelism of an algebraic variety. This is a sort of algebraic version of the third Lie theorem. Notwithstanding, there is one main problem: given an algebraic variety with a parallelism, how far is it from being an algebraic group? Is it possible to conjugate this parallelism with the canonical parallelism of invariant vector fields on an algebraic group? In the analytic context, from the Darboux–Cartan theorem [@sharpe p. 212], a $\mathfrak{g}$-parallelized complex manifold $M$ has a natural $(G,G)$ structure where $G$ is a Lie group with $\mathfrak{lie}(G) = \mathfrak{g}$. The obstruction to be a covering of $G$, as manifold with a $(G,G)$ structure, is contained in a monodromy group [@sharpe p. 130]. In [@Wang], Wang proved that parallelized compact complex manifolds are biholomorphic to quotients of complex Lie groups by discrete cocompact subgroups. This result has been extended by Winkelmann in [@Winkelmann1; @Winkelmann2] for some open complex manifolds. In this article we address the problem of classification of rational parallelisms on algebraic varieties up to birational transformations. Such a classification seems impossible in the algebraic category but we prove a criterion to ensure that a parallelized algebraic variety is isogenous to an algebraic group. Summarizing, we pursue the following plan: We regard infinitesimal symmetries of a rational parallelism as horizontal sections of a connection that we call the reciprocal Lie connection. This connection has a Galois group which is represented as a group of internal automorphisms of a Lie algebra. The obstruction to the algebraic conjugation to an algebraic group, under some assumptions, appear in the Lie algebra of this Galois group. In Section \[section\_parallelisms\] we introduce the basic definitions; several examples of parallelisms are given here. In Section \[section\_lie\] we study the properties of connections on the tangent bundle whose local analytic horizontal sections form a sheaf of Lie algebras of vector fields. We call them [*Lie connections*]{}. They always come by pairs, and they are characterized by having vanishing curvature and constant torsion (Proposition \[prop\_Lie\_char\]). We see that each rational parallelism has an attached pair of Lie connections, one of them with trivial Galois group. We compute the Galois groups of some parallelisms given in examples (Proposition \[prop\_example\]), and prove that any algebraic subgroup of ${\rm PSL}_2(\mathbf C)$ appears as the differential Galois group of a $\mathfrak{sl}_2(\mathbf C)$-parallelism (Theorem \[thm\_SL2\]). Section \[section\_DC\] is devoted to the construction of the isogeny between a $\mathfrak g$-parallelized variety and an algebraic group $G$ whose Lie algebra is $\mathfrak g$. In order to do this, we introduce the Darboux–Cartan connection, a $G$-connection whose horizontal sections are parallelism conjugations. We prove that if $\mathfrak g$ is centerless then the Darboux–Cartan connection and the reciprocal Lie connection have isogenous Galois groups. We prove that the only centerless Lie algebras $\mathfrak{g}$ such that there exists a $\mathfrak{g}$-parallelism with a trivial Galois group are algebraic Lie algebras, i.e., Lie algebras of algebraic groups. In particular this allows us to give a criterion for a parallelized variety to be isogenous to an algebraic group. The vanishing of the Lie algebra of the Galois group of the reciprocal connection is a necessary and sufficient condition for a parallelized variety to be isogenous to an algebraic group: Let $\mathfrak g$ be a centerless Lie algebra. An algebraic variety $(M,\omega)$ with a rational parallelism of type $\mathfrak g$ is isogenous to an algebraic group if and only if $\mathfrak{gal}(\nabla^{\rm rec}) = \{0\}$. The notion of [*isogeny*]{} can be extended beyond the simply-transitive case. Let us consider a complex Lie algebra $\mathfrak g$. An [*infinitesimally homogeneous variety*]{} of type $\mathfrak g$ is a pair $(M,\mathfrak s)$ consisting of a complex smooth irreducible variety $M$ and a finite-dimensional Lie algebra isomorphic to $\mathfrak g$ that spans the tangent bundle of $M$ on the generic point. We are interested in conjugation by rational or by algebraic maps, so that, whenever necessary, we replace $M$ by a suitable Zariski open subset. In this context, we say that a dominant rational map $f\colon M_1 \dasharrow M_2$ between varieties of the same dimension conjugates the infinitesimally homogeneous varieties $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ if $f^*(\mathfrak s_2) = \mathfrak s_1$. We say that $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ are [*isogenous*]{} if they are conjugated to the same infinitesimally homogeneous space of type $\mathfrak g$. Under some hypothesis on the Lie algebra $\mathfrak s\subset \mathfrak X(M)$ one can prove that $(M,\mathfrak s)$ is isogenous to a homogeneous space $(G/H,\mathfrak{lie}(G)^{\rm rec})$ with the action of right invariant vector fields. These hypothesis are satisfied by transitive actions of $\mathfrak{sl}_{n+1}(\mathbf C)$ on $n$-dimensional varieties. As a particular case of Theorem \[homogeneous\] one has Let $(M,\mathfrak s)$ be an infinitesimally homogeneous variety of complex dimension $n$ such that $\mathfrak s$ is isomorphic to $\mathfrak{sl}_{n+1}(\mathbf C)$. Then there exists a dominant rational map $M \dasharrow \mathbf{CP}_n$ conjugating $\mathfrak s$ with the Lie algebra $\mathfrak{sl}_{n+1}(\mathbf C)$ of projective vector fields in $\mathbf{CP}_n$. Appendix \[ApA\] is devoted to a geometrical presentation of Picard–Vessiot theory for linear and principal connections. Finally, Appendix \[apB\] contains a detailed proof of Deligne’s theorem of the realization of a regular parallelism modeled over any finite-dimensional Lie algebra. This includes also a computation of the Galois group that turns out to be, for this particular construction, an algebraic torus. Parallelisms {#section_parallelisms} ============ Let $M$ be a smooth connected affine variety over $\mathbf C$ of dimension $r$. We denote by $\mathbf C[M]$ its ring of regular functions and by $\mathbf C(M)$ its field of rational functions. Analogously, we denote by $\mathfrak X[M]$ and $\mathfrak X(M)$ respectively the Lie algebras of regular and rational vector fields in $M$, and so on. Let $\mathfrak g$ be a Lie algebra of dimension $r$. We fix a basis $A_1,\ldots,A_r$ of $\mathfrak g$, and the following notation for the associated structure constants $[A_i,A_j] = \sum_{k}\lambda_{ij}^kA_k$. A parallelism of type $\mathfrak g$ of $M$ is a realization of the Lie algebra $\mathfrak g$ as a Lie algebra of pointwise linearly independent vector fields in $M$. More precisely: A regular parallelism of type $\mathfrak g$ in $M$ is a Lie algebra morphism, $\rho\colon \mathfrak g \to \mathfrak X[M]$ such that $\rho A_1(x), \ldots, \rho A_r(x)$ form a basis of $T_xM$ for any point $x$ of $M$. \[ex:AGP\]Let $G$ be an algebraic group and $\mathfrak g$ be its Lie algebra of left invariant vector fields. Then the natural inclusion $\mathfrak g\subset \mathfrak X[G]$ is a regular parallelism of $G$. The Lie algebra $\mathfrak g^{\rm rec}$ of right invariant vector fields is another regular parallelism of the same type. Let invariant and right invariant vector fields commute, hence, an algebraic group is naturally endowed with a pair of commuting parallelisms of the same type. From Example \[ex:AGP\], it is clear that any *algebraic* Lie algebra is realized as a parallelism of some algebraic variety. On the other hand, Theorem \[TDeligne\] due to P. Deligne and published in [@Malgrange], ensures that any Lie algebra is realized as a regular parallelism of an algebraic variety. Analogously, we have the definitions of rational and local analytic parallelism. Note that a rational parallelism in $M$ is a regular parallelism in a Zariski open subset $M^\star \subseteq M$. There is dual definition, equivalent to that of parallelism. This is more suitable for calculations. A regular parallelism form (or coparallelism) of type $\mathfrak g$ in $M$ is a $\mathfrak g$-valued $1$-form $\omega\in\Omega^1[M]\otimes_{\mathbb C} \mathfrak g$ such that: - For any $x\in M$, $\omega_x\colon T_x M \to \mathfrak g$ is a linear isomorphism. - If $A$ and $B$ are in $\mathfrak g$ and $X$, $Y$ are vector fields such that $\omega(X) = A$ and $\omega (Y) = B$ then $\omega[X,Y] = [A,B]$. Analogously, we define local analytic and rational coparallelism of type $\mathfrak g$ in $M$. It is clear that each coparallelism induces a parallelism, and reciprocally, by the relation $\omega (\rho (A)) = A$. Thus, there is a natural equivalence between the notions of parallelism and coparallelism. From now on we fix $\rho$ and $\omega$ equivalent parallelism and coparallelism of type $\mathfrak g$ on $M$. The Lie algebra structure of $\mathfrak g$ forces $\omega$ to satisfy Maurer–Cartan structure equations $$\begin{gathered} {\rm d}\omega + \frac{1}{2}[\omega,\omega] = 0.\end{gathered}$$ Taking components $\omega = \sum_{i}\omega_iA_i$ we have $$\begin{gathered} {\rm d}\omega_i +\sum_{j,k=1}^r \frac{1}{2}\lambda_{jk}^i\omega_j\wedge \omega_k = 0.\end{gathered}$$ \[ex:group\] Let $G$ be an algebraic group and $\mathfrak g$ be the Lie algebra of left invariant vector fields in $G$. Then the structure form $\omega$ is the coparallelism corresponding to the parallelism of Example \[ex:AGP\]. \[ex:B\]Let $\mathfrak g = \langle A_1,A_2\rangle$ be the 2-dimensional Lie algebra with commutation relation $$\begin{gathered} = A_1.\end{gathered}$$ The vector fields $$\begin{gathered} X_1 = \frac{\partial}{\partial x},\qquad X_2 = x\frac{\partial}{\partial x}+ \frac{\partial}{\partial y},\end{gathered}$$ define a regular parallelism via $\rho (A_i)= X_i$ of $\mathbf C^2$. The associated parallelism form is $$\begin{gathered} \omega = A_1{\rm d}x + (A_2 - x A_1){\rm d}y.\end{gathered}$$ \[ex:MD\] Let $\mathfrak g = \langle A_1,A_2,A_3 \rangle$ be the 3-dimensional Lie algebra with commutation relations $$\begin{gathered} = \alpha A_2,\qquad [A_1,A_3]=\beta A_3, \qquad [A_2,A_3] = 0,\end{gathered}$$ with $\alpha$, $\beta$, non zero complex numbers. In particular, if $\alpha/\beta$ is not rational then $\mathfrak g$ is not the Lie algebra of an algebraic group. The vector fields $$\begin{gathered} X_1 = \frac{\partial}{\partial x} + \alpha y \frac{\partial}{\partial y} + \beta z\frac{\partial}{\partial z},\qquad X_2 = \frac{\partial}{\partial y},\qquad X_3 = \frac{\partial}{\partial z},\end{gathered}$$ define a regular parallelism via $\rho (A_i)= X_i$ of $\mathbf C^3$. The associated parallelism form is $$\begin{gathered} \omega = (A_1 - A_2\alpha y - A_3\beta z){\rm d}x + A_2{\rm d}y + A_3{\rm d}z.\end{gathered}$$ \[isogenous\] Let $(M,\omega)$ and $(N,\theta)$ be algebraic manifolds with coparallelisms of type $\mathfrak g$. We say that they are isogenous if there is an algebraic manifold $(P,\eta)$ with a coparallelism of type $\mathfrak g$ and dominant maps $f\colon P\to M$ and $g\colon P \to N$ such that $f^*(\omega) = g^*(\theta) = \eta$. Clearly, the notion of isogeny of parallelized varieties extends that of isogeny of algebraic groups. \[ex:cover\] Let $f\colon M\dasharrow G$ be a dominant rational map with values in an algebraic group with $\dim_{\mathbf{C}}M =\dim_{\mathbf{C}}G$. Then $\theta =f^*(\omega)$ is a rational parallelism form in $M$. \[ex:finite\]Let $H$ be a finite subgroup of the algebraic group $G$ and $$\begin{gathered} \pi\colon \ G\to M = H\setminus G = \{Hg \colon g\in G\}\end{gathered}$$ be the quotient by the action of $H$ on the left side. The structure form $\omega$ in $G$ is left-invariant and then it is projectable by $\pi$. Then, $\theta = \pi_*(\omega)$ is a regular parallelism form in $M$. \[ex:coverfinite\] Combining Examples \[ex:cover\] and \[ex:finite\], let $H\subset G$ be a finite subgroup and $f\colon M\to H \setminus G$ be a dominant rational map between manifolds of the same dimension. Then $\theta = f^*(\pi_*(\omega))$ is a rational parallelism form in $M$. By application of Example \[ex:coverfinite\] to the case of the multiplicative group we obtain rational multiples of logarithmic forms in $\mathbf{CP}_1$, $\frac{p}{q}\frac{{\rm d}f}{f}$ where $f\in\mathbf C(z)$. Thus, rational multiples of logarithmic forms in $\mathbf{CP}_1$ are the rational coparallelisms isogenous to that of the multiplicative group. By application of Example \[ex:coverfinite\] to the case of the additive group we obtain the exact forms in $\mathbf{CP}_1$, ${\rm d}F$ where $F\in\mathbf C(z)$. Thus, the exact forms in $\mathbf{CP}_1$ are the rational coparallelisms isogenous to that of the additive group. \[ex:quotient\_c\]Let $H$ be a subgroup of the algebraic group $G$, with Lie algebra $\mathfrak h\subset \mathfrak g$. Let us assume that $\mathfrak h$ admits a supplementary Lie algebra $\mathfrak h'$ $$\begin{gathered} \mathfrak g = \mathfrak h \oplus \mathfrak h' \qquad \mbox{(as vector spaces).}\end{gathered}$$ We consider the left quotient $M= H \setminus G$ of $G$ by the action of $H$ and the quotient map $\pi\colon G\to M$. It turns out that $\mathfrak h'$ is a Lie algebra of vector fields in $G$ projectable by $\pi$, and thus $\pi_*|_{\mathfrak h'} \colon \mathfrak h' \to \mathfrak X[M]$ gives a parallelism of $M$ that is regular in the open subset $$\begin{gathered} \{Hg\in M \colon \operatorname{Adj}_g(\mathfrak h) \cap \mathfrak h' = \{0\} \}.\end{gathered}$$ It turns out to be regular in $M$ if $H\lhd G$. Examples \[ex:B\] and \[ex:MD\] are particular cases where $G$ is $\operatorname{Af\/f}(2,\mathbf C)$ and $\operatorname{Af\/f}(3,\mathbf C)$ respectively. We can see also Example \[ex:quotient\_c\] as a coparallelism. Let $\pi'\colon\mathfrak g\to \mathfrak h'$ be the projection given by the vector space decomposition $\mathfrak g = \mathfrak h \oplus \mathfrak h'$. Since $\pi'\circ \omega$ is left invariant form in $G$, it is projectable by $\pi$. Hence, there is a form $\omega'$ in $M$ such that $\pi^*\omega' = \pi'\circ \omega$. This form $\omega'$ is the corresponding coparallelism. Associated Lie connection {#section_lie} ========================= Reciprocal connections ---------------------- Let $\nabla$ be a linear connection (rational or regular) on $TM$. The reciprocal connection is defined as $$\begin{gathered} \nabla^{\rm rec}_{\vec X}\vec Y = \nabla_{\vec Y} \vec X + \big[\vec X,\vec Y\big].\end{gathered}$$ From this definition it is clear that the difference $\nabla - \nabla^{\rm rec} = \operatorname{Tor}_{\nabla}$ is the torsion tensor, $\operatorname{Tor}_{\nabla} = -\operatorname{Tor}_{\nabla^{\rm rec}}$ and $(\nabla^{\rm rec})^{\rm rec} = \nabla$. Connections and parallelisms ---------------------------- Let $\omega$ be a coparallelism of type $\mathfrak g$ in $M$ and $\rho$ its equivalent parallelism. Denote by $\vec X_i$ the basis of vector fields in $M$ such that $\omega(\vec X_i)=A_i$ is a basis of $\mathfrak g$. The connection $\nabla$ associated to the parallelism $\omega$ is the only linear connection in $M$ for which $\omega$ is a $\nabla$-horizontal form. Clearly $\nabla$ is a flat connection and the basis $\{\vec X_i\}$ is a basis of the space of $\nabla$-horizontal vector fields. In this basis $\nabla$ has vanishing Christoffel symbols $$\begin{gathered} \nabla_{\vec X_i} \vec X_j = 0.\end{gathered}$$ Let us compute some infinitesimal symmetries of $\omega$. A vector field $\vec Y$ is an infinitesimal symmetry of $\omega$ if ${\rm Lie}_{\vec Y}\omega = 0$, or equivalently, if it commutes with all the vector fields of the parallelism $$\begin{gathered} \big[\vec X_i, \vec Y\big] = 0, \qquad i = 1,\ldots, r.\end{gathered}$$ \[Lemma1\]Let $\nabla$ be the connection associated to the parallelism $\omega$. Then for any vector field $\vec Y$ and any $j=1,\ldots,r$ $$\begin{gathered} \big[\vec X_j, \vec Y\big] = \nabla^{\rm rec}_{\vec X_j}{\vec Y}.\end{gathered}$$ Thus, $\vec Y$ is an infinitesimal symmetry of $\omega$ if and only if it is a horizontal vector field for the reciprocal connection $\nabla^{\rm rec}$. A direct computation yields the result. Take $\vec Y = \sum\limits_{k=1}^r f_k\vec X_k$, for each $j$ we have $$\begin{gathered} \nabla^{\rm rec}_{\vec X_j} \vec Y = \sum_{k=1}^r\big( \big(\vec X_jf_k\big)\vec X_k + f_k\big[\vec X_j, \vec X_k\big]\big) = \big[\vec X_j,\vec Y\big]. \tag*{\qed}\end{gathered}$$ The above considerations also give us the Christoffel symbols for $\nabla^{\rm rec}$ in the basis $\{\vec X_i\}$ $$\begin{gathered} \nabla^{\rm rec}_{\vec X_i}\vec X_j = \big[\vec X_i, \vec X_j\big] = \sum_{k=1}^r \lambda_{ij}^k \vec X_k,\end{gathered}$$ i.e., the Christoffel symbols of $\nabla^{\rm rec}$ are the structure constants of the Lie algebra $\mathfrak g$. \[Lemma2\] Let $\nabla$ be the connection associated to a coparallelism in $M$. Then, $\nabla^{\rm rec}$ is flat, and the Lie bracket of two $\nabla^{\rm rec}$-horizontal vector fields is a $\nabla^{\rm rec}$-horizontal vector field. The flatness and the preservation of the Lie bracket by $\nabla^{\rm rec}$ are direct consequences of the Jacobi identity. Let us compute the curvature $$\begin{gathered} R\big(\vec X_i,\vec X_j,\vec X_k\big) = \nabla^{\rm rec}_{\vec X_i}\big(\nabla^{\rm rec}_{\vec X_j} X_k\big) - \nabla^{\rm rec}_{\vec X_j}\big(\nabla^{\rm rec}_{\vec X_i} \vec X_k\big) - \nabla^{\rm rec}_{[\vec X_i,\vec X_j]}\vec X_k\\ \hphantom{R\big(\vec X_i,\vec X_j,\vec X_k\big)}{} = \rho ([A_i,[A_j,A_k]] - [A_j,[A_i, A_k]] - [[A_i,A_j],A_k]) = 0.\end{gathered}$$ Let us compute the Lie bracket for $\vec Y$ and $\vec Z$ $\nabla^{\rm rec}$-horizontal vector fields $$\begin{gathered} \nabla^{\rm rec}_{\vec X_i}\big[\vec Y, \vec Z\big]\! = \big[\vec X_i,\big[\vec Y, \vec Z\big]\big]\! = \big[\big[\vec X_i, \vec Y\big], \vec Z\big]\! + \big[ \vec Y, \big[\vec X_i, \vec Z \big]\big]\! = \big[\nabla^{\rm rec}_{\vec X_i}\vec Y, \vec Z\big]\! +\big[\vec Y, \nabla^{\rm rec}_{\vec X_i}\vec Z\big]\! = 0.\!\!\!\!\!\!\tag*{{}}\end{gathered}$$ \[Lemma3\] Let $x\in M$ be a regular point of the parallelism form $\omega$. The space of germs at $x$ of horizontal vector fields for $\nabla^{\rm rec}$ is a Lie algebra isomorphic to $\mathfrak g$. Moreover, let $\vec Y_1,\ldots, \vec Y_r$ be horizontal vector fields with initial conditions $\vec Y_i(x) = \vec X_i(x)$, then $\big[\vec Y_i, \vec Y_j\big] = - \sum\limits_{k=1}^r \lambda_{ij}^k \vec Y_k$, where the $\lambda_{i,j}$ are the structure constants of the Lie algebra generated by the $\vec X_i$. We can write the vector fields $\vec Y_i$ as linear combinations of the vector fields $\vec X_i$: $\vec Y_i = \sum\limits_{j=1}^r a_{ji}\vec X_j$. The matrix $(a_{ij})$ satisfies the differential equation $$\begin{gathered} \vec X_{k}a_{ij} = - \sum_{\alpha = 1}^r \lambda_{k \alpha}^i a_{\alpha j}, \qquad a_{ij}(x) = \delta_{ij}.\end{gathered}$$ On the other hand, we have $\big[\vec Y_i, \vec Y_j\big](x) = \sum\limits_{k=1}^r \hat\lambda_{ij}^k \vec Y_k (x)$, for certain unknown structure constants $\hat\lambda_{ij}^k$. Let us check that $\hat\lambda_{ij}^k = \lambda_{ji}^k = -\lambda_{ij}^k$, $$\begin{gathered} \big[\vec Y_i,\vec Y_j\big] = \left[\sum_{\alpha=1}^r a_{\alpha i} \vec X_\alpha, \sum_{\beta = 1}^r a_{\beta j} \vec X_{\beta} \right] \\ \hphantom{\big[\vec Y_i,\vec Y_j\big]}{} = \sum_{\alpha, \beta, \gamma =1}^r -a_{\alpha i}\lambda_{\alpha \gamma}^\beta a_{\gamma j}\vec X_\beta + \sum_{\alpha, \beta, \gamma =1}^r a_{\beta j}\lambda_{\beta\gamma}^\alpha a_{\gamma i} \vec X_{\alpha} + \sum_{\alpha, \beta, \gamma =1}^r a_{\beta j} a_{\alpha i} \lambda_{\alpha \beta}^\gamma \vec X_\gamma.\end{gathered}$$ Taking values at $x$, we obtain $$\begin{gathered} \big[\vec Y_i, \vec Y_j\big](x) = \sum_{\beta =1}^r - \lambda_{i j}^\beta \vec Y_\beta(x) + \sum_{\alpha =1}^r \lambda_{j i}^\alpha \vec Y_{\alpha}(x) + \sum_{\gamma =1}^r \lambda_{i j}^\gamma \vec Y_\gamma(x) = \sum_{\alpha =1}^r \lambda_{j i}^k \vec Y_{k}(x).\tag*{{}}\end{gathered}$$ Let $G$ be an algebraic group with Lie algebra $\mathfrak g$. As seen in Example \[ex:group\] the Maurer–Cartan structure form $\omega$ is a coparallelism in $G$. Let $\nabla$ be the connection associated to this coparallelism. There is another canonical coparallelism, the right invariant Maurer–Cartan structure form $\omega_{\rm rec}$, let us consider ${\bf i}\colon G\to G$ the inversion map, $$\begin{gathered} \omega_{\rm rec} = - {\bf i}^*(\omega).\end{gathered}$$ As may be expected, the connection associated to the coparallelism $\omega_{\rm rec}$ is $\nabla^{\rm rec}$. Right invariant vector fields in $G$ are infinitesimal symmetries of left invariant vector fields and vice versa. In this case, the horizontal vector fields of $\nabla$ and $\nabla^{\rm rec}$ are regular vector fields. As shown in the next three examples, symmetries of a rational parallelism are not in general rational vector fields. Let us consider the Lie algebra $\mathfrak g$ and the coparallelism $\omega = A_1{\rm d}x + (A_2 - x A_1){\rm d}y$, of Example \[ex:B\]. Let $\nabla$ be its associated connection. In cartesian coordinates, the only non-vanishing Christoffel symbol of the reciprocal connection is $\Gamma_{21}^1 = -1$. A basis of $\nabla^{\rm rec}$-horizontal vector fields is $$\begin{gathered} \vec Y_1 = e^y\frac{\partial}{\partial x}, \qquad \vec Y_2 = \frac{\partial}{\partial y}.\end{gathered}$$ Note that they coincide with $\vec X_1$, $\vec X_2$ at the origin point and $\big[\vec Y_1,\vec Y_2\big] = - Y_1$. Let us consider the Lie algebra $\mathfrak g$ and the coparallelism $\omega = (A_1 - \alpha y A_2 - \beta z A_3){\rm d}x + A_2{\rm d}y + A_3{\rm d}z$ of Example \[ex:MD\]. Let $\nabla$ be its associated connection. In cartesian coordinates, the only non-vanishing Christoffel symbols of the reciprocal connection are $$\begin{gathered} \Gamma_{11}^2 = -\alpha,\qquad \Gamma_{11}^3 = -\beta.\end{gathered}$$ A basis of $\nabla^{\rm rec}$-horizontal vector fields is $$\begin{gathered} \vec Y_1 = \frac{\partial}{\partial x}, \qquad \vec Y_2 = e^{\alpha x}\frac{\partial}{\partial y},\qquad \vec Y_3 = e^{\beta x}\frac{\partial}{\partial z}.\end{gathered}$$ Note that they coincide with $\vec X_1$, $\vec X_2$, $\vec X_3$ at the origin point and $$\begin{gathered} \big[\vec Y_1,\vec Y_2\big] = - \alpha Y_2,\qquad \big[\vec Y_1,\vec Y_3\big] = -\beta \vec Y_3.\end{gathered}$$ Let us consider the Lie algebra $\mathfrak g$ of Example \[ex:MD\] and the coparallelism $$\begin{gathered} \omega = (A_1 - \alpha y A_2 - \beta z A_3)\frac{{\rm d}x}{x} + A_2{\rm d}y + A_3{\rm d}z.\end{gathered}$$ Let $\nabla$ be its associated connection. In cartesian coordinates, the only non-vanishing Christoffel symbols of the reciprocal connection are $$\begin{gathered} \Gamma_{11}^2 = -\alpha,\qquad \Gamma_{11}^3 = -\beta.\end{gathered}$$ A basis of $\nabla^{\rm rec}$-horizontal vector fields on a simply connected open subspace $U\subset \mathbb C^\ast \times \mathbb C^2$ is $$\begin{gathered} \vec Y_1 = x\frac{\partial}{\partial x}, \qquad \vec Y_2 = x^{\alpha}\frac{\partial}{\partial y}, \qquad \vec Y_3 = x^{\beta}\frac{\partial}{\partial z}.\end{gathered}$$ Lie connections --------------- The connections $\nabla$ and $\nabla^{\rm rec}$ associated to a coparallelism $\omega$ of type $\mathfrak g$ are particular cases of the following definition. A Lie connection (regular or rational) in $M$ is a flat connection $\nabla$ in $TM$ such that the Lie bracket of any two horizontal vector fields is a horizontal vector field. Given a Lie connection $\nabla$ in $M$, there is a $r$-dimensional Lie algebra $\mathfrak g$ such that the space of germs of horizontal vector fields at a regular point $x$ is a Lie algebra isomorphic to $\mathfrak g$. We will say that $\nabla$ is a Lie connection of type $\mathfrak g$. The following result gives several algebraic characterizations of Lie connections: \[prop\_Lie\_char\] Let $\nabla$ be a linear connection in $TM$, the following statements are equivalent: - $\nabla$ is a Lie connection; - $\nabla^{\rm rec}$ is a Lie connection; - $\nabla$ is flat and has constant torsion, $\nabla \operatorname{Tor}_{\nabla} = 0$; - $\nabla$ and $\nabla^{\rm rec}$ are flat. Let us first see (1)$\Leftrightarrow$(2). Let $\nabla$ be a Lie connection. Around each point of the domain of $\nabla$ there is a parallelism, by possibly transcendental vector fields, such that $\nabla$ is its associated connection. Then, Lemma \[Lemma2\] states (1)$\Rightarrow$(2). Taking into account that $(\nabla^{\rm rec})^{\rm rec} = \nabla$ we have the desired equivalence. Let us see now that (1)$\Leftrightarrow$(3). Let us assume that $\nabla$ is a flat connection. For any three vector fields $X$, $Y$, $Z$ in $M$ we have $$\begin{gathered} (\nabla_X \operatorname{Tor}_{\nabla})(Y,Z) = - \operatorname{Tor}_{\nabla}(\nabla_X Y,Z ) - \operatorname{Tor}_{\nabla}(Y,\nabla_X Z) + \nabla_X \operatorname{Tor}_{\nabla}(Y,Z).\end{gathered}$$ Let us assume that $Y$ and $Z$ are $\nabla$-horizontal vector fields. Then, we have $$\begin{gathered} \operatorname{Tor}_{\nabla}(Y,Z) = \nabla_Y Z - \nabla_Z Y - [Y,Z] = - [Y,Z]\end{gathered}$$ and the previous equality yields $$\begin{gathered} (\nabla_X \operatorname{Tor}_{\nabla})(Y,Z) = - \nabla_X[Y,Z].\end{gathered}$$ Thus, we have that $\nabla \operatorname{Tor}_{\nabla}$ vanishes if and only if the Lie bracket of any two $\nabla$-horizontal vector fields is also $\nabla$-horizontal. This proves (1)$\Leftrightarrow$(3). Finally, let us see (1)$\Leftrightarrow$(4). It is clear that (1) implies (4) so we only need to see (4)$\Rightarrow$(1). Assume $\nabla$ and $\nabla^{\rm rec}$ are flat. Then, locally, there exist a basis $\{\vec X_i\}$ of $\nabla$-horizontal vector fields and a basis $\{\vec Y_i\}$ of $\nabla^{\rm rec}$-horizontal vector fields. By the definition of the reciprocal connection, we have that a vector field $\vec X$ is $\nabla$-horizontal if and only if it satisfies $[\vec X, \vec Y_i]=0$ for $i=1,\ldots,r$. By the Jacobi identity we have $$\begin{gathered} \big[\big[\vec X_i,\vec X_j\big],\vec Y_k\big]= 0.\end{gathered}$$ The Lie brackets $\big[\vec X_i,\vec X_j\big]$ are also $\nabla$-horizontal and $\nabla$ is a Lie connection. Let $\nabla$ be a Lie connection on $M$. Let $x$ be a regular point and $\vec X_1,\ldots, \vec X_r$ and $\vec Y_1,\ldots, \vec Y_r$ be basis of horizontal vector field germs on $M$ for $\nabla$ and $\nabla^{\rm rec}$ respectively with same initial conditions $\vec X_i(x) = \vec Y_i(x)$. Then $$\begin{gathered} \big[\vec X_i,\vec X_j\big](x) = - \big[\vec Y_i,\vec Y_j\big](x).\end{gathered}$$ It follows that $\nabla$ and $\nabla^{\rm rec}$ are of the same type $\mathfrak g$. By definition $\nabla$ is the connection associated to the local analytic parallelism given by the basis $\{\vec X_i\}$ of horizontal vector fields. Then we apply Lemma \[Lemma3\] in order to obtain the desired conclusion. Some results on Lie connections by means of Picard–Vessiot theory ----------------------------------------------------------------- Definitions and general results concerning the Picard–Vessiot theory of connections are given in Appendix \[ApA\]. Let $\nabla$ be a rational Lie connection in $TM$. The $\nabla$-horizontal vector fields are the symmetries of a rational parallelism of $M$ if and only if $\operatorname{Gal}(\nabla^{\rm rec}) = \{1\}$. We will use the notations of Section \[A6\]: $R^1(TM)$ is the ${\rm GL}_n(\mathbb{C})$-principal bundle associated to $TM$ and $\mathcal F'$ is the ${\rm GL}_n(\mathbb{C})$-invariant foliation on $R^1(TM)$ given by graphs of local basis of $\nabla$-horizontal sections. The Galois group $\operatorname{Gal}(\nabla^{\rm rec})$ can be computed as soon as we know the Zariski closure $\overline{{\mathscr{L}}}$ of a leaf ${\mathscr{L}}$ of the induced foliation $\mathcal F'$ on $R^1(TM)$. $\operatorname{Gal}(\nabla^{\rm rec})$ is finite is and only if $\overline{{\mathscr{L}}} = {\mathscr{L}}$ and is $\{1\}$ if and only if ${\mathscr{L}}$ is the graph of a rational section $M \to R^1(TM)$. This means that there exists a basis of rational $\nabla^{\rm rec}$-horizontal sections. These sections give the desired parallelism. For any Lie connection $\nabla$, $\operatorname{Gal}(\nabla)\subseteq \operatorname{Aut}(\mathfrak g)$. Let us choose a point $x \in M$ regular for $\nabla$ and a basis $A_1,\ldots, A_r$ of $\mathfrak{g}$, i.e., a basis $Y_1,\ldots, Y_r$ of local $\nabla$-horizontal section of $TM$ at $x$. Using notation of Section \[A6\], we will identify $R^1(T_xM)$ with the set of isomorphisms of linear spaces $ \sigma\colon \mathfrak g \to T_xM$; now $\operatorname{Gal}(\nabla)\subseteq {\rm GL}(\mathfrak g)$. Because of the construction of $\mathfrak g$, we have a canonical point in $R^1(TM)$ corresponding to the identity $ \sigma_o\colon \mathfrak g \to T_xM$. For $m \in M$, if $\sigma$ is an isomorphism from $\mathfrak g$ to $T_{m}M$ then one defines $H^k_{{i,j}}(\sigma)$ to be $$\begin{gathered} \frac{[X_{i},X_{j}] \wedge X_{1}\wedge \cdots \wedge \widehat{X_{k}}\wedge \cdots \wedge X_{r}}{X_{k} \wedge X_{1}\wedge \cdots \wedge \widehat{X_{k}} \wedge \cdots \wedge X_{r}} \Big|_m, \end{gathered}$$ where $X_i$ is the horizontal section such that $X_i(m)= \sigma A_i$. These functions are regular functions on $R^1(TM)$. Moreover they are constant and equal to the constant structures on the Zariski closure of the leaf passing through $\sigma_o$. The Galois group is the stabilizer of this leaf then the functions $H^k_{{i,j}}$ are invariant under the action of the Galois group, i.e., the Galois group preserves the Lie bracket. \[prop\_example\] Let $\mathfrak h'$ be a Lie sub-algebra of the Lie algebra of some algebraic group and let $G$ be the smallest algebraic subgroup such that ${\mathfrak{lie}}(G) = \mathfrak g \supset \mathfrak h'$. Assume the existence of an algebraic subgroup $H$ of $G$ whose Lie algebra $\mathfrak h$ is supplementary to $\mathfrak h'$ in $\mathfrak g$, $\mathfrak g = \mathfrak h \oplus \mathfrak h'$. Let us consider the following objects: - the quotient map $\pi\colon G \to M$ where $M$ is the variety of cosets $H\setminus G$, and $\nabla$ the Lie connection associated to the parallelism $\pi_*\colon \mathfrak h' \to \mathfrak X[M]$ in $M$ $($as given in Example [\[ex:quotient\_c\])]{}; - its reciprocal Lie connection $\nabla^{\rm rec}$ on $M$; - the Lie algebras of right invariant vector fields $$\begin{gathered} \mathfrak g^{\rm rec} = {\bf i}_*(\mathfrak g), \qquad \mathfrak h'^{\rm rec} = {\bf i}_*(\mathfrak h'),\end{gathered}$$ where ${\bf i}$ is the inverse map on $G$. Then, the following statements are true: - $\mathfrak h'$ is an ideal of $\mathfrak g$ $($equivalently $\mathfrak h'^{\rm rec}$ is an ideal of $\mathfrak g^{\rm rec})$; - $\mathfrak h$ is commutative $($equivalently $H$ is virtually abelian$)$; - the adjoint action of $G$ on $\mathfrak g^{\rm rec}$ preserves $\mathfrak h'^{\rm rec}$ and thus gives, by restriction, a morphism $\overline{\operatorname{Adj}}\colon G \to \operatorname{Aut}(\mathfrak h'^{\rm rec})$; - The Galois group of the connection $\nabla^{\rm rec}$ is ${\overline{\operatorname{Adj}}}(H) \subseteq \operatorname{Aut}(\mathfrak h'^{\rm rec})$ and thus virtually abelian. We have that $\mathfrak g$ is the algebraic hull of $\mathfrak h'$. From Lemma \[ap2\_2\] in Appendix \[apB\] we obtain $[\mathfrak g,\mathfrak g] \subseteq \mathfrak h'$. Statement (i) follows straightforwardly. Let us consider $A$ and $B$ in $\mathfrak h$. Then $[A,B]$ is in $\mathfrak h$ and also in $\mathfrak h'$ by the previous argument. Thus, $[A,B]=0$ and this finishes the proof of statement (ii). Let us denote by $H'$ the subgroup of $G$ spanned by the image of $\mathfrak h'$ by the exponential map. For each element $h\in H'$, the adjoint action of $h$ preserves the Lie algebra $\mathfrak h'$. By continuity of the adjoint action in the Zariski topology, we have that $\mathfrak h'$ is preserved by the adjoint action of all elements of $G$. This proves statement (iii). In order to prove the last statement in the proposition we have to construct a Picard–Vessiot extension for the connection $\nabla^{\rm rec}$. Let us consider a basis $\{A_1,\ldots, A_m\}$ of $\mathfrak h'$ and let $\bar A_i$ be the projection $\pi_*(A_i)$. We have an extension of differential fields $$\begin{gathered} (\mathbf C(M), \bar{\mathcal D}) \subseteq (\mathbf C(G), \mathcal D),\end{gathered}$$ where $\bar{\mathcal D}$ stands for the $\mathbf C(M)$-vector space of derivations spanned by $\bar A_1,\ldots, \bar A_m$ and $\mathcal D$ stands for the $\mathbf C(G)$-vector space of derivations spanned by $A_1,\ldots,A_m$ (see Appendix \[ApA\] for our conventions on differential fields). The projection $\pi$ is a principal $H$-bundle. Any rational first integral of $\{A_1,\ldots,A_m\}$ is constant along $H'$ and thus it is necessarily a complex number. Thus, the above extension has no new constants and it is strongly normal in the sense of Kolchin, with Galois group $H$. Note that the differential field automorphism corresponding to an element $h\in H$ is the pullback of functions by the left translation $L_{h}^{-1}$, that is, $(hf)(g) = f\big(h^{-1}g\big)$. The horizontal sections for the connection $\nabla^{\rm rec}$ are characterized by the differential equations $$\begin{gathered} \label{eq_proof_H} [\bar A_i, X] = 0.\end{gathered}$$ Let us consider $\{B_1,\ldots,B_m\}$ a basis of $\mathfrak h'^{\rm rec}$. From the Zariski closedness of $H$ in $G$ it follows that there are regular functions $f_{ij}\in\mathbf C[G]$ such that $B_i = \sum\limits_{j=1}^m f_{ij} A_j$. Thus let us define $\bar B_i = \sum\limits_{j=1}^m f_{ij} \bar A_j$. Those objects are vector fields in $M$ with coefficients in $\mathbf C[G]$, and clearly satisfy equation . Thus, the Picard–Vessiot extension of $\nabla^{\rm rec}$ is spanned by the functions $f_{ij}$ and it is embedded, as a differential field, in $\mathbf C(G)$. Let us denote such extension by $\mathbf L$. We have a chain of extensions $$\begin{gathered} \mathbf C(M) \subseteq \mathbf L \subseteq \mathbf C(G).\end{gathered}$$ By Galois correspondence, the Galois group of $\nabla^{\rm rec}$ is a quotient $H/K$ where $K$ is the subgroup of elements of $H$ that fix, by left translation, the functions $f_{ij}$. In order to prove statement (iv) we need to check that this group $K$ is the kernel of the morphism $\overline{\operatorname{Adj}}$. Let us note that the image under the adjoint action by $g\in G$ of an element $B\in \mathfrak h'^{\rm rec}$ is given by the left translation, $\overline{\operatorname{Adj}}(g)(B) = L_{g*}(B)$. This transformation makes sense for any derivation of $\mathbf C[G]$, and thus we have an action of $G$ on $\mathfrak X(G)$. Let us take $h$ in the kernel of $\overline{\operatorname{Adj}}$, thus, $\overline{\operatorname{Adj}}(h)(B_j) = B_j$ for any index $j$. Applying the transformation $L_{h*}$ to the expression of $B_i$ as linear combination of the left invariant vector fields $A_j$ we obtain $B_i = \sum\limits_{j=1}^m L_{h*}(f_{ij} A_j) = \sum\limits_{j=1}^m h(f_{ij})A_j$. The coefficients of $B_i$ as linear combination of the $A_j$ are unique, and thus, $h(f_{ij}) = f_{ij}$ we conclude that $h$ is an automorphism fixing $\mathbf L$. On the other hand, let us take $h\in H$ fixing $\mathbf L$. Then $L_{h*}\big(\sum f_{ij}A_j\big) = \sum f_{ij} A_j$ thus $\overline{\operatorname{Adj}}(h)(B_i) = B_i$ and then $h$ is in the kernel of $\overline{\operatorname{Adj}}$. Some examples of $\boldsymbol{\mathfrak{sl}_2}$-parallelisms {#sl2} ------------------------------------------------------------ We will construct some parallelized varieties as subvarieties of the arc space of the affine line ${\mathbb A}^1_\mathbf C$. This family of examples show how to realize every subgroup of ${\rm PSL}_2(\mathbf C)$ as the Galois group of the reciprocal Lie connection. ### The arc space of the affine line and its Cartan 1-form {#sl2parallelisms} In our special case, the arc space of the affine line ${\mathbb A}^1_\mathbf C$ with affine coordinate $z$, is the space of all formal power series $\widehat{z} = \sum z^{(i)} \frac{x^i}{i !}$. It will be denoted by ${\mathscr{L}}$, its ring of regular functions is $\mathbf C[ {\mathscr{L}}] = \mathbf C\big[z^{(0)},z^{(1)}, z^{(2)},\ldots \big]$. For an open subset $U\subset \mathbf C$ one denotes by ${\mathscr{L}}U$ the set of power series $\widehat z$ with $z^{(0)} \in U$. A biholomorphism $f\colon U \to V$ between open sets of $\mathbf C$ can be lift to a biholomorphism $ {\mathscr{L}}f\colon {\mathscr{L}}U \to {\mathscr{L}}V$ by composition $\widehat{z} \to f\circ\widehat{z}$. Let $\widehat{\mathfrak X}$ be the Lie algebra of formal vector fields $\mathbf C[[x]]\frac{\partial}{\partial x}$. One can build a rational form $\sigma\colon T{\mathscr{L}} \to \widehat{\mathfrak X}$ in following way (see [@guillemin-sternberg Section 2]). Let $v = \sum a_i \frac{\partial}{\partial z^{(i)}}$ be a tangent vector at the formal coordinate $\widehat{p}$, i.e., an arc in the Zariski open subset $\{z^{(1)} \not = 0\}$. The local coordinate $\widehat{p}$ can be used to have formal coordinates $p_0,p_{1}, p_{2}, \ldots $, on $\mathscr{L}$ and $v$ can be written $v = \sum b_i \frac{\partial}{\partial p_{i}}$. The form $\sigma$ is defined by $\sigma(v) = \sum b_i \frac{x^i}{i !} \frac{\partial}{\partial x} $. This form is rational and is an isomorphism between $T_p {\mathscr{L}}$ and $\widehat{\mathfrak X}$ satisfying $d\sigma = - \frac{1}{2}[\sigma, \sigma]$ and $({\mathscr{L}}f)^\ast \sigma = \sigma$ for any biholomorphism $f$. This means that $\sigma$ provides an action of $\widehat{\mathfrak X}$ commuting with the lift of biholomorphisms. This form seems to be a coparallelism but it is not compatible with the natural structure of pro-variety of ${\mathscr{L}}$ and $\widehat{\mathfrak X}$: $\sigma^{-1}\big(\frac{\partial}{\partial x}\big) = \sum\limits_{i \geq 0} z^{(i+1)}\frac{\partial}{\partial z^{(i)}}$ is a derivation of degree $+1$ with respect to the pro-variety structure of ${\mathscr{L}}$. The total derivation above will be denoted by $E_{-1}$. This gives a differential structure to the ring $\mathbf C[ {\mathscr{L}}]$. ### The parallelized varieties Let $\nu\in \mathbf C(z)$ be a rational function, $f$ be the rational function on the arc space given by the Schwarzian derivative $$\begin{gathered} f\big(z^{(0)},z^{(1)},z^{(2)},z^{(3)}\big) = \frac{z^{(3)}}{z^{(1)}}-\frac{3}{2} \left(\frac{z^{(2)}}{z^{(1)}}\right) ^2+ \nu\big(z^{(0)}\big)\big(z^{(1)}\big)^2,\end{gathered}$$ and $I \subset \mathbf C[{\mathscr L}]$ be the $E_{-1}$-invariant ideal generated by $p\big(z^{(0)}\big) {z^{(1)}}^2f\big(z^{(0)},z^{(1)},z^{(2)},z^{(3)}\big)$ where $p$ is a minimal denominator of $\nu$. The zero set $V$ of $I$ is a dimension $3$ subvariety of ${\mathscr{L}}$ and $\omega (TV) = {\mathfrak{sl}}_2(\mathbf C) \subset \widehat{\mathfrak X}$. This provides a ${\mathfrak{sl}}_2$-parallelism on $V$. One can compute explicitly this parallelism using $z^{(0)}$, $z^{(1)}$ and $z^{(2)}$ as étale coordinates on a Zariski open subset of $V$. Let us first compute the $\mathfrak{sl}_2$ action on ${\mathscr{L}}$. The standard inclusion of $\mathfrak{sl}_2$ in $\widehat{\mathfrak X}$ is given by $E_{-1} = \frac{\partial}{\partial x}$, $E_0 = x\frac{\partial}{\partial x}$ and $E_{1} = x^2\frac{\partial}{\partial x}$. Their actions on ${\mathscr{L}}$ are given by $E_{-1} = \sum z^{(i+1)}\frac{\partial}{\partial z^{(i)}}$, $E_0 = \sum iz^{(i)}\frac{\partial}{\partial z^{(i)}}$ and $E_1= \sum i(i-1) z^{(i-1)}\frac{\partial}{\partial z^{(i)}}$. The ideal $I$ is generated by the functions $E_{-1}^n\cdot f$. By definition $E_{-1}\cdot f \in I$, a direct computation gives that $E_0\cdot f = 2f \in I$, $E_1\cdot f = 0 \in I$. The relations in $\mathfrak{sl}_{2}$ give that $E_{-1}\cdot I \subset I$, $E_0\cdot I \subset I$ and $E_1\cdot I \subset I$, i.e., the vector fields $E_{-1}$, $E_0$ and $E_1$ are tangent to $V$. Now parameterizing $V$ by $z^{(0)}$, $z^{(1)}$ and $z^{(2)}$ one gets $$\begin{gathered} E_{-1}|_{\mathbf C ^3} = z^{(1)}\frac{\partial}{\partial z^{(0)}} + z^{(2)}\frac{\partial}{\partial z^{(1)}} + \left(-\nu\big(z^{(0)}\big)\big(z^{(1)}\big)^3 + \frac{3}{2}\frac{(z^{(2)})^2}{z^{(1)}}\right) \frac{\partial}{\partial z^{(2)}},\nonumber\\ E_0|_{\mathbf C^3} = z^{(1)}\frac{\partial}{\partial z^{(1)}} + 2 z^{(2)}\frac{\partial}{\partial z^{(2)}}, \qquad E_1|_{\mathbf C^3} = 2 z^{(1)}\frac{\partial}{\partial z^{(2)}}.\label{parrallel}\end{gathered}$$ They form a rational $\mathfrak{sl}_{2}$-parallelism on $\mathbf C^3$ depending on the choice of a rational function in one variable. ### Symmetries and the Galois group of the reciprocal connection \[thm\_SL2\]Any algebraic subgroup of ${\rm PSL}_{2}(\mathbf C)$ can be realized as the Galois group of the reciprocal connection of a parallelism of $\mathbf C^3$. A direct computation shows that $z\mapsto \varphi(z)$ is an holomorphic function satisfying $$\begin{gathered} \frac{\varphi'''}{\varphi'}-\frac{3}{2} \left(\frac{\varphi''}{\varphi'}\right)^2 + \nu(\varphi)(\varphi')^2 = \nu(z),\end{gathered}$$ if and only if its prolongation ${\mathscr{L}}\varphi\colon \widehat{z} \mapsto \varphi (\widehat{z})$ on the space ${\mathscr{L}}$ preserves $V$ and preserves each of the vector fields $E_{-1}$, $E_0$ and $E_1$. Taking infinitesimal generators of this pseudogroup, one gets for any local analytic solution of the linear equation $$\begin{gathered} \label{lin} a''' + 2 \nu a' +\nu 'a =0,\end{gathered}$$ a vector field $X = a(z)\frac{\partial}{\partial z}$ whose prolongation on ${\mathscr{L}}$ is $$\begin{gathered} {\mathscr{L}}X = a\big(z^{(0)}\big)\frac{\partial}{\partial z^{(0)}} + a'\big(z^{(0)}\big)z^{(1)}\frac{\partial}{\partial z^{(1)}} + \big( a''\big(z^{(0)}\big)\big(z^{(1)}\big)^2 + a'\big(z^{(0)}\big) z^{(2)}\big)\frac{\partial}{\partial z^{(2)}} + \cdots.\end{gathered}$$ The equation (\[lin\]) ensures that ${\mathscr{L}}{X}$ is tangent to $V$. The invariance of $\sigma$, $({\mathscr{L}}X)_{\ast}\sigma =0$, ensures that ${\mathscr{L}}X$ commutes with the ${\mathfrak{sl}}_{2}$-parallelism given above. This means that for any solution $a$ of (\[lin\]) the vector field $$\begin{gathered} a\big(z^{(0)}\big)\frac{\partial}{\partial z^{(0)}} + a'\big(z^{(0)}\big)z^{(1)}\frac{\partial}{\partial z^{(1)}}+ \big( a''\big(z^{(0)}\big)\big(z^{(1)}\big)^2 + a'\big(z^{(0)}\big) z^{(2)}\big)\frac{\partial}{\partial z^{(2)}}, \end{gathered}$$ commutes with $E_{-1}|_{\mathbf C ^3}$, $E_0|_{\mathbf C ^3}$ and $E_{1}|_{\mathbf C ^3}$. Then the linear differential system of flat section for the reciprocal connection reduces to the linear equation (\[lin\]). This equation is the second symmetric power of $y'' = \nu(z)y$. If $G \subset {\rm SL}_2(\mathbf C)$ is the Galois group of $y'' = \nu(z)y$ then the image of its second symmetric power representation $s^2\colon G \to \operatorname{Sym}^2(\mathbf C^2)$ is the Galois group of (\[lin\]). The kernel of this representation is $\{{\rm Id}, -{\rm Id}\}$ then the Galois group of (\[lin\]) is an algebraic subgroup of ${\rm PSL}_{2}(\mathbf C)$. Let us remark that, as it follows from its definition, the Galois group of an equation contains the monodromy group. Moreover one can determine the monodromy group of classical differential equations. Hypergeometric equations depend on three complex numbers $(a,b,c)$ $$\begin{gathered} z(1-z) F'' + (c-(a+b+1)z)F' - abF = 0, $$ and is equivalent to $$\begin{gathered} y'' = \nu(\ell, n,m ; z) y,\end{gathered}$$ with $$\begin{gathered} \nu(\ell,m,n ;z) = \frac{\big(1-\ell^2\big)}{4z^2} +\frac{1-m^2}{4(1-z)^2} +\frac{1-\ell^2-m^2+n^2}{4 z(1-z)},\end{gathered}$$ and $$\begin{gathered} F = z^{-c/2}(1-z)^{(c-a-b-1)/2} y, \qquad \ell = 1-c,\qquad m =c-a-b,\qquad n=a-b.\end{gathered}$$ These two equations have the same projectivized Galois group in ${\rm PGL}_2(\mathbf C)$. Any algebraic subgroup of ${\rm PGL}_2(\mathbf C)$ will be realized by an appropriate choice of $(a,b,c)$. ### The whole group For $a=b=1/2$, $c = 1$, the hypergeometric equation is the Picard–Fuchs equation of Legendre family. Its monodromy group is $\Gamma(2) \subset {\rm SL}_2(\mathbf Z)$ and is Zariski dense in ${\rm SL}_2(\mathbf C)$. ### The triangular subgroups For $b=0$ and $a=-1$ one can compute a basis of solutions of the equation: $1$ and $\int \big( \frac{1-z}{z} \big)^c {\rm d}z$. If $c$ is not rational, the Galois group is the group of invertible matrices $\left[\begin{smallmatrix} u & v\\ 0 & 1 \end{smallmatrix}\right]$. When $c$ is rational then $u$ must be a root of the unity of order the denominators of $c$. When $c\in \mathbf Z$, the Galois group is the group of matrices $\left[\begin{smallmatrix} 1 & v\\ 0 & 1 \end{smallmatrix}\right]$. For $b=0 $ and $c= a+1$ a basis of solution is given by $z^{-a}$ and $1$. Its Galois group is a subgroup of the group of matrices $\left[\begin{smallmatrix} u& 0 \\ 0 & 1 \end{smallmatrix}\right]$. The parameter $a$ is rational if and only if it is a finite subgroup. ### The dihedral subgroups For $c=1/2$ and $a+b=0$, a basis of solution is given by $\big(\sqrt{z}+\sqrt{1-z}\big)^a$ and $\big(\sqrt{z}-\sqrt{1-z}\big)^a$. The monodromy group is a dihedral group in ${\rm GL}_2(\mathbf C)$ whose quotients give dihedral subgroups of ${\rm PGL}_2(\mathbf C)$. ### The tetrahedral subgroup This group is the monodromy group of hypergeometric equation for $\ell = 1/3$, $m= 1/2$ and $n=1/3$. A basis of solution is given by $$\begin{gathered} (z-1)^{-1/12}\Big(\sqrt{3}\big(z^{1/3}+1\big) \pm 2\sqrt{z^{2/3} + z^{1/3} +1} \Big)^{1/4}. \end{gathered}$$ ### The octahedral subgroup This group is the monodromy group of hypergeometric equation for $\ell = 1/2$, $m= 1/3$ and $n=1/4$. A basis of solution is given by $$\begin{gathered} (z-1)^{-1/24}\Big[\sqrt{3} \big( \big(\sqrt{z}-1\big)^{1/3} +\big(\sqrt{z}+1\big)^{1/3}\big)^{1/3}\\ \qquad{} \pm 2\sqrt{\big(\sqrt{z}-1\big)^{2/3} + (z-1)^{1/3} +\big(\sqrt{z}+1\big)^{2/3}}\Big]^{1/4}.\\\end{gathered}$$ ### The icosaedral subgroup This group is the monodromy group of hypergeometric equation for $\ell = 1/2$, $m= 1/3$ and $n=1/5$. As icosahedral group is not solvable, the solution space is not described using formulas as simple as in preceding examples. Darboux–Cartan connections {#section_DC} ========================== Connection of parallelism conjugations {#DC conjugation} -------------------------------------- Let $\omega$ be a rational coparallelism on $M$ of type $\mathfrak g$ and $G$ an algebraic group with Lie algebra of left invariant vector fields $\mathfrak g$ and Maurer–Cartan form $\theta$. Denote by $M^\star$ the open subset of $M$ in wich $\omega$ is regular. We will study the contruction of conjugating maps between the parallelisms $(M,\omega)$ and $(G,\theta)$. Let us consider the trivial principal bundle $\pi\colon P = G\times M \to M$. In this bundle we consider the action of $G$ by right translations $(g, x) * g' = (gg', x)$. Let $\Theta$ be the $\mathfrak g$-valued form $\Theta = \theta - \omega$ in $P$. \[DCconnection\] The kernel of $\Theta$ is a rational flat invariant connection on the principal bundle $\pi\colon P \to M$. We call it the Darboux–Cartan connection of parallelism conjugations from $(M,\omega)$ to $(G,\theta)$. The equation $\Theta = 0$ defines a foliation on $P$ transversal to the fibers at regular points of $\omega$. The leaves of the foliation are the graphs of analytic parallelism conjugations from $(M,\omega)$ to $(G,\theta)$. By means of differential Galois theory the Darboux–Cartan connection has a Galois group $\operatorname{Gal}(\Theta)$ with Lie algebra $\mathfrak{gal}(\Theta)$. The following facts are direct consequences of the definition of the Galois group: - there is a regular covering map $c\colon (M^\star, \omega) \to (U,\theta)$ with $U$ an open subset of $G$, and $c^*(\theta) = \omega$ if and only if $\operatorname{Gal}(\Theta) = \{1\}$; - there is a regular covering map $c\colon (M^\star, \omega) \to (U,q_*\theta)$ with $U$ an open subset of $G/H$, $H$ a group of finite index, and $c^*(q_*\theta|_U) = \omega$ if and only if $\mathfrak{gal}(\Theta) = \{0\}$. In any case, the necessary and sufficient condition for $(M,\omega)$ and $(G,\theta)$ to be isogenous parallelized varieties is that $\mathfrak{gal}(\Theta) = \{0\}$. Darboux–Cartan connection and Picard–Vessiot -------------------------------------------- Note that the coparallelism $\omega$ gives a rational trivialization of $TM$ as the trivial bundle of fiber $\mathfrak g$. In $TM$ we have defined the connection $\nabla^{\rm rec}$ whose horizontal vector fields are the symmetries of the parallelism. On the other hand, $G$ acts in $\mathfrak g$ by means of the adjoint action. The Cartan-Darboux connection induces then a connection $\nabla^{\rm adj}$ in the associated trivial bundle $\mathfrak g\times M$ of fiber $\mathfrak g$. \[adj\_conjugation\]The map $$\begin{gathered} \tilde\omega\colon \ \big(TM, \nabla^{\rm rec}\big) \to \big(\mathfrak g \times M, \nabla^{\rm adj}\big), \qquad X_x \mapsto (\omega_x(X_x), x)\end{gathered}$$ is a birational conjugation of the linear connections $\nabla^{\rm rec}$ and $\nabla^{\rm adj}$. It is clear that the map $\tilde\omega$ is birational. Let us consider $\{A_1,\ldots,A_m\}$ a basis of $\mathfrak g$. Let $\rho\colon \mathfrak g\to \mathfrak X(M)$ be the parallelism associated to the parallelism $\omega$ and let us define $X_i = \rho(A_i)$. Then $\{X_1,\ldots, X_n\}$ is a rational frame in $M$ and the map $\tilde\omega$ conjugates the vector field $X_i$ with the constant section $A_i$ of the trivial bundle of fiber $\mathfrak g$. By definition of the reciprocal connection $$\begin{gathered} \nabla^{\rm rec}_{X_i}X_j = [X_i,X_j].\end{gathered}$$ On the other hand, by definition of the adjoint action and application of the covariant derivative as in equation  of Appendix \[ap7\_associated\] we obtain $$\begin{gathered} \nabla^{\rm adj}_{X_i}A_j = [A_i,A_j].\end{gathered}$$ Therefore we have that $\tilde\omega$ is a rational morphism of linear connections that conjugates $\nabla^{\rm rec}$ with $\nabla^{\rm adj}$. The following facts follow directly from Proposition \[adj\_conjugation\], and basic properties of the Galois group. \[cor:rec\] Let us consider the adjoint action $\rm{ Adj}\colon G \to {\rm GL}(\mathfrak g)$ and its derivative ${\rm adj}\colon \mathfrak{g}\to \operatorname{End}(\mathfrak g)$. The following facts hold: - $\operatorname{Gal}(\nabla^{\rm rec}) = \operatorname{Adj}({\rm Gal(\Theta)})$; - $\mathfrak{gal}(\nabla^{\rm rec}) = {\rm adj}(\mathfrak{gal}(\Theta))$; - if $\mathfrak g$ is centerless then $\mathfrak{gal}(\nabla^{\rm rec})$ is isomorphic to $\mathfrak{gal}(\Theta)$; - assume $\mathfrak g$ is centerless, then the necessary and sufficient condition for $(M,\omega)$ and $(G,\theta)$ to be isogenous is that $\mathfrak{gal}(\nabla^{\rm rec})=\{0\}$. \(a) and (b). First, by Proposition \[adj\_conjugation\] we have that $\operatorname{Gal}(\nabla^{\rm rec}) = \operatorname{Gal}(\nabla^{\rm adj})$ and so $\mathfrak{gal}(\nabla^{\rm rec}) = \mathfrak{gal}(\nabla^{\rm adj})$. By definition $\nabla^{\rm adj}$ is the associated connection induced by $\Theta$ in the associated bundle $\mathfrak g\times M$. This trivial bundle is the associated bundle induced by the adjoint representation $\operatorname{Adj}\colon G\to \operatorname{End}(\mathfrak g)$. Then, by Theorem \[associated galois\], we have $\operatorname{Gal}(\nabla^{\rm rec}) = \operatorname{Adj}({\rm Gal(\Theta)})$ and $\operatorname{Gal}(\nabla^{\rm rec}) = \operatorname{Adj}({\rm Gal(\Theta)})$. \(c) It is a direct consequence of (b). The kernel of ${\rm adj}\colon \mathfrak g \to \operatorname{End}(\mathfrak g)$ is the center of $\mathfrak g$. \(d) It follows from the definition of Darboux–Cartan connection (see remarks after Definition \[DCconnection\]) that the necessary and sufficient condition for $(M,\omega)$ and $(G,\theta)$ to be isogenous is that $\mathfrak{gal}(\nabla^{\rm rec})=\{0\}$. By point (b) we conclude. Algebraic Lie algebras ---------------------- Let us consider $(M,\omega)$ a rational coparallelism of type $\mathfrak g$ with $\mathfrak g$ a centerless Lie algebra. We do not assume *a priori* that $\mathfrak g$ is an algebraic Lie algebra. The connection $\nabla^{\rm rec}$ is, as said in Proposition \[adj\_conjugation\], conjugated to the connection in $\mathfrak g\times M$ induced by the adjoint action. Note that, in order to define this connection we do not need the group operation but just the Lie bracket in $\mathfrak g$. We have an exact sequence $$\begin{gathered} 0\to \mathfrak g' \to \mathfrak g \to \mathfrak g^{ab}\to 0,\end{gathered}$$ where $\mathfrak g'$ is the derived algebra $[\mathfrak g, \mathfrak g]$. Since the Galois group acts by adjoint action, we have that $\mathfrak g'\times M$ is stabilized by the connection $\nabla^{\rm rec}$ and thus we have an exact sequence of connections $$\begin{gathered} 0\to (\mathfrak g'\times M, \nabla')\to \big(\mathfrak g \times M,\nabla^{\rm rec}\big)\to \big(\mathfrak g^{ab}\times M, \nabla^{ab}\big)\to 0.\end{gathered}$$ The Galois group of $\nabla^{ab}$ is the identity, therefore $\nabla^{ab}$ has a basis of rational horizontal sections. By definition, the action of $\mathfrak g$ in $\mathfrak g^{ab}$ vanishes. Thus, the constant functions $M\to \mathfrak g^{ab}$ are rational horizontal sections. \[th:algebraic\] Let $\omega$ be a rational coparallelism of $M$ of type $\mathfrak g$ with $\mathfrak g$ a centerless Lie algebra. If $\mathfrak{gal}(\nabla^{\rm rec}) = \{0\}$ then $\mathfrak g$ is an algebraic Lie algebra. Assume $\mathfrak g$ is a linear Lie algebra and et $E$ be the smallest algebraic subgroup such that $ \operatorname{Lie} (E) = \mathfrak e\supset \mathfrak g$. We may assume that $E$ is also centerless. Let $A_{1}, \ldots, A_{r}$ be a basis of $\mathfrak g$, for $i=1,\ldots,r$, $X_{i} = \omega^{-1}(A_{i})$. Complete with $B_{1}, \ldots, B_{p}$ in such way that $A_1,\ldots,A_r,B_1,\ldots,B_p$ is a basis of $\mathfrak e$. We consider in $E \times M$ the distribution spanned by the vector fields $A_{i}+X_{i}$. This is a $E$-principal connection called $\nabla$. Let $\overline{\nabla}$ be the induced connection via the adjoint representation on $\mathfrak e \times M$ then 1. $\overline{\nabla}$ preserves $\mathfrak g$ and $\overline{\nabla}|_{\mathfrak g} = \nabla^{\rm rec}$, by hypothesis $\mathfrak{gal}(\overline{\nabla}|_{\mathfrak g}) = \{0\}$; 2. if $\widetilde{\nabla}$ is the quotient connection on ${\mathfrak e}/ \mathfrak g$ then $\mathfrak{gal}(\widetilde{\nabla}) =\{0\}$. If $\varphi \in \mathfrak{gal}(\overline{\nabla})$ then for any $X \in \mathfrak g$, $[X,B_{i}] \in \mathfrak g$ thus $0 = \varphi [X,B_{i}] = [ X, \varphi B_{i}]$ and $\varphi B_i$ commute with $\mathfrak g$. From the second point above $\varphi B \in \mathfrak g$. By hypothesis $\varphi B_i =0$ and $\mathfrak{gal}(\overline{\nabla}) =\{0\}$. The projection on $E$ of an algebraic leaf of $\nabla$ gives an algebraic leaf for the foliation of $E$ by the left translation by $\mathfrak g$. This proves the lemma. \[th\_criteria\] Let $\mathfrak g$ be a centerless Lie algebra. An algebraic variety $(M,\omega)$ with a rational parallelism of type $\mathfrak g$ is isogenous to an algebraic group if and only if $\mathfrak{gal}(\nabla^{\rm rec}) = \{0\}$. It follows directly from Lemma \[th:algebraic\] and Corollary \[cor:rec\]. \[th:pair\]Let $\mathfrak g$ be a centerless Lie algebra. Any algebraic variety endowed with a pair of commuting rational parallelisms of type $\mathfrak g$ is isogenous to an algebraic group endowed with its two canonical parallelisms of left and right invariant vector fields. Just note that to have a pair of commuting parallelism is a more restrictive condition than having a parallelism with vanishing Lie algebra of the Galois group of its reciprocal connection. This result can be seen as an algebraic version of Wang result in [@Wang]. It gives the classification of algebraic varieties endowed with pairs of commuting parallelisms. Assuming that the Lie algebra is centerless is not a superfluous hypothesis, note that the result clearly does not hold for abelian Lie algebras. There are rational $1$-forms in $\mathbf{CP}_1$ that are not exact (isogenous to $(\mathbf C, {\rm d}z)$) nor logarithmic (isogenous to $(\mathbf C^*,{\rm d}\log(z))$). In these examples, the pair of commuting parallelisms is given by twice the same parallelism. Let $(M,\omega,\omega')$ be a manifold endowed with a pair of commuting parallelism forms of type $\mathfrak g$, a centerless Lie algebra. From Lemma \[th:algebraic\] we have that $\mathfrak g$ is an algebraic Lie algebra. We can construct the algebraic group enveloping $\mathfrak g$ as follows. We consider the adjoint action $$\begin{gathered} {\rm adj}\colon \ \mathfrak g\hookrightarrow \operatorname{End}(\mathfrak g).\end{gathered}$$ The algebraic group enveloping $\mathfrak g$ is identified with the algebraic subgroup $G$ of $\operatorname{Aut}(\mathfrak g)$ whose Lie algebra is ${\rm adj}(\mathfrak g)$. From Corollary \[cor:rec\](a), we have that $\operatorname{Gal}(\Theta)=\{e\}$. Thus, there is a rational map $f\colon M\to G$ such that $f^*(\theta) = \omega$, where $\theta$ is the Maurer–Cartan form of $G$. We can express explicitly this map in terms of the commuting parallelism forms. For each $x\in M$ in the domain of regularity of the parallelisms, $\omega(x)$ and $\omega'(x)$ are isomorphisms of $T_xM$ with $\mathfrak g$. We define $$\begin{gathered} f(x) = - \omega(x)\circ \omega'(x)^{-1}.\end{gathered}$$ In virtue of Corollary \[th:pair\], if $\mathfrak g$ is a non-algebraic centerless Lie algebra, there is no algebraic variety endowed with a pair of regular commuting parallelisms of type $\mathfrak g$. This limits the possible generalizations of Theorem \[TDeligne\]. B. Malgrange has given in [@malgrange-P] another criterion: If $(M, \omega)$ is a parallelized variety and $\mathcal F$ is the foliation on $M \times M$ given by $\operatorname{pr}_1^\ast \omega - \operatorname{pr}_2^\ast \omega = 0$. Then $(M,\omega)$ is birational to an algebraic group if and only if leaves of $\mathcal F$ are graphs of rational maps. The relations with Theorem \[th\_criteria\] and Corollary \[th:pair\] are the following. One can identify $TM$ with the vertical tangent (i.e., the kernel of ${\rm d} \operatorname{pr}_2$) along the diagonal in $M\times M$. The diagonal is a leaf of $\mathcal{F}$ and the linearization of $\mathcal F$ along the diagonal defines a connection $\nabla_{\mathcal F}$ on $TM$. By construction: - $\nabla_{\mathcal F}$-horizontal sections commute with the parallelism, it is the reciprocal Lie connection; - if leaves of $\mathcal F$ are algebraic then $\nabla_{\mathcal F}$-horizontal section are algebraic. Some homogeneous varieties ========================== The notion of [*isogeny*]{} can be extended beyond the simply-transitive case. Let us consider a complex Lie algebra $\mathfrak g$. An [*infinitesimally homogeneous variety*]{} of type $\mathfrak g$ is a pair $(M,\mathfrak s)$ consisting of a complex smooth irreducible variety $M$ and a finite-dimensional Lie algebra isomorphic to $\mathfrak g$. As before, we are interested in conjugation by rational and algebraic maps so that, whenever necessary, we replace $M$ by a suitable Zariski open subset. In this context, we say that a dominant rational map $f\colon M_1 \dasharrow M_2$ between varieties of the same dimension conjugates the infinitesimally homogeneous varieties $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ if $f^*(\mathfrak s_2) = \mathfrak s_1$. We say that $(M_1,\mathfrak s_1)$ and $(M_2,\mathfrak s_2)$ are [*isogenous*]{} if they are conjugated to the same infinitesimally homogeneous space of type $\mathfrak g$. Let $G$ be an algebraic group over $\mathbf C$, $K$ an algebraic subgroup, $\mathfrak{lie}(G)$ its Lie algebra of left invariant vector fields and $\mathfrak{lie}(G)^{\rm rec}$ its Lie algebra of right invariant vector fields. A natural example of infinitesimally homogeneous space are the homogeneous spaces $G/H$ endowed with the induced action of the Lie algebra $\mathfrak{lie}(G)^{\rm rec}$. We want to recognize when a infinitesimally homogeneous space is isogenous to an homogeneous space. We prove that if $\mathfrak s \subset \mathfrak X(M)$ is a [*normal*]{} Lie algebra of vector fields then $(M,\mathfrak s)$ is isogenous to a homogeneous space. In particular, we prove that any $n$-dimensional infinitesimally homogeneous space of type $\mathfrak{sl}_{n+1}(\mathbf C)$ is isogenous to the projective space. Our answer is based on a generalization of the computations done in Section \[sl2\]. The $\boldsymbol{\mathfrak{sl}_2}$ case --------------------------------------- Let ${\mathscr{C}}$ be a curve with $X$, $Y$, $H$ three rational vector fields such that $[X,Y] = H$, $[H,X] = -X$ and $[H,Y] = Y$. Then there exists a rational dominant map $h \colon {\mathscr{C}}\dasharrow \mathbf {CP}_1$ such that $X = h^\ast\big(\frac{\partial}{\partial z}\big)$, $H = h^\ast \big(z\frac{\partial}{\partial z}\big)$ and $Y = h^\ast \big(z^2\frac{\partial}{\partial z}\big)$. Their proof is elementary. We outline here a more sophisticated proof in the case ${\mathscr{C}}= A^1_{\mathbf C}$ that will be generalized in the next section. Notations are the ones introduced in Section \[sl2\]. ${\mathscr{L}}$ is the space of parameterized arcs $\widehat{z} = \sum_i z^{(i)} \frac{x^i}{i!}$ on ${\mathscr{C}}$. The vector space $\mathbf C X + \mathbf C H + \mathbf C Y$ is denoted by $\mathfrak g$. Let $r_o\colon (\mathbf C, 0) \to A^1_\mathbf C$ be an arc with $r_o'(0) \not = 0$ and consider $V \subset {\mathscr{L}}$ defined by $$\begin{gathered} V =\{ \widehat{z} \in {\mathscr{L}}\, | \, \widehat{z}^\ast \mathfrak g =r_o^\ast \mathfrak g \}.\end{gathered}$$ This is a $3$-dimensional algebraic variety. The prolongations ${\mathscr{L}}X$, ${\mathscr{L}}Y$ and ${\mathscr{L}}H$ define a $\mathfrak{sl}_2$-parallelism on $V$. Let us describe the canonical structure of ${\mathscr{L}}$ (see [@guillemin-sternberg pp. 11–12] or next section for a different presentation). For $k$ an integer greater or equal to $-1$, let us consider the vector field on ${\mathscr{L}}$ $$\begin{gathered} E_k = \sum_{i\geq k} \frac{i !}{(i-k-1)!} z^{(i-k)}\frac{\partial}{\partial z^{(i)}}.\end{gathered}$$ We define a morphism of Lie algebra $\rho\colon \widehat{\mathfrak X} \to \mathfrak X({\mathscr{L}})$ by $x^{k+1}\frac{\partial}{\partial x} \mapsto E_k$ and the adic continuity. The Cartan form $\sigma$ $($as defined in Section [\[sl2parallelisms\])]{} restricted to $V$ takes values in the Lie algebra $r_0^*(\mathfrak g)$. It is the parallelism form reciprocal to the parallelism ${\mathscr{L}}X$, ${\mathscr{L}}H$ and ${\mathscr{L}}Y$ of $V$. Using Corollary \[th:pair\], $V$ is isogeneous to ${\rm PSL}_2(\mathbf C)$ as defined in Definition \[isogenous\]. For $p\in M$, $V_p = \{ \widehat{z} \in V \ |\ \widehat{z}(0)=p\}$ are homogeneous spaces for the action of $\widetilde{K} = \{\varphi \colon (\mathbf C,0) \to (\mathbf C,0) \, |\, r_o\circ \varphi \in V\}$, i.e., ${\mathscr{C}}= V/\widetilde{K}$. Let $K$ be the subgroup of ${\rm PSL}_2(\mathbf C)$ of upper triangular matrices. The actions of $\widetilde{K}$ on $V$ and the right action of $K$ on ${\rm PSL}_2(\mathbf C)$ are conjugated by the isogeny. This induces an isogeny between ${\mathscr{C}}$ and $\mathbf {CP}_1$. Let $\pi_1$ and $\pi_2$ be the two maps of the isogeny. A local transformation $\varphi$ such that $\pi_1\circ \varphi = \pi_1$ satisfies $\varphi^\ast \pi_1^\ast(X,H,Y) = \pi_1^\ast(X,H,Y)$ and the same is true for the push-forward $(\pi_2)_\ast \varphi$ of $\varphi$ on $\mathbf {CP}_1$. Then $(\pi_2)_\ast \varphi$ preserves $\frac{\partial}{\partial z}$ and $z\frac{\partial}{\partial z}$. It is the identity. This finishes the proof. Some jet spaces {#some_jet_spaces} --------------- Let $M$ be a $n$-dimensional affine variety. The space of parameterized subspaces of $M$ is the set of formal maps: $ M^{[n]} = \{ r\colon (\mathbf C^n,0) \to M \}$. Like the arc space, it has a natural structure of pro-algebraic variety. We will give the construction of its coordinate ring following [@beilinson-drinfeld Section 2.3.2, p. 80]. Let $\mathbf C[\partial_1, \ldots, \partial_n]$ be the $\mathbf C$-vector space of linear partial differential operators with constant coefficients. The coordinate ring of $M^{[n]}$ is $\operatorname{Sym}(\mathbf C[M]\otimes \mathbf C[\partial_1,\ldots,\partial_n]) / \mathcal L$ where - the tensor product is a tensor product of $\mathbf C$-vector spaces; - $\operatorname{Sym}( V )$ is the $\mathbf C$-algebra generated by the vector space $V$; - $\mathbf C[M]\otimes \mathbf C[\partial_1,\ldots, \partial_n]$ has a structure of $\mathbf C[\partial_1,\ldots,\partial_n]$-module [*via*]{} the right composition of differential operators; - $\operatorname{Sym}(\mathbf C[M]\otimes \mathbf C[\partial_1,\ldots, \partial_n])$ has the induced structure of $\mathbf C[\partial_1,\ldots,\partial_n]$-algebra; - the Leibniz ideal $\mathcal L$ is the $\mathbf C[\partial_1,\ldots, \partial_n]$-ideal generated by $fg\otimes 1 - (f\otimes1)(g\otimes1)$ for all $(f,g) \in \mathbf C[M]^2$ and by $1 - 1\otimes 1$. Local coordinates $(z_1, \ldots, z_n)$ on $M$ induce local coordinates on $M^{[n]}$ [*via*]{} the Taylor expansion of maps $r$ at $0$ $$\begin{gathered} r(x_1\ldots, x_n) = \left( \sum_{\alpha \in \mathbf N^n} r_1^{\alpha} \frac{x^\alpha}{\alpha!}, \ldots,\sum_{\alpha \in \mathbf N^n} r_n^{\alpha} \frac{x^\alpha}{\alpha!} \right).\end{gathered}$$ One denotes by $z_i^{\alpha}\colon M^{[n]} \to \mathbf C$ the function defined by $z_i^{\alpha}(r) = r_i^{\alpha}$. This function is the element $z_i\otimes \partial^\alpha$ in $\mathbf C[M^{[n]}]$. ### Prolongation of vector fields Any derivation $Y$ of $\mathbf C[M]$ can be trivially extended to a derivation of $\operatorname{Sym}(\mathbf C[M]\otimes \mathbf C[\partial_1, \ldots, \partial_n])$. It preserves the ideal generated by $fg\otimes 1 - (f\otimes1)(g\otimes1)$ for all $(f,g) \in \mathbf C[M]^2$ and by $1 - 1\otimes 1$ and commutes with the action of $\mathbf C[\partial_1, \ldots, \partial_n]$ then it preserves the Leibniz ideal and defines a derivation of $\mathbf C[M^{[n]}]$. This derivation is called the prolongation of $Y$, and it is denoted by $Y^{[n]}$. The same procedure can be used to define the prolongation of analytic or formal vector fields on $M$ to $M^{[n]}$. ### The canonical structure {#canonicalst} The jet space $M^{[n]}$ is endowed with a differential structure on its coordinate ring and with a group action by “reparameterizations”. The compatibility condition between these two structures is well-known (see [@guillemin-sternberg pp. 11–23]) and is easily obtained using the construction above. The action of $\partial_j\colon \mathbf C[M^{[n]}] \to \mathbf C[M^{[n]}]$ can be written in local coordinates and gives the total derivative operator $\sum_{i,\alpha} z_i^{\alpha + 1_j} \frac{\partial}{\partial z_i^{\alpha}}$. It is the differential structure of the jet space. The pro-algebraic group $$\begin{gathered} \Gamma = \big\{ \gamma\colon (\mathbf C^n,0) \overset{\sim}{\rightarrow} (\mathbf C^n,0); \text{ formal invertible}\big\}\end{gathered}$$ acts on $M^{[n]}$.This action is denoted by $S \gamma (r) = r\circ \gamma$. These two actions arise from the action of the Lie algebra $\widehat{\mathfrak{X}} = \bigoplus \mathbf C[[x_1,\ldots,x_n]]\partial_i$ on $M^{[n]}$. This action is described on the coordinate ring in the following way. For $\xi \in \widehat{\mathfrak{X}}$, $f\in \mathbf C[M]$ and $P \in \mathbf C[\partial_1,\ldots,\partial_n]$, we define $\xi \cdot ( f \otimes P) = f\otimes (P\circ \xi)|_0$ where the composition is evaluated in $0$ in order to get an element of $\mathbf C[\partial_1,\ldots,\partial_n]$. The action of $\bigoplus \mathbf C \partial_i$ is the differential structure. The action of $\widehat{\mathfrak{X}}^0 = \mathfrak{lie}(\Gamma)$, the Lie subalgebra of vector fields vanishing at $0$ is the infinitesimal part of the action of $\Gamma$. \[canonique\] Let $M^{[n]\ast}$ be the open subset of submersions. The action above gives a canonical form $\sigma\colon T M^{[n]\ast} \to \widehat{\mathfrak{X}}$ satisfying: - for any $r \in M^{[n]\ast}$, $\sigma$ is a isomorphism from $T_r M^{[n]\ast}$ to $\widehat{\mathfrak{X}}$; - for any $\gamma \in \Gamma$, $(S\gamma)^\ast \sigma = \gamma^\ast \circ \sigma$; - $d\sigma = -\frac{1}{2}[\sigma, \sigma]$. These equalities are [*not*]{} compatible with the projective systems. Normal Lie algebras of vectors fields ------------------------------------- Without lost of generality, we should 1. identify $\mathfrak g$ with its image in $\mathfrak X(M)$; 2. replace $M$ by a Zariski open subvariety on which $\mathfrak g$ is defined and of maximal rank at any point. If $p\in M$ one can identify $\mathfrak g$ with a Lie subalgebra of $\widehat{\mathfrak X}(M,p)$, the Lie algebra of formal vector fields on $M$ at $p$. For a Lie subalgebra $\mathfrak g \subset \mathfrak X[M]$, its normalizer at $p\in M$ is $$\begin{gathered} \widehat{N}(\mathfrak g,p) = \big\{ Y \in \widehat{\mathfrak X}(M,p) \, |\, Y,\mathfrak g] \subset \mathfrak g\big\}.\end{gathered}$$ A Lie subalgebra $\mathfrak g \subset \mathfrak X[M]$ is said to be normal if for generic $p \in M$ on has $ \widehat{N}(\mathfrak g,p) = \mathfrak g$. If $\mathfrak g$ is transitive then the Lie algebra $ \widehat{N}(\mathfrak g,p) $ is finite-dimensional. Let $k$ be an integer large enough so that the only element of $\mathfrak g$ vanishing at order $k$ at $p$ is $0$. If $\widehat{N}(\mathfrak g,p) $ is not finite-dimensional then there exists a non-zero $Y \in \widehat{N}(\mathfrak g,p)$ vanishing at order $k+1$ at $p$. For $X \in \mathfrak g$, the Lie bracket $[Y,X]$ is an element of $\mathfrak g$ vanishing at order $k$ at $p$. It is zero meaning that $Y$ is invariant under the flows of vector fields in $\mathfrak g$. The transitivity hypothesis together with $Y(p)=0$ proves the lemma. If there exists a point $p\in M$ such that $\mathfrak g$ is maximal among finite-dimensional Lie subalgebra of $\widehat{\mathfrak X}(M,p)$ then $\mathfrak g$ is normal. Because of the preceding lemma, if such a point exists then $\mathfrak g = \widehat{N}(\mathfrak g,p)$ in $\widehat{\mathfrak X}(M,p)$. By transitivity, for any couple of points $(p_1,p_2) \in M^2$ there is a composition of flows of elements of $\mathfrak g$ sending $p_1$ on $p_2$. These flows preserve $\mathfrak g$ thus the equality holds at any $p$. Let $M$ be $n$-dimensional and $\mathfrak g$ be a transitive Lie subalgebra of rational vector fields isomorphic to $\mathfrak{sl}_{n+1}(\mathbf C)$. Then $\mathfrak g$ is normal (see [@cartan]). Centerless, transitive and normal $\boldsymbol{\Rightarrow}$ isogenous to a homogeneous space --------------------------------------------------------------------------------------------- \[homogeneous\] Let $M$ be a smooth irreducible algebraic variety over $\mathbf C$ and $\mathfrak g$ be a transitive, centerless, normal, finite-dimensional Lie subalgebra of $\mathfrak X(M)$. Then there exists an algebraic group $G$, an algebraic subgroup $H \subset G$ and an isogeny between $(M,\mathfrak g)$ and $(G/H, \mathfrak{lie}(G))$. Moreover, if $N_G(\mathfrak{lie}(H)) = H$ then the isogeny is a dominant rational map $M \dasharrow G/H$. Because of the finiteness and the transitivity, there exists an integer $k$ such that at any $p \in M$ and for any $Y \in \widehat{N}(\mathfrak g,p)$, $j_k(Y)(p) \not = 0$, unless $Y=0$. Let $r_o\colon (\mathbf C^n,0) \to M$ be an invertible formal map with $r_o(0) =p$ a regular point. Let us consider the subspace of $M^{[n]}$ defined by $$\begin{gathered} V = \{ r \colon (\mathbf C^n,0) \to M \, |\, r^\ast \mathfrak g = r_o^\ast \mathfrak g\}.\end{gathered}$$ $V$ is finite-dimensional. If $r_o^{-1}\circ r$ is tangent to the identity at order $k$ then the induced automorphism of $\mathfrak g$ is the identity. The map $r_o^{-1}\circ r$ fixes $p$, thus it is the identity. This proves the lemma. Using $r_o$ one can identify the Lie algebra $\widehat{N}(\mathfrak g,p)$ with a Lie subalgebra of $\widehat{\mathfrak{X}}$. The latter acts on $M^{[n]}$ as described in Section \[canonicalst\]. As an application of the Theorem \[canonique\], one gets: The restriction of the canonical structure of $M^{[n]}$ gives an parallelism $$\begin{gathered} TV = r_o^\ast(\widehat{N}(\mathfrak g,p)) \times V,\end{gathered}$$ called the canonical parallelism. The horizontal sections of the reciprocal Lie connection of the canonical parallelism are $Y^{[n]}$ for $Y \in \widehat{N}(\mathfrak g,q)$ for $q\in M$. Under the hypothesis of normality of $\mathfrak g$, $V$ has two commuting parallelisms of type $\mathfrak g$. Using Corollary \[th:pair\], $\mathfrak g$ is the Lie algebra of an algebraic group $G$ isogeneous to $V$. $V$ is foliated by the orbits of the subgroup $K$ of $\Gamma$ stabilizing $V$. This group is algebraic with Lie algebra $\mathfrak k = r_o^\ast (\mathfrak g) \cap \widehat{\mathfrak{X}}^0$. Let $\mathfrak h \subset \mathfrak{lie}(G)$ be the Lie algebra corresponding to $\mathfrak k$ by the isogeny. Then the orbits of $\mathfrak h$ are algebraic. This means that $\mathfrak h$ is the Lie algebra of an algebraic subgroup $H$ of $G$, and that $V/K$ and $G/H$ are isogenous. Assume that $N_{G}(\mathfrak{lie}(H)) = H$. If $W$ is the isogeny between $V$ and $G$. The push-forward of a local analytic deck transformation of $W \to V$ is a transformation of $G$ preserving each element of $\mathfrak g$, it is a right translation. A deck transformation preserves the orbits of the pull-back of $\mathfrak k$ on $W$. Its push-forward preserves the orbits of a group containing $H$ with the same Lie algebra. By hypothesis the push-forward is in $H$ and then the isogony obtained by taking the quotient under $K$ and $H$ is the graph of a dominant rational map. Picard–Vessiot theory of a principal connection {#ApA} =============================================== In the previous reasoning we have used the concept of differential Galois group of a connection. Here, we present a dictionary between invariant connection and strongly normal differential field extension (in the sense of Kolchin). In our setting a differential field is a pair $(\mathcal K, \mathcal D)$ where $\mathcal K$ is a finitely generated field over $\mathbf C$ and $\mathcal D$ is a $\mathcal K$ vector space of derivations of $\mathcal K$ stable by Lie bracket. The dimension of $\mathcal D$ is called the rank of the differential field. Note that we can adapt this notion easily to that of a finite number of commuting derivations by taking a suitable basis of $\mathcal D$. However we prefer to consider the whole space of derivations. With our definition a differential field extension $(\mathcal K, \mathcal D) \to (\mathcal K', \mathcal D')$ is a field extension $\mathcal K \subset \mathcal K'$ such that each element of $\mathcal D$ extends to a unique element of $\mathcal D'$ and such extensions span the space $\mathcal D'$ as $\mathcal K'$-vector space. Differential field extensions and foliated varieties ---------------------------------------------------- First, let us see that there is a natural dictionary between finitely generated differential fields over $\mathbf C$ and irreducible foliated varieties over $\mathbf C$ modulo birational equivalence. Let $(M,\mathcal F)$ be an irreducible foliated variety of dimension $n$. The distribution $T\mathcal F \subset TM$ is of rank $r\leq n$. We denote by $\mathfrak X_{\mathcal F}$ the space of rational vector fields in $T\mathcal F$; it is a $\mathbf C(M)$-Lie algebra of dimension $r$. Hence, the pair $(\mathbf C(M),\mathfrak X_{\mathcal F})$ is a differential field. The field of constants is the field $\mathbf C(M)^{\mathcal F}$ of rational first integrals of the foliation. Let $(M,\mathcal F)$ and $(M',\mathcal F')$ be foliated varieties. A regular (rational) map $\phi\colon (M',\mathcal F')\dasharrow (M,\mathcal F)$ is a regular (rational) morphism of foliated varieties if ${\rm d}\phi$ induces an isomorphism between $T_x\mathcal F'$ and $T_{\phi(x)}\mathcal F$ for (generic values of) $x\in M'$. It is clear that $\mathcal F'$ and $\mathcal F$ have the same rank. A differential field extension, correspond here to a dominant rational map of irreducible foliated varieties $\phi\colon (M',\mathcal F')\dasharrow (M,\mathcal F)$. It induces the extension $\phi^*\colon (\mathbf C(M),\mathfrak X_{\mathcal F})\to (\mathbf C(M'),\mathfrak X_{\mathcal F'})$ by composition with $\phi$. Let $\mathcal F$ the foliation of $\mathbf C^2$ defined by $\{{\rm d}y-y{\rm d}x = 0\}$. It corresponds to the differential field $\big(\mathbf C(x,e^x), \big\langle\frac{{\rm d}}{{\rm d}x}\big\rangle\big) $. Throughout this appendix “connection” means “flat connection”. Invariant $\boldsymbol{\mathcal F}$-connections ----------------------------------------------- Let us consider from now a foliated manifold of dimension $n$ and rank $r$ without rational first integrals $(M,\mathcal F)$, an algebraic group $G$ and a principal irreducible $G$-bundle $\pi\colon P\to M$. A $G$-invariant connection in the direction of $\mathcal F$ is a foliation $\mathcal F'$ of rank $r$ in $P$ such that: - $\pi\colon (P,\mathcal F')\to (M,\mathcal F)$ is a dominant regular map of foliated varieties; - The foliation $\mathcal F'$ is invariant by the action of $G$ in $P$. With this definition $(\mathbf C(M), \mathfrak X_{\mathcal F})\to (\mathbf C(P), \mathfrak X_{\mathcal F'})$ is a differential field extension. Also, each element $g\in G$ induces a differential field automorphism of $(\mathbf C(P), \mathfrak X_{F'})$ that fixes $(\mathbf C(M), \mathfrak X_F)$ by setting $(g\cdot f)(x) = f(x\cdot g)$. Let $\mathfrak g$ be the Lie algebra of $G$. There is a way of defining a $G$-equivariant form $\Theta_{\mathcal F'}$ with values in $\mathfrak g$, and defined in ${\rm d}\pi^{-1}(T\mathcal F)$ in such way that $T\mathcal F'$ is the kernel of $\Theta_{\mathcal F'}$. First, there is a canonical form $\Theta_0$ defined in $\ker(d\pi)$ that sends each vertical vector $X_p\in \ker d_p\pi\subset T_pP$ to the element $\mathfrak g$ that verifies, $$\begin{gathered} \left.\frac{{\rm d}}{{\rm d}\varepsilon}\right|_{\varepsilon = 0} p \cdot \exp{\varepsilon A} = X_p.\end{gathered}$$ This form is $G$-equivariant in the sense that $R_g^*(\Theta_0) = \operatorname{Adj}_{g^{-1}} \circ \omega$. We have a decomposition of the vector bundle ${\rm d}\pi^{-1}(T\mathcal F) = \ker({\rm d}\pi) \oplus T\mathcal F'$. This decomposition allows to extend $\Theta_0$ to a form $\Theta_{\mathcal F'}$ defined for vectors in ${\rm d}\pi^{-1}(T\mathcal F)$ whose kernel is precisely $T\mathcal F$. We call *horizontal frames* to those sections $s$ of $\pi$ such that $s^*(\Theta_{\mathcal F}) = 0$. Picard–Vessiot bundle --------------------- We say that the principal $G$-bundle with invariant $\mathcal F$-connection $\pi\colon (P,\mathcal F')\to (M,\mathcal F)$ is a *Picard–Vessiot* bundle if there are no rational first integrals of $\mathcal F'$. The notion of Picard–Vessiot bundle corresponds exactly to that of primitive extension of Kolchin. In such case $G$ is the group of differential field automorphisms of $(\mathbf C(P), \mathfrak X_{F'})$ that fix $(\mathbf C(M), \mathfrak X_F)$ and $(\mathbf C(M), \mathfrak X_F)\to (\mathbf C(P), \mathfrak X_{F'})$ is a strongly normal extension. Moreover, any strongly normal extension with constant field $\mathbf C$ can be constructed in this way (see [@Kolchin Chapter VI, Section 10, Theorem 9]). One of the most remarkable properties of strongly normal extensions is the Galois correspondence (from [@Kolchin Chapter VI, Section 4]). Assume that $(\mathbf C(M),\mathfrak X_{\mathcal F})\to (\mathbf C(P),\mathfrak X_{\mathcal F'})$ is strongly normal with group of automorphisms $G$. Then, there is a bijection between the set of intermediate differential field extensions and algebraic subgroups of $G$. To each intermediate differential field extension, it corresponds the group of automorphisms that fix such an extension point-wise. To each subgroup of automorphisms it corresponds its subfield of fixed elements. The Picard–Vessiot bundle of an invariant $\boldsymbol{\mathcal F}$-connection ------------------------------------------------------------------------------ Let us consider an irreducible principal $G$-bundle $\pi\colon (P,\mathcal F') \to (M,\mathcal F)$ endowed with an invariant $\mathcal F$-connection $\mathcal F'$. We assume that $\mathcal F$ has no rational first integrals. A result of Bonnet (see [@Bonnet Theorem 1.1]) ensures that for a very generic point in $M$ the leaf passing through such point is Zariski dense in $M$. Let us consider such a Zariski-dense leaf $\mathcal L$ of $\mathcal F$ in $M$. Let us consider any leaf $\mathcal L'$ of $\mathcal F'$ in $P$ that projects by $\pi$ onto $\mathcal L$. Its Zariski closure is unique in the following sense: \[uniqueness\]Let $\mathcal L'$ and $\mathcal L''$ two leaves of $\mathcal F'$ whose projections by $\pi$ are Zariski dense in $M$. Then, there exist an element $g\in G$ such that $\overline{\mathcal L'} \cdot g = \overline{\mathcal L''}$. By construction, there is some $x\in\pi(\mathcal L')\cap\pi(\overline{\mathcal L''})$. Let us consider $p\in \pi^{-1}(\{x\})\cap \mathcal L'$ and $q\in\pi^{-1}(\{x\})\cap \overline{\mathcal L''}$. Since $p$ and $q$ are in the same fiber, there is a unique element $g\in G$ such that $p\cdot g = q$. By the $G$-invariance of the connection $\mathcal L'\cdot g$ is the leaf of $\mathcal F'$ that passes through $q$. The set $\overline{\mathcal L''}$ is, by construction, union of leaves of $\mathcal F'$ and contains the point $q$. Thus, $\overline{\mathcal L' \cdot g} \subseteq \overline{\mathcal L''}$, and $\overline{\mathcal L'}\cdot g \subseteq \overline{\mathcal L''}$. Now, by exchanging the roles of $\mathcal L'$ and $\mathcal L''$, we prove that there is an element $h$ such that $\overline{\mathcal L''}\cdot h\subseteq \overline{\mathcal L'}$. It follows $h = g^{-1}$. This finishes the proof. Let $L$ be the Zariski closure of $\mathcal L'$. Let us consider the algebraic subgroup $$\begin{gathered} H = \{g\in G \colon L \cdot g = L\}\end{gathered}$$ stabilizing $L$. The projection $\pi$ restricted to $L$ is dominant, thus there is a Zariski open subset $M^\star$ such that $\pi^\star \colon L^\star \to M^\star$ is surjective. Let us call $\mathcal F^\star$ the restriction of $\mathcal F'$ to $L^\star$. It follows that the bundle: $\pi^\star \colon (L^\star,\mathcal F^\star) \to (M^\star, \mathcal F|_{M^\star})$ is a principal bundle of structure group $H$ called Picard–Vessiot bundle. The differential field extension $(\mathbf C(M),\mathfrak X_{\mathcal F}) \to (\mathbf C(L^\star),\mathfrak X_{\mathcal F^\star})$ is the so-called Picard–Vessiot extension associated to the connection. The algebraic group $H$ is the differential Galois group of the connection. Split of a connection --------------------- Let us consider a pair of morphisms of foliated varieties $$\begin{gathered} \phi_j\colon \ (M_j,\mathcal F_j)\to(M,\mathcal F),\qquad \mbox{for} \quad j=1,2.\end{gathered}$$ Then, we can define in $M_1\times_M M_2$ a foliation $\mathcal F_1\times_{\mathcal F} \mathcal F_2$ in the following way. A vector $X = (X_1,X_2)$ is in $T(\mathcal F_1\times_{\mathcal F}\mathcal F_2)$ if and only if ${\rm d}\phi_1(X_1)= {\rm d}\phi_2(X_2)\in T\mathcal F$. Let us consider $(P,\mathcal F')$ a principal $\mathcal F$ connection. Note that the projection $$\begin{gathered} \pi_1 \colon \ (M_1\times_M P, \mathcal F_1\times_{\mathcal F} \mathcal F')\to (M_1,\mathcal F_1)\end{gathered}$$ is a principal $G$-bundle endowed of a $\mathcal F_1$-connection. We call this bundle the pullback of $(P,\mathcal F')$ by $\phi_1$. We also may consider the trivial $G$-invariant connection $\mathcal F_0$ in the trivial principal $G$-bundle $$\begin{gathered} \pi_0\colon \ (M\times G,\mathcal F_0) \to (M,\mathcal F),\end{gathered}$$ for what the leaves of $\mathcal F_0$ are of the form $(\mathcal L, g)$ where $\mathcal L$ is a leaf of $\mathcal F$ and $g$ a fixed element of $G$. We say that the $G$-invariant connection $(P,\mathcal F')$ is rationally trivial if there is a birational $G$-equivariant isomorphism of foliated manifolds between $(P,\mathcal F)$ and $(M\times G, \mathcal F_0)$. Invariant connections are always trivialized after pullback; there is a universal $G$-equivariant isomorphism defined over $P$ $$\begin{gathered} (P \times G, \mathcal F'\times_{\mathcal F} \mathcal F_0) \to (P\times_M P, \mathcal F'\times_{\mathcal F}\mathcal F'), \qquad (p, g) \mapsto (p, p\cdot g),\end{gathered}$$ that trivializes any $G$-invariant connection. However, the differential field $(\mathbf C(P),\mathfrak X_{\mathcal F'})$ may have new constant elements. To avoid this, we replace the pullback to $P$ by a pullback to the Picard–Vessiot bundle $L^\star$ $$\begin{gathered} (L^\star \times G, \mathcal F^\star\times_{\mathcal F} \mathcal F_0) \to (L^\star\times_M P, \mathcal F^\star\times_{\mathcal F}\mathcal F'), \qquad (p, g) \mapsto (p, p\cdot g).\end{gathered}$$ The Picard–Vessiot bundle has some minimality property. It is the smallest bundle on $M$ that trivializes the connection. We have the following result. \[uniqueness2\] Let us consider $\pi\colon (P,\mathcal F') \to (M,\mathcal F)$ be as above, $\pi^\star \colon (L^\star,\mathcal F^\star)\to (M,\mathcal F)$ the Picard–Vessiot bundle, and and $\phi\colon \big(\tilde M, \tilde{\mathcal F}\big) \to (M, \mathcal F)$ any dominant rational map of foliated varieties such that: - $\tilde{\mathcal F}$ has no rational first integrals in $\tilde M$; - the pullback $\big(\tilde M \times_M P,\tilde F \times_{\mathcal F}\mathcal F'\big)\to \big(\tilde M, \tilde{\mathcal F}\big)$ is rationally trivial. There is a dominant rational map of foliated varieties $\psi\colon \tilde M \dasharrow L^\star$ such that $\pi^\star\circ\psi = \phi$ in their common domain. Let us take $\tau \colon \tilde M \times G \dasharrow \tilde M \times_M P$ a birational trivialization, $\pi_2 \colon \tilde M \times_M P \to P$ be the projection in the second factor, and $\iota \colon \tilde M \to \tilde M \times G$ the inclusion $p \mapsto (p,e)$. Then, $\tilde\psi = \pi_2 \circ \tau \circ \iota$ is a rational map from $\tilde M$ to $P$ whose differential sends $T\tilde{\mathcal F}$ to $T\mathcal F$. By Bonnet theorem, $\tilde M$ is the Zariski closure of a leaf of $\tilde{\mathcal F}$ that projects by $\phi$ into a Zariski dense leaf of $\mathcal F$. From this, $\tilde\psi$ contains a dense leaf of $\mathcal F'$ in $P$. By applying a suitable right translation in $P$ and the uniqueness Theorem \[uniqueness\], we obtain the desired conclusion. Linear connections {#A6} ------------------ Let $(M,\mathcal F)$ be as above, of dimension $n$ and rank $r$. Let $\xi\colon E\to M$ be a vector bundle of rank $k$. A linear integrable $\mathcal F$-connection is a foliation $\mathcal F_E$ of rank $r$ which is compatible with the structure of vector bundle in the following sense: the point-wise addition of two leaves of any dilation of a leaf is also a leaf. This can also be stated in terms of a covariant derivative operator $\nabla$ wich is defined only in the direction of $\mathcal F$. First, the kernel of ${\rm d}\xi$ is naturally projected onto $E$ itself $$\begin{gathered} {\rm vert}_0 \colon \ \ker({\rm d}\xi) \to E, \qquad X_v \mapsto w,\end{gathered}$$ where $\left.\frac{{\rm d}}{{\rm d}\varepsilon}\right|_{\varepsilon = 0} v + \varepsilon w = X_v$. Then, the decomposition of ${\rm d}\xi^{-1}(T\mathcal F)$ as $\ker({\rm d}\xi)\oplus T\mathcal F_E$ allows us to extend ${\rm vert_0}$ to a projection $$\begin{gathered} {\rm vert}\colon \ {\rm d}\xi^{-1}(T\mathcal F) \to E.\end{gathered}$$ Thus, we define for each section $s$ its covariant derivative $\nabla s = s^*({\rm vert}\circ {\rm d}s|_{T\mathcal F})$. This is a $1$-form on $M$ defined only for vectors in $T \mathcal F$. This covariant derivative has the desired properties, it is additive and satisfies the Leibniz formula $$\begin{gathered} \nabla (fs) = {\rm d}f|_{T\mathcal F}\otimes s + f \nabla s.\end{gathered}$$ In general, we write for $X$ a vector in $T\mathcal F$, $\nabla_X s$ for the contraction of $\nabla s$ with the vector $X$. It is an element of $E$ over the same base point in $M$ that the vector $X$. We call *horizontal sections* to those sections $s$ of $\xi$ such that $\nabla s = 0$. Let $\pi\colon R^1(E)\to M$ be the bundle of linear frames in $E$. It is a principal linear ${\rm GL}_k(\mathbf C)$-bundle. The foliation $\mathcal F_E$ induces a foliation $\mathcal F'$ in $R^1(E)$ that is a $G$-invariant $\mathcal F$-connection. Let us consider the Picard–Vessiot bundle, $(L^\star,\mathcal F^\star)$. The uniqueness Theorem \[uniqueness2\] on the Picard–Vessiot bundle, can be rephrased algebraically in the following way. The Picard–Vessiot extension $(\mathbf C(M), \mathfrak X_{\mathcal F}) \to (\mathbf C(L^\star), \mathfrak X_{\mathcal F^\star})$ is characterized by the following properties (cf. [@SingerVanderput Section 1.3]): - there are no new constants, $\mathbf C(L^\star) = \mathbf C$; - it is spanned, as a field extension of $\mathbf C(M)$, by the coefficients of a fundamental matrix of solutions of the differential equation of the horizontal sections. Associated connections {#ap7_associated} ---------------------- Let $\pi\colon (P,\mathcal F')\to (M,\mathcal F)$ be a $G$-invariant connection, as before, where $\mathcal F$ is a foliation in $M$ without rational first integrals. Let us consider $\phi\colon G\to {\rm GL}(V)$ a finite-dimensional linear representation of $G$. It is well known that the associated bundle $\pi_P \colon V_P \to M$, $$\begin{gathered} V_P = P\times_G V = (P\times V)/G, \qquad (p\cdot g,v) \sim (p,g\cdot v),\end{gathered}$$ is a vector bundle with fiber $V$. Here we represent the action of $G$ in $V$ by the same operation symbol than before. The $G$-invariant connection $\mathcal F'$ rises to a foliation in $P\times G$ and then it is projected to a foliation $\mathcal F_V$ in $V_P$. In this way, the projection $$\begin{gathered} \pi_P\colon \ (V_P,\mathcal F_V)\to (M,\mathcal F),\end{gathered}$$ turns out to be a linear $\mathcal F$-connection. It is called the *Lie–Vessiot* connection induced in the associated bundle. The Galois group of the principal and the associated Lie–Vessiot connection are linked in the following way. \[associated galois\]Let $H\subset G$ be the Galois group of the principal connection $\mathcal F'$. Then, the Galois group of the associated Lie–Vessiot connection $\mathcal F_V$ is $\phi(H)\subseteq {\rm GL}(V)$. Let us consider the bundle of frames $R^1(V_P)$, with its induced invariant connection $\mathcal F''$. Let us fix a basis $\{v_1,\ldots,v_r\}$ of $V$. Then, we have a map $$\begin{gathered} \tilde \pi \colon \ P \to R^{1}(V_P), \qquad p\mapsto ([p, v_1],\ldots, [p, v_r]),\end{gathered}$$ where the pair $[p, v]$ represents the class of the pair $(p,v)\in P\times V$. By construction, $\tilde\pi$ sends $T\mathcal F'$ to $T\mathcal F''$. It implies that, if $\mathcal L^{\star}$ is a Picard–Vessiot bundle for $\mathcal F'$ then $\tilde\pi(L^\star)$ is a Picard–Vessiot bundle for $\mathcal F''$. Second, if $\mathcal L^\star$ is a principal $H$ bundle, then $\tilde\pi(L^\star)$ is a principal $H/K$ bundle where $K$ is the subgroup of $H$ that stabilizes the basis $\{v_1,\ldots,v_r\}$. Let us discuss how the covariant derivative operator in $\nabla$ is defined in terms of $\Theta_{\mathcal F'}$ and the action of $G$ in $V$. Let us denote by $\phi'\colon \mathfrak g\to \mathfrak{gl}(V)$ the induced Lie algebra morphism. Let $s$ be a local section of $\xi$. Let us consider the canonical projection $\bar\pi \colon P \times V \to V(P)$. This turns out to be also a principal bundle, here the action on pairs is $(p,v)\cdot g = \big(p \cdot g, g^{-1} \cdot v\big)$. Now we can take any section $r$ of this bundle, and define $\tilde s = r\circ s$. As $r$ takes values in a cartesian product, we obtain $\tilde s = (s_1, s_2)$ where $s_1$ is a section of $\pi$ and $s_2$ is a function with values in $V$. Finally we obtain $$\begin{gathered} \label{covariant_associated} \nabla s = {\rm d}s_2|_{T\mathcal F} - \phi'(s_1^*(\Theta_{\mathcal F'}))(s_2).\end{gathered}$$ A calculation shows that it does not depend of the choice of $r$ and it is the covariant derivative operator associated to $\mathcal F_V$. In particular, if $s_2$ is already an horizontal frame, then the covariant differential is given by the first term ${\rm d}s_s|_{T\mathcal F}$. Deligne’s realization of Lie algebra {#apB} ==================================== The proof of the existence of a regular parallelism for any complex Lie algebra $\mathfrak g$ is written in a set of two letters from P. Deligne to B. Malgrange (dated from November of 2005 and February of 2010 respectively) that are published verbatim in [@Malgrange]. We reproduce here the proof with some extra details. \[TDeligne\]Given any complex Lie algebra $\mathfrak g$ there exist an algebraic variety endowed with a regular parallelism of type $\mathfrak g$. \[ap2\_1\]Let $T$ be an algebraic torus acting regularly by automophisms in some algebraic group $H$ and let $\mathfrak t$ be the Lie algebra of $T$. Let us consider the semidirect product $$\begin{gathered} \mathfrak t \ltimes H, \qquad (t,h)(t',h') = (t+t', (\exp(t')\cdot h)h')\end{gathered}$$ as an algebraic variety and analytic Lie group. Its left invariant vector fields form a regular parallelism of $\mathfrak t \ltimes H$. The Galois group of this parallelism is a torus. Let us denote by $\alpha$ the action of $T$ in $H$ and $\alpha'\colon \mathfrak t \mapsto \mathfrak X[H]$ the Lie algebra isomorphism given by the infinitesimal generators $$\begin{gathered} (\alpha'X)_h= \left.\frac{{\rm d}}{{\rm d}\varepsilon}\right|_{\varepsilon=0} \alpha_{\exp(\varepsilon t)}(h).\end{gathered}$$ Let $X$ be an invariant vector field in $\mathfrak t$. Let us compute the left invariant vector field in $t\ltimes H$ whose value at the identity is $(X_0,0)$. In order to perform the computation we write the vector as an infinitesimally near point to $(0,e)$. $$\begin{gathered} L_{(t,h)}(0 + \varepsilon X_0, e) = (t + \varepsilon X_t, \alpha_{\exp(\varepsilon X)}(h) ) = (t + \varepsilon X_t, h + \varepsilon (\alpha'X)_h).\end{gathered}$$ And thus $dL_{(t,h)}(X_0,0) = (X_t, (\alpha'X)_h)$. We conclude that $(X,\alpha'X)\in\mathfrak X[\mathfrak t\ltimes H]$ is the left invariant vector field whose value at $(0,e)$ is $(X_0,0)$. Let us consider now $Y$ a left invariant vector field in $H$. Let us compute, as before, the left invariant vector field whose value at $(t,h)$ is $(0,Y_h)$ $$\begin{gathered} L_{(t,h)}(0,e+\varepsilon Y_e) = (t, L_h(e + \varepsilon Y_e)) = (t, h + \varepsilon Y_h).\end{gathered}$$ And thus $(0,Y)$ is the left invariant vector field whose value at $(0,e)$ is $(0,Y_e)$. These vector fields of the form $(X,\alpha'X)$ and $(0,Y)$ are regular and span the Lie algebra of left invariant vector fields in $\mathfrak t\ltimes H$. Hence, they form a regular parallelism. In order to compute the Galois group of the parallelism, let us compute its reciprocal parallelism. It is formed by the right invariant vector fields in the analytic Lie group $\mathfrak t \ltimes H$. A similar computation proves that if $X$ is an invariant vector field in $\mathfrak t$ then $(X,0)$ is right invariant in $\mathfrak t \ltimes H$. For each element $\tau\in T$, $\alpha_\tau$ is a group automorphism of $H$. Thus, it induces a derived automorphism $\alpha_{\tau*}$ of the Lie algebra of regular vector fields in $H$. Let $Y$ be now a right invariant vector field in $H$. Let us compute the right invariant vector field $Z$ in $\mathfrak t\ltimes H$ whose value at $(0,e)$ is $(0,Y_e)$: $$\begin{gathered} R_{(t,h)}(0,e+ \varepsilon Y_e) = (t,\alpha_{\exp(t)}(e+ \varepsilon Y_e)h) = (t, h + \varepsilon (\alpha_{\exp(t)*}Y)_h)\end{gathered}$$ and $Z_{t,h} = (0, (\alpha_{\exp(t)*} Y)_h)$. Those analytic vector fields depend on the exponential function in a torus thus we can conclude, by a standard argument of differential Galois theory, that the associated differential Galois group is a torus. Let us consider $\mathfrak g$ an arbitrary, non algebraic, finite-dimensional complex Lie algebra. We consider an embedding of $\mathfrak g$ in the Lie algebra of general linear group and $E$ the smallest algebraic subgroup whose Lie algebra $\mathfrak e$ contains $\mathfrak g$. $E$ is a connected linear algebraic group. \[ap2\_2\] With the above definitions and notation $[\mathfrak e, \mathfrak e] = [\mathfrak g, \mathfrak g]$. Let $H$ be the group of matrices that stabilizes $\mathfrak g$ and acts trivially on $\mathfrak g/[\mathfrak g, \mathfrak g]$. Its Lie algebra $\mathfrak h$ contains $\mathfrak g$ and thus $H\supseteq E$ and $\mathfrak h \supseteq \mathfrak e$. By definition of $H$ we have $[\mathfrak h, \mathfrak g] = [\mathfrak g, \mathfrak g]$, therefore $[\mathfrak e, \mathfrak g]\subseteq [\mathfrak g,\mathfrak g]$. Let us now consider the group $H_1$ that stabilizes $\mathfrak e$ and $\mathfrak g$ and that acts trivially in $\mathfrak e/[\mathfrak g, \mathfrak g]$. This is again an algebraic group containing $E$, and its Lie algebra $\mathfrak h_1$ satisfies $[\mathfrak h_1,\mathfrak e] \subseteq [\mathfrak g, \mathfrak g]$. Taking into account $\mathfrak e \subseteq \mathfrak h_1$ we have $[\mathfrak e, \mathfrak e] \subseteq [\mathfrak g, \mathfrak g]$. The other inclusion is trivial. Because of Lemma \[ap2\_2\], the abelianized Lie algebra $\mathfrak g^{ab} = \mathfrak g / [\mathfrak g, \mathfrak g]$ is a subspace of $\mathfrak e^{ab} = \mathfrak e/ [\mathfrak e, \mathfrak e]$. Moreover, if we consider the quotient map, $\pi\colon \mathfrak e \to \mathfrak e^{ab}$, then $\mathfrak g = \pi^{-1}(\mathfrak g^{ab})$. Let us consider an algebraic Levy decomposition $E\simeq L \ltimes U$ (see [@Onishchik Chapter 6]). Here, $L$ is reductive and $U$ is the unipotent radical, consisting in all the unipotent elements of $E$. The semidirect product structure is produced by an action of $L$ in $U$, so that, $(l_1,u_1)(l_2,u_2) = (l_1l_2, (l_2\cdot u_1)u_2)$. Since $L$ is reductive, its commutator subgroup $L'$ is semisimple. Let $T$ be the center of $L$, which is a torus, the map $$\begin{gathered} \varphi\colon \ T\times L' \to L, \qquad (t,l) \mapsto tl,\end{gathered}$$ is an isogeny. The isogeny defines an action of $T\times L'$ in $U$ by $(t,l)\cdot u = tl\cdot u$. We have found an isogeny $$\begin{gathered} (T\times L')\ltimes U \to E.\end{gathered}$$ The Lie algebra $\mathfrak u$ of $U$ is a nilpotent Lie algebra, so that the exponential map $\exp\colon \mathfrak u \to U$ is regular and bijective. In general, if $V$ is an abelian quotient of $U$ with Lie algebra $\mathfrak v$ then the exponential map conjugates the addition law in $\mathfrak v$ with the group law in $V$. \[ap2\_3\] With the above definitions and notation, let $\bar{\mathfrak u}$ be the biggest quotient of $\mathfrak u^{ab}$ in which $L$ acts by the identity. We have a Lie algebra isomorphism $\mathfrak e^{ab} \simeq \mathfrak t \times \bar{\mathfrak u}$. Let us compute $\mathfrak e^{ab}$. We compute the commutators $\mathfrak e$ by means of the isomorphism $\mathfrak e \simeq (\mathfrak t \times \mathfrak l') \ltimes \mathfrak u$. We obtain $$\begin{gathered} = (0,[l_1,l_2], a(t_2,l_2)u_1 +[u_1,u_2]),\end{gathered}$$ where $a$ represents the derivative at $(e,e)$ of the action of $L$ in $U$. From this we obtain that $[\mathfrak e, \mathfrak e]$ is spanned by $(\{0\}\times \mathfrak l')\ltimes \mathfrak \{0\}$, $\{0\}\ltimes [\mathfrak u, \mathfrak u]$ and $\{0\}\times \langle a(\mathfrak l)\mathfrak u \rangle$. Taking into account that $\bar{\mathfrak u}/ ( \langle a(\mathfrak l)\mathfrak u \rangle + [\mathfrak u, \mathfrak u] )$ is the biggest quotient of $\mathfrak u^{ab}$ in which $L$ acts trivially, we obtain the result of the lemma. Let $\mathfrak t$ be the Lie algebra of $T$. Its exponential map is an analytic group morphism and thus we may consider the analytic action of $\mathfrak t \times L'$ in $U$ given by $(t,l)\cdot u = (\exp(t) l )\cdot u$. Let $\tilde E$ be the algebraic variety and analytic Lie group $(\mathfrak t \times L') \ltimes U$. By application of Lemma \[ap2\_1\], and taking into account that $\tilde E \simeq \mathfrak t \ltimes H$, where $H$ is the group $L'\cdot U$, we have that the left invariant vector fields in $\tilde E$ are regular. Let us consider the projection $$\begin{gathered} \pi_1\colon \ \tilde E \to \mathfrak e^{ab} = \mathfrak t \times \bar{\mathfrak u}, \qquad (t,l,u) \mapsto (t,[\log(u)]),\end{gathered}$$ this projection is algebraic by construction, and also a morphism of Lie groups. By Lemmas \[ap2\_2\] and \[ap2\_3\], $\mathfrak g^{ab}$ is a vector subspace of the image. Then, let us take $\tilde G$ the fiber $\pi^{-1}_1(\mathfrak g^{ab})$. It is an algebraic submanifold of $\tilde E$ and an analytic Lie group. The derivative at the identity of $\pi_1$ is precisely the abelianization $\pi$ and it follows that the Lie algebra of $\tilde G$ is precisely $\mathfrak g$. Finally $\tilde G$ is an algebraic variety with a regular $\mathfrak g$-parallelism. This finishes the proof of Theorem \[TDeligne\]. The right invariant vector fields in $\tilde G$ are constructed as in Lemma \[ap2\_1\] by means of the exponential function in the torus. Hence, Galois groups of the parallelisms obtained via this construction are always tori. Acknowledgements {#acknowledgements .unnumbered} ---------------- The authors thank the ECOS-Nord program C12M01 and the project “IsoGalois” ANR-13-JS01-0002-01. They also thank the “Universidad Nacional de Colombia”(project HERMES code 37243) and the “Université de Rennes 1” (Actions Internationales 2016) for supporting this reseach, and also the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment. The authors thank Juan Diego Vélez for his help with the final redaction of the manuscript and the anonymous referees who gave relevant contributions to improve the paper. [99]{} Beilinson A., Drinfeld V., Chiral algebras, [*American Mathematical Society Colloquium Publications*](https://doi.org/10.1090/coll/051), Vol. 51, Amer. Math. Soc., Providence, RI, 2004. Bonnet P., Minimal invariant varieties and first integrals for algebraic foliations, [*Bull. Braz. 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Malgrange B., Leon [E]{}hrenpreis: some old souvenirs, in The Mathematical Legacy of [L]{}eon [E]{}hrenpreis, [*Springer Proc. Math.*](https://doi.org/10.1007/978-88-470-1947-8_1), Vol. 16, Springer, Milan, 2012, 3–6. Onishchik A.L., Vinberg E.B., Lie groups and algebraic groups, [*Springer Series in Soviet Mathematics*](https://doi.org/10.1007/978-3-642-74334-4), Springer-Verlag, Berlin, 1990. Sharpe R.W., Differential geometry. Cartan’s generalization of Klein’s Erlangen program, *Graduate Texts in Mathematics*, Vol. 166, Springer-Verlag, New York, 1997. van der Put M., Singer M.F., Galois theory of linear differential equations, [*Grundlehren der Mathematischen Wissenschaften*](https://doi.org/10.1007/978-3-642-55750-7), Vol. 328, Springer-Verlag, Berlin, 2003. Wang H.C., Complex parallisable manifolds, [*Proc. Amer. Math. Soc.*](https://doi.org/10.2307/2031863) **5** (1954), 771–776. Winkelmann J., On manifolds with trivial logarithmic tangent bundle, *Osaka J. 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ArXiv

--- abstract: | We prove the analogue for continuous space-time of the quenched LDP derived in Birkner, Greven and den Hollander [@BiGrdHo10] for discrete space-time. In particular, we consider a random environment given by Brownian increments, cut into pieces according to an independent continuous-time renewal process. We look at the empirical process obtained by recording both the length of and the increments in the successive pieces. For the case where the renewal time distribution has a Lebesgue density with a polynomial tail, we derive the quenched LDP for the empirical process, i.e., the LDP conditional on a typical environment. The rate function is a sum of two specific relative entropies, one for the pieces and one for the concatenation of the pieces. We also obtain a quenched LDP when the tail decays faster than algebraic. The proof uses coarse-graining and truncation arguments, involving various approximations of specific relative entropies that are not quite standard. In a companion paper we show how the quenched LDP and the techniques developed in the present paper can be applied to obtain a variational characterisation of the free energy and the phase transition line for the Brownian copolymer near a selective interface. *MSC2010:* 60F10, 60G10, 60J65, 60K37.\ *Keywords:* Brownian environment, renewal process, annealed vs. quenched, empirical process, large deviation principle, specific relative entropy.\ *Acknowledgment:* The research in this paper is supported by ERC Advanced Grant 267356 VARIS of FdH. MB is grateful for hospitality at the Mathematical Institute in Leiden during a sabbatical leave from September 2012 until February 2013, supported by ERC. author: - | M. Birkner\ F. den Hollander date: 9th December 2013 title: A quenched large deviation principle in a continuous scenario --- Introduction and main result {#intro} ============================ When we cut an i.i.d. sequence of letters into words according to an independent integer-valued renewal process, we obtain an i.i.d. sequence of words. In the *annealed* LDP for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. Birkner, Greven and den Hollander [@BiGrdHo10] considered the *quenched* LDP, i.e., conditional on a typical letter sequence. The rate function of the quenched LDP turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal time distribution. The goal of the present paper is to derive the analogue of the quenched LDP for the case where the i.i.d. sequence of letters is replaced by the process of Brownian increments, and the renewal process has a length distribution with a Lebesgue density that has a polynomial tail. In Section \[setting\] we define the continuous space-time setting, in Section \[LDPs\] we state both the annealed and the quenched LDP, while in Section \[disc\] we discuss these LDPs and indicate some further extensions. In Section \[proof\] we prove the quenched LDP subject to three propositions. In Sections \[props\]–\[removeass\] we give the proof of these propositions. In Section \[proofalpha1infty\] we prove the extensions. Appendix \[metrics\] recalls a few basic facts about metrics on path space, while Appendices \[entropy\]–\[contrelentr\] prove a few basic facts about specific relative entropy that are needed in the proof and that are not quite standard. Continuous space-time {#setting} --------------------- Let $X=(X_t)_{t \geq 0}$ be the standard one-dimensional Brownian motion starting from $X_0=0$. Let ${\mathscr{W}}$ denote its law on path space: the Wiener measure on $C([0,\infty))$, equipped with the $\sigma$-algebra generated by the coordinate projections. Let $T=(T_i)_{i \in {\mathbb{N}}_0}$ ($T_0=0$) be an independent continuous-time renewal process, with interarrival times $\tau_i=T_i-T_{i-1}$, $i\in{\mathbb{N}}$, whose common law $\rho=\mathscr{L}(\tau_1)$ is absolutely continuous with respect to the Lebesgue measure on $(0,\infty)$, with density $\bar{\rho}$ satisfying $$\label{ass:rhodensdecay} \lim_{x\to\infty} \frac{\log\bar{\rho}(x)}{\log x} = - \alpha, \qquad \alpha \in (1,\infty).$$ In addition, assume that $$\label{ass:rhobar.reg0} \begin{minipage}{0.85\textwidth} $\mathrm{supp}(\rho) = [s_*,\infty)$ with $0 \leq s_* < \infty$, and $\bar{\rho}$ is continuous and strictly positive on $(s_*,\infty)$, and varies regularly near $s_*$. \end{minipage}$$ (0,0) rectangle (361.35,361.35); ( 2.40, 2.40) rectangle (358.95,358.95); ( 15.61,162.33) – (345.74,162.33); (338.42,166.36) – (341.56,164.65) – (345.14,163.38) – (349.03,162.60) – (353.07,162.33) – (353.07,162.33) – (349.03,162.07) – (345.14,161.29) – (341.56,160.02) – (338.42,158.31) – (338.42,158.31) – (338.76,158.41) – (339.08,158.71) – (339.37,159.19) – (339.61,159.82) – (339.79,160.59) – (339.90,161.44) – (339.94,162.33) – (339.90,163.23) – (339.79,164.08) – (339.61,164.84) – (339.37,165.48) – (339.08,165.96) – (338.76,166.26) – (338.42,166.36) – cycle; ( 15.61,162.33) – ( 15.94,163.78) – ( 16.27,161.59) – ( 16.60,159.54) – ( 16.93,160.84) – ( 17.26,158.90) – ( 17.59,159.60) – ( 17.92,157.91) – ( 18.25,162.99) – ( 18.58,169.37) – ( 18.91,171.53) – ( 19.24,172.96) – ( 19.57,172.49) – ( 19.90,174.03) – ( 20.23,181.44) – ( 20.56,182.39) – ( 20.89,182.86) – ( 21.22,182.87) – ( 21.55,176.96) – ( 21.88,181.97) – ( 22.21,190.43) – ( 22.54,186.60) – ( 22.87,187.90) – ( 23.20,188.38) – ( 23.53,190.46) – ( 23.86,189.69) – ( 24.19,191.41) – ( 24.52,188.72) – ( 24.85,189.63) – ( 25.18,192.42) – ( 25.51,193.43) – ( 25.84,187.48) – ( 26.17,187.08) – ( 26.50,185.68) – ( 26.83,184.93) – ( 27.16,176.03) – ( 27.49,177.69) – ( 27.82,172.43) – ( 28.15,178.64) – ( 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(213.69,358.95); (299.52, 2.40) – (299.52,358.95); ( 15.61, 55.96) – ( 15.94, 57.40) – ( 16.27, 55.21) – ( 16.60, 53.16) – ( 16.93, 54.46) – ( 17.26, 52.52) – ( 17.59, 53.22) – ( 17.92, 51.53) – ( 18.25, 56.61) – ( 18.58, 62.99) – ( 18.91, 65.15) – ( 19.24, 66.58) – ( 19.57, 66.11) – ( 19.90, 67.65) – ( 20.23, 75.06) – ( 20.56, 76.02) – ( 20.89, 76.48) – ( 21.22, 76.49) – ( 21.55, 70.58) – ( 21.88, 75.59) – ( 22.21, 84.06) – ( 22.54, 80.22) – ( 22.87, 81.52) – ( 23.20, 82.00) – ( 23.53, 84.09) – ( 23.86, 83.31) – ( 24.19, 85.04) – ( 24.52, 82.34) – ( 24.85, 83.26) – ( 25.18, 86.04) – ( 25.51, 87.06) – ( 25.84, 81.11) – ( 26.17, 80.70) – ( 26.50, 79.30) – ( 26.83, 78.55) – ( 27.16, 69.65) – ( 27.49, 71.31) – ( 27.82, 66.05) – ( 28.15, 72.26) – ( 28.48, 66.84) – ( 28.81, 58.15) – ( 29.14, 63.21) – ( 29.47, 65.26) – ( 29.80, 64.72) – ( 30.13, 59.88) – ( 30.46, 61.74) – ( 30.79, 63.97) – ( 31.12, 66.02) – ( 31.45, 64.26) – ( 31.78, 63.78) – ( 32.11, 62.12) – ( 32.44, 61.83) – ( 32.77, 65.59) – ( 33.10, 64.43) – ( 33.43, 60.69) – ( 33.76, 58.66) – ( 34.09, 61.45) – ( 34.42, 65.99) – ( 34.75, 68.72) – ( 35.08, 62.61) – ( 35.41, 64.70) – ( 35.74, 67.43) – ( 36.07, 69.22) – ( 36.40, 61.70) – ( 36.73, 66.47) – ( 37.06, 68.99) – ( 37.39, 64.67) – ( 37.72, 64.98) – ( 38.05, 62.89) – ( 38.39, 64.10) – ( 38.72, 66.43) – ( 39.05, 63.83) – ( 39.38, 61.89) – ( 39.71, 58.76) – ( 40.04, 55.85) – ( 40.37, 53.80) – ( 40.70, 53.60) – ( 41.03, 53.31) – ( 41.36, 52.88) – ( 41.69, 51.43) – ( 42.02, 56.66) – ( 42.35, 55.05) – ( 42.68, 50.24) – ( 43.01, 42.59) – ( 43.34, 44.28) – ( 43.67, 45.20) – ( 44.00, 44.98) – ( 44.33, 44.12) – ( 44.66, 33.77) – ( 44.99, 36.60) – ( 45.32, 32.95) – ( 45.65, 38.52) – ( 45.98, 30.53) – ( 46.31, 32.61) – ( 46.64, 25.92) – ( 46.97, 25.70) – ( 47.30, 29.21) – ( 47.63, 29.76) – ( 47.96, 25.95) – ( 48.29, 27.42) – ( 48.62, 30.47) – ( 48.95, 29.26) – ( 49.28, 31.01) – ( 49.61, 27.05) – ( 49.94, 22.79) – ( 50.27, 27.55) – ( 50.60, 28.95) – ( 50.93, 31.24) – ( 51.26, 30.10) – ( 51.59, 29.70) – ( 51.92, 28.52) – ( 52.25, 29.41) – ( 52.58, 24.45) – ( 52.91, 23.73) – ( 53.24, 26.35) – ( 53.57, 26.98) – ( 53.90, 36.55) – ( 54.23, 35.57) – ( 54.56, 35.26) – ( 54.89, 36.87) – ( 55.22, 43.03) – ( 55.55, 45.37) – ( 55.88, 46.68) – ( 56.21, 56.28) – ( 56.54, 62.26) – ( 56.87, 60.34) – ( 57.20, 58.00) – ( 57.53, 49.53) – ( 57.86, 45.25) – ( 58.19, 54.29) – ( 58.52, 55.62) – ( 58.85, 53.32) – ( 59.18, 51.34) – ( 59.51, 52.77) – ( 59.84, 55.18) – ( 60.17, 53.53) – ( 60.50, 49.74) – ( 60.83, 47.23) – ( 61.16, 48.94) – ( 61.49, 49.74) – ( 61.82, 51.65) – ( 62.16, 54.47) – ( 62.49, 54.73) – ( 62.82, 53.35) – ( 63.15, 54.26) – ( 63.48, 50.97) – ( 63.81, 54.22) – ( 64.14, 54.47) – ( 64.47, 53.97) – ( 64.80, 54.52) – ( 65.13, 62.32) – ( 65.46, 59.00) – ( 65.79, 60.46) – ( 66.12, 62.95) – ( 66.45, 60.88) – ( 66.78, 64.38) – ( 67.11, 62.44) – ( 67.44, 58.20) – ( 67.77, 63.75) – ( 68.10, 70.99) – ( 68.43, 69.77) – ( 68.76, 73.24) – ( 69.09, 73.34) – ( 69.42, 73.81) – ( 69.75, 72.54) – ( 70.08, 79.94) – ( 70.41, 77.46) – ( 70.74, 73.50) – ( 71.07, 73.62) – ( 71.40, 82.34) – ( 71.73, 75.73) – ( 72.06, 75.19) – ( 72.39, 73.74) – ( 72.72, 70.48) – ( 73.05, 70.92) – ( 73.38, 73.24) – ( 73.71, 76.17) – ( 74.04, 76.33) – ( 74.37, 76.93) – ( 74.70, 75.89) – ( 75.03, 75.84) – ( 75.36, 75.12) – ( 75.69, 77.84) – ( 76.02, 77.27) – ( 76.35, 72.20) – ( 76.68, 76.31) – ( 77.01, 82.56) – ( 77.34, 82.84) – ( 77.67, 80.41) – ( 78.00, 78.82) – ( 78.33, 76.25) – ( 78.66, 74.42) – ( 78.99, 70.78) – ( 79.32, 68.68) – ( 79.65, 63.24) – ( 79.98, 66.37) – ( 80.31, 68.98) – ( 80.64, 68.78) – ( 80.97, 69.18) – ( 81.30, 70.67) – ( 81.63, 71.11) – ( 81.96, 68.87) – ( 82.29, 67.70) – ( 82.62, 66.06) – ( 82.95, 65.90) – ( 83.28, 63.80) – ( 83.61, 67.42) – ( 83.94, 64.59) – ( 84.27, 65.46) – ( 84.60, 61.64) – ( 84.93, 64.97) – ( 85.26, 67.75) – ( 85.59, 65.75) – ( 85.93, 65.77) – ( 86.26, 70.67) – ( 86.59, 66.31) – ( 86.92, 63.75) – ( 87.25, 57.79) – ( 87.58, 62.60) – ( 87.91, 59.99) – ( 88.24, 61.93) – ( 88.57, 63.66) – ( 88.90, 66.59) – ( 89.23, 76.39) – ( 89.56, 77.24) – ( 89.89, 80.11) – ( 90.22, 81.25) – ( 90.55, 83.79) – ( 90.88, 82.33) – ( 91.21, 75.87) – ( 91.54, 74.49) – ( 91.87, 74.39) – ( 92.20, 74.22) – ( 92.53, 74.65) – ( 92.86, 73.22) – ( 93.19, 76.54) – ( 93.52, 70.83) – ( 93.85, 72.07) – ( 94.18, 72.83) – ( 94.51, 73.59) – ( 94.84, 74.23) – ( 95.17, 73.74) – ( 95.50, 74.15) – ( 95.83, 73.96) – ( 96.16, 84.20) – ( 96.49, 88.10) – ( 96.82, 84.84) – ( 97.15, 83.96) – ( 97.48, 78.22) – ( 97.81, 80.58) – ( 98.14, 75.42) – ( 98.47, 76.84) – ( 98.80, 75.45) – ( 99.13, 76.93) – ( 99.46, 75.20) – ( 99.79, 77.43) – (100.12, 77.53) – (100.45, 80.92) – (100.78, 85.38) – (101.11, 89.04) – (101.44, 89.26) – (101.77, 90.06) – (102.10, 88.35) – (102.43, 86.86) – (102.76, 85.22) – (103.09, 91.74) – (103.42, 90.14) – (103.75, 94.31) – (104.08, 95.16) – (104.41, 95.52) – (104.74, 95.89) – (105.07, 92.98) – (105.40, 89.00) – (105.73, 91.64) – (106.06, 89.94) – (106.39, 92.40) – (106.72, 96.36) – (107.05, 99.52) – (107.38, 94.21) – (107.71, 94.02) – (108.04, 91.58) – (108.37, 97.41) – (108.70,103.44) – (109.03,109.80) – (109.36,112.02) – (109.70,113.49) – (110.03,105.77) – (110.36,107.84) – (110.69,101.08) – (111.02,102.59) – (111.35,100.82) – (111.68, 99.17) – (112.01, 95.41) – (112.34, 91.19) – (112.67, 90.90) – (113.00, 93.42) – (113.33, 95.81) – (113.66, 93.24) – (113.99, 95.10) – (114.32, 92.57) – (114.65, 92.50) – (114.98, 88.63) – (115.31, 92.01) – (115.64, 87.83) – (115.97, 90.08) – (116.30, 89.49) – (116.63, 92.54) – (116.96, 98.94) – (117.29, 96.01) – (117.62, 99.53) – (117.95, 95.04) – (118.28, 95.06) – (118.61, 90.63) – (118.94, 85.95) – (119.27, 89.91) – (119.60, 90.64) – (119.93, 94.23) – (120.26, 92.67) – (120.59, 89.48) – (120.92, 90.71) – (121.25, 86.27) – (121.58, 86.76) – (121.91, 92.65) – (122.24, 95.34) – (122.57, 90.56) – (122.90, 93.29) – (123.23, 89.91) – (123.56, 90.56) – (123.89, 97.92) – (124.22,106.75) – (124.55,106.87) – (124.88,106.09) – (125.21,109.81) – (125.54,114.99) – (125.87,106.79) – (126.20,106.68) – (126.53,106.83) – (126.86,107.82) – (127.19,116.46) – (127.52,110.13) – (127.85,116.72) – (128.18,114.72) – (128.51,115.16) – (128.84,113.63) – (129.17,114.73) – (129.50,117.59) – (129.83,122.53) – (130.16,121.58) – (130.49,127.85) – (130.82,128.23) – (131.15,124.90); at ( 73.38, 5.77) [$\scriptstyle Y^{(1)}$]{}; (131.15, 55.96) – (131.48, 53.00) – (131.81, 53.25) – (132.14, 53.91) – (132.47, 63.70) – (132.80, 64.42) – (133.13, 66.55) – (133.47, 59.10) – (133.80, 56.82) – (134.13, 62.43) – (134.46, 61.75) – (134.79, 60.43) – (135.12, 58.53) – (135.45, 56.42) – (135.78, 58.34) – (136.11, 50.61) – (136.44, 51.41) – (136.77, 52.20) – (137.10, 55.44) – (137.43, 51.68) – (137.76, 59.47) – (138.09, 56.80) – (138.42, 59.34) – (138.75, 60.48) – (139.08, 59.39) – (139.41, 57.12) – (139.74, 59.12) – (140.07, 59.49) – (140.40, 49.03) – (140.73, 57.21) – (141.06, 63.64) – (141.39, 62.04) – (141.72, 55.14) – (142.05, 55.52) – (142.38, 55.96) – (142.71, 59.38) – (143.04, 58.98) – (143.37, 56.76) – (143.70, 57.38) – (144.03, 60.26) – (144.36, 59.75) – (144.69, 58.11) – (145.02, 52.80) – (145.35, 53.05) – (145.68, 57.66) – (146.01, 55.45) – (146.34, 58.26) – (146.67, 61.49) – (147.00, 65.79) – (147.33, 59.13) – (147.66, 60.25) – (147.99, 61.24) – (148.32, 59.69) – (148.65, 57.05) – (148.98, 62.66) – (149.31, 65.70) – (149.64, 70.83) – (149.97, 66.97) – (150.30, 54.20) – (150.63, 54.29) – (150.96, 53.07) – (151.29, 56.70) – (151.62, 51.95) – (151.95, 45.70) – (152.28, 38.04) – (152.61, 36.17) – (152.94, 35.00) – (153.27, 35.62) – (153.60, 31.65) – (153.93, 33.70) – (154.26, 35.07) – (154.59, 32.28) – (154.92, 32.17) – (155.25, 31.78) – (155.58, 33.00) – (155.91, 30.15) – (156.24, 33.43) – (156.57, 38.26) – (156.90, 37.29) – (157.24, 37.15) – (157.57, 42.66) – (157.90, 39.51) – (158.23, 36.70) – (158.56, 35.91) – (158.89, 42.46) – (159.22, 36.62) – (159.55, 39.15) – (159.88, 41.44) – (160.21, 45.42) – (160.54, 40.03) – (160.87, 41.68) – (161.20, 45.36) – (161.53, 50.75) – (161.86, 51.93) – (162.19, 49.26) – (162.52, 51.29) – (162.85, 50.50) – (163.18, 52.59) – (163.51, 48.91) – (163.84, 45.70) – (164.17, 48.65) – (164.50, 49.08) – (164.83, 53.49) – (165.16, 56.11) – (165.49, 62.44) – (165.82, 60.60) – (166.15, 55.14) – (166.48, 54.32) – (166.81, 60.34) – (167.14, 59.20) – (167.47, 59.52) – (167.80, 61.39) – (168.13, 62.20) – (168.46, 61.42) – (168.79, 56.54) – (169.12, 62.45) – (169.45, 59.68) – (169.78, 58.06) – (170.11, 55.22) – (170.44, 51.90) – (170.77, 53.94); at (150.96, 5.77) [$\scriptstyle Y^{(2)}$]{}; (170.77, 55.96) – (171.10, 55.79) – (171.43, 55.86) – (171.76, 63.15) – (172.09, 59.89) – (172.42, 62.83) – (172.75, 65.93) – (173.08, 63.35) – (173.41, 70.87) – (173.74, 75.34) – (174.07, 66.24) – (174.40, 64.14) – (174.73, 65.47) – (175.06, 62.83) – (175.39, 67.44) – (175.72, 67.37) – (176.05, 67.88) – (176.38, 64.59) – (176.71, 69.68) – (177.04, 74.41) – (177.37, 73.68) – (177.70, 73.97) – (178.03, 74.72) – (178.36, 75.11) – (178.69, 72.00) – (179.02, 78.76) – (179.35, 79.93) – (179.68, 78.81) – (180.01, 75.24) – (180.34, 70.84) – (180.67, 73.54) – (181.01, 78.64) – (181.34, 81.05) – (181.67, 79.85) – (182.00, 72.64) – (182.33, 66.16) – (182.66, 68.39) – (182.99, 63.40) – (183.32, 61.31) – (183.65, 63.56) – (183.98, 57.68) – (184.31, 58.74) – (184.64, 49.91) – (184.97, 50.38) – (185.30, 45.72) – (185.63, 40.69) – (185.96, 39.36) – (186.29, 40.67) – (186.62, 41.19) – (186.95, 47.84) – (187.28, 49.35) – (187.61, 52.60) – (187.94, 54.32) – (188.27, 53.05) – (188.60, 51.22) – (188.93, 51.22) – (189.26, 50.49) – (189.59, 51.15) – (189.92, 52.13) – (190.25, 52.60) – (190.58, 52.78) – (190.91, 52.13) – (191.24, 45.39) – (191.57, 38.64) – (191.90, 40.47) – (192.23, 45.59) – (192.56, 49.76) – (192.89, 47.32) – (193.22, 48.90) – (193.55, 46.27) – (193.88, 49.28) – (194.21, 48.46) – (194.54, 48.25) – (194.87, 53.27) – (195.20, 47.49) – (195.53, 48.80) – (195.86, 47.74) – (196.19, 46.62) – (196.52, 42.10) – (196.85, 34.23) – (197.18, 36.31) – (197.51, 33.39) – (197.84, 32.25) – (198.17, 32.84) – (198.50, 29.07) – (198.83, 33.86) – (199.16, 30.99) – (199.49, 33.21) – (199.82, 26.04) – (200.15, 21.48) – (200.48, 27.66) – (200.81, 28.85) – (201.14, 28.49) – (201.47, 30.24) – (201.80, 26.19) – (202.13, 32.25) – (202.46, 33.17) – (202.79, 31.87) – (203.12, 29.65) – (203.45, 32.99) – (203.78, 34.32) – (204.11, 29.31) – (204.44, 27.40) – (204.78, 23.48) – (205.11, 31.10) – (205.44, 28.49) – (205.77, 27.43) – (206.10, 32.06) – (206.43, 30.70) – (206.76, 24.32) – (207.09, 20.83) – (207.42, 20.76) – (207.75, 25.42) – (208.08, 23.10) – (208.41, 20.68) – (208.74, 18.69) – (209.07, 17.20) – (209.40, 19.14) – (209.73, 28.38) – (210.06, 27.86) – (210.39, 32.34) – (210.72, 35.44) – (211.05, 33.98) – (211.38, 32.89) – (211.71, 32.07) – (212.04, 28.57) – (212.37, 27.59) – (212.70, 27.98) – (213.03, 20.93) – (213.36, 22.02) – (213.69, 24.18); at (192.23, 5.77) [$\scriptstyle Y^{(3)}$]{}; (213.69, 55.96) – (214.02, 53.00) – (214.35, 58.32) – (214.68, 56.65) – (215.01, 54.14) – (215.34, 59.62) – (215.67, 57.99) – (216.00, 66.17) – (216.33, 63.80) – (216.66, 69.37) – (216.99, 72.16) – (217.32, 70.94) – (217.65, 69.90) – (217.98, 68.27) – (218.31, 70.42) – (218.64, 74.79) – (218.97, 77.43) – (219.30, 74.71) – (219.63, 80.97) – (219.96, 75.62) – (220.29, 69.86) – (220.62, 73.41) – (220.95, 75.31) – (221.28, 70.46) – (221.61, 70.47) – (221.94, 72.04) – (222.27, 81.01) – (222.60, 83.28) – (222.93, 87.06) – (223.26, 82.88) – (223.59, 88.48) – (223.92, 91.10) – (224.25, 97.19) – (224.58,103.79) – (224.91,103.68) – (225.24,107.70) – (225.57,109.25) – (225.90,104.85) – (226.23,108.30) – (226.56,112.46) – (226.89,109.55) – (227.22,112.43) – (227.55,111.33) – (227.88,105.61) – (228.21,103.88) – (228.55,102.42) – (228.88,106.74) – (229.21,103.27) – (229.54,106.88) – (229.87,108.55) – (230.20,105.19) – (230.53,104.02) – (230.86,106.85) – (231.19,106.57) – (231.52,105.48) – (231.85,107.46) – (232.18,110.11) – (232.51,108.60) – (232.84,108.63) – (233.17,103.17) – (233.50,103.20) – (233.83, 94.64) – (234.16, 92.13) – (234.49, 92.71) – (234.82, 97.61) – (235.15,108.95) – (235.48,103.81) – (235.81, 98.83) – (236.14, 99.17) – (236.47,102.39) – (236.80,104.87) – (237.13,101.75) – (237.46,102.49) – (237.79,109.67) – (238.12,112.72) – (238.45,113.51) – (238.78,113.48) – (239.11,117.52) – (239.44,117.61) – (239.77,124.61) – (240.10,122.68) – (240.43,127.69) – (240.76,125.10) – (241.09,122.24) – (241.42,125.29) – (241.75,122.61) – (242.08,122.62) – (242.41,113.40) – (242.74,110.18) – (243.07,104.63) – (243.40,100.60) – (243.73,100.44) – (244.06, 99.09) – (244.39, 99.50) – (244.72,108.25) – (245.05,114.99) – (245.38,117.33) – (245.71,120.91) – (246.04,118.51) – (246.37,120.79) – (246.70,115.51) – (247.03,114.62) – (247.36,108.98) – (247.69,111.48) – (248.02,116.94) – (248.35,113.77) – (248.68,117.47) – (249.01,120.41) – (249.34,121.71) – (249.67,121.16) – (250.00,125.63) – (250.33,126.42) – (250.66,129.75) – (250.99,136.48) – (251.32,131.72) – (251.65,129.28) – (251.98,128.77) – (252.32,129.32) – (252.65,128.09) – (252.98,126.13) – (253.31,129.60) – (253.64,128.07) – (253.97,121.75) – (254.30,121.75) – (254.63,119.36) – (254.96,125.26) – (255.29,123.80) – (255.62,134.00) – (255.95,136.43) – (256.28,130.60) – (256.61,134.30) – (256.94,137.77) – (257.27,137.14) – (257.60,133.66) – (257.93,129.08) – (258.26,132.23) – (258.59,130.82) – (258.92,123.82) – (259.25,120.12) – (259.58,116.50) – (259.91,120.91) – (260.24,113.10) – (260.57,111.01) – (260.90,109.40) – (261.23,109.65) – (261.56,109.18) – (261.89,105.38) – (262.22,110.05) – (262.55,112.64) – (262.88,114.72) – (263.21,107.81) – (263.54,104.98) – (263.87,106.29) – (264.20,106.39) – (264.53,109.48) – (264.86,110.61) – (265.19,107.25) – (265.52,109.92) – (265.85,110.26) – (266.18,108.07) – (266.51,107.06) – (266.84,103.18) – (267.17, 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(338.48,140.88) – (338.81,139.00) – (339.14,140.47) – (339.47,140.61) – (339.80,143.65) – (340.13,148.65) – (340.46,151.75) – (340.79,152.39) – (341.12,148.67) – (341.45,149.87) – (341.78,139.82) – (342.11,135.70) – (342.44,131.61) – (342.77,130.69) – (343.10,129.94) – (343.43,128.61) – (343.76,130.56) – (344.09,128.03) – (344.42,125.73) – (344.75,122.06) – (345.08,122.53) – (345.41,122.89) – (345.74,124.68); Define the *word sequence* $Y = (Y^{(i)})_{i\in{\mathbb{N}}}$ by putting (see Fig. \[fig-wordsequence\]) $$\qquad Y^{(i)} = \Big(T_i-T_{i-1}, \big(X_{(s+T_{i-1}) \wedge T_i} -X_{T_{i-1}}\big)_{s\geq 0}\Big),$$ which takes values in the *word space* $$\label{eq:defF} F = \bigcup_{t>0} \Big(\{t\} \times \big\{ f \in C([0,\infty))\colon\, f(0)=0, f(s)=f(t) \; \text{for} \; s > t \big\}\Big)$$ equipped with a Skorohod-type metric (see Appendix \[metrics\]). Let $$Y^{N\text{-}\mathrm{per}} = \big(\,\underbrace{Y^{(1)},Y^{(2)},\dots,Y^{(N)}},\, \underbrace{Y^{(1)},Y^{(2)},\dots,Y^{(N)}},\,\dots\big)$$ denote the $N$-periodisation of $Y$, and let $$\label{RNdef} R_N = \frac1N \sum_{i=0}^{N-1} \delta_{\widetilde{\theta}^i Y^{N\text{-}\mathrm{per}}}$$ be the *empirical process of words*, where $\widetilde{\theta}$ is the left-shift acting on $F^{\mathbb{N}}$. Note that $R_N$ takes values in $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, the set of shift-invariant probability measures on $F^{\mathbb{N}}$. Endow $F^{\mathbb{N}}$ with the product topology and $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with the corresponding weak topology. When averaged over $X$ and $T$, the law of $Y$ is ($\mathscr{L}$ denotes law) $$\label{qrhoWdef} Q_{\rho,{\mathscr{W}}}=(q_{\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}} \quad \text{with} \quad q_{\rho,{\mathscr{W}}} = \int_{(0,\infty)} \rho(dt)\, \mathscr{L}\big((t, (X_{s \wedge t})_{s \geq 0})\big).$$ By the ergodic theorem, ${\mathop{\text{\rm w-lim}}}_{N\to\infty} R_N = Q_{\rho,{\mathscr{W}}}$ a.s., where ${\mathop{\text{\rm w-lim}}}$ denotes the weak limit. Large deviation principles {#LDPs} -------------------------- For definitions and properties of specific relative entropy, we refer the reader to Appendix \[entropy\]. The following theorem is standard (see e.g. Dembo and Zeitouni [@DeZe98 Section 6.5.3]). \[thm0:contaLDP\] [[**\[Annealed LDP\]**]{}]{}\ The family $\mathscr{L}(R_N)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}} (F^{\mathbb{N}})$ with rate $N$ and with rate function $$\label{eq:Iann} I^{\mathrm{ann}}(Q)= H(Q \mid Q_{\rho,{\mathscr{W}}}),$$ the specific relative entropy of $Q$ w.r.t. $Q_{\rho,{\mathscr{W}}}$. This rate function is lower semi-continuous, has compact level sets, is affine, and has a unique zero at $Q=Q_{\rho,{\mathscr{W}}}$. To state the quenched LDP, we need to look at the reverse of cutting out words, namely, glueing words together. Let ${y}=(y^{(i)})_{i\in{\mathbb{N}}}=((t_i,f_i))_{i\in{\mathbb{N}}} \in F^{\mathbb{N}}$. Then the *concatenation* of ${y}$, written $\kappa({y}) \in C([0,\infty))$, is defined by $$\begin{aligned} &\kappa({y})(s) = f_1(t_1)+\dots+f_{i-1}(t_{i-1}) +f_i\big(s-(t_1+\cdots+t_{i-1})\big),\\ &t_1+\cdots+t_{i-1} \leq s < t_1+\cdots+t_{i}, \;\; i \in {\mathbb{N}}. \end{aligned}$$ Write $\tau_i({y})=t_i$ to denote the length of the $i$-th word. For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with finite mean word length $m_Q = {\mathbb{E}}_Q[\tau_1] ={\mathbb{E}}_Q[\tau_1(Y)]$, put $$\label{eq:PsiQcont} \Psi_Q(A) = \frac{1}{m_Q} {\mathbb{E}}_Q\left[ \int_0^{\tau_1} {1}_A(\theta^s \kappa(Y)) \, ds\right], \quad A \subset C([0,\infty))\;\;\mbox{measurable},$$ where $\theta^s$ is the shift acting on $f \in C([0,\infty))$ as $\theta^s f(t) = f(s+t)-f(s)$, $t \geq 0$. Note that $\Psi_Q$ is a probability measure on $C([0,\infty))$ with stationary increments, i.e., $\Psi_Q = \Psi_Q \circ (\theta^s)^{-1}$ for all $s \ge 0$. We can think of $\Psi_Q$ as the “stationarised” version of $\kappa(Q)$. In fact, if $m_Q<\infty$, then $$\label{eq:PsiQ} \Psi_Q = {\mathop{\text{\rm w-lim}}}_{T\to\infty} \frac{1}{T} \int_0^T \kappa(Q) \circ (\theta^s)^{-1}\,ds,$$ and $\kappa(Q)$ is asymptotically mean stationary (AMS) with stationary mean $\Psi_Q$. In fact, the convergence in also holds in total variation norm (see Lemma \[lemma:PsiQ:TVlim\] in Appendix \[entropy\]). Note that $\Psi_{Q_{\rho,{\mathscr{W}}}}={\mathscr{W}}$. To state the quenched LDP, we also need to define word *truncation*. For $(t,f) \in F$ and ${{\rm tr}}> 0$, let $$[(t,f)]_{{\rm tr}}= \big(t \wedge {{\rm tr}}, (f(s \wedge {{\rm tr}})_{s\ge 0}\big)$$ be the word $(t,f)$ truncated at length ${{\rm tr}}$. Analogously, for ${y}=(y^{(i)})_{i\in{\mathbb{N}}} \in F^{\mathbb{N}}$ set $[{y}]_{{\rm tr}}=([y^{(i)}]_{{{\rm tr}}})_{i\in{\mathbb{N}}} \in F^{\mathbb{N}}$, and denote by $[Q]_{{\rm tr}}\in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}}) \subset \mathcal{P}^{\mathrm{inv}} (F^{\mathbb{N}})$ with $F_{0,{{\rm tr}}}=[F]_{{\rm tr}}$ the image measure of $Q \in \mathcal{P}^{\mathrm{inv}} (F^{\mathbb{N}})$ under the map ${y} \mapsto [{y}]_{{\rm tr}}$. \[thm0:contqLDP\] [[**\[Quenched LDP\]**]{}]{}\ Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{}. Then, for ${\mathscr{W}}$ a.e. $X$, the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with rate $N$ and with deterministic rate function $I^{\mathrm{que}}(Q)$ given by $$\label{eq:Iquelimitform} I^{\mathrm{que}}(Q) = \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}),$$ where $$\label{def:Ique.tr} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) = H\big([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}\big) + (\alpha-1) m_{[Q]_{{\rm tr}}} H\big(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}\big).$$ This rate function is lower semi-continuous, has compact level sets, is affine, and has a unique zero at $Q=Q_{\rho,{\mathscr{W}}}$. Theorem \[thm0:contqLDP\] is proved in Sections \[proof\]–\[removeass\]. Let $\mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})=\{Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}) \colon\,m_Q<\infty\}$. We will show that the limit in exists for all $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, and that $$\label{eq:Ique} I^{\mathrm{que}}(Q) = H(Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}), \qquad Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}}).$$ We will also see that $I^{\mathrm{que}}(Q)$ is the lower semi-continuous extension to $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ of its restriction to $\mathcal{P}^{\mathrm{inv,fin}} (F^{\mathbb{N}})$. Discussion {#disc} ---------- [**0.**]{} A *heuristic* behind Theorem \[thm0:contqLDP\] is as follows. Let $$\label{RNdefind} R^N_{t_1,\dots,t_N}(X), \qquad 0<t_1<\dots<t_N<\infty,$$ denote the empirical process of $N$-tuples of words when $X$ is cut at the points $t_1, \dots,t_N$ (i.e., when $T_i=t_i$ for $i=1,\dots,N$). Fix $Q \in \mathcal{P}^{\mathrm{inv,fin}} (F^{\mathbb{N}})$ and suppose that $Q$ is shift-ergodic. The probability ${\mathbb{P}}(R_N \approx Q \mid X)$ is an integral over all $N$-tuples $t_1,\dots,t_N$ such that $R^N_{t_1,\dots,t_N}(X) \approx Q$, weighted by $\prod_{i=1}^N \bar{\rho}(t_i-t_{i-1})$ (with $t_0=0$). The fact that $R^N_{t_1,\dots,t_N}(X) \approx Q$ has three consequences: 1. The $t_1,\dots,t_N$ must cut $\approx N$ substrings out of $X$ of total length $\approx N m_Q$ that look like the concatenation of words that are $Q$-typical, i.e., that look as if generated by $\Psi_Q$ (possibly with gaps in between). This means that most of the cut-points must hit atypical pieces of $X$. We expect to have to shift $X$ by $\approx\exp[N m_Q H(\Psi_Q \mid {\mathscr{W}})]$ in order to find the first contiguous substring of length $N m_Q$ whose empirical shifts lie in a small neighbourhood of $\Psi_Q$. By (\[ass:rhodensdecay\]), the probability for the single increment $t_1-t_0$ to have the size of this shift is $\approx \exp[-N\alpha\,m_Q H(\Psi_Q \mid {\mathscr{W}})]$. 2. The “number of local perturbations” of $t_1,\dots,t_N$ preserving the property $R^N_{t_1,\dots,t_N}(X)\approx Q$ is $\approx \exp[NH_{\tau|K}(Q)]$, where $H_{\tau|K}$ stands for the *conditional specific entropy (density) of word lengths under the law $Q$*. 3. The statistics of the increments $t_1-t_0,\dots,t_N-t_{N-1}$ must be close to the distribution of word lengths under $Q$. Hence, the weight factor $\prod_{i=1}^N \bar{\rho}(t_i-t_{i-1})$ must be $\approx \exp[N {\mathbb{E}}_Q[\log\bar{\rho}(\tau_1)]]$ (at least, for $Q$-typical pieces). Since $$\label{eqnsre1} m_Q H(\Psi_Q \mid {\mathscr{W}}) - H_{\tau|K}(Q) - {\mathbb{E}}_Q[\log\bar{\rho}(\tau_1)] = H(Q \mid q_{\rho,{\mathscr{W}}}),$$ the observations made in (1)–(3) combine to explain the shape of the quenched rate function in . For further details, see [@BiGrdHo10 Section 1.5]. *Note:* We have not defined $H_{\tau|K}(Q)$ rigorously here, nor do we prove . Our proof of Theorem \[thm0:contqLDP\] uses the above heuristic only very implicitly. Rather, it starts from the discrete-time quenched LDP derived in [@BiGrdHo10] and draws out Theorem \[thm0:contqLDP\] via control of exponential functionals through a coarse-graining approximation. [**1.**]{} We can include the cases $\alpha=1$ and $\alpha=\infty$ in . \[mainthmboundarycases\] Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{}.\ [(a)]{} If $\alpha=1$, then the quenched LDP holds with $I^\mathrm{que}=I^\mathrm{ann}$ given by .\ [(b)]{} If $\alpha=\infty$, then the quenched LDP holds with rate function $$\label{eq:ratefctalphainfty} I^\mathrm{que}(Q) = \begin{cases} H(Q \mid Q_{\rho,{\mathscr{W}}}) & \mbox{if} \;\; \lim\limits_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} H( \Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) = 0, \\ \infty & \mbox{otherwise}. \end{cases}$$ Theorem \[mainthmboundarycases\] is the continuous analogue of Birkner, Greven and den Hollander [@BiGrdHo10 Theorem 1.4] and is proved in Section \[proofalpha1infty\]. [**2.**]{} We can also include the case where $\bar{\rho}$ has an exponentially bounded tail: $$\label{ass:rhoexp} \bar{\rho}(t) \leq e^{-\lambda t} \mbox{ for some } \lambda >0 \mbox{ and } t \mbox{ large enough}.$$ \[thmexp\] Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{} and . Then, for ${\mathscr{W}}$ a.e. $X$, the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with rate $N$ and with deterministic rate function $I^{\mathrm{que}}(Q)$ given by $$\label{eq:ratefctexptail} I^\mathrm{que}(Q) = \left\{\begin{array}{ll} H(Q \mid Q_{\rho,{\mathscr{W}}}) &\mbox{if } Q \in {{\mathcal R}}_{\mathscr{W}},\\ \infty &\mbox{otherwise}, \end{array} \right.$$ where $${{\mathcal R}}_{\mathscr{W}}= \left\{Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})\colon\, {\mathop{\text{\rm w-lim}}}_{T\to\infty} \frac{1}{T} \int_0^T \delta_{\kappa(Y)} \circ (\theta^s)^{-1}\,ds = {\mathscr{W}}\;\; \text{for $Q$-a.e.\ $Y$}\right\}.$$ Theorem \[thmexp\] is the continuous analogue of Birkner [@Bi08 Theorem 1] and is proved in Section \[proofalpha1infty\]. On the set $\mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ the following holds: $$\label{eq:calRchar} \Psi_Q = {\mathscr{W}}\quad \mbox{ if and only if } \quad Q \in {{\mathcal R}}_{\mathscr{W}}.$$ The equivalence in is the continuous analogue of [@Bi08 Lemma 2] (and can be proved analogously). [**3.**]{} By applying the contraction principle we obtain the quenched LDP for single words. Let $\pi_1\colon\,F^{\mathbb{N}}\to F$ be the projection onto the first word, and let $\pi_1R_N = R_N \circ (\pi_1)^{-1}$. \[cor:marginal\] Suppose that $\rho$ satisfies [(\[ass:rhodensdecay\]–\[ass:rhobar.reg0\])]{}. For ${\mathscr{W}}$-a.e. $X$, the family $\mathscr{L}(\pi_1 R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}(F)$ with rate $N$ and with deterministic rate function $I^\mathrm{que}_1$ given by $$\label{eq:contractedratefct} I^\mathrm{que}_1(q) = \inf\big\{ I^\mathrm{que}(Q)\colon\, Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}),\,\pi_1 Q = q \big\}.$$ This rate function is lower semi-continuous, has compact levels sets, is convex, and has a unique zero at $q=q_{\rho,{\mathscr{W}}}$. For general $q$ it is not possible to evaluate the infimum in (\[eq:contractedratefct\]) explicitly. For $q$ with $m_q={\mathbb{E}}_q[\tau]={\mathbb{E}}_{q^{\otimes{\mathbb{N}}}}[\tau_1]=m_{q^{\otimes{\mathbb{N}}}}<\infty$ and $\Psi_{q^{\otimes{\mathbb{N}}}}={\mathscr{W}}$, we have $I^\mathrm{que}_1(q)=h(q \mid q_{\rho,{\mathscr{W}}})$, the relative entropy of $q$ w.r.t.  $q_{\rho,{\mathscr{W}}}$. [**4.**]{} We expect assumption to be redundant. In any case, it can be relaxed to (see Section \[prop1\]): $$\label{ass:rhobar.reg-mg} \begin{minipage}{0.85\textwidth} $\mathrm{supp}(\rho)= \cup_{i=1}^M [a_i,b_i] \cup [a_{M+1},\infty)$ with $M \in {\mathbb{N}}$ and $0 \leq a_1 < b_1 \leq a_2 < \cdots < b_M \leq a_{M+1}<\infty$, and $\bar{\rho}$ is continuous and strictly positive on $\cup_{i=1}^M (a_i,b_i) \cup (a_{M+1},\infty)$ and varies regularly near each of the finite endpoints of these intervals. \end{minipage}$$ [**5.**]{} It is possible to extend Theorem \[thm0:contqLDP\] to other classes of random environments, as stated in the following theorem whose proof will not be spelled out in the present paper. Theorems [\[thm0:contqLDP\]–\[thmexp\]]{} and Corollary [\[cor:marginal\]]{} carry over verbatim when the Brownian motion $X$ is replaced by a $d$-dimensional Lévy process $\bar{X}$ with the property that ${\mathbb{E}}[e^{\langle \lambda, \bar{X}_1 \rangle}] < \infty$ for all $\lambda \in {\mathbb{R}}^d$ (where $\langle\cdot\rangle$ denotes the standard inner product), ${\mathscr{W}}$ is replaced by the law of $\bar{X}$, and in the definition of $F$ in continuous paths are replaced by càdlàg paths. [**6.**]{} In the companion paper [@BidHo13b] we apply Theorem \[thm0:contqLDP\] and the techniques developed in the present paper to the Brownian copolymer. In this model a càdlàg path, representing the configuration of the polymer, is rewarded or penalised for staying above or below a linear interface, separating oil from water, according to Brownian increments representing the degrees of hydrophobicity or hydrophilicity along the polymer. The reference measure for the path can be either the Wiener measure or the law of a more general Lévy process. We derive a variational formula for the quenched free energy, from which we deduce a variational formula for the slope of the quenched critical line. This critical line separates a *localized phase* (where the copolymer stays close to the interface) from a *delocalized phase* (where the copolymer wanders away from the interface). This slope has been the object of much debate in recent years. The Brownian copolymer is the unique attractor in the limit of weak interaction for a whole universality class of discrete copolymer models. See Bolthausen and den Hollander [@BodHo97], Caravenna and Giacomin [@CaGi10], Caravenna, Giacomin and Toninelli [@CaGiTo12] for details. Proof of Theorem \[thm0:contqLDP\] {#proof} ================================== The proof proceeds via a *coarse-graining* and *truncation* argument. In Section \[cogrtrun\] we set up the coarse-graining and the truncation, and state a quenched LDP for this setting that follows from the quenched LDP in [@BiGrdHo10] and serves as the starting point of our analysis (Proposition \[thm00:contqLDP\] and Corollary \[prop:qLDPhtr\] below). In Section \[3prop\] we state three propositions (Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] below), involving expectations of exponential functionals of the coarse-grained truncated empirical process as well as approximation properties of the associated rate function, and we use these propositions to complete the proof of Theorem \[thm0:contqLDP\] with the help of Bryc’s inverse of Varadhan’s lemma. In Section \[3lem\] we state and prove two lemmas that are used in Section \[3prop\], involving approximation estimates under the coarse-graining. The proof of the three propositions is deferred to Sections \[props\]–\[removeass\]. Preparation: coarse-graining and truncation {#cogrtrun} ------------------------------------------- ### Coarse-graining {#cogr} Suppose that, instead of the absolutely continuous $\rho$ introduced in Section \[setting\], we are given a discrete $\hat{\rho}$ with $\mathrm{supp}(\hat\rho) \subset h {\mathbb{N}}$ for some $h>0$. Let $$\label{def:Eh} E_h = \{ f \in C([0,h])\colon\,f(0)=0 \}.$$ Path pieces of length $h$ in a continuous-time scenario can act as “letters” in a discrete-time scenario, and therefore we can use the results from [@BiGrdHo10]. Note that $(E_h)^{\mathbb{N}}$ as a metric space is isomorphic to $\{f \in C([0,\infty))\colon\,f(0)=0\}$ via the obvious glueing together of path pieces into a single path, provided the latter is given a suitable metric that metrises locally uniform convergence. Similarly, we can identify $\mathcal{P}^{\mathrm{inv}}(E_h^{\mathbb{N}})$ with $$\mathcal{P}^{h\text{-}\mathrm{inv}}(C([0,\infty))) = \big\{ Q \in \mathcal{P}(C([0,\infty))) \colon\, Q = Q \circ (\theta^{h})^{-1} \big\},$$ which is the set of laws on continuous paths that are invariant under a time shift by $h$. Note that the set $$\label{def:Fh} F_h = \bigcup_{t \in h{\mathbb{N}}} \Big(\{t\} \times \big\{ f \in C([0,\infty))\colon\,f(0)=0, f(s)=f(t) \; \text{for} \; s > t \big\}\Big)$$ is isomorphic to $\widetilde{E_h} = \cup_{n \in {\mathbb{N}}} \left(E_h\right)^n$ via the map $\iota_h\colon\, F_h \to \widetilde{E_h}$ defined by $$\label{iotahdef} \iota_h\big( (nh, f)\big) = \Big( \big(f\big((\,\cdot+(i-1)h) \wedge ih\big) -f((i-1)h)\big)\Big)_{i=1,\dots,n}, \qquad (nh, f) \in F_h.$$ For $Q \in \mathcal{P}^{\mathrm{inv, fin}}(F_h^{\mathbb{N}})$, define $$\label{eq:definitionPsiQh} \Psi_{Q,h}(A) = \frac1{m_Q} {\mathbb{E}}_Q\left[ \sum_{i=0}^{\tau_1-1} {1}_A\big(\theta^i \iota_h \kappa(Y)\big)\right] = \frac1{h \, m_Q} {\mathbb{E}}_Q\left[ \int_0^{h \tau_1} {1}_A\big( \kappa(Y)(h \lfloor u/h\rfloor +s))_ {s \geq 0} \big) \, du\right]$$ for $A \subset C([0,\infty))$ measurable, where $\tau_1$ is the length of the first word (counted in letters, so that the length of the first word viewed as an element of $F_h$ is $h\tau_1$) and $\theta$ is the left-shift acting on $(E_h)^{\mathbb{N}}$. The right-most expression in can be viewed as a coarse-grained version of (\[eq:PsiQcont\]). The following coarse-grained version of the quenched LDP serves as our starting point. \[thm00:contqLDP\] Fix $h>0$. Suppose that $\mathrm{supp}(\hat\rho) \subset h{\mathbb{N}}$ and $\lim_{n\to\infty} \log \hat\rho(\{nh\})/\log n = -\alpha$ with $\alpha \in (1,\infty)$. Then, for ${\mathscr{W}}$ a.e.  $X$, the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}((\widetilde{E_h})^{\mathbb{N}})$ with rate $N$ and with deterministic rate function given by $$\label{eq:ratefctfixedh} I^{\mathrm{que}}_h(Q) = H(Q \mid Q_{\hat\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_{Q,h} \mid {\mathscr{W}}), \qquad Q \in \mathcal{P}^{\mathrm{inv,fin}}((\widetilde{E_h})^{\mathbb{N}}),$$ and $$I^{\mathrm{que}}_h(Q) = \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_h([Q]_{{\rm tr}}), \qquad Q \notin \mathcal{P}^{\mathrm{inv,fin}}((\widetilde{E_h})^{\mathbb{N}}),$$ where $Q_{\hat\rho,{\mathscr{W}}}=(q_{\hat\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}}$ with $q_{\hat\rho,{\mathscr{W}}}$ defined as in , and $\Psi_{Q,h}$ defined via . The claim follows from [@BiGrdHo10 Corollary 1.6] by using $E_h$ as letter space and observing that $\widetilde{E_h}=\iota_h(F_h)$. Note that $F_h^{\mathbb{N}}$ is a closed subspace of $F^{\mathbb{N}}$. Since $\mathrm{supp}(\hat\rho) \subset h{\mathbb{N}}$ by assumption, we have $I^{\mathrm{que}}_h(Q) \geq H(Q \mid Q_{\hat\rho,{\mathscr{W}}}) = \infty$ for any $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $Q\big(F^{\mathbb{N}}\setminus F_h^{\mathbb{N}}\big)>0$. Therefore we can consider the random variable $R_N$ as taking values in $\mathcal{P}^{\mathrm{inv}}((\widetilde{E_h})^{\mathbb{N}})$, $\mathcal{P}^{\mathrm{inv}} (F_h^{\mathbb{N}})$ or $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, without changing the statement of Proposition \[thm00:contqLDP\]. Note that $I^{\mathrm{que}}_h$ is finite only on $\mathcal{P}^{\mathrm{inv}}(F_h^{\mathbb{N}})\subset\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$. We want to pass to the limit $h \downarrow 0$ and deduce Theorem \[thm0:contqLDP\] from Proposition \[thm00:contqLDP\]. However, an immediate application of a projective limit at the level of letters appears to be impossible. Indeed, when we replace $h$ by $h/2$, each “$h$-letter” turns into two “$(h/2)$-letters”, so the word length changes, and even diverges as $h \downarrow 0$. This does not fit well with the way the projective limit was set up in [@BiGrdHo10 Section 8], where the internal structure of the letters was allowed to become increasingly richer, but the word length had to remain the same. In some sense, the problem is that we have finite words but only infinitesimal letters (i.e., there is no fixed letter space). To remedy this, we proceed as follows. For fixed discretisation length $h>0$ we have a fixed letter space, and so Proposition \[thm00:contqLDP\] applies. We will handle the limit $h \downarrow 0$ via Bryc’s inverse of Varadhan’s lemma. This will require several intermediate steps. ### Truncation {#trun} It will be expedient to work with a *truncated* version of Proposition \[thm00:contqLDP\]. For $h>0$, let $\lceil t \rceil_h =h\lceil t/h \rceil$ for $t \in (0,\infty)$ and put $\lceil\rho\rceil_h =\rho\,\circ (\lceil\cdot\rceil_h)^{-1}$, i.e., $$\label{def:rho.h.trunc} \lceil \rho \rceil_h = \sum_{i\in{\mathbb{N}}} w_{h,i} \delta_{ih} \, \in \mathcal{P}(h{\mathbb{N}}) \subset \mathcal{P}((0,\infty)),$$ where $$w_{h,i} = \rho\big(((i-1)h,ih]\big) = \int_{(i-1)h}^{ih} \bar{\rho}(x)dx$$ is the coarse-grained version of $\rho$ from Section \[setting\]. It is easily checked that implies $$\lim_{n\to\infty} \frac{\log \lceil \rho\rceil_h(\{nh\})}{\log n} = -\alpha.$$ Write$ \mathscr{L}_{\lceil \rho \rceil_h}([R_N]_{{\rm tr}}\mid X)$ for the law of the truncated empirical process $[R_N]_{{\rm tr}}$ conditional on $X$ when the $\tau_i$’s are drawn according to $\lceil \rho \rceil_h$. \[prop:qLDPhtr\] For ${\mathscr{W}}$-a.e. $X$, the family $\mathscr{L}_{\lceil \rho \rceil_h}([R_N]_{{\rm tr}}\mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^\mathrm{inv}(F_h^{\mathbb{N}})$ with rate $N$ and with deterministic rate function given by $$\label{eq:Ique.h.tr} I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) = H\bigl(Q \mid Q_{\lceil\rho\rceil_h,{\mathscr{W}},{{\rm tr}}}\bigr) + (\alpha-1) m_Q H(\Psi_{Q,h} \mid {\mathscr{W}})$$ with $Q_{\lceil\rho\rceil_h,{\mathscr{W}},{{\rm tr}}} = ([q_{\lceil \rho \rceil_h, {\mathscr{W}}}]_{{\rm tr}})^{\otimes {\mathbb{N}}}$. This follows from Proposition \[thm00:contqLDP\] and the contraction principle. Alternatively, it follows from the proofs of [@BiGrdHo10 Theorem 1.2 and Corollary 1.6]. Note that $I^{\mathrm{que}}_{h,{{\rm tr}}}(Q)=\infty$ when under $Q$ the word lengths are not supported on $h{\mathbb{N}}\cap (0,{{\rm tr}}]$. Application of Bryc’s inverse of Varadhan’s lemma {#3prop} ------------------------------------------------- In this section we state three propositions (Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] below) and show that these imply Theorem \[thm0:contqLDP\]. The proof of these propositions is deferred to Sections \[props\]–\[removeass\]. ### Notations {#subsect:notations} In what follows we obtain the quenched LDP for the truncated empirical process $[R_N]_{{{\rm tr}}}$ by letting $h \downarrow 0$ in the coarse-grained and truncated empirical process $[R_{N,h}]_{{\rm tr}}$ with ${{\rm tr}}\in {\mathbb{N}}$ fixed (for a precise definition, see in Section \[prop1\]) and afterwards letting ${{\rm tr}}\to\infty$. (We assume that ${{\rm tr}}\in {\mathbb{N}}$ and $h=2^{-M}$ for some $M \in {\mathbb{N}}$, in particular, ${{\rm tr}}$ is an integer multiple of $h$.) In the coarse-graining procedure, it may happen that a very short continuous word $y=(t,f) \in F$ disappears, namely, when $0<t<h$. We remedy this by formally allowing “empty” words, i.e., by using $$\begin{aligned} \label{def:Fhat} \widehat{F} = F \cup \big\{ (0,0) \big\} = \bigcup_{t \geq 0} \Big(\{t\} \times \big\{ f \in C([0,\infty))\colon\,f(0)=0, f(s)=f(t) \; \text{for} \; s > t \big\}\Big)\end{aligned}$$ as word space instead of $F$. The metric on $F$ defined in Appendix \[metrics\] extends in the obvious way to $\widehat{F}$. Before we proceed, we impose *additional regularity assumptions* on $\bar{\rho}$ that will be required in the proof of Proposition \[prop:LambdaPhilimit1tr\]. Recall from that $\mathrm{supp}(\rho) = [s_*,\infty)$. Let $$\begin{aligned} \label{eq:Vbarrhodef} V_{\bar{\rho}}(t,h) = \sup_{v \in (0,2h)} \left| \log \frac{\int_t^{t+h} \bar{\rho}(s)\,ds}{\int_{t+v}^{t+h+v} \bar{\rho}(s)\,ds} \right|, \qquad t,h>0.\end{aligned}$$ We assume that there exist monotone sequences $(\eta_n)_{n \in {\mathbb{N}}}$ and $(A_n)_{n\in{\mathbb{N}}}$, with $\eta_n \in (0,1)$ and $A_n \subset (s_*,\infty)$ satisfying $\lim_{n\to\infty} \eta_n = 0$ and $\lim_{n\to\infty} A_n = (s_*,\infty)$, such that $(s_*,\infty) \setminus A_n$ is a (possibly empty) union of finitely many bounded intervals whose endpoints lie in $2^{-n} {\mathbb{N}}_0$, and $$\begin{aligned} \label{ass:rhobar.reg2} \sup_{t \in A_n} V_{\bar{\rho}}(t,2^{-n}) \leq \eta_n \qquad \forall\,n \in{\mathbb{N}}. \end{aligned}$$ In addition, we assume that there exists an $\eta_0 < \infty$ such that $$\begin{aligned} \label{ass:rhobar.reg1} \sup_{n \in {\mathbb{N}}} \sup_{t \in (s_*,\infty)} V_{\bar{\rho}}(t,2^{-n}) \leq \eta_0. \end{aligned}$$ These assumptions will be removed only in Section \[removeass\]. Note that – are satisfied when $\bar{\rho}$ is continuous and strictly positive on $(s_*,\infty)$ and varies regularly near $s_*$ and at $\infty$. ### Proof of Theorem \[thm0:contqLDP\] subject to (\[ass:rhobar.reg2\]–\[ass:rhobar.reg1\]) and three propositions A function $g$ on $\widehat{F}^\ell$ is Lipschitz when it satisfies $$\begin{aligned} \label{eq:g_Lipschitz} \big| g(y^{(1)},\dots,y^{(\ell)}) - g(y^{(1)}{}',\dots,y^{(\ell)}{}') \big| \leq C_g \sum_{j=1}^\ell d_F(y^{(j)},y^{(j)}{}') \quad \mbox{ for some } C_g < \infty. \end{aligned}$$ Consider the class $\mathscr{C}$ of functions $\Phi\colon\,\mathcal{P}(\widehat{F}^{\mathbb{N}}) \to {\mathbb{R}}$ of the form $$\label{eq:Phiform1} \Phi(Q) = \int_{\widehat{F}^{\ell_1}} g_1 \, d\pi_{\ell_1} Q \wedge \cdots \wedge \int_{\widehat{F}^{\ell_m}} g_m \, d\pi_{\ell_m}Q, \quad Q \in \mathcal{P}^{\mathrm{inv}}(\widehat{F}^{\mathbb{N}}),$$ where $m \in N$, $\ell_1,\dots, \ell_m \in {\mathbb{N}}$, and $g_i$ is a bounded Lipschitz function on $\widehat{F}^{\ell_i}$ for $i=1,\dots,m$. This class is well-separating and thus is sufficient for the application of Bryc’s lemma (see Dembo and Zeitouni [@DeZe98 Section 4.4]. Our first proposition identifies the exponential moments of $[R_N]_{{\rm tr}}$. \[prop:LambdaPhilimit1tr\] The families $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, and $\mathscr{L}([R_N]_{{{\rm tr}}} \mid X)$, ${{\rm tr}}\in {\mathbb{N}}$, are exponentially tight $X$-a.s. Moreover, for $\Phi \in \mathscr{C}$, $$\label{eq:LambdaPhilimit1tr} \Lambda_{0,{{\rm tr}}}(\Phi) = \lim_{N\to\infty} \frac1N \log {\mathbb{E}}\Big[ \exp\big( N \Phi([R_N]_{{{\rm tr}}})\big) ~\Big|~ X \Big] = \lim_{h \downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi) \quad \text{exists $X$-a.s.},$$ where $\Lambda_{h,{{\rm tr}}}$ is the generalised convex transform of $I^{\mathrm{que}}_{h,{{\rm tr}}}$ given by $$\label{eq:Phihtrform} \Lambda_{h,{{\rm tr}}}(\Phi) = \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}((\widetilde{E_h})^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}.$$ Furthermore, for $\Phi\in \mathscr{C}$, $$\label{eq:LambdaPhilimit3} \Lambda(\Phi) = \lim_{N\to\infty} \frac1N \log {\mathbb{E}}\Big[ \exp\big( N \Phi(R_N)\big) ~\Big|~ X \Big] = \lim_{{{\rm tr}}\to\infty} \Lambda_{0,{{\rm tr}}}(\Phi) \quad \text{exists $X$-a.s.}$$ Our second proposition identifies the limit in as the generalised convex transform of $I^{\mathrm{que}}_{{{\rm tr}}}$ defined in , $$I^{\mathrm{que}}_{{{\rm tr}}}(Q) = \begin{cases} H\bigl( Q \mid Q_{\rho,{\mathscr{W}},{{\rm tr}}} \bigr) + (\alpha-1) m_Q H\left( \Psi_Q \mid {\mathscr{W}}\right) & \text{if} \; Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}}), \\[1ex] \infty & \text{otherwise}, \end{cases}$$ and implies that the latter is the rate function for the truncated empirical process $[R_N]_{{\rm tr}}$. \[prop:qLDPtrunc1\] For $\Phi\in \mathscr{C}$, $$\label{eq:Lambda0tr} \Lambda_{0,{{\rm tr}}}(\Phi) = \sup_{Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{{{\rm tr}}}(Q) \big\}.$$ Furthermore, for ${\mathscr{W}}$-a.e. $X$, the family $\mathscr{L}([R_N]_{{{\rm tr}}} \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP on $\mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$ with deterministic rate function $I^{\mathrm{que}}_{{{\rm tr}}}$. Note that the family of truncation operators $[\cdot]_{{{\rm tr}}}$ forms a projective system as the truncation level ${{\rm tr}}$ increases. Hence we immediately get from Proposition \[prop:qLDPtrunc1\] and the Dawson-Gärtner projective limit LDP (see [@DeZe98 Theorem 4.6.1]) that the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, satisfies the LDP with rate function $Q \mapsto \sup_{{{\rm tr}}\in {\mathbb{N}}} I^{\mathrm{que}}_{{{\rm tr}}}([Q]_{{{\rm tr}}})$. Furthermore, since the projection can start at any initial level of truncation, we also know that the rate function is given by $Q \mapsto \sup_{{{\rm tr}}\geq n} I^{\mathrm{que}}_{{{\rm tr}}}([Q]_{{{\rm tr}}})$ for any $n\in{\mathbb{N}}$. Thus, Proposition \[prop:qLDPtrunc1\] in fact implies that the rate function is given by $$\begin{aligned} \label{eq:Ique-DGform} \tilde{I}^{\mathrm{que}}(Q) = \limsup_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{{\rm tr}}}([Q]_{{{\rm tr}}}).\end{aligned}$$ At this point, it remains to prove that $\tilde{I}^{\mathrm{que}}$ from actually equals $I^{\mathrm{que}}$ from and has the form claimed in . This is achieved via the following proposition, note that is the continuous analogue of [@BiGrdHo10 Lemma A.1]. \[prop:Ique.tr.cont\] [(1)]{} For $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{{\mathbb{N}}})$, $$\begin{aligned} \label{eq:lemma:Ique.tr.cont1} \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) = H(Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ [(2)]{} For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{{\mathbb{N}}})$ with $m_Q = \infty$ and $H(Q \mid Q_{\rho, {\mathscr{W}}})<\infty$ there exists a sequence $(\widetilde{Q}_{{\rm tr}})_{{{\rm tr}}\in {\mathbb{N}}}$ in $\mathcal{P}^{\mathrm{inv,fin}}(F^{{\mathbb{N}}})$ such that ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \widetilde{Q}_{{\rm tr}}= Q$ and $$\begin{aligned} \label{eq:lemma:Ique.tr.approx2} \tilde{I}^{\mathrm{que}}(\widetilde{Q}_{{\rm tr}}) \leq I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) + o(1), \qquad {{\rm tr}}\to \infty.\end{aligned}$$ Proposition \[prop:Ique.tr.cont\] (1) implies that for $Q \in \mathcal{P}^{\mathrm{inv,fin}} (F^{{\mathbb{N}}})$ the $\limsup$ in is a limit, i.e., it implies on $\mathcal{P}^{\mathrm{inv,fin}}(F^{{\mathbb{N}}})$ and also . To prove for $Q \in \mathcal{P}^{\mathrm{inv}}(F^{{\mathbb{N}}})$ with $m_Q = \infty$ and $H(Q \mid Q_{\rho, {\mathscr{W}}})<\infty$, consider $\widetilde{Q}_{{\rm tr}}$ as in Proposition \[prop:Ique.tr.cont\] (2). Then $$\begin{aligned} \tilde{I}^{\mathrm{que}}(Q) \leq \liminf_{{{\rm tr}}\to\infty} \tilde{I}^{\mathrm{que}}(\widetilde{Q}_{{\rm tr}}) \leq \liminf_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}),\end{aligned}$$ where the first inequality uses that $\tilde{I}^{\mathrm{que}}$ is lower semi-continuous (being a rate function by the Dawson-Gärtner projective limit LDP), and the second inequality is a consequence of . For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{{\mathbb{N}}})$ with $H(Q \mid Q_{\rho, {\mathscr{W}}})=\infty$ we have $$\begin{aligned} \liminf_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) \geq \liminf_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) = H(Q \mid Q_{\rho, {\mathscr{W}}}) = \infty,\end{aligned}$$ i.e., also in this case the $\limsup$ in is a limit and holds. It remains to prove the properties of $I^{\mathrm{que}}$ claimed in Theorem \[thm0:contqLDP\]: lower semi-continuity of $I^{\mathrm{que}}=\tilde{I}^{\mathrm{que}}$ follows from the representation via the Dawson-Gärtner projective limit LDP in ; compactness of the level sets of $I^{\mathrm{que}}$ and the fact that $Q_{\rho,{\mathscr{W}}}$ is the unique zero of $Q \mapsto I^{\mathrm{que}}(Q)$ are inherited from the corresponding properties of $I^{\mathrm{ann}}$ because $I^{\mathrm{que}} \leq I^{\mathrm{ann}}$; affineness of $Q \mapsto I^{\mathrm{que}}(Q)$ can be checked as in [@BiGrdHo10 Proof of Theorem 1.3]. [**Remark.**]{}  Theorem \[thm0:contqLDP\] together with Varadhan’s lemma implies that $$\begin{aligned} \label{eq:LambdaPhilimit2} \Lambda(\Phi) = \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}(F^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}(Q) \big\}, \qquad \Phi\in \mathscr{C}, \end{aligned}$$ and identifies $I^{\mathrm{que}}(Q)$ as the generalised convex transform $$\label{eq:Iquetrafo} I^{\mathrm{que}}(Q) = \sup_{\Phi \in \mathscr{C}} \big\{ \Phi(Q) - \Lambda(\Phi)\big\}, \qquad Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$$ (see [@DeZe98 Theorems 4.4.2 and 4.4.10]). The supremum in can also be taken over all continuous bounded functions on $\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$. Continuity of the empirical process under coarse-graining {#3lem} --------------------------------------------------------- Before embarking on the proof of Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] in Section \[props\], we state and prove two approximation lemmas (Lemmas \[obs:dSclose1\]–\[obs:Rdiscdiff\] below) that will be needed along the way. For $N \in {\mathbb{N}}$, $0=t_0 < t_1 < \cdots < t_N$ and $\varphi \in C([0,\infty))$, let $y_\varphi = (y_\varphi^{(i)})_{i\in{\mathbb{N}}}$ with $$\label{def:yphii} y_\varphi^{(i)} = \Big(t_i-t_{i-1}, \big(\varphi((t_{i-1}+s) \wedge t_i) -\varphi(t_{i-1})\big)_{s\geq 0}\Big) \in F, \qquad i=1,\dots,N,$$ and define $$\label{eq:defRNphi} R_{N;t_1,\dots,t_N}(\varphi) = \frac1N \sum_{i=0}^{N-1} \delta_{\widetilde{\theta}^i y_\varphi^{N\text{-}\mathrm{per}}} \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}).$$ We need a Skorohod-type distance $d_S$ on paths, which is defined in Appendix \[metrics\]. \[obs:dSclose1\] Let $i, j \in {\mathbb{N}}$, $i \leq j$, and $t, t' \in (0,\infty)$, $t<t'$, be such that $(i-1)h < t \leq ih$, $(j-1)h < t' \leq jh$. Then, for any $\varphi \in C([0,\infty))$ and $k \in {\mathbb{N}}$, $$\begin{aligned} \label{eq:dS_wishful} & d_S\big( \varphi((ih+\cdot) \wedge jh), \varphi((t+\cdot) \wedge t')\big) \notag \\ & \leq \log\tfrac{k+1}{k} + 2 \sup_{(i-1)h \leq s \leq (i+k)h} |\varphi(s)-\varphi((i-1)h)| + 2 \sup_{(j-1)h \leq s \leq jh} |\varphi(s)-\varphi((j-1)h)|.\end{aligned}$$ The same bound holds for $d_S([\varphi((ih+\cdot) \wedge jh)]_{{\rm tr}}, [\varphi((t+\cdot) \wedge t')]_{{\rm tr}})$ for any truncation length ${{\rm tr}}> 0$. Without loss of generality we may assume that $j \geq i+k$ (otherwise, employ the trivial time transform $\lambda(s)=s$ and estimate the left-hand side of by the second term in the right-hand side of ), and use the time transformation $$\begin{aligned} \lambda(s) = \begin{cases} s \, \frac{(i+k)h-t}{kh} & \text{if} \; s < kh, \\ s + ih -t & \text{if} \; s \geq kh. \end{cases}\end{aligned}$$ In that case $\lambda(s)+t=s+ih$ for $s \geq kh$ and $\gamma(\lambda)=|\log[((i+k)h-t)/kh]| \leq \log\frac{k+1}{k}$. The same argument applies to the truncated paths $[\varphi((ih+\cdot) \wedge jh)]_{{\rm tr}}$ and $[\varphi((t+\cdot) \wedge t')]_{{\rm tr}}$ (in fact, we can drop the third term in the right-hand side of when $(j-1)h>{{\rm tr}}$). \[obs:Rdiscdiff\] Let $\varphi \in C([0,\infty))$, $h>0$, $N\in{\mathbb{N}}$ and $t_0=0<t_1<\cdots<t_N$. Let $\ell \in {\mathbb{N}}$, and let $g\colon\,\widehat{F}^\ell \to {\mathbb{R}}$ be bounded Lipschitz with Lipschitz constant $C_g$. Then, for $k \in {\mathbb{N}}$ with $k \geq \ell$, $$\begin{aligned} &N \Big| \int_{\widehat{F}^{\ell}} g\, d\pi_\ell R_{N;t_1,\dots,t_N}(\varphi) - \int_{\widehat{F}^{\ell}} g\, d\pi_\ell R_{N;\lceil t_1 \rceil_h,\dots,\lceil t_N \rceil_h}(\varphi) \Big|\\ &\qquad \leq 4\ell \|g\|_\infty + C_g \ell N \bigl(2h + \log{\textstyle\frac{k+1}{k}}\bigr) + 4 C_g \ell \sum_{i=1}^N \sup_{\lceil t_i \rceil_h-h \leq s \leq \lceil t_i \rceil_h+kh} \big| \varphi(s) -\varphi({\lceil t_i \rceil_h-h}) \big|, \end{aligned}$$ where $\pi_\ell\colon\,\widehat{F}^{\mathbb{N}}\to \widehat{F}^\ell$ denotes the projection onto the first $\ell$ coordinates. The same bound holds for the truncated versions $[R_{N;t_1,\dots,t_N} (\varphi)]_{{\rm tr}}$ and $[R_{N;\lceil t_1 \rceil_h,\dots,\lceil t_N \rceil_h}(\varphi) ]_{{\rm tr}}$ for any truncation length ${{\rm tr}}>0$. For $i=1,\dots,N$, recall $y_\varphi^{(i)}$ from , i.e., $y_\varphi^{(i)}$ is the $i$-th word obtained by cutting the continuous path $\varphi$ along the time points $t_1,\ldots,t_n$, and let $$\begin{aligned} \tilde{y}_\varphi^{(i,h)} & = \Bigl(\lceil t_i \rceil_h-\lceil t_{i-1} \rceil_h, \bigl(\varphi((\lceil t_{i-1} \rceil_h+s) \wedge \lceil t_i \rceil_h) -\varphi(\lceil t_{i-1} \rceil_h)\bigr)_{s\geq 0}\Bigr), \end{aligned}$$ be the analogous quantity when the $h$-discretised time points $\lceil t_1 \rceil_h,\ldots,\lceil t_N \rceil$ are used. By Lemma \[obs:dSclose1\] we have $$\begin{aligned} d_F\bigl(y_\varphi^{(i)}, \tilde{y}_\varphi^{(i,h)}\bigr) &\leq \bigl(2h + \log{\textstyle\frac{k+1}{k}}\bigr) + 2 \sup_{\lceil t_{i-1} \rceil_h-h \leq s \leq \lceil t_{i-1} \rceil_h+kh} \big| \varphi(s) -\varphi({\lceil t_{i-1} \rceil_h}-h) \big|\\ &\qquad\qquad + 2 \sup_{\lceil t_{i} \rceil_h-h \leq s \leq \lceil t_{i} \rceil_h} \big| \varphi(s) -\varphi({\lceil t_{i} \rceil_h}-h) \big|. \end{aligned}$$ Writing $\tilde{y}^{(h)}=(\tilde{y}^{(i,h)})_{i\in{\mathbb{N}}}$ and putting, similarly as in , $$\label{eq:defRNphi_hdisc} R_{N;\lceil t_1 \rceil_h,\dots,\lceil t_N \rceil_h}(\varphi) = \frac1N \sum_{i=0}^{N-1} \delta_{\widetilde{\theta}^i(\tilde{y}^{(h)})^{N\text{-}\mathrm{per}}},$$ we see that the claim follows from in combination with Lemma \[obs:dSclose1\]. Note that possible boundary effects due to the periodisation are estimated by the term $4\ell\|g\|_\infty$. The observation about the truncated versions of the empirical process follow analogously from Lemma \[obs:dSclose1\]. Proof of Propositions \[prop:LambdaPhilimit1tr\]–\[prop:Ique.tr.cont\] {#props} ====================================================================== Proof of Proposition \[prop:LambdaPhilimit1tr\] {#prop1} ----------------------------------------------- The proof comes in 3 Steps. #### Step 1. A.s. exponential tightness of the family $\mathscr{L}(R_N \mid X)$, $N\in{\mathbb{N}}$, is standard, because the family of unconditional distributions $\mathscr{L}(R_N)$ satisfies the LDP with a rate function that has compact level sets. Indeed, let $M > 0$, and pick a compact set $K \subset \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ such that $\limsup_{N\to\infty} \tfrac1N \log {\mathbb{P}}(R_N \not\in K) \leq -2M$. Then ${\mathbb{P}}({\mathbb{P}}(R_N \not\in K \mid X) > e^{-MN}) \leq e^{MN} {\mathbb{E}}[{\mathbb{P}}(R_N \not\in K\mid X)] \leq \exp(MN -2MN +o(N))$, which is summable in $N$. Hence we have $\limsup_{N\to\infty} \tfrac1N \log {\mathbb{P}}(R_N \not\in K \mid X) \leq -M$ a.s. by the Borel-Cantelli lemma. The same argument applies to $[R_N]_{{{\rm tr}}}$ (alternatively, use the fact that $[\cdot]_{{\rm tr}}$ is a continuous map). #### Step 2a. We next verify that the limits in exist. In Step 2a we consider the case $\mathrm{supp}(\rho)=[0,\infty)$, in Step 2b the case $\mathrm{supp}(\rho)=[s_*,\infty)$ with $s_*>0$. Let ${{\rm tr}}\in {\mathbb{N}}$ and $h = 2^{-n}$. Let $Y^{(i,h)}=(\lceil T_i \rceil_h -\lceil T_{i-1} \rceil_h, (X_{(s+\lceil T_{i-1} \rceil_h) \wedge \lceil T_i \rceil_h}-X_{\lceil T_{i-1} \rceil_h})_{s\geq 0}) \in \widehat{F}$ be the $h$-discretised $i$-th word, and let $$\label{def:RNh} R_{N,h} = \frac1N \sum_{i=0}^{N-1} \delta_{\widetilde{\theta}^i (Y^{(h)})^{N\text{-}\mathrm{per}}}$$ be the $h$-discretised empirical process, where $Y^{(h)}=(Y^{(i,h)})_{i\in{\mathbb{N}}}$. Put $\ell = \ell_1 \vee \cdots \vee \ell_m$, $C_g=C_{g_1} \vee \cdots \vee C_{g_m}$. Let $$\label{eq:defDjh} D_{j,h} = \sup_{(j-1)h \leq s \leq j h} |X_s-X_{j h}|, \qquad A_{\varepsilon,k,h}(N) = \left\{ \sum_{i=1}^N \sum_{j=0}^k D_{\lceil T_i/h \rceil+j,h} \leq N \varepsilon \right\}.$$ By Lemma \[obs:Rdiscdiff\], on the event $A_{\varepsilon,k,h}(N)$ we have $$N \big| \Phi([R_N]_{{\rm tr}}) - \Phi([R_{N,h}]_{{\rm tr}}) \big| \leq 4\ell \|\Phi\|_\infty + N C_g \ell m \Big(2h+ \log{\textstyle\frac{k+1}{k}} + 4\varepsilon \Big),$$ and hence $$\begin{aligned} \label{eq:EeNPhiRNtr.ub1} {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \big| X \big] \leq & \, \exp\big[N C_g \ell m \big(2h+ \log{\textstyle\frac{k+1}{k}} + 4\varepsilon \big) + 4\ell \|\Phi\|_\infty \big]\, {\mathbb{E}}\big[ e^{N\Phi([R_{N,h}]_{{\rm tr}})} \big| X \big] \notag \\ & \, {} + e^{N \|\Phi\|_\infty} {\mathbb{P}}\big(A_{\varepsilon,k,h}(N)^c \mid X\big), \end{aligned}$$ For $\lambda >0$, estimate $$\begin{aligned} {\mathbb{P}}([A_{\varepsilon,k,h}(N)]^c | X) \leq e^{-N \lambda \varepsilon} \, {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N \sum_{m=0}^k D_{\lceil T_i/h \rceil+m,h} \Big] \,\, \Big| \, X \right],\end{aligned}$$ so that, by Lemma \[lem:expmomentsDsum\] in Step 4 below, $$\begin{aligned} \label{eq:limsuplogPrAN} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big([A_{\varepsilon,k,h}(N)]^c \mid X\big) \leq -\varepsilon \lambda + \frac12 \log \chi\big(2 k \lambda \sqrt{h}\big). \end{aligned}$$ Since $\lim_{u\downarrow 0} \chi(u)= 1$, we have, for all $\varepsilon > 0$ and $k\in{\mathbb{N}}$, $$\label{eq_Aepskh.unlikely} \limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big([A_{\varepsilon,k,h}(N)]^c \mid X\big) = - \infty \quad \text{a.s.}$$ (pick $\lambda=\lambda(h)$ in (\[eq:limsuplogPrAN\]) in such a way that $\lambda\to\infty$ and $\lambda\sqrt{h}\to 0$). Next, observe that $$\begin{aligned} {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \mid X \big] &= \int\cdots\int_{0<t_1<\cdots<t_N} \bar{\rho}(t_1) dt_1\, \bar{\rho}(t_2-t_1) dt_2 \times\cdots\times \bar{\rho}(t_N-t_{N-1}) dt_N \notag \\ &\qquad \times \exp\big[ {N\Phi\big([R_{N;t_1,\dots,t_N}(X)]_{{\rm tr}}\big)} \big],\\[0.5ex] \label{eq:expPhiRNhtr} {\mathbb{E}}\big[ e^{N\Phi([R_{N,h}]_{{\rm tr}})} \mid X \big] &= \sum_{1 \leq j_1 \leq \cdots \leq j_N} w_h(j_1,\dots,j_N) \exp\big[ {N\Phi\big([R_{N;h j_1,\dots,h j_N}(X)]_{{\rm tr}}\big)} \big], \end{aligned}$$ where $$\begin{aligned} \label{eq:wh.weights} w_h(j_1,\dots,j_N) &= \int\cdots\int_{0<t_1<\cdots<t_N} \bar{\rho}(t_1) dt_1\, \bar{\rho}(t_2-t_1) dt_2 \times\cdots\times \bar{\rho}(t_N-t_{N-1}) dt_N\\ &\qquad\qquad \times \prod_{k=1}^N {1}_{(h(j_k-1), h j_k]}(t_k). \end{aligned}$$ The idea is to replace the right-hand side of by $\prod_{k=1}^N \lceil \rho \rceil_h(h(j_k-j_{k-1}))$, which is the corresponding weight for a discrete-time renewal process with waiting time distribution $\lceil \rho \rceil_h$. The rigorous implementation of this idea requires some care, since the coarse graining can produce “empty” words. For $\underline{j}=(j_1,\dots,j_N)$ appearing in the sum in , let $R(\underline{j}) = \# \{ 1 \leq i \leq N \colon j_i = j_{i-1}\}$ be the total number of repeated values and $\underline{\hat{\jmath}}=(\hat \jmath_1,\dots,\hat \jmath_M)$ with $M=M(\underline{j}) =N-R(\underline{j})$, $1 \leq \hat \jmath_1 < \cdots < \hat \jmath_M$, the unique elements of $\underline{j}$. Note that any given $\underline{\hat{\jmath}}$ with $M=\lceil (1-\varepsilon) N \rceil$ can be obtained in this way from at most ${N \choose \lceil \varepsilon N \rceil}$ different $\underline{j}$’s. In the following, we write $\eta(h)=\eta_n$ and $A(h) = A_n$ with $\eta_n$ and $A_n$ from when $h=2^{-n}$. Let us parse through the right-hand side of successively for $k=N,N-1,\dots,1$. When $j_k=j_{k-1}$, we integrate $t_k$ out over $(h(j_k-1), h j_k]$ and estimate the (multiplicative) contribution of this integral from above by $1$. When $j_k>j_{k-1}$, we replace $\bar{\rho}(t_k-t_{k-1})$ by $\bar{\rho}(t_k- h j_{k-1})$ and integrate $t_k$ out over $(h(j_k-1), h j_k]$. For $h(j_k-j_{k-1}) \in A(h)$ we can estimate the contribution of this integral from above by $e^{\eta(h)} \lceil \rho \rceil_h(h(j_k-j_{k-1}))$) by using , while for $h(j_k-j_{k-1}) \not \in A(h)$ we can estimate it by $e^{\eta_0} \lceil \rho \rceil_h(h(j_k-j_{k-1}))$ by using with $s_*=0$. Thus, for $\underline{j}$ with $R(\underline{j}) \leq \varepsilon N$ and $\#\{ 1 \leq i < N \colon h (j_i - j_{i-1}) \not\in A(h) \} \leq \varepsilon N$, we have $$\begin{aligned} \label{eq:estwhbyrhohgr} w_h(\underline{j}) \leq e^{\varepsilon \eta_0 N} e^{\eta(h) N} \prod_{i=1}^{M} \lceil \rho \rceil_h \big(h(\hat \jmath_i- \hat \jmath_{i-1})\big) \end{aligned}$$ with $M=N-R(\underline{j})$. Furthermore, $$\begin{aligned} \label{eq:estlast} \Big| N\Phi\big([R_{N;h j_1,\dots,h j_N}(X)]_{{\rm tr}}\big) - M\Phi\big([R_{M;h \hat \jmath_1,\dots,h \hat \jmath_M}(X)]_{{\rm tr}}\big) \Big| \le (N-M) \ell \| \Phi \|_\infty \le \varepsilon N \ell \| \Phi \|_\infty.\end{aligned}$$ Combining (\[eq:expPhiRNhtr\]–\[eq:estlast\]), we find $$\begin{aligned} \label{eq:EeNPhiRNhtr.ub2} &{\mathbb{E}}\big[e^{N\Phi([R_{N,h}]_{{\rm tr}})} \mid X \big] \notag \\ &\leq e^{N \| \Phi \|_\infty} \Big\{ {\mathbb{P}}\Big( R(\lceil T_1 \rceil_h,\dots, \lceil T_N \rceil_h) \geq \varepsilon N \, \Big| \, X \Big) \notag \\ &\hspace{5em} {} + {\mathbb{P}}\Big( \#\big\{ 1 \leq i < N \colon \lceil T_i \rceil_h - \lceil T_{i-1} \rceil_h \not\in A(h) \big\} \geq \varepsilon N \, \Big| \, X \Big) \Big\} \notag \\ &\quad + e^{[\varepsilon \eta_0 + \eta(h)]N} {N \choose \varepsilon N} \sum_{M=\lceil (1-\varepsilon) N \rceil}^N \sum_{1 \leq \hat{\jmath}_1 < \cdots < \hat{\jmath}_M} e^{M \Phi\big([R_{M;h \hat{\jmath}_1,\dots,h \hat{\jmath}_M}(X)]_{{\rm tr}}\big)} \prod_{k=1}^M \lceil \rho \rceil_h(h(\hat{\jmath}_k-\hat{\jmath}_{k-1})) . \end{aligned}$$ But $$\begin{aligned} \sum_{1 \leq \hat{\jmath}_1 < \cdots < \hat{\jmath}_M} e^{M \Phi\big([R_{M;h \hat{\jmath}_1,\dots,h \hat{\jmath}_M}(X)]_{{\rm tr}}\big)} \prod_{k=1}^M \lceil \rho \rceil_h(h(\hat{\jmath}_k-\hat{\jmath}_{k-1})) = {\mathbb{E}}_{\lceil \rho \rceil_h}\big[ & e^{M\Phi([R_M]_{{\rm tr}})} \mid X \big],\end{aligned}$$ where ${\mathbb{E}}_{\lceil \rho \rceil_h}$ denotes expectation w.r.t. the reference measure $Q_{\lceil \rho \rceil_h, {\mathscr{W}}}$, and so we can apply Corollary \[prop:qLDPhtr\] and Varadhan’s lemma to obtain $$\begin{aligned} \label{eq:limEeMPhi} \lim_{M\to\infty} \frac1M \log {\mathbb{E}}_{\lceil \rho \rceil_h}\big[ & e^{M\Phi([R_M]_{{\rm tr}})} \mid X \big] = \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}(\widetilde{E_h}^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}.\end{aligned}$$ By elementary large deviation estimates for binomials we have, for any $\varepsilon>0$, $$\begin{aligned} \label{eq:toomanywrongloops1} &\limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big( R(\lceil T_1 \rceil_h,\dots, \lceil T_N \rceil_h) \geq \varepsilon N \, \big| \, X \big) = - \infty,\\ \label{eq:toomanywrongloops2} &\limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\Big( \#\big\{ 1 \leq i < N \colon\, \lceil T_i \rceil_h - \lceil T_{i-1} \rceil_h \not\in A(h) \big\} \geq \varepsilon N \, \Big| \, X \Big) = - \infty. \end{aligned}$$ (Note that the events in (\[eq:toomanywrongloops1\]–\[eq:toomanywrongloops2\]) are independent of $X$.) Combining , and , and noting that $\lim_{N\to\infty} \frac1N \log {N \choose \varepsilon N} = -\varepsilon\log\varepsilon - (1-\varepsilon)\log(1-\varepsilon)$, we find $$\label{eq:eNPhiRNasympt_upper0} \begin{aligned} &\limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \mid X \big] \\ &\qquad \leq \bigg\{ \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}((\widetilde{E_h})^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\} \\ &\qquad \qquad + C_g \ell m \big(2h+ \log{\textstyle\frac{k+1}{k}} + 4\varepsilon \big) + \varepsilon \eta_0 + \eta(h) + \varepsilon\log\tfrac {1}{\varepsilon} + (1-\varepsilon)\log\tfrac{1}{1-\varepsilon} \bigg\}\\ &\qquad \vee \bigg( \|\Phi\|_\infty + \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big(A_{\varepsilon,k,h}(N)^c \mid X\big)\bigg\} \\ &\qquad \vee \bigg\{ \|\Phi\|_\infty + \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\Big( R(\lceil T_1 \rceil_h,\dots, \lceil T_N \rceil_h) \geq \varepsilon N \, \Big| \, X \Big) \bigg\} \\ &\qquad \vee \bigg\{ \|\Phi\|_\infty + \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\Big( \#\{ 1 \leq i < N \colon \lceil T_i \rceil_h - \lceil T_{i-1} \rceil_h \not\in A(h) \} \geq \varepsilon N \, \Big| \, X \Big) \bigg\}, \end{aligned}$$ and hence $$\begin{aligned} \label{eq:eNPhiRNasympt_upper} \limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \big| X \big] \leq \liminf_{h\downarrow 0} \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}(\widetilde{E_h}^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}\end{aligned}$$ (let $h\downarrow 0$ along a suitable subsequence, followed by $\varepsilon \downarrow 0$ and $k\to\infty$, and use and (\[eq:toomanywrongloops1\]–\[eq:toomanywrongloops2\])). Analogous arguments yield $$\begin{aligned} \label{eq:eNPhiRNasympt_lower} \liminf_{N\to\infty} \frac1N \log {\mathbb{E}}\big[ e^{N\Phi([R_N]_{{\rm tr}})} \mid X \big] \geq \limsup_{h\downarrow 0} \sup_{Q \in \mathcal{P}^{\mathrm{inv, fin}}((\widetilde{E_h})^{\mathbb{N}})} \big\{ \Phi(Q) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q) \big\}.\end{aligned}$$ Indeed, we can simply restrict the sum in to $\underline{j}$’s with $j_1 < \cdots < j_N$, so that the approximation argument is in fact a little easier because we need not pass to the $\underline{\hat\jmath}$’s. Finally, combine (\[eq:eNPhiRNasympt\_upper\]–\[eq:eNPhiRNasympt\_lower\]) to obtain . #### Step 2b. Next we consider the case $\mathrm{supp}(\rho)=[s_*,\infty)$ with $s_*>0$ and indicate the changes compared to Step 2a. To some extent this case is easier than the case $s_*=0$, since for coarse-graining level $h<s_*$ no “empty” word can appear in the coarse-graining scheme. On the other hand, when implementing a replacement similar to , it can happen that an integral $\int \bar{\rho}(t_k-t_{k-1}) {1}_{(h(j_k-1),h j_k]}(t) \,dt_k$ gets mapped to $\lceil \rho \rceil_h(h(j_k-j_{k-1}))=0$ even though the true contribution of that integral to is strictly positive (namely, when $h (j_k-j_{k-1}) \leq s_* \leq h(j_k-j_{k-1}+1)$). The idea to remedy this problem is to replace $\lceil \rho \rceil_h(h(j_k-j_{k-1}))$ by a sum of “neighbouring” weights of $\lceil \rho \rceil_h$ and to suitably control the overcounting incurred by this replacement. The details are as follows. Fix $h>0$ and $s_{*,h} = \lceil s_* \rceil_h$. For $N\in N$, consider $\underline{j}=(j_1,\dots,j_N)$ as appearing in the sum in . We say that $k \in \{1,\dots,N\}$ is “problematic” when $h(j_k-j_{k-1}) \in \{ s_{*,h}-1, s_{*,h}, s_{*,h}+1\}$, and “relaxable” when $j_k-j_{k-1} \geq 2$ and $$\max_{m=-1,0,1} \left| \log \frac{\lceil \rho \rceil_h(h(j_k-j_{k-1}+m))}{\lceil \rho \rceil_h(h(j_k-j_{k-1}))} \right| \leq 2.$$ Write $K_{\text{pro}}(\underline{j}) = \{ 1 \leq k \leq N \colon\, k\; \text{problematic}\}$ and $K_{\text{rel}} (\underline{j}) = \{ 1 \leq k \leq N \colon\,k\; \text{relaxable}\}$. Try to construct an injection $f_{\text{rel}, \underline{j}}\colon\,K_{\text{pro}} \to K_{\text{rel}}$ with the property $f_{\text{rel},\underline{j}}(k) > k$ as follows: - Start with an empty “stack” ${\sf s}$. For $k=1,\dots,N$ successively: when $k$ is problematic, push $k$ on ${\sf s}$; when $k$ is relaxable and ${\sf s}$ is not empty, pop the top element, say $k'$, from ${\sf s}$ and put $f_{\text{rel},\underline{j}}(k')=k$; when $k$ is neither problematic nor relaxable, proceed with the next $k$. We say that $\underline{j}$ is “good” when the above procedure terminates with an empty stack (in particular, $f_{\text{rel},\underline{j}}(k')$ is defined for all $k' \in K_{\text{pro}}$) and $$\sum_{k \in K_{\text{pro}}} \big( f_{\text{rel},\underline{j}}(k) - k \big) \leq \varepsilon N$$ (in particular, $\# K_{\text{pro}}(\underline{j}) \leq \varepsilon N$), and also $\# \{ 1 \leq k \leq N \colon\, j_k-j_{k-1} \not\in A(h)\} \leq \varepsilon N$. For a given good $\underline{j}$, consider the set of all $\underline{\tilde\jmath}=(\tilde\jmath_1,\dots,\tilde\jmath_N)$ obtainable by setting $$\begin{aligned} \tilde\jmath_k=j_k+\Delta_k, \; \tilde\jmath_{f_{\text{rel},\underline{j}}(k)} = j_{f_{\text{rel},\underline{j}}(k)}-\Delta_k \quad \text{with}\;\: \Delta_k \in \{-1,0,1\} \quad \text{for}\;\:k \in K_{\text{pro}},\end{aligned}$$ and $\tilde\jmath_k=j_k$ for $k \not\in(K_{\text{pro}} \cup f_{\text{rel},\underline{j}}(K_{\text{pro}}))$. Note that a given good $\underline{j}$ corresponds to at most $3^{\varepsilon N}$ different $\underline{\tilde\jmath}$’s and that, for any such $\underline{\tilde\jmath}$, $$\begin{aligned} &\Big| N\Phi\big([R_{N;h j_1,\dots,h j_N}(X)]_{{\rm tr}}\big) - N\Phi\big([R_{N;h \tilde \jmath_1,\dots,h \tilde \jmath_M}(X)]_{{\rm tr}}\big) \Big| \notag \\ &\qquad \leq \ell \| \Phi \|_\infty \sum_{k \in K_{\text{pro}}} \big( f_{\text{rel},\underline{j}}(k) - k \big) \leq \varepsilon N \ell \| \Phi \|_\infty .\end{aligned}$$ With $w_h(j_1,\dots,j_N)$ defined in , we now see that (analogously to the argument prior to ) for any good $\underline{j}$, $$\begin{aligned} \label{eq:estwhbyrhohgr.c2} w_h(\underline{j}) \leq e^{\varepsilon \eta_0 N} e^{\eta(h) N} 2^{\varepsilon N} \sum_{\underline{\tilde\jmath} \; \text{corresp.\ to}\; \underline{j}} \; \prod_{i=1}^{N} \lceil \rho \rceil_h \big(h(\tilde \jmath_i - \tilde \jmath_{i-1})\big). \end{aligned}$$ Moreover, we have $$\begin{aligned} \label{eq:toomanywrongloops3} \limsup_{h \downarrow 0} \limsup_{N\to\infty} \frac1N \log {\mathbb{P}}\big( (\lceil T_1 \rceil_h, \dots, \lceil T_N \rceil_h) \; \text{not good} \, \big| \, X \big) = - \infty. \end{aligned}$$ To check , let $S_k$ be the size of the stack ${\sf s}$ in the $k$-th step of the above construction when we use $j_k=\lceil T_k \rceil_h$, and note that $(\lceil T_1 \rceil_h, \dots, \lceil T_N \rceil_h)$ is good when $\sum_{k=1}^N S_k < \varepsilon N$. A comparison of $(S_k)_{k\in{\mathbb{N}}}$ with a (reflected) random walk on ${\mathbb{N}}_0$ that draws its steps from $\{0, \pm 1 \}$, where $(+1)$-steps have a very small probability ($\leq \int_{s_*}^{s_*+2h} \bar{\rho}(t)\, dt$) and $(-1)$-steps have a very large probability ($\rho(A_h)$) when not from $0$, shows that $\limsup_{h \downarrow h} \frac1N \log {\mathbb{P}}(\sum_{k=1}^N S_k \geq \varepsilon N) = -\infty$ for every $\varepsilon >0$. We can then estimate similarly as in , to obtain for the case $s_*>0$ as well. Analogous arguments also yield the lower bound in . #### Step 3. We next verify that the limits in exist. Note that $$| \Phi(R_N) - \Phi([R_N]_{{\rm tr}})| \leq \|\Phi\|_\infty \frac1N \#\big\{ \text{loops among the first $N$ loops that are longer than ${{\rm tr}}$} \big\},$$ which can be made arbitrarily small (also on the exponential scale, via a suitable annealing argument that uses that loop lengths are i.i.d.). A similar estimate holds for $| \Phi([R_N]_{{{\rm tr}}}) - \Phi([R_N]_{{{\rm tr}}'})|$ with ${{\rm tr}}< {{\rm tr}}'$. This shows that $\Lambda_{0,{{\rm tr}}}(\Phi)$ forms a Cauchy sequence as ${{\rm tr}}\to\infty$. The arguments in Steps [2a]{} and [2b]{} can be combined to yield the same results when assumption is relaxed to assumption . Indeed, for a given coarse-graining level $h$, gives rise to finitely many types of “problematic points” that can be handled similarly as in Step [2b]{} (combined with arguments from Step [2a]{} when $a_1=0$). #### Step 4. We close by deriving the estimate on Brownian increments over randomly drawn short time intervals that was used in in Step 2. The intuitive idea is that even though there are arbitrarily large increments over short time intervals somewhere on the Brownian path, it is extremely unlikely to hit these when sampling along an independent renewal process. The proof employs a suitable annealing argument. Recall $D_{j,h}$ from (\[eq:defDjh\]). For $h>0$ fixed, the $D_{j,h}$’s are i.i.d.  and equal in law to $\sqrt{h}D_{1,1} = \sqrt{h} \sup_{0\leq s\leq 1} |X_s|$ by Brownian scaling. \[lem:expmomentsDsum\] Let $T=(T_i)_{i\in{\mathbb{N}}}$ be a continuous-time renewal process with interarrival law $\rho$ satisfying $\mathrm{supp}(\rho) \subset [h,\infty)$. For $\lambda \geq 0$ and $k \in {\mathbb{N}}_0$, define $$\begin{aligned} \xi(\lambda,h) = \limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N \sum_{m=0}^k D_{\lceil T_i/h \rceil+m,h} \Big] \, \Big| \, \sigma(D_{j,h}, j \in {\mathbb{N}})\right],\end{aligned}$$ which is $\geq 0$ and a.s. constant by Kolmogorov’s $0$-$1$-law. Then $$\begin{aligned} \lim_{h \downarrow 0} \xi(\lambda,h) = 0 \qquad \forall\,\lambda \geq 0. \end{aligned}$$ We consider only the case $k=0$, the proof for $k\in{\mathbb{N}}$ being analogous. Abbreviate $\mathscr{G}_h=\sigma(D_{j,h}, j \in {\mathbb{N}})$, and let $$\chi(u) = {\mathbb{E}}\Big[ \exp \big[ u \, {\textstyle\sup_{\,0 \leq t \leq 1} |X_t|} \big] \Big], \quad u \in {\mathbb{R}}.$$ Note that $\chi(\cdot)$ is finite and satisfies $\lim_{u\to 0} \chi(u) = 1$. We have $$\begin{aligned} {\mathbb{E}}\bigg[ {\mathbb{E}}\bigg[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \, \Big| \, \mathscr{G}_h \bigg]^2 \bigg] &\leq {\mathbb{E}}\bigg[ \exp\Big[ 2\lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \bigg]\\ &= {\mathbb{E}}\big[ \exp[2\lambda D_{1,h}] \big]^N = \chi\big(2\lambda \sqrt{h}\big)^N. \end{aligned}$$ Thus, for any $\epsilon>0$, $$\begin{aligned} & {\mathbb{P}}\left( {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \, \Big| \, \mathscr{G}_h \right]^2 \geq \big( \chi\big(2\lambda \sqrt{h}\big) + \epsilon \big)^N \right) \\ & \leq \, \big( \chi\big(2\lambda \sqrt{h}\big) + \epsilon \big)^{-N} {\mathbb{E}}\left[ {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \, \Big| \, \mathscr{G}_h \right]^2 \right] \leq \left( \frac{\chi\big(2\lambda \sqrt{h}\big)}{\chi\big(2\lambda \sqrt{h}\big) + \epsilon} \right)^N, \end{aligned}$$ which is summable in $N$. The Borel-Cantelli lemma therefore yields $$\limsup_{N\to\infty} \frac1N \log {\mathbb{E}}\left[ \exp\Big[ \lambda \sum_{i=1}^N D_{\lceil T_i/h \rceil,h} \Big] \, \Big| \, \mathscr{G}_h \right] \leq \frac12 \log \chi\big(2\lambda \sqrt{h}\big).$$ Proof of Proposition \[prop:qLDPtrunc1\] {#ss:prop2} ---------------------------------------- \[lemma:Iquetrregularised\] For ${{\rm tr}}\in {\mathbb{N}}$ and $Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$, $$\begin{aligned} \label{eq:Iquetrregularised} I^{\mathrm{que}}_{{\rm tr}}(Q) = \lim_{\varepsilon \downarrow 0} \, \limsup_{h \downarrow 0} \, \inf\Big\{ I^{\mathrm{que}}_{h,{{\rm tr}}}(Q')\colon\, Q' \in B_\varepsilon(Q) \cap \mathcal{P}^{\mathrm{inv}}((\widetilde{E}_{h,{{\rm tr}}})^{\mathbb{N}}) \Big\},\end{aligned}$$ where $h \downarrow 0$ along $2^{-m}$, $m\in {\mathbb{N}}$. Note that after $\widetilde{E}_{h,{{\rm tr}}}$ is identified with a subset of $F_{0,{{\rm tr}}}$ (see ), states that $I^{\mathrm{que}}_{h,{{\rm tr}}}$ converges to $I^{\mathrm{que}}_{{\rm tr}}$ as $h \downarrow 0$ in the sense of Gamma-convergence. Note that, when restricted to $\mathcal{P}^\mathrm{inv}(F_{0,{{\rm tr}}}^{\otimes{\mathbb{N}}})$, $$\label{eq:PsiQcont1} \text{both } Q \mapsto m_Q \text{ and } Q \mapsto \Psi_Q \text{ are continuous}$$ (by dominated convergence), while this is not true when $Q$ is allowed to vary over the whole of $\mathcal{P}^\mathrm{inv}(F^{\otimes{\mathbb{N}}})$. A more general statement is the following: if ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q_n = Q$ and $\{ \mathscr{L}_{Q_n}(\tau_1)\colon\, n \in {\mathbb{N}}\}$ are uniformly integrable, then $\lim_{n\to\infty} m_{Q_n} = m_Q$ and ${\mathop{\text{\rm w-lim}}}_{n\to\infty} \Psi_{Q_n} = \Psi_Q$. In the proof we use several properties of specific relative entropy derived in Appendix \[entropy\]. Let $Q \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$, and abbreviate the right-hand side of by $\widetilde{I}^\mathrm{que}_{{\rm tr}}(Q)$. Note that, by and the lower semi-continuity of $\Psi \mapsto H(\Psi \mid {\mathscr{W}})$, the map $$\mathcal{P}^\mathrm{inv}(F_{0,{{\rm tr}}}^{{\mathbb{N}}}) \ni Q' \mapsto m_{Q'} H(\Psi_{Q'} \mid {\mathscr{W}})$$ is lower semi-continuous. Hence, for any $\delta>0$, we have $m_{Q'} H(\Psi_{Q'} \mid {\mathscr{W}}) \geq m_Q H(\Psi_Q \mid {\mathscr{W}}) - \delta$ for all $Q' \in B_\varepsilon(Q) \cap \mathcal{P}^{\mathrm{inv}} (\widetilde{E}_{h,{{\rm tr}}}^{\mathbb{N}})$ when $\varepsilon$ is sufficiently small (depending on $\delta$). Combine this with in Lemma \[lemma:hregularised1\] in Appendix \[entropy\], and note that ${\mathop{\text{\rm w-lim}}}Q_{h,{{\rm tr}}} = Q_{{\rm tr}}$ as $h\downarrow0$, to obtain $\widetilde{I}^\mathrm{que}_{{\rm tr}}(Q) \geq I^{\mathrm{que}}_{{{\rm tr}}}(Q)$. For the reverse direction, we need to find $h_n>0$ with $\lim_{n\to\infty} h_n = 0$ and $Q'_n \in \mathcal{P}^{\mathrm{inv}}((\widetilde{E}_{h_n,{{\rm tr}}})^{\mathbb{N}})$ with ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q'_n = Q$ such that $\liminf_{n\to\infty} I^{\mathrm{que}}_{h_n,{{\rm tr}}} (Q'_n) \le I^{\mathrm{que}}_{{\rm tr}}(Q)$. Here a complication stems from the fact that we must ensure that both parts of $I^{\mathrm{que}}_{h_n,{{\rm tr}}}(Q'_n)$, namely, $H(Q'_n \mid Q_{\lceil \rho \rceil_{h_n},{\mathscr{W}},{{\rm tr}}})$ and $H(\Psi_{Q'_n,h_n} \mid {\mathscr{W}})$, converge simultaneously. The proof is deferred to Lemma \[lem:cg.2lev.blockapprox\] in Appendix \[entropy\]. We are now ready to give the proof of Proposition \[prop:qLDPtrunc1\]. Fix ${{\rm tr}}\in{\mathbb{N}}$. Denote the right-hand side of by $\tilde{\Lambda}_{{\rm tr}}(\Phi)$. Let $\Phi\colon\,\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}) \to {\mathbb{R}}$ be of the form (\[eq:Phiform1\]). For every $\delta > 0$ we can find a $Q^* \in \mathcal{P}^{\mathrm{inv}}(F_{0,{{\rm tr}}}^{\mathbb{N}})$ such that $\Phi(Q^*) - I_{{\rm tr}}^{\mathrm{que}}(Q^*) \geq \tilde{\Lambda}_{{\rm tr}}(\Phi) - \delta$. For $\varepsilon>0$ sufficiently small (depending on $\delta$) we have $\big| \Phi(Q') - \Phi(Q^*)\big| \leq \delta$ for all $Q' \in B_\varepsilon(Q^*)$ and, by Lemma \[lemma:Iquetrregularised\], $$\begin{aligned} \liminf_{h\downarrow 0} \, \inf\Big\{ I^{\mathrm{que}}_{h,{{\rm tr}}}(Q')\colon\, Q' \in B_\varepsilon(Q^*) \cap \mathcal{P}^{\mathrm{inv}}((\widetilde{E_h}_{,{{\rm tr}}})^{\mathbb{N}}) \Big\} \leq I^{\mathrm{que}}_{{\rm tr}}(Q^*) + \delta.\end{aligned}$$ Thus $$\begin{aligned} \liminf_{h\downarrow 0} \, \sup \Big\{ \Phi(Q') - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q')\colon\, Q' \in B_\varepsilon(Q^*) \cap \mathcal{P}^{\mathrm{inv}}((\widetilde{E_h}_{,{{\rm tr}}})^{\mathbb{N}}) \Big\} \geq \tilde{\Lambda}_{{\rm tr}}(\Phi) - 3\delta. \end{aligned}$$ Let $\delta\downarrow 0$ to obtain $\liminf_{h\downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi) = \Lambda_{0,{{\rm tr}}}(\Phi) \geq \tilde{\Lambda}_{{\rm tr}}(\Phi)$. For the reverse direction, pick for $h \in (0,1)$ a maximiser $Q^*_h \in \mathcal{P}^{\mathrm{inv}} ((\widetilde{E_h}_{,{{\rm tr}}})^{\mathbb{N}})$ of the variational expression appearing in the right-hand side of , i.e., $\Phi(Q^*_h) - I^{\mathrm{que}}_{h,{{\rm tr}}}(Q^*_h) = \Lambda_{h,{{\rm tr}}}(\Phi)$. This is possible because $\Phi-I^{\mathrm{que}}_{h,{{\rm tr}}}$ is upper semi-continuous and bounded from above, and $I^{\mathrm{que}}_{h,{{\rm tr}}}$ has compact level sets. We claim that $$\label{claim:Q*htight} \text{the family} \; \big\{ Q^*_h : h \in (0,1) \big\} \subset \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})\; \text{is tight}.$$ Assuming (\[claim:Q\*htight\]), we can choose a sequence $h(n) \downarrow 0$ such that $$\begin{aligned} &\lim_{n\to\infty} \Big[ \Phi(Q^*_{h(n)}) - I^{\mathrm{que}}_{h(n),{{\rm tr}}}(Q^*_{h(n)}) \Big] = \limsup_{h\downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi),\\ &{\mathop{\text{\rm w-lim}}}_{n\to\infty} Q^*_{h(n)} = \widetilde{Q} \;\; \text{for some} \; \widetilde{Q} \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}}). \end{aligned}$$ Then $\lim_{n\to\infty}\Phi(Q^*_{h(n)})=\Phi(\widetilde{Q})$ because $\Phi$ is continuous, and $\liminf_{n\to\infty} I^{\mathrm{que}}_{h(n),{{\rm tr}}}(Q^*_{h(n)}) \geq I^{\mathrm{que}}_{{\rm tr}}(\widetilde{Q})$ by Lemma \[lemma:Iquetrregularised\]. Hence $$\begin{aligned} \Lambda_{0,{{\rm tr}}}(\Phi) = \limsup_{h\downarrow 0} \Lambda_{h,{{\rm tr}}}(\Phi) = \lim_{n\to\infty} \Big[ \Phi(Q^*_{h(n)}) - I^{\mathrm{que}}_{h(n),{{\rm tr}}}(Q^*_{h(n)}) \Big] \leq \Phi(\widetilde{Q}) - I^{\mathrm{que}}_{{\rm tr}}(\widetilde{Q}) \leq \tilde{\Lambda}_{{\rm tr}}(\Phi).\end{aligned}$$ It remains to prove , which follows once we show that for each $N \in {\mathbb{N}}$ the family of projections $\pi_N(Q^*_h) \in \mathcal{P}^{\mathrm{inv}}(F^N)$, $h\in(0,1)$, is tight (because $F^{\mathbb{N}}$ carries the product topology; see Ethier and Kurtz [@EK86 Chapter 3, Proposition 2.4]). Let $M= \|\Phi\|_\infty+1$. Then necessarily $H( Q^*_h \mid [Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}}) \leq M$, and hence $h(\pi_N(Q^*_h) \mid \pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}}) \leq N M$ for all $h \in (0,1)$. Since $\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}}) = ([q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})^{\otimes N}$ converges weakly to $\pi_N ([Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) = ([q_{\rho,{\mathscr{W}}}]_{{\rm tr}})^{\otimes N}$ as $h \downarrow 0$, the family $\{\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})\colon\,h \in (0,1)\}$ is tight, and so for any $\varepsilon > 0$ we can find a compact $\mathcal{C} \subset F^N$ such that $\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})(\mathcal{C}^c) \leq \exp[-(NM + \log 2)/\varepsilon]$ uniformly in $h\in(0,1)$. By a standard entropy inequality (see in Appendix \[entropy\]), for all $h\in (0,1)$ we have $$\begin{aligned} \pi_N(Q^*_h)(\mathcal{C}^c) \leq \frac{\log 2 + h\big(\pi_N(Q^*_h) \mid \pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})\big)} {\log\Big(1+\big(\pi_N([Q_{\lceil \rho \rceil_h,{\mathscr{W}}}]_{{\rm tr}})(\mathcal{C}^c)\big)^{-1} \Big)} \leq \frac{\log 2 + MN}{\log\big(1+\exp[(N M + \log 2)/\varepsilon]\big)} \leq \varepsilon. \end{aligned}$$ This proves the representation of the limit $\Lambda_{0,{{\rm tr}}}(\Phi)$ from . From and , plus the exponential tightness in Proposition \[prop:LambdaPhilimit1tr\], we obtain the LDP via Bryc’s inverse of Varadhan’s lemma. Proof of Proposition \[prop:Ique.tr.cont\] {#prop3} ------------------------------------------ ### Proof of part (1) We first verify , i.e., for $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$, $$\begin{aligned} \label{eq:relentrsumlim} \lim_{{{\rm tr}}\to\infty} I^{\mathrm{que}}_{{\rm tr}}([Q]_{{\rm tr}}) & = \lim_{{{\rm tr}}\to\infty} \Big[ H([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) + (\alpha-1) m_{[Q]_{{\rm tr}}} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) \Big] \notag \\ & = H(Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ The proof comes in 5 Steps. [**Step 1.**]{} Note that $\lim_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [Q_{\rho,{\mathscr{W}}}]_{{\rm tr}}) = H(Q \mid Q_{\rho,{\mathscr{W}}})$ by the projective property of word truncations, $\lim_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} = m_Q<\infty$ by dominated convergence, and $$\liminf_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) \geq H(\Psi_Q \mid {\mathscr{W}})$$ by the lower semi-continuity of specific relative entropy together with ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \Psi_{[Q]_{{\rm tr}}} = \Psi_Q$. Hence, to obtain it remains to prove that $$\begin{aligned} \label{ineq:HPsiQtr.upper} \limsup_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) \leq H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ [**Step 2.**]{} \[prop:Ique.tr.cont.part1step2\] To prove , we use coarse-graining. For every $h>0$ we can identify $\widetilde{E_h}$ with $F_h \subset F$ (recall ). In order to represent $Q\in\mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ by a shift-invariant law on $(F_h)^{\mathbb{N}}$, we discretise the cut-points onto a *uniformly shifted* grid of width $h$, as follows. For $t \in {\mathbb{R}}$, $h>0$ and $u\in [0,1)$, define (compare with Section \[trun\]) $$\begin{aligned} \label{eq:t.hu} \lceil t \rceil_{h,u} = \min\big\{ (k+u)h \colon k \in {\mathbb{Z}}, (k+u)h \geq t \big\} \quad \big(= \lceil t -uh \rceil_h + uh \big).\end{aligned}$$ Draw $Y=(Y^{(i)})_{i\in{\mathbb{N}}}=((\tau_i, f_i))_{i\in{\mathbb{N}}}$ from law $Q$, and let $U$ be an independent random variable with uniform distribution on $[0,1]$. Put $T_0=0$, $T_n=\tau_1+\cdots+\tau_n$, $n \in {\mathbb{N}}$, $$\tilde{T}_i = \lceil T_i \rceil_{h,U}, \quad i \in {\mathbb{N}}_0, \qquad \tilde{\tau}_i = \tilde{T}_i - \tilde{T}_{i-1}, \; \tilde{f}_i = \big(\theta^{\tilde{T}_{i-1}} \kappa(Y)\big)(\, \cdot \wedge \tilde{\tau}_i), \quad i\in{\mathbb{N}}.$$ (Note that it may happen that $\tilde{\tau}_i=0$. We can remedy this by allowing “empty words”, i.e., by formally passing to $\widehat{F}$ as in Section \[subsect:notations\].) Write $\lceil Q \rceil_h$ for the distribution of $\tilde{Y}=(\tilde{Y}^{(i)})_{i \in {\mathbb{N}}}=((\tilde{\tau}_i, \tilde{f}_i))_{i \in {\mathbb{N}}}$ obtained in this way. We view $\lceil Q \rceil_h$ as an element of $\mathcal{P}^{\mathrm{inv,fin}}((F_h)^{\mathbb{N}})$. To check the shift-invariance of $\lceil Q \rceil_h$, note that by construction an initial part of length $S_1=\tilde{T}_0-T_0 = U h$ of the content of the first word is removed (in a two-sided situation, this part would be added at the end of the zero-th word). The corresponding quantity for the second word is $S_2=\tilde{T}_1-T_1 = \lceil T_1 \rceil_{h,U} - T_1$. Observe that, for measurable $A \subset [0,h)$ and $B \subset [0,\infty)$, $${\mathbb{P}}(S_2 \in A, T_1 \in B) = \int_B {\mathbb{P}}(T_1 \in dt) \int_{[0,1]} du\, {1}_A\big( \lceil t -uh \rceil_h - (t-uh) \big) = \frac1h\, {\mathbb{P}}(T_1 \in B)\, \lambda(A),$$ i.e., $S_2$ is distributed as $U h$ and independent of $Y$, and so $(\tilde{Y}^{(i+1)})_{i\in{\mathbb{N}}}$ again has law $\lceil Q \rceil_h$. This settles the shift-invariance. The key feature of the construction of $\lceil Q \rceil_h$ is that $\kappa(\tilde{Y}) = (\theta^{U h} \kappa)(Y)$, so that $$\label{Psihrel} \Psi_{\lceil Q \rceil_h, h} = \Psi_Q,$$ and therefore $$\begin{aligned} \label{HhHrel} H(\Psi_{\lceil Q \rceil_h, h} \mid {\mathscr{W}}) = H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ Thus, gives us a coarse-grained version of the right-hand of . [**Step 3.**]{} If ${{\rm tr}}$ is an integer multiple of $h$, then the coarse-graining $\lceil Q \rceil_h \in \mathcal{P}^{\mathrm{inv,fin}}((F_h)^{\mathbb{N}})$ of $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ defined in Step 2 commutes with the word length truncation $[ \cdot ]_{{\rm tr}}$, i.e., $[ \lceil Q \rceil_h ]_{{\rm tr}}= \lceil [ Q ]_{{\rm tr}}\rceil_h$. This is a deterministic property of the construction in . Indeed, fix $u \in [0,1)$ and $h$ with ${{\rm tr}}= Mh$ for some $M\in {\mathbb{N}}$, consider $t_{i-1}<t_i$ with $t_i-t_{i-1} > {{\rm tr}}$ (so that in the un-coarse-grained truncation procedure the $i$-th loop length would be replaced by ${{\rm tr}}$), let $k_{i-1}, k_i \in {\mathbb{N}}$ be such that $\lceil t_{i-1} \rceil_{h,u} = (k_{i-1}+u)h$ and $\lceil t_i \rceil_{h,u} = (k_i+u)h$. When we first truncate and then coarse-grain, the $i$-th point becomes $\lceil t_{i-1} + {{\rm tr}}\rceil_{h,u} = (k_{i-1}+M+u)h$. When we first coarse-grain and then truncate, the $i$-th point becomes $\lceil t_{i-1} \rceil_{h,u} + \big( (\lceil t_i \rceil_{h,u} - \lceil t_{i-1} \rceil_{h,u}) \wedge M h \big) = (k_{i-1}+u)h + M h$, which is the same. [**Step 4.**]{} Let $h=2^{-M}$, define $\lceil Q \rceil_h \in \mathcal{P}^{\mathrm{inv,fin}}((F_h)^{\mathbb{N}})$ as in Step 2, and write $Q'_h = \lceil Q \rceil_h \circ \iota_h^{-1}$ for the same object considered as an element of $\mathcal{P}^{\mathrm{inv,fin}}((\widetilde{E_h})^{\mathbb{N}})$ (recall (\[def:Eh\]–\[iotahdef\])). Write $\nu_h=\mathscr{L}\big((X_{\cdot \wedge h})\big)$ for the Wiener measure on $E_h$. Then $m_{Q'_h} = m_{\lceil Q \rceil_h}/h$ (the mean word length counted in $h$-letters), while $$\begin{aligned} \label{HPsihrel} H(\Psi_{Q'_h} \mid \nu_h^{\otimes {\mathbb{N}}}) = H(\Psi_{\lceil Q \rceil_h,h} \mid {\mathscr{W}}),\end{aligned}$$ by construction, and $$\begin{aligned} \label{Qtrhrel} \lceil [ Q ]_{{\rm tr}}\rceil_h = [ \lceil Q \rceil_h ]_{{\rm tr}}= [Q'_h]_{({{\rm tr}}/h)} \circ \iota_h,\end{aligned}$$ where the first equality follows from the commutation property in Step 3 and the second equality is a truncation of the words from $Q'_h$ as elements of $\widetilde{E_h}$. [**Step 5.**]{} Fix $\varepsilon>0$ and let ${{\rm tr}}_0 = {{\rm tr}}_0(Q,\varepsilon)$ be so large that $$\begin{aligned} {\mathbb{E}}_Q\big[ \big( |Y^{(1)}|-{{\rm tr}}\big)_+ \big] < \tfrac13 \varepsilon m_Q, \qquad {{\rm tr}}\geq {{\rm tr}}_0.\end{aligned}$$ Then, for $0<h<\tfrac{1}{24} \varepsilon m_Q$, we have $$\begin{aligned} \label{eq:justso} {\mathbb{E}}_{\lceil Q \rceil_h}\big[ h\big( \tfrac{|Y^{(1)}|}{h}-\tfrac{{{\rm tr}}}{h} \big)_+ \big] < \tfrac13 \varepsilon m_Q + 2h < \tfrac12 \varepsilon m_{\lceil Q \rceil_h}. \end{aligned}$$ Divide both sides of by $h$, and observe that the continuum word of length $|Y^{(1)}|$ under $\lceil Q \rceil_h$ corresponds to the discrete word of $|Y^{(1)}|/h$ $h$-letters under $Q_h'$, to obtain $$\begin{aligned} {\mathbb{E}}_{Q_h'}\big[ \big( |Y^{(1)}|-\tfrac{{{\rm tr}}}{h} \big)_+ \big] < \tfrac12 \varepsilon m_{Q_h'}.\end{aligned}$$ This estimate allows us to use Lemma \[lem:trcontinuous\] in Appendix \[entropy\], which says that for every $0<\varepsilon<\tfrac12$, $$\begin{aligned} \label{bepsest} (1-\varepsilon) \Big[ H(\Psi_{[Q_h']_{({{\rm tr}}/h)}} \mid \nu_h^{\otimes {\mathbb{N}}}) + b(\varepsilon) \Big] \leq H(\Psi_{Q_h'} \mid \nu_h^{\otimes {\mathbb{N}}})\end{aligned}$$ with $b(\varepsilon)= - 2\varepsilon + [\varepsilon \log \varepsilon + (1-\varepsilon) \log (1-\varepsilon)]/(1-\varepsilon)$. However, by (\[HPsihrel\]–\[Qtrhrel\]) we have $$H(\Psi_{[Q'_h]_{({{\rm tr}}/h)}} \mid \nu_h^{\otimes {\mathbb{N}}}) = H(\Psi_{\lceil [Q]_{{\rm tr}}\rceil_h,h} \mid {\mathscr{W}}) = H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}).$$ Substitute this relation into and use (\[HhHrel\]–\[HPsihrel\]), to obtain $$(1-\varepsilon) \Big[ H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) + b(\varepsilon) \Big] \leq H(\Psi_Q \mid {\mathscr{W}}).$$ Now let $\varepsilon \downarrow 0$ and use that $\lim_{\varepsilon\downarrow 0} b(\varepsilon)=0$, to obtain . ### Proof of part (2) {#subsect:prop:Ique.tr.cont.part2} Fix $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $m_Q= \infty$ and $H(Q \mid Q_{\rho,{\mathscr{W}}}) < \infty$. We construct $\widetilde{Q}_{{\rm tr}}\in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$, ${{\rm tr}}\in{\mathbb{N}}$, satisfying via a “smoothed truncation” that has the same concatenated word content as its “hard truncation” equivalent. The proof comes in 5 Steps. [**Step 1.**]{} It will we be convenient to consider the two-sided scenario, i.e., we regard $Q$ as a shift-invariant probability measure on $F^{\mathbb{Z}}$. Define $$\chi_{{\rm tr}}\colon\,F_{0,{{\rm tr}}}^{\mathbb{Z}}\times [0,1]^{\mathbb{Z}}\to F^{\mathbb{Z}}, \qquad \chi_{{\rm tr}}\colon\,\big( (f_i,\tau_i)_{i\in{\mathbb{Z}}}, (u_i)_{i\in{\mathbb{Z}}} \big) \mapsto (\tilde f_i, \tilde \tau_i)_{i\in{\mathbb{Z}}},$$ as follows. Put $t_0=0$, $t_i=t_{i-1}+\tau_i$, $t_{-i}=t_{-i+1}-\tau_{-i+1}$ for $i\in{\mathbb{N}}$, and $\varphi = \kappa\big( (f_i,\tau_i)_{i\in{\mathbb{Z}}}\big)$, set $$\begin{aligned} \tilde{t}_i = \begin{cases} t_i-u_i & \text{if} \; \tau_i={{\rm tr}},\\ t_i & \text{if} \; \tau_i <{{\rm tr}}, \end{cases}\end{aligned}$$ $\tilde\tau_i=t_i-t_{i-1}$ and $\tilde f_i(\cdot)=\varphi( (\,\cdot \wedge \tilde\tau_i)+t_{i-1})$ for $i\in{\mathbb{Z}}$. In words, the total concatenated word content remains unchanged, and if the length of the $i$-th word $\tau_i$ equals ${{\rm tr}}$, then its end-point $t_i$ is moved $u_i$ to the left. Put $\widetilde{Q}_{{\rm tr}}= ([Q]_{{\rm tr}}\otimes \mathrm{Unif}[0,1]^{\otimes {\mathbb{Z}}}) \circ \chi_{{\rm tr}}^{-1} \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{Z}})$. By construction, $\Psi_{\widetilde{Q}_{{\rm tr}}} = \Psi_{[Q]_{{\rm tr}}}$ and $m_{\widetilde{Q}_{{\rm tr}}} = m_{[Q]_{{\rm tr}}}$. In particular, $$\begin{aligned} m_{\widetilde{Q}_{{\rm tr}}} H(\Psi_{\widetilde{Q}_{{\rm tr}}} \mid {\mathscr{W}}) = m_{[Q]_{{\rm tr}}} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) .\end{aligned}$$ [**Step 2.**]{} Write $\widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} = ([q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes{\mathbb{Z}}} \otimes \mathrm{Unif} [0,1]^{\otimes {\mathbb{Z}}}) \circ \chi_{{\rm tr}}^{-1}$ for the result of the analogous operation on the reference measure $(q_{\rho,{\mathscr{W}}})^{\otimes {\mathbb{Z}}}$. We have ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \widetilde{Q}_{{\rm tr}}= Q$ and ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} ([q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes{\mathbb{Z}}} \otimes \mathrm{Unif}[0,1]^{\otimes {\mathbb{Z}}}) \circ \chi_{{\rm tr}}^{-1} = (q_{\rho,{\mathscr{W}}})^{\otimes {\mathbb{Z}}}$, and hence $$\begin{aligned} \label{eq:string} \liminf_{{{\rm tr}}\to\infty} H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) & \geq \sup_{\varepsilon > 0} \liminf_{{{\rm tr}}\to\infty} \inf_{Q' \in B_\varepsilon(Q)} H( Q' \mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) \notag \\ & \geq H(Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes {\mathbb{Z}}}) = \lim_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes {\mathbb{Z}}}),\end{aligned}$$ where we use Lemma \[lemma:hregularised1\] (2) in the second inequality. (Note: Inspection of the proof of Lemma \[lemma:hregularised1\] (2) shows that the inequality “$\leq$” in also holds for $Q$’s that are not product.) The last equality in holds because the truncations $[\,\cdot\,]_{{\rm tr}}$ form a projective family (see [@BiGrdHo10 Lemma 8.1]). As specific relative entropy can only decrease under the operation of taking image measures, we have $H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) \leq H([Q]_{{\rm tr}}\mid [q_{\rho, {\mathscr{W}}}]_{{\rm tr}}^{\otimes {\mathbb{Z}}}) \leq H(Q \mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}})$, so $\limsup_{{{\rm tr}}\to\infty} H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) \leq H(Q \mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}})$ and, indeed, $$\begin{aligned} \lim_{{{\rm tr}}\to\infty} H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) = \lim_{{{\rm tr}}\to\infty} H([Q]_{{\rm tr}}\mid [q_{\rho,{\mathscr{W}}}]_{{\rm tr}}^{\otimes {\mathbb{Z}}}) = H(Q \mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}).\end{aligned}$$ The proof of is complete once we show that $$\begin{aligned} \label{eq:HtildeQtr.bd} H( \widetilde{Q}_{{\rm tr}}\mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}) \leq H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) + o(1),\end{aligned}$$ since, by part (1), $$\begin{aligned} \widetilde{I}^{\mathrm{que}}(\widetilde{Q}_{{\rm tr}}) = H( \widetilde{Q}_{{\rm tr}}\mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}) + m_{\widetilde{Q}_{{\rm tr}}} H(\Psi_{\widetilde{Q}_{{\rm tr}}} \mid {\mathscr{W}}).\end{aligned}$$ [**Step 3.**]{} It remains to verify . Note that $$\begin{aligned} H( \widetilde{Q}_{{\rm tr}}\mid q_{\rho,{\mathscr{W}}}^{\otimes {\mathbb{Z}}}) - H( \widetilde{Q}_{{\rm tr}}\mid \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}} ) & = \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}}\bigg[ \log\frac{d\pi_N \widetilde{Q}_{{\rm tr}}}{d q_{\rho,{\mathscr{W}}}^{\otimes N}} - \log\frac{d\pi_N \widetilde{Q}_{{\rm tr}}}{d\pi_N \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}} \bigg] \notag \\ & = \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}} \bigg[ \log\frac{d\pi_N \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}}{d q_{\rho,{\mathscr{W}}}^{\otimes N}} \bigg],\end{aligned}$$ and that, by construction, ${d\pi_N \widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}}/{dq_{\rho,{\mathscr{W}}}^{\otimes N}}$ is a function of the word lengths $\tilde{\tau}_1,\dots,\tilde{\tau}_N$ only (indeed, because of the i.i.d. property of Brownian increments it easy to see that under both laws the word contents given their lengths are the same, namely, independent pieces of Brownian paths). Write $\widetilde{R}_{{\rm tr}}^{\mathrm{ref}}$ for the law of the sequence of word lengths under $\widetilde{Q}_{{\rm tr}}^{\mathrm{ref}}$. Then we must show that $$\begin{aligned} \label{eq:ElogdRtildedrho.bd} \limsup_{{{\rm tr}}\to\infty} \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}} \bigg[ \log\frac{d\pi_N \widetilde{R}_{{\rm tr}}^{\mathrm{ref}}}{d \rho^{\otimes N}} (\tilde\tau_1,\dots,\tilde\tau_N) \bigg] \leq 0. \end{aligned}$$ [**Step 4.**]{} Denote the density of $\pi_N \widetilde{R}_{{\rm tr}}^{\mathrm{ref}}$ with respect to Lebesgue measure on ${\mathbb{R}}_+^N$ by $\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}$. Consider fixed $\tilde\tau_1,\dots,\tilde\tau_N$, and decompose into maximal stretches of $\tilde\tau_i$’s with values in $({{\rm tr}}-1,{{\rm tr}}+1)$ (note that under $\chi_{{\rm tr}}$ no word can become longer than ${{\rm tr}}+1$, while when $\tilde\tau_i < {{\rm tr}}-1$ the corresponding word is not truncated, i.e., $\tilde{t}_i=t_i$). Thus, there are $0 \leq M < N$, $i'_1 \leq j'_2 < i'_2 \leq j'_2 < \cdots < i'_M \leq j'_M \leq N$ such that $\{ 1 \leq i \leq N\colon\,\tilde\tau_i \in ({{\rm tr}}-1,{{\rm tr}}+1) \} = \cup_{k=1}^M [i'_k, j'_k] \cap {\mathbb{N}}$. Observe that, by construction, $\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}(\tilde\tau_1,\dots,\tilde\tau_N)$ can be decomposed into a product of $\prod_{j \colon \tilde\tau_j\leq {{\rm tr}}-1} \bar{\rho} (\tilde\tau_j)$ and $M$ further factors involving the $\tilde\tau_i$’s from these stretches, where the $k$-th factor depends only on $(\tilde\tau_i\colon\, i'_k \leq i \leq j'_k)$. We claim that $$\begin{aligned} \label{eq:ftrN.ref.bd} \frac{\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}(\tilde\tau_1,\dots,\tilde\tau_N)}{ \prod_{j=1}^N \bar{\rho}(\tilde\tau_j)} \leq \prod_{k=1}^M \big( C_1 {{\rm tr}}^{1+\epsilon} \big)^{j'_k-i'_k+1} = \big( C_1 {{\rm tr}}^{1+\epsilon} \big)^{\# \{ 1 \leq i \leq N\colon\, \tilde\tau_i > {{\rm tr}}-1 \}}\end{aligned}$$ for some $C_1=C_1(\rho) <\infty$ and $\epsilon=\epsilon(\rho) \in [0,1]$ uniformly in ${{\rm tr}}$ for ${{\rm tr}}$ sufficiently large. To see why holds, consider for example the first stretch and assume for simplicity that $i'_1=1<j'_1$ and that we know that the $0$-th word is not truncated (i.e., $\tilde{t}_0=t_0=0$). Let $\ell \leq j'_1+1$, and pretend we know that the first $\ell-1$ words are truncated (i.e., $\tau_1=\cdots=\tau_{\ell-1}={{\rm tr}}$), while the $\ell$-th word is not ($\tau_\ell<{{\rm tr}}$). Then $\tilde\tau_1={{\rm tr}}-u_1$ and $\tilde\tau_i={{\rm tr}}-u_i+u_{i-1}$ for $2 \leq i \leq \ell-1$, and so $u_i=\sum_{j=1}^i ({{\rm tr}}-\tilde\tau_j)$ for $1 \leq i \leq \ell-1$ and $\tau_\ell =\tilde\tau_\ell-u_{\ell-1} = \tilde\tau_\ell-\sum_{j=1}^{\ell-1} ({{\rm tr}}-\tilde\tau_j)$. This case contributes to $\widetilde{f}_{{{\rm tr}},\ell}^{\mathrm{ref}}$ the term $$\begin{aligned} \label{eq:termf.ell.ref} \rho([{{\rm tr}},\infty))^{\ell-1} \bar\rho\Big(\tilde\tau_\ell- {\textstyle \sum_{j=1}^{\ell-1} ({{\rm tr}}-\tilde\tau_j)}\Big) \prod_{i=1}^{\ell-1} {1}_{[0,1]}\Big( {\textstyle \sum_{j=1}^{i} ({{\rm tr}}-\tilde\tau_j)}\Big).\end{aligned}$$ Note that, by , we have $\eqref{eq:termf.ell.ref}/\prod_{j=1}^\ell \bar{\rho}(\tilde\tau_j) \le C_2 (C_3 {{\rm tr}}^{1+\epsilon})^{\ell-1}$ for some $C_2=C_2(\rho), C_3=C_3(\rho) <\infty$ and $\epsilon=\epsilon(\rho) \in [0,1]$ uniformly in ${{\rm tr}}$ for ${{\rm tr}}$ sufficiently large. The contribution of any given stretch of length $j'_k-i'_k+1$ can be written as a sum of at most $2^{j'_k-i'_k+1}$ cases where the indices of the truncated words are specified. Each such case can be estimated by a suitable product of terms as in . Furthermore, outside the stretches the words are necessarily untruncated and thus contribute $\bar{\rho}(\tilde\tau_i)$ to $\widetilde{f}_{{{\rm tr}},N}^{\mathrm{ref}}$, which cancels with the corresponding term in $\rho^{\otimes N}$. [**Step 5.**]{} From and the shift-invariance of $\widetilde{Q}_{{\rm tr}}$ we obtain that $$\begin{aligned} \label{eq:ElogdRtildedrho.bd2} \lim_{N\to\infty} \frac1N {\mathbb{E}}_{\widetilde{Q}_{{\rm tr}}} \bigg[ \log\frac{d\pi_N \widetilde{R}_{{\rm tr}}^{\mathrm{ref}}}{d \rho^{\otimes N}} (\tilde\tau_1,\dots,\tilde\tau_N) \bigg] \leq C(1+\log {{\rm tr}}) Q(\tau_1>{{\rm tr}}-1). \end{aligned}$$ Now, $h(\mathscr{L}_Q(\tau_1) \mid \rho) \leq H(Q \mid q_{\rho,{\mathscr{W}}}^{\mathbb{N}}) < \infty$ by assumption. Because of , this implies that ${\mathbb{E}}_Q[\log(\tau_1)] < \infty$, and hence that $Q(\tau_1>{{\rm tr}}) = o(1/\log {{\rm tr}})$. Therefore implies . $\qed$ Removal of Assumptions – {#removeass} ======================== We give a brief sketch of the proof only, leaving the details to the reader. Assumptions – are satisfied when $\bar{\rho}$ satisfies and varies regularly at $\infty$ with index $\alpha$. The latter condition is stronger than . To prove the claim under alone, note that for every $\delta>0$ and $\alpha'<\alpha$ there exists a probability density $\bar{\rho}'=\bar{\rho}' (\delta,\alpha')$ such that $\bar{\rho} \leq (1+\delta)\bar{\rho}'$, $\bar{\rho}'$ varies regularly at $\infty$ with index $\alpha'$, and $\bar{\rho}'(t)dt$ converges weakly to $\bar{\rho}(t)dt$ as $\delta \downarrow 0$ and $\alpha' \uparrow \alpha$. Since the quenched LDP holds for $\bar{\rho}'$, we can proceed similarly as in [@BiGrdHo10 Sections 3.6 and 5] to get the quenched LDP for $\bar{\rho}$. More precisely, for $B \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$ we may write (recall and ) $$\begin{aligned} P(R_N \in B \mid X) &= \int_{0 \leq t_1 < \cdots < t_N < \infty} dt_1 \cdots dt_N\, \bar{\rho}(t_1)\,\bar{\rho}(t_2-t_1) \cdots \bar{\rho}(t_N-t_{N-1})\\[-2ex] &\hspace{18em} \times 1_B\big(R_{N;t_1,\ldots,t_N}(X)\big), \notag\end{aligned}$$ and estimate $\bar{\rho}(t_1) \leq (1+\delta)\bar{\rho}'(t_1)$, etc., to get $P(R_N \in B \mid X) \leq (1+\delta)^N\,P'(R_N \in B \mid X)$, where $P,P'$ have $\bar{\rho},\bar{\rho}'$ as excursion length distributions. Let $\mathcal{C} \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$ be a closed set, and let $\mathcal{C}^{(\varepsilon)}$ be its $\varepsilon$-blow-up. Then the LDP upper bound for $\bar{\rho}'$ gives $$\limsup_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X) \leq \log (1+\delta) - \inf_{Q \in \mathcal{C}^{(\varepsilon)}} I^\mathrm{que}_{\bar{\rho}'}(Q) \qquad X\text{-a.s.},$$ where the lower index $\bar{\rho}'$ indicates the excursion length distribution. Let $\delta \downarrow 0$ and $\alpha' \uparrow \alpha$, and use Lemma \[lemma:hregularised1\] (2), to get $$\limsup_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X) \leq - \inf_{Q \in \mathcal{C}^{(2\varepsilon)}} I^\mathrm{que}_{\bar{\rho}}(Q) \qquad X\text{-a.s.}$$ Finally, let $\varepsilon \downarrow 0$ and use the lower semi-continuity of $I^\mathrm{que}_{\bar{\rho}}$ to get the LDP upper bound for $\bar{\rho}$. An analogous argument works for the LDP lower bound: Now we pick $\alpha' > \alpha$, $\delta > 0$ and a probability density $\bar{\rho}'=\bar{\rho}' (\delta,\alpha')$ such that $\bar{\rho} \geq (1-\delta)\bar{\rho}'$, and $\bar{\rho}'$ satisfies the same conditions as above. Arguing as before, we obtain for any open $\mathcal{O} \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$, $$\liminf_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X) \geq - \inf_{Q \in \mathcal{O}} I^\mathrm{que}_{\bar{\rho}}(Q) \qquad X\text{-a.s.}$$ Proof of Theorems \[mainthmboundarycases\]–\[thmexp\] {#proofalpha1infty} ===================================================== We again give a brief sketch of the proofs only, leaving many details to the reader. Theorem \[mainthmboundarycases\](a), which says that for $\alpha=1$ the quenched rate function coincides with the annealed rate function, can be proved as follows: Since the claimed LDP upper bound holds automatically by the annealed LDP, it suffices to verify the matching lower bound. For this we can argue as in the proof of the lower bound in Section \[removeass\]. For any $\alpha'>1$ and $\delta>0$ we can approximate $\bar{\rho}$ by a suitable $\bar{\rho}'=\bar{\rho}' (\delta,\alpha')$ such that $\bar{\rho} \geq (1-\delta)\bar{\rho}'$. Then, using Theorem \[thm0:contqLDP\] with $\bar{\rho}'$ and taking $\delta \downarrow 0$, $\alpha' \downarrow 1$, we see that for any open $\mathcal{O} \subset \mathcal{P}^\mathrm{inv}(F^{\mathbb{N}})$, $$\liminf_{N\to\infty} \frac{1}{N} \log P(R_N \in \mathcal{C}^{(\varepsilon)}\mid X) \geq - \inf_{Q \in \mathcal{O} \cap \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})} I^\mathrm{ann}(Q) \qquad X\text{-a.s.}$$ (recall ). Finally note that any $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $H(Q \mid Q_{\rho,{\mathscr{W}}}) < \infty$ can be approximated by a sequence $(Q_n) \subset \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$ in such a way that $H(Q_n \mid Q_{\rho,{\mathscr{W}}}) \to H(Q \mid Q_{\rho,{\mathscr{W}}})$ to obtain the claim (using for example a “smoothed truncation” operation similar to Section \[subsect:prop:Ique.tr.cont.part2\]). Theorem \[mainthmboundarycases\](b), which says that for $\alpha=\infty$ the quenched rate function coincides with the annealed rate function on the set $\{Q\in\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})\colon\,\lim_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} H( \Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) = 0\}$ and is infinite elsewhere, follows from arguments analogous to [@BiGrdHo10 Section 7, Part (b)]: For the upper bound, we can pick arbitrarily large $\alpha'>1$ and approximate $\bar{\rho} \leq (1+\delta) \bar{\rho}'$ with the help of a suitable probability density $\bar{\rho}'$ which has decay exponent $\alpha'$. Using Theorem \[thm0:contqLDP\] with $\bar{\rho}'$ and taking $\alpha' \uparrow \infty$, $\delta \downarrow 0$, we see that the upper bound holds with the claimed form of the rate function. For the matching lower bound we can trace through the proof of the lower bound contained in Theorem \[thm0:contqLDP\] but replacing our “coarse-graining work horses” Proposition \[thm00:contqLDP\] and Corollary \[prop:qLDPhtr\] (which rely on [@BiGrdHo10 Cor. 1.6]) by versions that are suitable for $\alpha=\infty$ (which rely on [@BiGrdHo10 Thm. 1.4 (b)] instead), still using a suitable truncation approximation of the quenched rate function analogous to the one proven in Proposition \[prop:Ique.tr.cont\]. This constitutes a way of rigorously implementing the “first long string strategy” from [@BiGrdHo10 Section 4], as explained in the heuristic given in item 0 of Section \[disc\], through the coarse-graining approximation. Theorem \[thmexp\] follows from Theorem \[mainthmboundarycases\](b) via an observation that is the analogue of [@Bi08 Lemma 6]: subject to the exponential tail condition in , any $Q\in\mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$ with $H(Q \mid Q_{\rho,{\mathscr{W}}}) <\infty$ necessarily has $m_Q<\infty$. Because of this observation we can argue as follows. If $m_Q<\infty$, then $\lim_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} = m_Q$ and $\lim_{{{\rm tr}}\to\infty} \Psi_{[Q]_{{\rm tr}}} = \Psi_Q$ by dominated convergence (recall ), which in turn imply that $\liminf_{{{\rm tr}}\to\infty} m_{[Q]_{{\rm tr}}} H(\Psi_{[Q]_{{\rm tr}}} \mid {\mathscr{W}}) = m_Q H(\Psi_Q \mid {\mathscr{W}})$, as shown in Lemma \[lem:trcontinuous\] in Appendix \[entropy\]. The limit is zero if and only if $\Psi_Q = {\mathscr{W}}$, which by holds if and only if $Q \in {{\mathcal R}}_{\mathscr{W}}$. This explains the link between and . Basic facts about metrics on path space {#metrics} ======================================= We metrise $F$, defined in (and $F_h \subset F$ defined in ) as follows. Let $d_S(\phi_1, \phi_2)$ be a metric on $C([0,\infty))$ that generates Skorohod’s $J_1$-topology on $D([0,\infty)) \supset C([0,\infty))$, allowing for a certain amount of “rubber time” (see e.g.  Ethier and Kurtz [@EK86 Section 3.5 and Eqs. (5.1–5.3)]) $$\label{def:dS} d_S(\phi_1, \phi_2) = \inf_{\lambda \in \Lambda} \bigg\{ \gamma(\lambda) \vee \int\nolimits_0^\infty e^{-u} \sup_{t\geq 0} \big| \phi_1(t \wedge u) - \phi_2(\lambda(t) \wedge u) \big| \, du \bigg\},$$ where $\Lambda$ is the set of Lipschitz-continuous bijections from $[0,\infty)$ into itself and $$\gamma(\lambda) = \sup_{0 \leq s < t < \infty} \Big| \log \frac{\lambda(t)-\lambda(s)}{t-s} \Big|.$$ With $$\label{eq:metriconF} d_F(y_1,y_2) = |t_1-t_2| + d_S(\phi_1,\phi_2)$$ for $y_i=(t_i,\phi_i) \in F$, $(F,d_F)$ becomes complete and separable, and the same holds for $(F_h,d_F)$ for any $h>0$. [**Remark. **]{} We might at first be inclined to metrise $F$ in a more straightforward way than (\[eq:metriconF\]), e.g. via $$\label{eq:firstmetriconF} d^{\mathrm{first}}_F(y_1,y_2) = |t_1-t_2| + \|\phi_1-\phi_2\|_\infty, \quad y_i=(t_i,\phi_i) \in F, \: i=1,2.$$ However, if we would use Lipschitz functions with $d_F$ replaced by $d^{\mathrm{first}}_F$ in (\[eq:g\_Lipschitz\]), then in the analogue of Lemma \[obs:Rdiscdiff\] we would be forced to use terms of the form $\sup_{s \geq 0} |\varphi(s+t \wedge t') - \varphi(s+ih \wedge jh)|$ in the right-hand side. When used for $\varphi=X$ (a realisation of Brownian motion as in Proposition \[prop:LambdaPhilimit1tr\]), this would in turn force us to control the increments of the Brownian motion not only locally near the beginning and the end of each loop, but uniformly inside loops. In fact, it seems plausible that an analogue of Proposition \[prop:LambdaPhilimit1tr\] where $d_F$ is replaced by $d^{\mathrm{first}}_F$ actually fails. Furthermore, note that we cannot arrange $d_S$ in such a way that, for $\phi \in C([0,\infty))$, $h>0$, $t_1 \leq t'_1 < t_2 \leq t_2'$ with $|t'_1-t_1| \leq h$, $|t'_2-t_2| \leq h$, $$\begin{aligned} \label{eq:dS_wishful1} d_S\big( \phi((t_1+\cdot) \wedge t_2), \phi((t'_1+\cdot) \wedge t'_2)\big) \leq 2h + \sup_{t_1 \leq s \leq t'_1} |\phi(s)-\phi(t'_1)| + \sup_{t_2 \leq s \leq t'_2} |\phi(s)-\phi(t'_2)|.\end{aligned}$$ This is why in Lemma \[obs:Rdiscdiff\] we need the freedom to use an extra $k$ and to “look in a neighbourhood of the cut-points of size $kh$”. Basic facts about specific relative entropy {#entropy} =========================================== In Section \[ss:definitions\] we recall the definition of (specific) relative entropy of two probability measures. In Section \[ss:approximations\] we prove various approximation results for (specific) relative entropy, which were used heavily in Sections \[props\]. Especially the parts with $\Psi_Q$ require care because of the delicate nature of the word concatenation map $Q \mapsto \Psi_Q$. The latter is looked at in closer detail in Section \[subs:towards.Ique.tr.cont\]. Definitions {#ss:definitions} ----------- For $\mu,\nu$ probability measures on a measurable space $(S,\mathscr{S})$, $$h(\mu \mid \nu) = \begin{cases} \int_S (\log\frac{d\mu}{d\nu})\,d\mu, &\text{if} \; \mu \ll \nu, \\ \infty, & \text{otherwise,} \end{cases}$$ is the relative entropy of $\mu$ w.r.t. $\nu$. When the measurable space is a Polish space $E$ equipped with its Borel-$\sigma$-algebra, we also have the representation (see e.g. [@DeZe98 Lemma 6.2.13]) $$\begin{aligned} \label{eq:relentraslegendretransf} h(\mu \mid \nu) = \sup_{f \in C_b(E)} \Big\{ \int f\,d\mu - \log \int e^f \, d\nu \Big\} = \sup_{\scriptstyle f\colon\, E \to {\mathbb{R}}\; \atop \scriptstyle \text{bounded measurable}} \Big\{ \int f\,d\mu - \log \int e^f \, d\nu \Big\} \end{aligned}$$ (and if $\mu \ll \nu$ with a bounded and uniformly positive density, then the supremum in the right-hand side is achieved by $f=\log d\mu/d\nu$). Equation implies the entropy inequality $$\label{ineq:entropy} \mu(A)\leq \frac{\log 2 + h(\mu \mid \nu)}{\log[1+1/\nu(A)]}$$ by choosing $f=\alpha {1}_A$ and $\alpha=\log[1+1/\nu(A)]$ (see e.g. Kipnis and Landim [@KiLa99 Appendix 1, Proposition 8.2]). For $Q \in \mathcal{P}^{\mathrm{inv}}(F^{\mathbb{N}})$, $$\begin{aligned} \label{eq:SREwrtProd} H(Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}}) = \lim_{N\to\infty} \frac1N h\big( \pi_N Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes N}\big) = \sup_{N\in{\mathbb{N}}} \frac1N h\big(\pi_N Q \mid (q_{\rho,{\mathscr{W}}})^{\otimes N}\big)\end{aligned}$$ with $\pi_N$ the projection onto the first $N$ words, is the specific relative entropy of $Q$ w.r.t. $(q_{\rho,{\mathscr{W}}})^{\otimes{\mathbb{N}}}$. Similarly, using the canonical filtration $(\mathscr{F}^C_t)_{t \ge 0}$ on $C([0,\infty))$, for a probability measure $\Psi$ on $C([0,\infty))$ with stationary increments we denote by $$\begin{aligned} \label{eq:SREwrtWM} H(\Psi \mid {\mathscr{W}}) = \lim_{t\to\infty} \frac{1}{t} h\big(\Psi_{|}{}_{\mathscr{F}^C_t} \mid {\mathscr{W}}_{|}{}_{\mathscr{F}^C_t}\big) = \sup_{t>0} \frac{1}{t} h\big(\Psi_{|}{}_{\mathscr{F}^C_t} \mid {\mathscr{W}}_{|}{}_{\mathscr{F}^C_t}\big)\end{aligned}$$ the specific relative entropy w.r.t. Wiener measure. See Appendix \[contrelentr\] for a proof of . Approximations {#ss:approximations} -------------- Let $E$ be a Polish space. Equip $\mathcal{P}(E)$ with the weak topology (suitably metrised). $E^{\mathbb{N}}$ carries the product topology, and the set of shift-invariant probability measures $\mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ carries the weak topology (also suitably metrised). ### Blocks For $M\in{\mathbb{N}}$ and $r \in \mathcal{P}(E^M)$, denote by $r^{\otimes {\mathbb{N}}} \in \mathcal{P}(E^{\mathbb{N}})$ the law of an infinite sequence obtained by concatenating $M$-blocks drawn independently from $r$ (i.e., we identify $(E^M)^{\mathbb{N}}$ and $E^{\mathbb{N}}$ in the obvious way), and write $$\label{def:blockmeas} {\mathsf{sblock}}_M(r) = \frac1M \sum_{j=0}^{M-1} r^{\otimes {\mathbb{N}}} \circ (\theta^j)^{-1} \; \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$$ for its stationary mean. \[lemma:HblockmeasQ\] For $Q = q^{\otimes {\mathbb{N}}} \in\mathcal{P}^\mathrm{inv}(E)$ and $r \in \mathcal{P}(E^M)$, $$\label{eq:HblockmeasQ} H\big({\mathsf{sblock}}_M(r) \mid Q \big) = \frac1M h\big(r \mid \pi_M Q \big).$$ Moreover, for any $R \in \mathcal{P}^\mathrm{inv}(E)$, $$\label{eq:reconstrfromblocks} {\mathop{\text{\rm w-lim}}}_{M\to\infty} {\mathsf{sblock}}_M\big(\pi_M R\big) = R.$$ This proof is standard. Equation follows from the results in Gray [@Gr09b Section 8.4, see Theorem 8.4.1] by observing that ${\mathsf{sblock}}_M(r)$ is the asymptotically mean stationary measure of $r^{\otimes{\mathbb{N}}}$. It is also contained in Föllmer[@Foe88 Lemma 4.8], or can be proved with “bare hands” by explicitly spelling out $d\pi_N {\mathsf{sblock}}_M(r)/dq^{\otimes N}$ for $N \gg M$ and using suitable concentration arguments under $q^{\otimes N}$ as $N\to\infty$. Equation is obvious from the definition of weak convergence. ### Change of reference measure \[lemma:hregularised1\] [(1)]{} Let $\nu, \nu_1,\nu_2,\ldots \in \mathcal{P}(E)$ with ${\mathop{\text{\rm w-lim}}}_{n\to\infty} \nu_n = \nu$. Then $$\begin{aligned} \label{eq:hregularised1} h(\mu \mid \nu) = \lim_{\varepsilon \downarrow 0} \limsup_{n \to \infty} \inf_{\mu' \in B_\varepsilon(\mu)} h(\mu' \mid \nu_n), \quad \mu \in \mathcal{P}(E). \end{aligned}$$ [(2)]{} Let $Q=q^{\otimes {\mathbb{N}}}, Q_1=q_1^{\otimes {\mathbb{N}}},Q_2=q_2^{\otimes {\mathbb{N}}},\ldots \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ be product measures with ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q_n$ $= Q$. Then $$\begin{aligned} \label{eq:Hregularised1} H(R \mid Q) = \lim_{\varepsilon \downarrow 0} \limsup_{n \to \infty} \inf_{R' \in B_\varepsilon(R)} H(R' \mid Q_n), \quad R \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}}). \end{aligned}$$ \(1) Denote the term in the right-hand side of (\[eq:hregularised1\]) by $\tilde{h}(\mu)$. Let $f \in C_b(E)$, $\delta > 0$. We can find $\varepsilon_0 > 0$ and $n_0 \in {\mathbb{N}}$ such that $$\begin{aligned} \forall \, 0 < \varepsilon \leq \varepsilon_0, \, \mu' \in B_\varepsilon(\mu)\colon\,\, &\int_E f\, d\mu' \geq \int_E f\, d\mu - \frac{\delta}{2}, \\ \forall \, n \geq n_0\colon\,\, &\log \int_E e^f \, d\nu_n \leq \log \int_E e^f \, d\nu + \frac{\delta}{2}. \end{aligned}$$ Therefore, for $0 < \varepsilon \leq \varepsilon_0$ and $n \geq n_0$, $$\begin{aligned} \inf_{\mu' \in B_\varepsilon(\mu)} h(\mu' \mid \nu_n) \geq \int_E f\, d\mu' - \log \int_E e^f \, d\nu_n \geq \int_E f\, d\mu \smallskip - \log \int_E e^f \, d\nu - \delta.\end{aligned}$$ Now optimise over $f$ and take $\delta \downarrow 0$, to obtain $\tilde{h}(\mu) \geq h(\mu \mid \nu)$ via . For the reverse inequality, we may without loss of generality assume that $h(\mu \mid \nu) = \int_E \varphi \log \varphi\, d\nu < \infty$, where $\varphi=d\mu/d\nu \geq 0$ is in $L^1(\nu)$. Then for any $\delta > 0$ we can find a $\widetilde{\varphi} \geq 0$ in $C_b(E) \cap L^1(\nu)$ such that $\int_E \widetilde{\varphi}\,d\nu = 1$ and $$\begin{aligned} \int_E \big| \widetilde{\varphi} - \varphi \big| \, d\nu < \delta, \;\; \int_E \big| \widetilde{\varphi}\log\widetilde{\varphi} - \varphi\log\varphi \big| \, d\nu < \delta.\end{aligned}$$ Note that $\lim_{n\to\infty} \int_E \widetilde{\varphi}\,d\nu_n = 1$, and let $\widetilde{\varphi}_n = \widetilde{\varphi}/\int \widetilde{\varphi}\,d\nu_n$ and $\mu_n = \widetilde{\varphi}_n \nu_n$. Then, for $g \in C_b(E)$, $$\begin{aligned} \Big| \int_E g \, d\mu_n - \int_E g \,d\mu \Big| &= \Big| \frac1{\int_E \widetilde{\varphi} \, d\nu_n} \int_E g \widetilde{\varphi}\, d\nu_n - \int_E g\varphi \,d\nu \Big| \\ &\leq \Big| \frac{1}{\int_E \widetilde{\varphi}\, d\nu_n} - 1 \Big| \, \| g \widetilde{\varphi} \|_\infty + \Big| \int_E g \widetilde{\varphi}\, d\nu_n - \int_E g \widetilde{\varphi}\, d\nu \Big| + \Big| \int_E g (\widetilde{\varphi} - \varphi) \,d\nu\Big|, \end{aligned}$$ which can be made arbitrarily small by choosing $\delta$ small enough and $n$ large enough. In particular, for any $\varepsilon > 0$ we can choose $\delta$, $\widetilde{\varphi}$ and $n_0$ such that $\mu_n \in B_\varepsilon(\mu)$ for $n \geq n_0$. Hence $$\begin{aligned} \limsup_{n\to\infty} \inf_{\mu' \in B_\varepsilon(\mu)} h(\mu' \mid \nu_n) &\leq \limsup_{n\to\infty} h(\mu_n \mid \nu_n)\\ &= \limsup_{n\to\infty} \int_E \widetilde{\varphi}_n \log \widetilde{\varphi}_n \, d\nu_n = \int_E \widetilde{\varphi} \log \widetilde{\varphi} \, d\nu \leq h(\mu \mid \nu) + \delta, \end{aligned}$$ and letting $\delta \downarrow 0$ we $\tilde{h}(\mu) \leq h(\mu \mid \nu)$. \(2) Recall that for $R \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ and $Q$ a product measure, $$\lim_{N\to\infty} \frac1N h\left( \pi_N R \mid \pi_N Q\right) = H(R \mid Q) = \sup_{N\in{\mathbb{N}}} \frac1N h\left( \pi_N R \mid \pi_N Q\right).$$ Denote the expression in the right-hand side of (\[eq:Hregularised1\]) by $\tilde{H}(R)$. Fix $N\in{\mathbb{N}}$. Since for each $\varepsilon>0$, we have $B_{\varepsilon'}(R) \subset \{ R'\colon\, \pi_N R' \in B_{\varepsilon}(\pi_N R) \}$ for $\varepsilon'$ sufficiently small we also have $$\begin{aligned} \lim_{\varepsilon' \downarrow 0} \limsup_{n\to\infty} \inf_{R' \in B_{\varepsilon'}(R)} H(R' \mid Q_n) \ge \limsup_{n \to \infty} \inf_{\mu' \in B_\varepsilon(\pi_N R)} \frac1N h(\mu' \mid \pi_N Q_n). \end{aligned}$$ Let $\varepsilon \downarrow 0$ and use Part (1), to see that $\tilde{H}(R) \ge \frac1N h(\pi_N R \mid \pi_N Q)$ for any $N$. Hence also $\tilde{H}(R) \geq H(R \mid Q)$. For the reverse inequality, we may w.l.o.g. assume that $H(R \mid Q) < \infty$. Fix $\varepsilon>0$ and $\delta>0$. There is an $N \in {\mathbb{N}}$ such that $H(R \mid Q) \leq \frac1N h \big( \pi_N R \mid \pi_N Q \big) + \delta$, and since $\pi_N R \ll \pi_N Q=q^ {\otimes N}$ we can find a continuous, bounded and uniformly positive function $f_N\colon\,E^N \to [0,\infty)$ such that $\int_E f_N \, dq^{\otimes N} = 1$, $\int_E f_N \log f_N \, dq^{\otimes N} \leq h \big( \pi_N R \mid \pi_N Q \big) + N\delta$ and $\tilde{R}_N \in B_{\varepsilon/2}(R)$, where $\tilde{R}_N = {\mathsf{sblock}}_N\big(f_N\,q^{\otimes N}\big) \in \mathcal{P}^\mathrm{inv}(E^{\mathbb{N}})$ (see Lemma \[lemma:HblockmeasQ\]). By , we have $$H(\tilde{R}_N \mid Q) = \frac1N \int_E f_N \log f_N \, dq^{\otimes N}.$$ Now put $f_{N,n} = f_N/\int_E f_N\,q_n^{\otimes N}$, and define $\tilde{R}_{N,n} = {\mathsf{sblock}}_N \big(f_{N,n} \, q^{\otimes N}\big)$ as the “stationary version” of $(f_{N,n}\,q_n^{\otimes N})^{\otimes {\mathbb{N}}}$. In particular, $H(\tilde{R}_{N,n} \mid Q_n) = \frac1N \int f_{N,n} \log f_{N,n} \, dq_n^{\otimes N}$. Since $f_N$ is continuous, we have $\tilde{R}_{N,n} \in B_{\varepsilon}(R)$ and $\int_E f_{N,n} \log f_{N,n}\,dq_n^{\otimes N} \le H(R \mid Q) + 3\delta$ for $n$ large enough. Hence $$\limsup_{n\to\infty} \inf_{R' \in B_{\varepsilon}(R)} H(R' \mid Q_n) \le \limsup_{n\to\infty} H(\tilde{R}_{N,n} \mid Q_n) \le H(R \mid Q) + 4\delta.$$ Now let $\delta \downarrow 0$ followed by $\lim_{\varepsilon\downarrow 0}$ to conclude the proof. ### Existence of sharp coarse-graining approximations to the quenched rate function The following lemma was used in the proof of Lemma \[lemma:Iquetrregularised\]. \[lem:cg.2lev.blockapprox\] Let $Q \in \mathcal{P}^{\mathrm{fin}}(F^{\mathbb{N}})$ with $H( Q \mid Q_{\rho,{\mathscr{W}}}) < \infty$. There exist a sequence $(h_n)_{n\in{\mathbb{N}}}$ with $h_n>0$ and $\lim_{n\to\infty} h_n = 0$ and a sequence $(Q'_n)_{n\in{\mathbb{N}}}$ with $Q'_n \in \mathcal{P}^{\mathrm{fin}}(\widetilde{E}_{h_n}^{\mathbb{N}})$ and ${\mathop{\text{\rm w-lim}}}_{n\to\infty} Q'_n = Q$ such that $\limsup_{n\to\infty} I^{\mathrm{que}}_{h_n}(Q'_n) \leq I^{\mathrm{que}}(Q)$. The same holds with $F$ replaced by $F_{0,{{\rm tr}}}$ and $\widetilde{E}_{h_n}$ replaced by $\widetilde{E}_{h_n,{{\rm tr}}}$. Recall the definition of $\lceil Q \rceil_h$ in Step 2 of the proof of part (1) of Proposition \[prop:Ique.tr.cont\] (see page ). For any $N\in{\mathbb{N}}$, we have $$\begin{aligned} \label{eq:anyway1} h(\pi_N \lceil Q \rceil_h \mid \pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h) \leq h(\pi_N Q \mid \pi_N Q_{\rho,{\mathscr{W}}}) \leq N \, H( Q \mid Q_{\rho,{\mathscr{W}}}).\end{aligned}$$ The second inequality follows from . For the first inequality, use the fact that the construction of $\lceil Q \rceil_h$ can be implemented as a deterministic function of the pair of random variables $(Y,U)$, together with the fact that relative entropy can only decrease when image measures are taken. Write $\hat{\tau}_i = (\tilde{T}_i-\tilde{T}_{i-1})/h$, $i \in {\mathbb{N}}$. Since letters both under $\lceil Q_{\rho,{\mathscr{W}}} \rceil_h$ and under $Q_{\lceil \rho \rceil_h,{\mathscr{W}}}$ are constructed from a Brownian path that is independent of the word lengths, we have (recall \[def:rho.h.trunc\]) $$\begin{aligned} {1}(\hat{\tau}_1=k_1,\dots,\hat{\tau}_N=k_N) \, \frac{d\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h}{d\pi_N Q_{\lceil \rho \rceil_h,{\mathscr{W}}}} = \frac{(\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h)\big(\hat{\tau}_1=k_1,\dots,\hat{\tau}_N=k_N\big)} {\prod_{\ell=1}^N \lceil \rho \rceil_h(h k_\ell)}\end{aligned}$$ with $$\begin{aligned} & (\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h)\big(\hat{\tau}_1=k_1,\dots,\hat{\tau}_N=k_N\big) \notag \\ & = \int_{[0,1]} du \, \int_0^\infty \bar{\rho}(t_1) dt_1 \int_{t_1}^\infty \bar{\rho}(t_2-t_1) d(t_2-t_1) \cdots \int_{t_{N-1}}^\infty \bar{\rho}((t_N-t_{N-1})) d(t_N-t_{N-1}) \notag\\ &\qquad\qquad \times \prod_{\ell=1}^N {1}_{(h(\bar{k}_\ell-1+u), h(\bar{k}_\ell+u)]}(t_\ell),\end{aligned}$$ where $\bar{k}_\ell = k_1+\cdots+k_\ell$. Thus, by (\[eq:Vbarrhodef\]–\[ass:rhobar.reg1\]), $$\begin{aligned} \label{eq:anyway2} \sup_{N \in {\mathbb{N}}} \frac1N E_{\lceil Q \rceil_h}\left[ \Big| \log \frac{d\pi_N \lceil Q_{\rho,{\mathscr{W}}} \rceil_h}{d\pi_N Q_{\lceil \rho \rceil_h,{\mathscr{W}}}}\Big|\right] \leq r_Q(h)\end{aligned}$$ with $$\begin{aligned} r_Q(h) = \eta_n \lceil Q \rceil_h(\hat\tau_1 \in \bar{A}_n) + \eta_0 \lceil Q \rceil_h(\hat\tau_1 \not\in \bar{A}_n), \qquad h=2^{-n},\end{aligned}$$ where $\bar{A}_n \subset (s_*,\infty)$ is the set obtained from $A_n$ by removing pieces of length $2^{-n}$ from its edges (i.e., $\bar{A}_n$ is the $2^{-n}$-interior of $A_n$). But $\lim_{n\to\infty} \lceil Q \rceil_{2^{-n}}(\hat\tau_1 \not\in \bar{A}_n)=0$ because $A_n$ fills up $(s_*,\infty)$ as $n\to\infty$. Since $\lim_{n\to\infty} \eta_n=0$, we get $\lim_{h \downarrow 0} r_Q(h) = 0$. Combining (\[eq:anyway1\]–\[eq:anyway2\]), we obtain that $$\begin{aligned} H(\lceil Q \rceil_h \mid Q_{\lceil \rho \rceil_h,{\mathscr{W}}}) = \sup_{N \in {\mathbb{N}}} \frac1N h(\pi_N \lceil Q \rceil_h \mid \pi_N Q_{\lceil \rho \rceil_h,{\mathscr{W}}}) \leq H( Q \mid Q_{\rho,{\mathscr{W}}}) + r_Q(h)\end{aligned}$$ and, finally, $$\begin{aligned} &\limsup_{h \downarrow 0} H(\lceil Q \rceil_h \mid Q_{\lceil \rho \rceil_h,{\mathscr{W}}}) + (\alpha-1) m_{\lceil Q \rceil_h} H(\Psi_{\lceil Q \rceil_h, h} \mid {\mathscr{W}}) \notag\\ &\qquad \leq H( Q \mid Q_{\rho,{\mathscr{W}}}) + (\alpha-1) m_Q H(\Psi_Q \mid {\mathscr{W}}).\end{aligned}$$ The truncated case, where $F$ is replaced by $F_{0,{{\rm tr}}}$, etc., can be handled analogously. ### Approximation of $\Psi_Q$ The approximation in is stronger than just weak convergence. \[lemma:PsiQ:TVlim\] For $Q \in \mathcal{P}^{\mathrm{inv,fin}}(F^{\mathbb{N}})$, $$\begin{aligned} \label{eq:PsiQTVlimit} \lim_{T\to\infty} \sup_{A \subset C[0,\infty)\; \text{measurable}} \bigg| \Psi_Q(A) - \frac{1}{T} \int_0^T \big(\kappa(Q) \circ (\theta^s)^{-1}\big)(A) \,ds\bigg| = 0,\end{aligned}$$ i.e., the convergence in holds in total variation. Note that, by shift-invariance, $$\begin{aligned} \Psi_Q(A) = \frac{1}{N m_Q} {\mathbb{E}}_Q \left[ \int_0^{\tau_N} {1}_A\big( \theta^s \kappa(Y) \big) \, ds \right], \qquad N\in{\mathbb{N}}.\end{aligned}$$ Suppose that $Q$ is also ergodic. Then $\lim_{N\to\infty} \tau_N/N = m_Q$ $Q$-a.s.  and in $L^1(Q)$. Hence, for given $\varepsilon > 0$ we can find a $T_0(\varepsilon)$ such that, for $T \geq T_0(\varepsilon)$, $$\begin{aligned} \label{eq:Qerg.cons1} {\mathbb{E}}_Q\Big[ \Big| \frac{\tau_{N(T)}-T}{m_Q N(T)} \Big| \Big] + \Big| \frac{T}{m_Q N(T)} - 1 \Big| \leq \varepsilon,\end{aligned}$$ where $N(T) = \lceil T/m_Q \rceil$. Thus, for $T \geq T_0(\varepsilon)$ and any measurable $A \subset C[0,\infty)$, we have $$\begin{aligned} & \bigg| \Psi_Q(A) - \frac{1}{T} \int_0^T \big(\kappa(Q) \circ (\theta^s)^{-1}\big)(A) \,ds\bigg| \notag \\ & \leq \frac{1}{m_Q N(T)} \bigg| {\mathbb{E}}_Q\left[ \int_0^{\tau_{N(T)}} {1}_A\big( \theta^s \kappa(Y) \big) \, ds - \int_0^{T} {1}_A\big( \theta^s \kappa(Y) \big) \, ds \right] \bigg| \notag \\ & \qquad + \bigg| \Big(\frac{1}{m_Q N(T)}-\frac1T \Big) \int_0^{T} {1}_A\big( \theta^s \kappa(Y) \big) \, ds \bigg| \leq {\mathbb{E}}_Q\left[ \Big| \frac{\tau_{N(T)}-T}{m_Q N(T)} \Big| \right] + \Big| \frac{T}{m_Q N(T)} - 1 \Big| \leq \varepsilon,\end{aligned}$$ i.e., holds. If $Q$ is not ergodic, then use the ergodic decomposition $$Q = \int_{\mathcal{P}^{\mathrm{erg,fin}}(F^{\mathbb{N}})} Q' \, W_Q(dQ')$$ and note that $$m_Q = \int_{\mathcal{P}^{\mathrm{erg,fin}}(F^{\mathbb{N}})} m_{Q'} \, W_Q(dQ'), \quad \Psi_Q = \int_{\mathcal{P}^{\mathrm{erg,fin}}(F^{\mathbb{N}})} \frac{m_{Q'}}{m_Q} \, \Psi_{Q'} \, W_Q(dQ')$$ (see also [@BiGrdHo10 Section 6]). We can choose $N(T)$ so large that the set of $Q'$s for which holds (with $Q$ replaced by $Q'$) has $W_Q$-measure arbitrarily close to $1$. Continuity of the “letter part” of the rate function under truncation: discrete-time {#subs:towards.Ique.tr.cont} ------------------------------------------------------------------------------------ In this section we consider a discrete-time scenario as in [@BiGrdHo10]: $\rho \in \mathcal{P}({\mathbb{N}})$, $E$ is a Polish space, $\nu \in \mathcal{P}(E)$, the sequence of words $(Y^{(i)})_{i\in{\mathbb{N}}}$ with discrete lengths has reference law $q_{\rho,\nu}^{\otimes{\mathbb{N}}}$ with $q_{\rho,\nu}$ as in [@BiGrdHo10 Eq. (1.4)]. The following lemma extends [@BiGrdHo10 Lemma A.1] to Polish spaces (in [@BiGrdHo10] it was only proved and used for finite $E$, and without explicit control of the error term). Via coarse-graining, this lemma was used in the proof of Proposition \[prop:Ique.tr.cont\]. \[lem:trcontinuous\] Let $Q \in \mathcal{P}^{\mathrm{fin}}(\widetilde{E}^{\mathbb{N}})$ and $0 < \varepsilon < \tfrac12$. Let ${{\rm tr}}\in {\mathbb{N}}$ be so large that $$\begin{aligned} \label{eq:mQtr.qb} {\mathbb{E}}_Q\Big[ \big( |Y^{(1)}|-{{\rm tr}}\big)_+ \Big] < \frac{\varepsilon}{2} m_Q.\end{aligned}$$ Then $$\begin{aligned} \label{eq:HPsiQtr.qb} (1-\varepsilon) \big( H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) + b(\varepsilon) \big) \leq H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}})\end{aligned}$$ with $b(\varepsilon) = -2\varepsilon + [\varepsilon \log\varepsilon + (1-\varepsilon) \log (1-\varepsilon)]/(1-\varepsilon) $, satisfying $\lim_{\varepsilon \downarrow 0} b(\varepsilon)=0$. In particular, $$\begin{aligned} \label{eq:HPsiQtrlim.disc} \lim_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) = H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}}). \end{aligned}$$ We can assume w.l.o.g. that $H(\Psi_Q \mid \nu^{\otimes{\mathbb{N}}}) < \infty$ for otherwise is trivial and follows from lower-semicontinuity of specific relative entropy. First, assume that $Q$ is ergodic, then $\Psi_Q$ is ergodic as well (see [@Bi08 Remark 5]). For $\Psi \in \mathcal{P}^{\mathrm{erg}}(E^{\mathbb{N}})$ and $\delta \in (0,1)$, $$\begin{aligned} \label{eq:HPsinu.typset0} H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) & = \lim_{L\to\infty} -\frac{1}{L} \log \Big( \inf\big\{ \nu^{\otimes L}(B) \colon\, B \subset E^L, (\pi_L \Psi)(B) \geq 1-\delta \big\} \Big), \\ \label{eq:HPsinu.typset1} & = \lim_{L\to\infty} \sup\Big\{ -\frac{1}{L} \log \nu^{\otimes L}(B) \, \colon\, B \subset E^L, (\pi_L \Psi)(B) \geq 1-\delta \Big\}.\end{aligned}$$ This replaces the asymptotics of the covering number and its relation to specific entropy for ergodic measures on discrete shift spaces that was employed in the proof of [@BiGrdHo10 Lemma A.1], and can be deduced with bare hands from the Shannon-McMillan-Breiman theorem. Indeed, asymptotically optimal $B$’s are of the form $\{ \frac1L \log \frac{d\pi_L\Psi}{d\nu^{\otimes L}} \in H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) \pm \epsilon\}$: Put $f_L = \frac{d\pi_L \Psi}{d\nu^{\otimes L}}$ and set $B_L = \{ \frac1L \log f_L > H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) - \epsilon \}$. Then $(\pi_L \Psi)(B_L) \to 1$ by the Shannon-McMillan-Breiman, and $\nu^{\otimes L}(B_L) = \int_{B_L} \frac1{f_L} d\pi_L\Psi \leq \exp[-L(H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) - \epsilon)]$, i.e., the right-hand side of is $\geq H(\Psi \mid \nu^{\otimes {\mathbb{N}}})$. For the reverse inequality, consider any $B \subset E^L$ with $(\pi_L\Psi)(B) \geq \tfrac12$, say. Set $B'=B \cap \{ \frac1L \log f_L < H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) + \epsilon\}$. Then $\pi_L\Psi(B') \geq \tfrac13$ for $L$ large enough and $\nu^{\otimes L}(B) \geq \nu^{\otimes L}(B') \geq \exp[-L(H(\Psi \mid \nu^{\otimes {\mathbb{N}}}) + \epsilon)] \pi_L\Psi(B')$. Hence the right-hand side of is also $\leq H(\Psi \mid \nu^{\otimes {\mathbb{N}}})$. To check , fix $\varepsilon>0$. For $L$ sufficiently large, we construct a set $B_L \subset E^L$ such that $\pi_L \Psi_Q(B_L) \geq \tfrac12$ and $\nu^{\otimes L}(B_L) \leq \exp[ - L(1-\varepsilon)(b_L(\varepsilon) + H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}))]$, i.e., $$\begin{aligned} -\frac1L \log \nu^{\otimes L}(B_L) \geq (1-\varepsilon) \big[ H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) + b_L(\varepsilon) \big], \end{aligned}$$ where $\lim_{L\to\infty} b_L(\varepsilon) = b(\varepsilon)$. Via applied to $\Psi=\Psi_Q$, this yields . To construct the sets $B_L$, we proceed as follows. Put $N= \lceil (1+2\varepsilon) L/m_Q \rceil$. By the ergodicity of $Q$ (see [@BiGrdHo10 Section 3.1] for analogous arguments), we can find a set $A \subset \widetilde{E}^N$ such that $$\begin{aligned} &\forall\, (y^{(1)},\dots,y^{(N)}) \in A \colon\, \notag \\ \label{eq:BL.prop.1} & \hspace{2em} | \kappa(y^{(1)},\dots,y^{(N)}) | \geq L(1+\varepsilon), \;\; |y^{(1)}| \leq {{\rm tr}}, \;\; \sum_{i=1}^N (|y^{(i)}|-{{\rm tr}})_+ < \varepsilon L, \\ \label{eq:BL.prop.2} & \hspace{2em} {\mathbb{E}}_Q\Big[ |Y^{(1)}| {1}_A(Y^{(1)},\dots,Y^{(N)}) \big] \geq (1-\varepsilon) m_Q, \end{aligned}$$ and the set $$\begin{aligned} B'_L = B'_L(A) &= \Big\{ \pi_L \big( \theta^i \kappa([y^{(1)}]_{{\rm tr}},\dots,[y^{(N)}]_{{\rm tr}})\big)\colon \,\notag\\ &\qquad (y^{(1)},\dots,y^{(N)}) \in A, i=0,1,\dots,|y^{(1)}|-1 \Big\} \subset E^L \end{aligned}$$ satisfies $$\begin{aligned} \pi_L\Psi_{[Q]_{{\rm tr}}}(B'_L) \geq \frac12, \quad \nu^{\otimes \lceil L(1-\varepsilon) \rceil}(\pi_{\lceil L(1-\varepsilon) \rceil} B'_L) \leq \exp\big[ -L(1-\varepsilon) \big(H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})-2\varepsilon\big)\big].\end{aligned}$$ Here, use in (\[eq:BL.prop.1\]–\[eq:BL.prop.2\]), and note that $N\big(1-\tfrac{\varepsilon}{2}\big)m_Q \sim (1+2\varepsilon)\big(1-\tfrac{\varepsilon}{2}\big)L \geq (1+\varepsilon) L$ and $N \tfrac{\varepsilon}{2} m_Q \sim (1+2\varepsilon)\tfrac{\varepsilon}{2} L < \varepsilon L$ as $L\to\infty$. For $I \subset \{1,\dots,L\}$, $x \in E^L$ and $y \in E^{|I|}$, write $\mathsf{ins}_I(x; y) \in E^{L+|I|}$ for the word of length $L+|I|$ consisting of the letters from $y$ at index positions in $I$ and the letters from $x$ at index positions not in $I$, with the order of letters preserved within $x$ and within $y$ (the word $y$ is inserted in $x$ at the positions in $I$). Put $$\begin{aligned} B_L = \pi_L\Big( \big\{ \mathsf{ins}_I(x; y) \colon \, x \in B'_L, I\subset \{1,\dots,L\}, |I| \leq \varepsilon L, y \in E^{|I|} \big\} \Big).\end{aligned}$$ Then $\pi_L\Psi_Q(B_L) \geq \frac12$ by construction. Furthermore, for fixed $I\subset \{1,\dots,L\}$ with $|I|=k \le \varepsilon L$, $$\begin{aligned} \nu^{\otimes L} \Big( \pi_L\big( \big\{ \mathsf{ins}_I(x; y)\colon \, x \in B'_L, y \in E^k \big\} \big)\Big) = \nu^{\otimes L} \big( \pi_{L-k}(B'_L) \big) \leq \nu^{\otimes \lceil L(1-\varepsilon \rceil]}(\pi_{\lceil L(1-\varepsilon) \rceil} B'_L),\end{aligned}$$ and hence $$\begin{aligned} \nu^{\otimes L}(B_L) & \leq [\varepsilon L] {L \choose [\varepsilon L]} \exp\big[ -L(1-\varepsilon) \big(H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})-2\varepsilon\big)\big] \notag \\ & = \exp\big[ - L(1-\varepsilon) \big(b_L(\varepsilon) + H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}})\big) \big]\end{aligned}$$ with $b_L(\varepsilon) = - \frac{1}{(1-\varepsilon) L}(\log [\varepsilon L] + \log {L \choose [\varepsilon L]}) - 2\varepsilon$, which satisfies $\lim_{\varepsilon \downarrow 0} b_L(\varepsilon)= b(\varepsilon)$. It remains to prove . Since ${\mathop{\text{\rm w-lim}}}_{{{\rm tr}}\to\infty} \Psi_{[Q]_{{\rm tr}}} = \Psi_Q$, we have $\liminf_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) \geq H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}})$, while the reverse inequality $ \limsup_{{{\rm tr}}\to\infty} H(\Psi_{[Q]_{{\rm tr}}} \mid \nu^{\otimes {\mathbb{N}}}) \leq H(\Psi_Q \mid \nu^{\otimes {\mathbb{N}}})$ follows from (\[eq:mQtr.qb\]–\[eq:HPsiQtr.qb\]) and the fact that $\lim_{{{\rm tr}}\to\infty} {\mathbb{E}}_Q[( |Y^{(1)}|-{{\rm tr}})_+] = m_Q$ by dominated convergence. For non-ergodic $Q$, decompose as in [@BiGrdHo10 Eqs.(6.1)–(6.3)], use the above argument on each of the ergodic components, and use the fact that specific relative entropy is affine. Existence of specific relative entropy {#contrelentr} ====================================== In this section we prove . For technical reasons, we consider the two-sided scenario. The argument is standard, but the fact that time is continuous requires us to take care. Let $\Omega = \tilde{C}({\mathbb{R}})$ be the set of continuous functions $\omega\colon\, {\mathbb{R}}\to {\mathbb{R}}$ with $\omega(0)=0$, which is a Polish space e.g. via the metric $d(\omega, \omega') = \int_{\mathbb{R}}e^{-|t|} \big(|\omega(t)-\omega'(t)| \wedge 1\big) dt$. The shifts on $\Omega$ are $\theta^t \omega(\cdot) = \omega(\cdot+t)-\omega(t)$. A probability measure $\Psi$ on $\Omega$ has stationary increments when $\Psi = \Psi \circ (\theta^t)^{-1}$ for all $t \in {\mathbb{R}}$. For an interval $I \subset {\mathbb{R}}$ denote $\mathcal{F}_I = \sigma(\omega(t)-\omega(s)\colon\, s,t \in I)$. $\Psi_I$ denotes $\Psi$ restricted to $\mathcal{F}_I$. Write ${\mathscr{W}}$ for the Wiener measure on $\Omega$, i.e., the law of a (two-sided) Brownian motion. Let $\Psi \in \mathcal{P}(\Omega)$ with stationary increments be given and assume that $h(\Psi_{[0,T]} \mid {\mathscr{W}}_{[0,T]}) < \infty$ for all $T>0$. To verify , we imitate well-known arguments from the discrete-time setup (see e.g.  Ellis [@El85 Section IX.2]). For $I_1$, $I_2$ disjoint intervals in ${\mathbb{R}}$, denote by $\kappa^\Psi_{I_1, I_2}\colon\, \Omega \times \mathcal{F}_{I_2} \to [0,1]$ a regular version of the conditional law of (the increments of) $\Psi$ on $I_2$, given the increments in $I_1$, i.e., for fixed $\omega$, $\kappa^\Psi_{I_1, I_2}(\omega, \cdot)$ is a probability measure on $\mathcal{F}_{I_2}$, for fixed $A \in \mathcal{F}_{I_2}$, $\kappa^\Psi_{I_1, I_2}(\cdot, A)$ is an $\mathcal{F}_{I_1}$-measurable function, and $\kappa^\Psi_{I_1, I_2}(\omega, A)$ is a version of ${\mathbb{E}}_{\Psi}[{1}_A | \mathcal{F}_{I_1}]$. When $I_1=\emptyset$, $\kappa^\Psi_{\emptyset, I_2}(\omega, A) = \Psi_{I_2}(A)$. Similarly, define $\kappa^{\mathscr{W}}_{I_1, I_2}$ (which is simply $\kappa^{\mathscr{W}}_{I_1, I_2}(\omega, A) = {\mathscr{W}}_{I_2}(A)$ by the independence of the Brownian increments). Put $$\begin{aligned} a_{I_1,I_2} = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \, \log\left[ \frac{d \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot)} {d \kappa^{\mathscr{W}}_{I_1, I_2}(\omega_1, \cdot)}(\omega_2)\right], \end{aligned}$$ the expected relative entropy of the conditional distribution under $\Psi$ on $\mathcal{F}_{I_2}$ given $\mathcal{F}_{I_1}$ w.r.t. Wiener measure on $\mathcal{F}_{I_2}$). We have $a_{I_1,I_2} < \infty$ for bounded intervals, because of the assumption of finite relative entropy of $\Psi$ w.r.t.  ${\mathscr{W}}$ on compact time intervals. By stationarity, $a_{I_1,I_2}=a_{t+I_1,t+I_2}$ for any $t$. Let $I_1' \subset I_1$, note that $\kappa^\Psi_{I_1, I_2}(\omega, \cdot) \ll \kappa^\Psi_{I_1', I_2} (\omega, \cdot)$ for $\Psi$-a.e. $\omega$, and $\kappa^{\mathscr{W}}_{I_1, I_2}(\omega, \cdot) = \kappa^{\mathscr{W}}_{I_1', I_2}(\omega, \cdot) = {\mathscr{W}}_{I_2}(\cdot)$. By the consistency property of conditional distributions, we have $$\begin{aligned} a_{I_1',I_2} = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \log \left[\frac{d \kappa^\Psi_{I_1', I_2}(\omega_1, \cdot)} {d \kappa^{\mathscr{W}}_{I_1', I_2}(\omega_1, \cdot)}(\omega_2)\right].\end{aligned}$$ Indeed, $$\int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) f(\omega_1,\omega_2) = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I'_1, I_2}(\omega_1, d\omega_2) f(\omega_1,\omega_2)$$ for any function $f(\omega_1,\omega_2)$ that is $\mathcal{F}_{I_1'} \otimes \mathcal{F}_{{\mathbb{R}}}$-measurable. Hence $$\begin{aligned} &a_{I_1,I_2} - a_{I_1',I_2} \\ & = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \bigg( \log \left[\frac{d \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot)} {d \kappa^{\mathscr{W}}_{I_1, I_2}(\omega_1, \cdot)}(\omega_2)\right] - \log \left[\frac{d \kappa^\Psi_{I_1', I_2}(\omega_1, \cdot)} {d \kappa^{\mathscr{W}}_{I_1', I_2}(\omega_1, \cdot)}(\omega_2)\right] \bigg) \notag \\ \label{eq:hdeccont1} & = \int_\Omega \Psi(d\omega_1) \int_\Omega \kappa^\Psi_{I_1, I_2}(\omega_1, d\omega_2) \, \log \left[\frac{d \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot)} {d \kappa^\Psi_{I_1', I_2}(\omega_1, \cdot)}(\omega_2)\right] \geq 0\end{aligned}$$ because the inner integral is $h( \kappa^\Psi_{I_1, I_2}(\omega_1, \cdot) \mid \kappa^\Psi_{I_1', I_2} (\omega_1, \cdot)) \geq 0$. Choosing $I_1'=\emptyset$, , we get $a_{I_1, I_2} \geq a_{\emptyset, I_2} = h( \Psi_{I_2} \mid {\mathscr{W}}_{I_2})$. Observe $$\begin{aligned} \frac{d \Psi_{(0,s+t]}}{d {\mathscr{W}}_{(0,s+t]}}(\omega) = \frac{d \Psi_{(0,t]}}{d {\mathscr{W}}_{(0,t]}}(\omega) \, \frac{d \kappa^\Psi_{(0,t],(t,s+t]}(\omega, \cdot)} {d \kappa^{\mathscr{W}}_{(0,t],(t,s+t]}(\omega, \cdot)}(\omega) \quad \Psi_{(0,s+t]}-\text{a.s.},\end{aligned}$$ take logarithms and integrate w.r.t. $\Psi$ (using consistency of conditional expectation on the right-hand side), to obtain $$\begin{aligned} h\big( \Psi_{(0,s+t]} \mid {\mathscr{W}}_{(0,s+t]} \big) = h\big( \Psi_{(0,t]} \mid {\mathscr{W}}_{(0,t]} \big) + a_{(0,t], (t,s+t]} \geq h\big( \Psi_{(0,t]} \mid {\mathscr{W}}_{(0,t]} \big) + h\big( \Psi_{(0,s]} \mid {\mathscr{W}}_{(0,s]} \big). \end{aligned}$$ Thus, the function $(0,\infty) \ni t \mapsto h( \Psi_{(0,t]} \mid {\mathscr{W}}_{(0,t]})$ is super-additive, and follows from Fekete’s lemma. Under $\kappa^\Psi_{(-\infty,0], (0,h]}$, the coordinate process will be a Brownian motion with a (possibly complicated) drift process $U_t = \int_0^t u_s\, ds$, where $(u_t)_{t \geq 0}$ can be chosen adapted, and $${\mathbb{E}}_\Psi\big[ h(\kappa^\Psi_{(-\infty,0], (0,h]} \mid {\mathscr{W}}_{(0,h]}) \big] = {\mathbb{E}}_\Psi\big[{\textstyle \int_0^h} u_s^2 \,ds \big]$$ (see Föllmer [@Foe86]). [AAA]{} M. Birkner, Conditional large deviations for a sequence of words, Stoch. Proc. Appl. 118 (2008), 703–729. M. Birkner, A. Greven and F. den Hollander, Quenched large deviation principle for words in a letter sequence, Probab. Theory Relat. Fields 148 (2010), 403–456. M. Birkner and F. den Hollander, A variational view on the Brownian copolymer, manuscript in preparation. E. Bolthausen and F. den Hollander, Localization transition for a polymer near an interface, Ann. Probab. 25 (1997), 1334–1366. F. Caravenna and G. Giacomin, The weak coupling limit of disordered copolymer models, Ann. Probab. 38 (2010), 2322–2378. F. Caravenna, G. Giacomin and F.L. Toninelli, Copolymers at selective interfaces: settled issues and open problems, In: *Probability in Complex Physical Systems. In honour of Erwin Bolthausen and Jürgen Gärtner* (eds. J.-D. Deuschel, B. Gentz, W. König, M. von Renesse, M. Scheutzow, U. Schmock), Springer Proceedings in Mathematics 11, 2012, pp. 289–312. A. Dembo and O. Zeitouni, *Large Deviations Techniques and Applications* (2nd. Ed.), Springer, 1998. R. Ellis, *Entropy, Large Deviations, and Statistical Mechanics*, Springer, 1985. S. Ethier and T. Kurtz, *Markov Processes: Characterization and Convergence*, Wiley, 1986. H. Föllmer, Time reversal on Wiener space, Lecture Notes in Math. 1158, Springer, 1986, pp. 119–129. H. Föllmer, Random fields and diffusion processes, in: École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, pp. 101–203, Lecture Notes in Math. 1362, Springer, 1988. R.M. Gray, *Entropy and Information Theory*, Springer, 1991.\ <http://ee.stanford.edu/~gray/it.html> C. Kipnis and C. Landim, *Scaling Limits of Interacting Particle Systems*, Springer, 1999.

ArXiv

--- abstract: | Lamperti’s maximal branching process is revisited, with emphasis on the description of the shape of the invariant measures in both the recurrent and transient regimes. A truncated version of this chain is exhibited, preserving the monotonicity of the original Lamperti chain supported by the integers. The Brown theory of hitting times applies to the latter chain with finite state-space, including sharp strong time to stationarity. Additional information on these hitting time problems are drawn from the quasi-stationary point of view. **Running title:** Lamperti’s MBP. **Keywords**: discrete probability; maximal branching process; recurrence/ transience transition; shape of invariant measures; tails; failure rate monotonicity; truncation; sharp strong time to stationarity; generating functions. **MSC 2000 Mathematics Subject Classification**: 60 J 10, 60 J 80, 92 D 25. address: | $^{1}$Laboratoire de Physique Théorique et Modélisation\ Université de Cergy-Pontoise\ CNRS UMR-8089\ Site de Saint Martin\ 2 avenue Adolphe-Chauvin\ 95302 Cergy-Pontoise, France\ $^{2}$Depto. Ingenieria Matematica and Centro Modelamiento Matematico\ Universidad de Chile\ UMI 2807, Uchile-Cnrs\ Casilla 170-3 Correo 3\ Santiago, Chile\ E-mail: [email protected], [email protected] author: - 'Thierry Huillet$^{1}$, Servet Martinez$^{2}$' title: 'Revisiting John Lamperti’s maximal branching process' --- Introduction ============ The Lamperti’s maximal branching process (mbp) is a modification of the Galton-Watson (GW) branching process selecting at each step the descendants of the most prolific ancestor, [@L1]. As a Markov chain on the full set of non-negative integers, Lamperti ([@L1]-[@L2]) gave sharp conditions on the tails of the branching number under which this process is recurrent (either positive or null) or transient. Our contribution is to describe the corresponding shape of the invariant measures and we proceed as follows: while fixing a target invariant measure (supported by the integers) of the mbp, we show (in Proposition $2$) how to compute in general the law of the branching mechanism that gives rise to it. Several classes of distributions are supplied both in the recurrent and transient setups. In Propositions $3$, $4$ and $5$, the target invariant measures are probabilities with tails getting larger and larger, ranging from geometric, power-law with index $\alpha \in \left( 0,1\right) $ and power-law with index $0$ (the target has no moments of any positive order). In Propositions $6$ (and $7$), it is shown that the null recurrent (respectively transient) Lamperti chain has a non trivial invariant infinite and positive measure. An important feature of the Lamperti chain we also emphasize on is its failure rate monotonicity (Proposition $1$). The Lamperti’s mbp also makes sense when the branching mechanism takes values in the finite subset $\left\{ 1,...,N\right\} $ and the question of computing the law of the branching mechanism giving rise to any finitely supported target distribution makes sense. We address this point in Proposition $8$. If the target distribution is in particular the restriction to $\left\{ 1,...,N\right\} $ of the invariant measure of a mbp with full state-space, this construction allows to design a truncated version of the latter chain preserving its failure rate monotonicity feature (Proposition $% 9 $ and Corollary $10$). For failure rate monotone Markov chains with finite state-space, Brown, [@Brown], designed a theory of hitting times which thus applies to the truncated Lamperti chain. The main concern is the relationship existing between the first hitting times of both state $\left\{ N\right\} $ and the restricted invariant measure of the truncated Lamperti chain. By monotonicity, state $\left\{ N\right\} $ is the largest possible value that the truncated chain can explore. Under some technical condition on the initial distribution, it is recalled that the former hitting time exceeds stochastically the latter (Proposition $11$) which has the structure of a compound geometric random variable (Proposition $13$). The excess time is a sharp strong time to stationarity allowing to estimate the distance between the current state of the truncated chain to its equilibrium distribution. Its cumulated probability mass function up to $n$ can be computed from the probability that the truncated chain is in state $\left\{ N\right\} $ after $n$ steps, (Proposition $12$). The alternative classical quasi-stationary point of view to this problem is also addressed. In Proposition $14$, we exhibit the rate of decrease of the hitting times to state $\left\{ N\right\} $ in terms of the quasi-stationary distribution. In Proposition $15$, we show that under Brown’s conditions on the initial distribution $\mathbf{\pi }_{0}$, the ratio of the large tail probabilities for the first hitting times of state $\left\{ N\right\} $ starting from $% \mathbf{\pi }_{0}$ against the quasi-stationary distribution exceeds $1$. Proposition $16$ deals with a question raised by Brown concerning asymptotic exponentiality of the hitting times which applies to the truncated Lamperti chain and its time-reversal. Lamperti’s model ================ The Lamperti maximal branching process (mbp) process may be described as an extremal analogue of the GW branching process, where the next generation is formed by the offspring of a most productive individual, [@L1]. As a result of some selection (or detection) mechanism, iteratively in each generation, only the offspring of one of the most productive individuals of the underlying GW process with branching number $\nu $ is kept (or detected), the other ones being wiped out (or missed by the detector). This output mechanism amounts to pruning Galton-Watson trees by iterative selection of a largest family size ending up with the sub-tree of the fittest individuals. In [@L1], Lamperti relates this model to a percolation problem. With $X_{n}$ the size of such a population at generation $n$, $F_{n}\left( j\right) =\mathbf{P}\left( X_{n}\leq j\right) $ and $\nu _{j,n+1}\overset{d}{% =}\nu $ for all $j$, the dynamics under concern is $$X_{n+1}=\max_{j=1,...,X_{n}}\nu _{j,n+1}\Rightarrow F_{n+1}\left( j\right) =\sum_{i\geq 0}\mathbf{P}\left( X_{n}=i\right) \mathbf{P}\left( \nu \leq j\right) ^{i}=\mathbf{E}z^{X_{n}}\mid _{z=\mathbf{P}\left( \nu \leq j\right) }.$$ with initial condition: $X_{0}\overset{d}{\sim }\mathbf{\pi }_{0}$ with $% \mathbf{P}\left( X_{0}\leq j\right) :=F_{0}\left( j\right) .$ We denote $\mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) =\mathbf{E}% \max_{j=1,...,i}\nu _{j}=\mathbf{E}\left( m_{i}\right) $ where $% m_{i}=\max_{j=1,...,i}\nu _{j}.$ Let $p\left( j\right) :=\mathbf{P}\left( \nu =j\right) $. We will assume that the set $\left\{ j:p\left( j\right) >0\right\} $ is either $\Bbb{N}% _{0}:=\left\{ 0,1,2,...\right\} $ or $\Bbb{N}:=\left\{ 1,2,...\right\} $ but, as we shall see, the finite case when $\left\{ j:p\left( j\right) >0\right\} =\left\{ 1,...,N\right\} $ for some integer $N\gg 1$, will also be of interest. We shall let $\phi \left( z\right) =\mathbf{E}z^{\nu }$ be the probability generating function (pgf) of $\nu .$ We shall distinguish two regimes for the branching number $\nu $: Branching number** **$\nu >0$**.** ---------------------------------- If $\nu >0$ ($p\left( 0\right) =\mathbf{P}\left( \nu =0\right) =0$ and $% \mathbf{E}\left( \nu \right) >1$), then $X_{n}>0,$ $\forall n\geq 0$ ($% X_{0}=1$), owing to $$F_{n+1}\left( 0\right) =\mathbf{P}\left( X_{n+1}=0\right) =\mathbf{E}% z^{X_{n}}\mid _{z=p\left( 0\right) =0}=\mathbf{P}\left( X_{n}=0\right) =0.$$ We can omit state $0$, being disconnected. One main concern in this context is whether $X_{n}\rightarrow \infty $ with probability (wp) $1$ (a case of transience) or to some limiting random variable (rv) $X_{\infty }$ (a case of recurrence): the tails of $\nu $ matter to decide. In the recurrent case, what is the shape of the invariant probability measure? In the null-recurrent and transient cases, what are the shapes of the invariant measure (no longer probability measures). In particular how are the tails of the invariant measure related to the tails of $\nu $. - **Transition matrix of** $\left\{ X_{n}\right\} $. With $F\left( j\right) =\mathbf{P}\left( \nu \leq j\right) $, $j\geq 1$, $\left\{ X_{n}\right\} $ is a time-homogeneous Markov chain (MC) on $\Bbb{N}$ with transition matrix ($\sum_{j\geq 1}P\left( i,j\right) =1-F\left( 0\right) ^{i}=1$) $$P\left( i,j\right) =F\left( j\right) ^{i}-F\left( j-1\right) ^{i}\text{, }% i,j\geq 1$$ equivalently $$\begin{aligned} \mathbf{P}\left( X_{n+1}>i\mid X_{n}=i\right) &=&1-F\left( i\right) ^{i} \\ \mathbf{P}_{X_{n}}\left( X_{n+1}>X_{n}\right) &=&1-F\left( X_{n}\right) ^{X_{n}}.\end{aligned}$$ Note $P\left( 1,j\right) =\mathbf{P}\left( \nu =j\right) .$ - **Some properties of** $\left\{ X_{n}\right\} $: - The Lamperti chain clearly is irreducible and aperiodic. - It holds that $\mathbf{P}\left( X_{n+1}\leq j\mid X_{n}=i\right) =:P^{c}\left( i,j\right) =F\left( j\right) ^{i}$ is a decreasing function of $i$, for all $j$: the Lamperti MC $\left\{ X_{n}\right\} $ is stochastically monotone (SM). Equivalently, with $\left\{ >j\right\} $ denoting the upper set $\left\{ j+1,...\right\} ,$ $\mathbf{P}\left( X_{n+1}>j\mid X_{n}=i\right) =:P\left( i,\left\{ >j\right\} \right) $ is an increasing function of $i$, for all $j$ and by induction $P^{n}\left( i,\left\{ >j\right\} \right) $ is an increasing function of $i$, for all $j$ and $n.$ In fact, it has a stronger monotonicity feature: The Lamperti Markov chain $\left\{ X_{n}\right\} $ is failure-rate monotone. *Proof:* The cumulated transition matrix : $P^{c}\left( i,j\right) :=\sum_{k=1}^{j}P\left( i,k\right) =:$ $P\left( i,\left\{ \leq j\right\} \right) $ satisfies: $$P^{c}\left( i_{1},j_{1}\right) P^{c}\left( i_{2},j_{2}\right) \geq P^{c}\left( i_{1},j_{2}\right) P^{c}\left( i_{2},j_{1}\right) ,$$ for all $i_{1}<i_{2}$ and $j_{1}<j_{2}$ (the matrix $P^{c}$ is totally positive of order $2$, viz TP$_{2}$): the MC $\left\{ X_{n}\right\} $ is failure rate monotone. Since if $P^{c}$ is TP$_{2}$, $P^{c}\left( i,j\right) $ is a decreasing function of $i$, for all $j$, (set $j_{2}=\infty $ in the last inequality to get $P^{c}\left( i_{1},j_{1}\right) \geq P^{c}\left( i_{2},j_{1}\right) $), TP$_{2}$ matrices $P^{c}$ form a subclass of SM matrices $P^{c}$. $\Box $ - **Generation:** As for all Markov chains, with $\left( \mathcal{U}% _{n};n\geq 1\right) $ a sequence of independent identically distributed (iid) uniform-$\left( 0,1\right) $ rvs: $$X_{n+1}=\sum_{j\geq 1}j\cdot \mathbf{1}\left( \mathcal{U}_{n+1}\in \left[ P^{c}\left( X_{n},j-1\right) ,P^{c}\left( X_{n},j\right) \right) \right) .$$ We can also check that, with $F^{-1}\left( y\right) =\inf \left( x:F\left( x\right) \geq y\right) $ the inverse function of $F$, one has $% X_{n+1}=F^{-1}\left( \mathcal{U}_{n+1}^{1/X_{n}}\right) .$ - **Transience versus recurrence:** Note that if $\mathbf{P}\left( \nu >i\right) \sim \lambda /i$, $\lambda >0$, for large $i$ ($\sim $ meaning that the ratio of the two terms appearing to the left and right of this symbol tend to $1$ as $i\rightarrow \infty $), $\mathbf{P}\left( X_{n+1}>i\mid X_{n}=i\right) \sim 1-e^{-\lambda }>0.$ In this case, $$\mathbf{P}\left( X_{n+1}\leq \left[ ix\right] \mid X_{n}=i\right) \sim F\left( ix\right) ^{i}\sim e^{-\lambda /x}.$$ and with $Z_{n}=\log X_{n}$$$\mathbf{P}\left( Z_{n+1}-Z_{n}\leq z\mid Z_{n}=\log i\right) \sim e^{-\lambda e^{-z}},$$ independent of $i$. This shows that for large $i$ and for this choice of $% \nu $, $\left\{ Z_{n}\right\} $ resembles a random walk with independent increments whose common law is a Gumbel distribution with mean $m=\log \lambda +\gamma $ ($\gamma $ the Euler constant). So $\left\{ Z_{n}\right\} $ (and $\left\{ X_{n}\right\} $) drifts to $\infty $ if $\lambda >e^{-\gamma }$ ($m>0$) and the basic results of Lamperti in [@L1], [@L2] are: $$\text{If }\lim \inf_{i}i\mathbf{P}\left( \nu >i\right) <c:=e^{-\gamma }\text{% , then }X_{n}\overset{a.s.}{\rightarrow }X_{\infty }\text{ (ergodicity),} \label{L1}$$ where $X_{\infty }$ is a non-degenerate rv and ergodicity means positive recurrence and aperiodicity. $$\text{If }\lim \sup_{i}i\mathbf{P}\left( \nu >i\right) >c:=e^{-\gamma }\text{% , then }X_{n}\rightarrow \infty \text{ wp }1\text{ (transience).} \label{L2}$$ In particular, if $\nu $ has tails heavier than $1/i$ ($i\mathbf{P}\left( \nu >i\right) \rightarrow \infty $), then $X_{n}\rightarrow \infty $ wp $1,$ (transience). $$\begin{array}{l} \text{Critical case, \cite{L2}: } \\ \text{If }\mathbf{P}\left( \nu >i\right) \sim e^{-\gamma }/i+d/\left( i\log i\right) \text{, the process }\left\{ X_{n}\right\} \text{ is:} \\ \text{- positive recurrent if }d<-e^{-\gamma }\pi ^{2}/12 \\ \text{- null recurrent if }d\in \left[ -e^{-\gamma }\pi ^{2}/12,e^{-\gamma }\pi ^{2}/12\right) \\ \text{- transient if }d>e^{-\gamma }\pi ^{2}/12. \end{array} \label{L3}$$ The case $d=e^{-\gamma }\pi ^{2}/12$ is left open and would require additional information on the tails of $\nu $ to decide whether here $% \left\{ X_{n}\right\} $ is transient or null recurrent$.$ Whenever the process $\left\{ X_{n}\right\} $ is ergodic, with $\Phi _{\infty }\left( z\right) :=\mathbf{E}z^{X_{\infty }}$, the functional equation $$F_{\infty }\left( j\right) =\mathbf{P}\left( X_{\infty }\leq j\right) =\Phi _{\infty }\left( \mathbf{P}\left( \nu \leq j\right) \right) \text{, }j\geq 1 \label{FE0}$$ admits a unique solution for the pair $\left( \mathbf{P}\left( X_{\infty }\leq j\right) ,\mathbf{P}\left( \nu \leq j\right) \right) $. Because $\Phi _{\infty }\left( z\right) $ is a pgf with $\Phi _{\infty }\left( 0\right) =0$, we have $\Phi _{\infty }\left( z\right) <z$ and so $X_{\infty }$ is stochastically larger than $\nu $: $$\text{For all }j\geq 1\text{: }\mathbf{P}\left( X_{\infty }\leq j\right) <% \mathbf{P}\left( \nu \leq j\right) . \label{SD}$$ Clearly, the maximal branching process asymptotically selects a family size $% X_{\infty }$ which is larger than the typical family size $\nu $ of the underlying Galton-Watson process. It is then of utmost interest to solve the functional equation (\[FE0\]). As we shall see, the position we will adopt is the following: suppose one has some initial guess of the limiting rv $% X_{\infty }$, we will identify the branching number $\nu $ of the Lamperti mbp realizing this task. An additional problem of interest: how long does it take for $\left\{ X_{n}\right\} $ to reach $X_{\infty }?$ To have an insight on this question, we shall ask how long it takes, for a suitably truncated version $% X_{n}^{\left( N\right) }$ of $X_{n}$, to reach height $N\gg 1$, which is intuitively more demanding than reaching the invariant measure of the truncated chain itself. We shall address these points. - **Time spent in the worst state.** Whenever the process $\left\{ X_{n}\right\} $ is ergodic, it visits infinitely often all the states, in particular the state $\left\{ 1\right\} $, and a sample path of it is made of iid successive non-negative excursions through that state$.$ State $% \left\{ 1\right\} $ is the worst case of the selection mechanism that the Lamperti chain realizes. By the ergodic theorem, the fraction of time spent by $\left\{ X_{n}\right\} $ in this state is $\pi \left( 1\right) =\mathbf{P}% \left( X_{\infty }=1\right) .$ The expected first return time ($\tau _{1,1}$) to state $\left\{ 1\right\} $ is $\mathbf{E}\left( \tau _{1,1}\right) =1/\pi \left( 1\right) $. Suppose $\left\{ X_{n}\right\} $ enters state $\left\{ 1\right\} $ from above at some time $n_{1}.$ The first return time $\tau _{1,1}:=\inf \left( n>n_{1}:X_{n}=1\mid X_{n_{1}}=1\right) $ to state $\left\{ 1\right\} $ is: - either $1$ if $X_{n_{1}}$ stays there with probability $P\left( 1,1\right) =F\left( 1\right) $ in the next step; this corresponds to a trivial excursion of length $1$ and height $0$. - or, with probability $1-F\left( 1\right) $, $\left\{ X_{n}\right\} $ starts a true excursion with positive height and length $\tau _{1,1}^{+}\geq 2.$ Thus $$\mathbf{E}\left( \tau _{1,1}\right) =\frac{1}{\pi \left( 1\right) }=F\left( 1\right) +\left( 1-F\left( 1\right) \right) \mathbf{E}\left( \tau _{1,1}^{+}\right) \text{ and}$$ $$\mathbf{E}\left( \tau _{1,1}^{+}\right) =\frac{1}{1-F\left( 1\right) }\left( \frac{1}{\pi \left( 1\right) }-F\left( 1\right) \right) >2,$$ entailing the relationship: $\frac{1}{\pi \left( 1\right) }>2-p\left( 1\right) $. Given $\left\{ X_{n}\right\} $ enters state $\left\{ 1\right\} $ from above at some time $n_{1}$, it stays there with probability $P\left( 1,1\right) =F\left( 1\right) $ in the next step, so $\left\{ X_{n}\right\} $ will quit state $\left\{ 1\right\} $ at time $n_{1}+G$ where $G$ is a shifted geometric random time with success probability $1-F\left( 1\right) .$ After $n_{1}+G$, the chain moves up before returning to state $\left\{ 1\right\} $ again and the time it takes is $\tau _{1,1}^{+}$. Considering two consecutive instants where $\left\{ X_{n}\right\} $ enters state $% \left\{ 1\right\} $ from above (defining an alternating renewal process), the fraction of time spent in state $\left\{ 1\right\} $ is: $$\rho =\frac{\mathbf{E}\left( G\right) }{\mathbf{E}\left( G\right) +\mathbf{E}% \left( \tau _{1,1}^{+}\right) }.$$ From the expression $\mathbf{E}\left( G\right) =F\left( 1\right) /\left( 1-F\left( 1\right) \right) $ and the value of $\mathbf{E}\left( \tau _{1,1}^{+}\right) $, we get: $$\rho =F\left( 1\right) \pi \left( 1\right) .\text{ }$$ - **Time reversal:** Suppose $\left\{ X_{n}\right\} $ is ergodic. Let $\pi \left( j\right) =% \mathbf{P}\left( X_{\infty }=j\right) $, $j\geq 1$. With $\mathbf{\pi }% ^{\prime }=\left( \pi \left( 1\right) ,\pi \left( 2\right) ,...\right) $ the transpose of the column-vector $\mathbf{\pi }$, $P^{\prime }$ the transpose of $P$ and $\pi \left( i\right) =\mathbf{P}\left( X_{\infty }=i\right) $ the stochastic matrix $$\overleftarrow{P}=D_{\mathbf{\pi }}^{-1}P^{\prime }D_{\mathbf{\pi }}$$ is the transition matrix of the time-reversed chain $\left\{ X_{n}^{\leftarrow }\right\} $. Since $\overleftarrow{P}\neq P$, there is no detailed balance. The process $\left\{ X_{n}^{\leftarrow }\right\} $ is such that its time-reversal $\left( X_{n}^{\leftarrow }\right) ^{\leftarrow }=X_{n}$ is stochastically monotone. The backward process $\left\{ X_{n}^{\leftarrow }\right\} $ can be generated as follows, with a time-reversal flavor: with $\left( J_{n};n\geq 1\right) $ an iid sequence with $J_{1}\overset{d}{\sim }\mathbf{\pi }$, independent of the $\nu $’s, consider the Markovian dynamics $$Y_{n+1}=J_{n+1}\cdot \mathbf{1}\left( \max_{k=1,...,J_{n+1}}\nu _{k,n+1}=Y_{n}^{{}}\right) , \label{TR}$$ giving $Y_{n+1}$ as a $\mathbf{\pi }-$mixture of the number of ancestors whose most productive individuals produce exactly $Y_{n}^{{}}$ descendants in a Galton-Watson process with branching number $\nu $. We have $$\begin{aligned} \mathbf{P}\left( Y_{n+1}=j\mid Y_{n}^{{}}=i\right) &=&\pi \left( j\right) \mathbf{P}\left( \max_{k=1,...,j}\nu _{k,n+1}=i\right) \\ &=&\pi \left( j\right) \left[ F\left( i\right) ^{j}-F\left( i-1\right) ^{j}\right] =\pi \left( j\right) \mathbf{P}\left( X_{n+1}=i\mid X_{n}=j\right) ,\end{aligned}$$ equivalently $$Q=P^{\prime }D_{\mathbf{\pi }}$$ where $Q\left( i,j\right) =\mathbf{P}\left( Y_{n+1}=j\mid Y_{n}^{{}}=i\right) .$ The process $Y_{n}^{{}}$ is substochastic (there is a positive probability that given $Y_{n}^{{}}$ no such index $Y_{n+1}$ exists) and a coffin state can be added to the state-space $\Bbb{N}$ where the system is sent to if $Y_{n+1}$ does not exist. Let $\tau _{i}$ be the first hitting time of the coffin state for $Y_{n}^{{}}$ started at $i$ with $% \mathbf{P}\left( \tau _{i}=1\right) =1-\pi \left( i\right) ,$ the mass defect in state $i$ of $Q$. Then $X_{n}^{\leftarrow }=Y_{n}^{{}}\mid \tau _{Y_{n}^{{}}}>1$ (upon conditioning $Y_{n}^{{}}$ stepwise on the event that the hitting time of the coffin state exceeds one time unit). The process $% \left\{ X_{n}^{\leftarrow }\right\} $ thus constructed has the transition matrix $\overleftarrow{P}$, as required. Branching number** **$\nu \geq 0$**.** -------------------------------------- If $p\left( 0\right) =\mathbf{P}\left( \nu =0\right) >0$ $:$ the above functional equation must be considered for $j\geq 0.$ We have $F_{n+1}\left( 0\right) =\sum_{i}\mathbf{P}\left( X_{n}=i\right) \mathbf{P}\left( \nu =0\right) ^{i}=\mathbf{E}z^{X_{n}}\mid _{z=p\left( 0\right) }>0.$ At each $n$, there is a positive probability that $X_{n}=0.$ If for some $n$, $X_{n}=0$, clearly $X_{n^{\prime }}=0$ for all $n^{\prime }>n:$ state $0$ is absorbing. $\left\{ X_{n}\right\} $ is again a Markov chain now on $\mathbf{N}_{0}$ with transition probability matrix $$P\left( i,j\right) =F\left( j\right) ^{i}-F\left( j-1\right) ^{i}\text{, }% i,j\geq 0 \label{P1}$$ in particular with $P\left( i,0\right) =F\left( 0\right) ^{i}>0$. Two cases arise: $\left( a\right) $ If $\mathbf{E}\left( \nu \right) \leq 1,$ there is almost sure (a.s.) extinction of the underlying branching process, say at $\tau _{% \mathbf{\pi }_{0},0},$ and also therefore of $\left\{ X_{n}\right\} $ at $% \tau _{\mathbf{\pi }_{0},0}^{X}\leq \tau _{\mathbf{\pi }_{0},0}$. We have $% \mathbf{P}\left( X_{n}=0\right) =\mathbf{P}\left( \tau _{0}^{X}\leq n\right) \rightarrow 1$ or $\mathbf{P}\left( X_{n}=0\right) =1$, $\forall n\geq \tau _{0}^{X}$ ($\tau _{\mathbf{\pi }_{0},0}^{X}$ is the absorption time of $% \left\{ X_{n}\right\} $ at $0$). In this case, $\Phi _{\infty }\left( z\right) =1$ for all $z\in \left[ 0,1\right] $ and one possible solution to the functional equation is $F_{\infty }\left( j\right) =1$, $j\geq 0.$ The only problem here is to fix the law of $\tau _{\mathbf{\pi }_{0},0}^{X}$ which (with $\mathbf{e}_{0}^{\prime }=\left( 1,0,0,...\right) $ with $1$ in position $0$), is: $$\mathbf{P}\left( X_{n}=0\right) =\mathbf{P}\left( \tau _{0}^{X}\leq n\right) =\mathbf{\pi }_{0}^{\prime }P^{n}\mathbf{e}_{0}\text{.}$$ $\left( b\right) $ If $\mathbf{E}\left( \nu \right) >1,$ there is extinction of the underlying branching process with probability $0<\rho _{e}<1$ ($\rho _{e}$ the smallest solution in $\left[ 0,1\right] $ to $\phi \left( z\right) =z$) entailing: - a.s. extinction: given the underlying branching process certainly goes extinct (an event with probability $\rho _{e}$), the branching process is generated by the branching number $\nu _{e}$ with $\mathbf{E}\left( z^{\nu _{e}}\right) =\phi \left( z\rho _{e}\right) /\rho _{e}$ and $\mathbf{E}% \left( \nu _{e}\right) \leq 1$, entailing: $X_{n}^{e}\rightarrow 0$ with probability (wp) $1$. The question is how fast and we are back to the case $% \left( a\right) $. - a.s. explosion: given the underlying branching process certainly explodes (an event wp $1-\rho _{e}$), the branching process is generated by $\nu _{% \overline{e}}$ characterized by $\mathbf{E}\left( z^{\nu _{\overline{e}% }}\right) =\left( \phi \left( \rho _{e}+z\left( 1-\rho _{e}\right) \right) -\rho _{e}\right) /\left( 1-\rho _{e}\right) $ with $\mathbf{P}\left( \nu _{% \overline{e}}=0\right) =0$ and $\mathbf{E}\left( \nu _{\overline{e}}\right) =% \mathbf{E}\left( \nu \right) >1$. We are back to the discussion of Subsection $2.1$ with $\left\{ X_{n}^{\overline{e}}\right\} $ either going to $\infty $ or to a limiting rv depending on the tails of $\nu _{\overline{e% }}$. The only two cases that really matter are thus the case developed in Subsection $2.1$ and case $\left( a\right) $ with state $0$ absorbing wp $1$, which was dealt with. We will therefore only consider the remaining first case when $\left\{ X_{n}\right\} $ has state-space $\Bbb{N}.$ Large $i$ estimates of $m_{i}\mathbf{=}\max_{j=1,...,i}\nu _{j}$ ================================================================ We will use ideas stemming from limit laws for maxima of a large sample of iid rvs in the continuum to give large $i$ estimates of $m_{i}\mathbf{=}% \max_{j=1,...,i}\nu _{j},$ [@EKM]. Let $X>0$ be some real-valued rv with density and no atom at $0$. Suppose $X$ has a finite mean $\mathbf{E}\left( X\right) $. Let $\overline{F}_{X}\left( x\right) =\mathbf{P}\left( X>x\right) ,$ $x>0,$ be its complementary probability distribution function (pdf). Define the law of some integral-valued rv $\nu \in \Bbb{N}$ by: $$\mathbf{P}\left( \nu >j\right) =\mathbf{P}\left( X>j\right) ,\text{ }% j=0,1,... \label{1}$$ Let $\overline{F}\left( j\right) =\mathbf{P}\left( \nu >j\right) $, $% j=0,1,...$. With $\mathbf{E}\left( X\right) =\int_{0}^{\infty }\mathbf{P}% \left( X>x\right) dx$ and $\mathbf{E}\left( \nu \right) =\sum_{j\geq 0}% \overline{F}\left( j\right) $ we have $\mathbf{E}\left( \nu \right) -1<% \mathbf{E}\left( X\right) <\mathbf{E}\left( \nu \right) $. This suggests that if $\mathbf{E}\left( X\right) $ is large, $\mathbf{E}\left( \nu \right) $ is very close to $\mathbf{E}\left( X\right) .$ Maxima of a large sample of iid rvs in the continuum ---------------------------------------------------- Let $M_{i}=\max \left( X_{1},...,X_{i}\right) $ with $\left( X_{i}\right) _{i\geq 1}$ iid with $X_{1}\overset{d}{=}X.$ Two cases arise: $\left( i\right) $ Von Mises case: With $a\left( x\right) >0,$ absolutely continuous (with respect to Lebesgue measure) with density $a^{\prime }\left( x\right) $ having $\lim a^{\prime }\left( x\right) =0$ as $% x\rightarrow \infty $, consider $$\mathbf{P}\left( X>x\right) =c\exp \left[ -\int^{x}\frac{dz}{a\left( z\right) }\right] \text{, }c>0.$$ Then $a\left( x\right) =\mathbf{E}\left( X-x\mid X>x\right) $ is the mean excess function with $a\left( x\right) /x\rightarrow 0$ as $x\rightarrow \infty .$ Define $d_{i}$ by $\overline{F}_{X}\left( c_{i}\right) =1/i$ and $d_{i}$ by $% c_{i}=a\left( c_{i}\right) .$ We have $$d_{i}^{-1}\left( M_{i}-c_{i}\right) \overset{d}{\rightarrow }G\text{ as }% i\rightarrow \infty ,$$ where $G$ has a Gumbel distribution $\mathbf{P}\left( G\leq x\right) =e^{-e^{-x}}$, $x$ real. The sequence $c_{i}$ is increasing with $i$ with $% c_{i}/i\rightarrow 0$ so at sublinear rate. With $\gamma $ the Euler constant, it then holds that $$d_{i}^{-1}\left( \mathbf{E}\left( M_{i}\right) -c_{i}\right) \rightarrow \mathbf{E}\left( G\right) =\gamma \text{ as }i\rightarrow \infty ,$$ so when $i$ gets large $\mathbf{E}\left( M_{i}\right) \sim c_{i}.$ $\left( ii\right) $ With $\alpha >0,$ suppose $$\mathbf{P}\left( X>x\right) =x^{-\alpha }L\left( x\right) ,$$ where $L\left( x\right) $ is some slowly varying function at $\infty $, with $L\left( tx\right) /L\left( x\right) \rightarrow 1$ as $x\rightarrow \infty $ for all $t>0$. Defining $c_{i}$ by $\overline{F}_{X}\left( c_{i}\right) =1/i$ we have $$c_{i}^{-1}M_{i}\overset{d}{\rightarrow }F\text{ as }i\rightarrow \infty ,$$ where $F$ has a Fréchet distribution $\mathbf{P}\left( F\leq x\right) =e^{-x^{-\alpha }}$, $x>0$ with $\mathbf{E}\left( F\right) =\Gamma \left( 1-1/\alpha \right) $ if $\alpha >1$, $=\infty $ if $\alpha \in \left( 0,1\right] $. If $\alpha >1,$ with $c_{i}=i^{1/\alpha }L_{1}\left( i\right) $ for some other slowly varying function $L_{1}$, it holds that $$c_{i}^{-1}\mathbf{E}\left( M_{i}\right) \rightarrow \mathbf{E}\left( F\right) =\Gamma \left( 1-1/\alpha \right) \text{ as }i\rightarrow \infty ,$$ so when $i$ gets large $\mathbf{E}\left( M_{i}\right) \sim \Gamma \left( 1-1/\alpha \right) c_{i}.$ And the sequence $c_{i}$ is increasing also at sublinear rate. Maxima of a large sample of discrete iid rvs: large $i$ estimation of $m_{i}$ ----------------------------------------------------------------------------- Let $m_{i}=\max \left( \nu _{1},...,\nu _{i}\right) $ with $\left( \nu _{i}\right) _{i\geq 1}$ iid with $\nu _{1}\overset{d}{=}\nu $ and $\nu $’s law given from $X$’s law as before$.$ Let $c_{i}$ be defined by $\overline{F}\left( c_{i}\right) =1/i$. In general, it is not true that, upon scaling $m_{i}$, there is a proper weak limit for this scaled rv, because in general, in the discrete setting $% \overline{F}\left( j\right) /\overline{F}\left( j-1\right) \nrightarrow 1$ ($% \mathbf{P}\left( \nu =j\right) /\mathbf{P}\left( \nu >j-1\right) \nrightarrow 0$) as $j\rightarrow \infty $. All that can be said is that $% m_{i}-c_{i}$ is tight (or bounded in probability), with $m_{i}/c_{i}% \rightarrow 1$ in probability as $i\rightarrow \infty $. Also, if we are interested in $$\mathbf{E}\left( m_{i}\right) =\sum_{j\geq 0}\left( 1-F\left( j\right) ^{i}\right) ,$$ using the latter argument, for large $i$, $\mathbf{E}\left( m_{i}\right) $ and $\mathbf{E}\left( M_{i}\right) $ are of the same order of magnitude. Ergodic case from (\[L1\])$:$ With $c_{i}$ defined by $\overline{F}% _{X}\left( c_{i}\right) =1/i$ in $\left( i\right) $ the Von Mises case or $% \left( ii\right) $ when $\mathbf{P}\left( X>x\right) =x^{-\alpha }L\left( x\right) $ and $\alpha >1$ in the domain of attraction of the Fréchet$% \left( \alpha \right) $ law $$\begin{aligned} \left( i\right) \text{ }\mathbf{E}\left( m_{i}\right) &\sim &c_{i} \\ \left( ii\right) \text{ }\mathbf{E}\left( m_{i}\right) &\sim &\Gamma \left( 1-1/\alpha \right) c_{i}=\Gamma \left( 1-1/\alpha \right) i^{1/\alpha }L_{1}\left( i\right) .\end{aligned}$$ In all these cases, $\mathbf{E}\left( m_{i}\right) $ grows at sublinear rate as $i$ gets large. Under the above assumptions on the law of $\nu $, there is thus an integer $% I $ such that $$\begin{aligned} \mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) &\leq &i-1\text{ for all }i\geq I \\ \mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) &<&\infty \text{ for all }i\text{ for which }\mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) >i-1,\end{aligned}$$ which by Foster theorem implies that $\left\{ X_{n}\right\} $ is ergodic, [@Fost]. The limit law of the MC is the unique integrable solution to the corresponding functional equation for $X_{\infty }.$ Transient case from (\[L2\]): If $\nu $ is in the domain of attraction of the Fréchet law with $\alpha \in \left( 0,1\right) $, $\mathbf{E}\left( m_{i}\right) =\mathbf{E}\left( X_{n+1}\mid X_{n}=i\right) $ grows at a superlinear rate which leads to a transience case ($X_{n}\rightarrow \infty $ wp $1$, as $n\rightarrow \infty $). Such $\nu $s have infinite mean. If $% \alpha =1$, the process is transient (positive recurrent) if $\lim \sup_{i}i% \mathbf{P}\left( \nu >i\right) >e^{-\gamma }$ (respectively $<e^{-\gamma }$). Whenever the tails of $\nu $ satisfy any one of these conditions, $% \mathbf{E}\left( \nu \right) =\infty .$ This shows that transience of $% \left\{ X_{n}\right\} $ does not necessarily mean $\mathbf{E}\left( \nu \right) =\infty $. *Example:* If $\alpha =1$, there are positive recurrent examples for which $\mathbf{E}\left( \nu \right) =\infty $, for instance those obtained from $$\mathbf{P}\left( \nu >i\right) =\frac{1}{i\log ^{\beta }\left( 1+i\right) }% \text{ with }0<\beta <1\text{, }$$ with $L\left( x\right) =\log ^{\beta }\left( 1+x\right) $ slowly varying at $% \infty $. $\diamondsuit $ General approach to find solutions of the functional equation ============================================================= In the ergodic case from (\[L1\]), the invariant probability measure $\pi \left( j\right) :=\mathbf{P}\left( X_{\infty }=j\right) $ solves $$\mathbf{\pi }^{\prime }=\mathbf{\pi }^{\prime }P.$$ However, here, the pdfs of $\nu >0$ and $X_{\infty }>0$ are related by the functional equation: $$F_{\infty }\left( j\right) =\Phi _{\infty }\left( F\left( j\right) \right) , \label{FE}$$ and we shall give many examples of explicit pairs $\left( F_{\infty }\left( j\right) ,F\left( j\right) \right) $ solving it. As indicated above, it participates to the general program of finding the branching number $\nu $ of the Lamperti mbp realizing an initial target guess of the limiting rv $% X_{\infty }$. The rv $X_{\infty }$ is taken from the classical (shifted) set of probability mass functions (pmfs) supported by the integers. We will then compute explicitly the law of $\nu $ corresponding to classical target pmfs such as geometric, Sibuya, Poisson. The obtained distributions are far from classical and somewhat surprising. *Remark:* With the idea of spanning trees in the background, there exists a determinantal Kirchoff formula stating that, [@Pit]: $$\pi \left( j\right) =\det \left[ \left( I-P\right) ^{\left( j,j\right) }\right] ,$$ where $\left( I-P\right) ^{\left( j,j\right) }$ is the Laplacian matrix $I-P$ to which row $j$ and column $j$ have been removed. In view of the expression (\[P1\] with $i,j\geq 1$) of the Lamperti matrix $P$, the Kirchoff formula shows that the computation of $\mathbf{\pi }$ from $P$ (and so from $F$) is not, in principle, a simple matter. Our approach being to find $F$ (and so $% P $), starting from the knowledge of $\mathbf{\pi },$ this leads in return to non trivial determinantal identities. $\diamondsuit $ - **Lagrange inversion formula:** In the sequel, we shall denote by: $\left( n\right) _{k}=n\left( n-1\right) ...\left( n-k+1\right) $ and $% \left[ n\right] _{k}=n\left( n+1\right) ...\left( n+k-1\right) $ the falling and rising factorials (of order $k$) of $n$. Take $\Phi _{\infty }\left( z\right) =z\Psi _{\infty }\left( z\right) $ for some new (given) pgf $\Psi _{\infty }$ obeying $\Psi _{\infty }\left( 0\right) \neq 0.$ The pgf $\Psi _{\infty }$ is the one of $X_{\infty }-1$. Apply Lagrange inversion formula to solve $z\Psi _{\infty }\left( z\right) =u $. It gives the inverse $\Phi _{\infty }^{-1}\left( z\right) $ of $\Phi _{\infty }\left( z\right) $ as a power series in $z,$ with $$\varphi _{n}:=\left[ z^{n}\right] \Phi _{\infty }^{-1}\left( z\right) =\frac{% 1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n}.$$ Then $$F\left( j\right) =\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\sum_{n\geq 1}\varphi _{n}F_{\infty }\left( j\right) ^{n}$$ gives the $F\left( j\right) $ consistent with the original choice of $% F_{\infty }\left( j\right) $. With $B_{n,k}\left( x_{1},x_{2},...\right) $ (respectively $\widehat{B}_{n,k}\left( x_{1},x_{2},...\right) $) the exponential (respectively ordinary) Bell polynomials in the indeterminates $% \left( x_{1},x_{2},...\right) $, obeying $B_{n,k}\left( x_{1},x_{2},...\right) =0$ if $k>n$ and $B_{n,0}\left( x_{1},x_{2},...\right) =\delta _{n,0},$ we have in principle ([@Com], p. $161$) ($x_{k}=k!\pi \left( k+1\right) $) $$\begin{aligned} \varphi _{n} &=&\frac{1}{n!}\sum_{k=0}^{n-1}\left( -n\right) _{k}\pi \left( 1\right) ^{-\left( n+k\right) }B_{n-1,k}\left( \frac{2!\pi \left( 2\right) }{% 2},\frac{3!\pi \left( 3\right) }{3},...\right) \\ &=&\frac{1}{n}\sum_{k=0}^{n-1}\frac{\left( -n\right) _{k}}{k!}\pi \left( 1\right) ^{-\left( n+k\right) }\widehat{B}_{n-1,k}\left( \pi \left( 2\right) ,\pi \left( 3\right) ,...\right) \\ &=&\frac{\pi \left( 1\right) ^{-n}}{n}\sum_{k=0}^{n-1}\frac{\left( -n\right) _{k}}{k!}\widehat{B}_{n-1,k}\left( \frac{\pi \left( 2\right) }{\pi \left( 1\right) },\frac{\pi \left( 3\right) }{\pi \left( 1\right) },...\right)\end{aligned}$$ $$\begin{aligned} &=&\frac{1}{n!}\sum_{k=0}^{n-1}\left( -1\right) ^{k}\pi \left( 1\right) ^{-\left( n+k\right) }B_{n+k-1,k}\left( 0,2!\pi \left( 2\right) ,3!\pi \left( 3\right) ,...\right) \\ &=&\frac{\pi \left( 1\right) ^{-n}}{n}\sum_{k=0}^{n-1}\left( -1\right) ^{k}% \binom{n+k-1}{k}\widehat{B}_{n+k-1,k}\left( 0,\frac{\pi \left( 2\right) }{% \pi \left( 1\right) },\frac{\pi \left( 3\right) }{\pi \left( 1\right) }% ,...\right) .\end{aligned}$$ Owing to $\left( -n\right) _{k}=\left( -1\right) ^{k}\left[ n\right] _{k}$ and (see [@Com], p. $145$) $$\begin{aligned} B_{n-1,k}\left( x_{1},x_{2},...\right) &=&\left( n-1\right) !\sum^{*}\prod_{m\geq 1}\frac{1}{k_{m}!}\left( \frac{x_{m}}{m!}\right) ^{k_{m}}, \\ \widehat{B}_{n-1,k}\left( x_{1},x_{2},...\right) &=&k!\sum^{*}\prod_{m\geq 1}% \frac{x_{m}^{k_{m}}}{k_{m}!}\end{aligned}$$ where the star sum runs over $k_{m}\geq 0$, obeying $\sum_{m\geq 1}k_{m}=k$ and $\sum_{m\geq 1}mk_{m}=n-1$, we have equivalently $\varphi _{1}=1/\pi \left( 1\right) $ and if $n\geq 2$$$\varphi _{n}=\frac{\pi \left( 1\right) ^{-n}}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\left[ n\right] _{k}C_{n-1,k},$$ where, with $C_{n-1,0}=\delta _{n-1,0},$$$C_{n-1,k}=\sum^{*}\prod_{m\geq 1}\frac{\left( \pi \left( m+1\right) /\pi \left( 1\right) \right) ^{k_{m}}}{k_{m}!}. \label{C}$$ To summarize, we obtained the pdf $F$ of $\nu $ corresponding to $\pi \left( j\right) :=\mathbf{P}\left( X_{\infty }=j\right) ,$ solving (\[FE\]), as The mapping $X_{\infty }\rightarrow \nu $ is one-to-one. With $C_{n-1,k}$ given by (\[C\]) and $h_{n}=\frac{1}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\left[ n\right] _{k}C_{n-1,k}$ ($h_{1}=1$), $$F\left( j\right) =\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\sum_{n\geq 1}h_{n}\cdot \left( F_{\infty }\left( j\right) /\pi \left( 1\right) \right) ^{n} \label{FES}$$ is the cumulated mass function of $\nu $ corresponding to any given $\mathbf{% \pi }$. The obtained expression (\[FES\]) only depends on the ratio $F_{\infty }\left( j\right) /F_{\infty }\left( 1\right) $. Note that $\Phi _{\infty }^{-1}\left( z\right) $ is increasing from $z=0$ to $z=1$ and concave. From the fact that it is increasing, we conclude that if $F_{\infty }\left( j\right) $ is a pdf, then so is $F\left( j\right) $. From the concavity, we conclude $F\left( j\right) \geq F_{\infty }\left( j\right) $ for all $j$ (as already mentioned, $X_{\infty }$ is stochastically larger than $\nu $). While proceeding in this way, we observe that, given we first fix the law of $X_{\infty }$, the one of the corresponding $\nu $ follows. Suppose we were able to find a suitable pair of pdfs $\left( F\left( j\right) ,F_{\infty }\left( j\right) \right) $ by the Lagrange inversion formula. Then, with $F_{0}\left( j\right) =\mathbf{1}\left( j\leq 1\right) $ ($X_{0}\overset{d}{\sim }\delta _{1}$) and $\Phi _{0}\left( z\right) =z$, $% F_{1}\left( j\right) =\Phi _{0}\left( F\left( j\right) \right) =F\left( j\right) $ is a pdf, the one of $\nu $. Let $\Phi _{1}\left( z\right) =\sum_{j\geq 1}z^{j}\left( F_{1}\left( j\right) -F_{1}\left( j-1\right) \right) $ be the pgf of $X_{1}\overset{d}{=}\nu $. Next, $F_{2}\left( j\right) =\Phi _{1}\left( F\left( j\right) \right) $ is a pdf because $\Phi _{1}$ is monotone increasing obeying $\Phi _{1}\left( 0\right) =0$, $\Phi _{1}\left( 1\right) =1$. By recurrence $F_{n+1}\left( j\right) =\Phi _{n}\left( F\left( j\right) \right) $ is the pdf of some rv $X_{n+1}$ obtained from the one of $X_{n}$ and $F_{n}\left( j\right) \rightarrow F_{\infty }\left( j\right) $ solution to $F_{\infty }\left( j\right) =\Phi _{\infty }\left( F\left( j\right) \right) $. We shall deal with special cases of $X_{\infty }.$ - **Infinite divisibility:** suppose $\Psi _{\infty }\left( z\right) $ is the pgf of an infinitely divisible (ID) rv (meaning $X_{\infty }-1$ is ID). Then, as a compound Poisson rv, $$\Psi _{\infty }\left( z\right) =e^{-\lambda \left( 1-h\left( z\right) \right) },$$ for some rate $\lambda >0$ and pgf $h\left( z\right) $ obeying $h\left( 0\right) =0.$ If $\mathbf{P}\left( X_{\infty }=j+1\right) =\left[ z^{j}\right] \Psi _{\infty }\left( z\right) =\pi _{\lambda }\left( j+1\right) $ is a known simple function of $\lambda $, then $\left[ z^{j}\right] \Psi _{\infty }\left( z\right) ^{-n}$ is readily obtained as $% \pi _{-n\lambda }\left( j+1\right) $, a useful identity to get the $h_{n}$ in (\[FES\]) and so $F$ from $F_{\infty }$. - **Complete monotonicity:** Suppose $\overline{F}_{\infty }\left( j\right) $ defines a in $\left[ 0,1\right] $-valued completely monotone sequence of complementary pdfs, meaning $$\begin{aligned} \left( -1\right) ^{k}\Delta ^{\left( k\right) }\overline{F}_{\infty }\left( j\right) &\geq &0\text{ for all }j,k\geq 0,\text{ equivalently} \\ \left( -1\right) ^{k}\Delta ^{\left( k\right) }\mathbf{P}\left( X_{\infty }=j\right) &\geq &0\text{ for all }j\geq 1,k\geq 0,\end{aligned}$$ where $\Delta :$ $\Delta h\left( j\right) =h\left( j+1\right) -h\left( j\right) $ is the right-shift operator and $\Delta ^{\left( k\right) }$ its $% k-$th iterate. Note $\mathbf{P}\left( X_{\infty }=j\right) =\Delta F_{\infty }\left( j-1\right) =-\Delta \overline{F}_{\infty }\left( j-1\right) $. By Hausdorff representation theorem, $\overline{F}_{\infty }\left( j\right) $ is completely monotone (CM) if and only if $$\overline{F}_{\infty }\left( j\right) =\int_{0}^{1}u^{j}\lambda \left( du\right) ,$$ for some probability measure $\lambda \left( du\right) $ on $\left[ 0,1\right] .$ Equivalently, with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}z^{j}% \mathbf{P}\left( X_{\infty }=j\right) $ the pgf of $X_{\infty }$, $$\sum_{j\geq 0}z^{j}\overline{F}_{\infty }\left( j\right) =\frac{1-\Phi _{\infty }\left( z\right) }{1-z}=\int_{0}^{1}\frac{1}{1-zu}\lambda \left( du\right) ,$$ as a Stieltjes transform of $\lambda \left( du\right) .$ Note that, with $U% \overset{d}{\sim }\lambda \left( du\right) $$$\Phi _{\infty }\left( z\right) =z\int_{0}^{1}\frac{1-u}{1-zu}\lambda \left( du\right) =\mathbf{E}\left( \frac{z\left( 1-U\right) }{1-zU}\right) ,$$ showing that $X_{\infty }-1$, with pgf $\Psi _{\infty }\left( z\right) =z^{-1}\Phi _{\infty }\left( z\right) =\mathbf{E}\left( \frac{1-U}{1-zU}% \right) $, is a $\lambda -$mixture of a shifted geometric rv, so that $% X_{\infty }-1$ is log-convex and infinitely divisible. As noted in [@Gupta], log-convex (log-concave) pmfs are decreasing (increasing) failure rate monotone, say DFR (IFR), meaning $\Delta r_{j}$ decreasing (increasing) where $r_{j}=\pi \left( j\right) /\overline{F}_{\infty }\left( j-1\right) =% \mathbf{P}\left( X_{\infty }=j\right) /\mathbf{P}\left( X_{\infty }\geq j\right) $ is a discrete failure ‘rate’. Explicit examples of $\left( \nu ,X_{\infty }\right) $ with support $\left\{ 1,...,\infty \right\} $. ----------------------------------------------------------------------------------------------------- In some cases, the computation of the pair $\left( F\left( j\right) ,F_{\infty }\left( j\right) \right) $ is obtained as a simple expression. - **Geometric** example: $X_{\infty }\overset{d}{\sim }$geom$\left( p\right) $ Suppose $X_{\infty }\overset{d}{\sim }$geom$\left( p\right) $, so with $% F_{\infty }\left( j\right) =1-q^{j}$. The sequence $\overline{F}_{\infty }\left( j\right) $ is of course CM as a result of $$\overline{F}_{\infty }\left( j\right) =q^{j}=\int_{0}^{1}u^{j}\lambda \left( du\right) ,\text{ with }\lambda \left( du\right) =\delta _{q}\left( du\right) ,$$ so $X_{\infty }-1$ is ID. $\left( i\right) $ The solution to (\[FE\]) is: $$\mathbf{P}\left( \nu \leq j\right) =F\left( j\right) =\frac{1-q^{j}}{% 1-q^{j+1}}\text{, }j=1,2,...$$ $\left( ii\right) $ The sequence $\overline{F}\left( j\right) $ is CM and so $\nu -1$ is ID. The distribution $F\left( j\right) $ has decreasing failure rate (DFR). $\left( iii\right) $ There are two ways to generate the corresponding branching number $\nu :$$$\left( iii-a\right) :\text{ }\nu =\inf \left( i\geq 1:\mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) ,$$ where $\left( \mathcal{B}_{i}\left( \alpha _{i}\right) ;\text{ }i\geq 1\right) $ is an independent sequence of Bernoulli rvs with success parameter $\alpha _{i}=1/\left( 1+q+...+q^{i}\right) .$ Or: $$\left( iii-b\right) :\text{ }\nu =\max_{i=1,...,G}\xi _{i}$$ where $G\overset{d}{\sim }$geom$\left( p\right) $ independent of the iid sequence $\left( \xi _{i}\text{, }i\geq 1\right) $ with $\xi _{1}\overset{d}{% \sim }$geom$\left( p\right) .$ $\left( iv\right) $ The tails of both $\left( \nu ,X_{\infty }\right) $ are geometric with: $\mathbf{P}\left( \nu >j\right) /\mathbf{P}\left( X_{\infty }>j\right) \rightarrow p<1$. *Proof:* $\left( i\right) $ We have $$\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =% \frac{pz}{1-qz}\text{ and so}$$ $$\Phi _{\infty }\left( F\left( j\right) \right) =\frac{p\frac{1-q^{j}}{% 1-q^{j+1}}}{1-q\left( \frac{1-q^{j}}{1-q^{j+1}}\right) }=1-q^{j}=\mathbf{P}% \left( X_{\infty }\leq j\right) =F_{\infty }\left( j\right) .$$ $\left( ii\right) $ With $\lambda \left( du\right) =p\sum_{j\geq 1}q^{j-1}\delta _{q^{j}}$, a probability measure, $$\overline{F}\left( j\right) =\frac{pq^{j}}{1-q^{j+1}}=\int_{0}^{1}u^{j}% \lambda \left( du\right) .$$ The rv $\nu \geq 1$ has finite mean $\mathbf{E}\left( \nu \right) =\sum_{j\geq 0}\mathbf{P}\left( \nu >j\right) =1+p\sum_{j\geq 1}q^{j}/\left( 1-q^{j+1}\right) <\infty $ and $\left\{ X_{n}\right\} $ is recurrent positive. For $j\geq 1$, we have $$\frac{\mathbf{P}\left( \nu =j\right) }{\mathbf{P}\left( \nu >j\right) }=% \frac{p}{q}\frac{1}{1-q^{j}}$$ which is decreasing with $j$. $\left( iii\right) $ The first statement $\left( iii-a\right) $ results from: $\mathbf{P}\left( \nu >i\right) =\prod_{j=1}^{i}\left( 1-\alpha _{j}\right) =\frac{pq^{i}}{1-q^{i+1}}.$ $\left( iii-b\right) $ results from $$\mathbf{P}\left( \max_{i=1,...,G}\xi _{i}>i\right) =\sum_{k\geq 1}pq^{k-1}q^{ik}=\frac{pq^{i}}{1-q^{i+1}}.$$ $\left( iv\right) $ The tails of $\nu $ are given by $\mathbf{P}\left( \nu >j\right) =1-\Phi _{\infty }^{-1}\left( \mathbf{P}\left( X_{\infty }\leq j\right) \right) =1-\frac{z}{p+qz}\mid _{1-p^{j}}\sim pq^{j}$ (for large $j$) with $\mathbf{P}\left( \nu >j\right) /\mathbf{P}\left( X_{\infty }>j\right) \rightarrow p<1$. For this model, $\nu $ and $X_{\infty }$ are geometric (power-law) and tail-equivalent but the tails of $\nu $ are thinner than the ones of $X_{\infty }.$ $\Box $ Related examples to the geometric one ($X_{\infty }$ having dominant geometric tails with an algebraic prefactor): - Suppose $X_{\infty }\overset{d}{\sim }$**negative-binomial ,**conditioned to be positive**:** With** **$\left[ \alpha \right] _{k}=\Gamma \left( \alpha +k\right) /\Gamma \left( \alpha \right) $, $\alpha >0$, suppose $$\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =% \frac{\left( \frac{p}{1-qz}\right) ^{\alpha }-p^{\alpha }}{1-p^{\alpha }}$$ is the pgf of a negative-binomial** **rv**,** conditioned to be positive. Then, by direct inversion $$F\left( j\right) =\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\frac{1-\left( 1+\sum_{k=1}^{j}\frac{\left[ \alpha \right] _{k}}{k!}% q^{k}\right) ^{-1/\alpha }}{q},$$ which defines the pdf of $\nu $. In this case, $\Psi _{\infty }\left( z\right) =z^{-1}\Phi _{\infty }\left( z\right) $ is not the pgf of an ID rv. Plugging $\alpha =1$ gives back the latter geometric case. The tails of $% X_{\infty }$ goes, up to a constant prefactor, like $j^{\alpha -1}q^{j}$. - Suppose $X_{\infty }\overset{d}{\sim }$shifted **negative-bin (**$% \Psi _{\infty }\left( z\right) $ now is the pgf of the ID rv $X_{\infty }-1$): then $$\begin{aligned} \Phi _{\infty }\left( z\right) &=&\mathbf{E}\left( z^{X_{\infty }}\right) =z\left( \frac{p}{1-qz}\right) ^{\alpha }\text{ and }\Psi _{\infty }\left( z\right) =p^{\alpha }\left( 1-qz\right) ^{-\alpha } \\ \left[ z^{j}\right] \Psi _{\infty }\left( z\right) &=&p^{\alpha }\frac{% \left[ \alpha \right] _{j}}{j!}q^{j}\Rightarrow \left[ z^{j}\right] \Psi _{\infty }^{-n}\left( z\right) =\frac{\left[ -n\alpha \right] _{j}}{j!}% p^{-n\alpha }q^{j} \\ \frac{1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n} &=&% \frac{\left[ -n\alpha \right] _{n-1}}{n!}p^{-n\alpha }q^{n-1} \\ F_{\infty }\left( j\right) &=&\sum_{k=1}^{j}\mathbf{P}\left( X_{\infty }=k\right) =p^{\alpha }\sum_{k=1}^{j}\frac{\left[ \alpha \right] _{k-1}}{% \left( k-1\right) !}q^{k-1} \\ F\left( j\right) &=&\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\sum_{n\geq 1}F_{\infty }\left( j\right) ^{n}\frac{1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n}.\end{aligned}$$ The negative-binomial distribution with pgf $\Psi _{\infty }\left( z\right) =p^{\alpha }\left( 1-qz\right) ^{-\alpha }$ is CM (and so log-convex, ID and DFR) only when $\alpha \leq 1$. When $\alpha \geq 1$ it is log-concave, ID and IFR. - $X_{\infty }\overset{d}{\sim }$**Fisher log-series**. With $p\in \left( 0,1\right) $ and $c=-\log \left( 1-p\right) $, suppose $$\begin{aligned} \Phi _{\infty }\left( z\right) &=&\mathbf{E}\left( z^{X_{\infty }}\right) =-c^{-1}\log \left( 1-pz\right) =:z\Psi _{\infty }\left( z\right) \\ \mathbf{P}\left( X_{\infty }=k\right) &=&c^{-1}p^{k}/k\text{, }k\geq 1\text{ and }\mathbf{P}\left( X_{\infty }\leq j\right) =c^{-1}\sum_{k=1}^{j}p^{k}/k% \text{ }\end{aligned}$$ involving a truncated logarithm. We have $$\begin{aligned} \Phi _{\infty }^{-1}\left( z\right) &=&p^{-1}\left( 1-e^{-cz}\right) \text{ and } \\ F\left( j\right) &=&\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =p^{-1}\left( 1-e^{-\sum_{k=1}^{j}p^{k}/k}\right) .\end{aligned}$$ For all $j\geq 1$, we have by construction $$F\left( j\right) >F_{\infty }\left( j\right) =c^{-1}\sum_{k=1}^{j}p^{k}/k.$$ The tails of $X_{\infty }$ goes, up to a constant prefactor, like $% j^{-1}p^{j}$. In addition, $$\begin{aligned} \mathbf{P}\left( X_{\infty }=i\right) &=&\int_{0}^{1}u^{i-1}\mu \left( du\right) \text{ where }\mu \left( du\right) =c^{-1}1_{u\in \left( 0,p\right) }du \\ \mathbf{P}\left( X_{\infty }>i\right) &=&\int_{0}^{1}u^{i}\lambda \left( du\right) \text{ where }\lambda \left( du\right) =c^{-1}\left( 1-u\right) ^{-1}1_{u\in \left( 0,p\right) }du,\end{aligned}$$ and both $X_{\infty }$ and $\nu $ are CM. Let us now look at situations when $X_{\infty }$ has heavy (algebraic) tails with index $\alpha >0$: - The power-law **Sibuya** example, [@Sibu]. With $\alpha \in \left( 0,1\right) $, suppose $X_{\infty }\overset{d}{\sim }$Sibuya$\left( \alpha \right) $, that is: $\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =1-\left( 1-z\right) ^{\alpha },$ with $\mathbf{P}\left( X_{\infty }=j\right) =\pi \left( j\right) =\alpha \left[ 1-\alpha \right] _{j-1}/j!$. Then: $\left( i\right) $ The sequence $\pi \left( j\right) $ is CM, so log-convex, DFR and $X_{\infty }-1$ is ID. $\left( ii\right) $ $$X_{\infty }=\inf \left( i\geq 1:\mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) , \label{ber}$$ where $\left( \mathcal{B}_{i}\left( \alpha _{i}\right) \right) _{i\geq 1}$ is a sequence of independent Bernoulli rvs obeying $\mathbf{P}\left( \mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) =\alpha /i.$ $\left( iii\right) $ The solution to (\[FE\]) is: $$\mathbf{P}\left( \nu \leq i\right) =1-\left( 1-\alpha \sum_{j=1}^{i}\left[ 1-\alpha \right] _{j-1}/j!\right) ^{1/\alpha },$$ $\left( iv\right) $ Both $\left( X_{\infty },\nu \right) $ have algebraic (power-law) tails, but with tail index $\alpha $ and $1$ respectively. $\left( v\right) $ We have $$\mathbf{P}\left( \nu >i\right) \sim \frac{1}{\Gamma \left( 1-\alpha \right) ^{1/\alpha }}i^{-1}\text{ as }j\rightarrow \infty$$ and $1/\Gamma \left( 1-\alpha \right) ^{1/\alpha }<e^{-\gamma }.$ For all $% \alpha \in \left( 0,1\right) $, the Lamperti chain generated by $\nu $ is positive recurrent, with invariant probability measure $\mathbf{\pi }$. *Proof:* $\left( i\right) $ It can be checked that, with $\mu \left( du\right) \overset{d}{\sim }$Beta$\left( 1-\alpha ,\alpha \right) $$$\mathbf{P}\left( X_{\infty }=j\right) =\int_{0}^{1}u^{j}\mu \left( du\right) .$$ $\left( ii\right) $ is obvious and a well-known property of Sibuya$\left( \alpha \right) $ distributed rvs, [@Sibu]. $\left( iii\right) $ We have $\Phi _{\infty }^{-1}\left( z\right) =1-\left( 1-z\right) ^{1/\alpha }$ and so $$\mathbf{P}\left( \nu \leq i\right) =1-\left( 1-\alpha \sum_{j=1}^{i}\left[ 1-\alpha \right] _{j-1}/j!\right) ^{1/\alpha }.$$ $\left( iv\right) $ We have $\mathbf{P}\left( X_{\infty }=j\right) \sim \frac{\alpha }{\Gamma \left( 1-\alpha \right) }j^{-\left( \alpha +1\right) }$ and $\mathbf{P}\left( X_{\infty }>j\right) \sim \frac{1}{\Gamma \left( 1-\alpha \right) }j^{-\alpha }.$ Therefore $\mathbf{P}\left( \nu \leq j\right) \sim \Phi _{\infty }^{-1}\left( \mathbf{P}\left( X_{\infty }\leq j\right) \right) \sim 1-\mathbf{P}\left( X_{\infty }>j\right) ^{1/\alpha }\sim 1-\frac{1}{\Gamma \left( 1-\alpha \right) ^{1/\alpha }}j^{-1}.$ And $% \nu $ has lighter tails (of index $1$) than $X_{\infty }$ (of index $\alpha $)$.$ This is a concrete manifestation in the tails of the fact that $% X_{\infty }$ is stochastically larger than $\nu $. $\left( v\right) $ To decide whether or not $\nu $ belongs to the ergodic family, (\[L1\]), we need to compare $1/\Gamma \left( 1-\alpha \right) ^{1/\alpha }$ with $e^{-\gamma }$, $\gamma =-\Gamma ^{\prime }\left( 1\right) $ being the Euler constant. Indeed, based on Lamperti’s criterion, the chain is recurrent if $1/\Gamma \left( 1-\alpha \right) ^{1/\alpha }<e^{-\gamma }$ or $\log \Gamma \left( 1-\alpha \right) /\alpha >\gamma $ for all $\alpha \in \left( 0,1\right) $. But this is always true because $% \log \Gamma \left( 1-\alpha \right) /\alpha $ is an increasing function of $% \alpha $ with $\log \Gamma \left( 1-\alpha \right) /\alpha \rightarrow \gamma $ as $\alpha \rightarrow 0$ ($\log \Gamma \left( 1-\alpha \right) \sim \log \left( 1-\alpha \Gamma ^{\prime }\left( 1\right) \right) \sim -\alpha \Gamma ^{\prime }\left( 1\right) $)$.$ The critical upper bound $% e^{-\gamma }$ for the coefficient $1/\Gamma \left( 1-\alpha \right) ^{1/\alpha }$ is attained for $\alpha \rightarrow 0.$ $\Box $ Related examples to the Sibuya one with power-law tails are: - **Pareto** ($\alpha >0$): Suppose $\mathbf{P}\left( X_{\infty }>i\right) =\left( i+1\right) ^{-\alpha }$. Clearly, $$X_{\infty }=\inf \left( i\geq 1:\mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) ,$$ where $\left( \mathcal{B}_{i}\left( \alpha _{i}\right) \right) _{i\geq 1}$ is a sequence of independent Bernoulli rvs obeying $\mathbf{P}\left( \mathcal{B}_{i}\left( \alpha _{i}\right) =1\right) =1-\left( 1+1/i\right) ^{-\alpha }$ where $\alpha >0$. Indeed, $$\mathbf{P}\left( X_{\infty }>i\right) =\prod_{j=1}^{i}\left( 1-\alpha _{j}\right) =\prod_{j=1}^{i}\left( 1+1/j\right) ^{-\alpha }=\left( i+1\right) ^{-\alpha }.$$ We have $\mathbf{P}\left( X_{\infty }=i\right) =i^{-\alpha }-\left( i+1\right) ^{-\alpha }=i^{-\alpha }\left( 1-\left( \left( i+1\right) /i\right) ^{-\alpha }\right) \sim \alpha i^{-\left( \alpha +1\right) }$ and so $\Phi _{\infty }\left( z\right) =\sum_{i\geq 1}z^{i}i^{-\alpha }-\sum_{i\geq 1}z^{i}\left( i+1\right) ^{-\alpha }=1-z^{-1}\left( 1-z\right) L_{\alpha }\left( z\right) =z\Psi _{\infty }\left( z\right) .$ When $\alpha \leq 1$, the polylog function $L_{\alpha }\left( z\right) =\sum_{i\geq 1}z^{i}i^{-\alpha }$ is not defined at $z=1$ but $z\Psi _{\infty }\left( z\right) =1-z^{-1}\left( 1-z\right) L_{\alpha }\left( z\right) $ is a true pgf taking the value $1$ at $z=1$. Lagrange inversion formula gives the power-series expansion of $\Phi _{\infty }^{-1}\left( z\right) $ giving $\mathbf{P}\left( \nu \leq j\right) =\Phi _{\infty }^{-1}\left( 1-\left( j+1\right) ^{-\alpha }\right) .$ The rv $X_{\infty }-1$ (with pgf $\Psi _{\infty }\left( z\right) $) is infinitely divisible. Indeed, the polylogarithm can be expressed in terms of the integral of the Bose-Einstein distribution $$L_{\alpha }\left( z\right) =\frac{1}{\Gamma \left( \alpha \right) }% \int_{0}^{\infty }\frac{x^{\alpha -1}}{z^{-1}e^{x}-1}dx=\frac{z}{\Gamma \left( \alpha \right) }\int_{0}^{1}\frac{\left( -\log u\right) ^{\alpha -1}}{% 1-uz}du$$ showing, by Hausdorff representation, that $$\mathbf{P}\left( X_{\infty }>i\right) =\left( i+1\right) ^{-\alpha }=\int_{0}^{1}u^{i}\lambda \left( du\right) \text{ where }\lambda \left( du\right) =\frac{1}{\Gamma \left( \alpha \right) }\left( -\log u\right) ^{\alpha -1}du\text{ }$$ is the probability density of $U=e^{-X}$, with $X\overset{d}{\sim }$Gamma$% \left( \alpha ,1\right) .$ The law of $X_{\infty }\geq 1$ is completely monotone (and $X_{\infty }-1$ is ID). Note $$\begin{aligned} \Phi _{\infty }\left( z\right) &=&1-z^{-1}\left( 1-z\right) L_{\alpha }\left( z\right) =z\int_{0}^{1}\frac{1-u}{1-uz}\lambda \left( du\right) =z\Psi _{\infty }\left( z\right) \\ \Psi _{\infty }\left( z\right) &=&\mathbf{E}\left( z^{X_{\infty }-1}\right) =\int_{0}^{1}\frac{1}{1-zu}\mu \left( du\right) \text{ where }\mu \left( du\right) =\left( 1-u\right) \lambda \left( du\right)\end{aligned}$$ - **Zipf** ($\alpha >1$): Suppose $\mathbf{P}\left( X_{\infty }=i\right) =i^{-\alpha }/\varsigma \left( \alpha \right) $ with associated pgf $\Phi _{\infty }\left( z\right) =L_{\alpha }\left( z\right) /L_{\alpha }\left( 1\right) ,$ $L_{\alpha }\left( 1\right) =\varsigma \left( \alpha \right) .$ Lagrange inversion formula gives the power-series expansion of $% \Phi _{\infty }^{-1}\left( z\right) $. We have $$\mathbf{P}\left( \nu \leq j\right) =\Phi _{\infty }^{-1}\left( 1-\mathbf{P}% \left( X_{\infty }>i\right) \right)$$ where, with $\lambda _{0}\left( du\right) =\frac{1}{\Gamma \left( \alpha \right) }\left( -\log u\right) ^{\alpha -1}du$$$\begin{aligned} \mathbf{P}\left( X_{\infty }>i\right) &=&\frac{1}{\varsigma \left( \alpha \right) }\sum_{j>i}j^{-\alpha }=\frac{1}{\varsigma \left( \alpha \right) }% \int_{0}^{1}\sum_{j>i}u^{j-1}\lambda _{0}\left( du\right) =\frac{1}{% \varsigma \left( \alpha \right) }\int_{0}^{1}u^{i}\left( 1-u\right) ^{-1}\lambda _{0}\left( du\right) \\ &=&\int_{0}^{1}u^{i}\lambda \left( du\right) \text{ where }\lambda \left( du\right) =\frac{\left( 1-u\right) ^{-1}}{\varsigma \left( \alpha \right) \Gamma \left( \alpha \right) }\left( -\log u\right) ^{\alpha -1}du\text{ }\end{aligned}$$ is the probability density of $U=e^{-X}$, with $X$ having density $$\frac{1}{\varsigma \left( \alpha \right) \Gamma \left( \alpha \right) }\frac{% e^{-x}x^{\alpha -1}}{1-e^{-x}}\text{, }x>0.$$ So $X_{\infty }$ (and $X_{\infty }-1$) is CM. Thus $X_{\infty }-1$ is infinitely divisible and even self-decomposable, say SD (see Example $12.18$ page $435$ of [@SH]). - **The critical case when** $X_{\infty }$ **has no moments of any positive order**: Suppose that with $\beta >0$ and $L_{1}\left( x\right) =\log \left( 1+x\right) >0$, slowly varying at $\infty $ $$\mathbf{P}\left( X_{\infty }=j\right) =\frac{C_{0}}{jL_{1}\left( j\right) ^{\beta +1}},\text{ }j\geq 1$$ where $C_{0}>0$ is the normalizing constant. Then $\mathbf{E}\left( X_{\infty }^{q}\right) =\infty $ for all $q>0.$ In this case, $\mathbf{P}% \left( X_{\infty }>j\right) \sim C_{0}\cdot L_{1}\left( j\right) ^{-\beta }$ as $j\rightarrow \infty $ with tails heavier than any power-law. Then: $\left( i\right) $ The rv $\nu $ whose distribution solves (\[FE\]) (as from Proposition $2$) is a well-defined rv obeying $j\mathbf{P}\left( \nu >j\right) \rightarrow e^{-\gamma }$ as $j\rightarrow \infty .$ $\left( ii\right) $ Furthermore $$\mathbf{P}\left( \nu >j\right) \underset{j\uparrow \infty }{\sim }e^{-\gamma }/j+d/\left( j\log j\right) +o\left( 1/\left( j\log j\right) \right)$$ with $$d=-\frac{\left( \beta +1\right) e^{-\gamma }\pi ^{2}}{12}<-\frac{e^{-\gamma }\pi ^{2}}{12}.$$ By (\[L3\]), the corresponding Lamperti chain is critical but it remains positive recurrent for all $\beta >0.$ *Proof:* $\left( i\right) $ This model for $X_{\infty }$ is indeed obtained in the limit $\alpha \rightarrow 0$ of the ansatz ($\alpha >0$) $$\mathbf{P}\left( X_{\infty }=j\right) =\frac{C_{0}}{j^{\alpha +1}L_{1}\left( j\right) ^{\beta +1}},\text{ }i\geq 1,$$ extending the previous Sibuya example with tail index $\alpha $. $\left( ii\right) $ In such an example of $X_{\infty }$ with logarithmic tails, we have more precisely $$\Phi _{\infty }\left( z\right) \underset{z\uparrow 1}{\sim }1-\frac{C_{0}}{% \left( -\log \left( 1-z\right) \right) ^{\beta }}$$ with local inverse: $\Phi _{\infty }^{-1}\left( z\right) \underset{z\uparrow 1}{\sim }1-e^{-\left( \frac{1-z}{C_{0}}\right) ^{-1/\beta }}$. We get $$\frac{1-\Phi _{\infty }\left( z\right) }{1-z}\underset{z\uparrow 1}{\sim }% \frac{1}{1-z}\frac{C_{0}}{\left( -\log \left( 1-z\right) \right) ^{\beta }}$$ so that [@Fla], with $C_{k}=\left( \frac{1}{\Gamma \left( \alpha \right) }\right) ^{\left( k\right) }\mid _{\alpha =1}$(in particular $C_{1}=\gamma $, $C_{2}=\gamma ^{2}-\pi ^{2}/6,$ with $C_{1}^{2}-C_{2}=\pi ^{2}/6$) $$\mathbf{P}\left( X_{\infty }>j\right) \underset{j\uparrow \infty }{\sim }% \frac{C_{0}}{\log ^{\beta }j}\left( 1-\frac{\beta C_{1}}{\log j}+\frac{\beta \left( \beta +1\right) C_{2}}{2\log ^{2}j}+o\left( \frac{1}{\log ^{2}j}% \right) \right) .$$ Observing $\left( 1-\frac{\beta C_{1}}{\log j}+\frac{\beta \left( \beta +1\right) C_{2}}{2\log ^{2}j}\right) ^{-1/\beta }\underset{j\uparrow \infty }{\sim }1+\frac{C_{1}}{\log j}+\frac{\left( \beta +1\right) }{2\left( \log j\right) ^{2}}\left( C_{1}^{2}-C_{2}\right) $, we are led to $$\begin{aligned} &&\mathbf{P}\left( \nu \leq j\right) \underset{j\uparrow \infty }{\sim }\Phi _{\infty }^{-1}\left( 1-\mathbf{P}\left( X_{\infty }>j\right) \right) \underset{j\uparrow \infty }{\sim }1-\left( \frac{1}{j}\right) ^{\left( 1-% \frac{\beta C_{1}}{\log j}+\frac{\beta \left( \beta +1\right) C_{2}}{2\log ^{2}j}\right) ^{-1/\beta }} \\ &&\underset{j\uparrow \infty }{\sim }1-e^{-\gamma }/j-d/\left( j\log j\right) +o\left( 1/\left( j\log j\right) \right)\end{aligned}$$ with $$d=-\frac{\left( \beta +1\right) e^{-\gamma }\pi ^{2}}{12}.$$ Because $d<-\pi ^{2}e^{-\gamma }/12$ for all $\beta >0,$ we conclude that $% \left\{ X_{n}\right\} $ generated by this $\nu $ just remains always positive-recurrent. $\Box $ **- Null-recurrent issues.*** * Irreducible aperiodic Markov chains may have or not a non-trivial invariant positive (infinite) measure, [@Harr]. In the null-recurrent case from (\[L2\]), the Lamperti model has a non trivial ($\neq \mathbf{0}$) invariant positive measure. *Proof:* To see a transition positive/null recurrence transition in the critical case, suppose $\delta \left( j\right) :=\Delta F_{\infty }\left( j\right) >0$ with $\Delta F_{\infty }\left( j\right) \rightarrow 0$ as $% j\rightarrow \infty ,$ $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}$ convergent for all $z\in \left[ 0,1\right) $, $\Phi _{\infty }\left( 0\right) =0$ and $\Phi _{\infty }\left( 1\right) =\infty .$ In this case $\Delta F_{\infty }\left( j\right) $ no longer is a probability mass at $j$. One can search solutions of (\[FE\]) in this case as well and Proposition $2$ applies simply while substituting $\delta \left( j\right) $ to $\pi \left( j\right) $ in the obtained expression of $\mathbf{P% }\left( \nu \leq j\right) .$ Because $\mathbf{P}\left( \nu \leq j\right) $ only depends on the ratio $F_{\infty }\left( j\right) /F_{\infty }\left( 1\right) $, regardless of any normalization, such a sequence $\delta \left( j\right) $ defines an invariant positive and infinite measure in the null-recurrent case. $\Box $ The simplest example is the following: $\Delta F_{\infty }\left( j\right) =1/j$, with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}$ obeying $\Phi _{\infty }\left( 1\right) =\infty .$ We have $\Phi _{\infty }\left( z\right) =-\log \left( 1-z\right) $ so that, upon inverting $\Phi _{\infty }$$$\mathbf{P}\left( \nu \leq j\right) =1-e^{-\sum_{k=1}^{j}\frac{1}{k}}$$ a true pdf. Recalling $\sum_{k=1}^{j}\frac{1}{k}-\gamma -\log j\sim 1/\left( 2j\right) $, we get $\mathbf{P}\left( \nu >j\right) \sim e^{-\gamma }/j+O\left( j^{-2}\right) .$ The constant $d$ in (\[L3\]) is $d=0$ and the Lamperti chain with a branching number $\nu $ distributed as such is null-recurrent. This is also true if $\Delta F_{\infty }\left( j\right) =1/\left[ j\log \left( 1+j\right) ^{\beta +1}\right] $ with $\beta <0$ or $% \Delta F_{\infty }\left( j\right) =j^{-\alpha },$ $\alpha \in \left( 0,1\right) ,$ both expressions leading to a diverging series $\Phi _{\infty }\left( 1\right) $. - **Transient issues: non-unicity of the invariant measure.** Whenever** **$\left\{ X_{n}\right\} $ is transient, one obvious solution to the invariant measure equation $\mathbf{\pi }^{\prime }=\mathbf{\pi }% ^{\prime }P$ is $\mathbf{\pi }=\mathbf{0}$. This corresponds to the fact that $X_{\infty }\overset{d}{\sim }\delta _{\infty }$. However this solution is not unique and there are other invariant positive measures. The question of the existence of a non-trivial invariant measure for transient chains was raised by Harris, [@Har]. To exhibit such an invariant measure, suppose $\delta \left( j\right) :=\Delta F_{\infty }\left( j\right) >0$ with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}$ convergent for all $% z\in \left[ 0,1\right) $, $\Phi _{\infty }\left( 0\right) =0$ and $\Phi _{\infty }\left( 1\right) =\infty .$ In this case $\Delta F_{\infty }\left( j\right) $ no longer is a probability mass at $j$ either but it is no longer required $\Delta F_{\infty }\left( j\right) \rightarrow 0$ as $j\rightarrow \infty .$ In the transient case from (\[L3\]), the Lamperti model has a non trivial ($\neq \mathbf{0}$) invariant positive measure. *Proof:* One can search solutions of (\[FE\]) in this case as well and Proposition $2$ applies simply while substituting $\delta \left( j\right) $ to $\pi \left( j\right) $ in the obtained expression of $\mathbf{P% }\left( \nu \leq j\right) .$ Because, from (\[FES\]), $\mathbf{P}\left( \nu \leq j\right) $ only depends on the ratio $F_{\infty }\left( j\right) /F_{\infty }\left( 1\right) $ regardless of any normalization, such a sequence $\delta \left( j\right) $ defines an invariant measure in the transient case as well. $\Box $ - The simplest explicit example is the following counting measure one: $% \delta \left( j\right) =\Delta F_{\infty }\left( j\right) =1$, $F_{\infty }\left( j\right) =j$, with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}=z/\left( 1-z\right) $ obeying $% \Phi _{\infty }\left( 1\right) =\infty .$ There is a solution to (\[FE\]) which is $$\mathbf{P}\left( \nu \leq j\right) =\Phi _{\infty }^{-1}\left( j\right) =% \frac{j}{1+j}.$$ We have: $\mathbf{P}\left( \nu >j\right) =1/\left( 1+j\right) $ so that $j% \mathbf{P}\left( \nu >j\right) \underset{j\rightarrow \infty }{\rightarrow }% 1>e^{-\gamma }$, indeed corresponding to a transient case. - Suppose now $\Delta F_{\infty }\left( j\right) =j$, $F_{\infty }\left( j\right) =j\left( j+1\right) /2,$ so with $\Phi _{\infty }\left( z\right) =\sum_{j\geq 1}\Delta F_{\infty }\left( j\right) z^{j}=z/\left( 1-z\right) ^{2}$ obeying $\Phi _{\infty }\left( 1\right) =\infty .$ There is a solution to (\[FE\]) which is $$\mathbf{P}\left( \nu \leq j\right) =\Phi _{\infty }^{-1}\left( \frac{j\left( j+1\right) }{2}\right) =\frac{j\left( j+1\right) +1-\sqrt{1+2j\left( j+1\right) }}{j\left( j+1\right) }.$$ When inverting $\Phi _{\infty }\left( z\right) $ we have chosen the branch for which $\Phi _{\infty }^{-1}\left( 0\right) =0$. We have: $\mathbf{P}% \left( \nu >j\right) =\left( \sqrt{1+2j\left( j+1\right) }-1\right) /\left( j\left( j+1\right) \right) $ so that $j\mathbf{P}\left( \nu >j\right) \rightarrow \sqrt{2}>e^{-\gamma }$, also corresponding to a transient case. Defining the reversed failure rate of the sequence $\delta \left( j\right) $ as $$\overline{r}\left( j\right) =\frac{\delta \left( j\right) }{% \sum_{k=1}^{j}\delta \left( k\right) }=\frac{\Delta F_{\infty }\left( j\right) }{F_{\infty }\left( j\right) }\text{, }j\geq 1,$$ we conclude that in both examples, $\overline{r}\left( j\right) \asymp 1/j$ so with decreasing reversed failure rate. *Remark:* By the ergodic theorem: - in case (\[L1\]): $$n^{-1}\sum_{m=1}^{n}\mathbf{1}\left( X_{m}=j\mid X_{0}\overset{d}{\sim }% \mathbf{\pi }_{0}\right) \rightarrow \pi \left( j\right) \text{ as }% n\rightarrow \infty ,$$ - in cases (\[L2\]) and (\[L3\]): For all states $i,j\geq 1$$$\frac{\sum_{m=1}^{n}\mathbf{1}\left( X_{m}=i\mid X_{0}\overset{d}{\sim }% \mathbf{\pi }_{0}\right) }{\sum_{m=1}^{n}\mathbf{1}\left( X_{m}=j\mid X_{0}% \overset{d}{\sim }\mathbf{\pi }_{0}\right) }\rightarrow \frac{\delta \left( i\right) }{\delta \left( j\right) }\text{ as }n\rightarrow \infty .\text{ }% \diamondsuit$$ - **Poisson target:** We finally develop some additional examples in the recurrent case, not in the latter classes and related to the fundamental Poisson distribution class: - **Shifted** **Poisson:** Suppose** **$\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =ze^{\lambda \left( z-1\right) }=z\Psi _{\infty }\left( z\right) ,$ **(**$\Psi _{\infty }\left( z\right) $ is the pgf of an ID Poisson rv which is log-concave). Then $$\begin{aligned} \frac{1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n} &=&% \frac{1}{n}\left[ z^{n-1}\right] e^{-n\lambda \left( z-1\right) }=\left( -1\right) ^{n-1}\frac{e^{\lambda n}}{n!}\left( n\lambda \right) ^{n-1} \\ F_{\infty }\left( j\right) &=&e^{-\lambda }\sum_{k=0}^{j-1}\frac{\lambda ^{k}% }{k!} \\ F\left( j\right) &=&\sum_{n\geq 1}\left( -1\right) ^{n-1}\frac{e^{\lambda n}% }{n!}\left( n\lambda \right) ^{n-1}F_{\infty }\left( j\right) ^{n} \\ &=&\sum_{n\geq 1}\left( -1\right) ^{n-1}\frac{\left( n\lambda \right) ^{n-1}% }{n!}\left( \sum_{k=0}^{j-1}\frac{\lambda ^{k}}{k!}\right) ^{n}=W_{\lambda }\left( \sum_{k=0}^{j-1}\frac{\lambda ^{k}}{k!}\right)\end{aligned}$$ The Lambert function, solving $x=W\left( x\right) e^{W\left( x\right) },$ is (by Lagrange inversion formula): $$\begin{aligned} W\left( x\right) &=&\sum_{n\geq 1}\left( -1\right) ^{n-1}\frac{n^{n-1}}{n!}% x^{n}\text{ hence} \\ W_{\lambda }\left( x\right) &:&=\lambda ^{-1}W\left( \lambda x\right) =\sum_{n\geq 1}\left( -1\right) ^{n-1}\frac{\left( n\lambda \right) ^{n-1}}{% n!}x^{n}.\end{aligned}$$ And $W_{\lambda }\left( x\right) $ solves: $x=W_{\lambda }\left( x\right) e^{\lambda W_{\lambda }\left( x\right) }$. It is positive and increasing when $x>0$, so $F\left( j\right) $ is a well-defined pdf if $F\left( \infty \right) =W_{\lambda }\left( e^{\lambda }\right) =1,$ which is the case. - **Poisson conditioned to be positive:** Suppose** **$\Phi _{\infty }\left( z\right) =\mathbf{E}\left( z^{X_{\infty }}\right) =\left( e^{\lambda z}-1\right) /\left( e^{\lambda }-1\right) $, leading directly to $\Phi _{\infty }^{-1}\left( z\right) =% \frac{1}{\lambda }\log \left( 1+z\left( e^{\lambda }-1\right) \right) .$ Then $$\begin{aligned} F_{\infty }\left( j\right) &=&\frac{1}{e^{\lambda }-1}\sum_{k=1}^{j}\frac{% \lambda ^{k}}{k!} \\ F\left( j\right) &=&\Phi _{\infty }^{-1}\left( F_{\infty }\left( j\right) \right) =\frac{1}{\lambda }\log \left( 1+F_{\infty }\left( j\right) \left( e^{\lambda }-1\right) \right) ,\end{aligned}$$ which defines a pdf with $F\left( \infty \right) =1$. In this case, although $\Psi _{\infty }\left( z\right) =z^{-1}\Phi _{\infty }\left( z\right) $ is not the pgf of an ID rv, the calculation of $F\left( j\right) $ is straightforward. Examples of $\nu \rightarrow \nu _{\left( N\right) }$ with finite support $\left\{ 1,...,N\right\} $. ----------------------------------------------------------------------------------------------------- In this Sub-section, we look at situations where both $\left( X_{\infty },\nu _{\left( N\right) }\right) $ have finite support $\left\{ 1,...,N\right\} $. Note that if $\nu $ has support $\left\{ 1,...,N\right\} $, so does $\left\{ X_{n}\right\} $ (defined recursively by $% X_{n+1}=\max_{j=1,...,X_{n}}\nu _{j,n+1}$) and then $X_{\infty }$. Conversely, if $X_{\infty }$ has support $\left\{ 1,...,N\right\} $, there exists $\nu $ with support $\left\{ 1,...,N\right\} $ such that $% X_{n+1}=\max_{j=1,...,X_{n}}\nu _{j,n+1}$ defines a sequence $\left( X_{n}\right) $ with finite support. In such cases, the Lamperti Markov chain will always be ergodic in view of its transition matrix $P_{\left( N\right) } $ being irreducible. We shall let $\pi _{\left( N\right) }\left( k\right) =% \mathbf{P}\left( X_{\infty }=k\right) .$ - **The general case:** Suppose $\Phi _{\infty }\left( z\right) =\sum_{k=1}^{N}\pi _{\left( N\right) }\left( k\right) z^{k}$, so that $\Psi _{\infty }\left( z\right) =\sum_{k=0}^{N-1}\pi _{\left( N\right) }\left( k+1\right) z^{k}$. We have $$\begin{aligned} \Psi _{\infty }\left( z\right) ^{-\alpha } &=&\pi _{\left( N\right) }\left( 1\right) ^{-\alpha }\left( 1+\sum_{k=1}^{N-1}\frac{\pi _{\left( N\right) }\left( k+1\right) }{\pi _{\left( N\right) }\left( 1\right) }z^{k}\right) ^{-\alpha } \\ &=&\pi _{\left( N\right) }\left( 1\right) ^{-\alpha }\sum_{l\geq 0}z^{l}\sum_{k=0}^{l}\left( -1\right) ^{k}\left[ \alpha \right] _{k}\sum_{{}}^{*}\prod_{m=1}^{N-1}\frac{\left( \pi _{\left( N\right) }\left( m+1\right) /\pi _{\left( N\right) }\left( 1\right) \right) ^{k_{m}}}{k_{m}!}\end{aligned}$$ where the star sum runs over $k_{m}\geq 0$, $m=1,...,N-1$ obeying $% \sum_{m=1}^{N-1}k_{m}=k$ and $\sum_{m=1}^{N-1}mk_{m}=l$. From this, we obtain the finite support version of (\[FES\]) as For any given $X_{\infty }$ with support $\left\{ 1,...,N\right\} $, the mapping $X_{\infty }\rightarrow \nu _{\left( N\right) }$ is one-to-one and onto. With $$C_{n-1,0}^{\left( N-1\right) }=\delta _{n-1,0}\text{ and }C_{n-1,k}^{\left( N-1\right) }:=\sum_{{}}^{*}\prod_{m=1}^{N-1}\frac{\left( \pi _{\left( N\right) }\left( m+1\right) /\pi _{\left( N\right) }\left( 1\right) \right) ^{k_{m}}}{k_{m}!}$$ where the star sum runs over $k_{m}\geq 0$, $m=1,...,N-1$ obeying $% \sum_{m=1}^{N-1}k_{m}=k$ and $\sum_{m=1}^{N-1}mk_{m}=n-1\geq k$, $$\varphi _{n}:=\left[ z^{n}\right] \Phi _{\infty }^{-1}\left( z\right) =\frac{% 1}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n}=\frac{\pi _{\left( N\right) }\left( 1\right) ^{-n}}{n}\sum_{k=0}^{n-1}\left( -1\right) ^{k}\left[ n\right] _{k}C_{n-1,k}^{\left( N-1\right) }. \label{FS1}$$ So, with $h_{1}=1$ and $h_{n}=\frac{1}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\left[ n\right] _{k}C_{n-1,k}^{\left( N-1\right) },$ $n\geq 2$$$\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) =\sum_{n\geq 1}h_{n}\cdot \left( \mathbf{P}\left( X_{\infty }\leq j\right) /\pi _{\left( N\right) }\left( 1\right) \right) ^{n} \label{FS2}$$ is the pdf of $\nu _{\left( N\right) }$ associated to any $\mathbf{P}\left( X_{\infty }\leq j\right) =\sum_{k=1}^{j}\pi _{\left( N\right) }\left( k\right) $, $j=1,...,N$ obeying $\mathbf{P}\left( X_{\infty }\leq N\right) =1.$ *Remark:* $\left( i\right) $ $\varphi _{1}=1/\pi _{\left( N\right) }\left( 1\right) $ ($h_{1}=1$) and for $n\geq 2$, the sum over $k$ giving the expression of $% \varphi _{n}$ (or of $h_{n}$) can start at $k=1$. $\left( ii\right) $ if (a separable case in $\left( k,N\right) $): $\pi _{\left( N\right) }\left( k\right) =a_{k}/A_{N},$ $a_{k}\geq 0$, where $% A_{N}=\sum_{k=1}^{N}a_{k}$ is a normalization factor, the law of $\nu _{\left( N\right) }$ does not depend on $A_{N}$ because it only depends on the ratios $\pi _{\left( N\right) }\left( k\right) /\pi _{\left( N\right) }\left( 1\right) =a_{k}/a_{1}.$ $\diamondsuit $ **Examples:** Just like in the infinite-dimensional case, there are examples amenable to a straightforward calculation. $\left( i\right) $ Suppose $$\Phi _{\infty }\left( z\right) =\frac{\left( q+pz\right) ^{N}-q^{N}}{1-q^{N}}% ,$$ corresponding to a binomial model restricted to $\left\{ 1,...,N\right\} $ with $$\begin{aligned} \mathbf{P}\left( X_{\infty }=k\right) &=&\left[ z^{k}\right] \Phi _{\infty }\left( z\right) =\frac{1}{1-q^{N}}\binom{N}{k}p^{k}q^{N-k} \\ \mathbf{P}\left( X_{\infty }\leq j\right) &=&\sum_{k=1}^{j}\mathbf{P}\left( X_{\infty }=k\right) .\end{aligned}$$ By direct inversion of $\Phi _{\infty }\left( z\right) $, we have that $$\Phi _{\infty }^{-1}\left( \mathbf{P}\left( X_{\infty }\leq j\right) \right) =\frac{q}{p}\left[ \left( 1+\sum_{k=1}^{j}\binom{N}{k}\left( \frac{p}{q}% \right) ^{k}\right) ^{1/N}-1\right]$$ is the pdf $\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) $ of some rv $\nu _{\left( N\right) }.$ Note $\mathbf{P}\left( \nu _{\left( N\right) }\leq N\right) =\Phi _{\infty }^{-1}\left( \mathbf{P}\left( X_{\infty }\leq N\right) \right) =1.$ $\left( ii\right) $ Suppose $$\Phi _{\infty }\left( z\right) =z\left( q+pz\right) ^{N-1}=z\Psi _{\infty }\left( z\right)$$ corresponding to a shifted binomial model supported $\left\{ 1,...,N\right\} $$$\begin{aligned} \mathbf{P}\left( X_{\infty }=k\right) &=&\left[ z^{k}\right] \Phi _{\infty }\left( z\right) =\binom{N-1}{k-1}p^{k-1}q^{N-k} \\ \mathbf{P}\left( X_{\infty }\leq j\right) &=&\sum_{k=1}^{j}\mathbf{P}\left( X_{\infty }=k\right) .\end{aligned}$$ With $n\geq 1$, we have that $\Phi _{\infty }^{-1}\left( z\right) =\sum_{n\geq 1}\frac{z^{n}}{n}\left[ z^{n-1}\right] \Psi _{\infty }\left( z\right) ^{-n}$ with $$\begin{aligned} \varphi _{n} &=&\frac{q^{-n\left( N-1\right) }}{n}\left[ z^{n-1}\right] \left( 1+\frac{p}{q}z\right) ^{-n\left( N-1\right) } \\ &=&\left( -1\right) ^{n-1}q^{-n\left( N-1\right) }\left( \frac{p}{q}\right) ^{n-1}\frac{\left[ n\left( N-1\right) \right] _{n-1}}{n!}\end{aligned}$$ $$\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) =\sum_{n\geq 1}\varphi _{n}\mathbf{P}\left( X_{\infty }\leq j\right) ^{n}$$ The rv $\nu _{\left( N\right) }$ has support $\left\{ 1,...,N\right\} $. $\left( iii\right) $ Truncation of the infinite-dimensional model. This situation occurs if, for $\left( \pi _{\left( N\right) }\left( k\right) ,k=1,...,N\right) $, we consider the normalized restriction of the invariant measure $\mathbf{\pi }$ with full support $\Bbb{N}$ to its $N$ first entries. For example, assuming $\left( \pi \left( k\right) =pq^{k-1},k\geq 1\right) $ is geometric, we get $$\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) =\sum_{n\geq 1}h_{n}\cdot \left( \frac{1-q^{j}}{p}\right) ^{n}$$ where $h_{n}=\frac{q^{n-1}}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\frac{% \left[ n\right] _{k}}{k!}\sum_{{}}^{*}\frac{k!}{\prod_{m=1}^{N-1}k_{m}!}=% \frac{q^{n-1}}{n}\sum_{k=1}^{n-1}\left( -1\right) ^{k}\frac{\left[ n\right] _{k}}{k!}\left( N-1\right) ^{k}$, so that with $A_{n,N}=\sum_{k=1}^{n-1}% \left( -1\right) ^{k}\frac{\left[ n\right] _{k}}{k!}\left( N-1\right) ^{k}$, $$\mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) =\frac{1}{q}% \sum_{n\geq 1}\frac{A_{n,N}}{n}\cdot \left( \frac{q\left( 1-q^{j}\right) }{p}% \right) ^{n}.\text{ }\diamondsuit$$ Take any probability measure $\mathbf{\pi }_{\left( N\right) }$ with support $\left\{ 1,...,N\right\} $. Compute $F_{\left( N\right) }\left( j\right) =% \mathbf{P}\left( \nu _{\left( N\right) }\leq j\right) $ from $\mathbf{\pi }% _{\left( N\right) }$ as from (\[FS2\]). Construct the $N\times N$ stochastic matrix $P_{\left( N\right) }$ with entries $P_{\left( N\right) }\left( i,j\right) =F_{\left( N\right) }\left( j\right) ^{i}-F_{\left( N\right) }\left( j-1\right) ^{i}$, $i,j\in \left\{ 1,...,N\right\} .$ The matrix $P_{\left( N\right) }$ is the transition matrix of some ergodic Lamperti chain $X_{n}^{\left( N\right) }$ with state-space $\left\{ 1,...,N\right\} ,$ having $\mathbf{\pi }_{\left( N\right) }$ as invariant probability measure and reproduction mechanism $\nu _{\left( N\right) }$. The MC $\left\{ X_{n}^{\left( N\right) }\right\} $ is failure rate monotone. Furthermore: $$\mathbf{P}_{\mathbf{\pi }_{0}}\left( X_{n}^{\left( N\right) }=j\right) \underset{n,N\rightarrow \infty }{\rightarrow }\pi \left( j\right)$$ *Proof:* The reasons are similar to the ones raised for the Lamperti chain taking values in $\Bbb{N}$. The probability $\mathbf{P}\left( X_{n+1}^{\left( N\right) }\leq j\mid X_{n}^{\left( N\right) }=i\right) =F_{\left( N\right) }\left( j\right) ^{i}$ is a decreasing function of $i$, for all $j$: the MC $\left\{ X_{n}^{\left( N\right) }\right\} $ is stochastically monotone. The cumulated transition matrix : $P_{\left( N\right) }^{c}\left( i,j\right) =\sum_{k=1}^{j}P_{\left( N\right) }\left( i,k\right) $ obeys: $$P_{\left( N\right) }^{c}\left( i_{1},j_{1}\right) P_{\left( N\right) }^{c}\left( i_{2},j_{2}\right) \geq P_{\left( N\right) }^{c}\left( i_{1},j_{2}\right) P_{\left( N\right) }^{c}\left( i_{2},j_{1}\right) ,$$ for all $i_{1}<i_{2}$ and $j_{1}<j_{2}$ ($P_{\left( N\right) }^{c}$ is totally positive of order $2$): the MC $\left\{ X_{n}^{\left( N\right) }\right\} $ is failure rate monotone. Note the induced Kirchoff determinantal identities for finite matrices: $\pi _{\left( N\right) }\left( j\right) =\det \left[ \left( I-P_{\left( N\right) }\right) ^{\left( j,j\right) }\right] .$ The last statement is obvious. $% \Box $ (Truncation of $X_{n}$) $\left( i\right) $ Take for $\mathbf{\pi }_{\left( N\right) }$ the restriction to $\left\{ 1,...,N\right\} $ of the invariant measure* *$% \mathbf{\pi }$ of the Lamperti model with countable state-space, so with: $% \mathbf{\pi }_{\left( N\right) }\left( k\right) =\pi \left( k\right) /\sum_{k=1,...,N}\pi \left( k\right) $*,* $k=1,...,N$*.* $\left( ii\right) $ Take for $\mathbf{\pi }_{\left( N\right) }$ the restriction to $\left\{ 1,...,N-1\right\} $ of the invariant measure* * $\mathbf{\pi }$ of the Lamperti model with countable state-space, so with: $% \mathbf{\pi }_{\left( N\right) }\left( k\right) =\pi \left( k\right) $*,* $k=1,...,N-1$, $\pi _{\left( N\right) }\left( N\right) =\sum_{k\geq N}\pi \left( k\right) $*.* Constructing the corresponding transition matrices $P_{\left( N\right) }$, in both cases, the truncations preserve the failure rate monotonicity of $P.$ The corresponding Lamperti chains $X_{n}^{\left( N\right) }$ with state-space $\left\{ 1,...,N\right\} ,$ having $\mathbf{\pi }_{\left( N\right) }$ as restricted invariant measure and reproduction mechanism $\nu _{\left( N\right) }$ are called the truncated Lamperti chains up to state $N$. *Remarks:* - The case $\left( i\right) $ is simpler because in this separable case, the corresponding law of $\nu _{\left( N\right) }$ does not depend on the normalization factor $\sum_{k=1,...,N}\pi \left( k\right) $. - Censored Markov chain** **([@ZL], [@GH]): with $% P_{11}=Q_{\left( N\right) }$ and $$P=\left[ \begin{array}{ll} P_{11} & P_{12} \\ P_{21} & P_{22} \end{array} \right] ,$$ define $$P_{\left( N\right) }=P_{11}+P_{12}\left( I-P_{22}\right) ^{-1}P_{21}.$$ Let $Q_{2,2}=\left( I-P_{22}\right) ^{-1}$ be the fundamental matrix of $% P_{22}$, with $Q_{2,2}\left( i,j\right) $ the mean number of visits to state $j$ in $\left\{ N+1,...,\infty \right\} $ starting from $i$ in $\left\{ N+1,...,\infty \right\} $, before visiting first $\left\{ 1,...,N\right\} $. The matrix element $\left( P_{12}Q_{2,2}P_{21}\right) \left( i,j\right) $ is the taboo probability of the paths from states $i$ to $j$ both in $\left\{ 1,...,N\right\} $ which are not allowed to visit $\left\{ 1,...,N\right\} $ in between. $P_{\left( N\right) }$ has invariant measure $\mathbf{\pi }% _{\left( N\right) }^{\prime }=\left( \pi _{1},...,\pi _{N}\right) /$norm (the restriction of $\mathbf{\pi }$ to its $N$ first entries). However, it is not clear that such a $P_{\left( N\right) }$ is SM (probably not) nor that $P_{\left( N\right) }^{c}$ is FRM$.$ Besides, $P_{\left( N\right) }$has a complicated structure in case of Lamperti. Truncating a Markov chain invariant measure while preserving the monotonicity properties of the original is not so straightforward.* *$\diamondsuit $ Brown’s analysis of the truncated Lamperti model ================================================ In this Section, we consider the truncated version $\left\{ X_{n}^{\left( N\right) }\right\} $ of the chain $\left\{ X_{n}\right\} $ corresponding to the one preserving the $N$ first entries of the full invariant measure $% \mathbf{\pi }$ of $\left\{ X_{n}\right\} $, meaning $\pi \left( i\right) \rightarrow \pi _{\left( N\right) }\left( i\right) =\pi \left( i\right) /\sum_{i=1}^{N}\pi \left( i\right) $, $i=1,...,N$ (the restriction to $% \left\{ 1,...,N\right\} $ of the full invariant measure supported by $\Bbb{N% }$). This MC has totally ordered state-space, with $\left\{ N\right\} $ as a maximal element. It is a separable case and this truncation preserves the failure-rate monotonicity of $P^{c}:$ $P_{\left( N\right) }^{c}$ remains FRM, else $P_{\left( N\right) }^{c}$ is TP$_{2}$. As in [@Brown], we shall be concerned by the relationship existing between the first hitting times of both state $\left\{ N\right\} $ and the restricted invariant measure $\mathbf{\pi }_{\left( N\right) },$ given $X_{0}^{\left( N\right) }% \overset{d}{\sim }\mathbf{\pi }_{0}$. We will assume $\pi _{0}\left( N\right) =0$, to ensure that $\left\{ X_{n}^{\left( N\right) }\right\} $ hits $\left\{ N\right\} $ for the first time with positive probability after at least one time unit. To illustrate his theory, Brown designs some ad hoc ($4\times 4$) FRM matrices; the truncated Lamperti chain is a more relevant example. The following general results for hitting times hold for the Lamperti truncated chain (see also [@Lorekth] for a survey). [@Brown]. Suppose $\mathbf{\pi }_{0}$ is such that $\pi _{0}\left( i\right) /\pi _{\left( N\right) }\left( i\right) $ decreases with $i$ and $% \pi _{0}\left( N\right) =0$. Then $\left( i\right) $ $\mathbf{P}\left( X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }\overset{d}{\sim }\mathbf{\pi }_{0}\right) $ is non-decreasing with $n.$ $\left( ii\right) $ Let $\tau _{i,j}=\inf \left( n\geq 1:X_{n}^{\left( N\right) }=j\mid X_{0}^{\left( N\right) }=i\right) $, with $\tau _{j,j}:=0$. With $\tau _{\mathbf{\pi }_{0},j}=\inf \left( n\geq 1:X_{n}^{\left( N\right) }=j\mid X_{0}^{\left( N\right) }\overset{d}{\sim }\mathbf{\pi }_{0}\right) :$$$\tau _{\mathbf{\pi }_{0},N}\overset{d}{=}T_{\left( N\right) }+\tau _{\mathbf{% \pi }_{\left( N\right) },N} \label{B0}$$ where $T_{\left( N\right) }\geq 1$ and $\tau _{\mathbf{\pi }_{\left( N\right) },N}\geq 0$ are independent. *Proof:* The condition that $\mathbf{\pi }_{0}$ is such that $\pi _{0}\left( i\right) /\pi _{\left( N\right) }\left( i\right) $ decreases with $i$ holds if $\pi _{0}\left( i\right) =\delta _{i,1}$ and also if $\pi _{0}\left( i\right) =z^{i}\pi _{\left( N\right) }\left( i\right) /$norm, $% i=1,...,N-1$ for some $z\in \left( 0,1\right) $). It says that the initial probability mass assigned to states near the bottom state $\left\{ 1\right\} $ should exceed the one assigned by $\mathbf{\pi }_{\left( N\right) }$. In particular: $\pi _{0}\left( 1\right) >\pi _{\left( N\right) }\left( 1\right) .$ The proof of this statement was derived in [@Brown] in a continuous-time setting and is easily adaptable to discrete-time. $\left( i\right) $ Let $\mathbf{e}_{N}^{\prime }=\left( 0,...,0,1\right) $ be an $N-$dimensional row vector, with $1$ in position $N$. Because $% P_{\left( N\right) }^{c}$ is FRM (in particular SM), $\mathbf{P}\left( X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }\overset{d}{\sim }% \mathbf{\pi }_{0}\right) =\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}% \mathbf{e}_{N}$ is non-decreasing with $n$ [@Brown]. See Lemma $4.2$ of [@Brown] where it is shown that this condition is fulfilled if $% \overline{P}_{\left( N\right) }^{n}\left( \mathbf{\pi }_{0},j\right) /% \overline{\pi }_{\left( N\right) }\left( j\right) $ is decreasing in $j$, which is the case for FRM Markov chains. Here $\overline{\pi }_{\left( N\right) }\left( j\right) =\sum_{k=j}^{N}\overline{\pi }_{\left( N\right) }\left( k\right) $ and $\overline{P}_{\left( N\right) }^{n}\left( \mathbf{% \pi }_{0},j\right) =\sum_{k=j}^{N}P_{\left( N\right) }^{n}\left( \mathbf{\pi }_{0},k\right) .$ It is needed in the proof that $\pi _{0}\left( i\right) /\pi _{\left( N\right) }\left( i\right) $ decreases with $i.$ $\left( ii\right) $ Owing to $P_{\left( N\right) }^{n}\left( \mathbf{\pi }% _{0},N\right) =\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}% _{N}\rightarrow \pi _{\left( N\right) }\left( N\right) $ as $n\rightarrow \infty $, $\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}% _{N}/\pi _{\left( N\right) }\left( N\right) $ is a probability distribution function of some rv $T_{\left( N\right) }$ with $$\mathbf{P}\left( T_{\left( N\right) }\leq n\right) =\mathbf{\pi }% _{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}_{N}/\pi _{\left( N\right) }\left( N\right) \text{, }n\geq 0.$$ With $G_{\mathbf{\pi }_{0},N}\left( z\right) =\mathbf{\pi }_{0}^{\prime }\sum_{n\geq 0}z^{n}P_{\left( N\right) }^{n}\mathbf{e}_{N}=\mathbf{\pi }% _{0}^{\prime }\left( I-zP_{\left( N\right) }\right) ^{-1}\mathbf{e}_{N}$, a Green kernel of $P_{\left( N\right) }$, we thus have $$\mathbf{E}\left( z^{T_{\left( N\right) }}\right) =\frac{1-z}{\pi _{\left( N\right) }\left( N\right) }G_{\mathbf{\pi }_{0},N}\left( z\right) . \label{B1}$$ Now $$\mathbf{\pi }_{\left( N\right) }\left( N\right) =\mathbf{P}_{\mathbf{\pi }% _{\left( N\right) }}\left( X_{n}^{\left( N\right) }=N\right) =\sum_{m=0}^{n}P_{\left( N\right) }^{n-m}\left( N,N\right) \mathbf{P}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}=m\right) ,$$ of convolution type. Taking the generating function of both sides $$\mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{\left( N\right) },N}}\right) =% \frac{\mathbf{\pi }_{\left( N\right) }\left( N\right) }{\left( 1-z\right) G_{N,N}\left( z\right) }, \label{B2}$$ where $G_{N,N}\left( z\right) =\sum_{m\geq 0}z^{m}P_{\left( N\right) }^{m}\left( N,N\right) =\left( I-zP_{\left( N\right) }\right) ^{-1}\left( N,N\right) $ is the Green kernel of $\left\{ X_{n}^{\left( N\right) }\right\} $ at $\left( N,N\right) .$ Similarly, $$\mathbf{P}_{\mathbf{\pi }_{0}}\left( X_{n}^{\left( N\right) }=N\right) =\sum_{m=0}^{n}P_{\left( N\right) }^{n-m}\left( N,N\right) \mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}=m\right)$$ leading to, $$G_{\mathbf{\pi }_{0},N}\left( z\right) =G_{N,N}\left( z\right) \mathbf{E}% \left( z^{\tau _{\mathbf{\pi }_{0},N}}\right) \label{B3}$$ Taking the product of (\[B2\]-\[B3\]), and recalling (\[B1\]), we get $$\phi _{\mathbf{\pi }_{0},N}\left( z\right) :=\mathbf{E}\left( z^{\tau _{% \mathbf{\pi }_{0},N}}\right) =\mathbf{E}\left( z^{T_{\left( N\right) }}\right) \mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{\left( N\right) },N}}\right) .\text{ }\Box \label{B3a}$$ The latter equation indicates that $\tau _{\mathbf{\pi }_{0},N}\geq 1$ is stochastically larger than $\tau _{\mathbf{\pi }_{\left( N\right) },N}$: it takes a shorter time for $\left\{ X_{n}^{\left( N\right) }\right\} $ to first hit $\left\{ N\right\} $ starting from $\mathbf{\pi }_{\left( N\right) }$ than starting from $\mathbf{\pi }_{0}.$ The time to first hit state $% \left\{ N\right\} $ is important in the Lamperti context because at this instant, the progeny after selection is the maximum possible. But of course the process will not remain in that state unless one forces the chain to have $\left\{ N\right\} $ absorbing. As a result also, $T_{\left( N\right) }$ interprets as $\tau _{\mathbf{\pi }% _{0},\mathbf{\pi }_{\left( N\right) }},$ the first hitting time of $\mathbf{% \pi }_{\left( N\right) }$ starting from $\mathbf{\pi }_{0}$. So with $X_{T_{\left( N\right) }}\overset{d}{\sim }\mathbf{\pi }_{\left( N\right) }$, $X_{T_{\left( N\right) }}$ independent of $T_{\left( N\right) }$ and $\mathbf{P}\left( X_{n}=N\text{ for some }n<T_{\left( N\right) }\right) =0$. The latter equation also indicates that $\tau _{\mathbf{\pi }% _{0},N}\geq 1$ is stochastically larger than $\tau _{\mathbf{\pi }_{0},% \mathbf{\pi }_{\left( N\right) }}\geq 1$ (statistically, $\left\{ X_{n}^{\left( N\right) }\right\} $ started from $\mathbf{\pi }_{0}$ enters $% \mathbf{\pi }_{\left( N\right) }$ before first hitting state $N$)$.$ As a consequence, ([@Brown], Corollary $4.1$) For all $n\geq 0$$$\begin{aligned} \text{sep}\left( \mathbf{P}_{\mathbf{\pi }_{0}}\left( X_{n}^{\left( N\right) }=\cdot \right) ,\mathbf{\pi }_{\left( N\right) }\right) &=&\underset{k}{% \max }\left( 1-\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}% _{k}/\pi _{\left( N\right) }\left( k\right) \right) \\ &=&1-\mathbf{\pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}_{N}/\pi _{\left( N\right) }\left( N\right) =\mathbf{P}\left( T_{\left( N\right) }>n\right) ,\end{aligned}$$ and $T_{\left( N\right) }$ is a minimal strong stationary time with separating state $N$. The separation distance sep$\left( \cdot ,\cdot \right) $ from $\mathbf{P}_{% \mathbf{\pi }_{0}}\left( X_{n}^{\left( N\right) }=\cdot \right) $ to $% \mathbf{\pi }_{\left( N\right) }$ gives an upper bound for the total variation norm between these two probability measures. $$\mathbf{E}\left( T_{\left( N\right) }\right) =1+\sum_{n\geq 1}\frac{\mathbf{% \pi }_{0}^{\prime }P_{\left( N\right) }^{n}\mathbf{e}_{N}}{\pi _{\left( N\right) }\left( N\right) }=1+\frac{1}{\pi _{_{\left( N\right) }}\left( N\right) }\mathbf{\pi }_{0}^{\prime }\left( I-P_{\left( N\right) }\right) ^{-1}P_{\left( N\right) }\mathbf{e}_{N}$$ There are some other facts pertaining to the fact that $\tau _{\mathbf{\pi }% _{\left( N\right) },N}$ has a geometric convolution representation. [@Brown] $\left( i\right) $ $\mathbf{P}\left( X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }=N\right) $ is non-increasing with $n$, so $$\mathbf{P}\left( W_{1}^{\left( N\right) }>n\right) =\frac{P_{\left( N\right) }^{n}\left( N,N\right) -\pi _{_{\left( N\right) }}\left( N\right) }{1-\pi _{_{\left( N\right) }}\left( N\right) }. \label{B40}$$ is a well defined complementary mass function of some rv $W_{1}^{\left( N\right) }$. $\left( ii\right) $$$\tau _{\mathbf{\pi }_{\left( N\right) },N}=\sum_{i=1}^{G_{N}}W_{i}^{\left( N\right) } \label{B4}$$ where $G_{N}\overset{d}{\sim }$geo$\left( \mathbf{\pi }_{_{\left( N\right) }}\left( N\right) \right) $ (viz $\mathbf{P}\left( G_{N}=j\right) =\pi _{_{\left( N\right) }}\left( N\right) \left( 1-\pi _{_{\left( N\right) }}\left( N\right) \right) ^{j},$ $j=0,1,...$), independent of $W_{i}^{\left( N\right) },$ $i\geq 1,$ an iid sequence with $W_{i}^{\left( N\right) }% \overset{d}{=}W_{1}^{\left( N\right) }.$ *Proof:* $\left( i\right) $ Because $P_{\left( N\right) }$ is SM as well, $P_{\left( N\right) }^{n}\left( N,N\right) \geq P_{\left( N\right) }^{n}\left( i,N\right) $ for all $i$ and $n$. Therefore, with $n_{2}>n_{1}$, $$P_{\left( N\right) }^{n_{2}}\left( N,N\right) =\sum_{i=1}^{N}P_{\left( N\right) }^{n_{2}-n_{1}}\left( N,i\right) P_{\left( N\right) }^{n_{1}}\left( i,N\right) \leq P_{\left( N\right) }^{n_{1}}\left( N,N\right) \sum_{i=1}^{N}P_{\left( N\right) }^{n_{2}-n_{1}}\left( N,i\right) =P_{\left( N\right) }^{n_{1}}\left( N,N\right) .$$ As a result, $\mathbf{P}\left( X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }=N\right) =\mathbf{e}_{N}^{\prime }P_{\left( N\right) }^{n}\mathbf{% e}_{N}=P_{\left( N\right) }^{n}\left( N,N\right) $ is non-increasing with $n$ so that the law of $W_{1}^{\left( N\right) }$ is well-defined. $\left( ii\right) $ $$\sum_{n\geq 0}z^{n}\mathbf{P}\left( W_{1}^{\left( N\right) }>n\right) =\frac{% 1-\mathbf{E}\left( z^{W_{1}^{\left( N\right) }}\right) }{1-z}=\frac{1}{1-\pi _{_{\left( N\right) }}\left( N\right) }\left( G_{N,N}\left( z\right) -\frac{% \pi _{_{\left( N\right) }}\left( N\right) }{1-z}\right)$$ Using (\[B2\]), we get $$\mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{\left( N\right) },N}}\right) =% \frac{1}{1+\frac{1-\pi _{_{\left( N\right) }}\left( N\right) }{\pi _{_{\left( N\right) }}\left( N\right) }\left( 1-\mathbf{E}\left( z^{W_{1}^{\left( N\right) }}\right) \right) } \label{B5}$$ which is the pgf of the geometric convolution $\sum_{i=1}^{G_{N}}W_{i}^{% \left( N\right) }$. $\Box $ Note $G_{N}=0$ entails $\tau _{\mathbf{\pi }_{\left( N\right) },N}=0$, an event with probability $\pi _{_{\left( N\right) }}\left( N\right) $. *Remark:* Stochastically monotone Markov chain have a real and simple second largest eigenvalue, [@Keilson]. Suppose $1=\lambda _{1}>\lambda _{2}>\left| \lambda _{3}\right| \geq ...\geq \left| \lambda _{N}\right| >0$ where $\lambda _{k}=\lambda _{k,\left( N\right) }$ are the $N-$dependent eigenvalues of $P_{\left( N\right) }$. Then, $$\forall i,j\in \left\{ 1,...,N\right\} \text{, }\forall n\in \Bbb{N}\text{, }% \exists c>0:\text{ }\left| P_{\left( N\right) }^{n}\left( i,j\right) -\pi _{_{\left( N\right) }}\left( j\right) \right| \leq c\lambda _{2,\left( N\right) }^{n}.$$ In particular, $\left| P_{\left( N\right) }^{n}\left( N,N\right) -\pi _{_{\left( N\right) }}\left( N\right) \right| \leq c\lambda _{2,\left( N\right) }^{n}$ and $P_{\left( N\right) }^{n}\left( N,N\right) $ is getting close to $\pi _{_{\left( N\right) }}\left( N\right) $ as $n$ gets large, useful for (\[B40\]). $\diamondsuit $ - **Quasi-stationary distribution (qsd).** An alternative point of view on $\tau _{\mathbf{\pi }_{0},N}$ and $\tau _{\mathbf{\pi }_{\left( N\right) },N}$ can also be seen from the classical theory of qsd’s, [@cmsm]. With $i\neq N$, let $\tau _{i,N}=\inf \left( n\geq 1:X_{n}^{\left( N\right) }=N\mid X_{0}^{\left( N\right) }=i\right) $. We have $$\mathbf{P}\left( \tau _{i,N}>1\right) =\mathbf{P}\left( X_{1}^{\left( N\right) }\leq N-1\mid X_{0}^{\left( N\right) }=i\right) =P^{c}\left( i,N-1\right) =F_{\left( N\right) }\left( N-1\right) ^{i}$$ $$\begin{aligned} \mathbf{P}\left( \tau _{i,N}>n+1\right) &=&\sum_{1\leq j<N}\mathbf{P}% _{i}\left( X_{n}^{\left( N\right) }=j,\tau _{i,N}>n+1\right) \\ &=&\sum_{1\leq j<N}F_{\left( N\right) }\left( N-1\right) ^{j}\mathbf{P}% _{i}\left( X_{n}^{\left( N\right) }=j,\tau _{i,N}>n\right)\end{aligned}$$ $$\mathbf{P}\left( \tau _{i,N}>n+1\mid \tau _{i,N}>n\right) =\sum_{1\leq j<N}F_{\left( N\right) }\left( N-1\right) ^{j}\mathbf{P}_{i}\left( X_{n}^{\left( N\right) }=j\mid \tau _{i,N}>n\right)$$ $$\underset{n\rightarrow \infty }{\rightarrow }\sum_{1\leq j<N}F_{\left( N\right) }\left( N-1\right) ^{j}\mu _{\left( N-1\right) }\left( j\right) =:% \mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }=:\rho _{N}.$$ In the latter displayed formula, $\mathbf{\mu }_{\left( N-1\right) }\left( \cdot \right) $ is the quasi-stationary limiting distribution of $\left\{ X_{n}^{\left( N\right) }\right\} $ when state $N$ has been removed and $% Z_{\left( N-1\right) }\overset{d}{\sim }\mathbf{\mu }_{\left( N-1\right) }$. Stated differently, $\mathbf{\mu }_{\left( N-1\right) }^{\prime }$ is the $% \left( N-1\right) -$dimensional left eigenvector (associated to the dominant eigenvalue $\rho _{N}<1$) of the substochastic matrix $P_{\left( N-1\right) } $ obtained while removing the $N-$th row and column $N-$th column of $% P_{\left( N\right) }$. We have used $\mathbf{P}_{i}\left( X_{n}^{\left( N\right) }=j\mid \tau _{i,N}>n\right) \underset{n\rightarrow \infty }{% \rightarrow }\mu _{\left( N-1\right) }\left( j\right) $, $j\in \left\{ 1,...,N-1\right\} .$ Consequently, With $\rho _{N}$ the value of the pgf of $Z_{\left( N-1\right) }$ evaluated at $F_{\left( N\right) }\left( N-1\right) $, independently of $i\in \left\{ 1,...,N-1\right\} $$$\underset{n\rightarrow \infty }{\lim }-\frac{1}{n}\log \mathbf{P}\left( \tau _{i,N}>n\right) =-\log \mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }=-\log \rho _{N}.$$ Equivalently, $$\rho _{N}=\mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }$$ is the rate of decay of $\mathbf{P}\left( \tau _{i,N}>n\right) $. Similarly, - With $\mathbf{\pi }_{0,0}$ defined by $\mathbf{\pi }_{0}^{\prime }=:\left( \mathbf{\pi }_{0,0}^{\prime },0\right) $, for any initial distribution $% \mathbf{\pi }_{0,0},$$$\underset{n\rightarrow \infty }{\lim }-\frac{1}{n}\log \mathbf{P}\left( \tau _{\mathbf{\pi }_{0,0},N}>n\right) =-\log \mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) },$$ giving the decay rate of $\mathbf{P}\left( \tau _{\mathbf{\pi }% _{0,0},N}>n\right) .$ - With $\mathbf{\pi }_{\left( N-1\right) }^{\prime }$ defined by $\mathbf{% \pi }_{\left( N\right) }^{\prime }=\left( \mathbf{\pi }_{\left( N-1\right) }^{\prime },\pi _{\left( N\right) }\left( N\right) \right) $, when starting from the invariant measure $$\underset{n\rightarrow \infty }{\lim }-\frac{1}{n}\log \mathbf{P}\left( \tau _{\mathbf{\pi }_{\left( N-1\right) }^{\prime },N}>n\right) =-\log \mathbf{E}% \left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }.$$ - Clearly also, when the initial distribution coincides with the quasi-stationary distribution: $\mathbf{\pi }_{0,0}=\mathbf{\mu }_{\left( N-1\right) },$ $$-\frac{1}{n}\log \mathbf{P}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}>n\right) =-\log \mathbf{E}\left( z^{Z_{\left( N-1\right) }}\right) \mid _{z=F_{\left( N\right) }\left( N-1\right) }=-\log \rho _{N}$$ for all $n.$ Letting $$\mathbf{\mu }_{\left( N-1\right) }^{\prime }P_{\left( N-1\right) }=\rho _{N}% \mathbf{\mu }_{\left( N-1\right) }^{^{\prime }}\text{ and }P_{\left( N-1\right) }\mathbf{\phi }_{\left( N-1\right) }=\rho _{N}\mathbf{\phi }% _{\left( N-1\right) },$$ be the $\left( N-1\right) $-dimensional left and right positive eigenvectors of $P_{\left( N-1\right) }$ chosen so as to satisfy: $\left| \mathbf{\mu }% _{\left( N-1\right) }\right| :=\sum_{j=1}^{N-1}\mu _{\left( N-1\right) }\left( j\right) =1$ and $\mathbf{\mu }_{\left( N-1\right) }^{\prime }% \mathbf{\phi }_{\left( N-1\right) }=1$, fixing the length $\left\| \mathbf{% \phi }_{\left( N-1\right) }\right\| _{2}^{1/2}$ of $\mathbf{\phi }_{\left( N-1\right) }$, then (by Perron-Frobenius theorem) $$\rho _{N}^{-n}P_{\left( N-1\right) }^{n}\rightarrow \mathbf{\phi }_{\left( N-1\right) }^{\prime }\mathbf{\mu }_{\left( N-1\right) }\text{ as }% n\rightarrow \infty .$$ Hence, with $\mathbf{\pi }_{0}^{\prime }=\left( \mathbf{\pi }_{0,0}^{\prime },0\right) ,$ $\left| \mathbf{\pi }_{0,0}\right| =1,$ and $\mathbf{\pi }% _{\left( N\right) }^{\prime }=\left( \mathbf{\pi }_{\left( N-1\right) }^{\prime },\pi _{\left( N\right) }\left( N\right) \right) $, $\left| \overline{\mathbf{\pi }}_{\left( N-1\right) }\right| <1$, and making use of $% \tau _{N,N}=0$$$\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) =\mathbf{\pi }% _{0,0}^{\prime }P_{\left( N-1\right) }^{n}\mathbf{1}\text{ and }\mathbf{P}% \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) =\mathbf{\pi }% _{\left( N-1\right) }^{\prime }P_{\left( N-1\right) }^{n}\mathbf{1<P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) \label{B6}$$ $$\mathbf{E}\left( \tau _{\mathbf{\pi }_{0},N}\right) =\mathbf{\pi }% _{0,0}^{\prime }\left( I-P_{\left( N-1\right) }\right) ^{-1}\mathbf{1}\text{ and }\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) =% \mathbf{\pi }_{\left( N-1\right) }^{\prime }\left( I-P_{\left( N-1\right) }\right) ^{-1}\mathbf{1<E}\left( \tau _{\mathbf{\pi }_{0},N}\right)$$ and $$\begin{aligned} \rho _{N}^{-n}\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) &\rightarrow &\mathbf{\pi }_{0,0}^{\prime }\mathbf{\phi }_{\left( N-1\right) }\text{ as }n\rightarrow \infty \\ \rho _{N}^{-n}\mathbf{P}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) &\rightarrow &\mathbf{\pi }_{\left( N-1\right) }^{\prime }% \mathbf{\phi }_{\left( N-1\right) }\text{ as }n\rightarrow \infty .\end{aligned}$$ Suppose $\mathbf{\pi }_{0}$ is such that $\pi _{0}\left( i\right) /\pi _{\left( N\right) }\left( i\right) $ decreases with $i$ and $\pi _{0}\left( N\right) =0$. Then $$\frac{\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) }{\mathbf{P}% \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) }\underset{% n\rightarrow \infty }{\rightarrow }\frac{\mathbf{\pi }_{0,0}^{\prime }% \mathbf{\phi }_{\left( N-1\right) }}{\mathbf{\pi }_{\left( N-1\right) }^{\prime }\mathbf{\phi }_{\left( N-1\right) }}\geq 1.$$ *Proof:* Due to the stochastic domination of $\tau _{\mathbf{\pi }% _{0},N}$ over $\tau _{\mathbf{\pi }_{\left( N\right) },N}$ stated in Proposition $11$, the positive sequence $$u_{n}:=\frac{\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) }{\mathbf{% P}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) }=\frac{\rho _{N}^{-n}\mathbf{P}\left( \tau _{\mathbf{\pi }_{0},N}>n\right) }{\rho _{N}^{-n}\mathbf{P}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}>n\right) }$$ is bounded below by $1$ ($u_{n}\geq 1$ for all $n$). The sequence $u_{n}$ is convergent with limit $u_{*}=\frac{\mathbf{\pi }_{0,0}^{\prime }\mathbf{\phi }_{\left( N-1\right) }}{\mathbf{\pi }_{\left( N-1\right) }^{\prime }\mathbf{% \phi }_{\left( N-1\right) }}$ and the limit obeys $u_{*}\geq 1$. We have $\rho _{N}^{-n}P_{\left( N-1\right) }^{n}\mathbf{1}\rightarrow \mathbf{\phi }_{\left( N-1\right) }$ as $n\rightarrow \infty $. The entries $% \phi _{\left( N-1\right) }\left( i\right) $ are decreasing with $i$, because it follows by induction that stochastic monotonicity of $P_{\left( N\right) } $ implies the one of $P_{\left( N-1\right) }^{n}$, so that $\mathbf{e}% _{i}^{\prime }P_{\left( N-1\right) }^{n}\mathbf{1}$ is decreasing with $i$. Because $\pi _{0,0}\left( i\right) /\pi _{\left( N-1\right) }\left( i\right) $ is decreasing with $i$, the initial probability mass assigned to states near the bottom state $\left\{ 1\right\} $ where $\mathbf{\phi }_{\left( N-1\right) }$ takes its largest values exceeds the one assigned by $\mathbf{% \pi }_{\left( N\right) }$. It is thus not that surprising that the numerator of $u_{*}$ exceeds its denominator. $\Box $ *Remark:* From (\[B6\]) $$\begin{aligned} \mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{0},N}}\right) &=&1-\left( 1-z\right) \mathbf{\pi }_{0,0}^{\prime }\left( I-zP_{\left( N-1\right) }\right) ^{-1}\mathbf{1}\text{ and } \\ \mathbf{E}\left( z^{\tau _{\mathbf{\pi }_{\left( N\right) },N}}\right) &=&1-\left( 1-z\right) \mathbf{\pi }_{\left( N-1\right) }^{\prime }\left( I-zP_{\left( N-1\right) }\right) ^{-1}\mathbf{1}\end{aligned}$$ Comparing the expression* *of the pgf of* *$\tau _{\mathbf{\pi }% _{\left( N\right) },N}$ in terms of the Green kernel of $P_{\left( N-1\right) }$ with (\[B5\]), yields and identity for $\mathbf{\pi }% _{\left( N-1\right) }^{\prime }\left( I-zP_{\left( N-1\right) }\right) ^{-1}% \mathbf{1}$. Note that in (\[B5\]), only the values of $\pi _{_{\left( N\right) }}\left( N\right) $ and $P_{\left( N\right) }^{n}\left( N,N\right) $ matter. Comparing the expression* *of the pgf of* *$\tau _{% \mathbf{\pi }_{0},N}$ in terms of the Green kernel of $P_{\left( N-1\right) } $ with (\[B3a\]), also yields and identity for $\mathbf{\pi }% _{0,0}^{\prime }\left( I-zP_{\left( N-1\right) }\right) ^{-1}\mathbf{1}$. $% \diamondsuit $ By the definition of quasi-stationary distributions, we had $$\mathbf{P}\left( X_{n}^{\left( N\right) }=j\mid \tau _{\mathbf{\pi }% _{0},N}>n\right) \underset{n\rightarrow \infty }{\rightarrow }\mu _{\left( N-1\right) }\left( j\right) \text{, }j\in \left\{ 1,...,N-1\right\} .$$ Because $P_{\left( N\right) }$ is stochastically monotone, Siegmund-Pollack theorem holds, stating [@PS] $$\mathbf{P}\left( X_{n}^{\left( N\right) }=j\mid \tau _{\mathbf{\pi }% _{0},N}>n\right) \underset{n,N\rightarrow \infty }{\rightarrow }\pi \left( j\right) \text{, }j\geq 1.$$ As $N$ gets large, the qsd $\mathbf{\mu }_{\left( N-1\right) }$ gets very close to $\mathbf{\pi }_{\left( N-1\right) }$. - **Asymptotic exponentiality.** - The rv $\tau _{\mathbf{\mu }_{\left( N-1\right) },N}$ is geometric with success parameter $1-\rho _{N}$, $$\mathbf{E}\left( z^{\tau _{\mathbf{\mu }_{\left( N-1\right) },N}}\right) =% \frac{z\left( 1-\rho _{N}\right) }{1-\rho _{N}z},$$  so with mean and variance $\mathbf{E}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) =1/\left( 1-\rho _{N}\right) $ and $\sigma ^{2}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) =\rho _{N}/\left( 1-\rho _{N}\right) ^{2}.$ Suppose $\rho _{N}\rightarrow 1$ as $% N\rightarrow \infty $. Then $\tau _{\mathbf{\pi }_{\left( N\right) },N}/% \mathbf{E}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) $becomes approximately exponential with mean $1$. We have $\mathbf{E}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) \rightarrow \infty $ while $\sigma \left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) /% \mathbf{E}\left( \tau _{\mathbf{\mu }_{\left( N-1\right) },N}\right) =\sqrt{% \rho _{N}}\rightarrow 1$, as* *$N\rightarrow \infty $*.* - Brown raised the question of asymptotic exponentiality of $\tau _{\mathbf{% \pi }_{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) $. If $\sigma ^{2}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) <\infty $, as a scaled geometric convolution, $\tau _{\mathbf{\pi }_{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) $ is approximately exponential if $\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) \rightarrow \infty $ while $% \sigma \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) /\mathbf{E}% \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) \rightarrow 1$, as* *$N\rightarrow \infty $ for the truncated Lamperti model with truncated target distribution $\mathbf{\pi }_{\left( N\right) }$. Error bounds can be obtained from the first two moments of $\tau _{\mathbf{\pi }% _{\left( N\right) },N}$ given by $$\begin{aligned} \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) &=&% \mathbf{E}\left( G_{N}\right) \mathbf{E}\left( W_{1}^{\left( N\right) }\right) =\frac{1-\pi _{_{\left( N\right) }}\left( N\right) }{\pi _{_{\left( N\right) }}\left( N\right) }\mathbf{E}\left( W_{1}^{\left( N\right) }\right) \\ &=&\frac{1}{\pi _{_{\left( N\right) }}\left( N\right) }\sum_{n\geq 0}\left( P_{\left( N\right) }^{n}\left( N,N\right) -\pi _{_{\left( N\right) }}\left( N\right) \right)\end{aligned}$$ $$\begin{aligned} \sigma ^{2}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) &=&% \mathbf{E}\left( G_{N}\right) \sigma ^{2}\left( W_{1}^{\left( N\right) }\right) +\left( \mathbf{E}\left( W_{1}^{\left( N\right) }\right) \right) ^{2}\sigma ^{2}\left( G,_{N}\right) \\ \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}^{2}\right) &=&2% \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) ^{2}+% \mathbf{E}\left( G_{N}\right) \mathbf{E}\left( \left( W_{1}^{\left( N\right) }\right) ^{2}\right) .\end{aligned}$$ The question of the approximation by an exponential distribution also arises for $\tau _{\mathbf{\pi }_{0},N}/\mathbf{E}\left( \tau _{\mathbf{\pi }% _{0},N}\right) $. In this direction indeed, ([@Brown], [@Brown2]) With $t\geq 0$ $$\begin{aligned} \sup_{t}\left| \mathbf{P}\left( \frac{\tau _{\mathbf{\pi }_{\left( N\right) },N}}{\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) }% >t\right) -e^{-t}\right| &\leq &\pi _{_{\left( N\right) }}\left( N\right) \frac{\mathbf{E}\left( \left( W_{1}^{\left( N\right) }\right) ^{2}\right) }{% \mathbf{E}\left( W_{1}^{\left( N\right) }\right) ^{2}} \\ &=&2\left( 1-\pi _{_{\left( N\right) }}\left( N\right) \right) \left[ \frac{% \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}^{2}\right) }{2% \mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) ^{2}}% -1\right] , \\ \sup_{t}\left| \mathbf{P}\left( \frac{\tau _{\mathbf{\pi }_{0},N}}{\mathbf{E}% \left( \tau _{\mathbf{\pi }_{0},N}\right) }>t\right) -e^{-t}\right| &\leq &% \frac{\mathbf{E}\left( T_{\left( N\right) }\right) }{\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) }+2\left( 1-\pi _{_{\left( N\right) }}\left( N\right) \right) \left[ \frac{\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}^{2}\right) }{2\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) ^{2}}-1\right]\end{aligned}$$ gives the sup-norm distance between respectively $\tau _{\mathbf{\pi }% _{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) $, $\tau _{\mathbf{\pi }_{0},N}/\mathbf{E}\left( \tau _{% \mathbf{\pi }_{0},N}\right) $ and an exponential rv with mean $1.$ This shows that if, as $N$ grows large, the mean and standard deviation of $% \tau _{\mathbf{\pi }_{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{% \pi }_{\left( N\right) },N}\right) $ behave like the one of an exponential distribution that is if $\sigma \left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) /\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) \rightarrow 1$, then $\mathbf{E}\left( \tau _{\mathbf{\pi }% _{\left( N\right) },N}^{2}\right) /\left( 2\mathbf{E}\left( \tau _{\mathbf{% \pi }_{\left( N\right) },N}\right) ^{2}\right) \rightarrow 1$ and the exponential approximation for the law of $\tau _{\mathbf{\pi }_{\left( N\right) },N}/\mathbf{E}\left( \tau _{\mathbf{\pi }_{\left( N\right) },N}\right) $ is valid. If in addition, as $N$ becomes large $$\mathbf{E}\left( T_{\left( N\right) }\right) /\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) \ll 2\left( 1-\pi _{_{\left( N\right) }}\left( N\right) \right) \left[ \frac{\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}^{2}\right) }{2\mathbf{E}\left( \tau _{% \mathbf{\pi }_{\left( N\right) },N}\right) ^{2}}-1\right] ,$$ then the same holds true for the law of $\tau _{\mathbf{\pi }_{0},N}/\mathbf{% E}\left( \tau _{\mathbf{\pi }_{0},N}\right) $. - **Time reversal.** Consider the time-reversed** **version $% X_{n,\left( N\right) }^{\leftarrow }$ of the truncated Lamperti chain, so with one-step transition matrix $$\overleftarrow{P}_{\left( N\right) }=D_{\mathbf{\pi }_{\left( N\right) }}^{-1}P_{\left( N\right) }^{\prime }D_{\mathbf{\pi }_{\left( N\right) }}.$$ Its time-reversed transition matrix being $P_{\left( N\right) }$ which is in particular stochastically monotone, the Brown theory for hitting times applies to the time-reversed process as well (see [@Brown]), with $% \overleftarrow{\tau }_{\mathbf{\pi }_{0},N}$ and $\overleftarrow{\tau }_{% \mathbf{\pi }_{\left( N\right) },N}$ standing for the hitting times of the time-reversed chain. The time-reversed process $X_{n,\left( N\right) }^{\leftarrow }$ thus constructed is a truncated version of the process defined from (\[TR\]). **Acknowledgments:** The authors are indebted for support of the CMM-Basal Conicyt project AFB170001 and I.E.A. Cergy. T. H. acknowledges partial support from the labex MME-DII (Modèles Mathématiques et Économiques de la Dynamique, de l’ Incertitude et des Interactions), ANR11-LBX-0023-01. This work also benefited from the support of the Chair “Modélisation Mathématique et Biodiversité” of Veolia-Ecole Polytechnique-MNHN-Fondation X. [99]{} Brown, M. (1990). Consequences of Monotonicity for Markov Transition Functions. City College, CUNY Report No MB89-03, unpublished manuscript. Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. The Annals of Probability. Vol. 18, N${{}^{\circ }}$ 3, 1388-1402. Collet, P., Martínez, S., San Martín, J. *Quasi-stationary distributions*. Probability and Its Applications, Springer, Ney York, (2013). Comtet, L. *Analyse combinatoire.* Tomes $1$ et $2$. Presses Universitaires de France, Paris, 1970. Embrechts, P., Klüppelberg, C., Mikosch, T. *Modelling Extremal Events for Insurance and Finance.* Stochastic Modelling and Applied Probability, Springer, (1997). Flajolet, P., Odlyzko, A. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3(2), 216-240. Foster, F. G. (1953). On the stochastic matrices associated with certain queuing processes. Ann. Math. Statistics 24, 355-360. Grassmann, W.K. and Heyman, D.P. (1990). Equilibrium Distribution of Block-Structured Markov Chains with Repeating Rows. Journal of Applied Probability, Vol. 27, No. 3, 557-576. Gupta, P. L., Gupta, R. C. and Tripathi, R. C. (1997). On the monotonic properties of discrete failure rates. Journal of Statistical Planning and Inference, 65, 255-268. Harris, T. E. The existence of stationary measures for certain Markov processes. (1956). Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. II, pp. 113-124. University of California Press, Berkeley and Los Angeles. Harris, T. E. (1957). Transient Markov chains with stationary measures. Proc. Amer. math. Soc. 8, 937-942. Harris, T. E. *The theory of branching processes.* Die Grundlehren der Mathematischen Wissenschaften, Bd. 119 Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, N.J. (1963). Keilson, J., Kester, A. (1977). Monotone matrices and monotone Markov processes. Stochastic Processes and their applications. 5, 231-241. Lamperti, J. (1970). Maximal Branching Processes and ’Long-Range Percolation’. Journal of Applied Probability, Vol. 7, No. 1, pp. 89-98. Lamperti, J. (1972). Remarks on maximal branching processes. Teor. Verojatnost. i Primenen. 17, 46-54. Lorek, P. (2007) Speed of convergence to stationarity for stochastically monotone Markov chains. Doctoral dissertation. Mathematical Institute, University of Wrowclaw. Pitman, J. and Tang, W. Tree formulas, mean first passage times and Kemenys constant of a Markov chain, arXiv preprint arXiv:1603.09017 (2016). Pollak, M., Siegmund, D. (1986). Convergence of quasistationary to stationary distributions for stochastically monotone Markov processes. J. Appl. Probab. 23, No. 1, 215-220. Sibuya, M. (1979). Generalized hypergeometric, digamma and trigamma distributions. Annals of the Institute of Statistical Mathematics, 31, 373-390. Steutel, F. W., van Harn, K. *Infinite divisibility of probability distributions on the real line.* Monographs and Textbooks in Pure and Applied Mathematics, 259. Marcel Dekker, Inc., New York, 2004. Zhao, Y.Q. and Liu, D. (1996). The Censored Markov Chain and the Best Augmentation. Journal of Applied Probability, Vol. 33, No. 3, pp. 623-629.

ArXiv

--- author: - 'K. Nandra[^1]' - 'P.M. O’Neill' - 'I.M. George' - 'J.N. Reeves' - 'T.J. Turner' title: 'An XMM-Newton survey of broad iron lines in AGN' --- Introduction ============ Observations with  showed complex, broad emission from iron to be very common in Seyfert galaxies (Nandra et al. 1997). These lines can be interpreted as emission from a relativistic accretion disk, in which case they represent a powerful probe of the strong gravity regime around black holes (Fabian et al. 1989; Stella 1990). The most celebrated case is MCG-6-30-15, where the broad, skewed line seen with  is of high signal-to-noise ratio, and the disk line interpretation is apparently robust (Tanaka et al. 1995; Fabian et al. 1995). Several other high quality profiles from [*ASCA*]{} also showed broad, relativistic lines (e.g. George et al. 1998; Nandra et al. 1999; Done et al. 2000). Since the launch of , it has been possible to obtain even higher quality data on these broad emission lines. Early results confirmed relativistic emission in some cases, including MCG-6-30-15 (e.g. Wilms et a. 2001; Fabian et al. 2002; Vaughan & Fabian 2004), but in others no broad line was detected (e.g. Gondoin et al. 2001; Pounds et al. 2003; Bianchi et al. 2004). In yet others, complexity has been observed around iron-$K$, but the interpretation as relativistic disk emission has been challenged. One specific suggestion is that absorption by a high column, high ionization warm absorber can mimic the “red wing” characteristic of an accretion disk line (Reeves et al. 2004). The absence of a comprehensive and systematic survey of the X-ray spectra of Seyferts observed by  prevents firm conclusions being drawn as to the prevalence of broad iron lines in AGN and the robustness of their interpretation. Here we present preliminary results from such a study, the full results of which will be presented in a forthcoming paper (Nandra et al. 2006, in preparation). The sample and data analysis ============================ Our sample is culled from pointed AGN observations in the  archive. We examine only local AGN($z<0.05$) and exclude Seyfert 2 galaxies and radio loud objects. Furthermore we choose only the objects with the highest number of counts in the 2-10 keV band, to maximise the signal-to-noise ratio around the iron line. The sample reported here consists of 41 observations of 30 objects. An important feature of our work is that we have performed a well-defined, uniform analysis, with conservative selection cirteria and using the latest available calibrations. The techniques are fully decribed in O’Neill et al. (2006, in preparation), but compared to much of the previous work the improvements include: a) consistent definition of source and background regions for each observation b) well defined and conservative background rejection c) precise definition of good-time intervals d) standardised spectral grouping related to the instrumental resolution. We restrict our analysis to the pn instrument. For observations with significant pileup we use only the pattern 0 events. Spectral fits are undertaken in the 2.5-10 keV range only, to minimize complications due to absorption and soft excess emission, and avoid the instrumental calibration feature around 2.2 keV. ------------ ---------- ---------------- ----------------- -------------- Fraction Energy Width EW (keV) (keV) (eV) (1) (2) (3) (4) [*ASCA*]{} 77% $6.34\pm 0.04$ $0.43 \pm 0.12$ $160 \pm 30$ [*XMM*]{} 73% $6.32\pm 0.05$ $0.36\pm 0.04$ $108\pm 12$ ------------ ---------- ---------------- ----------------- -------------- : Comparison between mean parameters for broad lines determined by  (Nandra et al. 1997) and  (this work). Note that that  fits did not account for a distant narrow component of the Fe K$\alpha$, nor did they include a warm absorber. The fraction of objects in which the F-test indicates a 99% improvement is given, along with the mean Energy, Gaussian $\sigma$ and equivalent width. \[tab:mpars\] Results ======= Base model ---------- While we have excluded the most heavily obscured objects (Seyfert 2s) from our sample, there remains a possibility that absorption can have a significant effect even on the spectra above 2.5 keV. We account for this by fitting an XSTAR (Kallman et al. 2004) photoionization model to the spectra,excluding the iron band (4.5-7.5 keV). Where the the fit improves significantly at 95% confidence, according to the F-test, this XSTAR component is included in all subsequent fits, with free $N_{\rm H}$ and ionization parameter. It is now also known that many AGN exhibit narrow cores to their iron K$\alpha$ lines (e.g. Yaqoob & Padhmanhaban 2004). These are thought to arise from very distant material such as the torus (e.g. Krolik & Kallman 1987; Awaki et al. 1991). If so they will be accompanied by continuum Compton reflection. We therefore include in all fits below a neutral Compton reflection component appropriate for a slab geometry (Magdziarz & Zdziarski 1995), with accompanying Fe K$\alpha$, Fe K$\beta$ and Ni K$\alpha$, line emission (George & Fabian 1991) and a Compton shoulder (Matt 2002). The emission lines and reflection are all incorporated in a single model with solar abundances. We assume an inclination of $60^{\circ}$ for the slab and hence the reflection is characterized by a single parameter, $R = \Omega/2\pi$, where $\Omega$ is the solid angle subtended by the slab at the illuminating source. ![Characteristic emission radius for the relativistic iron K$\alpha$ lines versus disk inclination. []{data-label="fig:rbreak"}](rbreak_inc.ps){width="80mm"} Simple parameterization of the broad emission --------------------------------------------- To provide a simple, model-independent characterization of further complexity in the iron band, we have added a broad Gaussian to the fits described above. A significant improvement to the fit was found in 30 of the 41 observations, and 22 of the 30 objects. Clearly, complexity at iron K$\alpha$ is extremely common in Seyferts. A comparison between the mean parameters of the broad Gaussian fits to the  data (Nandra et al. 1997) and our new  sample, is given in Table \[tab:mpars\]. There is remarkable agreement in all cases, with the exception that the line equivalent widths in the  sample are about $50$ % higher. This difference can be attributed to the fact that the narrow line cores were deconvolved in the  fits, but not with . The energies of the broad lines seen with  clearly indicate that they are associated with iron, as they are very close to the expected energy, but there is some evidence that the typical energy is redshifted compared to the neutral value. The lines are usually quite broad, with $<\sigma>=0.36$ keV or 40,000 km s$^{-1}$ FWHM. Significant dispersion is seen in all the measured quantities, however, which confirms the result from  that there is a wide variety of line profiles, and takes this further in that the variation from object-to-object cannot be attributed solely to varying relative contributions of the narrow core and broader emission. It should also be noted that in 5 of the fits, the width of the “broad" gaussian component is $<10,000$ km s$^{-1}$. These lines could plausibly arise from the optical broad line region (BLR), rather than the inner disk. Disk line models ---------------- We have tested explicitly whether the complex line shapes seen in the  spectra can be accounted for with a relativistic accretion disk. We do this by adding an additional, neutral reflection component with Fe and Ni line emission as above, but this time apply relativistic blurring (Laor 1991; Fabian et al. 2002). Rather than leave all the parameters free, we initially chose to fix the inner and outer radii at $R_{\rm i}=6 R_{\rm g}$ and $R_{\rm o}=400 R_{\rm g}$. We adopt an emissivity law appropriate for a point source above a slab in a Newtonian geometry, which can be approximated as a broken power law. The adopted emissivity depends on $R^{-q}$, with $q=0$ within and $q=3$ outside some break radius $R_{\rm br}$. This represents the characteristic radius where the majority of the line emission originates, so can be used to assess whether relativistic effects are important. The inclination and reflection fraction are left as free parameters too. The relativistically blurred model improves the fits significantly in $\sim 75$% of the observations and indeed gives markedly better fits than a Gaussian in several cases. The characteristic emission radius ($R_{\rm br}$) is plotted against the inclination in Fig. \[fig:rbreak\]. The bottom left part of this diagram is where we expect “classic” disk lines to occur. Here the emission is concentrated in the innermost regions ($<50$ R$_{\rm g}$) and the inclination is relatively low, such that much of the emission is redshifted. The upper left portion is where we expect weak and very broad lines from highly inclined disks. It is sparsely populated, which is expected as such lines are difficult to detect. The upper right portion shows several strong disk lines with apparently high inclinations but at relatively large radii. This indicates that the lines are broad but predominantly towards the blue rather than the red. These are likely candidates for a highly ionized disk, which is in reality at lower inclination than inferred in fits which assume the disk is neutral. Finally, at the bottom right of the diagram we see emission at low inclination and large radius. In these objects the lines will be relatively narrow and not strongly shifted. For these, there is no requirement for the line to arise in the inner accretion disk and they may come from more distant material, such as the optical BLR. Using this model, we can also assess the evidence for black hole spin. A simple test is to repeat the fits using an inner radius of $1.235 R_{\rm g}$, appropriate for a Kerr Black hole with $a/M=0.998$, as opposed to the Schwarzschild value of $6 R_{\rm g}$. Only two of the spectra showed an appreciable improvement with such a model. In both, NGC 3783 and NGC 4151, there is complex absorption which strongly affects the spectrum above 2.5 keV (Reeves et al. 2004; Schurch et al. 2004). We therefore consider the evidence for maximal Kerr black holes to be tentative, leaving it an open question as to whether black holes in AGN are generally rotating. Comparison with alternative models ---------------------------------- Some recent studies have suggested alternatives to the relativistic disk model for broad iron lines in AGN. A number of objects show no evidence for broad emission at all, including some in our sample. In others, it may be possible to model the “red wing” as a high ionization warm absorber (Reeves et al. 2004), and the “blue wing” with blends of narrow lines. To test this, we have fitted a model comprising a high ionization warm absorber (in [*addition*]{} lower ionization gas already included), with three narrow emission lines, two fixed at the energies appropriate to helium and hydrogen–like iron and another intermediate (6.4-6.7 keV) line with free energy. Once again neutral, unblurred reflection is also included to account for any narrow emission at 6.4 keV. ![Difference in $\chi^{2}$ between the relativistic disk line model and an alternative model comprising a high ionization warm absorber, and a blend of narrow lines. All fits include both line and continuum from a distant neutral reflector and a soft X-ray warm absorber where needed. The disk line model has 3 fewer free parameters than the alternative, but provides a dramatically better fit in a large number of cases (see Fig. \[fig:dream\])[]{data-label="fig:dchi"}](xmm_delchi_hist.ps){width="80mm"} A comparison of the $\chi^{2}$ values is given in Fig. \[fig:dchi\]. The alternative model has 3 more free parameters than the disk line model, but provides a substantially worse fit in a large number of objects. In a few cases the alternative model fits a little better, but not substantially so considering the larger number of free parameters. From the consideration of this alternative model, and the results from Fig \[fig:rbreak\], we can define a sample of robust relativistic lines for which disk models both indicate a small characteristic radius, and fit much better than the alternative. There are 11 spectra of 9 objects satisfying the criteria that $R_{br}<20$ $R_{\rm g}$ and $\Delta\chi^{2}>10$ for the relativistic model compared to the alternative model (despite having 3 [*fewer*]{} free parameters). The line profiles are shown in Fig. \[fig:dream\]. Discussion and conclusions ========================== Our systematic  survey should serve to clear up some of the controversy about how often broad emission lines from an accretion disk can be claimed robustly in AGN. For Seyferts, at least, complexity at iron K$\alpha$ is seen in about 3/4 of objects and this complexity is always interpretable in terms of an accretion disk model. In about 1/3 of our sample, that interpretation is clearly preferred over competing models. In a few cases with high signal-to-noise ratio the relativistic emission appears to be absent, but great caution needs to be exercised before this can be concluded definitively, as even with  very good statistics are required (Guainazzi et al., this volume). In cases where broad emission appears to be absent, the disk may simply be highly inclined, such that the line is very broad and weak. Alternatively, the inner disk may be hot and/or highly ionized (Nayakshin 2000), which can also account for cases where broad emission is present, but indicative of a relative large characteristic radius ($\sim 100$ $R{\rm g}$). Alternatively, the lack of broad emission seen in a given observation may be due to line profile variability (Longinotti et al. 2004). Perhaps surprisingly, we have not yet found any strong evidence for black hole spin. This contrasts with some previous studies indicating maximally rotating holes in MCG-6-30-15 (Wilms et al. 2001) and some black hole binaries (e.g. Miller et al. 2002). This is probably due to our conservative approach in consideration of distant reflection and complex absorption. On the other hand, our observations provide no evidence [*against*]{} rapidly rotating black holes in AGN and while it has been pointed out in several previous studies that complex absorption can mimic very broad lines (e.g. Done & Gierlinski 2006), it is important to bear in mind that the converse is also true. Our main conclusion, however, is that the accretion disk interpretation for broad iron K$\alpha$ lines in AGN appears to be robust. The implication is that the potential for X-ray observations, particularly with [*XEUS*]{} and [*Con–X*]{}, to reveal new information about the innermost regions of accreting black holes may well be realised. We thank Tim Kallman for help with XSTAR; PPARC and the Leverhulme Trust for financial support and gratefully acknowledge those who built and operate the  satellite. 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ArXiv

--- abstract: 'Counter-rotating vortices in miscible two-component Bose-Einstein condensates, in which superflows counter-rotate between the two components around the overlapped vortex cores, are studied theoretically in a pancake-shaped potential. In a linear stability analysis with the Bogoliubov–de Gennes model, we show that counter-rotating vortices are dynamically unstable against splitting into multiple vortices. The instability shows characteristic behaviors as a result of countersuperflow instability, which causes relaxation of relative flows between the two components in binary condensates. The characteristic behaviors are completely different from those of multiquantum vortices in single-component Bose-Einstein condensates; the number of vortices generated by the instability can become larger than the initial winding number of the counter-rotating vortex. We also investigate the nonlinear dynamics of the instability by numerically solving the Gross-Pitaevskii equations. The nonlinear dynamics drastically changes when the winding number of counter-rotating vortices becomes larger, which lead to nucleation of vortex pairs outside of the vortex core. The instability eventually develops into turbulence after the relaxation of the relative rotation between the two components.' author: - 'Shungo Ishino $^1$, Makoto Tsubota $^{1,2}$, and Hiromitsu Takeuchi $^1$' title: 'Counter-rotating vortices in miscible two-component Bose-Einstein condensates' --- Introduction ============ Quantized vortices are one of the remarkable consequences of Bose-Einstein condensation and superfluidity in quantum fluids and are found in superfluids $^4$He and $^3$He and Bose-Einstein condensates (BECs) of atomic gas. In the context of hydrodynamics, quantized vortices often appear and play an important role in the understanding of various phenomena, such as the rotating of superfluid He [@Donnelly_Book1991], thermal counterflow in superfluid $^4$He [@Vinen1957; @Adachi_PRB2010], and quantum turbulence [@Adachi_PRB2010; @Kobayashi_PRL2005; @Kobayashi_PRA2007]. Among the many types of physics of quantized vortices, multiquantum vortices, whose winding number is larger than unity, are an interesting and important subject. Multiquantum vortices have never been realized in superfluid $^4$He except in some transient states [@Karn_PRB1980]. This is chiefly because an $L$-charged vortex, whose winding number is $L$, is unstable and easily splits into $L$ single-quantum vortices, whose winding number is unity. Atomic BECs form another subject in the study of multiquantum vortices. In experiments, optical technology enables us to make a multiquantum vortex and visualize the vortex directly [@Matthews_PRL1999; @Shibayama_JPB2011]. Furthermore, because of the weak interaction between the atoms, it is relatively easy to perform a theoretical analysis by using the Gross-Pitaevskii (GP) model and the Bogoliubov–de Gennes (BdG) model [@Pethick_book]. Thus, the splitting of multiquantum vortices has been experimentally observed [@Isoshima_PRL2007; @Shin_PRL2004] and theoretically studied [@Mottonen_PRA2003; @Kawaguchi_PRA2004; @Huhtamaki_PRL2006; @Isoshima_PRL2007]. An $L$-charged vortex essentially has unstable modes with $l(\leqq L)$-fold symmetry and splits into $L$ single-quantum vortices. Some studies also discuss multiquantum vortices in two-component BECs [@Skryabin_PRA2000; @Park_PRAR2004; @Brtka_PRA2010; @Wen_PRA2013]. Such vortex splitting instabilities are understood as a dynamic instability in the BdG model [@Pethick_book]. Hydrodynamic instability has been actively studied, independent of these topics, in two-component BECs, for example, the Kelvin-Helmholtz instability [@Takeuchi_PRB2010; @Suzuki_PRA2010] and the Rayleigh-Taylor instability [@Sasaki_PRA2009; @Gautam_PRA2010]. As another type of hydrodynamic instability, we previously studied instability in uniform countersuperflow, miscible two-component BECs with relative superfluid velocity between the two components [@Takeuchi_PRL2010; @Ishino_PRA2011]. It is well known that uniform, miscible two-component BECs are stable when the intraspecies interaction coefficients $g_{11}$ and $g_{22}$ and interspecies interaction coefficient $g_{12}$ satisfy the condition $g_{11}g_{22}>g_{12}^2$ [@Pethick_book]. However, when the relative superfluid velocity between the two components exceeds a critical value, the system becomes dynamically unstable, causing a characteristic density pattern and vortex nucleation [@Takeuchi_PRL2010; @Ishino_PRA2011]. The nucleated vortices are stretched so as to reduce the relative superflows between the two components. Then, reconnection frequently occurs between the vortices, leading to binary quantum turbulence. CSI has been recently observed in experiments [@Hamner_PRL2011]. In this paper, we discuss counter-rotating (CR) vortices in miscible two-component BECs trapped by a harmonic oscillator potential. We consider that the first and second components simultaneously have an $L$-charged vortex and a $-L$-charged one at the center of the BECs, respectively. The winding numbers of the two vortices have the same magnitude but opposite sign. Therefore, the two BECs relatively rotate. For the following discussion, we denote an $L$-charged vortex in the first and second components of the two-component BEC as $(L,0)$-vortex and $(0,L)$-vortex, respectively. Thus, a CR vortex that is overlapped by an $L$-charged vortex and a $-L$-charged vortex is written as an $(L,-L)$-vortex. The BECs with a CR vortex are expected to be closely related to countersuperflow because the BECs with a CR vortex and countersuperflow have similarity, such as relative motion. Counter-rotating binary BECs have been theoretically studied in a toroidal trap [@Suzuki_PRA2010; @Abad_arXiv2013]. Our work focuses on natures of counter-rotating systems as a vortex in a harmonic oscillator potential. This paper is organized as follows. In Sec. II, we formulate a system of two-component BECs with a CR vortex in the GP model at zero temperature. Section III is devoted to a linear stability analysis of CR vortices in the BdG model. We show that the instability of CR vortices is characterized by countersuperflow instability (CSI) by numerically solving the BdG equations. In Sec. IV, we reveal the nonlinear development of the instability of CR vortices by numerically solving the time-dependent GP equations. The results are summarized in Sec. V. Formulation {#sec:formulations} =========== We consider miscible two-component BECs described by the condensate wave functions $\Psi _j({\bm r},t)=\sqrt{n_j({\bm r},t)}e^{i\phi _j({\bm r},t)}$ in the mean-field approximation at zero temperature, where the index $j$ refers to each component ($j=1,2$). The wave functions are governed by the coupled GP equations [@Pethick_book] $$\begin{aligned} i \hbar \frac{\partial}{\partial t} \Psi _j = \left(-\frac{\hbar^2}{2m_j}{\bm \nabla}^2+V_j({\bm r})+\sum_{k=1,2} g_{jk}|\Psi _k|^2\right)\Psi _j, \label{eq:GP} \end{aligned}$$ where $m_j$ is the mass of the $j$th component and the coefficient $g_{jk}=2\pi\hbar^2a_{jk}/m_{jk}$ represents the atomic interaction with $m_{jk}^{-1}=m_{j}^{-1}+m_{k}^{-1}$ and the $s$-wave scattering length $a_{jk}$ between the $j$th and $k$th components. Our analysis supposes the conditions $g_{11}g_{22}>g_{12}^2$ and $g_{jj}>0$, indicating that the static, miscible two-component BECs are stable [@Pethick_book]. For simplicity, we set the mass and the $s$-wave scattering length of the two components to the same value, namely, $m_1=m_2=m$, $a_{11}=a_{22}=a$, and $g_{11}=g_{22}=g$. The particle numbers of the two components are also the same: $N_1=N_2=N$. The external trapping potential is a harmonic oscillator potential, given by $V_j({\bm r})=\frac{1}{2}m(\omega_r^2r^2+\omega_z^2z^2)$ with $r^2=x^2+y^2$. The BECs may be treated as a two-dimensional system when we use the “pancake" trap geometry with $\omega_r \ll \omega_z$. Therefore, we separate the degrees of freedom of the wave functions as $\Psi_j(x,y,z,t)=\psi_j(x,y,t)\phi_j(z)$. When the potential energy in the $z$ direction is sufficiently larger than the interaction energy, $\phi_j(z)$ is approximated by the one-particle ground-state wave function in a harmonic oscillator potential: $\phi_j(z)=\left(N_z/\sqrt{\pi}a_{hz}\right)^{1/2}\exp\left(-{z^2}/{2a_{hz}^2}\right)$, where $N_z$ is a normalization constant and $a_{hz}=\sqrt{\hbar/m\omega_z}$. Then the GP equations are reduced to the dimensionless form $$\begin{aligned} i \frac{\partial}{\partial t} \psi _j = \left(-\frac{1}{2}{\bm \nabla}_r^2+\frac{1}{2}r^2+\sum_{k=1,2} C_{jk}|\psi _k|^2\right)\psi _j \label{eq:2DGP} ,\end{aligned}$$ with ${\bm \nabla}_r^2=\partial_r^2+r^{-1}\partial_r-r^{-2}\partial_\theta^2$, where the length, time, and wave functions are scaled as $$\begin{aligned} x=a_{hr}\tilde{x},\ t=\frac{\tilde{t}}{\omega_r},\ \psi_j=\frac{\sqrt{N_{2D}}}{a_{hr}}\tilde{\psi}_j. \label{eq:scale} \end{aligned}$$ Here, $a_{hr}=\sqrt{\hbar/m\omega_r}$ and the two-dimensional particle number $N_{2D}$ relates to $N_z$ through $N=N_{2D}N_z$. The tildes in Eq. (\[eq:2DGP\]) are omitted for simplicity. The nondimensional interaction coefficient $$\begin{aligned} C_{jk}=\frac{2\sqrt{2\pi}Na_{jk}}{a_{hz}} \label{eq:Cjk} \end{aligned}$$ includes all parameters of this system. Because the parameters of the two components are the same, the intraspecies interaction coefficients are the same: $C_{11}=C_{22}=C$. The chemical potential $\mu_{\rm b}$ and the healing length $\xi_{\rm b}$ of this system are calculated in the Thomas-Fermi approximation [@Pethick_book] as $$\begin{aligned} \mu_{\rm b} &= \frac{15^{2/5}}{2}\left(\frac{N(a+a_{12})}{{\bar a}_h}\right)^{2/5}\hbar{\bar \omega}, \\ \xi_{\rm b}&=\frac{\hbar}{\sqrt{m(g+g_{12})n_{\rm b}}}, \end{aligned}$$ where ${\bar a}_h\equiv(a_{hr}^2a_{hz})^{1/3}$, ${\bar \omega}\equiv(\omega_r^2\omega_z)^{1/3}$, and $n_{\rm b}$ is the density in bulk. The stationary state of two BECs that have an $(L,-L)$-vortex at the center is described by the cylindrically symmetric functions $$\begin{aligned} \psi_1^0&=\sqrt{n^0_{1}(r)}e^{i\left( L\theta-\mu_{1}t/\hbar\right)}, \label{eq:stationary1} \\ \psi_2^0&=\sqrt{n^0_{2}(r)}e^{i\left(-L\theta-\mu_{2}t/\hbar\right)}, \label{eq:stationary2} \end{aligned}$$ where $\theta$ is the polar angle and $\mu_j$ is the chemical potential of the $j$th component. The square of the amplitude $n^0_{j}(r)$ gives the radial density profile. The amplitudes are obtained through the imaginary time propagation method by inserting Eqs. (\[eq:stationary1\]) and (\[eq:stationary2\]) into the GP equations. The densities of the two BECs must vanish at $r=0$ and $r=\infty$ because of the vortices and the trapping potential. Note that $\mu_1=\mu_2=\mu$ and the densities have the same function, $n^0_{1}(r)=n^0_{2}(r)=n^0(r)$, because the two BECs have the same parameters and the same winding number magnitude. In this paper, the nondimensional interaction parameters are $C\simeq 5500$ and $C_{12}=0.9C$, which causes repulsive interspecies interaction. For example, the parameters are realized in a system with $10^5$ atoms of $^{87}$Rb for each component. The $s$-wave scattering lengths are $a=5.3$ nm and $a_{12}=0.9a$, and also the trapping frequencies are $\omega_r=2\pi\times 5$ Hz and $\omega_z=2\pi\times 500$ Hz. Then the healing length is $\xi_{\rm b} = 0.54\; {\mu\rm m}$, where the density $n_{\rm b}$ in bulk is estimated by the Thomas-Fermi approximation without vortices. We investigate the cases of small and large winding numbers of CR vortices. Linear Stability {#sec:linear} ================ Here, we study the linear stability of the CR vortices in the BdG model. After the formulation of the BdG equations, we first discuss the dynamic instability of $(L,-L)$-vortices for small $L$. Then we investigate the instability for large $L$ and its relation to CSI. Bogoliubov–de Gennes analysis {#sec:bdg} ----------------------------- We consider a collective excitation above the stationary state written by Eqs. (\[eq:stationary1\]) and (\[eq:stationary2\]) as $\psi_j=\psi_j^0+\delta\psi_j$. Because the system has rotational symmetry, we write the excitation wave functions $\delta\psi _{j}$ with $$\begin{aligned} \delta\psi_1&=\ \ e^{i\left( L\theta-\mu t/\hbar \right)}\{u_1(r)e^{i\left( l\theta-\omega t \right)}-v^*_1(r)e^{-i\left( l\theta-\omega^* t \right)}\}, \nonumber\\ \label{eq:delta1} \\ \delta\psi_2&=e^{i\left(-L\theta-\mu t/\hbar \right)}\{u_2(r)e^{i\left( l\theta-\omega t \right)}-v^*_2(r)e^{-i\left( l\theta-\omega^* t \right)}\}, \nonumber\\ \label{eq:delta2} \end{aligned}$$ where $l$ is the angular momentum quantum number. By inserting $\psi_j=\psi_j^0+\delta\psi_j$ to linearize the GP equations with respect to $\delta\psi _{j}$, we obtain the BdG equations. In matrix notation, these are $$\begin{aligned} \sigma {\cal M}{\it W}=\omega{\it W}, \label{eq:bdg} \end{aligned}$$ where $$\begin{aligned} {\cal M}=\left( \begin{array}{cccc} h_{+} & Cn^0(r) & C_{12}n^0(r) & C_{12}n^0(r) \\ Cn^0(r) & h_{-} & C_{12}n^0(r) & C_{12}n^0(r) \\ C_{12}n^0(r) & C_{12}n^0(r) & h_{-} & Cn^0(r) \\ C_{12}n^0(r) & C_{12}n^0(r) & Cn^0(r) & h_{+} \\ \end{array} \right) ,\end{aligned}$$ with $$\begin{aligned} h_{\pm}=&-\frac{1}{2}\left\{\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}-\frac{(l\pm L)^2}{r^2}\right\} \nonumber \\ &+\frac{1}{2}r^2+2Cn^0(r)+C_{12}n^0(r)-\mu, \label{eq:h} \end{aligned}$$ $$\begin{aligned} {\it W}=\left\{u_1(r),-v_1(r),u_2(r),-v_2(r)\right\}^T, \label{eq:w} \end{aligned}$$ and $\sigma={\rm diag}(1,-1,1,-1)$. Here, the parameters are scaled as Eqs. (\[eq:scale\]) and the tildes are omitted. Because the operator $\sigma {\cal M}$ in the BdG equations \[Eq. (\[eq:bdg\])\] is non-Hermitian, the frequency $\omega$ may have an imaginary part. The linear stability of the system is investigated by numerically diagonalizing Eq. (\[eq:bdg\]). The system is dynamically unstable when the frequency of excitations has an imaginary part ${\rm Im}\, \omega >0$ because the excitations are amplified exponentially with time. Here, we do not take into account the thermodynamic instability by neglecting energy dissipation, which causes spontaneous amplification of the collective modes with negative energy. Because a solution $(\omega,\ l,\ u_j, v_j)$ has its conjugate solution $(-\omega^*,\ -l,\ u_j^*, v_j^*)$, we present here only the results for $l \geqq 0$ without loss of generality. Figure \[fig:bdg2\](a) shows the imaginary part of the frequencies of unstable modes with ${\rm Im}\, \omega >0$ for the cases of small $L$: $(1,-1)$-, $(2,-2)$-, and $(3,-3)$-vortices. Although some unstable modes appear for each value of $l$, we show the largest imaginary part among them. First, we explain the case of $L=1$. This problem is connected to the interaction between vortices in different components in miscible two-component BECs. The interaction between vortices in different components is repulsive (attractive) for repulsive (attractive) interspecies interaction when the distance between the vortices is large compared to the size of the vortex core [@Eto_PRA2011]. Here, whether the intervortex interaction becomes repulsive or attractive is independent of the sign of the winding number of each vortex since the interaction results from the density nonuniformity through the term $g_{12}|\Psi_1|^2|\Psi_2|^2$ in the energy functional. However, our results show that the short-range interaction depends on the signs of the vortex winding numbers. $(1,-1)$-vortices have an unstable mode with $l=1$ \[Fig. \[fig:bdg2\](a)\]. For an attractive ($g_{12}<0$) interspecies interaction, $(1,-1)$-vortices have an unstable mode with $l=1$, too. We found also that, for $(1,1)$- or $(-1,-1)$-vortices, there were unstable modes for $g_{12}>0$ but not for $g_{12}<0$. These results mean that the sign of the short-range interaction depends on the signs of the winding numbers of vortices for $g_{12}<0$; thus, the interaction between a $(1, 0)$-vortex and a $(0, 1)$-vortex is attractive but that between a $(1, 0)$-vortex and a $(0, -1)$-vortex is repulsive for attractive interspecies interaction. In fact, the amplitude of the unstable mode of a $(1,-1)$-vortex is localized around the vortex cores and its amplification makes the vortex split into a $(1,0)$-vortex and a $(0,-1)$-vortex. This effect is nontrivial compared to the long-range interaction between vortices in different components [@Eto_PRA2011]. Nontrivial effects also occur for $(L, -L)$-vortices with $L>1$. $(2,-2)$-vortices have unstable modes with $l=1,$ 2, and $3$. The mode with $l=3$ has the largest imaginary part and is thus the most unstable. Because the modes with $l$ cause the density profiles with $l$-fold symmetry, it is expected that a density pattern with three-fold symmetry appears after onset of the instability. This situation differs from the density patterns that appear in the splitting process of an $L$-charged vortex in single-component BECs, where an $L$-charged vortex splits into $L$ single-quantum vortices. Then the $L$-charged vortex has unstable modes with $l \leqq L$ that make a density pattern with $l$-fold symmetry. However, the number $l$ of the most unstable mode is larger than $L$ in the case of the CR vortices. Figure \[fig:bdg2\](b) shows the radial distribution of the most unstable mode for the $(2,-2)$-vortex, where we plotted the density fluctuation $$\delta n_i(r) = |\psi_i^0(\theta=0,t=0)-\delta\psi_i^0(\theta=0,t=0)|^2 - n^0_i(r).$$ The most unstable mode is localized in the vortex core and its amplitude decreases outside of the core. The amplitude vanishes at the center $r=0$ because of the divergence of the term $\propto (l\pm L)^2/r^2$ in Eq. (\[eq:h\]) with $l\neq L$. The zero amplitude at $r=0$ makes it possible to cause vortices at $r=0$ after the amplification of the mode. In the case of the $(2,-2)$-vortex, the amplification makes a vortex with a winding number opposite to that of the original vortex in each component, as is discussed in Sec. \[sec:dynamics\]. In Fig. \[fig:bdg2\](a), we plot the maximum values of the imaginary part ${\rm Im}\, \omega$ of the modes with $0 \leqq l \leqq 6$ for $L=1$, 2, and $3$ in BECs with a repulsive interspecies interaction. We also investigate the unstable modes for BECs with an attractive interspecies interaction. We observe that the imaginary parts of odd (even) $l$ are typically larger than those of even (odd) $l$ for $g_{12}>0$ ($g_{12}<0$). There is no physical explanation for this behavior at this time, and this is an open problem for the future. Next, we show a typical example of the instability of CR vortices with large $L$. $(10,-10)$-vortices have unstable modes with $l=1,2,\dots,19$ in Fig. \[fig:bdg10\](a). The most unstable mode has $l=15$. We show the change of the density caused by the most unstable mode (solid lines) and the typical unstable mode with $l=15$ for the cases of large $L$ (dotted lines) in Fig. \[fig:bdg10\](b). The amplitude of the most unstable mode is localized around the vortex core and decreases outside of the core as in the case of small $L$. However, the peak of the amplitude being almost outside of the vortex core differs from the case of small $L$. In addition to the localized modes, in this case, there appear unstable modes with large ${\rm Im}\,\omega$ whose amplitude is distributed broadly outside of the core. Typically, amplitudes of such modes oscillate spatially over a wide range. Aspects of countersuperflow instability {#sec:csi} --------------------------------------- We discuss here the relation between the instability of CR vortices and CSI. In Refs. [@Takeuchi_PRL2010; @Ishino_PRA2011], CSI has been discussed in the bulk where condensate densities are uniform. Characteristic aspects of CSI are expected to appear in our CR vortex systems because of the relative rotation between the two components. To explain the nontrivial problem of the angular number $l$ of some unstable modes being larger than the winding number $L$ of the $(L,-L)$-vortex, we further proceed with the local density approximation. Let us introduce the local wave number vector ${\bm q}=(q_r, l/r)$ in polar coordinates, where $q_r$ is the pseudo-wave number in the radial direction. When the relative velocity $V_R$ is much larger than the critical velocity of CSI in uniform systems, the wave numbers $q_\parallel$ and $q_\bot$, which are, respectively, parallel and normal to the relative velocity, are characterized by the relation [@Ishino_PRA2011] $$\begin{aligned} (q_\parallel-V_R/2)^2+q_\bot^2=V_R^2/4. \end{aligned}$$ As a first step of the analysis, we evaluate approximately the instability of CR vortices with the local density of the BECs. According to the form of the critical relative velocity of CSI in uniform systems, we define the critical relative velocity $V_c$ in the local density approximation as $$\begin{aligned} V_{c}(r)=2\sqrt{Cn^0(r)(1-|\gamma|)}, \label{eq:critical} \end{aligned}$$ where $C$ is the nondimensional intraspecies interaction coefficient, $n^0(r)$ is the density profile of the stationary state, and $\gamma = C_{12}/C$. Equation (\[eq:critical\]) and the local relative velocity $V_R(r)\equiv 2L/r$ are plotted for the $(2,-2)$-vortex and the $(10,-10)$-vortex in Fig. \[fig:velocity\]. We have shown that an unstable mode has a certain amount of its amplitude even far from the vortex core for the $(10,-10)$-vortex. This must be interpreted in relation to CSI by the fact that the local relative velocity $V_R$ is larger than the critical velocity $V_c$ in the whole region for the $(10,-10)$-vortex \[Fig. \[fig:velocity\](b)\]; CSI can occur locally in the bulk far from the $(10,-10)$-vortex because of the large relative velocity. In contrast, for the $(2,-2)$-vortex \[Fig. \[fig:velocity\](a)\], we have $V_R > V_c$ only near the vortex core and the surface of the BECs, where the local density approximation is inapplicable in the presence of a large gradient of the density. The fact that unstable modes are strongly localized in the vortex core in Figs. \[fig:bdg2\](b) and \[fig:bdg10\](b) is consistent with the large difference between $V_R$ and $V_c$ around the center $r \sim 0$, although we have found that the unstable modes do not appear near the surface. This argument can be applicable to our system qualitatively, because $V_R(r)$ is much larger than the criterion $V_c(r)$ in a broad area around the density peak for the $(10,-10)$-vortex, as shown in Fig. \[fig:velocity\](b). By replacing $q_\parallel$ and $q_\bot$ by $l/r$ and $q_r$, one obtains the characteristic numbers $$\begin{aligned} l &\sim r V_R=2L, \\ q_r& \sim V_R/2=L/r. \end{aligned}$$ The former number comes from the maximum value of $q_\parallel \sim V_R$ for unstable modes and the latter is the maximum wave number normal to the relative velocity. In fact, the former relation, $l=2L,$ is almost consistent with the maximum number $l=19$ of unstable modes for $L=10$ in Fig. \[fig:bdg10\](a). This relation also roughly describes the maximum $l$ number even for the cases of small $L$ in Fig. \[fig:bdg2\](a). This consistency shows that the instability of CR vortices is dominated by CSI. In this way, the nontrivial unstable modes with $l>L$ obtained in the BdG model are qualitatively understood by CSI. Additionally, the radial wave number $q_r=L/r$ is roughly consistent with the wave number of the characteristic unstable modes for $(10,-10)$-vortices, which we show as dotted lines in Fig. \[fig:bdg10\](b). In an area around the density peak, $r\sim 30\xi_{\rm b}$, the radial wavelength is $\lambda_r = 2\pi/q_r\simeq 19\xi_{\rm b}$. This wavelength is consistent with the wavelength estimated from the characteristic unstable mode in Fig. \[fig:bdg10\](b). Nonlinear development {#sec:dynamics} ===================== , \[fig:dy\_el2-1\](d), \[fig:dy\_el2-2\](b), and \[fig:dy\_el2-2\](d), respectively. The vertical axis shows the magnitude of the velocity normalized by $c_{\rm b}=\sqrt{\mu_b/m}$. []{data-label="fig:vr2"}](Vr2.eps){width="1.0\linewidth"} To reveal the nonlinear development of the instability of CR vortices, we numerically solved Eq. (\[eq:2DGP\]). We consider a feasible case of binary BECs with repulsive interspecies interaction by using the same parameters as in Sec. \[sec:formulations\]. We investigated the time developments from the stationary states with small and large numbers of $L$ in Eqs. (\[eq:stationary1\]) and (\[eq:stationary2\]). To trigger the instability, a small white noise is added to the initial states. We do not demonstrate the time development of the instability from a $(1,-1)$-vortex, because the dynamics is simple; the amplification of the unstable mode with $l=1$ leads to the splitting of a $(1,-1)$-vortex into a $(1,0)$-vortex and a $(0,-1)$-vortex. We found that the splitting occurs even for $g_{12}<0$. Therefore, the short-range interaction between $(1,0)$- and $(0,-1)$-vortices is considered to be repulsive for both $g_{12}>0$ and $g_{12}<0$. {width="1.0\linewidth"} , \[fig:dy\_el10\](c), \[fig:dy\_el10\](f), and \[fig:dy\_el10\](g), respectively. The vertical axis shows the magnitude of the velocity normalized by $c_{\rm b}=\sqrt{\mu_b/m}$. []{data-label="fig:vr10"}](Vr10.eps){width="1.0\linewidth"} We will show first the instability dynamics of a $(2,-2)$-vortex as a typical example for the case of small $L$. Figure \[fig:dy\_el2-1\] represents the time development of the density and phase profiles of each component. In the early stage of the instability, a density pattern with three-fold symmetry appears owing to the strong amplification of the unstable mode with $l=3$ \[Figs. \[fig:dy\_el2-1\](a)–\[fig:dy\_el2-1\](c)\]. Then, three single-quantum vortices move away from the center and a single-quantum vortex remains at $r=0$ in each component. The sign of the winding number of the vortex at $r=0$ is opposite to that of the three vortices. Thus, the total winding number is conserved throughout this process. Consequently, a $(2,-2)$-vortex splits into three $(1,0)$-vortices, three $(0,-1)$-vortices, and a $(-1,1)$-vortex. To understand the unique dynamics qualitatively, we calculated the quantity $$V_{\theta,j}(r) = \left\langle{\bm v}_j\cdot{\bm e}_\theta \right\rangle_\theta,$$ where ${\bm v}_{j} = (\psi_j^*\nabla\psi_j-\psi_j\nabla\psi_j^*)/2i|\psi_j|^2$ and ${\bm e}_\theta$ is the unit vector in the rotation direction. The brackets $\langle\cdots\rangle_\theta$ denote average over a circle of radius $r$. This quantity characterizes the radial profile of the mean local velocity in the rotational direction for the $j$th component. We have $V_{\theta,1}=L/r$ and $V_{\theta,2}=-L/r$ in the initial state and $V_{\theta,1} \approx -V_{\theta,2}$ throughout the instability development because of the symmetric parameter setting between the two components. Figure \[fig:dy\_el2-2\] shows the time development after the process of Fig. \[fig:dy\_el2-1\]. The three pairs of $(1,0)$- and $(0,-1)$-vortices move outward further \[Figs. \[fig:dy\_el2-2\](a) and \[fig:dy\_el2-2\](b)\]. Then, as shown in Fig. \[fig:vr2\], the relative rotational velocity between the two components is suppressed around the center, although its sign turns negative there in the presence of a $(-1,1)$-vortex at $r=0$. The $(-1,1)$-vortex at the center is dynamically unstable, splitting into a $(-1,0)$-vortex and a $(0,1)$-vortex, both of which move outward \[Figs. \[fig:dy\_el2-2\](b) and \[fig:dy\_el2-2\](c)\]. After that, the relative velocity is suppressed and is almost zero in the center region (see Fig. \[fig:vr2\]; $286.1 {\rm ms}$). We have observed that all vortices survive without pair annihilation until $286.1 \;{\rm ms}$ in the numerical simulation. The instability of CR vortices gradually becomes more complex when $L$ increases. The number of vortices that appear after the vortex splitting process increases monotonically with $L$ according to the results of the linear stability analysis in Sec. \[sec:linear\]. For example, a $(3,-3)$-vortex splits into seven vortices in each component, where we observed five $(1,0)$-vortices, five $(0,-1)$-vortices, two $(-1,0)$-vortices, and two $(0,1)$-vortices after the splitting process. If $L$ is large enough, the instability develops qualitatively different from that for small $L$. We have shown that some unstable modes can be distributed broadly far from the center $r=0$ for large $L$. These modes cause nucleation of vortices in the bulk region in addition to the vortex multiplication caused by vortex splitting in the center. In a three-dimensional homogeneous system, CSI causes nucleation of vortex rings after the characteristic density pattern formation [@Takeuchi_PRL2010; @Ishino_PRA2011]. In our quasi-two-dimensional system, the instability causes pair nucleation of vortices in the bulk. Figure \[fig:dy\_el10\] shows the instability development from a $(10,-10)$-vortex. The most unstable mode in this case is $l=15$. The density pattern in the early stage \[Fig. \[fig:dy\_el10\](b)\] is much more complex compared to that for $L=2$ in Fig. \[fig:dy\_el2-1\]. We can see in Fig. \[fig:dy\_el10\](c) that vortex pairs are nucleated in the region far from $r=0$. Since the direction of superfluid velocity between a vortex and an antivortex of the vortex pairs is opposite to that of the initial rotational superflow in each component, the pair nucleation locally reduces the relative velocity $V_{\theta,1}-V_{\theta,2}$ around $r \sim 15 \xi_b$ in Fig. \[fig:vr10\]. Because of the numerous vortices from pair nucleation in addition to vortex splitting, a highly turbulent region around the center appears \[Fig. \[fig:dy\_el10\](d)\]. The relative velocity is strongly suppressed in the turbulent region and the region becomes larger with time \[Figs. \[fig:dy\_el10\](e) and \[fig:dy\_el10\](f)\]. Eventually, the relative rotational velocity vanishes by the two components exchanging their angular momentum, and then the turbulent region spreads out to the whole system \[Fig. \[fig:dy\_el10\](g)\]. Summary ======= We studied the linear stability and the instability development of CR vortices in miscible two-component BECs. We found that a CR vortex has unstable modes whose angular number is larger than the winding number of the CR vortex. The appearance of such modes is a unique feature of this system, which is dominated by CSI. The number of vortices appearing in the vortex splitting process owing to the amplification of these modes is larger than the winding number of the initial vortex. The total winding number is conserved in this process by nucleating vortices with opposite winding number. When the winding number becomes larger, the unstable modes become more broadly distributed so as to nucleate vortex pairs in the bulk region. The vortices spread over the cloud, leading to binary quantum turbulence. The instability of CR vortices is one of the tools for creating binary quantum turbulence in BEC experiments. A CR vortex can be realized experimentally by applying the topological phase imprinting method [@Matthews_PRL1999; @Williams_Nature1999; @Shibayama_JPB2011]. We can imprint a phase that causes opposite rotations between two components by using two BECs with different hyperfine states. Experimental evidence of the instability of CR vortices can be observed as the characteristic density pattern or the multiplication of vortices. Additionally, we observe the drastic difference between the expansions of the cloud during the time of flight before and after the instability, because the centrifugal force on the atoms is reduced by the relaxation of relative rotation caused by the instability. Experimental observation of the instability of CR vortices is valuable in terms of the physics of quantized vortices, hydrodynamic instability, and quantum turbulence. S.I. acknowledges the support of a Grant-in-Aid for JSPS Fellows (Grant No. 244499). H.T. acknowledges the support of the “Topological Quantum Phenomena” (No. 22103003) Grant-in Aid for Scientific Research on Innovative Areas from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. [99]{} R. J. Donnelly, [*Quantized Vortices in Helium II* ]{} (Cambridge University Press, Cambridge, England, 1991). W. F. Vinen, Proc. R. Soc. London, Ser. A [**240**]{}, 114 (1957); [**240**]{}, 128 (1957); [**242**]{}, 493 (1957); [**243**]{}, 400 (1958). H. Adachi, S. Fujiyama, and M. Tsubota, Phys. Rev. B. [**81**]{}, 104511(2010). M. Kobayashi and M. Tsubota, Phys. Rev. Lett. [**94**]{}, 065302 (2005). M. Kobayashi and M. Tsubota, Phys. Rev. A [**76**]{}, 045603 (2007). P. W. Karn, D. R. Starks, and W. 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ArXiv

--- abstract: 'We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the presence of multiplicity by employing the perturbed Bloch eigenvalues. We show that all the crossing Bloch modes contribute to the homogenization at the internal edge and that higher and lower modes do not contribute to the homogenization process.' address: 'Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India.' author: - Sivaji Ganesh Sista - Vivek Tewary bibliography: - 'mylit.bib' title: Generic Simplicity of Spectral Edges and Applications to Homogenization --- Bloch eigenvalues ,Genericity ,Periodic Operators ,Homogenization 47A55 ,35J15 ,35B27 Introduction ============ The goal of the paper is to study regularity properties of spectral edges of a periodic second-order elliptic operator in divergence form, given by $$\mathcal{A}u:=-\frac{\partial}{\partial y_k}\left(a_{kl}(y)\frac{\partial u}{\partial y_l}\right),\label{eq1:operator}$$ where summation over repeated indices is assumed. We make the following assumptions on the coefficients of the operator : The coefficients $a_{kl}(y)$ are measurable bounded real-valued periodic functions defined on $\mathbb{R}^d$. Let $Y=[0,2\pi)^d$ be a basic cell for its lattice of periods in the $d$-dimensional euclidean space $\mathbb{R}^d$. The space of measurable bounded periodic real-valued functions in $Y$ is denoted by $L^\infty_\sharp(Y,\mathbb{R})$. Hence, $a_{kl}\in L^\infty_\sharp(Y,\mathbb{R})$. In many instances, we will identify $Y$ with a torus $\mathbb{T}^d$ and the space $L^\infty_\sharp(Y,\mathbb{R})$ with $L^\infty(\mathbb{T}^d,\mathbb{R})$, in the standard way. The matrix $A=(a_{kl})$ is symmetric, i.e., $a_{kl}(y)=a_{lk}(y)$. Further, the matrix $A$ is [*coercive*]{}, i.e., there exists an $\alpha>0$ such that $$\label{coercivity} \forall\, v\in\mathbb{R}^d\mbox{ and } a.e.\, y\in\mathbb{R}^d,\langle A(y)v, v\rangle\geq \alpha||v||^2.$$ Let $Y^{'}=\left[-\dfrac{1}{2},\dfrac{1}{2}\right)^d$ be a basic cell for the dual lattice in $\mathbb{R}^d$. Then, the spectrum of $\mathcal{A}$ can be studied by evaluating, for $\eta\in Y'$, the spectrum of the shifted operator $$\begin{aligned} \mathcal{A}(\eta)=e^{-i\eta\cdot y}\mathcal{A}e^{i\eta\cdot y}=-\left(\frac{\partial}{\partial y_k}+i\eta_k\right)a_{kl}(y)\left(\frac{\partial}{\partial y_l}+i\eta_l\right).\label{shiftedoperator}\end{aligned}$$ This is an unbounded operator in $L^2_\sharp(Y)$, the space of all $L^2_{loc}(\mathbb{R}^d)$ functions that are $Y$-periodic. The operator $\mathcal{A}$ in $L^2(\mathbb{R}^d)$ is unitarily equivalent to the fibered operator $$\int^{\bigoplus}_{Y^{'}}\mathcal{A}(\eta)d\eta$$ in the Bochner space $L^2(Y^{'},L^2_{\sharp}(Y))$. As a consequence of this fact, the spectrum of $\mathcal{A}$ is the union of the spectra of $\mathcal{A}(\eta)$ in $L^2_{\sharp}(Y)$ as $\eta$ varies in $Y^{'}$. For a proof, see [@Reed1978 p. 284]. Let $(\lambda_n(\eta))_{n=1}^\infty$ denote the sequence of increasing eigenvalues for $\mathcal{A}(\eta)$, counting multiplicity. The functions $\eta\mapsto\lambda_n(\eta)$ are known as the Bloch eigenvalues of the operator $\mathcal{A}$. Let $\sigma_n^-=\displaystyle\min_{\eta\in Y^{'}}\lambda_n(\eta)$ and $\sigma_n^+=\displaystyle\max_{\eta\in Y^{'}}\lambda_n(\eta)$, then, the spectrum of the operator $\mathcal{A}$ is given by $\bigcup_{n\in\mathbb{N}}[\sigma_n^-,\sigma_n^+]$. Therefore, it is a union of closed intervals, which may overlap. However, it may also be written as $[0,\infty)\setminus\sqcup_{j=1}^N (\mu_j^-,\mu_j^+)$, where $N$ takes values in $\mathbb{N}\cup\{\infty\}$. The pairwise disjoint intervals $(\mu_j^-,\mu_j^+)$ are known as spectral gaps and $(\mu_j^{\pm})_{j=1}^N$ are known as spectral edges. As depicted in Fig. \[figure1\], $\sigma_n^\pm$ may not be spectral edges, even though the corresponding Bloch eigenvalue is simple. In the first part of the paper, we will study regularity of Bloch eigenvalues near the points where the spectral edge is attained in the dual parameter space. Regularity properties of the Bloch eigenvalues in the parameter are important in applications in the theory of effective mass [@AllairePiatnitski2005] and Bloch wave method in homogenization [@Conca1997]. For periodic Schrödinger operators, Wilcox [@Wilcox78] proved that outside a set of measure zero in the dual parameter space, the Bloch eigenvalues are analytic and the Bloch eigenfunctions may be chosen to be analytic. However, this measure zero set might intersect the spectral edge, which would limit applicability of such a result. For the linear elasticity operator, the Bloch eigenvalues are not analytic near the bottom spectral edge, which poses a major difficulty in the passage to limit in the Bloch wave method of homogenization [@SivajiGanesh2005]. Study of parametrized eigenvalue problems is an active area of research, even in finite dimensions [@michor98], [@Rainer2014]. Broadly speaking, regularity results for parametrized eigenvalues of selfadjoint operators are available in two cases: (i) for one parameter eigenvalue problems [@Rellich1969], [@Kato1995]. (ii) for simple eigenvalues, regardless of the number of parameters. Multiple parameters are unavoidable in most applications of interest. Examples include propagation of singularities for hyperbolic systems of equations with multiple characteristics leading to novel phenomena such as conical refraction [@lax1982], [@dencker1988], stability of hyperbolic initial-boundary-value problems [@Zumbrun2005] and Bloch waves for elasticity system [@SivajiGanesh2005]. Hence, an assumption of simplicity is useful in applications [@AllaireVanni2004], [@AllaireRauch2011], [@AllaireRauch2013]. In the literature, it has been shown that under perturbations of some relevant parameters like domain shape, coefficients, potentials etc, a multiple eigenvalue can be made simple. In a well-known paper [@Albert1975], Albert proves that, for a compact manifold $M$, the set of all smooth potentials $V\in C^{\infty}(M)$ for which the operator $-\Delta+V$ has only simple eigenvalues is a residual set in the space of all smooth admissible potentials. Similar results were proved by Uhlenbeck [@Uhlenbeck1976] using topological methods. Generic simplicity of the spectrum with respect to domain has been established and applied in proving stabilizability and controllability results for the plate equation [@Zuazua2000] and the Stokes system in two dimensions [@Zuazua2001] by Ortega and Zuazua. We intend to generalize Albert’s method to the spectrum of periodic operators. Albert’s result is applicable to operators with discrete spectrum, whereas a periodic operator typically has no eigenvalues. The symmetries of the periodic operator allow us to write it as a direct integral of operators with compact resolvent. Hence, the method of Albert may be applied in a fiberwise manner. However, the fiber  is an operator with complex-valued coefficients. Further, the perturbation is sought in the second-order term as opposed to the zeroth-order term in [@Albert1975]. In this paper, we overcome these difficulties and prove that the Bloch eigenvalues can be made simple locally in the parameter through a perturbation in the coefficients of the operator $\mathcal{A}$. Further, by applying fiberwise perturbation on the fibered operator $\int^{\bigoplus}_{Y^{'}}\mathcal{A}(\eta)d\eta$, we can make sure that the corresponding eigenvalue of interest is simple for all parameter values. ![Bloch eigenvalues $\lambda_4$ and $\lambda_5$ are simple, but have no spectral gap between them.[]{data-label="figure1"}](1.pdf) The latter part of the paper is concerned with the theory of Bloch wave homogenization. In homogenization, one studies the limits of solutions to equations with highly oscillatory coefficients, such as $$\begin{aligned} \label{anotherequation} -\nabla\cdot\left(A\left(\frac{x}{\epsilon}\right)\nabla u^{\epsilon}\right)+\varkappa^2u^{\epsilon}=f~~\mbox{ in }\mathbb{R}^d,\end{aligned}$$ for $f\in L^2(\mathbb{R}^d)$ and $\varkappa>0$. Suppose that $u^\epsilon$ converges weakly in $H^1(\mathbb{R}^d)$ to $u^{*}$. Then, the theory of homogenization [@Tartar2009], [@Bensoussan2011] shows that the limit $u^{*}$ solves an equation of the same type and identifies the matrix $A^*$: $$\begin{aligned} -\nabla\cdot\left(A^*\nabla u^{*}\right)+\varkappa^2u^{*}=f~~\mbox{ in }\mathbb{R}^d.\end{aligned}$$ Bloch wave method of homogenization achieves this characterization through regularity properties of Bloch eigenvalues at the bottom of the spectrum. In particular, the homogenized matrix $A^{*}$ is characterized by the Hessian of the lowest Bloch eigenvalue at $0\in Y^{'}$ [@Conca1997]. Similarly, in the theory of internal edge homogenization [@Birman2006], the following regularity properties of Bloch eigenvalues near the spectral edge play an important role in obtaining operator error estimates: 1. The spectral edge must be simple, i.e., it is attained by a single Bloch eigenvalue. 2. The spectral edge must be attained at finitely many points by a Bloch eigenvalue. 3. The spectral edge must be non-degenerate, i.e., for some $m,r\in\mathbb{N}$, if the Bloch eigenvalue $\lambda_m(\eta)$ attains the spectral edge $\lambda_0$ at the points $\{\eta_j\}_{j=1}^r$, then the Bloch eigenvalue must satisfy, for $j=1,2,\ldots,r$, $$\begin{aligned} \lambda_{m}(\eta)-\lambda_0=(\eta-\eta_j)^T B_j (\eta-\eta_j)+{O}(|\eta-\eta_j|^3), \mbox{ for } \eta \mbox{ near } \eta_j, \end{aligned}$$ where $B_j$ are positive definite matrices. While these features are readily available for the lowest Bloch eigenvalue corresponding to the divergence-type scalar elliptic operator, these properties may not be available for other spectral gaps of the same operator [@Kuchment]. However, the following results are available regarding these properties: Klopp and Ralston [@KloppRalston00] proved the simplicity of a spectral edge of Schrödinger operator $-\Delta+V$ under perturbation of the potential term. In two dimensions, spectral edges are known to be isolated [@Filonov15]. Also, in two dimensions, a degenerate spectral edge can be made non-degenerate through a perturbation with a potential having a larger period [@ParShteren17]. The validity of hypotheses (A), (B), (C) is usually assumed in the literature [@Kuchment]; for example, in establishing Green’s function asymptotics [@KuchmentRaich2012], [@KhaKuchmentRaich2017], for internal edge homogenization [@Birman2006] and to establish localization for random Schrödinger operators [@Veselic2002]. Local simplicity of Bloch eigenvalues is assumed in the study of diffractive geometric optics [@AllaireRauch2011], [@AllaireRauch2013] and homogenization of periodic systems [@AllaireVanni2004]. Following Klopp and Ralston [@KloppRalston00], we apply a perturbation to the coefficients of the operator $\mathcal{A}$ so that a multiple spectral edge becomes simple, under the condition that the coefficients are in $W^{1,\infty}$. However, if the coefficients of $\mathcal{A}$ are in $L^\infty$, a multiple spectral edge can be made simple through a small perturbation of the coefficients with the added assumption that the spectral edge is attained at only finitely many points. Thus, our results suggest a possible interplay between the validity of these assumptions and the regularity of the coefficients. Further, these spectral tools also allow us to achieve homogenization at an internal edge in the presence of multiplicty. More details on the spectrum of elliptic periodic operators may be found in Reed and Simon [@Reed1978] and for state of the art on periodic differential operators, see the review by Kuchment [@Kuchment]. Main Results ------------ Let $S\!ym(d)$ denote the space of all real symmetric matrices, i.e., if $A=(a_{kl})\in S\!ym(d)$, then $a_{kl}=a_{lk}$. Let $$M_B^{>}=\{A:\mathbb{R}^d\to{S\!ym}(d):a_{kl}\in L^{\infty}_\sharp(Y,\mathbb{R}) \mbox{ and $A$ is coercive }\}.$$ $M_B^{>}$ may be identified as a subset of the space of $d(d+1)/2$-tuples of $L^{\infty}_\sharp$ functions and we shall use the norm-topology on this space in our further discussion. A Baire space is a topological space in which the countable intersection of dense open sets is dense. Note that $M_B^{>}$ is an open subset of the space of all symmetric matrices with $L^\infty_\sharp(Y,\mathbb{R})$ entries, which forms a complete metric space, and hence $M_B^{>}$ is a Baire Space. We shall call a property [*generic*]{} in a topological space $X$, if it holds on a set whose complement is of first category in $X$. In particular, a property that is generic on a Baire space holds on a dense set. The rest of the subsection will be devoted to the statements of the main results. \[theorem:1\] Let $\eta_0\in Y^{'}$. The eigenvalues of the shifted operator $\mathcal{A}(\eta_0)$ are generically simple with respect to the coefficients $A=\left(a_{kl}\right)_{k,l=1}^d$ in $M_B^{>}$. Theorem \[theorem:1\] is an extension of the theorem of Albert [@Albert1975] which proves that the eigenvalues of $-\Delta+V$ are generically simple with respect to $V\in C^{\infty}(M)$ for a compact manifold $M$. The potential $V$ is the quantity of interest for Schrödinger operator, $-\Delta+V$. For the applications that we have in mind, for example, the theory of homogenization, the periodic matrix $A$ in the divergence type elliptic operator $-\nabla\cdot(A\nabla)$ is of physical importance. The spectrum of such operators is not discrete, and is analyzed through Bloch eigenvalues, which introduces an extra parameter $\eta\in Y^{'}$ to the problem. The determination of real-valued perturbation for the shifted operator $\mathcal{A}(\eta)$, which has complex-valued coefficients, poses additional difficulties, when coupled with the lack of regularity of the coefficients which the applications demand. \[theorem:2\] Let $m\in\mathbb{N}$, then for the Bloch eigenvalue $\lambda_m(\eta)$ of the periodic operator $\mathcal{A}=-\nabla\cdot(A\nabla)$, where $A\in M_B^>$, there exists a perturbation of $\mathcal{A}$ such that the perturbed eigenvalue $\tilde{\lambda}_m(\eta)$ is simple for all $\eta\in Y^{'}$. A spectral edge $\lambda_0$ is said to be [*simple*]{} if the set $\displaystyle\{m\in\mathbb{N}:\exists\,\eta\in Y^{'}\mbox{ such that }\lambda_m(\eta)=\lambda_0\}$ is a singleton. A spectral edge is said to be multiple if it is not simple. \[theorem:3\] Let $A\in M_B^>$. Further, suppose that its entries $A=\left(a_{kl}\right)_{k,l=1}^d$ belong to the class $W^{1,\infty}_\sharp(Y,\mathbb{R})$. Then, a multiple spectral edge of the operator $\mathcal{A}=\displaystyle -\nabla\cdot (A\nabla)$ can be made simple by a small perturbation in the coefficients. \[theorem:4\] Let $A\in M_B^>$. Further, suppose that its entries $A=\left(a_{kl}\right)_{k,l=1}^d$ belong to the class $L^{\infty}_\sharp(Y,\mathbb{R})$. Let $\lambda_0$ correspond to the upper edge of a spectral gap of $\mathcal{A}$ and let $m$ be the smallest index such that the Bloch eigenvalue $\lambda_{m}$ attains $\lambda_0$. Assume that the spectral edge is attained by $\lambda_m(\eta)$ at finitely many points. Then, there exists a matrix $B=(b_{kl})_{k,l=1}^d$ with $L^\infty_\sharp(Y,\mathbb{R})$-entries and $t_0>0$ such that for every $t\in(0,t_0]$, a spectral edge is achieved by the Bloch eigenvalue $\lambda_m(\eta;A+tB)$ of the operator $\mathcal{A}=\displaystyle -\nabla\cdot (A+tB)\nabla$ and the spectral edge is simple. 1. While Theorem \[theorem:2\] achieves global simplicity for a Bloch eigenvalue, the perturbed operator is no longer a differential operator, i.e., it is non-local. In the theory of homogenization, non-local terms usually appear as limits of non-uniformly bounded operators [@Briane2002], [@BrianeCalc2002]. In the presence of crossing modes, non-locality appears in the theory of effective mass [@Chabu2018]. 2. Theorem \[theorem:3\] is an adaptation of the theorem of Klopp and Ralston [@KloppRalston00] to divergence-type operators. Their proof relies heavily on the Hölder regularity for weak solutions of divergence-type operators. In our proof, we require Hölder continuity of the solutions as well as their derivatives. Hence, we have to impose $W^{1,\infty}$ condition on the coefficients. 3. In Theorem \[theorem:4\], we weaken the $W^{1,\infty}$ requirement on the coefficients under assumption of finiteness on the number of points at which the spectral edge is attained. This is essential for the applications that we have in mind, in the theory of homogenization, where only $L^\infty$ regularity is available on the coefficients. We shall also prove a theorem on internal edge homogenization, whose complete statement is deferred to Section \[homogen\]. Let $A\in M_B^>$. Birman and Suslina [@Birman2006] propose an effective operator and prove operator error estimates with respect to the operator norm in $L^2(\mathbb{R}^d)$ for the limit as $\epsilon\to 0$ of the operator $\mathcal{A}^\epsilon\coloneqq-\nabla\cdot\left(A(\frac{x}{\epsilon})\nabla\right)$, at a non-zero spectral edge $\lambda_0$ under the regularity hypotheses $(A), (B), (C)$. Under appropriate modifications of the regularity hypotheses on the spectral edge, an effective operator is proposed as an approximation of the operator $\mathcal{A}^\epsilon$ in the limit $\epsilon\to 0$ at a multiple spectral edge, and operator error estimates with respect to the operator norm in $L^2(\mathbb{R}^d)$ are proved. Multiplicity of Bloch eigenvalues is a crucial difficulty in Bloch wave homogenization. The internal edge homogenization result is an attempt at circumventing this issue. Previously, this was handled by use of directional analyticity of Bloch eigenvalues for the linear elasticity operator whose lowest Bloch eigenvalue has multiplicty $3$ [@SivajiGanesh2005]. 1. Bloch wave method belongs to the family of multiplier techniques in partial differential equations. In particular, exponential type multipliers, $e^{\tau\phi}$, with real exponents, are used in obtaining Carleman estimates for elliptic operators [@rousseau12]. 2. Any operator of the form $-\nabla\cdot A\nabla$ in $L^2(\mathbb{R}^d)$ may be written in direct integral form, provided $A$ is periodic. A satisfactory spectral theory for such operators is available for real symmetric $A$. However, non-selfadjoint operators are becoming increasingly important in physics [@Sjoestrand09]. For non-symmetric $A$, the eigenvalues of the fibers $\mathcal{A}(\eta)$ may no longer be real and the eigenfunctions may not form a complete set. These difficulties were surmounted in proving the Bloch wave homogenization theorem for non-selfadjoint operators in [@Sivaji2004]. Nevertheless, the generalized eigenfunctions form a complete set for a large class of elliptic operators of even order [@agmon62]. However, we are not aware of physical interpretations of complex-valued Bloch-type eigenvalues. 3. Most of the results of this paper would have similar analogues for internal edges of an elliptic system of equations, for example, the elasticity system. It would be interesting to consider these problems for the spectrum of non-elliptic operators such as the Maxwell operator. The plan of this paper is as follows; in Section \[local\_simplicity\], we prove Theorem \[theorem:1\] on generic simplicity of Bloch eigenvalues at a point. In Section \[global\_simplicity\], we prove Theorem \[theorem:2\] and in subsequent sections \[simplicity\_edge\_1\] and \[simplicity\_edge\_2\], we prove Theorems \[theorem:3\] and \[theorem:4\] concerning generic simplicity of spectral edges. In the final section \[homogen\], we give a short introduction to internal edge homogenization and furnish an application of perturbation theory to Bloch wave homogenization by proving Theorem \[theorem:5\]. Local Simplicity of Bloch eigenvalues {#local_simplicity} ===================================== Let $\eta_0\in Y^{'}$. Let $P$ be the set defined by $$P\coloneqq\{A\in M_B^{>}\mbox{ : the eigenvalues of $\mathcal{A}(\eta_0)$ are simple} \}.$$ We can write the set $P$ as an intersection of countably many sets as follows: Let $P_0:=M_B^{>}$, and $$\begin{aligned} P_n&:=\{A\in M_B^{>}: \mbox{ the first $n$ eigenvalues of $\mathcal{A}(\eta_0)$ } \mbox{are simple} \}.\\ &=\{A\in M_B^>: \lambda_1(\eta_0)<\ldots<\lambda_n(\eta_0)<\lambda_{n+1}(\eta_0)\leq\lambda_{n+2}(\eta_0)\leq\ldots\}.\end{aligned}$$ Note that, $$\begin{aligned} P \subseteq \ldots \subseteq P_n\subseteq P_{n-1}\subseteq\ldots\subseteq P_1\subseteq P_0 &\qquad\mbox{and}\qquad P=\bigcap_{n=0}^{\infty}P_n.\end{aligned}$$ We shall require the following two lemmas. \[lemma:11\] $P_n$ is open in $M_B^{>}$ for all $n\in\mathbb{N}\cup\{0\}$. \[lemma:22\] $P_{n+1}$ is dense in $P_n$, for all $n\in \mathbb{N}\cup\{0\}$. (of Theorem \[theorem:1\]) We recall that a property is said to be generic in a topological space $X$, if it holds on a set whose complement is of first category in $X$. We can write $P$ as the countable intersection $P=\displaystyle\bigcap_{n=0}^\infty P_n$, where $P_n$ is an open and dense set in $M_B^>$ for all $n\in\mathbb{N}\cup\{0\}$. Hence, the complement of $P$ is a set of first category. Therefore, the simplicity of eigenvalues of $\mathcal{A}(\eta_0)$ is a generic property in $M_B^>$. The rest of this section is devoted to the proofs of Lemmas \[lemma:11\] and \[lemma:22\]. Proof of Lemma \[lemma:11\] --------------------------- In this subsection, we begin by proving continuous dependence of the eigenvalues of the shifted operator $\mathcal{A}(\eta)$ on its coefficients. The main tool in this proof is Courant-Fischer min-max principle, which states that $$\begin{aligned} \lambda_m(\eta_0)=\min_{\dim F =m}\max_{v\in F}\frac{\int_{Y} A(\nabla+i\eta_0) v.\overline{(\nabla+i\eta_0) v}~dx}{\int_{Y} v^2~dx},\end{aligned}$$ where $F$ ranges over all subspaces of $H^1_\sharp(Y)$ of dimension $m$. Let $A_1,A_2\in M_B^{>}$ and let $\eta\mapsto\lambda_n^1(\eta), \eta\mapsto\lambda^2_n(\eta)$ be the n-th Bloch eigenvalues of the operators $\mathcal{A}_1$ and $\mathcal{A}_2$ respectively. Then $$\begin{aligned} |\lambda_n^1(\eta_0)-\lambda^2_n(\eta_0)|\leq d c_n(\eta_0) ||A_1-A_2||_{L^\infty},\end{aligned}$$ where $c_n(\eta_0)$ is the $n^{th}$ eigenvalue of the shifted Laplacian $-(\nabla+i\eta_0)^2$ on $Y$ with periodic boundary conditions. Let $a_1(v)=\int_{Y}A_1(\nabla+i\eta_0) v.\overline{(\nabla+i\eta_0) v}~dy$ and $a_2(v)=\int_{Y}A_2(\nabla+i\eta_0) v.\overline{(\nabla+i\eta_0) v}~dy$ be the quadratic forms that appear in the min-max principle. $$\begin{aligned} |a_1(v)-a_2(v)|&=\left|\int_{Y}(A_1-A_2)(\nabla+i\eta_0) v.\overline{(\nabla+i\eta_0) v}~dy\right|\\ &\leq d||A_1-A_2||_{L^\infty}\int_{Y}|(\nabla+i\eta_0) v|^2~dy, \end{aligned}$$ Therefore, $$\begin{aligned} a_1(v)\leq a_2(v)+d||A_1-A_2||_{L^\infty}\int_{Y}|(\nabla+i\eta_0) v|^2~dy. \end{aligned}$$ Now, divide both sides by $\int_Y |v|^2~dy$, the $L^2_\sharp(Y)$ inner product of $v$ with itself and apply the appropriate min-max to obtain $$\begin{aligned} \lambda^1_m(\eta_0)\leq \lambda^2_m(\eta_0)+d c_m(\eta_0)||A_1-A_2||_{L^\infty}.\end{aligned}$$ Notice that the constant $c_m(\eta_0)$ is precisely the $m^{th}$ eigenvalue of the shifted Laplacian $-(\nabla+i\eta_0)^2$ on $Y$ with periodic boundary conditions. By interchanging the role of $A_1$ and $A_2$, the inequality $$\begin{aligned} \lambda^2_m(\eta_0)\leq \lambda^1_m(\eta_0)+d c_m(\eta_0)||A_2-A_1||_{L^\infty},\end{aligned}$$ is obtained, which completes the proof of this proposition. In [@Conca1997], the Bloch eigenvalues have been proved to be Lipschitz continuous in $\eta\in Y^{'}$. Indeed, one may prove that the Bloch eigenvalues are jointly continuous in $\eta\in Y^{'}$ and the coefficients of the operator. Let $A\in P_n$ and $$\begin{aligned} \delta=\min\{\lambda_{j+1}(\eta_0)-\lambda_{j}(\eta_0):j=1,2,\ldots,n\}.\end{aligned}$$ Let $\displaystyle c=\max_{1\leq j\leq n}d c_j(\eta_0)$, where $c_j(\eta_0)$ is the $j^{th}$ eigenvalue of the shifted Laplacian $-(\nabla+i\eta_0)^2$ on $Y$ with periodic boundary conditions. Let $$\begin{aligned} U=\left\{A'\in M_B^>:||A-A'||_{L^\infty}< \frac{\delta}{4c}\right\}.\end{aligned}$$ $U$ is an open set in $M_B^>$ containing $A$. We shall show that $U$ is a subset of $P_n$. Let $A'\in U$. Let $\{\lambda'_j(\eta),~j=1,2,\ldots\}$ be the Bloch eigenvalues of operator $\mathcal{A}'$ associated to $A'$. For $j=1,2,\ldots,n$, we have: $$\begin{aligned} |\lambda'_j(\eta_0)-\lambda_j(\eta_0)| & \leq dc_j(\eta_0)||A-A'||_{L^\infty} \leq dc_j(\eta_0)\frac{\delta}{4c}\leq\frac{\delta}{4}. \end{aligned}$$ Hence, $$\begin{aligned} \delta & \leq \lambda_{j+1}(\eta_0)-\lambda_j(\eta_0)\\ & \leq |\lambda'_{j+1}(\eta_0)-\lambda_{j+1}(\eta_0)|+|\lambda'_j(\eta_0)-\lambda'_{j+1}(\eta_0)|+|\lambda'_j(\eta_0)-\lambda_j(\eta_0)|\\ & \leq \frac{\delta}{4}+ |\lambda'_j(\eta_0)-\lambda'_{j+1}(\eta_0)|+\frac{\delta}{4}\\ & = \frac{\delta}{2}+\lambda'_{j+1}(\eta_0)-\lambda'_j(\eta_0). \end{aligned}$$ Therefore, $\lambda'_{j+1}(\eta_0)-\lambda'_j(\eta_0)\geq\frac{\delta}{2}>0$ for $j=1,2,\ldots,n.$ Therefore, the first $n$ Bloch eigenvalues of $\mathcal{A}'$ are simple at $\eta_0$, as required. Proof of Lemma \[lemma:22\] --------------------------- In this section, we shall use perturbation theory of selfadjoint operators to prove Lemma \[lemma:22\]. Let $A\in M_B^>$ and $B$ be a symmetric matrix with $L^\infty_\sharp(Y,\mathbb{R})$-entries. For $|\tau|<\sigma_0\coloneqq\frac{\alpha}{2d||B||_{L^\infty}}$, $A+\tau B\in M^>_B$, where $\alpha$ is a coercivity constant for $A$ as in . Consider the operator $\mathcal{A}(\eta_0)+\tau\mathcal{B}(\eta_0)$ in $L^2_\sharp(Y)$. We shall prove in Appendix \[PerturbationTheory\], that the operator family $\mathcal{F}(\tau)=\mathcal{A}(\eta_0)+\tau\mathcal{B}(\eta_0)$ is a selfadjoint holomorphic family of type $(B)$ for $|\tau|<\sigma_0$. For its definition and related notions, see Kato [@Kato1995]. We shall make use of the following theorem which asserts the existence of a sequence of eigenpairs associated with a selfadjoint holomorphic family of type $(B)$, analytic in $\tau\in(-\sigma_0,\sigma_0)$. The proof of this theorem dates back to Rellich, hence we shall call these eigenvalue branches as Rellich branches. (Kato-Rellich)\[katorellich\] Let $\mathcal{A}(\eta_0)(\tau)$ be a selfadjoint holomorphic family of type $(B)$, defined for $\tau\in R$ where $R=\{z\in\mathbb{C}:|\operatorname{Re}({z})|<\sigma_0,|\operatorname{Im}({z})|<\sigma_0\}$ and $\sigma_0\coloneqq \frac{\alpha}{2d||B||_{L^\infty}}$. Let $\mathcal{A}(\eta_0)(\tau)+C_*I$ have compact resolvent for some $C_*\in \mathbb{R}$. Then, there exists a sequence of scalar-valued functions $(\lambda_j(\tau;\eta_0))_{j=1}^\infty$ and $L^2_\sharp(Y)$-valued functions $(u_j(\tau;\eta_0))_{j=1}^\infty$ defined on $I=(-\sigma_0,\sigma_0)$, such that 1. For each fixed $\tau\in I$, the sequence $(\lambda_j(\tau;\eta_0))_{j=1}^\infty$ represents all the eigenvalues of $\mathcal{A}(\eta_0)(\tau)$ counting multiplicities and the functions $(u_j(\tau;\eta_0))_{j=1}^\infty$ represent the corresponding eigenvectors. 2. For each $j\in\mathbb{N}$, the functions $(\lambda_j(\tau;\eta_0))_{j=1}^\infty$ and $(u_j(\tau;\eta_0))_{j=1}^\infty$ are analytic on $I$ with values in $\mathbb{R}$ and $L^2_\sharp(Y)$ respectively. 3. The sequence $(u_j(\tau;\eta_0))_{j=1}^\infty$ is orthonormal in $L^2_\sharp(Y).$ 4. Suppose that the $m^{th}$ eigenvalue of $\mathcal{A}(\eta_0)(\tau)$ at $\tau=0$ has multiplicity $p$, i.e., $$\begin{aligned} \lambda_m(0;\eta_0)=\lambda_{m+1}(0;\eta_0)=\ldots=\lambda_{p+m-2}(0;\eta_0)=\lambda_{p+m-1}(0;\eta_0).\end{aligned}$$ For each interval $K\subset\mathbb{R}$ with $\overline{K}$ containing the eigenvalue $\lambda_m(0;\eta_0)$ and no other eigenvalue, $\lambda_m(\tau;\eta_0), $ $\lambda_{m+1}(\tau;\eta_0),\ldots, \lambda_{p+m-1}(\tau;\eta_0)$ are the only eigenvalues of $\mathcal{A}(\eta_0)(\tau)$, counting multiplicities, lying in the interval $K$. By Kato-Rellich Theorem, an eigenvalue $\lambda(\eta_0)$ of $\mathcal{F}(0)$ of multiplicity $h$, splits into $h$ analytic functions $(\lambda_m(\tau;\eta_0))_{m=1}^h$. Further, the corresponding eigenfunctions $(u_m(\tau;\eta_0))_{m=1}^h$ are also analytic. Let the $h$ eigenvalues and eigenvectors of $\mathcal{F}(\tau)$ have the following power series expansions at $\tau=0$ for $m=1,2,\ldots,h$: $$\begin{aligned} \lambda_m(\tau;\eta_0)=\lambda(\eta_0)+\tau a_m(\eta_0)+\tau^2\beta_m(\tau,\eta_0)\notag\\ u_{m}(\tau;\eta_0)=u_m(\eta_0)+\tau v_m(\eta_0)+\tau^2 w_m(\tau,\eta_0).\end{aligned}$$ The proof of Lemma \[lemma:22\] will rely on the fact that we may choose $B$ in such a way that $a_m(\eta_0)\neq a_n(\eta_0)$ for some $m,n\in\{1,2,\ldots,h\}$. Then, for sufficiently small $\tau$, $\lambda_m(\tau;\eta_0)\neq\lambda_n(\tau;\eta_0)$. In that case, the multiplicity of the perturbed Bloch eigenvalue at $\eta_0$ will be less than $h$. The eigenpairs satisfy the following equation: $$\begin{aligned} \left(-(\nabla+i\eta_0)\cdot (A+\tau B)(\nabla+i\eta_0)-\lambda_m(\tau,\eta_0)\right)u_m(\tau,\eta_0)=0.\end{aligned}$$ Differentiating the above with respect to $\tau$ and setting $\tau$ to $0$, we obtain: $$\begin{aligned} -(\nabla+i\eta_0)\cdot A(\nabla+i\eta_0)v_m(\eta_0)-(\nabla+i\eta_0)\cdot B(\nabla+i\eta_0)u_m(\eta_0)-\lambda(\eta_0)v_m(\eta_0)\notag-a_m(\eta_0)u_m(\eta_0)=0\end{aligned}$$ Finally, multiply by $u_n(\eta_0)$ and integrate over $Y$ to conclude that $$\begin{aligned} \label{hellmanfeynman} \int_Y B(\nabla+i\eta_0)u_m(\eta_0)\cdot(\nabla-i\eta_0)\overline{u_n(\eta_0)}~dy=a_m(\eta_0)\delta_{mn}.\end{aligned}$$ Equation  suggests the following construction. Given a perturbation $B$ and a basis $F=\{f_1,f_2,\ldots,f_h\}$ for the unperturbed eigenspace $N(\eta_0)\coloneqq ker(\mathcal{A}(\eta_0)-\lambda(\eta_0)I)$, we can define a selfadjoint operator $G_B$ on $N(\eta_0)$ whose matrix in the basis $F$ is given by $$\begin{aligned} \left([G_B]_F\right)_{m,n}\coloneqq\int_Y B(\nabla+i\eta_0)f_m\cdot(\nabla-i\eta_0)\overline{f_n}~dy.\end{aligned}$$ In particular, it follows from equation  that in the basis of unperturbed eigenfunctions $E=\{u_1(\eta_0), u_2(\eta_0),\ldots,u_h(\eta_0)\}$, $[G_B]_E$ is a diagonal matrix, $$\begin{aligned} _E=diag(a_1(\eta_0),a_2(\eta_0),\ldots,a_h(\eta_0)).\end{aligned}$$ If $[G_B]_E$ is a scalar matrix, then the operator $G_B$ is a scalar multiple of identity operator. However, if we can find a basis $F$ for the eigenspace and a matrix $B$, corresponding to which, the matrix $[G_B]_F$ has a non-zero off-diagonal entry, then for that choice of $B$, $[G_B]_E$ will not be a scalar matrix, and hence, $a_m(\eta_0)\neq a_n(\eta_0)$ for some $m,n\in\{1,2,\ldots,h\}$. \[find B\] There exists a symmetric matrix $B$ with $L^\infty_\sharp(Y,\mathbb{R})$-entries such that the operator $G_B$ is not a scalar multiple of identity. As noted earlier, the proposition will be proved if we can find a basis $F$ and a matrix $B$ with $L^\infty_\sharp(Y,\mathbb{R})$ entries, such that the matrix $[G_B]_F$ has a non-zero off-diagonal entry. Let $F=\{f_1,f_2,\ldots,f_h\}$ be any basis of $ker(\mathcal{A}(\eta_0)-\lambda(\eta_0)I)$. Suppose that for some $j\in\{1,2,\ldots,d\}$, $$\begin{aligned} \label{alternative1} (\partial_j+i\eta_{0,j})f_1(\partial_j-i\eta_{0,j})\overline{f_2}\not\equiv 0, \end{aligned}$$ where $\eta_0=(\eta_{0,1},\eta_{0,2},\ldots,\eta_{0,d})$. Since, $f_i\in H_\sharp^1(Y)$, $g\coloneqq(\partial_j+i\eta_{0,j})f_1(\partial_j-i\eta_{0,j})\overline{f_2}\in L^1_\sharp(Y)$. Hence, by Hahn-Banach Theorem, there is a continuous linear functional $\kappa\in(L^1_\sharp(Y))^*$, such that $\kappa(g)=||g||\neq 0.$ However, by duality, there exists a $\beta\in L^\infty_\sharp(Y)$, such that $\kappa(g)=\int_Y \beta\,gdy=||g||\neq 0.$ Now, either $\int_Y \operatorname{Re}(\beta)g\neq 0$ or $\int_Y \operatorname{Im}(\beta)g\neq 0$. Suppose, without loss of generality that $\int_Y \operatorname{Re}(\beta)g\neq 0$ and define $$\begin{aligned} B=diag(0,0,\ldots,0,\operatorname{Re}(\beta),0,\ldots,0)\end{aligned}$$ with $\operatorname{Re}(\beta)$ in the $j^{th}$ place, then $$\begin{aligned} ([G_B]_F)_{1,2}&=\int_Y B(\nabla+i\eta_0)f_1\cdot(\nabla-i\eta_0)\overline{f_2}~dy\\ &=\int_Y\operatorname{Re}(\beta)(\partial_j+i\eta_{0,j})f_1(\partial_j-i\eta_{0,j})\overline{f_2}~dy\\ &=\int_Y\operatorname{Re}(\beta) g~dy\neq 0. \end{aligned}$$ Alternatively, if $(\nabla+i\eta_0)f_1\cdot(\nabla-i\eta_0)\overline{f_2}\equiv 0$, then there exists $j\in\{1,2,\ldots,d\}$, such that $$\begin{aligned} \label{alternative2} |(\partial_j+i\eta_{0,j})f_1|^2-|(\partial_j+i\eta_{0,j})f_2|^2\not\equiv 0. \end{aligned}$$ It is easy to see that if  and  do not hold, then $f_1$ and $f_2$ are both a scalar multiple of $\exp(i\eta_0\cdot y)$, which contradicts the fact that they are distinct elements of basis of $N=ker(\mathcal{A}(\eta_0)-\lambda(\eta_0)I)$. Since, for all $m=1,2,\ldots,h$, $f_m\in H_\sharp^1(Y)$, $g{'}\coloneqq|(\partial_j+i\eta_{0,j})f_1|^2-|(\partial_j+i\eta_{0,j})f_2|^2\in L^1_\sharp(Y,\mathbb{R})$. Hence, by Hahn-Banach Theorem, there is a continuous linear functional $\kappa{'}\in(L^1_\sharp(Y,\mathbb{R}))^*$, such that $\kappa{'}(g{'})=||g{'}||\neq 0.$ However, by duality, there exists a $\beta{'}\in L^\infty_\sharp(Y,\mathbb{R})$, such that $\kappa{'}(g{'})=\int_Y \beta{'}g{'}=||g{'}||\neq 0.$ Define $$\begin{aligned} B=diag(0,0,\ldots,0,\beta{'},0,\ldots,0)\end{aligned}$$ with $\beta{'}$ in the $j^{th}$ place, then in the new basis $F^{'}=\{f_1+f_2,f_1-f_2,f_3,\ldots,f_h\}$, the $(1,2)^{th}$ entry of $[G_B]_{F^{'}}$ is given by $$\begin{aligned} \int_Y B(\nabla+i\eta_0)(f_1+f_2)\cdot(\nabla-i\eta_0)(\overline{f_1}-\overline{f_2})~dy = \int_Y \beta{'}|(\partial_j+i\eta_{0,j})f_1|^2-|(\partial_j+i\eta_{0,j})f_2|^2~dy\neq 0. \end{aligned}$$ Thus, either way, we have found a basis in which an off-diagonal entry of $[G_B]_F$ is non-zero. Hence, the operator $G_B$ is not a scalar multiple of identity. In particular, the matrix $[G_B]_E$ cannot be a scalar matrix. Let $A\in P_n$. Given $\epsilon>0$, we want to find $A'\in P_{n+1}$ such that $||A-A'||_{L^\infty}<\epsilon$. We shall construct $A'$ in the form $A'=A+\tau B$, where $B$ is a symmetric matrix with $L^\infty_\sharp(Y,\mathbb{R})$-entries and $\tau\in\mathbb{R}$. By Lemma \[lemma:11\], we can choose $\tau_0$ so that $A+\tau B\in P_n$ for $|\tau|<\tau_0$. Hence, the first $n$ eigenvalues of the operator $-(\nabla+i\eta_0)\cdot(A+\tau B)(\nabla+i\eta_0)$ are simple for $|\tau|<\tau_0$. Subsequently, we must choose $\tau$ such that $|\tau|<\sigma_0=\frac{\alpha}{2d||B||_{L^\infty}}$, in order to apply the Kato-Rellich Theorem. Now, suppose that the $(n+1)^{th}$ eigenvalue of $\mathcal{A}(\eta_0)$ has multiplicity $h$. By Kato-Rellich Theorem (Theorem \[katorellich\]), the $h$ eigenvalue branches of the perturbed operator $\mathcal{A}(\eta_0)+\tau\mathcal{B}(\eta_0)$ are given by the following power series at $\tau=0$, for $r=1,2,\ldots,h$: $$\begin{aligned} \lambda_r(\tau;\eta_0)=\lambda(\eta_0)+\tau a_r(\eta_0)+\tau^2\beta_r(\tau;\eta_0). \end{aligned}$$ If there are $m,n\in\{1,2,\ldots,h\}$ such that $a_m(\eta_0)\neq a_n(\eta_0)$, then there is a $\tau_1$ such that, $\lambda_m(\tau;\eta_0)\neq\lambda_n(\tau;\eta_0)$ for $|\tau|<\tau_1$. Since two of the $h$ eigenvalue branches are distinct for small $\tau$, the multiplicity of the perturbed eigenvalue, which can only go down for small $\tau$, must be less than or equal to $h-1$. This can be achieved through an application of Proposition \[find B\] which gives us a matrix $B_1$ such that at least two of $(a_r(\eta_0))_{r=1}^h$ are distinct. Now, starting from the matrix $A+\tau_1 B_1$, we repeat the procedure above so that the multiplicity of the $(n+1)^{th}$ eigenvalue is further reduced. The perturbed matrix is now labelled $A+\tau_1B_1+\tau_2B_2$. Finally, after a finite number of such steps, we can reduce the multiplicity of the $(n+1)^{th}$ eigenvalue to $1$. At the end of this procedure, we obtain a matrix of the form $A'=A+\sum_{j=1}^N\tau_jB_j$, for some $N\in\mathbb{N}$. Each perturbation must be chosen so that $\sum_{j=1}^{N}\tau_j||B_j||_{L^\infty}<\epsilon$. \[simple\_in\_neighborhood\] Theorem \[theorem:1\] proves that an eigenvalue $\lambda(\eta_0)$ of the shifted operator $\mathcal{A}(\eta_0)$ can be made simple by a perturbation of the matrix $A\in M_B^>$. However, since the Bloch eigenvalues are Lipschitz continuous functions of the parameter $\eta\in Y^{'}$ [@Conca1997], the perturbed eigenvalue $\tilde{\lambda}(\eta)$ will continue to remain simple in some neighborhood of $\eta_0$. 1. The perturbation formula  may be thought of as a variation of the Hellmann-Feynman theorem in the physics literature. The coefficients of the differential operator  are real-valued functions, in as much as they are related to properties of materials. The presence of complex-valued coefficients in the perturbation formula complicates the choice of the real-valued perturbation $B$. 2. In the theory of homogenization, the coefficients of the second order divergence-type periodic elliptic operator are usually only measurable and bounded. By regularity theory [@Ladyzhenskaya68], the eigenfunctions of the shifted operator $\mathcal{A}(\eta)$ are known to be Hölder continuous. However, derivatives of eigenfunctions, which may not be bounded, appear in the perturbation formula . Therefore, the perturbation $B$ is chosen using the Hahn-Banach Theorem. Global Simplicity {#global_simplicity} ================= In the previous section, we have proved that a given Bloch eigenvalue $\lambda_m(\eta)$ of the operator $\mathcal{A}$ can be made simple locally in $Y^{'}$ through a small perturbation in the coefficients. In this section, we shall perform perturbation on the operator $\mathcal{A}$ in such a way that its spectrum still retains the fibered character, i.e., $\sigma(\tilde{A})=\cup_{\eta\in Y^{'}}\sigma(\tilde{A}(\eta))$ and the $m^{th}$ eigenvalue function $\eta\mapsto\tilde{\lambda}_m(\eta)$ is simple for all $\eta\in Y^{'}$. However, the perturbed operator $\tilde{\mathcal{A}}$ may no longer be a differential operator. The operator  has a direct integral decomposition $\mathcal{A}=\int_{\eta\in\mathbb{T}^d}^{\bigoplus}\mathcal{A}(\eta)d\eta$ where $\mathcal{A}(\eta)=-(\nabla+i\eta)\cdot A(\nabla+i\eta)$ is an unbounded operator in $L^2_\sharp(Y)$. We would like to point out that $Y^{'}$ is understood to parametrize the torus, $\mathbb{T}^d$. Consider the $m^{th}$ Bloch eigenvalue $\lambda_m(\eta)$ of $\mathcal{A}$. By Lemma \[lemma:22\], at any point $\eta_0\in Y^{'}$, we can find a perturbation of the coefficients $A=(a_{kl})$ of $\mathcal{A}(\eta_0)$ so that the perturbed eigenvalue $\tilde{\lambda}_m(\eta_0)$ is simple. By Remark \[simple\_in\_neighborhood\], there is a neighborhood of $\eta_0$, $\mathcal{G}_{\eta_0}$ in which the perturbed eigenvalue $\tilde{\lambda}_m(\eta)$ of the perturbed shifted operator $\tilde{\mathcal{A}}(\eta)$ is simple. In this manner, for each $\xi\in \mathbb{T}^d$, we obtain a perturbation $B_{\xi}$ and a neighborhood, $\mathcal{G}_{\xi}$ in which the eigenvalue of the perturbed operator $\tilde{\mathcal{A}}(\eta)=-(\nabla+i\eta)\cdot (A+B_{\xi})(\nabla+i\eta)$ is simple. These sets form an open cover of the torus. By compactness of $\mathbb{T}^d$, there is a finite subcover having the property that in each member $\mathcal{G}_{\xi}$ of the subcover, the corresponding perturbation $B_{\xi}$ causes the perturbed eigenvalue $\tilde{{\lambda}}_m(\eta)$ to be simple in $\mathcal{G}_{\xi}$. Let $\{\mathcal{G}_1,\mathcal{G}_2,\ldots,\mathcal{G}_n\}$ be the finite subcover of the torus obtained above. Define $\mathcal{O}_1=\mathcal{G}_1$. For $r\geq 1$, define $\mathcal{O}_{r+1}=\displaystyle\mathcal{G}_{r+1}\setminus\bigcup_{j=1}^r\mathcal{G}_j$. Suppose that $B_j$ is the perturbation corresponding to the set $\mathcal{O}_j$. Now, define the parametrized operator $$\begin{aligned} \tilde{\mathcal{A}}(\eta)=-(\nabla+i\eta)\cdot (A+\sum_{j=1}^n B_{j}\,\chi_{\mathcal{O}_j})(\nabla+i\eta)\end{aligned}$$ which depends measurably on $\eta\in\mathbb{T}^d$. Finally, define the direct integral $\tilde{\mathcal{A}}=\int_{\eta\in\mathbb{T}^d}^{\bigoplus}\tilde{\mathcal{A}}(\eta)d\eta$, where each of the fibers is a differential operator in $L^2_\sharp(Y)$. Then, it is known [@Reed1978 p.284] that, $$\sigma(\tilde{A})=\bigcup_{\eta\in Y^{'}}\sigma(\tilde{A}(\eta)).$$ Hence, we may define an $m^{th}$ eigenvalue function $\eta\mapsto\tilde{\lambda}_m(\eta)$ with the property that $$\begin{aligned} |\lambda_m(\eta)-\tilde{\lambda}_m(\eta)|\leq C\max_{1\leq j\leq n}||B_j||_{L^{\infty}},\end{aligned}$$ where $\lambda_m(\eta)$ is the $m^{th}$ Bloch eigenvalue of $\mathcal{A}$. 1. Although the $m^{th}$ eigenvalue of the perturbed operator is simple for all parameter values, $\tilde{\lambda}_m(\eta)$ may only be measurable in $\eta\in Y^{'}$. However, $\tilde{\lambda}_m(\eta)$ is analytic in each $\mathcal{O}_j\subset \mathbb{T}^d$. 2. The perturbed operator $\tilde{\mathcal{A}}$ is no longer a differential operator, even though each fiber $\tilde{\mathcal{A}}(\eta)$ is a differential operator. In fact. we shall prove in Theorem \[notdiffop\] that $\tilde{\mathcal{A}}$ is a differential operator if and only if $B_1=B_2=\ldots=B_n.$ 3. A rigorous account of direct integral decomposition of operators, such as the one employed above for periodic operators, may be found in [@schmudgen90] and [@maurin68]. \[notdifferential\] Let $B$ be a symmetric matrix with $L^\infty_\sharp(Y,\mathbb{R})$-entries. Define $\mathcal{B}(\eta)=-(\nabla+i\eta)\cdot B(\nabla+i\eta)$. Let $\mathcal{O}\subset Y^{'}$ be a proper subset of $Y^{'}$. Then, the direct integral defined by $\mathcal{B}=\int^{\bigoplus}_{\eta\in\mathbb{T}^d}\mathcal{B}(\eta)\chi_{\mathcal{O}}$ is not a differential operator. By Peetre’s Theorem [@peetre60], [@duistermaatkolk2010 p. 236], a linear operator $\mathcal{B}:\mathcal{D}(\mathbb{R}^d)\to\mathcal{D}^{'}(\mathbb{R}^d)$ is a differential operator if and only if $supp(Pu)\subset supp(u)$ for all $u\in\mathcal{D}(\mathbb{R}^d)$. Here, $\mathcal{D}(\mathbb{R}^d)$ denotes the space of compactly supported smooth functions on $\mathbb{R}^d$ with the topology of test functions. Also, let $\mathcal{S}(\mathbb{R}^d)$ denote the Schwartz class of rapidly decreasing smooth functions on $\mathbb{R}^d$. In order to show that $\mathcal{B}$ is not a differential operator, we will show that it does not preserve supports. Given $g\in\mathcal{D}(\mathbb{R}^d)$, we define its Gelfand transform as $$\begin{aligned} g_\sharp(y,\eta)=\sum_{p\in\mathbb{Z}^d}g(y+2\pi p)e^{-i(y+2\pi p)\cdot\eta}.\end{aligned}$$ This is a function in $L^2(Y^{'},L^2_\sharp(Y))$. The map from $g\mapsto g_\sharp$ is an isometry on $\mathcal{D}(\mathbb{R}^d)$ in the $L^2$-inner product and hence it may be extended to a unitary isomorphism from $L^2(\mathbb{R}^d)$ to $L^2(Y^{'},L^2_\sharp(Y))$. We shall show that $\mathcal{B}(g)$ is not compactly supported. $\mathcal{B}(g)$ is a tempered distribution defined as: $$(\mathcal{B}(g),\phi)=\int_{\mathcal{O}}\int_Y B(\nabla+i\eta)g_\sharp(y,\eta)\cdot(\nabla-i\eta)\overline{\phi}_\sharp(y,\eta)~dyd\eta.$$ We may define the Fourier transform of $\mathcal{B}(g)$ in $\mathcal{S}^{'}(\mathbb{R}^d)$ as $$\begin{aligned} (\widehat{\mathcal{B}(g)},\phi)=(\mathcal{B}(g),\mathcal{F}^{-1}(\phi)),\end{aligned}$$ where $\mathcal{F}^{-1}(\phi)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}\phi(\eta)e^{iy\cdot\eta}~d\eta$ is the inverse Fourier transform of $\phi$. Since $\phi\in\mathcal{S}(\mathbb{R}^d)$, there exists a $\psi\in\mathcal{S}(\mathbb{R}^d)$ such that $\phi=\widehat{\psi}$. Therefore, $$\begin{aligned} (\widehat{\mathcal{B}(g)},\phi)=(\mathcal{B}(g),\mathcal{F}^{-1}(\phi))=(\mathcal{B}(g),\psi).\end{aligned}$$ By Poisson Summation Formula [@grafakosclassical2008 p. 171], we conclude that $$\begin{aligned} \label{poissonsum} \psi_\sharp(y,\eta)&=\sum_{p\in\mathbb{Z}^d}\psi(y+2\pi p)e^{-i(y+2\pi p)\cdot\eta}=\frac{1}{(2\pi)^{d/2}}\sum_{q\in\mathbb{Z}^d}\widehat{\psi}(\eta+q)e^{iq\cdot y}\notag\\&\quad=\frac{1}{(2\pi)^{d/2}}\sum_{q\in\mathbb{Z}^d}{\phi}(\eta+q)e^{iq\cdot y}. \end{aligned}$$ Now, suppose that $\phi\in\mathcal{S}(\mathbb{R}^d)$ vanishes on $\bigcup_{q\in\mathbb{Z}^d}(\mathcal{O}+q)$, then $\psi_\sharp$, as obtained in , vanishes on $\mathcal{O}$. Hence, $$\begin{aligned} (\widehat{\mathcal{B}(g)},\phi)&=(\mathcal{B}(g),\psi)=\int_Y\int_{\mathcal{O}}B(\nabla+i\eta)g_\sharp(y,\eta)\cdot(\nabla-i\eta)\overline{\psi}_\sharp(y,\eta)~d\eta\,dy=0. \end{aligned}$$ Therefore, $\widehat{\mathcal{B}(g)}$ vanishes on the open set $\bigcup_{q\in\mathbb{Z}^d}(\mathcal{O}+q)$. By Schwartz-Paley-Wiener Theorem [@rudinfunctional1991 p. 191], $\widehat{\mathcal{B}(g)}$ cannot be the Fourier transform of a compactly supported distribution, i.e., $\mathcal{B}(g)$ is not compactly supported. Let $\{\mathcal{O}_1,\mathcal{O}_2,\ldots,\mathcal{O}_n\}$ be a partition of $Y^{'}$ up to a set of measure zero, i.e., $Y^{'}\setminus\bigcup_{j=1}^n\mathcal{O}_j$ is a set of measure zero. Define $\mathcal{B}:\mathcal{D}(\mathbb{R}^d)\to\mathcal{D}^{'}(\mathbb{R}^d)$ by $\mathcal{B}(g)=\sum_{j=1}^n\int^{\bigoplus}_{\eta\in\mathcal{O}_j}\mathcal{B}_j(\eta)g_\sharp(y,\eta)$ where $\mathcal{B}_j(\eta)=-(\nabla+i\eta)\cdot B_j(\nabla+i\eta)$ where for all $j\in\{1,2,\ldots,n\}$, $B_j$ are matrices with $L^\infty_\sharp(Y,\mathbb{R})$-entries, then $\mathcal{B}$ is a differential operator if and only if $B_1=B_2=\ldots=B_n$.\[notdiffop\] If $B\coloneqq B_1=B_2=\ldots=B_n$, then $\mathcal{B}(g)=-\nabla\cdot B\nabla(g)$ which is a differential operator. Conversely, without loss of generality, assume that $B_1\neq B_2$ and suppose that $\mathcal{B}$ is a differential operator. Then, $$\begin{aligned} \mathcal{B}(g)=&\int^{\bigoplus}_{\eta\in Y^{'}}\mathcal{B}_1(\eta)d\eta+\int^{\bigoplus}_{\eta\in\mathcal{O}_2}(\mathcal{B}_2-\mathcal{B}_1)(\eta)d\eta+\int^{\bigoplus}_{\eta\in\mathcal{O}_3}(\mathcal{B}_3-\mathcal{B}_1)(\eta)d\eta+\ldots\\&\qquad +\int^{\bigoplus}_{\eta\in\mathcal{O}_n}(\mathcal{B}_n-\mathcal{B}_1)(\eta)d\eta. \end{aligned}$$ Hence, $$\begin{aligned} \mathcal{B}(g)-\int^{\bigoplus}_{\eta\in Y^{'}}\mathcal{B}_1(\eta)d\eta=\sum_{j=2}^{n}\int^{\bigoplus}_{\eta\in\mathcal{O}_j}(\mathcal{B}_j-\mathcal{B}_1)(\eta)d\eta \end{aligned}$$ The left hand side of the above equation is a differential operator. We will show that the right hand side is not a differential operator to obtain a contradiction. We proceed as in Lemma \[notdifferential\]. Define $\mathcal{C}:\mathcal{D}(\mathbb{R}^d)\to\mathcal{D}^{'}(\mathbb{R}^d)$ by $$(\mathcal{C}(g),\phi)=\sum_{j=2}^{n}\int_{\mathcal{O}_j}\int_Y (B_j-B_1)(\nabla+i\eta)g_\sharp(y,\eta)\cdot(\nabla-i\eta)\overline{\phi}_\sharp(y,\eta)~dy\,d\eta$$ It is easy to see that $\mathcal{C}(g)\in\mathcal{S}^{'}(\mathbb{R}^d)$. Therefore, we may define its Fourier transform by $$\begin{aligned} (\widehat{\mathcal{C}(g)},\phi)=(\mathcal{C}(g),\mathcal{F}^{-1}(\phi)),\end{aligned}$$ where $\mathcal{F}^{-1}(\phi)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}\phi(\eta)e^{iy\cdot\eta}~d\eta$ is the inverse Fourier transform of $\phi$. Since $\phi\in\mathcal{S}(\mathbb{R}^d)$, there exists $\psi\in\mathcal{S}(\mathbb{R}^d)$ such that $\phi=\widehat{\psi}$. Therefore, $$\begin{aligned} (\widehat{\mathcal{C}(g)},\phi)=(\mathcal{C}(g),\mathcal{F}^{-1}(\phi))=(\mathcal{C}(g),\psi).\end{aligned}$$ By Poisson Summation Formula [@grafakosclassical2008 p. 171], we conclude that $$\begin{aligned} \label{poissonsum2} \psi_\sharp(y,\eta)&=\sum_{p\in\mathbb{Z}^d}\psi(y+2\pi p)e^{-i(y+2\pi p)\cdot\eta}= \frac{1}{(2\pi)^{d/2}}\sum_{q\in\mathbb{Z}^d}\widehat{\psi}(\eta+q)e^{iq\cdot y}\notag\\ &=\frac{1}{(2\pi)^{d/2}}\sum_{q\in\mathbb{Z}^d}{\phi}(\eta+q)e^{iq\cdot y}. \end{aligned}$$ Now, suppose that $\phi\in\mathcal{S}(\mathbb{R}^d)$ vanishes on $\bigcup_{q\in\mathbb{Z}^d}(\bigcup_{j=2}^n\mathcal{O}_j+q)$, then $\psi_\sharp$, as obtained in , vanishes on $\bigcup_{j=2}^n\mathcal{O}_j$. Hence, $$\begin{aligned} (\widehat{\mathcal{C}(g)},\phi)&=(\mathcal{C}(g),\psi)\\ &=\sum_{j=2}^{n}\int_Y\int_{\mathcal{O}_j}(B_j-B_1)(\nabla+i\eta)g_\sharp(y,\eta)\cdot(\nabla-i\eta)\overline{\psi}_\sharp(y,\eta)~d\eta\,dy=0. \end{aligned}$$ Therefore, $\widehat{\mathcal{C}(g)}$ vanishes on the open set $\bigcup_{q\in\mathbb{Z}^d}(\bigcup_{j=2}^n\mathcal{O}_j+q)$. By Schwartz-Paley-Wiener Theorem [@rudinfunctional1991 p. 191], $\widehat{\mathcal{C}(g)}$ cannot be the Fourier transform of a compactly supported distribution, i.e., $\mathcal{C}(g)$ is not compactly supported. Therefore, $\mathcal{C}$ is not a differential operator. Proof of Theorem \[theorem:3\] {#simplicity_edge_1} ============================== In this section, we prove that a spectral edge of a periodic elliptic differential operator can be made simple through a perturbation in the coefficients. The proof essentially follows Klopp and Ralston [@KloppRalston00], with the straightforward modification that the coefficients must come from $W^{1,\infty}_\sharp(Y,\mathbb{R})$. This condition is required to ensure that the eigenfunctions and their derivatives are Hölder continuous functions. We produce the proof here for completeness. Suppose that the coefficients of the operator , $a_{kl}\in W^{1,\infty}_\sharp(Y)$. Note that the Bloch eigenvalues which are defined for $\eta\in Y^{'}$ are Lipschitz continuous in $\eta$ and may be extended as periodic functions to $\mathbb{R}^d$. In the sequel, we shall treat the Bloch eigenvalues as functions on $\mathbb{T}^d$, which is identified with $Y^{'}$ in a standard way. Also, we shall write $\lambda_j(\eta,A)$ to specify that a Bloch eigenvalue corresponds to a particular matrix $A$, appearing in the operator $\mathcal{A}$. We shall prove the theorem for an upper endpoint of a spectral gap. The proof for a lower endpoint is identical. We shall require the following lemma. Consider the operator $\mathcal{A}$ as in , with $A\in M_B^>$. Let $\lambda_0$ correspond to the upper edge of a spectral gap of $\mathcal{A}$ and let $m$ be the smallest index such that the Bloch eigenvalue $\lambda_{m}$ attains $\lambda_0$, then 1. There exist numbers $a,b\in\mathbb{R}$ such that $\lambda_{m-1}(\eta)<a<\lambda_0<\lambda_m(\eta)<b$ for all $\eta\in Y^{'}$. Further, there exists $M\in\mathbb{N}$ such that $M>m$ and the Bloch eigenvalue $\lambda_M$ satisfies $\lambda_M(\eta)>b$ for all $\eta\in Y^{'}$.\[lemma.1\] 2. Let $B$ be a symmetric matrix with $L^\infty_\sharp(Y,\mathbb{R})$-entries. There is a finite open cover of $Y^{'}$, $\{\mathcal{G}_1,\mathcal{G}_2,\ldots,\mathcal{G}_n\}$ such that for each $\mathcal{G}_j$, we have an orthonormal set in $L^2_\sharp(Y)$ of functions analytic for $\eta\in\mathcal{G}_j$ and for sufficiently small $t$, $$\label{orthonormal set} \{ \phi_m^{(j)}(\eta,A-tB), \phi_{m+1}^{(j)}(\eta,A-tB),\ldots, \phi_{R_j}^{(j)}(\eta,A-tB) \}.$$ Further, for each fixed $t$, the linear subspace generated by the functions in  contains the eigenspaces corresponding to eigenvalues of $-\nabla\cdot(A-tB)\nabla$ between $a$ and $b$.\[lemma.2\] 3. The functions in  may be chosen such that the following equation is satisfied $$\left\langle\frac{d{\phi}^{(j)}_r}{dt},\phi^{(j)}_s\right\rangle=0,\label{orthogonality1}$$ where $\langle\cdot,\cdot\rangle$ denotes the $L^2_\sharp(Y)$ inner product. \[lemma.5\] **Proof of \[lemma.1\]** As noted in Remark \[simple\_in\_neighborhood\], the Bloch eigenvalues are Lipschitz continuous functions on a compact set $\mathbb{T}^d$. Hence, the function $\eta\mapsto\lambda_m(\eta)$ is bounded above, say by $b$. Since, $\lambda_0$ is a spectral edge, $\displaystyle\delta\coloneqq\min_{\eta\in Y^{'}}\lambda_m(\eta)-\max_{\eta\in Y^{'}}\lambda_{m-1}(\eta)$ is positive. Choose $a=\lambda_0-\frac{\delta}{2}$. These choices of $a$ and $b$ satisfy our requirements. By Weyl’s law [@Reed1978], the eigenvalues of the periodic Laplacian on $Y$ satisfy the following inequality, for some $s>0$ and $C_1>0$, for large $M$, $$\begin{aligned} \label{BlochBoundedBelow1} \lambda_M(0,I)\geq\lambda_M^{N}\geq C_1 M^s, \end{aligned}$$ where $\lambda_M^{N}$ denotes the $M^{th}$ eigenvalue of the Neumann Laplacian on $Y$. By Lipschitz continuity of Bloch eigenvalues in the dual parameter, we have $$\begin{aligned} |\lambda_M(\eta,I)-\lambda_M(0,I)|\leq C|\eta|\leq C_2.\end{aligned}$$ Therefore, for all $\eta\in Y^{'}$, $$\label{BlochBoundedBelow2}\lambda_M(\eta,I)\geq \lambda_M(0,I)-C_2.$$ On combining  and , for all $\eta\in Y^{'}$, we obtain $$\begin{aligned} \lambda_M(\eta,I)\geq C_1M^s-C_2.\end{aligned}$$ It follows from a standard argument involving min-max principle, that $\lambda_M(\eta,I)\leq C_3||A^{-1}||_{L^\infty}\lambda_M(\eta,A)$. Therefore, for all $\eta\in Y^{'}$, $$\begin{aligned} \lambda_M(\eta,A)&\geq \dfrac{1}{C_3||A^{-1}||_{L^\infty}}\lambda_M(\eta,I)\notag\\ &\quad\geq \dfrac{C_1M^s}{C_3||A^{-1}||_{L^\infty}}-\dfrac{C_2}{C_3||A^{-1}||_{L^\infty}}. \end{aligned}$$ Finally to prove \[lemma.2\], choose $M$ large enough so that $$\begin{aligned} \dfrac{C_1M^s}{C_3||A^{-1}||_{L^\infty}}-\dfrac{C_2}{C_3||A^{-1}||_{L^\infty}}>b.\end{aligned}$$ **Proof of \[lemma.2\]** For each $\xi\in\mathbb{T}^d$, there is a circle $\Gamma_\xi$ in the complex plane containing the eigenvalues of $\mathcal{A}(\xi)$ between $a$ and $b$. Let $B$ be a $d\times d$ real symmetric matrix with $W^{1,\infty}_\sharp(Y)$ entries. Observe that the operator $P_\xi$ defined by $$\label{ProjectionOperator} P_\xi(\eta;A-tB)\coloneqq-\frac{1}{2\pi i}\int_{\Gamma_\xi} \left(\mathcal{A}(\eta;A-tB)-zI\right)^{-1}~dz$$ is real-analytic in a neighborhood $R_\xi$ of $\xi$ and for small $t$, where $$\begin{aligned} \mathcal{A}(\eta;A-tB)\coloneqq -(\nabla+i\eta)\cdot(A-tB)(\nabla+i\eta).\end{aligned}$$ The operator $P_\xi$ is an orthogonal projection onto the eigenspace of $\mathcal{A}(\eta;A-tB)$ corresponding to the eigenvalues between $a$ and $b$. The analyticity of the projection operator follows from the analyticity of the integrand, which is a consequence of the operator family $\mathcal{A}(\eta;A-tB)$ being a holomorphic family of type $(B)$. A proof of this fact is available in [@Sivaji2004] for perturbation in $\eta$. For a perturbation in $t$, a proof is given in Appendix \[PerturbationTheory\]. Therefore, in a neighborhood of $\eta=\xi, t=0$, we obtain an orthonormal basis for the range of $P_{\xi}(\eta,A-tB)$. In this manner, we obtain an open cover of $\mathbb{T}^d$. By compactness of $\mathbb{T}^d$, the open cover has a finite subcover $\{\mathcal{G}_1,\mathcal{G}_2,\ldots,\mathcal{G}_n\}$ with the following properties. 1. For each $\mathcal{G}_j$, we have an orthonormal set in $L^2_\sharp(Y)$ $$\begin{aligned} \{\phi^{(j)}_{m}(\eta,A-tB),\ldots,\phi^{(j)}_{R_j}(\eta,A-tB)\}\end{aligned}$$ whose elements are analytic for $\eta\in\mathcal{G}_j$ and $|t|<\delta$. 2. The linear subspace generated by $$\begin{aligned} \{\phi^{(j)}_{m}(\eta,A-tB),\ldots,\phi^{(j)}_{R_j}(\eta,A-tB)\}\end{aligned}$$ contains the eigenspaces corresponding to eigenvalues of $\mathcal{A}(\eta;A-tB)$ that lie between $a$ and $b$. **Proof of \[lemma.5\]** Let $\tilde\phi_r=\sum_{p=m} u_{rp}\phi_p$, then $$\begin{aligned} \left\langle\frac{d{\tilde{\phi_r}}}{dt},\tilde\phi_s\right\rangle=\sum_p\frac{d{u}_{rp}}{dt}\overline{u_{sp}}+\sum_{p,q}u_{rp}\overline{u_{sq}}\left\langle\frac{d{\phi}_p}{dt},\phi_q\right\rangle. \end{aligned}$$ If we set $U$ to be the matrix with entries $u_{rs}$ and $A$ to be the matrix with entries $-\left\langle\phi_r,\frac{d{\phi}_s}{dt}\right\rangle$, will hold if $$\begin{aligned} \frac{dU}{dt}=UA. \end{aligned}$$ This is solved with the initial condition $U(0)=I$. The matrix $A$ is skew-symmetric, therefore, $U(t)$ is unitary and analytic for $\eta\in\mathcal{G}$. Replace $\phi_r$ with $\tilde{\phi}_r$ to complete the proof of \[lemma.5\]. Consider the sesquilinear form $$\begin{aligned} a(\eta,t)(u,v)\coloneqq\int_Y(A-tB)(\nabla+i\eta)u\cdot(\nabla-i\eta)\overline{v}.\end{aligned}$$ For the functions constructed in \[lemma.2\], $\langle\frac{d\phi^{(j)}_r}{dt},\phi^{(j)}_s\rangle=0$ for all $r,s$. Thus, $$\begin{aligned} \frac{d}{dt}\left(a(\eta,t)(\phi_r^{(j)}(\eta,t),\phi_s^{(j)}(\eta,t))\right)=-\int_Y B(\nabla+i\eta)\phi_r^{(j)}(\eta,t)\cdot(\nabla-i\eta)\overline{\phi_s^{(j)}(\eta,t)}. \end{aligned}$$ A function $f$ defined on $\mathbb{R}^d$ is said to be $(\eta,Y)$-periodic if for all $p\in\mathbb{Z}^d$, $y\in\mathbb{R}^d$, $u(y+2\pi p)=e^{2\pi i p\cdot \eta}u(y)$. The eigenfunctions of $\mathcal{A}(\eta,A)$ with periodic boundary conditions, when multiplied by $\exp(-i\eta\cdot y)$, become eigenfunctions of $\mathcal{A}\coloneqq-\nabla\cdot A\nabla$ with $(\eta,Y)$-periodic boundary conditions, i.e., there are $\lambda$ and $u$ such that $-\nabla\cdot(A\nabla)u=\lambda u$, where $u$ is $(\eta,Y)$-periodic. Since $u$ is a complex-valued function, the regularity theorem [@Ladyzhenskaya68 Chapter 3, Section 15], cannot be applied directly. However, since the operator is linear, we may write $u=v+iw$ and express the eigenvalue equation for $u$ as two equations for the real-valued functions $v$ and $w$. In particular, $v$ and $w$ satisfy $-\nabla\cdot(A\nabla)v=\lambda v$ and $-\nabla\cdot(A\nabla)w=\lambda w$ in the interior of $Y$. Hence, by the regularity theory for elliptic equations with $W^{1,\infty}$ coefficients, $v$ and $w$ and their first-order derivatives are Hölder continuous in the interior of $Y$. Further, the Hölder estimates in the interior of $Y$ are independent of $\eta\in Y^{'}$. Consequently, $u$ and its derivatives are Hölder continuous in the interior of $Y$. Choose $\widehat{\eta}$ and $\phi_0$ such that $\mathcal{A}(\widehat{\eta},A)\phi_0=\lambda_0\phi_0.$ Choose $\phi_0\neq \exp(-i\hat{\eta}\cdot y)$. This can be achieved because the multiplicity of the Bloch eigenvalue at $\hat{\eta}$ is greater than one. Therefore, $(\nabla+i\widehat{\eta})\phi_0$ is non-zero. Consequently, there exist a $y_0$ in the interior of $Y$, an $l$ with $1\leq l\leq d$, and a $\theta>0$ such that $\left|\left(\frac{\partial}{\partial x_l}+i\widehat{\eta}_l\right)\phi_0(y_0,\widehat{\eta})\right|^2\geq\theta$. Since $\phi_0$ and its derivatives are Hölder continuous in the interior of $Y$, there is a small $\epsilon_0>0$ such that, $$\begin{aligned} \label{continuityofderivatives1}\mbox{for }|y-y_0|<\epsilon_0,\qquad\left|\left(\frac{\partial}{\partial x_l}+i{\widehat{\eta}_l}\right)\phi_0(y,\widehat{\eta})\right|^2>\frac{2\theta}{3}.\end{aligned}$$ Additionaly, since $\phi_r^{(j)}$ obtained earlier in \[lemma.2\] are linear combinations of eigenfunctions, by the Hölder continuity of the eigenfunctions and their derivatives, an $\epsilon_0$ may be chosen so that $$\begin{aligned} \label{continuityofderivatives2} \sum_{p=m}^{R_j} \left|\left(\frac{\partial}{\partial x_l}+i{\eta}_l\right)\phi_p^{(j)}(y,\eta,A)-\left(\frac{\partial}{\partial x_l}+i{\eta}_l\right)\phi_p^{(j)}(y_0,\eta,A)\right|^2<\frac{\theta}{3}, \end{aligned}$$ for $\eta\in\mathcal{G}_j$ and $|y-y_0|<\epsilon_0$. Define the matrix $B=diag(0,\ldots,0,b_l,0,\ldots,0)$ all of whose diagonal entries are zero other than $b_{l}$ which is chosen as a function $b_{l}\in C_0^{\infty}(|y-y_0|<\epsilon_0)$ such that $b_{l}\geq 0$ and $\int_Y b_{l}=1$. Extend $B$ periodically to $\mathbb{R}^d$. There is an index $q$ such that $\widehat{\eta}\in\mathcal{G}_q$. Therefore, $\displaystyle\phi_0(y,\widehat{\eta})=\sum_{r=m}^{R_q} c_r\phi_r^{(q)}(y,\widehat{\eta},A)$. Define $\phi_0(y,\widehat{\eta},t)=\displaystyle\sum_{r=m}^{R_q} c_r\phi_r^{(q)}(y,\widehat{\eta},A-tB)$. Then, by , $$\begin{aligned} \frac{d}{dt}\left(a(\widehat{\eta},t)(\phi_0(\cdot,\widehat{\eta},t),\phi_0(\cdot,\widehat{\eta},t))\right)|_{t=0}&=-\int_Y b_l\left(\frac{\partial}{\partial y_l}+i\widehat{\eta_l}\right)\phi_0(y,\widehat{\eta})\left(\frac{\partial}{\partial y_l}-i\widehat{\eta_l}\right)\overline{\phi_0(y,\widehat{\eta})}~dy\notag\\ &\qquad\leq\frac{-2\theta}{3}. \end{aligned}$$ Hence, $$\begin{aligned} \label{eq:var3} a(\widehat{\eta},t)(\phi_0(\cdot,\widehat{\eta},t),\phi_0(\cdot,\widehat{\eta},t))\leq\lambda_0-\frac{2\theta}{3}t+t^2\beta(t). \end{aligned}$$ For each $\eta\in\mathcal{G}_j$, we define the function $$\begin{aligned} \label{stareq} \phi_{*}^{(j)}(y,\eta,t)=\sum_{r=m}^{R_j}\overline{\left({\partial_l}+i{\eta}_l\right)\phi_r^{(j)}(y_0,\eta,A)}\phi_r^{(j)}(y,\eta,t). \end{aligned}$$ For $\phi(\cdot,\eta,t)=\displaystyle\sum_{k=r}^{R_j} a_r\phi_r^{(j)}(\cdot,\eta,A-tB)$, $\phi(\cdot,\eta,t)$ is perpendicular to $\phi_*^{(j)}(\cdot,\eta,t)$ if and only if $$\begin{aligned} \label{eq:perp1} \sum_{r=1}^{R_j} a_r \left({\partial_l}+i{\eta}_l\right)\phi_r^{(j)}(y_0,\eta,A) = 0. \end{aligned}$$ For $\phi(\cdot,\eta,t)$ satisfying  and $||\phi||_{L^2_\sharp(Y)}=1$, the following holds for $\eta\in\mathcal{G}_j$, $$\begin{aligned} {}&\frac{d}{dt}\left(a(\eta,t)(\phi(\cdot,\eta,t),\phi(\cdot,\eta,t))\right)|_{t=0}\\ &\quad=-\int_Y b_{l}\left(\frac{\partial}{\partial y_l}+i{\eta_l}\right)\phi(y,\eta,0)\left(\frac{\partial}{\partial y_l}-i{\eta_l}\right)\overline{\phi(y,\eta,0)}~dy\notag\\ &\quad=-\int_{B_{\epsilon_0}(y_0)}\left|\sum_{r=m}^{R_j} a_r\left(\left({\partial_l}+i{\eta}_l\right)\phi_r^{(j)}(y,\eta,A)- \left({\partial_l}+i{\eta}_l\right)\phi_r^{(j)}(y_0,\eta,A)\right)\right|^2b_{l}~dy\notag\\ &\quad\geq-\frac{\theta}{3}, \end{aligned}$$where the last inequality follows from . Therefore, the following holds true, uniformly for $\eta\in\mathcal{G}_j$ and $||\phi||_{L^2_\sharp(Y)}=1$, $$\begin{aligned} \label{eq:var4} a(\eta,t)(\phi(\cdot,\eta,t),\phi(\cdot,\eta,t))\geq\lambda_0-\frac{\theta}{3}t+t^2\gamma(t). \end{aligned}$$ To find an upper bound for $\lambda(\widehat{\eta},t)$, we apply the following variational characterization of the eigenvalues of $\mathcal{A}(\eta,t)$ to . If $\phi_1,\phi_2,\ldots,\phi_{m-1}$ are the first $m-1$ eigenfunctions corresponding to the selfadjoint operator $\mathcal{A}(\eta,t)$, then the $m^{th}$ eigenvalue of $\mathcal{A}(\eta,t)$ is given by the formula $$\begin{aligned} \lambda_{m}(\eta,t)=\min_{\phi\perp\{\phi_1,\phi_2,\ldots,\phi_{m-1}\},~||\phi||_{L^2_\sharp(Y)}=1}~a(\eta,t)(\phi,\phi).\end{aligned}$$ Therefore, $$\begin{aligned} \label{estimate1} \lambda_{m}(\widehat{\eta},t)<\lambda_0-\frac{7\theta}{12}t, \end{aligned}$$ for $t$ sufficiently small. To find a lower bound for $\lambda_{m+1}(\eta,t)$, we apply another variational characterization for the eigenvalues to  , viz., $$\begin{aligned} \lambda_{m+1}(\eta,t)=\max_{{\dim V = m}}~\min_{{\phi\perp V,~||\phi||_{L^2_\sharp(Y)}=1}}~a(\eta,t)(\phi,\phi), \end{aligned}$$ where $V$ varies over $m$-dimensional subspaces of $H^1_\sharp(Y)$. For each fixed $\eta$ and $t$, take the $m$-dimensional subspace $V$ spanned by the first $m-1$ eigenfunctions of $\mathcal{A}(\eta,t)$ and $\phi_*^{(j)}$ as defined in , i.e., $$\begin{aligned} V=\{\phi_1(\eta,t), \phi_2(\eta,t),\ldots,\phi_{m-1}(\eta,t),\phi_*^{(j)}(\eta,t)\}.\end{aligned}$$ Then, $\phi(\eta,t)$ satisfying the equation  is perpendicular to $V$ and allows us to conclude that $$\begin{aligned} \label{estimate2} \lambda_{m+1}(\eta,t)>\lambda_0-\frac{5\theta}{12}t, \end{aligned}$$ for small $t$. The two estimates obtained above  and  together imply that the perturbed spectral edge is attained by a single Bloch eigenvalue. The proof of Theorem \[theorem:3\] depends crucially on the interior Hölder continuity of the Bloch eigenfunctions and their derivatives. This requires the coefficients of the elliptic operator to have $W^{1,\infty}_\sharp(Y)$ entries. We attempt to reduce this regularity requirement to $L^\infty$ in Section \[simplicity\_edge\_2\]. Proof of Theorem \[theorem:4\] {#simplicity_edge_2} ============================== We shall prove Theorem \[theorem:4\] for an upper endpoint of a spectral gap. The proof for a lower endpoint is identical. Let $\lambda_0$ be the upper endpoint of a spectral gap of $\mathcal{A}\coloneqq -\nabla\cdot(A\nabla)$, which is achieved by the Bloch eigenvalue $\lambda_m(\eta)$ at finitely many points $\eta_1,\eta_2,\ldots,\eta_N$ in $Y^{'}$. The proof uses ideas from Parnovski and Shterenberg [@ParShteren17] and is divided into the following steps: 1. By Proposition \[corollarymultiple\], there is a single perturbation $B$ of the coefficients so that the Bloch eigenvalue $\lambda_m(\eta;A+tB)$ is simple at the points $\eta_1,\eta_2,\ldots,\eta_N$. 2. However, the perturbation creates new points at which the new spectral edge has been attained. We shall prove that given $\delta>0$, we can find perturbation parameter $t$ such that all the points at which the spectral edge is attained are within $\delta$-distance of the old spectral edge (Lemma \[continuity\_spectral\_edge\]). 3. We prove that these new spectral edges are not multiple. We shall require the following preliminaries. The multiplicity of a Bloch eigenvalue can be reduced at a finite number of points in the dual parameter by application of the same perturbation. This will be the content of the next proposition. \[corollarymultiple\] Fix $m\in\mathbb{N}$. Let $S=\{\eta_1,\eta_2,\ldots,\eta_N\}$ be a finite collection of points in $Y^{'}$. Then, there exists a matrix $B$ with $L^\infty_\sharp(Y,\mathbb{R})$-entries and a $t_0$ positive such that for all $t\in(0,t_0]$, the Bloch eigenvalue $\lambda_m(t,\eta)$ of the operator $\mathcal{A}+t\mathcal{B}=-\nabla\cdot(A+tB)\nabla$ is simple for all $\eta_n\in S$, $1\leq n\leq N$. To this end, we require the following lemma. \[hahnbanach\] Let $N\in\mathbb{N}$. Let $X$ be a normed linear space over $\mathbb{K}~(\mathbb{R} \mbox{ or } \mathbb{C})$ and let $x_1,x_2,\ldots,x_N$ be non-zero elements of $X$. Then there exists an $x^*\in X^*$ such that $\forall$ $n=1,2,\ldots,N$, $\langle x^*,x_n\rangle\neq 0$ Consider the finite dimensional subspace $F$ of $X$ spanned by $x_1,x_2,\ldots,x_N$. For each $n=1,2,\ldots,N$, let $F_n^*$ denote the subspace of $F^*$ containing $x^*\in F^*$ such that $\langle x^*,x_n \rangle=0$. Then, $F^*\neq \cup_{n=1}^N F^*_n$ since a vector space cannot be written as a finite union of its proper subspaces. Hence, there exists an $x^*\in F^*$ such that $x^*\not\in\cup_{n=1}^N F^*_n$. Hence, for all $n=1,2,\ldots,N$, $\langle x^*,x_n\rangle\neq 0$. Finally, extend $x^*$ to $X^*$ using the Hahn-Banach Theorem. As a part of the proof of Lemma \[lemma:22\], we prove that for a given $m\in\mathbb{N}$ and $\eta_0\in Y^{'}$, there exists a $t_0$ positive such that for all $t\in(0,t_0]$, the Bloch eigenvalue $\lambda(t,\eta)$ of the perturbed operator $\mathcal{A}+t\mathcal{B}$ is simple at $\eta_0$. In the present proposition, we shall make a Bloch eigenvalue $\lambda_m(\eta)$ of the operator $\mathcal{A}$ simple at a finite number of points in $Y^{'}$ through a perturbation in the coefficients. As in the proof of Proposition \[find B\], the perturbation at any $\eta_n \in S$ gives rise to a selfadjoint holomorphic family of type $(B)$, analytic in $\tau\in(-\sigma_0,\sigma_0)$, where $\sigma_0=\frac{\alpha}{2d||B||_{L^\infty}}$. Suppose that the eigenvalue $\lambda_m(\eta_n)$ of the operator $\mathcal{A}(\eta_n)$ has multiplicity $h_n$. For the perturbed operator $-\nabla\cdot(A+\tau B)\nabla$, the eigenvalue $\lambda_m(\eta_n)$ splits into $h_n$ branches. Suppose that the $h_n$ eigenvalues and eigenvectors are given as follows. For $n=1,2,\ldots,N$ and $r=1,2,\ldots,h_n$: $$\begin{aligned} \lambda_m^{r}(\tau;\eta_n)=\lambda_m(\eta_n)+\tau a_m^r(\eta_n)+\tau^2\beta_m^r(\tau,\eta_n)\\ u_m^{r}(\tau;\eta_n)=u_m^r(\eta_n)+\tau v_m^r(\eta_n)+\tau^2 w_m^r(\tau,\eta_n).\end{aligned}$$ As before, the following system of equations holds true for $n=1,2,\ldots,N$ and $r=1,2,\ldots,h_n$: $$\int_{Y}B(\nabla+i\eta_n) u_m^r(\eta_n)\cdot\overline{(\nabla+i\eta_n) u_m^s(\eta_n)}~dy=a_m^r(\eta_n)\delta_{rs}.$$ The above equations define operators that act on the unperturbed eigenspaces at each $\eta_n$. The multiplicity would go down if we find $B$ and bases for the unperturbed eigenspaces in which some off-diagonal entry, in particular, the $(1,2)$-entry is non-zero. To achieve this, we proceed as in the proof of Proposition \[find B\]. For any choice of basis of the unperturbed eigenspace at $\eta_n$, we find that either  or  holds. However, we cannot use this idea anymore, since, different $\eta_n$ would have different matrices $B$. To remedy this, we notice that, at each $\eta_n=(\eta_{n,1},\eta_{n,2},\ldots,\eta_{n,d})$, for a basis given by $\{f_n^1,f_n^2,\ldots,f_n^{h_n}\}$ either $$\label{alternative3} \sum_{l=1}^d(\partial_l+i\eta_{n,l}){f_n^1}(\partial_l-i\eta_{n,l})\overline{f_n^2}\not\equiv0,$$ or, if the above sum is zero, then in the modified basis $\{f_n^1,f_n^1+f_n^2,f_n^3,\ldots,f_n^{h_n}\}$, $$\label{alternative4} \sum_{l=1}^d(\partial_l+i\eta_{n,l}){f_n^1}(\partial_l-i\eta_{n,l})\overline{(f_n^1+f_n^2)}=\sum_{l=1}^d|(\partial_l+{i}\eta_{n,l}){f_n^1}|^2\not\equiv0,$$ provided that $f_n^1\neq \exp(-i\eta_n\cdot y)$. We can always choose $f_n^1$ to be a function different from $\exp(-i\eta_n\cdot y)$ since at any of the $\eta_n$, we have an eigenspace of dimension greater than $1$. For each $\eta_n$, call the non-zero sum between  and  as $p_n$. Further, take either $\Re(p_n)$ or $\Im(p_n)$ depending on whichever is non-zero. If both are non-zero, we may take either one. This will make sure that we have a collection of only real-valued functions. By the above procedure, we have $N$ elements of $L^1_\sharp(Y,\mathbb{R})$, again labelled as $\{p_1,p_2,\ldots,p_N\}$. By Lemma \[hahnbanach\], there is an $\alpha\in (L^1_\sharp(Y,\mathbb{R}))^*$ such that $\alpha(p_n)\neq 0$ for all $n=1,2,\ldots,N$. By duality, there exists a $\beta\in L^\infty_\sharp(Y,\mathbb{R})$ such that $\alpha(p_n)=\int_Y \beta p_n~dy\neq 0$. Define $B=diag(\beta,\beta,\ldots,\beta)$, then either, $$\Re{\int_{Y}B(\nabla+i\eta_n) f_n^1\cdot\overline{(\nabla+i\eta_n) f_n^2}~dy}\neq 0,$$ or $$\Re{\int_{Y}B(\nabla+i\eta_n) f_n^1\cdot(\nabla-i\eta_n)(\overline{f_n^1}+\overline{f_n^2})~dy}\neq 0,$$ depending on $\eta_n$. At the end of this step, the multiplicity of $\lambda_m(\eta)$ at each of the points $\eta_n$ will reduce at least by $1$. We repeat the procedure with the points among $\{\eta_1,\eta_2,\ldots,\eta_N\}$ where the eigenvalue is still multiple. Finally, we require at most $M$ steps to make the Bloch eigenvalue simple at each of these points, where $M=\displaystyle\max_{1\leq n\leq N}{h_n}$. In the next lemma, we shall prove that a spectral edge does not move very far for small perturbations in the coefficients of the periodic operator $\mathcal{A}$. We shall denote the operator $-\nabla\cdot(A+tB)\nabla$ as $\mathcal{A}+t\mathcal{B}$, where $\mathcal{A}=-\nabla\cdot(A\nabla)$ and $\mathcal{B}=-\nabla\cdot(B\nabla)$. Let $S_t$ denote the set of points at which the new spectral edge is attained, i.e., $$\begin{aligned} S_t \coloneqq \{\eta\in Y^{'} : \mbox{ The Bloch eigenvalue } \lambda_{m}(\eta;A+tB)\mbox{ attains the spectral edge at } \eta.\}\end{aligned}$$ \[continuity\_spectral\_edge\] Let $N\in\mathbb{N}$. Let $A\in M_B^>$ and let $B$ be a real symmetric matrix with $L^\infty_\sharp(Y)$-entries. Let $\mathcal{A}=-\nabla\cdot(A\nabla)$ be a periodic elliptic differential operator. Let $\lambda_0$ be the upper endpoint of a spectral gap, which is attained by the Bloch eigenvalue $\lambda_m(\eta)$ at finitely many points $\eta_1,\eta_2,\ldots,\eta_N$ in $Y^{'}$. Given a $ \delta$ belonging to the open interval $(0,1)$, there is a $t_0$ such that $$\begin{aligned} \mbox{for }t\in(0,t_0],\qquad S_t\subset \bigcup_{j=1}^{N}B(\eta_j,\delta).\end{aligned}$$ We prove this lemma by contradiction. Assume that there is a $\delta\in(0,1)$ and sequences $(t_n)$ and $(\xi_n)$ such that $t_n\to 0$ and $\xi_n\in S_{t_n}$ such that $$\forall~1\leq j\leq N,\qquad|\xi_n-\eta_j|\geq \delta.\label{farpoint}$$ Let $\lambda_0(A+tB)$ denote the spectral edge associated to the operator $\mathcal{A}+t\mathcal{B}$. The perturbed spectral edge satisfies the following inequality. $$\begin{aligned} |\lambda_0(A)-\lambda_0(A+t_nB)|& = |\min_{\eta\in Y^{'}}\lambda_m(\eta;A)-\min_{\eta\in Y^{'}}\lambda_m(\eta;A+t_nB)|\notag\\ &\quad = |-\max_{\eta\in Y^{'}}(-\lambda_m(\eta;A))+\max_{\eta\in Y^{'}}(-\lambda_m(\eta;A+t_nB))|\notag\\ &\quad\leq\max_{\eta\in Y^{'}}|\lambda_m(\eta;A)-\lambda_m(\eta;A+t_nB)|\notag\\ &\quad\leq Ct_n.\label{edgeconvergence} \end{aligned}$$ Since $(\xi_n)$ is a bounded sequence in $Y^{'}$, a subsequence of $(\xi_n)$ converges to $\hat{\xi}$, which we continue to denote by $(\xi_n)$. We shall prove that $$\begin{aligned} \label{diagconvergence} \lambda_m(\xi_n;A+t_nB)\to\lambda_m(\hat{\xi};A). \end{aligned}$$ Observe that, $$\begin{aligned} {}&\left|\int (A+t_n B)(\nabla+i\xi_n)u(\nabla-i\xi_n)\bar{u}dy-\int A(\nabla+i\hat{\xi})u(\nabla-i\hat{\xi})\bar{u}dy\right|\leq\\ &\quad\left|\int A(\nabla+i\xi_n)u(\nabla-i\xi_n)\bar{u}dy-\int A(\nabla+i\hat{\xi})u(\nabla-i\hat{\xi})\bar{u}dy\right|+t_n||B||_{L^\infty}\int (\nabla+i\xi_n)u(\nabla-i\xi_n)\bar{u}dy \end{aligned}$$ Divide throughout by $||u||^2_{L^2(Y)}$ and apply the min-max principle to obtain the following inequality. $$\begin{aligned} \label{triangle1} |\lambda_m(\xi_n;A+t_nB)-\lambda_m(\hat{\xi};A)|\leq|\lambda_m(\xi_n)-\lambda_m(\hat{\xi})|+t_n||B||_{L^\infty}|\lambda_m(\xi_n;I)|. \end{aligned}$$ In order to establish , notice that the first and second part of  converge to $0$ by the Lipschitz continuity of $\lambda_m(\cdot)$ and the boundedness of $\lambda_m(\xi_n;I)$, respectively. It follows from  and  that $\lambda_0(A)=\lambda_m(\hat{\xi};A)$ and hence, $\hat{\xi}$ is also a spectral edge. By , this contradicts the initial assumption that there are only $N$ points at which the spectral edge is attained. The spectral edge of the operator $\mathcal{A}$ is attained at finitely many points $\eta_1,\eta_2,\ldots,\eta_N$ in $Y^{'}$. Choose among $\eta_1,\eta_2,\ldots,\eta_N$ the points where the Bloch eigenvalue $\lambda_m(\eta)$ is not a simple eigenvalue. Now, apply Proposition \[corollarymultiple\] to these points, so that for the perturbed operator $\mathcal{A}+t\mathcal{B}$, the corresponding Bloch eigenvalue becomes simple at these points. The points which were simple to begin with, will remain simple for sufficiently small $t$. There is a neighborhood $\mathcal{O}_j$ of each of the points $(\eta_j)_{j=1}^N$ in which the Bloch eigenvalue is simple for a range of $t$. Each of these neighborhoods contain a ball, $B(\eta_j,\delta_j)$ of radius $\delta_j$ centered at $\eta_j$. Let $\delta\coloneqq\displaystyle\min_{1\leq j\leq N}\delta_j$, then by Lemma \[continuity\_spectral\_edge\], there exists a $t_0$ positive such that for all $t\in(0,t_0]$, the spectral edge of the perturbed operator $\mathcal{A}+t\mathcal{B}$ is contained in the union of the balls $\displaystyle\bigcup_{j=1}^N B(\eta_j,\delta)$. Hence, we have obtained a perturbation of the operator $\mathcal{A}$ such that its spectral edge is simple. An Application to the Theory of Homogenization {#homogen} ============================================== Birman and Suslina [@BirmanSuslina2003] have described homogenization as a [*spectral threshold effect*]{}. Their analysis focuses on finding norm resolvent estimates of different orders. For the operator $\mathcal{A}$, it is known that $\inf \sigma(\mathcal{A})=0$. This corresponds to the bottom edge of its spectrum. A non-zero spectral edge is called an internal edge. The notion of homogenization has been extended to internal edges in [@Birman2004], [@Birman2006]. Internal Edge Homogenization {#internal1} ---------------------------- In this subsection, we review the internal edge homogenization theorem of Birman and Suslina [@Birman2006]. Consider the equation  corresponding to the operator $\mathcal{A}$ . Let $\lambda_0$ denote an internal edge, corresponding to the upper endpoint of a spectral gap of $\mathcal{A}$ and let $m$ be the smallest index such that the Bloch eigenvalue $\lambda_{m}$ attains $\lambda_0$, then $$\begin{aligned} \lambda_0=\min_{\eta\in Y^{'}}\lambda_{m}(\eta).\end{aligned}$$ Birman and Suslina [@Birman2006] make the following regularity assumptions on $\lambda_0$. These are exactly the properties of spectral edge that are required in order to define effective mass in the theory of motion of electrons in solids [@Filonov15]. 1. \[assumptions1:1\] $\lambda_0$ is attained by the $m^{th}$ Bloch eigenvalue $\lambda_{m}(\eta)$ at finitely many points $\eta_1,\eta_2,\ldots,\eta_N$. 2. \[assumptions1:2\]For $j=1,2,\ldots,N$, $\lambda_{m}(\eta)$ is simple in a neighborhood of $\eta_j$, therefore, $\lambda_{m}(\eta)$ is analytic in $\eta$ near $\eta_j$. 3. \[assumptions1:3\] For $j=1,2,\ldots,N$, $\lambda_{m}(\eta)$ is non-degenerate at $\eta_j$, i.e., $$\begin{aligned} \lambda_{m}(\eta)-\lambda_0=(\eta-\eta_j)^T B_j (\eta-\eta_j)+{O}(|\eta-\eta_j|^3),\mbox{ for } \eta \mbox{ near } \eta_j, \end{aligned}$$ where $B_j$ are positive definite matrices. Under these assumptions, the internal edge homogenization theorem is proved. \[BirmanSuslina\] Let $\mathcal{A}$ be the operator in $L^2(\mathbb{R}^d)$ defined by  and let $\lambda_0$ be an internal edge of the spectrum of $\mathcal{A}$. Assume conditions \[assumptions1:1\], \[assumptions1:2\], \[assumptions1:3\] and let $\varkappa^2>0$ be small enough so that $\lambda_0-\varkappa^2$ is in the spectral gap. Let $\mathcal{A}^\epsilon$ denote the unbounded operator $-\nabla\cdot\left(A(\frac{x}{\epsilon})\nabla\right)$ defined in $L^2(\mathbb{R}^d)$. For $1\leq j\leq N$, let $\psi_j(y,\eta_j)\coloneqq\exp(iy\cdot\eta_j)\phi_j(y)$, where $\phi_j$ is the eigenvector corresponding to the eigenvalue $\lambda_{0}=\lambda_{m}(\eta_j)$ of the operator $\mathcal{A}(\eta_j)=-(\nabla+i\eta_j)\cdot A(\nabla+i\eta_j)$. Then, $$\begin{aligned} ||R(\epsilon)-R^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(\epsilon)\quad\mbox{as}\quad\epsilon\to 0\quad&\mbox{where},\\ R(\epsilon)=\left(\mathcal{A}^\epsilon-(\epsilon^{-2}\lambda_{0}-\varkappa^2)I\right)^{-1} \mbox{and}\quad R^0(\epsilon)\coloneqq&|Y|\sum_{j=1}^N [\psi_j^\epsilon]\left(B_j\nabla^2+\varkappa^2I\right)^{-1}[\overline{\psi_j^\epsilon}]\end{aligned}$$ are bounded operators on $L^2(\mathbb{R}^d)$ and $||\cdot||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}$ denotes the operator norm. Here, $[f]$ denotes the operation of multiplication by the function $f$. Internal Edge Homogenization for a multiple spectral edge {#internal2} --------------------------------------------------------- In this section, we shall prove a theorem corresponding to internal edge homogenization of the operator $\mathcal{A}^\epsilon=-\nabla\cdot\left(A(\frac{x}{\epsilon})\nabla\right)$ in $L^2(\mathbb{R}^d)$ in the presence of multiplicity. We shall interpret the three assumptions \[assumptions1:1\], \[assumptions1:2\], \[assumptions1:3\] that have been made on the spectral edge as hypotheses on the shape and structure of the spectral edge. Without knowledge of the shape and structure of the spectral edge, it is not possible to obtain any explicit homogenization result. Starting with a spectral edge which is not simple, we shall appeal to Theorem \[theorem:4\] to modify the spectral edge so that it becomes simple. We shall make the following assumptions on the spectral edge. We assume the finiteness of the number of points at which the spectral edge is attained, however, since the contributions from different points are added up, we may as well assume that the spectral edge is attained at one point. Therefore, suppose that for the operator , a spectral gap exists. Let $\lambda_0$ denote the upper endpoint of this spectral gap of $\mathcal{A}$ and let $m$ be the smallest index such that the Bloch eigenvalue $\lambda_{m}$ attains $\lambda_0$, then $\displaystyle\lambda_0=\min_{\eta\in Y^{'}}\lambda_{m}(\eta)$. Suppose that the spectral edge is attained at a unique point $\eta_0\in Y^{'}$. Also suppose that the eigenvalue $\lambda_0$ has multiplicity $2$. Therefore, there exists a neighborhood $\mathcal{O}$ of $\eta_0$, on which the Bloch eigenvalue $\lambda_{m}(\eta)$ is simple except at $\eta_0$. Now, a perturbation matrix $B$ with $L^\infty_\sharp(Y,\mathbb{R})$ entries, as in Theorem \[theorem:4\], is applied to the coefficients of operator $\mathcal{A}$, so that the new operator $\tilde{\mathcal{A}}(t)=\mathcal{A}+t\mathcal{B}$, has a simple spectral edge $\tilde{\lambda}_0(t)$ for sufficiently small $t$. However, the perturbed Bloch eigenvalues $\tilde{\lambda}_{m}(\eta,t)$ and $\tilde{\lambda}_{m+1}(\eta,t)$ are simple in the neighborhood $\mathcal{O}$ for small enough $t$. These properties follow from the analyticity of the projection operator , $P(\eta;A+tB)$, which is a consequence of the operator family $\tilde{\mathcal{A}}(t)$ being a holomorphic family of type $(B)$. For more details, see Appendix \[PerturbationTheory\]. For the perturbed spectral edge, we assume the following hypothesis 1. \[assumptions2:1\] $\tilde{\lambda}_{m}(\eta;t)$ attains minimum $\tilde{\lambda}_0(t)$ at a unique point $\eta_0(t)\in\mathcal{O}$ and is non-degenerate on $\mathcal{O}$, i.e., $$\begin{aligned} \tilde{\lambda}_{m}(\eta;t)-\tilde{\lambda}_0(t)=(\eta-\eta_0(t))^T \tilde{B}_0(t) (\eta-\eta_0(t))+{O}(|\eta-\eta_0(t)|^3), \end{aligned}$$ for $\eta\in\mathcal{O}$, where $\tilde{B}_0(t)$ is positive definite, i.e., there is $\alpha_0>0$, independent of $t$, such that $\tilde{B}_0(t)>\alpha_0 I$. Further, the order above holds uniformly for sufficiently $t$. 2. \[assumptions2:2\] $\tilde{\lambda}_{m+1}(\eta;t)$ attains minimum $\tilde{\lambda}_1(t)$ at a unique point $\eta_1(t)\in\mathcal{O}$ and is non-degenerate on $\mathcal{O}$, i.e., $$\begin{aligned} \tilde{\lambda}_{m+1}(\eta;t)-\tilde{\lambda}_1(t)=(\eta-\eta_1(t))^T \tilde{B}_1(t) (\eta-\eta_1(t))+{O}(|\eta-\eta_1(t)|^3), \end{aligned}$$ for $\eta\in\mathcal{O}$, where $\tilde{B}_1(t)$ is positive definite, i.e., there is $\alpha_1>0$, independent of $t$, such that $\tilde{B}_1(t)>\alpha_1 I$. Further, the order above holds uniformly for sufficiently $t$. In essence, we are asking for the Bloch eigenvalues to have the shapes before and after the perturbation as in Fig. \[figure2\] and Fig. \[figure3\]. ![Spectral Edge before perturbation.[]{data-label="figure2"}](2.pdf) ![Spectral Edge after perturbation.[]{data-label="figure3"}](3.pdf) We will now set up notation for the internal edge homogenization theorem that we intend to prove. For $j=0,1$, let $\tilde{\psi}_{m+j}(y,\eta_j(t))=\exp(iy\cdot\eta_j(t))\tilde{\phi}_{m+j}(y;t)$, where $\tilde{\phi}_{m+j}$ is a normalized eigenvector corresponding to the eigenvalue $\tilde{\lambda}_{j}(t)=\tilde{\lambda}_{m+j}(\eta_j(t))$ of $\tilde{\mathcal{A}}(\eta_j;t)=-(\nabla+i\eta_j)\cdot (A+tB)(\nabla+i\eta_j)$. In what follows, we shall choose $t={O}(\epsilon^4)$. Define the following operators $$\begin{aligned} \label{resolvent1} {R}(\epsilon)\coloneqq\left(\mathcal{A}^\epsilon-(\epsilon^{-2}\lambda_{0}-\varkappa^2)I\right)^{-1},\mbox{ and}\end{aligned}$$ $$\begin{aligned} \label{resolvent2} \tilde{R}^0(\epsilon)\coloneqq|Y|[\tilde{\psi}_m^\epsilon]\left(\tilde{B}_0(t)\nabla^2+\varkappa^2I\right)^{-1}[\overline{\tilde{\psi}_m^\epsilon}]+|Y|[\tilde{\psi}_{m+1}^\epsilon]\left(\tilde{B}_1(t)\nabla^2+\varkappa^2I\right)^{-1}[\overline{\tilde{\psi}_{m+1}^\epsilon}].\end{aligned}$$ We shall require the following two lemmas. \[resolventlemma1\] Let $$\begin{aligned} \label{resolvent3} \tilde{{R}}(\epsilon)\coloneqq\left(\tilde{\mathcal{A}}^\epsilon(t)-(\epsilon^{-2}\tilde{\lambda}_{0}(t)-\varkappa^2)I\right)^{-1}, \end{aligned}$$ where $\tilde{\mathcal{A}}^\epsilon(t)=-\nabla\cdot\left(A(\frac{x}{\epsilon})+tB(\frac{x}{\epsilon})\right)\nabla$ is an unbounded operator in $L^2(\mathbb{R}^d)$, satisfying assumptions \[assumptions2:1\] and \[assumptions2:2\]. Choose $t={O}(\epsilon^4)$. Then, $$||R(\epsilon)-\tilde{R}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(\epsilon)\quad\mbox{as}\quad \epsilon\to 0.$$ \[resolventlemma2\] With the same notation as in Lemma \[resolventlemma1\], it holds that $$||\tilde{R}(\epsilon)-\tilde{R}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(\epsilon)\quad\mbox{as}\quad \epsilon\to 0.$$ The proofs of these lemmas will be the content of subsections \[continuity\_of\_resolvents\] and \[internal\_homogenization\_result\]. Now, we state the internal edge homogenization theorem for a multiple spectral edge. \[theorem:5\] Let $\mathcal{A}$ be the operator defined in $L^2(\mathbb{R}^d)$ as $\mathcal{A}\coloneqq -\nabla\cdot(A\nabla)$. Suppose that the entries of the matrix $A$ belong to $M_B^>$. Let $\lambda_0$ be the upper edge of a spectral gap associated to operator $\mathcal{A}$. Suppose that $\lambda_0$ is attained at one point $\eta_0\in Y^{'}$ and its multiplicity is $2$. Let $\varkappa^2>0$ be small enough so that $\lambda_0-\varkappa^2$ remains in the spectral gap. Let $\mathcal{A}^\epsilon$ be defined as $\mathcal{A}^\epsilon=-\nabla\cdot\left(A(\frac{x}{\epsilon})\nabla\right)$ in $L^2(\mathbb{R}^d)$. Let $\tilde{\mathcal{A}}(t)=\mathcal{A}+t\mathcal{B}$ be a perturbation of $\mathcal{A}$ such that the perturbed operator has a simple spectral edge at $\tilde{\lambda}_0(t)$. Let $\tilde{\mathcal{A}}^\epsilon(t)=-\nabla\cdot\left(A(\frac{x}{\epsilon})+tB(\frac{x}{\epsilon})\right)\nabla$. Choose $t={O}(\epsilon^4)$. Assume conditions \[assumptions2:1\], \[assumptions2:2\] on the perturbed eigenvalues. Then, $$\begin{aligned} \label{resolventinequality0} ||R(\epsilon)-\tilde{R}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(\epsilon)\quad\mbox{as}\quad \epsilon\to 0, \end{aligned}$$ where $R(\epsilon)$ and $\tilde{R}^0(\epsilon)$ are defined in  and , respectively. Observe that $$\begin{aligned} \label{intermediateinequality} {}&||R(\epsilon)-\tilde{R}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)} \notag\\ &\quad\quad\leq||R(\epsilon)-\tilde{R}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}+||\tilde{R}(\epsilon)-\tilde{R}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}. \end{aligned}$$ Applying Lemmas \[resolventlemma1\] and \[resolventlemma2\] to , we obtain . 1. Theorem \[theorem:5\] allows the computation of the homogenized coefficients through perturbed Bloch eigenvalues. Both the crossing modes contribute to homogenization, even though the spectral edge is simple after the perturbation. 2. A perturbation of the form $\tilde{\mathcal{A}}(t)$, as mentioned in Theorem \[theorem:5\], exists for sufficiently small $t$ by Theorem \[theorem:4\]. 3. If the spectral edge is attained at finitely many points, the contribution to the effective operator from each of those points, are merely added up, as in Theorem \[BirmanSuslina\]. Hence, our assumption that the spectral edge is attained at one point is not restrictive. Further, the assumption that multiplicity of the spectral edge is $2$ can also be relaxed, since our method allows successive reduction of multiplicity of Bloch eigenvalues at multiple points. Proof of Lemma \[resolventlemma1\] {#continuity_of_resolvents} ---------------------------------- The aim of this section is to prove Lemma \[resolventlemma1\]. We begin by introducing some notation. Define the two resolvents $S(\epsilon)$ and $\tilde{S}(\epsilon)$ by $$\begin{aligned} \label{scaledresolvents} S(\epsilon)=\left(\mathcal{A}-(\lambda_{0}-\epsilon^2\varkappa^2)I\right)^{-1} \quad\mbox{and}\quad \tilde{S}(\epsilon)=\left(\tilde{\mathcal{A}}(t)-(\tilde{\lambda}_{0}(t)-\epsilon^2\varkappa^2)I\right)^{-1}\end{aligned}$$ Define $$\begin{aligned} \mathfrak{h}[u]\coloneqq\int_{\mathbb{R}^d}A\nabla u\cdot\nabla \overline{u}~dy-\lambda_0\int_{\mathbb{R}^d}|u|^2~dy.\end{aligned}$$ Then, $\mathfrak{h}$ is a closed sectorial form with domain $H^1(\mathbb{R}^d)$. Consider another form $\mathfrak{p}(t)$ with domain $H^1(\mathbb{R}^d)$ defined by $$\begin{aligned} \mathfrak{p}(t)[u]\coloneqq\int_{\mathbb{R}^d}tB\nabla u\cdot\nabla \overline{u}~dy-(\tilde{\lambda}_0-\lambda_0)\int_{\mathbb{R}^d}|u|^2~dy.\end{aligned}$$ To the sectorial forms $\mathfrak{h}$ and $\mathfrak{p}$, we shall apply the following theorem about continuity of resolvents which can be found in [@Kato1995 p. 340]. \[KLMN\] Let $\mathfrak{h}$ be a densely defined, closed sectorial form bounded from below and let $\mathfrak{p}$ be a form relatively bounded with respect to $\mathfrak{h}$, so that $D(\mathfrak{h})\subset D(\mathfrak{p})$ and $$\begin{aligned} \label{eq:hypothesis1} |\mathfrak{p}[u]|\leq a||u||^2+b\mathfrak{h}[u], \end{aligned}$$ where $0\leq b<1$, but $a$ may be positive, negative or zero. Then $\mathfrak{h}+\mathfrak{p}$ is sectorial and closed. Let $H, K$ be the operators associated with $\mathfrak{h}$ and $\mathfrak{h}+\mathfrak{p}$, respectively. If $\zeta$ is not in the spectrum of $H$ and $$\begin{aligned} \label{eq:hypothesis2} ||(a+bH)R(\zeta,H)||<1, \end{aligned}$$ then $\zeta$ is not in the spectrum of $K$ and $$\begin{aligned} \label{inequality5} ||R(\zeta,K)-R(\zeta,H)||\leq \frac{4||(a+bH)R(\zeta,H)||}{(1-||(a+bH)R(\zeta,H)||)^2}||R(\zeta,H)||. \end{aligned}$$ In order to apply the theorem, we must verify the hypotheses  and . We shall prove that $\mathfrak{p}$ is relatively bounded with respect to $\mathfrak{h}$, i.e., there exist $a,b\in\mathbb{R}$, such that: $$\begin{aligned} |\mathfrak{p}[u]|\leq a||u||^2+b\mathfrak{h}[u],\end{aligned}$$ Observe that $$\begin{aligned} \mathfrak{h}[u]\geq \alpha\int_{\mathbb{R}^d}|\nabla u|^2~dy-\lambda_0\int_{\mathbb{R}^d}|u|^2~dy,\end{aligned}$$ and $$\begin{aligned} \mathfrak{p}[u]&\leq t||B||_{L^\infty}\int_{\mathbb{R}^d}|\nabla u|^2~dy+|\tilde{\lambda}_0-\lambda_0|\int_{\mathbb{R}^d}|u|^2~dy\notag\\ &\quad=\frac{t||B||_{L^\infty}}{\alpha}\left\{\int_{\mathbb{R}^d}\alpha|\nabla u|^2~dy-\lambda\int_{\mathbb{R}^d}|u|^2~dy\right\}+\left\{|\tilde{\lambda}_0-\lambda_0|+\frac{t||B||_{L^\infty}}{\alpha}\lambda_0\right\}\int_{\mathbb{R}^d}|u|^2~dy\notag\\ &\quad= b\mathfrak{h}[u]+a||u||^2,\end{aligned}$$ where $a=\left\{|\tilde{\lambda}_0-\lambda_0|+\frac{t||B||_{L^\infty}}{\alpha}\lambda_0\right\}\approx c_1t$ and $b=\frac{t||B||_{L^\infty}}{\alpha}=c_2t$ for some constants $c_1$ and $c_2$. Next, observe that for selfadjoint operator $H$, the resolvent $R(\zeta,H)$ is a normal operator, therefore, we have (see [@Kato1995 p. 177]) $$\begin{aligned} ||R(\zeta,H)||\leq\dfrac{1}{dist(\zeta,\sigma(H))}.\end{aligned}$$ Further, $$\begin{aligned} ||(a+bH)R(\zeta,H)||&\leq||aR(\zeta,H)||+||bHR(\zeta,H)||&\notag\\ &\quad\leq\frac{a}{dist(\zeta,\sigma(H))}+||b(I-\zeta R(\zeta,H))||&\notag\\ &\quad\leq\frac{a}{dist(\zeta,\sigma(H))}+b||I||+b||\zeta R(\zeta,H)||&\notag\\ &\quad\leq\frac{a}{dist(\zeta,\sigma(H))}+b+b\frac{|\zeta|}{dist(\zeta,\sigma(H))}.\end{aligned}$$ The operator corresponding to the sectorial form $\mathfrak{h}$ is $H\coloneqq-\nabla\cdot A\nabla-\lambda_0I$, therefore, $0\in\sigma(H)$, so that, for $\zeta=-\epsilon^2\varkappa^2$ $$\begin{aligned} ||(a+bH)R(\zeta,H)||&\leq\frac{a}{\epsilon^2\varkappa^2}+2b.&\end{aligned}$$ Notice that $R(\zeta,H)=S(\epsilon)$ and $R(\zeta,K)=\tilde{S}(\epsilon)$. Let us assume that $t$ is small enough so that Theorem \[KLMN\] can be applied to the resolvents in . In particular, we have $$\begin{aligned} ||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}&=~||R(\zeta,H)-R(\zeta,K)||&\notag\\&\leq\dfrac{4(c_1t+2c_2t\epsilon^2\varkappa^2)}{(\epsilon^2\varkappa^2-c_1t-2c_2t\epsilon^2\varkappa^2)^2}.&\end{aligned}$$ Choose $t$ so that $c_1t=\epsilon^4\varkappa^2$, then, $$\begin{aligned} ||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}\leq\frac{4(1+2c_3\epsilon^2\varkappa^2)}{\varkappa^2(1-\epsilon^2-2c_3\epsilon^4\varkappa^2)^2}.\end{aligned}$$ Further, for $\epsilon^2<1/2$, $$\begin{aligned} \label{some_inequality_1} ||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}\leq\frac{16(1+c_3\varkappa^2)}{\varkappa^2(1-c_3\varkappa^2)^2}.\end{aligned}$$ Define the scaling transformation $T_\epsilon$ by $$\begin{aligned} T_\epsilon:u(y)\mapsto \epsilon^{d/2}u(\epsilon y). \end{aligned}$$ These are unitary operators on $L^2(\mathbb{R}^d)$. For the operators  and , it holds that $$\begin{aligned} R(\epsilon)=\epsilon^2T^*_\epsilon S(\epsilon)T_\epsilon\quad\mbox{and}\quad \tilde{R}(\epsilon)=\epsilon^2T^*_\epsilon \tilde{S}(\epsilon)T_\epsilon. \end{aligned}$$ Proving Lemma \[resolventlemma1\] is equivalent to proving that $$\begin{aligned} ||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}\left(\dfrac{1}{\epsilon}\right). \end{aligned}$$ In fact, in , we proved $$\begin{aligned} ||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(1). \end{aligned}$$ Proof of Lemma \[resolventlemma2\] {#internal_homogenization_result} ---------------------------------- The aim of this section is to prove Lemma \[resolventlemma2\]. Let $\left(\tilde{\lambda}_l(\eta;t)\right)_{l=1}^\infty$ and $\left(\tilde{\phi}_l(y,\eta;t)\right)_{l=1}^\infty$ be the Bloch eigenvalues and the corresponding orthonormal Bloch eigenvectors for the operator $\tilde{\mathcal{A}}(t)$, defined in Theorem \[theorem:5\]. Let, $\tilde{\psi}_l(y,\eta;t)=e^{iy\cdot\eta}\tilde{\phi}_l(y,\eta;t)$. In the sequel, we shall suppress the dependence on $t$ for notational convenience. The operator $\tilde{\mathcal{A}}$ may be decomposed in terms of the Bloch eigenvalues as in the theorem below, a proof of which may be found in [@Bensoussan2011]. Let $g\in L^2(\mathbb{R}^d)$. Define $l^{\,th}$ Bloch coefficient of $g$ as follows: $$\begin{aligned} (\tilde{\mathcal{B}}_l g)(\eta)=\int_{\mathbb{R}^d}\overline{\tilde{\psi}_l(y,\eta)}g(y)~dy,~l\in\mathbb{N},\eta\in Y^{'}. \end{aligned}$$ Then, the following inverse formula holds. $$\begin{aligned} g(y)=\sum_{l=1}^{\infty}\int_{Y^{'}}(\tilde{\mathcal{B}}_l g)(\eta){\psi}_l(y,\eta)~d\eta=&\sum_{l=1}^{\infty}(\tilde{\mathcal{B}}_l^*)(\tilde{\mathcal{B}}_l g),\quad\mbox{where},\\ (\tilde{\mathcal{B}}_l^*h)(y)=\int_{Y^{'}}h(\eta)\tilde{\psi}_l(y,\eta)\,d\eta&\quad\mbox{for}\quad h\in L^2(Y^{'}).\end{aligned}$$ In particular, the following representation holds for the operator $\tilde{\mathcal{A}}$: $$\begin{aligned} \tilde{\mathcal{A}}=\sum_{l\in\mathbb{N}}\tilde{\mathcal{B}}_l^*\tilde{\lambda}_l\tilde{\mathcal{B}}_l. \end{aligned}$$ Also, $$\begin{aligned} R(\zeta,\tilde{\mathcal{A}})=\left(\tilde{\mathcal{A}}-\zeta I\right)^{-1}=\sum_{l\in\mathbb{N}}\tilde{\mathcal{B}}_l^*(\tilde{\lambda}_l-\zeta)^{-1}\tilde{\mathcal{B}}_l. \end{aligned}$$ Define the Fourier Transform and the inverse Fourier Transform $$\begin{aligned} \left({\mathcal{F}} u\right)(\eta)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{-iy\cdot\eta}u(y)~dy,\qquad\left({\mathcal{F}^{-1}} u\right)(\eta)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{iy\cdot\eta}u(y)~dy.\end{aligned}$$ Define the operator $$\begin{aligned} \tilde{S}^0(\epsilon)\coloneqq|Y|[\tilde{\psi}_m]\left(\tilde{{B}}_0\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}[\overline{\tilde{\psi}_m}]+|Y|[\tilde{\psi}_{m+1}]\left(\tilde{{B}}_1\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}[\overline{\tilde{\psi}_{m+1}}] \end{aligned}$$ For the operators  and , it holds that $$\begin{aligned} \tilde{R}(\epsilon)=\epsilon^2T^*_\epsilon \tilde{S}(\epsilon)T_\epsilon\quad\mbox{and}\quad \tilde{R}^0(\epsilon)=\epsilon^2T^*_\epsilon \tilde{S}^0(\epsilon)T_\epsilon. \end{aligned}$$ Therefore, to prove Lemma \[resolventlemma2\], it is sufficient to prove that $$\begin{aligned} \label{tobeproved1} ||\tilde{S}(\epsilon)-\tilde{S}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}\left(\dfrac{1}{\epsilon}\right). \end{aligned}$$ By making $\mathcal{O}$ smaller if required (see \[assumptions2:1\] and \[assumptions2:2\]), we may assume that $$\begin{aligned} 2(\tilde{\lambda}_{m}(\eta)-\tilde{\lambda}_0)\geq \tilde{{B}}_0(\eta-\eta_0)^2,~\eta\in\mathcal{O},\quad \mbox{ and }\\ 2(\tilde{\lambda}_{m+1}(\eta)-\tilde{\lambda}_1)\geq \tilde{{B}}_1(\eta-\eta_1)^2,~\eta\in\mathcal{O}. \end{aligned}$$ Let $\chi$ be the characteristic function of $\mathcal{O}$, then the projections $F=\tilde{\mathcal{B}}_m^*\,\chi\,\tilde{\mathcal{B}}_m+\tilde{\mathcal{B}}_{m+1}^*\,\chi\,\tilde{\mathcal{B}}_{m+1}$ and $F^\perp=I-F$ commute with $\tilde{\mathcal{A}}$. Now, observe that $$\begin{aligned} ||\tilde{S}(\epsilon)-\tilde{S}^0(\epsilon)||_{L^2\to L^2}&=||\tilde{S}(\epsilon)F^{\perp}+\tilde{S}(\epsilon)F-\tilde{S}^0(\epsilon)F-\tilde{S}^0(\epsilon)F^\perp||_{L^2\to L^2}\notag\\ &\leq||\tilde{S}(\epsilon)F^{\perp}||_{L^2\to L^2}+||\tilde{S}(\epsilon)F-\tilde{S}^0(\epsilon)F||_{L^2\to L^2}+||\tilde{S}^0(\epsilon)F^\perp||_{L^2\to L^2} \end{aligned}$$ Thus, in order to prove , it is sufficient to prove the following: $$\begin{aligned} ||\tilde{S}(\epsilon)F^{\perp}||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(1),\label{tobeproved3}\\ ||\tilde{S}^0(\epsilon)F^\perp||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(1),\label{tobeproved4} \end{aligned}$$$$\begin{aligned} ||\tilde{S}(\epsilon)F-\tilde{S}^0(\epsilon)F||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}\left(\dfrac{1}{\epsilon}\right) \label{tobeproved2}. \end{aligned}$$ [*Proof of *]{}: Notice that the Bloch wave decomposition of $\tilde{S}(\epsilon)$ is given by $$\begin{aligned} \tilde{S}(\epsilon)=\sum_{l=1}^{\infty}\tilde{\mathcal{B}}_l^* \left(\tilde{\lambda}_l-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\tilde{\mathcal{B}}_l.\end{aligned}$$ We may write, $$\begin{aligned} \tilde{S}(\epsilon)&=\tilde{S}(\epsilon)F+\tilde{S}(\epsilon)F^\perp,& \end{aligned}$$ where $$\begin{aligned} \tilde{S}(\epsilon)F=\tilde{\mathcal{B}}_m^* \left(\tilde{\lambda}_m-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\chi \tilde{\mathcal{B}}_m+\tilde{\mathcal{B}}_{m+1}^* \left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\chi \tilde{\mathcal{B}}_{m+1}, \end{aligned}$$ and $$\begin{aligned} \label{awayfromedge1} \tilde{S}(\epsilon)F^\perp&=\sum_{l\neq m,m+1}\tilde{\mathcal{B}}_l^* \left(\tilde{\lambda}_l-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1} \tilde{\mathcal{B}}_l+\tilde{\mathcal{B}}_m^*\left(\tilde{\lambda}_m-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\left(1-\chi\right) \tilde{\mathcal{B}}_m\notag\\ &\quad\quad+\tilde{\mathcal{B}}_{m+1}^*\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\left(1-\chi\right) \tilde{\mathcal{B}}_{m+1}. \end{aligned}$$ To prove , notice that in the first term of , the sum does not include indices $m$ and $m+1$, therefore, the Bloch eigenvalues $\tilde{\lambda}_l$ are bounded away from the spectral edge $\tilde{\lambda}_0$, uniformly in $\epsilon$ and hence, the expression $\left(\tilde{\lambda}_l-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}$ is bounded independent of $\epsilon$, for $l\neq m, m+1$. Due to the non-degeneracy conditions assumed in \[assumptions2:1\] and \[assumptions2:2\], the Bloch eigenvalues $\tilde{\lambda}_m$ and $\tilde{\lambda}_{m+1}$ are bounded away from $\tilde{\lambda}_0$ outside $\mathcal{O}$, independent of $\epsilon$. Hence, the last two terms in  are bounded independent of $\epsilon$. [*Proof of *]{}: Similarly, we may write $$\begin{aligned} \tilde{S}^0(\epsilon)&=\tilde{S}^0(\epsilon)F+\tilde{S}^0(\epsilon)F^\perp,\quad\mbox{where}\\ \tilde{S}^0(\epsilon)F&=|Y|[\tilde{\psi}_m]\left(\tilde{{B}}_0\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}\chi[\overline{\tilde{\psi}_m}]+|Y|[\tilde{\psi}_{m+1}]\left(\tilde{{B}}_1\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}\chi[\overline{\tilde{\psi}_{m+1}}],\quad\mbox{and}, \end{aligned}$$ $$\begin{aligned} \label{awayfromedge2} \tilde{S}^0(\epsilon)F^\perp&=|Y|[\tilde{\psi}_m]\left(\tilde{{B}}_0\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}\left(1-\chi\right)[\overline{\tilde{\psi}_m}]&\notag\\ &\quad+|Y|[\tilde{\psi}_{m+1}]\left(\tilde{{B}}_1\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}\left(1-\chi\right)[\overline{\tilde{\psi}_{m+1}}].& \end{aligned}$$ $\tilde{S}^0(\epsilon)F^\perp$ may be further written as $$\begin{aligned} \tilde{S}^0(\epsilon)F^\perp&=|Y|[\tilde{\phi}_m]\mathcal{F}^{-1}\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}\left(1-\chi\right)\mathcal{F}[\overline{\tilde{\phi}_m}]\notag\\ &\quad+|Y|[\tilde{\phi}_{m+1}]\mathcal{F}^{-1}\left(\tilde{{B}}_1(\eta-\eta_1)^2+\epsilon^2\varkappa^2I\right)^{-1}\left(1-\chi\right)\mathcal{F}[\overline{\tilde{\phi}_{m+1}}].\notag \end{aligned}$$ The proof of  follows from the positive-definiteness of $\tilde{B}_0$ and $\tilde{B}_1$ assumed in \[assumptions2:1\] and \[assumptions2:2\], which makes the operator norm of the terms in  independent of $\epsilon$. Now, it only remains to prove . [*Proof of *]{}: Write $\tilde{S}(\epsilon)F=S_0+S_1$, where, for $j=0,1$, $$\begin{aligned} \label{res1} S_j\coloneqq\tilde{\mathcal{B}}_{m+j}^*\left(\tilde{\lambda}_{m+j}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\,\chi\, \tilde{\mathcal{B}}_{m+j}\notag\\ =X_{m+j}^* \left(\tilde{\lambda}_{m+j}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1} X_{m+j}, \end{aligned}$$ and, for $j=0,1$, $$\begin{aligned} \left(X_{m+j} u\right)(\eta)=\int_{\mathbb{R}^d}\chi\overline{\tilde{\psi}_{m+j}}(y,\eta)u(y)~dy\quad\mbox{and}\quad\left(X_{m+j}^* u\right)(y)=\int_{\mathbb{Y}^{'}}\chi\tilde{\psi}_{m+j}(y,\eta)u(y)~d\eta. \end{aligned}$$ Write $\tilde{S}^0(\epsilon)F=S_0^0+S_1^0$, where, for $j=0,1$, $$\begin{aligned} \label{res2} S_j^0&=|Y|[\tilde{\phi}_{m+j}]\mathcal{F}^{-1}\left(\tilde{{B}}_j(\eta-\eta_j)^2+\epsilon^2\varkappa^2I\right)^{-1}\left(\chi\right)\mathcal{F}[\overline{\tilde{\phi}_{m+j}}]\notag\\ &= (X^0_{m+j})^*\left(\tilde{{B}}_j(\eta-\eta_j)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m+j}, \end{aligned}$$ and, for $j=0,1$, $$\begin{aligned} \left(X^0_{m+j} u\right)(\eta)=\int_{\mathbb{R}^d}\chi e^{-iy\cdot\eta}\overline{\tilde{\phi}_{m+j}}(y,\tilde{\eta}_j)u(y)~dy\quad\mbox{and}\quad\left(X^0_{m+j} u\right)^*(\eta)=\int_{\mathbb{R}^d}\chi e^{-iy\cdot\eta}\tilde{\phi}_{m+j}(y,\tilde{\eta}_j)u(y)~dy. \end{aligned}$$ Observe that, $$\begin{aligned} ||\tilde{S}(\epsilon)F-\tilde{S}^0(\epsilon)F||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}\leq||S_0-S_0^0||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}+||S_1-S_1^0||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}. \end{aligned}$$ Therefore, to prove , it remains to prove that for $j=0,1$, $$\begin{aligned} ||{S}_j-{S}_j^0||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}\left(\dfrac{1}{\epsilon}\right). \end{aligned}$$ where $S_j, S_j^0$ are defined in , . Consider, $$\begin{aligned} \epsilon||{S}_0-{S}_0^0||&=\epsilon||X_{m}^* [\left(\tilde{\lambda}_{m}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}] X_{m}-(X^0_{m})^*\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m}||. \end{aligned}$$ Therefore, $$\begin{aligned} \label{final1} \epsilon||{S}_0-{S}_0^0||& \leq \epsilon||X_{m}^* [\left(\tilde{\lambda}_{m}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}] X_{m}-X_{m}^*\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X_{m}||&\notag\\ &\qquad+\epsilon||X_{m}^*\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X_{m}-(X^0_{m})^*\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m}||.& \end{aligned}$$ The first of the two terms on the right hand side (RHS) in the inequality  is estimated by using the following chain of inequalities. $$\begin{aligned} &\epsilon|\left(\tilde{\lambda}_{m}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1} -\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}|&\notag\\ &\qquad\leq c\epsilon|\eta-\eta_0|^3\left(\tilde{\lambda}_{m}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}&\notag\\ &\qquad\leq\left(c|\eta-\eta_0|^2\left(\tilde{{B}}_0(\eta-\eta_0)^2\right)^{-1}\right)\left(2\epsilon|\eta-\eta_0|\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}\right)\leq C_1.& \end{aligned}$$ The proof of the boundedness of the second term on the RHS in inequality  hinges on the analyticity of the Bloch eigenfunctions, and may be found in [@Birman2006]. Finally, consider $$\begin{aligned} \epsilon||{S}_1-{S}_1^0||&=\epsilon||X_{m+1}^* [\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}] X_{m+1}-(X^0_{m+1})^*\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m+1}||. \end{aligned}$$ Therefore, $$\begin{aligned} \label{final2} \epsilon||{S}_1-{S}_1^0||&\leq \epsilon||X_{m+1}^* [\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}] X_{m+1}-X_{m+1}^*\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X_{m+1}||&\notag\\ &~+\epsilon||X_{m+1}^*\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X_{m+1}-(X^0_{m+1})^*\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m+1}||.& \end{aligned}$$ The first of the two terms on RHS in inequality  is estimated by using the following chain of inequalities. $$\begin{aligned} &\epsilon|\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1} -\left(\tilde{{B}}_1(\eta-\eta_1)^2+\epsilon^2\varkappa^2I\right)^{-1}|&\notag\\ &\qquad\leq\epsilon|\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_1+\epsilon^2\varkappa^2\right)^{-1} -\left(\tilde{{B}}_1(\eta-\eta_1)^2+\epsilon^2\varkappa^2I\right)^{-1}|&\notag\\ &\qquad\leq c\epsilon|\eta-\eta_1|^3\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_1+\epsilon^2\varkappa^2\right)^{-1}\left(\tilde{{B}}_0(\eta-\eta_1)^2+\epsilon^2\varkappa^2I\right)^{-1}&\notag\\ &\qquad\leq\left(c|\eta-\eta_1|^2\left(\tilde{{B}}_1(\eta-\eta_0)^2\right)^{-1}\right)\left(2\epsilon|\eta-\eta_1|\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}\right)\leq C_2.& \end{aligned}$$ As before, the proof of the boundedness of the second term on RHS in inequality  may be found in [@Birman2006]. Perturbation Theory of holomorphic family of type $(B)$ {#PerturbationTheory} ======================================================= In this section, we show that a perturbation in the coefficients of the operator $\mathcal{A}$ gives rise to a corresponding holomorphic family of sectorial forms of type $(a)$. Further, the selfadjointness of the forms coupled with the compactness of the resolvent for the operator family ensures that it is a selfadjoint holomorphic family of type $(B)$. For definition of these notions, see Kato [@Kato1995]. Let $A\in M_B^{>}$ and $B=(b_{kl})$ be a symmetric matrix with $L^\infty_{\sharp}(Y,\mathbb{R})$ entries. Then, for $\sigma<\frac{\alpha}{d||B||_{L^\infty}}$, $A+\sigma B$ belongs to $M_B^{>}$, where $\alpha$ is a coercivity constant for $A$, as in . For a fixed $\eta_0 \in Y^{'}$ and for $\sigma_0\coloneqq \frac{\alpha}{2d||B||_{L^\infty}}$, let us define the operator family $$\begin{aligned} \mathcal{A}(\eta_0)(\tau)=-(\nabla+i\eta_0)\cdot(A+\tau B)(\nabla+i\eta_0),\quad\tau\in R,\end{aligned}$$ where $R=\{z\in\mathbb{C}:|\operatorname{Re}(z)|<\sigma_0,|\operatorname{Im}(z)|<\sigma_0\}.$ For real $\tau$, $-\sigma_0<\tau<\sigma_0$, $A+\tau B$ is coercive with a coercivity constant $\alpha/2$. The holomorphic family of sesquilinear forms $\mathfrak{t}(\tau)$ associated to operator $\mathcal{A}+\tau\mathcal{B}$, with the $\tau$-independent domain $\mathfrak{D}(\mathfrak{t}(\tau))=H^1_\sharp(Y)$, is defined as $$\begin{aligned} \mathfrak{t}(\tau)[u,v] &\coloneqq \int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{v}}{\partial y_k}~dy+i\eta_{0,l}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) u\frac{\partial \overline{v}}{\partial y_k}~dy\\ &\quad-i\eta_{0,k}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) \overline{v} \frac{\partial u}{\partial y_l}~dy+\eta_{0,l}\eta_{0,k}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) u\overline{v}~dy,\end{aligned}$$ where $\eta_0\coloneqq(\eta_{0,1},\eta_{0,2},\ldots,\eta_{0,d})$ and summation over repeated indices is assumed. $\mathfrak{t}(\tau)$ is a holomorphic family of type $(a)$. The quadratic form associated with $\mathfrak{t}(\tau)$ is as follows: $$\begin{aligned} \mathfrak{t}(\tau)[u] & \coloneqq \int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy+i\eta_{0,l}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy \\ & \quad-i\eta_{0,k}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) \overline{u} \frac{\partial u}{\partial y_l}~dy+\eta_{0,k}\eta_{0,l}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) u\overline{u}~dy. \end{aligned}$$ [**$(i)$**]{} $\mathfrak{t}(\tau)$ is sectorial. Let us write $\tau=\rho+i\gamma$, then the quadratic form $\mathfrak{t}(\tau)$ can be written as the sum of its real and imaginary parts: $$\begin{aligned} \mathfrak{t}(\tau)= \Re{\mathfrak{t}(\tau)[u]}+ i\Im{\mathfrak{t}(\tau)[u]} \end{aligned}$$ where the real part is $$\begin{aligned} \label{realpart} \Re{\mathfrak{t}(\tau)[u]} & \coloneqq \int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy+i\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy\notag \\ &\quad -i\eta_{0,k}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) \overline{u} \frac{\partial u}{\partial y_l}~dy+\eta_{0,k}\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\overline{u}~dy, \end{aligned}$$ and the imaginary part is $$\begin{aligned} \label{imaginarypart} \Im{\mathfrak{t}(\tau)[u]} & \coloneqq \int_Y \left(\gamma b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy+2\Im{\left(\eta_{0,l}\int_Y \left(\gamma b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy\right)}\notag \\ &\quad +\eta_{0,k}\eta_{0,l}\int_Y \left(\gamma b_{kl}(y)\right) u\overline{u}~dy. \end{aligned}$$ The real part  of ${\mathfrak{t}(\tau)[u]}$ may also be written as $$\begin{aligned} \label{realpart2} \Re{\mathfrak{t}(\tau)[u]} & \coloneqq \int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy+2\Re{\left(i\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy\right)}\notag \\ &\quad +\eta_{0,k}\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\overline{u}~dy. \end{aligned}$$ The first term in  is estimated from below as follows: $$\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy \geq \frac{\alpha}{2}\int_{Y}|\nabla u|^2~dy.\label{eq:111}$$ The second term in  may be bounded from above as follows: $$\begin{aligned} \Re{\left(i\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy\right)} & \leq C_1||u||_{L^2_{\sharp}(Y)}||\nabla u||_{L^2_{\sharp}(Y)}\notag \\ & \leq C_1C_*||u||^2_{L^2_{\sharp}(Y)}+\frac{C_1}{C_*}||\nabla u||^2_{L^2_{\sharp}(Y)}\notag \\ & = C_2||u||^2_{L^2_{\sharp}(Y)}+\frac{\alpha}{4}||\nabla u||^2_{L^2_{\sharp}(Y)},\label{eq:222} \end{aligned}$$ where $C_*=\frac{4C_1}{\alpha}$ and $C_1, C_2$ are some constants independent of $u$ and $\rho$. The last term in  is estimated as $$\begin{aligned} \eta_{0,k}\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\overline{u}~dy \leq C_3 ||u||^2_{L^2_{\sharp}(Y)},\label{eq:333} \end{aligned}$$ for some $C_3>0$. Finally, combining , and , we obtain $$\begin{aligned} \Re{\mathfrak{t}(\tau)[u]} \geq \frac{\alpha}{4}||u||^2_{H^1_{\sharp}(Y)}-C_4||u||^2_{L^2_{\sharp}(Y)},\label{eq:444} \end{aligned}$$ for some $C_4>0$. Estimating the imaginary part  from above, we obtain $$\begin{aligned} |\Im{\mathfrak{t}(\tau)[u]}| \leq C_5||\nabla u||^2_{L^2_\sharp(Y)}+C_6||u||^2_{L^2_\sharp(Y)},\label{eq:999} \end{aligned}$$ for some positive $C_5, C_6$. Now, choose a scalar $C_7$ so that $C_7=\frac{4C_5}{\alpha}$. The inequality may be written as $$\begin{aligned} \Re{\mathfrak{t}(\tau)[u]} + C_4||u||^2_{L^2_{\sharp}(Y)} + \frac{C_6}{C_7}||u||^2_{L^2_{\sharp}(Y)} \geq \frac{\alpha}{4}||u||^2_{H^1_{\sharp}(Y)}+\frac{C_6}{C_7}||u||^2_{L^2_{\sharp}(Y)}.\label{eq:555} \end{aligned}$$ Now, we define a new quadratic form $\tilde{\mathfrak{t}}[u]\coloneqq \mathfrak{t}[u]+(C_4+\frac{C_6}{C_7})||u||^2_{L^2_\sharp{Y}},$ then inequality becomes $$\begin{aligned} \Re{\tilde{\mathfrak{t}}(\tau)[u]} \geq \frac{\alpha}{4}||u||^2_{H^1_{\sharp}(Y)}+\frac{C_6}{C_7}||u||^2_{L^2_{\sharp}(Y)}.\label{eq:666} \end{aligned}$$ This may be further written as $$\begin{aligned} \Re{\tilde{\mathfrak{t}}(\tau)[u]} - \frac{\alpha}{4}||u||^2_{L^2_\sharp(Y)} \geq \frac{\alpha}{4}||\nabla u||^2_{L^2_{\sharp}(Y)}+\frac{C_6}{C_7}||u||^2_{L^2_{\sharp}(Y)}.\label{eq:777} \end{aligned}$$ On multiplying throughout by $C_7$, the inequality becomes $$\begin{aligned} C_7\left\{\Re{\tilde{\mathfrak{t}}(\tau)[u]} - \frac{\alpha}{4}||u||^2_{L^2_\sharp(Y)}\right\} \geq C_5||\nabla u||^2_{L^2_{\sharp}(Y)}+{C_6}||u||^2_{L^2_{\sharp}(Y)}.\label{eq:888} \end{aligned}$$ Since $\Im{\tilde{\mathfrak{t}}[u]}=\Im{\mathfrak{t}[u]}$, combining the inequalities and , we obtain $$\begin{aligned} |\Im{\tilde{\mathfrak{t}}(\tau)[u]}| \leq C_7\left\{\Re{\tilde{\mathfrak{t}}(\tau)[u]} - \frac{\alpha}{4}||u||^2_{L^2_\sharp(Y)}\right\}. \end{aligned}$$ This proves that the form $\tilde{\mathfrak{t}}$ is sectorial. However, the property of sectoriality is invariant under a shift. Therefore, $\mathfrak{t}$ is sectorial, as well. [**$(ii)$**]{} $\mathfrak{t}(\tau)$ is closed. This follows from the inequality . If $u_n\xrightarrow{{\mathfrak{t}-convergence}}u$ then $\Re{\mathfrak{t}[u_n-u_m]}\to 0$ as $n,m\to\infty$. By , $(u_n)$ is a Cauchy sequence in $H^1_\sharp(Y)$. By completeness, there is $v\in H^1_\sharp(Y)$ to which the sequence converges. However, $\mathfrak{t}$-convergence implies $L^2$ convergence, and therefore, $u=v$. Clearly, $\mathfrak{t}[u_n-u]\to 0$. [**$(iii)$**]{} $\mathfrak{t}(\tau)$ is a holomorphic family of type $ (a) $. We have proved that $\mathfrak{t}(\tau)[u]$ is sectorial and closed. It remains to prove that the form is holomorphic. This is easily done since $\mathfrak{t}(\tau)[u]$ is linear in $\tau$ for each fixed $u\in H^1_\sharp(Y)$. The first representation theorem of Kato ensures that there exists a unique m-sectorial operator with domain contained in $H^1_\sharp(Y)$ associated with each $\mathfrak{t}(\tau)$. A proof may be found in [@Kato1995 p.322]. The family of such operators associated with a holomorphic family of sesquilinear forms of type $(a)$ is called a holomorphic family of type $( B )$. The aforementioned m-sectorial operator is given by $$\begin{aligned} \mathcal{A}(\eta_0)(\tau)=-(\nabla+i\eta_0)(A+\tau B)(\nabla+i\eta_0).\end{aligned}$$ It follows from the symmetry of the matrix $A+\tau B$ that the family $\mathcal{A}(\eta_0)(\tau)$ is a selfadjoint holomorphic family of type $(B)$. Moreover, by the compact embedding of $H^1_\sharp(Y)$ in $L^2_\sharp(Y)$, the operator $\mathcal{A}(\eta_0)(\tau)+C^*I$ has compact resolvent for each $\tau\in R$ for some appropriate constant $C^*$, independent of $\tau\in R$. Hence, by Kato-Rellich Theorem, there exists a complete orthonormal set of eigenvectors associated with the operator family $\mathbb{A}(\eta_0)(\tau)$ which are analytic for the whole interval $-\sigma_0<\tau<\sigma_0$.

ArXiv

--- abstract: | The one-dimensional partially asymmetric simple exclusion process with open boundaries is considered. The stationary state, which is known to be constructed in a matrix product form, is studied by applying the theory of $q$-orthogonal polynomials. Using a formula of the $q$-Hermite polynomials, the average density profile is computed in the thermodynamic limit. The phase diagram for the correlation length, which was conjectured in [@me99](J. Phys. A [**32**]{} (1999) 7109), is confirmed. \[Keywords: asymmetric simple exclusion, exact solution, density profile, $q$-orthogonal polynomials\] author: - | Tomohiro SASAMOTO\ [*Department of Physics, Graduate School of Science,*]{}\ [*University of Tokyo,*]{}\ [*Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan*]{} title: 'Density Profile of the One-Dimensional Partially Asymmetric Simple Exclusion Process with Open Boundaries' --- Introduction ============ \[intro\] The one-dimensional asymmetric simple exclusion process (ASEP) [@L; @Sp] is a system of particles which hop preferentially in one direction on a one-dimensional lattice with hard-core exclusion interaction. The ASEP has been studied extensively since it is one of the few models which show rich non-equilibrium behaviors and is exactly solvable [@Derrida98]. Besides, the ASEP has applications to many interesting problems such as the hopping conductivity, growth processes and the traffic flows [@SZ]. In this article, we consider the stationary state of the ASEP with open boundary conditions. That is, the system is connected to particle resevoirs at boundaries. The case where particles can hop only in one direction, which we refer to as the “totally asymmetric” case in the sequel, was solved in [@DEHP; @SD]. The current and the density profile were calculated exactly in the thermodynamic limit. The phase diagram for the current and the correlation length were identified. The system exhibits phase transitions depending on the parameters at the boundaries. Recently the obtained phase diagram was discussed from the point of view of the domain wall dynamics [@KSKS]. The partially asymmetric case with the open boundary conditions was partially solved in [@me99]. The current was evaluated in the thermodynamic limit. The phase diagram for the current was identified. It turned out to be the same as the one obtained by mean-field approximation [@ER] or by employing a plausible assumption [@Sandow]. The phase diagram for the correlaiton length was also obtained by assuming that the correlation length is given by the logarithm of the ratio of the largest and the second largest eigenvalue of a certain matrix which plays a similar role as a transfer matrix does in equilibrium statistical mechanical models. It was shown that the phase diagram has a richer structure than that for the totally asymmetric case. The average density profile was, however, not calculated in [@me99]. In this sence the obtained phase diagram for the correlation length has remained a conjecture. The purpose of this paper is the confirmation of this phase diagram. By using the explicit formula for the Poisson kernel of the $q$-Hermite polynomials, the average density profile in the thermodynamic limit is calculated for the partially asymmetric case. It turns out that the phase diagram was correctly predicted in [@me99]. In this article, we only consider the case where hoppings of particles at the boundaries and those at the bulk part of the system are compatible. In other words, when we allow the particle input at the left boundary and the particle output at the right boundary, the hopping rate to the right is assumed to be larger than that to the left. When hoppings at the boundaries and those at bulk is imcompatible, the current becomes zero in the thermodynamic limit. The situation seems to be similar to the closed boundary condition where particles can not enter or go out of the system [@SS]. Off course, when we consider the finite chain, the current remains to be positive. We remark that the asymptotic current for this case was evaluated in [@BECE]. The paper is organized as follows. In the next section, the definition of the model is given in terms of the master equation. The so-called matrix product ansatz, which gives the stationary state in the form of matrix product, is also explained. Some properties of the $q$-Hermite polynomials and the relationship to the matrix product ansatz are explained in section \[q-H\]. The section \[density\] is the main section of this article. First, the one-point funciton is represented in the form of double integrals. Second, the average density profile in the thermodynamic limit are summerlized whereas the evaluation of the integrals are relegated to Appendices. The phase diagram for the correlation length is identified. The concluding remarks are given in the last section. Definition of Model and Matrix Product Ansatz ============================================= The one-dimensional asymmetric simple exclusion process (ASEP) is defined as follows. During the infinitesimal time interval $\d t$, each particle jumps to the right nearest neighboring site with probability $p_R \d t$ and to the left nearest neighboring site with probability $p_L \d t$. If the chosen site is already occupied, the particle does not move due to the exclusion rule. More than one particle can not be on the same site. Each site can be either empty or occupied. The case where particles can hop only in one direction, i.e., the case where either $p_L = 0$ or $p_R=0$ is called the “totally asymmetric” case. The $p_R=p_L$ case is called the “symmetric” case whereas the case where particles hop in both directions with different rates will be referred to as the “partially asymmetric” case. In addition, we allow the particle input at the left end of the chain with rate $\alpha$ and allow the particle output at the right end of the chain with rate $\beta$ (Fig. 1). Here the length of the chain is denoted by $L$. In this article, we restrict our attention to the partially asymmetric case since the totally asymmetric case and the symmetric case was already solved in [@DEHP; @SD] and in [@me96] respectively. The restrictions on the parameters are $0 < p_L < p_R$ and $\alpha,\beta>0$. More formally, the process is defined in terms of the master equation. Each configuration of the system is indicated by $\{\tau_1,\tau_2,\ldots,\tau_L\}$ where $\tau_j$ $(j=1,2,\ldots,L)$ denotes the particle number at site $j$. Namely $\tau_j=0$ if the site $j$ is empty whereas $\tau_j=1$ if the site $j$ is occupied. Let $P(\tau_1,\tau_2,\ldots,\tau_L;t)$ denote the probability that the system has the configulation $\{ \tau_1,\tau_2,\ldots,\tau_L\}$ at time $t$. Then the time evolution of the ASEP is described by the following master equation, $$\begin{aligned} &\quad \frac{\d}{\d t} P(\tau_1,\tau_2,\ldots,\tau_L;t) \notag \\ &= \alpha (2\tau_1-1) P(0,\tau_2,\ldots,\tau_L;t) \notag \\ &\quad + \sum_{j=1}^{L-1} (\tau_j-\tau_{j+1}) \left[ p_L P(\tau_1,\tau_2,\ldots,0,1,\ldots,\tau_L;t) \right. \notag \\ &\quad \left. - p_R P(\tau_1,\tau_2,\ldots,1,0,\ldots,\tau_L;t) \right] \notag \\ &\quad + \beta (1-2\tau_L) P(\tau_1,\tau_2,\ldots,\tau_{L-1},1;t). \label{mas-eq}\end{aligned}$$ For instance, the master equation for $L=2$ case reads $$\label{mas-eq-L2} \frac{\d}{\d t} \begin{bmatrix} P(00;t)\\ P(01;t)\\ P(10;t)\\ P(11;t) \end{bmatrix} = - \begin{bmatrix} \alpha & -\beta & 0 & 0\\ 0 & \alpha + p_L +\beta & -p_R & 0\\ -\alpha & -p_L & p_R & -\beta\\ 0 & -\alpha & 0 & \beta \end{bmatrix} \begin{bmatrix} P(00;t)\\ P(01;t)\\ P(10;t)\\ P(11;t) \end{bmatrix}.$$ One can confirm himself that the dynamics of the ASEP is correctly encoded in the master equation (\[mas-eq\]). When time $t$ goes to infinity, the system is expected to reach the stationary state. The probability distribution in the stationary state will be denoted as $P(\tau_1,\tau_2,\ldots,\tau_L)$. For instance, except for the normalization, the stationary state for $L=2$ case is the eigenvector of the $4\times 4$ matrix in the right hand side of (\[mas-eq-L2\]) with the eigenvalue zero. Explicitly, it reads $$\label{stationary-2} \begin{bmatrix} P(00)\\ P(01)\\ P(10)\\ P(11) \end{bmatrix} = \text{Const.} \begin{bmatrix} \frac{1}{\alpha^2} \\ \frac{1}{\alpha\beta}\\ \frac{1}{p_R} \left( \frac{p_L}{\alpha\beta} + \frac{1}{\alpha} +\frac{1}{\beta} \right) \\ \frac{1}{\beta^2} \end{bmatrix}.$$ In [@DEHP], it was shown that the probability distribution of the ASEP in the stationary state for general $L$ can be written in the form of the matrix product as $$\label{originalMPA} P(\tau_1,\tau_2,\ldots,\tau_L) = \frac{1}{Z_L} \langle W| \prod_{j=1}^{L}(\tau_j D + (1-\tau_j) E)| V\rangle,$$ where $D$ and $E$ are square matrices and $\langle W|$ and $|V\rangle$ are vectors satisfying following relations, $$\begin{gathered} \label{mat-cond1} p_R DE-p_L ED = \zeta (D+E), \\ \label{mat-cond2} \alpha \langle W| E = \zeta\langle W|, \quad \beta D |V\rangle = \zeta |V\rangle.\end{gathered}$$ Here $\zeta$ is an arbitrary number. If one defines the matrix $C$ by $$\label{eq:def-c} C=D+E,$$ the normalization $Z_L$ is given by $$Z_L = \langle W| C^L |V\rangle.$$ Here, for the case of $L=2$, we check that the state (\[originalMPA\]) indeed gives the stationary state of the process by using the algebraic relations (\[mat-cond1\]) and (\[mat-cond2\]). For $L=2$, (\[originalMPA\]) reads $$\begin{bmatrix} P(00) \\ P(01) \\ P(10) \\ P(11) \end{bmatrix} = \frac{1}{Z_2} \begin{bmatrix} \langle W| E^2 |V\rangle \\ \langle W| ED |V\rangle \\ \langle W| DE |V\rangle \\ \langle W| D^2 |V\rangle \end{bmatrix}.$$ Three components, $P(00),P(01),P(11)$ can be calculated by simply using (\[mat-cond2\]). On the other hand, one computes $P(10)$ first changing the order of matrices $D,E$ by (\[mat-cond1\]) and then using (\[mat-cond2\]). Hence we get $$\begin{bmatrix} P(00) \\ P(01) \\ P(10) \\ P(11) \end{bmatrix} = \frac{1}{Z_2} \begin{bmatrix} \frac{\zeta^2}{\alpha^2}\\ \frac{\zeta^2}{\alpha\beta}\\ \frac{\zeta^2}{p_R} \left( \frac{p_L}{\alpha\beta} + \frac{1}{\alpha} +\frac{1}{\beta} \right)\\ \frac{\zeta^2}{\beta^2} \end{bmatrix}.$$ One can compare this expression with (\[stationary-2\]) to see that this expression indeed gives the stationary state for $L=2$ case. One also sees that the arbitrary parameter $\zeta$ appears in the same way for all components, $P(00),P(01),P(10),P(11)$. Changing the parameter $\zeta$ only changes the normalization $Z_2$. This is true for general $L$ as well. In this article, the proof that the state (\[originalMPA\]) gives the stationary state for general $L$ is not given. See [@DEHP]. We express several physical quantities in the form of matrix products. The one-point function $\langle n_j \rangle_L$ is defined as the probability that the site $j$ is occupied. In other words, $\langle n_j \rangle_L$ is the average density at site $j$. The two-point function $\langle n_j n_k\rangle_L$ is defined as the probability that the sites $j$ and the site $k$ are both occupied. Higher correlation functions are defined similarly. In the matrix language, they are computed by $$\begin{aligned} \label{def-1pt} \langle n_j \rangle_L &= \langle W| C^{j-1} D C^{L-j} |V\rangle /Z_L , \\ \label{def-2pt} \langle n_j n_k\rangle_L &= \langle W| C^{j-1} D C^{k-j-1} D C^{L-k}|V\rangle /Z_L ,\end{aligned}$$ and so on. The current through the bond between site $j$ and site $j+1$ is defined by $ J_L^{(j)} = p_R \langle n_j (1-n_{j+1})\rangle - p_L \langle n_j (1-n_{j-1})\rangle. $ In the steady state, the current is independent of $j$ and hence is denoted by $J_L$. It is given by $$\label{def-current} J_L = \zeta \frac{\langle W|C^{L-1}|V\rangle} {\langle W|C^{L}|V\rangle} = \zeta \frac{Z_{L-1}} {Z_L}.$$ Once one finds a representation of these algebraic relations, by using the above formula, one can in principle calculate the physical quantities such as the particle current $J_L$, the one-point function $\langle n_j \rangle_L$, the two-point function $\langle n_j n_k\rangle_L$ and the higher correlation functions. We note that the process has an obvious particle-hole symmetry. When we look at holes instead of particles, they tend to hop to the left with rate $p_R$ and to the right with rate $p_L$ with hard-core exclusion. In addition, they are injected at right end with rate $\beta$ and they are removed at the left end with rate $\alpha$. In other words, the process is invariant under the changes, $$\begin{aligned} \notag \text{particle} &\leftrightarrow \text{hole} \\ \label{symmetry} \alpha &\leftrightarrow \beta \\ \notag \text{site number} \,\, j &\leftrightarrow \text{site number} \,\, L-j+1.\end{aligned}$$ Due to this symmetry, it is sufficient to obtain the density for the right half of the system. The density for the left half of the system is obtained by using the above symmetry as $$\label{ri-le} \langle n_j \rangle_L (\alpha,\beta) = 1-\langle n_{L-j+1} \rangle_L (\beta,\alpha),$$ where the dependence of $\langle n_j \rangle_L$ on the parameters $\alpha$ and $\beta$ are eplicitly indicated. Before closing the section, we present some simulation results (Fig. 2 and Fig. 3). Figure 2 shows the space-time diagrams for several choices of parameters. It is clear that the properties of the system crutially depend on the values of the boundary parameters. The differeces become more transparent when we consider the particle current or the the average density profile of the stationary state. In principle, the stationary state is achieved only in the infinite time limit. However, we see from Fig. 2 that the system practically goes into a stationary state after some transient time. Hence if we average the density over a long time after the transient time, it would be regarded as the average density profile of the stationary state practically. The results are shown in Fig. 3. When $\alpha$ is small and $\beta$ is large, the bulk density is low. It decays sharply near the right boundary. This is called the low density phase. Conversely, when $\alpha$ is large and $\beta$ is small, the bulk density is high. It decays sharply near the left boundary. This is called the high density phase. When $\alpha=\beta$ is small, the low density region and the high density region coexist (coexistence line). Finally, when both $\alpha$ and $\beta$ are large enough, the density takes the value $1/2$ at bulk and decays slowly near both the boundaries. This is called the maximal current phase. Our main tasks in the following are to obtain the average density profiles in Fig. 4 exactly. Representation of Algebra and $q$-Hermite Polynomials ===================================================== \[q-H\] First we introduce some notations for later convenience. We introduce the $q$-number, $$\{n\} = 1-q^n,$$ and the $q$-shifted factorial, $$\begin{aligned} (a;q)_n &= \label{q-shi-fac-1} (1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}), \\ (a;q)_0 &= 1. \label{q-shi-fac-2}\end{aligned}$$ We also define $$\label{q-prod-inf} (a;q)_{\infty} = \prod_{j=0}^{\infty}(1-aq^j),$$ for $|q|<1$. Since products of $q$-shifted factorials appear so often, we use the notations, $$\begin{aligned} (a_1,a_2,\cdots,a_k;q)_{\infty} &= (a_1;q)_{\infty}(a_2;q)_{\infty}\cdots (a_k;q)_{\infty}, \\ (a_1,a_2,\cdots,a_k;q)_{n} &= (a_1;q)_{n}(a_2;q)_{n}\cdots (a_k;q)_{n} .\end{aligned}$$ Next a representation of the algebraic relaiton (\[mat-cond1\]) and (\[mat-cond2\]) is given. If we define $$D = 1+d, \hspace{10mm} E = 1+e,$$ we see that these relations become $$\begin{gathered} \label{q-b} d e - q e d = 1-q, \\ \label{w-coh} \langle W| e = a \langle W|, \hspace{10mm} d |V\rangle = b |V\rangle,\end{gathered}$$ where we put $$\begin{gathered} q\ = p_L/p_R, \\ a = \frac{1-\tilde{\alpha}} {\tilde{\alpha}}, \quad b = \frac{1-\tilde{\beta}} {\tilde{\beta}},\end{gathered}$$ with $\tilde{\alpha}=\alpha/(p_R-p_L), \tilde{\beta}=\beta/(p_R-p_L)$. Since $0< p_R<p_L$, we have $0<q<1$. In this article, we take the following representation for the matrices $d, e$ and the vectors $\langle W|, |V\rangle$, $$\begin{gathered} \label{de-qb} d = \begin{bmatrix} 0 & \{1\}^{\frac{1}{2}} & 0 & 0 & \cdots \\ 0 & 0 & \{2\}^{\frac{1}{2}} & 0 & \\ 0 & 0 & 0 & \{3\}^{\frac{1}{2}} & \\ \vdots & & & \ddots &\ddots \\ \end{bmatrix}, \hspace{10mm} e = \begin{bmatrix} 0 & 0 & 0 & 0 & \cdots \\ \{1\}^{\frac{1}{2}} & 0 & 0 & 0 & \\ 0 & \{2\}^{\frac{1}{2}} & 0 & 0 & \\ 0 & 0 & \{3\}^{\frac{1}{2}} & 0 & \\ \vdots & & & \ddots & \ddots \end{bmatrix}, \\ \label{WV-qb} \langle W| = \kappa\,_c\langle a| = \kappa \left( 1, \frac{a}{\sqrt{(q;q)_1} }, \frac{a^2}{ \sqrt{(q;q)_2} },\ldots \right), \hspace{10mm} |V \rangle = \kappa \, |b\rangle_c = \kappa \left( \begin{matrix} 1\\ \displaystyle\frac{b}{ \sqrt{(q;q)_1}}\\ \displaystyle\frac{b^2}{ \sqrt{(q;q)_2} }\\ \vdots \end{matrix} \right).\end{gathered}$$ The constant $\kappa$ is takes as $\kappa^2=(ab;q)_{\infty}$ so that $\langle W|V\rangle=1$. It should be noticed that there exists another useful representaion of the algebraic relations (\[mat-cond1\]) and (\[mat-cond2\]). It was first given in [@DEHP] and was used to obtain the phase diagram of the correlation length in [@me99]. The advantage of the represetation (\[de-qb\]) and (\[WV-qb\]) is that the commutation relation of the matrices $d$ and $e$ turns out to be a simple diagonal matrix. We have $$\label{de-com} de-ed = (1-q) \begin{bmatrix} 1 & 0 & 0 & 0 & \cdots \\ 0 & q & 0 & 0 & \\ 0 & 0 & q^2 & 0 & \\ 0 & 0 & 0 & q^3 & \\ \vdots & & & & \ddots \end{bmatrix}.$$ This fact will play an important role for the calculation of the average density profile in the next section. Next we list some properties of the continuous $q$-Hermite polynomials . The proofs can be found for instance in [@AAR; @GR]. The continuous $q$-Hermite polynomials $\{ H_n(x|q) |\, n=0,1,2,\ldots\}$ are defined by the three term recurrence relation, $$\label{rec-qh} H_{n+1}(x;q) + (1-q^n)H_{n-1}(x;q) = 2 x H_n(x;q),$$ with the initial condition, $$\label{init-cond} H_{-1}(x;q) = 0, \hspace{10mm} H_0(x;q) = 1.$$ They are explicitly given by the formula, $$H_n(\cos\theta|q) = \sum_{k=0}^{n} \frac{(q;q)_n }{ (q;q)_k (q;q)_{n-k} } e^{i(n-2k)\theta}.$$ The orthogonality relation reads $$\label{ortho} \int_{0}^{\pi} H_n(\cos\theta|q) H_m(\cos\theta|q) (e^{2i\theta},e^{-2i\theta};q)_{\infty}\d \theta = 2\pi \frac{ (q;q)_n }{ (q;q)_{\infty} } \delta_{mn}.$$ The generating function is also known and is given by $$\label{gen} \sum_{n=0}^{\infty} \frac{ H_n(\cos\theta|q) }{ (q;q)_n } \lambda^n = \frac{1}{(\lambda e^{i\theta},\lambda e^{-i\theta};q)_{\infty} }$$ for $|\lambda| < 1$. To calculate the average density profile, we also need the so-called Poisson kernel, $$\label{Mehler} \sum_{n=0}^{\infty} \frac{ H_n(\cos\theta|q) H_n(\cos\varphi|q) r^n } {(q;q)_n} = \frac{ (r^2;q)_{\infty} } { (r e^{i(\theta+\varphi)}, r e^{-i(\theta+\varphi)}, r e^{i(\theta-\varphi)}, r e^{-i(\theta-\varphi)} ;q)_{\infty} }.$$ This formula is called $q$-Mehler formula in the mathematics literature. Here we notince that, if we introduce $$\label{p-Hermite} p_n(x) = H_n(x|q)/\sqrt{(q;q)_n},$$ the three-term recurrence relation is rewritten into the form, $$\label{p-eigen} \begin{bmatrix} 0 & \{1\}^{\frac{1}{2}} & 0 & 0 & \cdots \\ \{1\}^{\frac{1}{2}} & 0 & \{2\}^{\frac{1}{2}} & 0 & \\ 0 & \{2\}^{\frac{1}{2}} & 0 & \{3\}^{\frac{1}{2}} & \\ \vdots & & \ddots & \ddots &\ddots \\ \end{bmatrix} \begin{bmatrix} p_ 0(x)\\ p_ 1(x)\\ p_ 2(x)\\ \vdots \end{bmatrix} = 2x \begin{bmatrix} p_ 0(x)\\ p_ 1(x)\\ p_ 2(x)\\ \vdots \end{bmatrix}.$$ In other words, $|p(x)\rangle =\,^t(p_0(x), p_1(x), \ldots)$ is formally an eigenvector of the matrix $d+e$ with eigenvalue $2x$. This is the basic relationship between the representaion of the algebra (\[mat-cond1\]),(\[mat-cond2\]) and the theory of $q$-orthogonal polynomials. Finally, the completeness of the continuous $q$-Hermite polynomials reads $$\label{complete} 1 = \frac{(q;q)_{\infty}} {2\pi} \int_0^{\pi} \d \theta (e^{i\theta},e^{-i\theta};q)_{\infty} |p(\cos\theta)\rangle \langle p(\cos\theta) |.$$ Calculation of Density Profile ============================== \[density\] In this section, the average density profile is calculated by using the formula (\[Mehler\]) of the continuous $q$-Hermite polynomials. We first recall the asymptotic behaviors of the normalization $Z_L$ and the current in the thermodynamic limit $J=\lim_{L\rightarrow\infty} J_L$. The asymptotic expressions for $Z_L$ were given in [@me99] and are summarized as follows: - For phase $A$ (low-density phase; $a>1$ and $a>b$ ; $\tilde{\alpha} < \frac12$ and $\tilde{\alpha} < \tilde{\beta}$) $$Z_L \simeq \frac{(a^{-2};q)_{\infty}} {(b/a;q)_{\infty}} [(1+a)(1+a^{-1})]^L ,$$ - For phase $B$ (high-density phase; $b>1$ and $a<b$ ; $\tilde{\beta} < \frac12$ and $\tilde{\alpha} > \tilde{\beta}$) $$Z_L \simeq \frac{(b^{-2};q)_{\infty}} {(a/b;q)_{\infty}} [(1+b)(1+b^{-1})]^L ,$$ - For phase $C$ (maximal current phase; $0<a,b<1$ ; $\tilde{\beta} > \frac12$ and $\tilde{\alpha} > \frac12$) $$Z_L \simeq \frac{(ab;q)_{\infty} (q;q)_{\infty}^3 4^{L+1}} {\sqrt{\pi}(a,b;q)_{\infty}^2 L^{ \frac{3}{2} } } ,$$ - On the coexistense line ($a=b>1$ ; $\tilde{\alpha} = \tilde{\beta} < \frac12$) $$Z_L \simeq \frac{(a-a^{-1}) (a^{-2};q)_{\infty} L} {(q;q)_{\infty}} [(1+a)(1+a^{-1})]^{L-1}.$$ Using (\[def-current\]), the current in the thermodynamic limit is readily computed as - For phase A ($\tilde{\alpha} < \frac12$ and $\tilde{\beta} >\tilde{\alpha}$) $$\label{current-A} J=(p_R-p_L)\tilde{\alpha} (1-\tilde{\alpha}),$$ - For phase B ($\tilde{\beta} < \frac12$ and $\tilde{\alpha} >\tilde{\beta}$) $$\label{current-B} J=(p_R-p_L)\tilde{\beta} (1-\tilde{\beta}),$$ - For phase C ($\tilde{\alpha} > \frac12$ and $\tilde{\beta} >\frac12$) $$\label{current-C} J=\frac{p_R-p_L}{4}.$$ The phase diagram for the current is depicted in Fig. 4. Now we turn to consider the average density profile. As you can see from the figures in Fig. 4, the average density is almost constant at bulk part except on the coexistence line. Hence we are interested in the average bulk density and how the density decays near the boundaries. When the average density decays like $e^{-r/\xi}$ with $r$ distance from the boundary, we refer to $\xi$ as the correlation length in this paper. As for the average density near the boundares, it is sufficient to compute the density near the right boundary due to the symmetry (\[symmetry\]). The relation (\[ri-le\]) enables us to know the average density profile near the left boundary. On the other hand, the coexistence line should be treated separately. Since it is easier to calculate the density difference than the density itself, we rewrite the density at site $j$ as $$\label{density-decompose} \langle n_j \rangle_L = \sum_{k=j}^{L-1} (\langle n_k \rangle_L - \langle n_{k+1} \rangle_L) + \langle n_L \rangle_L .$$ At the right boundary, we have $$\begin{aligned} \label{d-right} \langle n_L \rangle_L &= \frac{1}{Z_L} \langle W| C^{L-1} D |V\rangle \notag\\ &= \frac{1}{\tilde{\beta}} \frac{Z_{L-1}}{Z_L} \notag\\ &\rightarrow \frac{J}{\beta}. \hspace{10mm} (L\rightarrow\infty)\end{aligned}$$ Using (\[current-A\])-(\[current-C\]), the density at the right boundary is easily calculated. Next we notice that $$\begin{aligned} \langle n_k \rangle_L - \langle n_{k+1} \rangle_L &= \frac{1}{Z_L} (\langle W| C^{k-1}DC^{L-k}|V\rangle - \langle W| C^{k}DC^{L-k-1}|V\rangle ) \notag\\ &= \frac{1}{Z_L} \langle W|C^{k-1}(DC-CD)C^{L-k-1}|V\rangle \notag\\ &= \frac{1}{Z_L} \langle W|C^{k-1}(DE-ED)C^{L-k-1}|V\rangle \notag\\ &= \frac{1}{Z_L} \langle W|C^{k-1}(de-ed)C^{L-k-1}|V\rangle.\end{aligned}$$ Now one can represent $\langle W|C^{k-1}(de-ed)C^{L-k-1}|V\rangle$ in the form of double integrals. The calculation proceeds as follows. First we notice that $$\begin{aligned} &\quad \langle W|C^{k-1}(de-ed)C^{L-k-1}|V\rangle \notag\\ &= \kappa^2 (q;q)_{\infty}^2 \int_0^{\pi} \frac{\d \theta}{2\pi} (e^{2i\theta},e^{-2i\theta};q)_{\infty} \int_0^{\pi} \frac{\d \varphi}{2\pi} (e^{2i\varphi},e^{-2i\varphi};q)_{\infty} \notag\\ &\quad \times \, _c \langle a| C^{k-1} |p(\cos\theta)\rangle \langle p(\cos\theta)| (de-ed) C^{L-k-1} |p(\cos\varphi)\rangle \langle p(\cos\varphi)| b\rangle_c \notag\\ &= (ab;q)_{\infty} (q;q)_{\infty}^2 \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} [2(1+\cos\theta)]^{k-1}[2(1+\cos\varphi)]^{L-k-1} \notag\\ &\quad\times \, _c \langle a| p(\cos\theta)\rangle \langle p(\cos\theta)| (de-ed) |p(\cos\varphi)\rangle \langle p(\cos\varphi)| b\rangle_c . \end{aligned}$$ Here the formula (\[gen\]) gives $$\label{ap-pb} _c \langle a| p(\cos\theta)\rangle = \frac{1}{(ae^{i\theta},a^{-i\theta};q)_{\infty}}, \quad \langle p(\cos\varphi)| b\rangle_c = \frac{1}{(be^{i\theta},b^{-i\theta};q)_{\infty}}.$$ The remaining term $ \langle p(\cos\theta)| (de-ed) |p(\cos\varphi)\rangle $ can also be computed by using the fact (\[de-com\]) and the formula (\[Mehler\]). We see that $$\begin{aligned} \langle p(\cos\theta)| (de-ed) |p(\cos\varphi)\rangle &= (1-q) \sum_{n=0}^{\infty} p_n(\cos\theta) p_n(\cos\varphi) q^n \notag\\ &= (1-q) \sum_{n=0}^{\infty} \frac{ H_n(\cos\theta|q) H_n(\cos\varphi|q) q^n } {(q;q)_n} \notag\\ &= \frac{ (q;q)_{\infty} } { (q e^{i(\theta+\varphi)}, q e^{-i(\theta+\varphi)}, q e^{i(\theta-\varphi)}, q e^{-i(\theta-\varphi)} ;q)_{\infty} }\end{aligned}$$ Hence we have $$\begin{gathered} \langle W|C^{k-1}(de-ed)C^{L-k-1}|V\rangle \notag\\ = (ab;q)_{\infty}(q;q)_{\infty}^3 \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} [2(1+\cos\theta)]^{k-1}[2(1+\cos\varphi)]^{L-k-1} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} }\end{gathered}$$ for $0<a,b<1$. Summing with respect to $k$ from $j$ to $L-1$, we get $$\begin{gathered} \sum_{k=j}^{L-1} (\langle n_k \rangle_L - \langle n_{k+1} \rangle_L) \notag\\ = \frac{(ab;q)_{\infty}(q;q)_{\infty}^3}{Z_L} \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} } \notag\\ \times \frac{ [2(1+\cos\theta)]^{j-1}[2(1+\cos\varphi)]^{L-j} -[2(1+\cos\theta)]^{L-1} } { 2\cos\varphi - 2 \cos\theta }, \label{diff}\end{gathered}$$ when $0<a,b<1$. Hence, substitution of this expression into (\[density-decompose\]) leads to the expression of the average density profile for $0<a,b<1$. Finally, the analytic continuation of (\[diff\]) gives the expression of the density porfile for other values of the parameters as well. This is conveniently done by writing (\[diff\]) as a contour integral of the two complex variables $z_1=e^{i\theta}$ and $z_2=e^{i\varphi}$. That is, we rewrite (\[diff\]) as $$\label{I1-I2} \sum_{k=j}^{L-1} (\langle n_k \rangle_L - \langle n_{k+1} \rangle_L) = \frac{1}{Z_L}(I_1+I_2),$$ where $$\begin{aligned} I_1 &= \frac14 (ab;q)_{\infty}(q;q)_{\infty}^3 \int\frac{\d z_1}{2\pi i z_1} \int\frac{\d z_2}{2\pi i z_2} \notag\\ &\quad \frac{(z_1^2,z_1^{-2},z_2^2,z_2^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{L-1}} {(az_1,a z_1^{-1},qz_1 z_2, q z_1^{-1} z_2^{-1}, q z_1 z_2^{-1},q z_1^{-1} z_2^{-1}, bz_2,b z_2^{-1};q)_{\infty} (z_2 + z_2^{-1} -z_1-z_1^{-1})}, \notag\\ I_2 &= \frac14 (ab;q)_{\infty}(q;q)_{\infty}^3 \int\frac{\d z_1}{2\pi i z_1} \int\frac{\d z_2}{2\pi i z_2} \notag\\ &\quad \frac{(z_1^2,z_1^{-2},z_2^2,z_2^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{j-1} [(1+z_2)(1+z_2^{-1})]^{L-j} } {(az_1,a z_1^{-1},q z_1 z_2, q z_1^{-1} z_2^{-1}, q z_1 z_2^{-1},q z_1^{-1} z_2^{-1}, bz_2,b z_2^{-1};q)_{\infty} (z_2 + z_2^{-1} -z_1-z_1^{-1})}.\end{aligned}$$ When $0<a,b<1$, the contours of $z_1$ and $z_2$ are both unit circles. For other valures of the parameters, the contours are deformed as poles in the integrands move in and out of the unit circles. The evaluation of the integrals $I_1$ and $I_2$ in the thermodynamic limit for each phase is relegated to Appendices. Basically, the integral $I_1$ gives the density at bulk region whereas the integral $I_2$ gives the density near the right boundary. The results are summerized in the following. We thus obtain the phase diagram shown in Fig. 5. Notice that this phase diagram was corrrectly predicted in [@me99]. Therein the correlation length for each phase was assumed to be given by the logarithm of the ratio of the largest and the second largest eigenvalue of the matrix $C$. - Phase $C$ ($\tilde{\alpha} > 1/2$ and $\tilde{\beta}>1/2$; $0<a,b<1$) In this phase, the average density at bulk is $1/2$. The average density decays near the right bounary as $$\label{density-max} \langle n_j \rangle_L = \frac12 - \frac{1}{2\sqrt{\pi} l^{\frac12}}.$$ Here and in the following, we set $l=L-j+1$. The density decays algebraically and hence the correlation length is infinite. This average density profile is exactly the same as that for the totally asymmetric case. The density decay at the left boundary can be obtained from the symmetry relation (\[ri-le\]). - Phase $A_1$ ($ \tilde{\alpha} < \tilde{\beta} < \tilde{\alpha}/[(1-\tilde{\alpha})q+\tilde{\alpha}]$ and $\tilde{\beta} < 1/2$; $aq<b<a$ and $b>1$ ) The average density at bulk is $\tilde{\alpha}$. The density near the right boundary decays exponentially as $$\label{density-A1} \langle n_j \rangle_L = \tilde{\alpha} - \frac{(b^{-2}q,q;q)_{\infty}} {(a^{-1}b^{-1}q,ab^{-1}q;q)_{\infty}} \left[ \frac{\tilde{\alpha}(1-\tilde{\alpha}) } {\tilde{\beta}(1-\tilde{\beta})} \right]^l (1-2\tilde{\beta}),$$ with the correlation length $$\xi^{-1} = \ln \frac{\tilde{\beta}(1-\tilde{\beta})} {\tilde{\alpha}(1-\tilde{\alpha})}.$$ On the other hand, the density near the left boundary takes the constant value $\tilde{\alpha}$. This fact is obtained by conbining the average density profile near the right boundary for the high-density phase below and the symmetry (\[symmetry\]). - Phase $A_2$ ($q/(1+q) < \tilde{\alpha} < 1/2$ and $\tilde{\beta} > 1/2$; $1<a<q^{-1}$ and $b<1$) In the bulk region, the average density takes the constant value $\tilde{\alpha}$. On the other hand, the density profile near the right boundary decays exponentially as $$\label{density-A2} \langle n_j \rangle_L = \tilde{\alpha} - \frac{(a-b)(1-ab)(abq,a^{-1}bq;q)_{\infty} (q;q)_{\infty}^4} {(a-1)^2(b-1)^2(aq,a^{-1}q,bq;q)_{\infty}} \frac{[4\tilde{\alpha}(1-\tilde{\alpha})]^l} {\sqrt{\pi} l^{\frac32}},$$ with the correlation length $$\xi^{-1} = -\ln 4[\tilde{\alpha}(1-\tilde{\alpha})].$$ But the decay is not purely exponential but with algebraic corrections. At the left boundary, the density takes the constant value $\tilde{\alpha}$. - Phase $A_3$ ( $\tilde{\beta} > \tilde{\alpha}/[(1-\tilde{\alpha})q+\tilde{\alpha}]$ and $\tilde{\alpha} < q/(1+q)$; $a>q^{-1}$ and $b<aq$) The average density at bulk is $\tilde{\alpha}$. Near the right boundary, the density decays exponentially as $$\label{density-A3} \langle n_j \rangle_L = \tilde{\alpha} - \frac{(1-ab)(1-(aq)^{-1})} {(1-b(aq)^{-1})(1+aq)} \left[\frac{(1+aq)(1+(aq)^{-1})} {(1+a)(1+a^{-1})} \right]^l$$ with the correlation length $$\xi^{-1} = \ln \frac{q} {[\tilde{\alpha}+(1-\tilde{\alpha})q]^2}.$$ This density profile has no correspondense for the totally asymmetric case. At the left boundary, the density takes the constant value $\tilde{\alpha}$. - Phase $B_1$ ($ \tilde{\alpha}q/[(1-\tilde{\alpha})+q\tilde{\alpha}] < \tilde{\beta} < \tilde{\beta} $ and $\tilde{\alpha} < 1/2$; $bq<a<b$ and $a>1$) The average density at bulk is $1-\tilde{\beta}$. As can be seen from the calculation in Appendix, The density takes the constant value near the right boundary. $$\label{density-B} \langle n_j \rangle_L = 1-\tilde{\beta}$$ Since this phase is related to the phase $A_1$ through the symmetry (\[symmetry\]), the denisty decays exponentially near the left boundary with the correlation length $$\xi^{-1} = \ln \frac{\tilde{\alpha}(1-\tilde{\alpha})} {\tilde{\beta}(1-\tilde{\beta})}.$$ - Phase $B_2$ ($q/(1+q) < \tilde{\beta} < 1/2$ and $\tilde{\alpha} > 1/2$; $a<1$ and $1<b<q^{-1}$) This phase is symmetric to phase $A_2$ through the symmetry (\[symmetry\]). The average density at bulk and near the right boundary is $1-\tilde{\beta}$. Near the left boundary, the density decays exponentially with the correlation length $$\xi^{-1} = -\ln 4[\tilde{\beta}(1-\tilde{\beta})].$$ - Phase $B_3$ ($a<bq$ and $b>q^{-1}$) ($\tilde{\beta} < \tilde{\alpha}q/[1-\tilde{\alpha}+\tilde{\alpha}q]$ and $\tilde{\beta} < q/(1+q)$; $a<bq$ and $b>q^{-1}$) This phase is symmetric to phase $A_3$ through the symmetry (\[symmetry\]). The average density at bulk and near the right boundary is $1-\tilde{\beta}$. The density decays exponentially with the correlation length $$\xi^{-1} = \ln \frac{q} {[\tilde{\beta}+(1-\tilde{\beta})q]^2},$$ near the left boundary. - Coexistence line ($\tilde{\alpha}=\tilde{\beta}<1/2$; $a=b>1$) The average density shows linear profile at bulk, $$\label{density-coex} \langle n_j \rangle_L = \tilde{\alpha} + (1-2\tilde{\alpha})\frac{j}{L}.$$ This is essentially the same as the result for the totally asymmetric case. Before closing the section, we present a simulation result for the correlation length to show the differece between the phase $A_2$ and $A_3$. The simulation was done on $\tilde{\beta}=1$ line. For the totally asymmetric case, the system is in the $A_2$ phase on this line. The the correlation length is given by $\xi_{A_2}=1/\ln4[\alpha(1-\alpha)]$. On the other hand, for the partially asymmetric case, the system is in the $A_3$ phase when $\tilde{\alpha}<q/(1+q)$. The correlation length is given by $\xi_{A_3}=1/\ln\frac{q}{[\tilde{\beta}+(1-\tilde{\beta})q]^2}$. The differences between these two expressions become large especially when $q$ and $\alpha$ are small. Especially, as $\alpha \rightarrow 0$, $\xi_{A_2}$ goes to zero whereas $\xi_{A_3}$ goes to $-1/\ln q$. In Fig. 6 the simulation result for the correlation length is shown for $p_R=1.0,\,P_L=0.9,\,\beta=10$ which corresponds to $q=0.9,\,\tilde{\beta}=1$. It is clear that the the correlatin lenght approaches the finite value as $\alpha \rightarrow 0$ and is well described by the formula for $\xi_{A_3}$. Concluding Remarks ================== \[conc\] In this article, we have computed the average density profile of the partially asymmetric simple exclusion process with open boundaries. The calculation has been done for a wide rage of parameters satisfying $0<p_L<p_R$ and $\alpha>0,\beta>0$. The phase diagram for the correlation length has been obtained. It has turned out that the phase diagram was correctly predicted in the earlier paper [@me99]. In [@me99], the phase diagram was obtained by assuming that the correlation length is given by the logarithm of the ratio of the largest and the second largest eigenvalues of the matrix $C$. The discussions were only for the phases with the exponentially decaying profile. In this article, we have not only confirmed this fact but also obtained the asymptotic expressions of the average density profile for all phases. There are two key facts which allowed us to calculate the average density profile exactly in the thermodaynamic limit. One is that the commuation relation of the matrices $D,E$ becomes a simple diagonal matrix and the other is the formula (\[Mehler\]) of the $q$-Hermite polynomials. There seems to be many possible applications and generalizations of the analysis of this article. First, it is possible to generalize the analysis in this paper to the partially asymmetric exclusion process on a ring with a single defect particle [@Mallick; @Jafa]. The corresponding totally asymmetric case was already solved in [@Mallick]. Second, the case where $p_L>p_R$ is also interesting. Although the current was evaluated in [@BECE], more exact results are desirable. Third, it would be interesting to apply the simialr analysis to the multi-species models [@EFGM; @EKKM; @AHR98-1; @ADR; @MMR]. Compared to the ASEP, much less is known about these models. Several investigatios are now in progress [@RSS]. The results about these will be reported elsewhere. Acknowledgment {#acknowledgment .unnumbered} ============== The author would like to thank P. Deift, E. R. Speer and N. Rajewsky for fruitful discussions and comments. He also thanks the continuous encouragement of M. Wadati. The author is a Research Fellow of the Japan Society for the Promotion of Science. Evaluation of Integral $I_1$ ============================ In this appendix, the integral $I_1$, $$\begin{aligned} I_1 &= \frac14 (ab;q)_{\infty}(q;q)_{\infty}^3 \int\frac{\d z_1}{2\pi i z_1} \int\frac{\d z_2}{2\pi i z_2} \notag\\ &\quad \frac{(z_1^2,z_1^{-2},z_2^2,z_2^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{L-1}} {(az_1,a z_1^{-1},qz_1 z_2, q z_1^{-1} z_2^{-1}, q z_1 z_2^{-1},q z_1^{-1} z_2^{-1}, bz_2,b z_2^{-1};q)_{\infty} (z_2 + z_2^{-1} -z_1-z_1^{-1})}, \label{I1def}\end{aligned}$$ is evaluated. First, for the case where $a,b<1$, both of the contours of $z_1$ and $z_2$ are unit circles. We have $$\begin{gathered} I_1 = I_1^{(0)} = \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} } \notag\\ \times \frac{ [2(1+\cos\theta)]^{L-1} } { 2\cos\varphi - 2 \cos\theta }. \label{I10}\end{gathered}$$ Second, consider the case where $a$ becomes larger one but $b$ is still smaller than one. We assume $$\label{na} a>aq>aq^2>\cdots >aq^{n^{(a)}} >1> aq^{n^{(a)}+1}>\cdots .$$ Then the contour of $z_2$ is still a unit circle but the contour of $z_1$ has to be modified to include all poles at $z_1=aq^k$ ($k=0,1,\ldots,n^{(a)}$) and to exclude all poles at $z_1=(a q^k)^{-1}$ ($k=0,1,\ldots,n^{(a)}$). Seperating the contributions from poles at $z_1=aq^{k}$ and $z_1=(aq^k)^{-1}$, the integral $I_1$ can be rewritten as $$\begin{aligned} \label{eq:I1-1} I_1 &= I_1^{(0)} - \frac{(ab;q)_{\infty} (q;q)_{\infty}^2}{2 a} \sum_{k=0}^{n^{(a)}} \frac{(-)^k q^{k(k-1)/2} (a^2 q^{2k},a^{-2} q^{-2k};q)_{\infty} [\lambda_k^{(a)}]^{L-1} } {(q;q)_k (a^2 q^k;q)_{\infty}} \notag\\ &\quad\times \int_{C_0} \frac{\d z_2}{2\pi i z_2} \frac{(z_2^2,z_2^{-2};q)_{\infty}} {(aq^{k+1}z_2,aq^{k+1}z_2^{-1},a^{-1}q^{-k}z_2, a^{-1}q^{-k}z_2^{-1},b z_2,bz_2^{-1};q)_{\infty}} . \end{aligned}$$ Here the contour $C_0$ denotes the unit circle. The $\lambda_k^{(c)}$’s are defined by $$\lambda_k^{(c)} = (1+c q^k)(1+c^{-1}q^{-k}),$$ for $c=a,b$ and $k=0,1,2,\ldots$. The intgral can be evaluated explicitly by using the general formula, $$\label{int-aw} \int_C \frac{\d z}{2\pi i z} \frac{(z^2,z^{-2};q)_{\infty}} {(az,az^{-1},bz,bz^{-1},cz,cz^{-1},dz,dz^{-1};q)_{\infty}} = \frac{2(abcd;q)_{\infty}} {(q,ab,ac,ad,bc,bd,cd;q)_{\infty}}.$$ The contour $C$ is such that it includes all poles of the type $f q^{k}$ and excludes all poles of the type $f^{-1}q^{-k}$ with $f=a,b,c,d$ and $k=0,1,2,\ldots$. The parameters $a,b,c,d$ in this formula has nothing to do with the $a,b,c,d$ which appear in the rest of this article. This formula plays a crutial role in proving the orthogonaliry relation of the Askey-Wilson polynomials. The proof can be found in [@AW85]. Now we get $$\begin{aligned} \label{I1a} I_1 &= I_1^{(0)} + I_1^{(a)}, \notag\\ I_1^{(a)} &= -\frac{(ab;q)_{\infty}}{a} \sum_{k=0}^{n^{(a)}} \frac{(-)^k q^{k(k-1)/2} (a^2 q^{2k},a^{-2}q^{-2k};q)_{\infty}} {(q;q)_k (a^2 q^k,ab q^{k+1}, a^{-1}b q^{-k};q)_{\infty}} [\lambda_k^{(a)}]^{L-1} . \end{aligned}$$ When $b$ also becomes larger than one, there appear the terms which come from poles at $z_2=bq^k$ and $z_2=(bq^k)^{-1}$ in (\[I1def\]). When (\[na\]) and $$\label{nb} b>bq>bq^2>\cdots >bq^{n^{(b)}} >1> bq^{n^{(b)}+1}>\cdots$$ hold, we have $$\begin{aligned} \label{I1ab} I_1 &= I_1^{(0)} + I_1^{(a)} + I_1^{(b,0)} + I_1^{(b,b)} + I_1^{(b,a)}, \\ I_1^{(b,0)} &= \frac{(ab;q)_{\infty}(q;q)_{\infty}^2}{2 b} \sum_{k=0}^{n^{(b)}} \frac{(-)^k q^{k(k-1)/2} (b^2q^{2k},b^{-2}q^{-2k};q)_{\infty} } {(q;q)_{k}(b^2q^k;q)_{\infty}} \notag\\ &\quad\times \int_{C_0}\frac{\d z_1}{2\pi i z_1} \frac{(z_1^2,z_1^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{L-1}} {( az_1,az_1^{-1}, bq^{k+1}z_1,bq^{k+1}z_1^{-1}, b^{-1}q^{-k}z_1,b^{-1}q^{-k}z_1^{-1};q)_{\infty}} , \\ I_1^{(b,b)} &= -\sum_{k=0}^{n^{(b)}} \sum_{m=k+1}^{n^{(b)}} \frac{(-)^k q^{k(k-1)/2+(m-k)(m-k-1)/2} [\lambda_m^{(b)}]^{L-1} } {b (q;q)_k (q;q)_{m-k-1}} \notag\\ &\quad\times \frac{(q,ab,b^2q^{2k},b^{-2}q^{-2k},b^2q^{2m},b^{-2}q^{-2m};q)_{\infty}} {(q^{m-k},b^2 q^k,b^2q^{m+k+1},b^{-2}q^{-k-m}, abq^m,ab^{-1}q^{-m};q)_{\infty} } , \\ I_1^{(b,a)} &= \sum_{k=0}^{n^{(b)}} \sum_{m:bq^k > aq^m} \frac{(-)^{k+m} q^{k(k+1)/2+m(m+1)/2} [\lambda_m^{(a)}]^{L-1} } {b (q;q)_k (q;q)_{m}} \notag\\ &\quad\times \frac{(q,ab,b^2q^{2k},b^{-2}q^{-2k},a^2q^{2m},a^{-2}q^{-2m};q)_{\infty}} {(b^2 q^k,a^2 q^m,abq^{k+m+1},a^{-1}bq^{k+1-m}, ab^{-1}q^{m-k},a^{-1}b^{-1}q^{-k-m};q)_{\infty}} .\end{aligned}$$ Lastly, when $a<1$ and (\[nb\]) holds, we have $$\label{I1b} I_1 = I_1^{(0)} + I_1^{(b,0)} + I_1^{(b,b)}.$$ Now we turn to the calculation of the asymptotic expression of the integral $I_1$. - [The case $a<1$ and $b<1$]{} In this case, $I_1=I_1^{(0)}$. We evaluate the asymptotic behavior of $I_1^{(0)}$ by employing the steepest decent method. First we change the variable from $\theta,\varphi$ to $u,y$ as $$\begin{gathered} \label{change1} 1+\cos\theta = 2 e^{-u/L}, \\ \label{change2} 1+\cos\varphi = 2 y e^{-u/L},\end{gathered}$$ to obtain $$\begin{aligned} \label{I1-1} I_1 = -\frac{4^{L+2}}{L^{\frac23}} \int_{0}^{\infty} \d u u^{\frac12} e^{-u} \sqrt{ \frac{1-e^{-u/L}} {u/L } } e^{-u/L} \int_{0}^{e^{u/L}} \d y \frac{ y^{\frac12}\sqrt{1-y e^{-u/L}} } {1-y} \notag\\ \times \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} }.\end{aligned}$$ Here $\theta$ and $\varphi$ are considerd as functions in $u$ and $y$ through (\[change1\]) and (\[change2\]) respectively. Now we can take the limit $L\rightarrow\infty$ in the integrand. Changing the variable $y$ back to $\varphi$ by $$1+\cos\varphi = 2 y,$$ we have $$I_1 \simeq -\frac{\sqrt{\pi} (q;q)_{\infty}^2 4^{L+1} } {2 (a;q)_{\infty}^2 L^{\frac32} } \int_{0}^{\pi} \d \varphi \frac{(e^{2i\varphi},e^{-2i\varphi};q)_{\infty}} {(qe^{i\varphi},qe^{i\varphi},qe^{-i\varphi},qe^{-i\varphi}, be^{i\varphi},be^{i\varphi};q)_{\infty}} .$$ Using the formula (\[int-aw\]), we get $$I_1 \simeq -\frac{\pi^{\frac32} (1-b) 4^{L+1} } {(a,b;q)_{\infty}^2 L^{\frac32}}.$$ - [The case $a<1$ and $b>1$]{} The integral $I_1$ is given by (\[I1b\]). The main contributions come from $I_1^{(0)}$ and $I_1^{(b,0)}$. Each contribution behaves as $4^L$. whilest the normalization $Z_L$ behaves as $[(1+b)(1+b^{-1})]^L$ for this case. Since $(1+b)(1+b^{-1})$ is larger than $4$, $I_1$ is negligible compared to $Z_L$. So we do not compute the explicit expression. - [The case $a>1$]{} The main contribution comes from the $k=0$ term in the summation of $I_1^{(a)}$. We have $$I_1 \simeq -\frac{a(1-ab)(a^{-2};q)_{\infty}[(1+a)(1+a^{-1})]^{L-1}} {(a^{-1}b;q)_{\infty}}.$$ Evaluation of Integral $I_2$ ============================ In this Appendix, the integral $I_2$, $$\begin{aligned} I_2 &= \frac14 (ab;q)_{\infty}(q;q)_{\infty}^3 \int\frac{\d z_1}{2\pi i z_1} \int\frac{\d z_2}{2\pi i z_2} \notag\\ &\quad \frac{(z_1^2,z_1^{-2},z_2^2,z_2^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{j-1} [(1+z_2)(1+z_2^{-1})]^{L-j} } {(az_1,a z_1^{-1},qz_1 z_2, a z_1^{-1} z_2^{-1}, q z_1 z_2^{-1},q z_1^{-1} z_2^{-1}, bz_2,b z_2^{-1};q)_{\infty} (z_2 + z_2^{-1} -z_1-z_1^{-1})}, \label{I2def}\end{aligned}$$ is evaluated. First, for the case where $a,b<1$, both of the contours of $z_1$ and $z_2$ are unit circles. We have $$\begin{gathered} I_2 = I_2^{(0)} = \int_0^{\pi} \frac{\d \theta}{2\pi} \int_0^{\pi} \frac{\d \varphi}{2\pi} \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} } \notag\\ \times \frac{ [2(1+\cos\theta)]^{j-1} [2(1+\cos\varphi)]^{L-j} } { 2\cos\varphi - 2 \cos\theta } . \label{I20}\end{gathered}$$ Since we will calculate the average density profile near the right boundary, we set $l=L-j+1$. Similarly to the case of the integal $I_1$, when $a$ or $b$ or both of them become larger than one, there apper other contributions besides $I_2^{(0)}$. When (\[na\]) and (\[nb\]) hold, the result is $$\begin{aligned} I_2 &= I_2^{(0)} + I_2^{(a,0)} + I_2^{(a,a)} + I_2^{(a,b)} + I_2^{(b,0)} + I_2^{(b,b)} + I_2^{(b,a)}, \label{I2all} \\ I_2^{(a,0)} &= -\frac{(ab;q)_{\infty}(q;q)_{\infty}^2}{2 a} \sum_{k=0}^{n^(a)} \frac{(-)^k q^{k(k-1)/2} (a^2q^{2k},a^{-2}q^{-2k};q)_{\infty} [\lambda_k^{(a)}]^{L-l} } {(q;q)_{k}(a^2q^k;q)_{\infty}} , \notag\\ &\quad\times \int_{C_0}\frac{\d z_2}{2\pi i z_2} \frac{(z_2^2,z_2^{-2};q)_{\infty} [(1+z_2)(1+z_2^{-1})]^{l-1}} {(aq^{k+1}z_2,aq^{k+1}z_2^{-1}, a^{-1}q^{-k}z_2,a^{-1}q^{-k}z_2^{-1}, bz_2,bz_2^{-1};q)_{\infty}} , \\ I_2^{(a,a)} &= \sum_{k=0}^{n^{(a)}} \sum_{m=k+1}^{n^{(a)}} \frac{(-)^k q^{k(k-1)/2+(m-k)(m-k-1)/2} [\lambda_k^{(a)}]^{L-l} [\lambda_m^{(a)}]^{l-l} } {a (q;q)_k (q;q)_{m-k-1}} \notag\\ &\quad\times \frac{(q,ab,a^2q^{2k},a^{-2}q^{-2k},a^2q^{2m},a^{-2}q^{-2m};q)_{\infty}} {(q^{m-k},a^2 q^k,a^2q^{m+k+1},a^{-2}q^{-k-m}, abq^m,ba^{-1}q^{-m};q)_{\infty} } , \\ I_2^{(a,b)} &= -\sum_{k=0}^{n^{(a)}} \sum_{m:aq^k > bq^m} \frac{(-)^{k+m} q^{k(k+1)/2+m(m+1)/2} [\lambda_k^{(a)}]^{L-l} [\lambda_m^{(b)}]^{l-l} } {a (q;q)_k (q;q)_{m}} \notag\\ &\quad\times \frac{(q,ab,a^2q^{2k},a^{-2}q^{-2k},b^2q^{2m},b^{-2}q^{-2m};q)_{\infty}} {(a^2 q^k,b^2 q^m,abq^{k+m+1},ab^{-1}q^{k+1-m}, a^{-1}bq^{m-k},a^{-1}b^{-1}q^{-k-m};q)_{\infty}} , \\ I_2^{(b,0)} &= \frac{(ab;q)_{\infty}(q;q)_{\infty}^2}{2 b} \sum_{k=0}^{n^{(b)}} \frac{(-)^k q^{k(k-1)/2} (b^2q^{2k},b^{-2}q^{-2k};q)_{\infty} [\lambda_k^{(b)}]^{l-1} } {(q;q)_{k}(b^2q^k;q)_{\infty}} \notag\\ &\quad\times \int_{C_0}\frac{\d z_1}{2\pi i z_1} \frac{(z_1^2,z_1^{-2};q)_{\infty} [(1+z_1)(1+z_1^{-1})]^{L-l}} {( az_1,az_1^{-1}, bq^{k+1}z_1,bq^{k+1}z_1^{-1}, b^{-1}q^{-k}z_1,b^{-1}q^{-k}z_1^{-1};q)_{\infty}} , \\ I_2^{(b,b)} &= -\sum_{k=0}^{n^{(b)}} \sum_{m=k+1}^{n^{(b)}} \frac{(-)^k q^{k(k-1)/2+(m-k)(m-k-1)/2} [\lambda_k^{(b)}]^{l-1} [\lambda_m^{(b)}]^{L-l} } {b (q;q)_k (q;q)_{m-k-1}} \notag\\ &\quad\times \frac{(q,ab,b^2q^{2k},b^{-2}q^{-2k},b^2q^{2m},b^{-2}q^{-2m};q)_{\infty}} {(q^{m-k},b^2 q^k,b^2q^{m+k+1},b^{-2}q^{-k-m}, abq^m,ab^{-1}q^{-m};q)_{\infty} } , \\ I_2^{(b,a)} &= \sum_{k=0}^{n^{(b)}} \sum_{m:bq^k > aq^m} \frac{(-)^{k+m} q^{k(k+1)/2+m(m+1)/2} [\lambda_k^{(b)}]^{l-1} [\lambda_m^{(a)}]^{L-l} } {b (q;q)_k (q;q)_{m}} \notag\\ &\quad\times \frac{(q,ab,b^2q^{2k},b^{-2}q^{-2k},a^2q^{2m},a^{-2}q^{-2m};q)_{\infty}} {(b^2 q^k,a^2 q^m,abq^{k+m+1},a^{-1}bq^{k+1-m}, ab^{-1}q^{m-k},a^{-1}b^{-1}q^{-k-m};q)_{\infty}} .\end{aligned}$$ Now we consider the asymptotic expression of the integral $I_2$. We take the limit $L\rightarrow\infty$ at first and then take the limit $l\rightarrow\infty$. - [The case $a,b<1$]{} In this case, $I_2=I_2^{(0)}$. The evaluation for this case proceeds analogously to the evaluation of $I_1^{(0)}$. Changing the variables $\theta,\varphi$ to $u,y$ as in (\[change1\]) and (\[change2\]), we get $$\begin{aligned} \label{I2-1} I_2 = -\frac{4^{L+2}}{L^{\frac23}} \int_{0}^{\infty} \d u u^{\frac12} e^{-u} \sqrt{ \frac{1-e^{-u/L}} {u/L } } e^{-u/L} \int_{0}^{e^{u/L}} \d y \frac{ y^{l-\frac12}\sqrt{1-y e^{-u/L}} } {1-y} \notag\\ \times \frac{ (e^{2i\theta},e^{-2i\theta},e^{2i\varphi},e^{-2i\varphi};q)_{\infty} } {(ae^{i\theta}, ae^{-i\theta}, qe^{i(\theta+\varphi)}, qe^{-i(\theta+\varphi)}, qe^{i(\theta-\varphi)}, qe^{-i(\theta-\varphi)}, be^{i\varphi}, be^{-i\varphi};q)_{\infty} }.\end{aligned}$$ We take the limit $L\rightarrow\infty$ in this expression and change the varialbe $y$ back to $\varphi$ to get $$I_2 \simeq -\frac{\sqrt{\pi} (q;q)_{\infty}^2 4^{L+2} } {2 (a;q)_{\infty}^2 L^{\frac23}} \int_{0}^{\pi} \d \phi \left[ \frac12 (1+\cos\varphi) \right]^{l} \frac{(qe^{2i\varphi},qe^{-2i\varphi};q)_{\infty}} {(qe^{i\varphi},qe^{i\varphi},qe^{-i\varphi},qe^{-i\varphi}, be^{i\varphi},be^{i\varphi};q)_{\infty}}.$$ We can consider the limit $l\rightarrow\infty$ by using the steepest descent method. We have $$I_2 \simeq -\frac{ 2 \cdot 4^{L+1}\pi } {(a,b;q)_{\infty}^2 L^{\frac23} l^{\frac12}}.$$ - [The case $1<a<q^{-1}$ and $b<1$]{} In this case, the integral $I_2$ is given by $I_2 = I_2^{(0)}+I_2^{(a,0)}$. The main contribution comes from the $k=0$ term in $I_2^{(a,0)}$. $$\begin{aligned} \label{I2a0} I_2 &\simeq -(ab,a^{-2};q)_{\infty} (q;q)_{\infty}^2 [(1+a)(1+a^{-1})]^{L-l} \notag\\ &\quad\times \int_{C_0} \frac{\d z_2}{2\pi i z_2} \frac{(z_2^2,z_2^{-2};q)_{\infty} [(1+z_2)(1+z_2^{-1})]^{l-1}} {(a^{-1}z_2,a^{-1}z_2^{-1},aqz_2, aqz_2^{-1},bz_2,bz_2^{-1};q)_{\infty}}.\end{aligned}$$ Taking the limit $l\rightarrow\infty$ by using the steeptest descent method, we get $$I_2 \simeq -\frac{4^l (ab,a^{-2};q)_{\infty}(q;q)_{\infty}^2 [(1+a)(1+a^{-1})]^{L-l} } {a \sqrt{\pi} l^{\frac32} (a^{-1},qa,b;)_{\infty}^2}.$$ - [The case $a>q^{-1}$ and $aq>b$]{} In general, the integral for this case has the expression (\[I2all\]). The main contribution comes from the $k=0,m=1$ term in $I_2^{(a,a)}$. We have $$\label{I2aa} I_2 \simeq -\frac{(1-ab)(1-a^{-2}q^{-2})(a^{-2};q)_{\infty} [(1+a)(1+a^{-1})]^{L-l} [(1+aq)(1+a^{-1}q^{-1})]^{l-1}} {(a^{-1}bq^{-1};q)_{\infty}}.$$ - [The case $a>b>aq$ and $b>1$]{} The main contribution for this case comes from the $k=0,m=1$ term in $I_2^{(a,b)}$. We have $$\label{I2ab} I_2 \simeq -\frac{(1-ab)(q,a^{-2},b^{-2};q)_{\infty} [(1+a)(1+a^{-1})]^{L-l} [(1+b)(1+b^{-1})]^{l-1} } {a (a^{-1}b,a^{-1}b^{-1},ab^{-1}q;q)_{\infty}} .$$ - [The case $b>1$ and $b>a$]{} In this case, the normalization $Z_L$ behaves as $[(1+b)(1+b^{-1})]^L$. All contributions to $I_2$ can be neglected compared to $Z_L$. Hence we do not compute the asymptotic expression explicitly for this case. Figure Captions Fig. 1 : One-dimensional partially asymmetric simple exclusion process with open boundaries. Particles have hard-core exclusion interaction and tend to hop to the right (resp. left) nearest neiboring site with rate $p_R$ (resp. $p_L$). There are also particle injection (resp. ejection) at the left (resp. right) edge. Fig. 2 : Space-time diagram of the ASEP from Monte-Carlo simulations. The holizontal axis represents the site number $j$ whereas the vertical axis represents time. The existence of pariticle is represented as a black point. The lattice length is taken to be $L=200$. The bulk hopping rates are taken to be $p_R=1,\,p_L=0$. After some transient time, the system practically goes to a steady state. The steady state depends crutially on the values of the boundary parameters. Fig. 3 : Average density profile of the ASEP from Monte-Carlo simulations. The holizontal axis represents the site number $j$ whereas the vertical axis represents the average density. The lattice length is taken to be $L=200$. The bulk hopping rates are taken to be $p_R=1,\,p_L=0$ Fig. 4 : The phase diagram of the current. Regions $A,B$ and $C$ are called the low-density phase, the high-density phase and the maximal current phase respectively. Fig. 5 : The phase diagram of the correlation length in the $\tilde{\alpha}$-$\tilde{\beta}$ plane for the partially asymmetric case. The low-density phase (resp. high-density phase) is divided into three phases, $A_1,A_2$ and $A_3$ (resp. $B_1,B_2$ and $B_3$). Fig. 6 : The correlation length $\xi$ for the case $p_R=1,p_L=0.9,\tilde{\beta}=1$. The solid line is the theoretical prediction given by $\xi=1/\ln\frac{q}{[\tilde{\beta}+(1-\tilde{\beta})q]^2}$ whereas the black dots are the simulation data. [99]{} T. Sasamoto, J. Phys. A [**32**]{} (1999) 7109. T. M. Ligget, [*Interacting Particle Systems*]{} (Springer-Verlag, New York, 1985). H. Spohn, [*Large Scale Dynamics of Interacting Particles*]{} (Springer-Verlag, New York, 1991). B. Derrida, Phys. Rep. 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(500,500) (20,305)[\[0.35\][]{}]{} (50,295)[(1,0)[50]{}]{} (70,280)[$j$]{} (15,380)[(0,1)[70]{}]{} (-3,410) (20,265)[($A$) low density phase]{} (32,253)[$p_R=1.0, \,p_L=0.0$]{} (37,240)[$\alpha=0.2,\,\beta=1.0$]{} (220,308)[\[0.35\][]{}]{} (250,295)[(1,0)[50]{}]{} (270,280)[$j$]{} (215,380)[(0,1)[70]{}]{} (197,410) (200,265)[($C$) maximal current phase]{} (232,253)[$p_R=1.0, \,p_L=0.0$]{} (237,240)[$\alpha=1.0,\,\beta=1.0$]{} (20,5)[\[0.35\][]{}]{} (50,-5)[(1,0)[50]{}]{} (70,-20)[$j$]{} (15,80)[(0,1)[70]{}]{} (-3,110) (40,-35)[coexistence line]{} (32,-47)[$p_R=1.0, \,p_L=0.0$]{} (37,-60)[$\alpha=0.2,\,\beta=0.2$]{} (220,0)[\[0.35\][]{}]{} (250,-5)[(1,0)[50]{}]{} (270,-20)[$j$]{} (215,80)[(0,1)[70]{}]{} (197,110) (220,-35)[($B$) high density phase]{} (232,-47)[$p_R=1.0, \,p_L=0.0$]{} (237,-60)[$\alpha=1.0,\,\beta=0.2$]{} (500,300) (0,190)[\[0.6\][]{}]{} (70,182)[(1,0)[50]{}]{} (90,170)[$j$]{} (-3,212)[(0,1)[70]{}]{} (-20,240) (30,155)[($A$) low density phase]{} (42,140)[$p_R=1.0, \,p_L=0.0$]{} (47,125)[$\alpha=0.2,\,\beta=0.25$]{} (200,190)[\[0.6\][]{}]{} (270,182)[(1,0)[50]{}]{} (290,170)[$j$]{} (220,155)[($C$) maximal current phase]{} (242,140)[$p_R=1.0, \,p_L=0.0$]{} (247,125)[$\alpha=1.0,\,\beta=1.0$]{} (0,0)[\[0.6\][]{}]{} (70,-5)[(1,0)[50]{}]{} (90,-20)[$j$]{} (50,-35)[coexistence line]{} (42,-50)[$p_R=1.0, \,p_L=0.0$]{} (47,-65)[$\alpha=0.2,\,\beta=0.2$]{} (200,0)[\[0.6\][]{}]{} (270,-5)[(1,0)[50]{}]{} (290,-20)[$j$]{} (230,-35)[($B$) high density phase]{} (242,-50)[$p_R=1.0, \,p_L=0.0$]{} (247,-65)[$\alpha=0.25,\,\beta=0.2$]{} (500,300) (0,100)[\[1.0\][]{}]{} (-20,270) (280,85)

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--- abstract: 'We determine the behavior of the general solution, small or large, of nonlinear first order ODEs in a neighborhood of an irregular singular point chosen to be infinity. We show that the solutions can be controlled in a ramified neighborhood of infinity using a finite set of asymptotic constants of motion; the asymptotic formulas can be calculated to any order by quadratures. These constants of motion enable us to obtain qualitative and accurate quantitative information on the solutions in a neighborhood of infinity, as well as to determine the position of their singularities. We discuss how the method extends to higher order equations. There are some conceptual similarities with a KAM approach, and we discuss this briefly.' address: - 'Mathematics Department The Ohio State University Columbus, OH 43210' - 'Department of Mathematics, The University of Chicago, 5734 S. University Avenue Chicago, Illinois 60637' - 'IRMA, Université de Strasbourg et CNRS, 67084 Strasbourg, France' author: - 'O. Costin, M. Huang and F. Fauvet' title: 'Global behavior of solutions of nonlinear ODEs: first order equations' --- $ $ -0.2cm Introduction ============ The point at infinity is most often an [*irregular singular point*]{} for equations arising in applications.[^1] Within this class of equations, there are essentially two types for which a global description of solutions exists: linear systems and integrable ones. However, in a stricter sense, even for some linear problems global questions such as explicit values of connection coefficients are still open. The behavior of the general solutions of [*linear*]{} ODEs has been thoroughly analyzed starting in the late 19th century, see [@Fabry] and [@Wasow] and references therein. After the pioneering work of Écalle, Ramis, Sibuya and others the description of their solutions in $\CC$ is by now quite well understood [@Ecalle; @Ecalle-book; @Balser; @Balser3; @Braaksma; @Ramis1; @Ramis2; @Duke]. [*Integrable*]{} systems provide another important class of systems allowing for global description of solutions. The ensemble of integrable systems is a zero measure set in the parameter space of general equations: a generic small perturbation of an integrable system destroys integrability. Nonetheless, integrable equations occur remarkably often in many areas of mathematics, such as orthogonal polynomials, the analysis of the Riemann-zeta function, random matrix theory, self-similar solutions of integrable PDEs and combinatorics, cf. [@Bleher],[@Deift3]–[@Deift1], [@Ablowitz; @Fokas3; @Calogero; @Conte; @Deift1], [@Fokas]–[@zak]. However, even in integrable systems, achieving global control of solutions in a [ *practical way*]{} is a challenging task, and it is one of the important aims of the emerging Painlevé project [@Painleve22]. In [*nonintegrable*]{} systems, particularly near irregular singularities, our understanding is much more limited. Small solutions are given by generalized Borel summable [*transseries*]{}; this was discovered by Écalle in the 1980s and proved rigorously in many contexts subsequently. Transseries are essentially formal multiseries in powers of $1/x^{k_i}$ $e^{-\lambda_j x}$, and possibly $x^{-1}\log x$; see again [@Ecalle; @Ecalle-book; @Balser; @Balser3; @Braaksma; @Ramis1; @Ramis2; @Duke] and [@OCBook]. Here $x$ is the independent variable and $\lambda_j$ are eigenvalues of the linearization with the property $\Re (\lambda_j x)>0$. In general, [*only*]{} small solutions are well understood. However, for generic nonlinear systems of higher order, small solutions form lower dimensional manifolds in the space of all solutions, see, e.g., [@Duke]. The present understanding of general nonlinear equations is thus quite limited. We introduce a new line of approach, combining ideas from generalized Borel summability and KAM theory (see, e.g. [@Arnold]) for the analysis near infinity, chosen to be an irregular singular point, of solutions of relatively general differential equations with meromorphic coefficients. Applying the method does not require knowledge of Borel summability, transseries or KAM theory. For small solutions, in [@Invent] it was shown that in a region adjacent to the sector where the solution, $y$, is small, $y(x)$ is almost periodic. In this sense $y$ becomes an approximately cyclic variable. In the $x$-complex plane, the singular points of $y$ are arranged in quasi-periodic arrays as well. The analysis in [@Invent] covers an angularly small region beyond the sector where $y$ is small. Looking directly at the asymptotics of $y$ beyond this region would require a multiscale approach: $y$ has a periodic behavior–the fast scale, with $O(1/x)$ changes in the quasi-period. Multiscale analysis is usually a quite involved procedure (see, e.g., [@Bender]). It is natural to make a hodograph transformation in which the dependent and independent variables are switched. As mentioned above, in the “nontrivial” regions, the dependent variable is an almost cyclic one. The setting becomes somewhat similar to a KAM one: there is an underlying completely integrable system, and one looks for persistence of invariant tori. Adiabatic invariants are simply the conserved quantities associated with these tori. Evidently there are many differences between the ODE setting and the KAM one, for instance the fact that the small parameter is “internal”, $1/x$. In this work we restrict the analysis to first order equations, mainly to ensure a transparent and concrete analysis. In theory however, the method generalizes to equations of any order, and we touch on these issues at the end of the paper. We look at equations which, after normalization, are of the form $dy/dx=F(z,y)$, $z=1/x$, with $g$ bi-analytic at ${\bf 0}$ and $F_y({\bf 0})=1$. We show that in any sector on Riemann surfaces towards infinity, the [*general*]{} solution is represented by transseries and/ or, in an implicit form, by some constant of motion. In fact, on large circles around $x=0$, the solution cycles among transseries representations and ones in which constants of motion describe it accurately. The regions where these behaviors occur overlap slightly to allow for asymptotic matching (cf. Corollary \[C1\]). The connection between the large $x$ behavior and the initial condition is relatively easy to obtain. Let $\beta=F_{zy}({\bf 0})$. The constants of motion have asymptotic expansions of the form $$\label{eq:eqinit} C(x,y)\sim x-\beta\log x+F_0(y)+x^{-1}F_1(y)+\cdots+x^{-j}F_j(y)+\cdots, \ \ x\to\infty .$$ Clearly, under the assumptions above, the solution $y$ can be obtained asymptotically from and the implicit function theorem. The requirement that $C$ is to leading order of the form $f_1(x)+f_2(y)$, determines $C$ up to trivial transformations, see Theorem \[T1\] and Note \[N1\]. The functions $F_j$ are shown in the proof of Theorem \[T1\] to solve first order autonomous ODEs, and thus they can always be calculated by quadratures. To illustrate this, we use a nonintegrable Abel equation, $$\label{eq:Abel} u'=u^3-t .$$ We note that there is no consensus on how nonintegrabilty should be defined; for (\[eq:Abel\]), it is the case that the equation passes no criterion of integrability, including the poly-Painlevé test, and that there are no solutions known, explicit or coming from, say, some associated Riemann-Hilbert reformulation. The Abel equation has the normal form (see §\[sable\], where further details about this example are given) $$\begin{gathered} \label{tr12} y'+3y^3-\frac{1}{9}+\frac{1}{5x}y=0 .\end{gathered}$$ Regions of smallness are those for which $y$ approaches a root of $3y^3-1/9$; in these regions, $y$ is given by a transseries [@Invent]. Otherwise, $y$ has an implicit representation of the form $$\begin{gathered} \label{newton0} y=\frac{1}{3}\exp\bigg(-C-x+\frac{1}{5}\log x+\left(\sqrt{3}-\frac{2\sqrt{3}}{5x}\right)\arctan \left(\frac{6y+1}{\sqrt{3}}\right)\\ -\log(3y-1)+\frac{1}{2}\log(9y^2+3y+1))+\frac{1}{x}\left(\frac{27y^2}{5(1-27y^3)}+\frac{1}{25}+O(1/x)\right)\bigg)+\frac{1}{3} ,\end{gathered}$$ obtained by inverting an appropriate constant of motion $C$ (see ); for the values of $\beta, F_0, F_1$ see §\[S4\]. While in a numerical approach to calculating solutions the precision deteriorates as $x$ becomes large, the accuracy of instead, [*increases*]{}. In examples, even when is truncated to two orders, is strikingly close to the actual solution even for relatively small values of the independent variable, see e.g. Figure\[fig:abel4\]. The procedure allows for a convenient way to link initial conditions to global asymptotic behavior, see e.g. . Solvability versus integrability -------------------------------- First order equations for which the associated second order autonomous system is Hamiltonian are in particular [ *integrable*]{}. Indeed, by their definition, there is a globally defined smooth $H$ with the property that $\dot{x}\frac{\partial H}{\partial x}+\dot{y}\frac{\partial H}{\partial y}=0$, that is $H(x(t),y(t))=const$, providing a closed form implicit, global representation of $y$. While the differential equation provides “infinitesimal” information, $H$ –effectively an integral– provides a global one. Conversely, clearly, if there exists an implicit solution of the equation or indeed a smooth enough conserved quantity, the equation comes from a Hamiltonian system. What we provide is a finite set of matching conserved quantities, analogous to an atlas of overlapping maps projecting the differential field onto the trivial one, $H'=0$. They give, in a sense, a [ *foliation of the phase space*]{} allowing for global control of solutions. With obvious adaptations, this picture extends to higher order systems. In integrable systems there is just one single-valued map and the field is globally rectifiable. In general, the conserved quantities may be branched and not globally defined. Normalization and definitions {#Sec11} ----------------------------- Many equations of the form $y'=F(y,1/x)$ with $F$ analytic for small $y$ and small $1/x$ can be brought to the normal form $y'=P_0(y)+Q(y,1/x)$ by systematic changes of variables, see [*e.g.*]{} [@Duke], [@OCBook]. The assumptions are that $Q(y,z)$ is entire in $y$ and analytic in $z$ for small $z$, and $O(y^2,yz^2,z)$ for small $y$ and $z$ and that $P_0$ is a polynomial. We assume that the roots of $P_0$ are [*simple*]{}. It will be seen from the analysis that a more general $P_0$ can be accommodated. We thus write the equation as $$\label{eq:eqy0} y'=\sum_{k=0}^{\infty}\frac{P_k(y)}{x^k}=Q_1(y,1/x)=P_0(y)+Q(y,1/x) .$$ \[D1\] \[def1\] $\bullet$ A formal constant of motion of for $x\to \infty$ in an unbounded domain $\mathcal{D}\subset\mathbb{C}^2$ or on a Riemann surface covering it, and in which to leading order in $1/x$ the variables $x$ and $y$ are separated additively is a formal series $$\label{eq:eq0} \tilde{C}(y,x)=A(x) +F_0(y)+\frac{F_1(y)}{x}+\cdots+\frac{F_j(y)}{x^j}+\cdots$$ such that we have $$\frac{d}{dx}\tilde{C}(y(x),x)=O(x^{-\infty})$$ in the sense that, for any $j$, $F_j$ and $H_j$ defined by $$\label{eq:defC} \frac{H_{j+1}(x,y)}{x^{j+1}}:= A'(x) +D_x \left(F_0(y)+\frac{F_1(y)}{x}+\cdots+\frac{F_j(y)}{x^j}\right)$$ are uniformly bounded in $\mathcal{D}$; here $D_x$ is the derivative along the field, $$D_x F(x,y)=\nabla F\cdot (1,Q_1)=F_x(x,y)+F_y(x,y)Q_1(y,1/x).$$ See also below. $\bullet$ An actual constant of motion associated to $\tilde{C}$ in $\mathcal{D}\subset\CC^2$ is a function $C$ so that $C(y,x)\sim\tilde{C}(y,x)$ as $x\to\infty$ and $\frac{d}{dx}C(y(x),x)=0$ for all solutions in $\mathcal{D}$. \[N1\] It will be seen that there is rigidity in the form of the constant of motion: if the variables in $\tilde{C}$ are, to leading order, separated additively as in , then, up to trivial transformations, we must have $$\label{eq:eqA} A(x)=-x+a\log x$$ where $a$ is the same as the one in the transseries expansion of the solution, see Proposition \[trans\]. Finding the terms in the expansion of $\tilde{C}$ ------------------------------------------------- Using (\[eq:eqA\]) and truncating at an arbitrary $n>2$, let $$\label{eq:def2c} {C}_n(y,x)=:-x+a \log x+F_0(y)+\sum_{k=1}^{n}\frac{F_k(y)}{x^k} .$$ We can check that $D_x C_n$ satisfies $$\begin{gathered} \label{formalexpans} D_x C_n=-1+P_0F_0'+\frac{a+P_1F_0'+P_0F_1'}{x}\\ +\sum_{k=2}^{n}\frac{(1-k)F_{k-1}+\sum_{j=0}^{k}P_{k-j}F'_{j}}{x^k} +\frac{-nF_n+\sum_{j=0}^{n}\sum_{k=0}^{\infty}P_{n+k+1-j}F'_{j}x^{-k}}{x^{n+1}}\end{gathered}$$ (cf. ) where the numerator of the last term is $H_n$ by definition. In order for $\tilde{C}$ to be a formal constant of motion, the coefficients of $x^{-j},j=0,1,2,\ldots$ must vanish, giving $$\begin{aligned} \label{refF} F_0'(y)&=\frac{1}{P_0(y)}\\ F_1'(y)&=-\frac{a+P_1(y)F_0'(y)}{P_0(y)} \label{refF1}\\ \label{dfk} F_k'(y)&=\frac{(k-1)F_{k-1}(y)-\sum_{j=0}^{k-1}P_{k-j}(y)F'_{j}(y)}{P_0(y)}\;\quad(2\leq k \leq n) .\end{aligned}$$ It follows in particular that $F'_0\ne 0$ and $F_0$ is bounded in $\mathcal{D}$. In solving the differential system, the constants of integration are chosen so that $F_k$ are indeed uniformly bounded in $y$, see . Solving for $y(x)$ ------------------ The expression $C_n$ is an approximate constant of motion ; we thus can find an approximate solution $y_n$ by fixing $C_n=K$. We then write $$\label{eq:eqyn} G(y;K):= F_0(y)-K-x+a\log x+\sum_{k=1}^n \frac{F_k(y)}{x^k}=0$$ and we note that in the domain relevant to us ($\mathcal{S}_1$, see Theorem \[regcom\] below) the analytic implicit function theorem applies since $$\label{eq:difK} \frac{\partial G}{\partial y}=\frac{1}{P_0(y)} +\frac{1}{x}E_1(y,x)$$ where $P_0$ is away from $0$ in our domain, and for some $E_1$ which is bounded in $\mathcal{D}$ by since $Y$ is bounded. Writing $y=y_n$ in and using the analytic implicit function theorem, treating $1/x$ as a small parameter, we get $$\label{eq:eqy} y_n=G_0(x;K)+\frac{G_1(x;K)}{x}+\cdots+\frac{G_n(x;K)}{x^n} + \frac{\tilde{H}_n(x;K)}{x^{n+1}}$$ where $\tilde{H}_n$ and the $G_j$’s are bounded. In the same way it is checked that $y_n$ is solution of up to corrections $R_n(x;K)/x^{n+1}$, that is, $y_n'-Q_1(y_n,1/x)=-R_n(x;K)/x^{n+1}$ where $R_n$ is bounded. Let $p_1,\ldots,p_m$ be the distinct roots of $P_0$. Let $\mathcal{R}_y$ be the universal cover of $Y=\mathbb{C}\backslash\{p_1,...,p_m\}$. Let $\pi:X\rightarrow Y$ be the covering map. \[Def4\] $\bullet$ An [**elementary $\bf y$-path**]{} of type $$\alpha=(\alpha_1,...,\alpha_m,\alpha_{m+1},\ldots,\alpha_{mk})\in\mathbb{Z}^{mk}, k\in\NN$$ is a piecewise smooth curve $\gamma$ in $\mathcal{R}_y$ whose image under $\pi$ turns $\alpha_{1}$ times around $p_1$, then $\alpha_{2}$ times around $p_2$, and so on, $\alpha_{m}$ times around $p_m$, then again $\alpha_{m+1}$ times around $p_1$ , etc. Note that $\alpha$ is in fact an element of the fundamental group. $\bullet$ A [**$\bf y$-path of type $\boldsymbol\alpha$** ]{} is a smooth curve $\gamma$ obtained as an arbitrary forward concatenation of elementary $ y$-paths of type $\alpha$. More precisely, a $ y$-path of type $\alpha$ is a map $\gamma:[0,\infty)\to \mathcal{R}_y$ so that, for any $N\in \mathbb{Z}^+$, $\gamma|_{[N,N+1]}$ is an elementary $ y$-path of type $\alpha$. We will naturally denote by $\gamma|_{[0,a]}$ subarcs of $\gamma$. We see that $y$-paths are compositions of [*closed loops*]{} in the [*complex $y$ domain.*]{} $\bullet$ $\mathcal{S}_r$ is [**a regular domain of type $\boldsymbol\alpha$**]{} or an $R$-domain of type $\alpha$, if it is an unbounded open subset of $\mathcal{R}_y$ that contains only images of $ y$-paths of type $\alpha$. Thus the image of any unbounded $y$-path of type $\alpha'\neq \alpha$ is not a subset of $\mathcal{S}_r$. [**Note.**]{} In our results we only need $ y$-paths with the additional property that $x(y)\to\infty$ along the path. To take a trivial illustration, in the equation $y'=y$ an example of a $ y$-path along which $x\to \infty$ is $t\mapsto \exp(it), t\ge 0$. Main results ============ Existence of formal constants of motion --------------------------------------- Under the assumptions at the beginning of §\[D1\] we have \[regcom\] Let $\mathcal{S}_y$ be an $R$-domain of type $\alpha$, and $$\mathcal{S}_1=\{y\in \mathcal{S}_y: |\pi(y)|< M_0 ~{\rm and} ~|\pi(y)-p_k|>\epsilon ~{\rm for~all~}k\}$$ where $M_0>0$ is an arbitrary constant. Let $\mathcal{C}$ be the union of $m$ circular paths surrounding $\alpha_k\in\ZZ$ times the root $p_k$, $k=1,\ldots,m$, chosen so that $$\label{eq:eqnontr} \int_{\mathcal{C}}\frac{1}{P_0(y)}dy\neq0 .$$ Then, if $R_0$ is large enough, there exists a formal constant of motion in $$\mathcal{D}_1=\{(x,y):|x|>R_0,y\in\mathcal{S}_1\}$$ of the form . The terms $F_k$ in the expansion of $\tilde{C}$ in can be calculated by quadratures. Actual constants of motion are obtained in Theorem \[T1\]. Consider now a set $\mathcal{S}$ of curves $\gamma$, $|\gamma(t)|\to\infty$ as $t\to\infty$, with the property that for all $t_1<t_2$ and all $n$ (which is in fact equivalent to for $n=0,1$ ) $$\label{eq:restrgam} \left| \Re \displaystyle \int_{\gamma(t_1)}^{\gamma(t_2)} \frac{\partial}{\partial y}Q_1(y,\gamma(t))|_{y=y_n(\gamma(t))}\gamma'(t)dt\right|\leqslant b\log(|\gamma(t_2)/\gamma(t_1)|+1) ,$$ where $b>0$ is a constant, and such that there is an $M$ so that for all $n$ we have $|y_n|<M$ along $\gamma$. Here $M$ can be chosen large if $x$ is large. Note that $\mathcal{S}$ contains the curves $\gamma(t)$ so that $y_n(\gamma(t))$ is an $\alpha$-path. Indeed, by , in this case, the integrand in is of the form $\frac{P_0'(y)}{P_0(y)}dy+O(1)\frac{d\gamma(t)}{\gamma(t)}$ and hence the integral equals $2\pi i N +O(\log(|N|+1))$ for large $N$ where $N$ is the number of loops. \[T1\] Assume $\tilde{C}$ in is a formal constant of motion in a region $\mathcal{D}=\mathcal{S}\cap \mathcal{D}_1$. Then there exists an actual constant of motion $C=C(x,y)$ defined in the same region, so that $C\sim \tilde{C}$ as $x\to\infty$. Regions where $P_0(u)$ is small ------------------------------- Assume $x_0$ is large and $|P_0(y(x_0))|<\epsilon$ is sufficiently small. This means that for some root $r_k$ of $P_0$ we have $|y(x_0)-r_k|<\epsilon_1$ where $\epsilon_1$ is also small. Without loss of generality we can assume that $r_k=0$ and $x\in \RR^+$ since the change of variables $y_1=y-r_k$, $x=x_1 e^{i\phi}$ does not change the form of the equation. Assume also that after normalization the stability condition $\Re P_0'(0)<0$ holds. Again without loss of generality, by taking $y_2=\alpha y_1$ we can arrange that $P_0'(0)=-1$. The new function $Q$ in will have the form $y^2Q_1(y,1/x)+x^{-2}Q_2(y,1/x)$ where $Q_1$ and $Q_2$ are analytic for small $y$ and $1/x$. As a result, the normalized equation assumes the form $$\label{nf} y'=-y+f_0(x)+\frac{ay}{x}+y^2Q_1(y,1/x)+x^{-2}Q_2(y,1/x) .$$ We also arrange that $f_0=O(x^{-M})$ as $x\to\infty$, for suitably large $M$; this is possible through a change of variables of the form $y_2=y_3+\sum_{k=1}^M c_k x^{-k}$, where the $c_k$’s are the coefficients of the formal power series solution for small $y$. \[trans\] \[see [@Duke] Theorem 3\] Any solution of that is $o(1)$ as $x\to\infty$ along some ray in the right half plane can be written as a Borel summed transseries, that is $$\label{eq:eqtrans} y(x)=\sum_{k=0}^{\infty}C^k x^{ka+1}e^{-kx} y_k$$ where $y_k$ are generalized Borel sums of their asymptotic series, and the decomposition is unique. There exist bounds, uniform in $n$ and $x$, of the form $|y_n(x)|<A^k$, and thereby the sum converges uniformly in a region $\mathcal{R}$ that contains any sector $\mathcal{S}_c:=\{x:|\arg\, x|<c<\pi/2\}$. Note that Theorem 3 in [@Duke] applies to general n-th order ODEs. \[ptranss\] \(i) If, after the normalization above, $y(x_0)$ is small (estimates can be obtained from the proof), then $y$ is given by . \(ii) $C(y(x),x)$, obtained by inversion of (\[eq:eqtrans\]) for large $x$ in the right half plane and small $y$, is a constant of motion defined for all solutions for which $y(x_0)$ is small (cf. (i)). \(i) We write the differential equation in the equivalent integral form $$\begin{gathered} \label{eq:intfor} y=F_0(x)+y_0 e^{-(x-x_0)}(x/x_0)^a\\ + e^{-x}x^a \int_{x_0}^x e^s s^{-a} \left[ y^2(s)Q_1(y(s),1/s)+s^{-2}Q_2(y(s),1/s) \right]ds ,\end{gathered}$$ where $F_0(x)=O(x^{-M})$ ($M$ can be chosen arbitrarily large in the normalization process, [@Duke]) and $F_0(x_0)=0$. It is straightforward to show that for (\[eq:intfor\]) is contractive in the norm $\|y\|=\sup_{x\in\mathcal{S}_c}|x^{M-1} y(x)|$ (see the beginning of this section) and thus it has a unique solution in this space. Hence, by uniqueness, the solution of the ODE with $y(x_0)=y_0$, has the property $y(x)\to 0$ as $x\to\infty$. The rest of (i) now follows from [@Duke]. \(ii) We see from Proposition \[trans\] that $y(x;C)$ is analytic in a domain of the form $\mathcal{S}_c\times \mathbb{D}_\rho$ (As usual, $\mathbb{D}_\rho$ denotes the disk of radius $\rho$.) We look at the rhs of as a function $H(x,C)$. It follows from [@Duke] that $y_1(x)=x^{-1}(1+o(1/x))$. By uniform convergence, we clearly have $$\label{eq:difh} \frac{\partial H}{\partial C}=\sum_{k=0}^{\infty}kC^{k-1} x^{ka+1}e^{-kx} y_k=e^{-x}x^a(1+o(1))\ne 0 .$$ The rest follows from the implicit function theorem. As a result of Theorem \[T1\] and Proposition \[ptranss\] we have the following: \[C1\] If $G_0$ in approaches a root of $P_0$ and $x$ is large enough, then $y$ enters a transseries region, where the new constant is given, after normalization, by Proposition \[ptranss\] (ii); thus the constants of motion in different regions match. Proofs and further results ========================== Proof of Theorem \[regcom\] --------------------------- Let $(x_0,y_0)\in\mathcal{D}_1$. Recalling (\[formalexpans\]), we see that has the solution $$F_0(y)=\int_{y_0}^y\frac{1}{P_0(s)}ds+c_0$$ (we take $c_0=0$ since it can be absorbed into the constant of motion). Eq. gives $$\label{aaa}F_1(y)=f_1(y)+c_1:=-\int_{y_0}^y\frac{a+\frac{P_1(s)}{P_0(s)}}{P_0(s)}ds+c_1,$$ where to ensure boundedness of $F_1(y)$ as the number of loops $\to\infty$, we let $$a=-\frac{\int_{\mathcal{C}}\frac{P_1(y)}{P_0(y)^2}dy}{\int_{\mathcal{C}}\frac{1}{P_0(y)}dy}$$ and $c_1$ is determined to ensure boundedness of $F_2$ (cf. ). Inductively we have $$\label{fk} F_{k+1}(y)=\int_{y_0}^{y}\frac{k(f_k(s)+c_k)-\sum_{j=0}^{k}P_{{k+1}-j}(s)F'_{j}(s)}{P_0(s)}ds+c_{k+1}=:f_{k+1}(y)+c_{k+1}$$ for $2\leq k+1\leq n$, and, to ensure boundedness of $F_{k+1}(y)$ as the number of loops $\to\infty$ we need to choose $$\label{gk} c_{k}=\dfrac{\int_{\mathcal{C}}\dfrac{-kf_{k}+\sum_{j=0}^{k}P_{k+1-j}(y)f'_{j}(y)}{P_0(y)}dy}{k\int_{\mathcal{C}}\dfrac{1}{P_0(y)}dy}$$ for $1\leq k\leq n-1$. It is clear by induction that every singularity of $F_k(y)$ is a root of $P_0$. To complete the proof we need to show that the $F_k$’s are bounded in $\mathcal{D}_1$: Assume $y\in\mathcal{S}_1$. For $\deg(P_0)\geq1$ and $1\leq k\leq n$ we have $$|F_k'(y)|\lesssim k!\ \ \ \ \text{and}\ \ \ \ |F_k(y)|\lesssim k!(|y|+1)$$ where, as usual, $\lesssim $ means $\le $ up to an irrelevant multiplicative constant. We prove the lemma by induction on $k$. Note that in (\[aaa\]) and (\[fk\]) the integration paths can be decomposed into finitely many circular loops $\mathcal{C}$ and a ray, slightly deformed around possible singularities, which implies $$|F_1(y)|\lesssim \log|y|+1\lesssim |y|+1$$ and $$|F_k(y)|\lesssim \left|\int_{y_0}^{y}|F_k'(s)|ds\right|$$ where the integration path is a straight line (possibly bent as above). We see from (\[dfk\]) that $$|F_k'(y)|\lesssim \frac{(k-1)|F_{k-1}(y)|}{|P_0(y)|}+\sum_{j=0}^{k-1}|F_j'(y)|\lesssim \frac{(k-1)|F_{k-1}(y)|}{|y|+1}+\sum_{j=0}^{k-1}|F_j'(y)|.$$ The conclusion then follows by induction. Note that the the last term of satisfies $$\left|-nF_n+\sum_{j=0}^{n}\sum_{k=0}^{\infty}P_{n+k+1-j}F'_{j}x^{-k}\right|\lesssim (n+1)!(|y|+1)|P_0(y)|$$ Proof of Theorem \[T1\] ----------------------- Let $y(x;K)=y_n(x;K)+\delta(x;K)$, where $y_n$ is given in . We seek $\delta$ so that $y$ is an exact solution of in $\mathcal{D}$. Let $\phi(y,\delta)$ be the polynomial satisfying $Q_1(y+\delta,x)-Q_1(y,x)=Q_{1,y}(y,x)\delta+\delta^2 \phi(y,\delta,x)$ where $Q_{1,y}(y,x):=\frac{\partial Q_1(y,x)}{\partial y}$. We obtain $$\label{eq:del} \delta'-\frac{b\delta}{x}-\frac{\partial Q_1(y,x)}{\partial y}\delta=\frac{R(x;K)}{x^{n+1}}-\frac{b\delta}{x}+\phi(y_n,\delta,x)\delta^2=:E(x;\delta(x);K) ,$$ where $R=:R_n$ is defined after ; both $R$ and $\phi$ are, by assumption, bounded. In integral form, reads $$\label{eq:deli} \delta(x)=\int_{\infty}^x \frac{x^b}{s^b}e^{\int_{s}^xQ_{1,y}(y_n(t),t)dt} E(s;\delta(s);K)ds$$ where the integrals are taken along curves in $\mathcal{D}$. Using we see that (\[eq:deli\]) is contractive in the norm $\|\delta\|=\ds \sup_{|x|\geqslant |x_1|; x\in\mathcal{D}}|x|^{n}|\delta(x)|$ in an arbitrarily large ball, if $|x_1|$ is large enough and $n>b2^{b+1}$. Thus has a unique solution and, of course, $\delta(x)$ is the limit of the Picard like iteration $$\begin{gathered} \label{eq:eqar} \delta_0=\int_{\infty}^x \frac{x^b}{s^b} e^{\int_{s}^xQ_{1,y}(y_n(t),t)dt} \frac{R(s;K)}{s^{n+1}}ds\\ \delta_1=\int_{\infty}^x \frac{x^b}{s^b} e^{\int_{s}^xQ_{1,y}(y_n(t),t)dt} E(s;\delta_0(s);K) ds\\ etc.\end{gathered}$$ By $\delta$ is a smooth function depending on $(x, K)$ only, and $\delta=O(x^{-n})$. Smoothness is shown as usual by bootstrapping the integral representation . Now we have, by , $\partial_K y_n(x;K)=P_0(y_n)(1+o(1))$. We can easily check that $\partial_K\delta(x,K)=O(x^{-n})$. This is done using essentially the same arguments employed to check contractivity of the integral equation for $\delta$ in the equation in variations for $\delta_K$, derived by differentiating with respect to $K$. We use the implicit function theorem to solve for $K$, giving $K=K(x,y)$, a smooth function of $(x,y)$. It has the following properties: $K(x,y(x))$ is by construction constant along admissible trajectories and by straightforward verification, i.e. comparing $K$ with $\tilde{C}$, we see that it is asymptotic to $\tilde{C}$ up to $O(x^{-n})$. It is known that if a function differs from the $n$th truncate of its series by $O(x^{-n})$ for large $n$, then in fact the difference is $o(x^{-n})$ (cf. [@OCBook] Proposition 1.13 (iii)). Position of singularities of the solution ----------------------------------------- It is convenient to introduce constants of motion specific to singular regions; they provide a practical way to determine the position of singularities, to all orders. We define a [**simple singular solution path**]{} $\gamma(s):[0,1)\rightarrow \mathcal{R}_y$ to be a piecewise smooth curve whose projection $\pi(\gamma[0,1))\in \CC$ is unbounded but turns around every $p_k$ only finitely many times. A [**simple singular solution domain**]{} $\mathcal{S}_s$ is the homotopy class of any simple singular solution path, in the sense that any two unbounded paths in $\mathcal{S}_s$ can be continuously deformed into each other without passing through any $p_k$. \[sincom\] Let $m_0=\deg(P_0)\geq 2$, $\mathcal{S}_s$ be a simple singular solution domain, and $\mathcal{D}_2=\{(x,y):|x|>R,y\in\mathcal{S}_s, ~{\rm and} ~|y-p_k|>\epsilon ~{\rm for~all~}k\}$. Assume that $$\frac{|P_k(y)|}{|P_0(y)|}\lesssim |y|^{-q}$$ for large $y$, for some $q\geq 0$ and all $k\geq 1$. Note that this needs only be true in $\mathcal{S}_s$, which could be an angular region. Then there exists in $\mathcal{D}_2$ a formal constant of motion of the form $$\label{eq:com1} \tilde{C}(y,x)=x+F_0(y)+\frac{F_1(y)}{x}+\cdots+\frac{F_j(y)}{x^j}+\cdots ,$$ where $F_k(y)$ are single valued as $y\to\infty$. Furthermore, any simple singular solution path passing through some arbitrary $(x_0,y_0)$ tends to a singularity, whose position $x_{sing}$ satisfies $$\label{sing} x_{sing}= C_n(y_0,x_0)+O\left(\frac{1}{x_0^{n+1}}\right)$$ for all $n\in\mathbb{N}$, where $C_n$ is $\tilde{C}$ truncated to $x^{-n}$. Moreover, if there are only finitely many nonzero $P_k$, then there exists in $\mathcal{D}_2$ a true constant of motion of the form (\[eq:com1\]), i.e. the sum is convergent for large $|x|$. The proof is similar to that of Theorem \[regcom\]. In order for $\tilde{C}$ to be a formal constant of motion, we must have $$\begin{aligned} F_0'(y)&=-\frac{1}{P_0(y)}\\ \label{dfk1} F_k'(y)&=\frac{(k-1)F_{k-1}(y)-\sum_{j=0}^{k-1}P_{k-j}(y)F'_{j}(y)}{P_0(y)}\;\quad(1\leq k\leq n).\end{aligned}$$ We solve successively for the $F_k$ and obtain $$F_0(y)=\int_{\infty}^y\frac{1}{P_0(s)}ds$$ where the integration path lies in $\mathcal{S}_s$. Clearly $F_0$ is bounded and single valued as $y\to\infty$. Inductively we have $$\label{fk1} F_k(y)=\int_{\infty}^{y}\frac{(k-1)F_{k-1}(s)-\sum_{j=0}^{k-1}P_{k-j}(s)F'_{j}(s)}{P_0(s)}ds$$ for $1\leq k\leq n$. To prove the rest of the proposition, we need the following lemma: Assume that $y\in\mathcal{S}_s$. For $1\leq k\leq n$ we have $$|F_k'(y)|\lesssim \frac{k!}{|y|^{m_0+q}}$$ $$|F_k(y)|\lesssim \frac{k!}{|y|^{m_0+q-1}}$$ as $y\rightarrow\infty$. Furthermore, if $P_k=0$ for $k>k_0>0$, then $$|F_k'(y)|\lesssim \frac{c^k}{|y|^{m_0+q}}$$ $$|F_k(y)|\lesssim \frac{c^k}{|y|^{m_0+q-1}}.$$ The estimates are obtained by induction on $k$. Note that (\[dfk1\]) implies $$|F_k'(y)|\lesssim \frac{(k-1)|F_{k-1}(y)|}{|y|^{m_0}}+|y|^{-q}\sum_{j=0}^{k-1}|F'_{j}(y)|$$ provided that the assumptions of the lemma hold for $1\leq j\leq k-1$. If $P_k=0$ for $k>k_0>0$, we again show the lemma by induction. Assume that for $0<l\leq k$ we have $$|F_l'(y)|\leq(c_0 l_0)^l \sum_{j=1}^{l+1}\binom {l}{j-1}|y|^{-1+j(1-m_0)-q}$$ (this is obviously true for $l=1$). This implies $$|F_k(y)|\leq(c_0 k_0)^k\sum_{j=1}^{k+1}\binom {k}{j-1}\frac{|y|^{j(1-m_0)-q}}{j(m_0-1)+q}$$ Thus it follows from (\[dfk1\]) that $$F_{k+1}'(y)=\frac{kF_{k}(y)-\sum_{j=\max\{k-k_0+1,0\}}^{k}P_{k+1-j}(y)F'_{j}(y)}{P_0(y)}$$ where, by the induction assumption, the first term satisfies the estimate $$\begin{gathered} \left|\frac{kF_{k}(y)}{P_0(y)}\right|\leq c_1 k \left|\frac{F_{k}(y)}{y^{m_0}}\right|\\ \leq c_0^k k_0^{k+1} c_1\sum_{j=2}^{k+2}\frac{(k+1)\binom {k}{j-2}}{(j-1)(m_0-1)}|y|^{-1+j(1-m_0)-q}\\ \leq c_0^k k_0^{k+1} c_1 \sum_{j=2}^{k+2}\binom {k+1}{j-1}|y|^{-1+j(1-m_0)-q}\end{gathered}$$ where $c_0>1+c_1$. Note that the last inequality follows from $$(k+1)\binom {k}{j-2}=\binom {k+1}{j-1}(j-1).$$ The second term is easy to estimate, since it is clearly bounded by $$k_0 (c_0 k_0)^k \sum_{j=1}^{k+1}\binom {k}{j-1}|y|^{-1+j(1-m_0)-q}.$$ Since $c_1$ is fixed, we can assume that $c_0>1+c_1$, and we have $$|F_{k+1}'(y)|\leq(c_0 k_0)^{k+1} \sum_{j=1}^{k+2}\binom {k+1}{j-1}|y|^{-1+j(1-m_0)-q}.$$ This shows the second part of the lemma. Now since $$|D_x C_n|=\Big|\frac{-nF_n+\sum_{j=0}^{n}\sum_{k=0}^{\infty}P_{n+k+1-j}F'_{j}x^{-k}}{x^{n+1}}\Big| \lesssim \frac{|P_0(y)|}{|x|^{n+1}|y|^{m_0+q}}$$ (cf. \[formalexpans\]), the estimate for $x_{sing}$ follows immediately from integrating $D_xC_n$ from $x_0$ to $x_{sing}=C_n(\infty,x_{sing})$ along the simple singular solution path. The condition $$\frac{|P_k(y)|}{|P_0(y)|}\lesssim |y|^{-q}$$ is not the most general one for which there exists a formal constant of motion in a simple singular domain. However, this condition is frequently satisfied by ODEs that occur in applications (see §\[sable\]). In such cases we can easily use (\[sing\]) to find the position of the singularity (see e.g. (\[asing\])). Example: the nonintegrable Abel equation {#sable} ========================================= To illustrate how to obtain information of the solution of a first order ODE using Theorem \[regcom\] and Proposition \[sincom\], we take as an example the nonintegrable Abel equation . Normalization is achieved by the transformation $x=-(9/5)A^2 t^{5/3}$, $A^3=1$, $u(t)=A^{3/5}(-135)^{1/5}x^{1/5}y(x)$ [@Invent], yielding $$\label{abel} y'=-3y^3+\frac{1}{9}-\frac{y}{5x}.$$ Obviously (\[abel\]) satisfies the assumptions in Theorem \[regcom\] and \[sincom\], with $P_0(y)=-3y^3+\ds\frac{1}{9}$ and $P_1(y)=-\ds\frac{y}{5}$. The three roots of $P_0$ are $\ds\frac{1}{3},~\ds\frac{(-1)^{2/3}}{3}$, and $\ds\frac{(-1)^{4/3}}{3}$. It is known [@Invent] that there exists a solution in the right half plane $\mathbb{H}$ that goes to the root $\ds\frac{1}{3}$ as $x\to\infty$. Similarly, there are solutions that go to the other two roots in other regions, which we will explore in §\[phase\]. In those cases, the behavior of the solution follows from Proposition \[trans\] (see also [@Invent]). However, there are also solutions that do not go to any of the three roots. In these cases, the formal constant of motion will be a useful tool to describe quantitatively the behavior of the solution. Constants of motion in $R$-domains {#S4} ---------------------------------- (cf. Definition \[Def4\]). First we choose an elementary solution path along which the solution $y$ to (\[abel\]) turns around the root $\ds\frac{1}{3}$ clockwise as shown in Fig \[fig:abel1\] and \[fig:abel2\]. ![Solution $y(x)$ with $y_0=1.1$ along the line segments from 1+5i to 1.5+50i to 1.6+120i[]{data-label="fig:abel1"}](abel1.eps) ![Real and imaginary parts of $y(x)$. The upper curve is the real part, the lower curve is the imaginary part, and the straight line is the root $1/3$.[]{data-label="fig:abel2"}](abel2.eps) For simplicity we calculate the first two terms of the expansion (\[eq:def2c\]). We have $$\begin{aligned} \label{f36} F_0(y)&=\int\frac{1}{-3y^3+\frac{1}{9}}dy=\sqrt{3}\arctan \left(\frac{6y+1}{\sqrt{3}}\right)-\log(3y-1)+\frac{1}{2}\log(9y^2+3y+1)\\ a&=\dfrac{\int_{\mathcal{C}}\frac{y}{5(-3y^3+\frac{1}{9})^2}dy}{\int_{\mathcal{C}}\frac{1}{-3y^3+\frac{1}{9}}dy}=\frac{1}{5}\\ F_1(y)&=-\int\dfrac{\frac{1}{5}-\frac{y}{5(-3y^3+\frac{1}{9})}}{-3y^3+\frac{1}{9}}dy =\frac{1}{10}\left(\frac{54y^2}{1-27y^3}-4\sqrt{3}\arctan \left(\frac{6y+1}{\sqrt{3}}\right)\right)+\frac{1}{25} ,\end{aligned}$$ where the constant $\ds\frac{1}{25}$ is found using (\[gk\]). We plot the first two orders of the formal constant of motion in Fig. \[fig:abel3\]. ![Formal constant of motion with $F_0$ and $F_1$.[]{data-label="fig:abel3"}](abel3.eps) Since this formal constant of motion is almost a constant along any path in the same $R$-domain, it can be used to find the solution asymptotically, writing $$C=-x+\frac{1}{5}\log x+F_0(y)+\frac{F_1(y)+O(1/x)}{x}.$$ Placing the term $\log(3y-1)$ (cf. ) in the equation above on the left side and $C$ on the right side, taking the exponential, and solving for $y$, we obtain $$\begin{gathered} \label{newton} y=\frac{1}{3}\exp\bigg(-C-x+\frac{1}{5}\log x+\left(\sqrt{3}-\frac{2\sqrt{3}}{5x}\right)\arctan \left(\frac{6y+1}{\sqrt{3}}\right)\\ +\frac{1}{2}\log(9y^2+3y+1)+\frac{1}{x}\left(\frac{27y^2}{5(1-27y^3)}+\frac{1}{25}+O(1/x)\right)\bigg)+\frac{1}{3} .\end{gathered}$$ The reason for taking the exponential in (\[newton\]) is to take care of the branching due to $\log x$, whereas the other $\log$ and $\arctan$ do not matter since the solution does not encircle their singularities. Equation contains, in an implicit form, the solution $y$ to two orders in $x$. $y$ can be determined from this implicit equation in a number of ways; we chose, for simplicity to numerically solve the implicit equation using Newton’s method. The solution is plotted in Fig. \[fig:abel4\], where we take $C=2.18-4.65i$ and calculate the solution for the second half of the path corresponding to $|x|>61.4$. Note that the relative error is within $1.5\%$. Since the accuracy of the formal constant of motion is unaffected by going along the solution path as long as $|x|$ is large, we can obtain quantitative behavior of the solution for very large $|x|$. By contrast, in a numerical approach, the further one integrates along the path, the less accurate the calculated solution becomes. ![Comparison of solutions obtained numerically by the Runge-Kutta method and using the formal constant of motion .[]{data-label="fig:abel4"}](abel4n.eps) ![A small section of the left-top plot in Fig. \[fig:abel4\].[]{data-label="fig:abel4z"}](abel4z.eps) Finding the positions of the singularities ------------------------------------------ We illustrate how to find singularities of the Abel’s equation using Proposition \[sincom\]. It is known [@Invent] that there are only square root singularities, and they appear in two arrays. For simplicity we choose a simple singular path along which $y$ goes to $+\infty$. According to Proposition \[sincom\] we have $$\begin{gathered} \label{F0} F_0(y)=-\int_{\infty}^{y}\frac{1}{-3s^3+\frac{1}{9}}ds\\=-\sqrt{3}\arctan (\frac{6y+1}{\sqrt{3}})+\log(3y-1)-\frac{1}{2}\log(9y^2+3y+1)+\frac{\sqrt{3}\pi}{2}\\ F_1(y)=-\frac{1}{5}\int\frac{y}{(-3y^3+\frac{1}{9})^2}dy\\ =\frac{-\frac{54y^2}{1-27y^3}+2\sqrt{3}\arctan (\frac{6y+1}{\sqrt{3}})+2\log(3y-1)-\log(9y^2+3y+1)}{10} .\end{gathered}$$ Thus the position of the singularity is given by the formula $$\begin{gathered} \label{asing} x_1 =C+o(1)=x_0-\sqrt{3}\left(1-\frac{1}{5x_0}\right)\left(\arctan \left(\frac{6y_0+1}{\sqrt{3}}\right)-\frac{\pi}{2}\right)\\ +\left(1+\frac{1}{5x_0}\right)\left(\log(3y_0-1)-\frac{1}{2}\log(9y_0^2+3y_0+1)\right)-\frac{27y_0^2}{5x_0(1-27y_0^3)}+o(1) ,\end{gathered}$$ where the initial condition $(x_0,y_0)$ satisfies $|x_0|$ is large and $y_0$ is not close to any of the three roots. We note that the presence of the arctan in the leading order implies that the solutions remain quasi-periodic beyond the domain accessible to the methods in [@Invent]. In (\[asing\]) we have the freedom of choosing branch of $\log$ and $\arctan$, which enables us to find arrays of singularities. For example, the position of a singularity corresponding to the initial condition $x_0=10+60i,~y_0=0.7+0.3i$, calculated using (\[asing\]) is $x_1=9.80628+60.2167i$, which is accurate with six significant digits, as checked numerically. The detailed behavior of the solution near the singularity can be found by expanding the right hand side of (\[asing\]). We omit the calculation here since there are many other methods to determine this behavior (cf. [@Invent]) and it is of lesser importance to the paper. Connecting regions of transseries {#phase} --------------------------------- We choose a path consisting of line segments The path in $x$ consists of line segments connecting $50i$, $50$, $-50i$, $-50$, $50i$, $50$, $-50i$, and $-50(\sqrt{3}+i)$. This corresponds to an angle of $2\pi$ in the original variable, with initial condition $y(50i)=0.6$. Along this path, the solution of (\[abel\]) approaches all three complex cube roots of $1/27$. For instance, the root $1/3$ is approached when $x$ traverses the first quadrant along the first segment, the root ${(-1)^{4/3}}/{3}$ is approached when $x$ goes to the lower half plane, and the root ${(-1)^{2/3}}/{3}$ is approached when $x$ goes back to the upper half plane. Some of these values are approached more than once along the entire path. This behavior can easily be shown using the phase portrait of $G_0$, cf. and Corollary \[C1\]. Note that along a straight line $x=x_0+x e^{t i}$ where the angle $t$ is fixed the leading term (with only $G_0$ on the right hand side) of the ODE (\[abel\]) can be written as $$\frac{d y}{d x}=e^{t i}\left(-3y^3-\frac{y}{5(x_0+x e^{t i})}+\frac{1}{9}\right).$$ Denoting $y_1=\Re{y}$ and $y_2=\Im{y}$, we have $$\left\{ \begin{array}{ll} \dfrac{d y_1}{d x}=-3 y_1^3\cos t-3 y_2^3\sin t+9 y_1 y_2^2\cos t+9 y_1^2 y_2\sin t+\ds\frac{\cos t}{9}\\ \dfrac{d y_2}{d x}=-3 y_1^3\sin t+3 y_2^3\cos t+9 y_1 y_2^2\sin t-9 y_1^2 y_2\cos t+\ds\frac{\sin t}{9} \end{array} \right.$$ We can then analyze the phase portraits. For the purpose of illustration, we show some of them in Fig \[fig:abel7\] and \[fig:abel8\]. ![Phase portrait of $\Re(y)$ and $\Im(y)$ for $t=-\pi/4$. The $``\times"$ marks are the three roots.[]{data-label="fig:abel7"}](abelode7b.eps) On the line segment connecting $50i$ and $50$, it is clear that the initial condition $0.6$ is in the basin of attraction of $1/3$ (cf. Fig. \[fig:abel7\]). ![Phase portrait of $\Re(y)$ and $\Im(y)$ for $t=5\pi/4$.[]{data-label="fig:abel8"}](abelode8b.eps) Since the only stable equilibrium is $a_0=\ds\frac{(-1)^{4/3}}{3}$, on the line segment connecting $50$ and $-50i$ the solution converges to $a_0$ (cf. Fig. \[fig:abel8\]). Numerical calculations confirm this, (cf. Fig \[fig:abel6\]). ![Behavior of the solution across transseries regions. Dotted horizontal lines are the imaginary parts of the three roots. The horizontal line is arclength. The path in $x$ consists of line segments connecting $50i$, $50$, $-50i$, $-50$, $50i$, $50$, $-50i$, and $-50(\sqrt{3}+i)$. This corresponds to an angle of $2\pi$ in the original variable.[]{data-label="fig:abel6"}](abel6n.eps) Finally, note that there cannot be a limit cycle in the phase portraits drawn if $x$ goes along a straight line. If the solution $y$ approaches a limit cycle, it must lie in an $R$-domain. Thus the formal constant motion formula (\[eq:def2c\]) is valid, and the first term $F_0$ specifies a direction for $x$. If $x$ goes strictly along this direction towards $\infty$ then the term $a\log x$, which does not vanish in our case, will go to $\infty$, contradicting the results about the constant of motion. On the other hand, if $x$ goes in a different direction, then $-x+F_0(y)$ goes to $\infty$ much faster than $a\log x$, again a contradiction. Extension to higher orders -------------------------- For higher orders, such as the Painlevé equations P1 and P2, a similar procedure works, though the details are quite a bit more complicated, and we leave them for a subsequent work. We illustrate, without proofs, the results for $P1$, $y''=6y^2+z$. Now, there are two asympotic constants of motion, as expected. The normal form we work with is $u''+u'x^{-1}-u-u^2/2-392x^{-4}/625=0$. Denoting by $s$ the “energy of elliptic functions” $s={u'}^2/2-u^3/3+u^2$ (it turns out that $s$ is one of the bicharacteristic variables of the sequence of now PDEs governing the terms of the expansion; thus the pair $(u,s)$ is preferable to $(u,u')$), one constant of motion has the asymptotic form $$C_1=x-L(s,u)+x^{-1}K_1(s,u)+\cdots$$ In the above, denoting $R=\sqrt{u^3/3+u^2+s}$, $L$ is an incomplete elliptic integral, $L=\int R^{-1}(s,u)du$ and the integration is following a path winding around the zeros of $R$. The functions $K_1$, $K_2$, $\cdots$ have similar but longer expressions. We note the absence of a term of the form $a\log x$ (the reason for this is easy to see once the calculation is performed). A second constant can now be obtained by reduction of order and applying the first order techniques, or better, by the “action-angle” approach described in the introduction. It is of the form $$C_2=xJ(s)+[L(s)J(u,s)-J(s)L(u,s)]+x^{-1}\tilde{K}_1+\cdots$$ where $J(u,s)=\int R(s,u)du$; when the variable $u$ is missing from $J(u,s)$ or $R(u,s)$, this simply means that we are dealing with complete elliptic integrals. There is directionality in the asymptotics, as the loops encircling the singularities need to be rigidly chosen according to the asymptotic direction studied. A slightly different representation allows us to calculate the constants to all orders. Because of directionality, a different asymptotic formula exists and is more useful for the “lateral connection”, that is, for calculating the solution along a circle of fixed but large radius, which will be detailed in a separate paper, as part of the Painlevé project, see e.g. [@Painleve22]. Acknowledgments =============== The authors are very grateful to R. Costin for a careful reading of the manuscript and numerous useful suggestions. 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J., Ramis, J.-P., Sibuya, Y., Multisummability of formal power series solutions of linear ordinary differential equations Asymptotic Anal. 5 no. 1, pp. 27–45. (1991). Bender, C. and Orszag, S. [*Advanced Mathematical Methods for scientists and engineers*]{}, McGraw-Hill, 1978, Springer-Verlag (1999). Bleher, P. and Its, A., Double scaling limit in the random matrix model: the Riemann-Hilbert approach. Comm. Pure Appl. Math. 56, no. 4, pp. 433–516. (2003). Bornemann, F., Clarkson, P., Deift, P. Edelman, A., Its, A.and Lozier, D., AMS notices, V. 57, 11 p. 1389 (2010). Braaksma, B. L. J., Multisummability of formal power series solutions of nonlinear meromorphic differential equations. Ann. Inst. Fourier no. 3, 42, pp. 517–540, (1992). Boutet de Monvel, A., Fokas, A. S., Shepelsky, D., Integrable nonlinear evolution equations on a finite interval. Comm. Math. Phys. 263 , no. 1, pp. 133–172 (2006). Calogero, F. A new class of solvable dynamical systems. J. Math. Phys. 49 no. 5, 052701, 9 (2008). Clarkson, P A and Kruskal, M.D., The Painlevé-Kowalevski and poly-Painlevé tests for integrability", Studies in Applied Mathematics 86 pp 87-165, (1992). Conte, R., Musette, M., Verhoeven, C. Painlevé property of the Hńon-Heiles Hamiltonians. Théories asymptotiques et équations de Painlevé, pp. 65–82, Sémin. Congr., 14, Soc. Math. France, Paris, (2006). Costin O., Asymptotics and Borel summability, Chapmann & Hall, New York (2009) Costin O., On Borel summation and Stokes phenomena of nonlinear differential systems [*Duke Math. J.*]{}, 93, No. 2, (1998). Costin O. and Costin R.D., On the formation of singularities of solutions of nonlinear differential systems in antistokes directions [*Inventiones Mathematicae*]{}, 145, 3, pp. 425–485, (2001). Deift, P. Four lectures on random matrix theory. Asymptotic combinatorics with applications to mathematical physics (St. Petersburg, 2001), pp. 21–52, Lecture Notes in Math., 1815, Springer, Berlin, (2003). Baik, J., Deift, P., Johansson, K. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12, no. 4, pp. 1119–1178 (1999). Deift, P.; Zhou, X. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 no. 2, pp. 295–368, (1993). Fokas, A. S. Soliton multidimensional equations and integrable evolutions preserving Laplace’s equation. Phys. Lett. A 372, no. 8, pp. 1277–1279, (2008). Écalle, J., [*Fonctions Resurgentes, Publications Mathematiques D’Orsay*]{}, (1981). Écalle, J., Six lectures on transseries, analysable functions and the constructive proof of Dulac’s conjecture, [*Bifurcations and periodic orbits of vector fields*]{}, NATO ASI Series, Vol. 408, pp. 75-184, (1993) Fabry, C. E. [*Thèse (Faculté des Sciences)*]{}, Paris, 1885 Fokas, A. S. Integrable nonlinear evolution equations on the half-line. Comm. Math. Phys., no. 1, pp. 1–39, 230 (2002). Goriely, A. Integrability and nonintegrability of dynamical systems. Advanced Series in Nonlinear Dynamics, 19. World Scientific Publishing Co., Inc., River Edge, NJ, (2001). Grammaticos, B., Ramani, A., Tamizhmani, K. M., Tamizhmani, T., Carstea, A. S. Do all integrable equations satisfy integrability criteria? Adv. Difference Equ., Art. ID 317520, (2008). Its, A., Jimbo, M. and Maillet, J-M Integrable quantum systems and solvable statistical mechanics models. J. Math. Phys. 50, no. 9, 095101, 1 pp. 81–06, (2009) Kruskal, M. D., Grammaticos, B., Tamizhmani, T. Three lessons on the Painlevé property and the Painlevé equations. Discrete integrable systems, 1–15, Lecture Notes in Phys., 644, Springer, Berlin (2004). Its, A. Connection formulae for the Painlevé transcendents. The Stokes phenomenon and Hilbert’s 16th problem (Groningen, 1995), pp. 139–165, World Sci. Publ., River Edge, NJ, (1996). Its, A. The Riemann-Hilbert problem and integrable systems. Notices Amer. Math. Soc. 50 no. 11, pp. 1389–1400 (2003). Iwano, M., Integration analytique d’un systeme d’equations differentielles non lineaires dans le voisinage d’un point singulier, [*Ann. Mat. Pura Appl. (4) [**44**]{} 1957, 261-292*]{} Ramis, J-P., Gevrey, Dévissage. (French) Journées Singulières de Dijon (Univ. Dijon, Dijon, 1978), pp. 4, 173–204, Astérisque, 59-60, Soc. Math. France, Paris, (1978). Ramis, J.-P. Les séries $k$-sommables et leurs applications. (French) Lecture Notes in Phys., 126, pp. 178–199, Springer, Berlin-New York, (1980). Takahashi, M., On completely integrable first order ordinary differential equations. Real and complex singularities, 388–418, World Sci. Publ., Hackensack, NJ, (2007). Tamizhmani, K. M., Grammaticos, B., Ramani, A. Do all integrable evolution equations have the Painlevé property? SIGMA Symmetry Integrability Geom. Methods Appl. 3, Paper 073, 6 (2007). Treves, F., Differential algebra and completely integrable systems. Hyperbolic problems and related topics, pp. 365–408, Grad. Ser. Anal., Int. Press, Somerville, MA, (2003). Wasow, W., [ Asymptotic expansions for ordinary differential equations, Interscience Publishers, (1968)]{} Yoshino, M., Analytic non-integrable Hamiltonian systems and irregular singularity. Ann. Mat. Pura Appl. (4) 187, no. 4, pp. pp. 555–562. (2008). Zakharov, V. E., (ed.): What is Integrability? Berlin etc., Springer-Verlag 1991 [^1]: A singular point of an equation is irregular if, for [*small*]{} solutions, the linearization is not of Frobenius type. By a small solution we mean one that tends to zero in some direction after simple changes of coordinates.

ArXiv

--- abstract: 'Our proximity and external vantage point make M31 an ideal testbed for understanding the structure of spiral galaxies. The Andromeda Optical and Infrared Disk Survey (ANDROIDS) has mapped M31’s bulge and disk out to R=40 kpc in $ugriJK_s$ bands with CFHT using a careful sky calibration. We use Bayesian modelling of the optical-infrared spectral energy distribution (SED) to estimate profiles of M31’s stellar populations and mass along the major axis. This analysis provides evidence for inside-out disk formation and a declining metallicity gradient. M31’s $i$-band mass-to-light ratio ($M/L_i^*$) decreases from 0.5 dex in the bulge to $\sim 0.2$ dex at 40 kpc. The best-constrained stellar population models use the full $ugriJK_s$ SED but are also consistent with optical-only fits. Therefore, while NIR data can be successfully modelled with modern stellar population synthesis, NIR data do not provide additional constraints in this application. Fits to the $gi$-SED alone yield $M/L_i^*$ that are systematically lower than the full SED fit by 0.1 dex. This is still smaller than the 0.3 dex scatter amongst different relations for $M/L_i$ via $g-i$ colour found in the literature. We advocate a stellar mass of $M_*(30~\mathrm{kpc}) =10.3^{+2.3}_{-1.7}\times 10^{10}~\mathrm{M}_\odot$ for the M31 bulge and disk.' author: - 'Jonathan Sick,$^1$ Stephane Courteau,$^1$ Jean-Charles Cuillandre,$^2$ Julianne Dalcanton,$^3$ Roelof de Jong,$^4$ Michael McDonald,$^5$ Dana Simard,$^1$' - 'R. Brent Tully$^6$' title: 'The Stellar Mass of M31 as inferred by the Andromeda Optical & Infrared Disk Survey' --- Introduction ============ The ANDROIDS programme has used the MegaCam and WIRCam cameras on the Canada-France-Hawaii Telescope (CFHT) to map M31’s bulge and disk homogeneously within $R=40$ kpc with $ugriJK_s$ bands and enable global studies of M31’s structure and stellar populations using both resolved stars and integrated spectral energy distributions (SEDs). In this contribution, we use ANDROIDS to estimate the stellar mass profile of the M31 disk with Bayesian modelling of the optical to near-IR (NIR) SED. This approach is more rigorous than the colour-$M/L^*$ prescriptions [e.g. @Zibetti:2009; @Taylor:2011; @Into:2013] often employed by pixel-by-pixel stellar mass estimation studies that use only a $g-i$ colour and marginalize over all likely star formation histories. By studying M31 in detail, an overall goal of ANDROIDS is to explore systematic uncertainties in studies of more distant and poorly resolved systems. M31 Surface Brightness Calibration ================================== Background subtraction is the most significant challenge for observational studies of M31’s structure since we cannot observe its disk and blank sky simultaneously. This is particularly acute in our NIR maps where skyglow is 3-dex brighter than the disk, while also having strong spatial and temporal variations. In [@Sick:2014], we describe our ANDROIDS/WIRCam sky-target nodding and background subtraction schemes and find that the NIR background cannot be known to better than 2% given the scale of sky-target nods required for M31. We overcome this uncertainty by solving for sky offsets that formally minimize surface brightness differences between overlapping pairs of images. Such sky offsets are $\sim1$% of the NIR brightness, but systematically uncertain up to a zeropoint normalization that is 0.16% of the sky level. In optical bands, the sky background is both more stable and somewhat dimmer, though we still employ sky-target nodding with the Elixir-LSB method for CFHT/MegaCam to build a real-time map of sky and scattered light backgrounds over one-hour sliding windows. With Elixir-LSB we easily identify low surface brightness features in M31’s outer disk, such as the Northern Spur, at levels below $\mu_g\sim26$ mag arcsec$^{-2}$ [@Sick:2013a]. The aforementioned sky offset zeropoint uncertainty requires that our surface brightness profiles be finely calibrated against external datasets. Resolved stellar catalogs transformed into surface brightness maps, such as our own WIRCam star catalog, and even Panchromatic Hubble Andromeda Treasury [PHAT; @Dalcanton:2012], provide a useful dataset up to the limit of completeness corrections. Extremely wide-field imaging is also useful as it enables a simultaneous mapping of the background and the disk light. We are currently using Dragonfly [@Abraham:2014] to image M31 as a replacement for the venerable wide-field plates of [@Walterbos:1987]. SED Stellar Mass Modelling ========================== ![Posterior stellar population profiles for different bandpass sets: $ugriJK_s$ (blue), $ugri$ (green), $gi$ (red). A declining metallicity gradient and inside-out disk formation (seen by an increase in the e-folding time, $\log \tau$, of the exponentially declining star formation history model) are clearly evident.[]{data-label="fig:pop_profile"}](figure1){width="0.7\columnwidth"} ![Posterior stellar $M/L_i^*$ (left) and stellar mass (right) major axis profiles. The $ugriJK_s$ (blue) and $ugri$ (green) fits are consistent, while fit of only $gi$ (red) are lower by 0.1 dex in $M/L_i^*$. Equivalent $gi$–$M/L_i^*$ relations in the literature can vary by 0.3 dex of $M/L_i^*$.[]{data-label="fig:mass_profile"}](figure2){width="0.7\columnwidth"} From the calibrated surface brightness profiles, we model the SED at each radial bin to estimate the stellar population, and hence the stellar mass-to-light ratio, $M/L_i^*$. Our modelling engine is the Flexible Stellar Population Synthesis (FSPS) software [@Conroy:2009; @Conroy:2010]. We chose FSPS for its reliable calibration and “lighter” AGB contribution than older SP models [e.g., @Bruzual:2003], and allowance for deep customization of the computed stellar populations.[^1] We use a Markov Chain Monte Carlo approach to modelling SEDs extracted along the northern major axis of the M31 disk implemented with the `emcee` python package [@Foreman-Mackey:2013]. We tested different star formation history parameterizations and found that a simple ‘$\tau$’ model, involving constant plus exponentially declining star formation rate components minimized residuals compared to more sophisticated ‘delayed $\tau$’ and late burst models. Of the dust attenuation treatments, the default power-law attenuation law with separate components for young and older stellar populations also minimized residuals compared to Milky Way or starburst attenuation models. We found that posterior SED residuals are minimized by fitting the entire $ugriJK_s$ SED. This contradicts [@Taylor:2011] and [@Zibetti:2009] who advocated against using NIR bands in mass estimation due to uncertain AGB treatments of the previous generation of stellar population synthesis models [e.g. @Bruzual:2003; @Maraston:2005]. Much like the NIR, the $griJK_s$-SED fit has little predictive power over the crucial $u$-band. This result should therefore encourage SED modellers to incorporate as many bandpasses as possible, including UV and IR, to obtain the best constraints on stellar populations and masses. We modelled SEDs extracted from a logarithmically-sized wedge [e.g. @Courteau:2011 their Fig. 2] to produce stellar population profiles (shown in Fig. \[fig:pop\_profile\]). Interestingly, the $ugri$-fit and $ugriJK_s$-fit SEDs produce statistically identical stellar population profiles, with the only exception being a slightly tighter posterior credible region from the full-SED fits. Although the consistency of optical and optical-NIR SED fits is reassuring from the perspective of NIR calibrations, it is also disappointing that the NIR data has not produced a remarkably improved posterior stellar population estimate. Clearly evident is that poorly sampled SEDs can bias results. Fitting only the $gi$ SED (that is, using an input information equivalent to those using colour-$M/L^*$ look-up-tables) clearly biases the posterior stellar population distribution, with significantly lower dust opacities and lower mass-to-light ratios. By comparison, we have also plotted mass-to-light ratios predicted by three colour-$M/L^*$ relations [@Zibetti:2009; @Taylor:2011; @Into:2013]. These fits systematically vary by 0.3 dex, far larger than the 0.1 dex of internal systematic uncertainty typically claimed by $g-i$ – $M/L^*$ fits [@Courteau:2013]. Compared to our full SED fits, modelling of the $gi$ SED is less biased than these other $M/L^*$ fits, which are based on other stellar population synthesis models. This serves as reminder that stellar mass estimates remain dominated by prior assumptions such as choices of IMF, dust, and details of AGB treatments, among other concerns. Discussion ========== We have used $ugriJK_s$ SEDs to map the stellar mass of M31’s disk and find a stellar mass, within $30$ kpc, of $M_{ugri}^{*} = 10.3^{+2.3}_{-1.7}\times 10^{10}~\mathrm{M}_\odot$. This result is consistent with the stellar bulge and disk masses quoted by [@Tamm:2012] ($10.1\times10^{10}~\mathrm{M}_\odot$). Future work will extend this analysis to a full 2D mapping of the M31 stellar mass distribution. We are matching these stellar mass maps with dynamical tracers of gas and stars to construct a mass model of M31’s stellar, gas, and dark matter components (Simard et al., in progress). The DiskFit code [@Spekkens:2007] allows us to correct the H<span style="font-variant:small-caps;">i</span> velocity fields of [@Saglia:2010] and [@Chemin:2009] for non-circular motions in M31’s central parts. The success of this mass model will be determined by the stability of a dynamical N-body realization. Acknowledgements {#acknowledgements .unnumbered} ================ J.S. and S.C. acknowledge support through respective Graduate Scholarship and Discovery grants from the Natural Sciences and Engineering Research Council of Canada. We thank the Canadian Advanced Network for Astronomical Research (CANFAR) for enabling the computing facilities needed for this work. [19]{} natexlab\#1[\#1]{} , R. G., & [van Dokkum]{}, P. G. 2014, [PASP]{}, 126, 55 , G., & [Charlot]{}, S. 2003, [MNRAS]{}, 344, 1000 , L., [Carignan]{}, C., & [Foster]{}, T. 2009, [ApJ]{}, 705, 1395 , C., [Gunn]{}, J. E., & [White]{}, M. 2009, [ApJ]{}, 699, 486 , C., [White]{}, M., & [Gunn]{}, J. E. 2010, [ApJ]{}, 708, 58 , S., [Widrow]{}, L. M., [McDonald]{}, M., [et al.]{} 2011, [ApJ]{}, 739, 20 , S., [Cappellari]{}, M., [de Jong]{}, R. S., [et al.]{} 2014, Rev. Mod. Phys., 86, 47 , J. J., [Williams]{}, B. F., [Lang]{}, D., [et al.]{} 2012, [ApJS]{}, 200, 18 , D., [Hogg]{}, D. W., [Lang]{}, D., & [Goodman]{}, J. 2013, [PASP]{}, 125, 306 , T., & [Portinari]{}, L. 2013, [MNRAS]{}, 430, 2715 , C. 2005, [MNRAS]{}, 362, 799 , R. P., [Fabricius]{}, M., [Bender]{}, R., [et al.]{} 2010, [A&A]{}, 509, A61 , J., [Courteau]{}, S., & [Cuillandre]{}, J.-C. 2013, 1310.4832 , J., [Courteau]{}, S., [Cuillandre]{}, J.-C., [et al.]{} 2014, [AJ]{}, 147, 109 , K., & [Sellwood]{}, J. A. 2007, [ApJ]{}, 664, 204 , A., [Tempel]{}, E., [Tenjes]{}, P., [Tihhonova]{}, O., & [Tuvikene]{}, T. 2012, [A&A]{}, 546, A4 , E. N., [Hopkins]{}, A. M., [Baldry]{}, I. K., [et al.]{} 2011, [MNRAS]{}, 418, 1587 , R. A. M., & [Kennicutt]{}, Jr., R. C. 1987, [A&AS]{}, 69, 311 , S., [Charlot]{}, S., & [Rix]{}, H. 2009, [MNRAS]{}, 400, 1181 [^1]: The lead author (J.S.) contributes to the maintenance of a Python-language wrapper for FSPS: <http://dan.iel.fm/python-fsps>

ArXiv

--- abstract: 'We show that in some cases, catalyst-assisted entanglement transformation cannot be implemented by multiple-copy transformation for pure states. This fact, together with the result we obtained in \[R. Y. Duan, Y. Feng, X. Li, and M. S. Ying, Phys. Rev. A 71, 042319 (2005)\] that the latter can be completely implemented by the former, indicates that catalyst-assisted transformation is strictly more powerful than multiple-copy transformation. For purely probabilistic setting we find, however, these two kinds of transformations are geometrically equivalent in the sense that the sets of pure states which can be converted into a given pure state with maximal probabilities not less than a given value have the same closure, no matter catalyst-assisted transformation or multiple-copy transformation is used.' address: 'State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology Tsinghua University, Beijing, China, 100084' author: - Yuan Feng - Runyao Duan - Mingsheng Ying bibliography: - 'relation.bib' title: 'Relation between catalyst-assisted transformation and multiple-copy transformation for bipartite pure states' --- Introduction ============ Quantum entanglement, which is essential in quantum information processing such as quantum cryptography [@BB84], quantum superdense coding [@BW92] and quantum teleportation [@BBC+93], has been extensively studied. One fruitful research direction on quantum entanglement is to discuss the possibility of transforming a bipartite entangled pure state into another one allowing only local operations on the separate subsystems respectively and classical communication between them (or LOCC for short). The asymptotic case when arbitrarily large number of copies are provided is considered by Bennett and his collaborators [@BBPS96]. While in deterministic and finite manner, the first and significant step was made by Nielsen [@NI99] who discovered the connection between the theory of majorization in linear algebra [@MO79] and entanglement transformation. Nielsen proved that a bipartite entangled pure state $|\psi_1\rangle$ can be transformed into another bipartite entangled pure state $|\psi_2\rangle$ by LOCC if and only if $\lambda_{\psi_1}\prec\lambda_{\psi_2}$, where the probability vectors $\lambda_{\psi_1}$ and $\lambda_{\psi_2}$ denote the Schmidt coefficient vectors of $|\psi_1\rangle$ and $|\psi_2\rangle$, respectively. Here the symbol $\prec$ stands for ‘majorization relation’. Generally, an $n$-dimensional real vector $x$ is said to be majorized by another $n$-dimensional real vector $y$, denoted by $x\prec y$, if the following relations hold: $$\sum_{i=1}^l x^\downarrow_i \leq \sum_{i=1}^l y^\downarrow_i {\rm \ \ \ for\ any\ \ \ }1\leq l\leq n,$$ with equality holding when $l=n$, where $x^\downarrow$ denotes the vector obtained by rearranging the components of $x$ in nonincreasing order. Nielsen’s theorem gives a necessary and sufficient condition when two entangled pure states are comparable in the sense that one can be transformed into another by LOCC. There exist, however, incomparable states such that any one cannot be transformed into another only using LOCC. To treat the case of transformations between incomparable states, Vidal [@Vi99] generalized Nielsen’s work by allowing probabilistic transformations. He found that the maximal probability of transforming $|\psi_1\rangle$ into $|\psi_2\rangle$ by LOCC can be calculated by $$\label{eq:Vidal} P(|\psi_1\rangle \rightarrow |\psi_2\rangle)=\min_{1\leq l\leq n} \frac{E_l(\lambda_{\psi_1})}{E_l(\lambda_{\psi_2})},$$ where $E_l(x)$ denotes $\sum_{i=l}^n x^{\downarrow}_i$. In Ref.[@JP99], Jonathan and Plenio discovered a very surprising phenomenon that sometimes an entangled state can enable otherwise impossible entanglement transformations without being consumed at all. A simple but well known example is $|\psi_1\rangle\nrightarrow |\psi_2\rangle$ but $|\psi_1\rangle\otimes|\phi\rangle \rightarrow |\psi_2\rangle\otimes|\phi\rangle$, where $|\psi_1\rangle=\sqrt{0.4}|00\rangle+\sqrt{0.4}|11\rangle+\sqrt{0.1}|22\rangle+\sqrt{0.1}|33\rangle,$ $|\psi_2\rangle=\sqrt{0.5}|00\rangle+\sqrt{0.25}|11\rangle+\sqrt{0.25}|22\rangle,$ and $|\phi\rangle=\sqrt{0.6}|44\rangle+\sqrt{0.4}|55\rangle.$ The role of the state $|\phi\rangle$ is just like a catalyst in a chemical process. Daftuar and Klimesh [@DK01] examined catalyst-assisted entanglement transformation and derived some interesting results. In [@FD05a], we investigated catalyst-assisted transformation in probabilistic setting. A necessary and sufficient condition was presented under which there exist partial catalysts that can increase the maximal transforming probability of a given entanglement transformation. The mathematical structure of catalyst-assisted probabilistic transformation was also carefully investigated. Another interesting phenomenon of entanglement transformation was noticed by Bandyopadhyay $et$ $al$. [@SRS02]. In some occasions, increasing the number of copies of the original state can also help entanglement transformations. Take the above example. Instead of introducing a catalyst state $|\phi\rangle$, providing 3 copies of $|\psi_{1}\rangle$ is also sufficient to transform these copies together into the same number of $|\psi_{2}\rangle$. A question naturally arises here is: what is the relation between catalyst-assisted entanglement transformation and multiple-copy transformation? In [@DF05b], we found that multiple-copy entanglement transformation can be completely implemented by catalyst-assisted one. Furthermore, the mixing of these two has also the same power as pure catalyst-assisted transformation. In other words, any transformation which can be realized collectively on multiple copies and with the aid of a catalyst can be exactly implemented by only providing some appropriate catalyst. Later on, we proved that these two kinds of transformations are asymptotically equivalent in the sense that they can simulate each other’s ability to implement a desired transformation with the same optimal success probability, when the dimension of catalysts and the number of copies provided tend to infinity [@DF05a]. The contribution of the current paper is twofold. First, we show that in some cases catalyst-assisted entanglement transformation is strictly more powerful than multiple-copy one by deriving a sufficient condition when the former cannot be implemented by the latter. Second, for purely probabilistic setting we find, however, these two kinds of transformations are geometrically equivalent. That is, no matter catalyst-assisted transformations or multiple-copy transformations are used, the sets of quantum states that can be converted into a given state with maximal probabilities not less than a given value have the same closure. It is worth noting that the geometrical equivalence between these two kinds of transformations proved in the current paper is different from the asymptotical equivalence shown in [@DF05a]. We will elaborate the difference at the end of Section III after necessary notations have been introduced. For simplicity, in what follows we denote a bipartite pure state by the probability vector of its Schmidt coefficients. This will not cause any confusion because it is well known that the fundamental properties of a bipartite pure state under LOCC are completely determined by its Schmidt coefficients. Therefore, from now on, we consider only probability vectors (sometimes we even omit the normalization of a nonnegative vector to be a probability one) instead of quantum states and always identify a probability vector with the bipartite pure state represented by it. Deterministic case ================== In this section, we study the relation between catalyst-assisted transformation and multiple-copy transformation in deterministic case. First, we introduce some notations. Denote by $V^n$ the set of all $n$-dimensional nonnegative vectors and let $x,y,\cdots$ range over $V^n$. Let $$S(y)=\{x\in V^n\ |\ x\prec y\}$$ be the set of states that can be transformed into $y$ by LOCC directly, $$T(y)=\{x\in V^n\ |\ \exists \mbox{ probability vector } c,\ x\otimes c\prec y\otimes c\}$$ be the set of states that can be transformed into $y$ by LOCC with the aid of some catalyst, and $$M(y)=\{x\in V^n\ |\ \exists \mbox{ integer }k\ \geq 1,\ x^{\otimes{k}}\prec y^{{\otimes{k}}}\}$$ the set of states which, when some appropriate number of copies are provided, can be transformed into the same number of $y$ by LOCC. Suppose $x\in T(y)$ and $x'\in T(y')$. Then $\bar{x}\in T(\bar{y})$ where $\bar{x}= x\oplus x'$ and $\bar{y}= y\oplus y'$. [*Proof.*]{} By definition, $x\in T(y)$ and $x'\in T(y')$ imply that there exist $c$ and $c'$ such that $x\otimes c\prec y\otimes c$ and $x'\otimes c'\prec y'\otimes c'$. It can be easily checked that the vector $c\otimes c'$ serves as a catalyst for the transformation from $\bar{x}$ to $\bar{y}$, that is, $\bar{x}\otimes c \otimes c'\prec \bar{y}\otimes c\otimes c'$. Thus $\bar{x}\in T(\bar{y})$.$\Box$ The following lemma, important in its own right, is a powerful tool which gives us a sufficient condition on $x$ and $y$ such that they are incomparable in $any$ multiple-copy transformations. In other words, $any$ number of $x$ cannot be collectively transformed into the same number of $y$ using LOCC. Suppose $x$ and $y$ are two nonincreasingly arranged $n$-dimensional probability vectors, $x_1=y_1$ but $x\nprec y$. Let $$d=\min\{l : 1\leq l \leq n,\ \sum_{i=1}^l x_i > \sum_{i=1}^l y_i\}.$$ Denote by $t_1$ the number of components in $x$ which are equal to $x_1$, while $t_2$ the number of components in $y$ which are equal to $y_1$. If $t_1=t_2=t$ and $$\label{eq:ass0} x_1x_d\geq x_{t+1}^2 \mbox{\ \ \ \ and\ \ \ \ } y_1y_d\geq y_{t+1}^2,$$ then $x\not\in M(y)$. [*Proof.*]{} First, it is obvious that $1\leq t< d<n$. From the assumption Eq.(\[eq:ass0\]), we have for any integer $k\geq 1$, the components of $x^{\otimes k}$ and $y^{\otimes k}$ can be arranged nonincreasingly as follows $$\begin{array}{rcl} (x^{\otimes k})^\downarrow&=& x_1^k\oplus (x_1^{k-1}x_2)^{\oplus k}\oplus\cdots\oplus (x_{t}^{k-1}x_{t-1})^{\oplus k}\oplus x_t^k\\ &&\oplus (x_1^{k-1}x_{t+1})^{\oplus k}\oplus\cdots\oplus (x_{t}^{k-1}x_{t+1})^{\oplus k}\oplus\cdots\\ &&\oplus(x_{1}^{k-1}x_{d})^{\oplus k}\oplus\cdots\oplus(x_{t}^{k-1}x_{d})^{\oplus k}\oplus\cdots \end{array}$$ and $$\begin{array}{rcl} (y^{\otimes k})^\downarrow&=& y_1^k\oplus (y_1^{k-1}y_2)^{\oplus k}\oplus\cdots\oplus (y_{t}^{k-1}y_{t-1})^{\oplus k}\oplus y_t^k\\ &&\oplus (y_1^{k-1}y_{t+1})^{\oplus k}\oplus\cdots\oplus (y_{t}^{k-1}y_{t+1})^{\oplus k}\oplus\cdots\\ &&\oplus(y_{1}^{k-1}y_{d})^{\oplus k}\oplus\cdots\oplus(y_{t}^{k-1}y_{d})^{\oplus k}\oplus\cdots \end{array}$$ where by $\alpha^{\oplus k}$ we denote the vector $$\begin{array}{c} \underbrace{(\alpha,\ \alpha,\ \cdots,\ \alpha)}.\\ k\mbox{\ times} \end{array}$$ Here we only write out explicitly the largest $t^k+kt(d-t)$ components of $x^{\otimes k}$ and $y^{\otimes k}$ since it is enough for our argument. Notice that $x_1=\cdots =x_t>x_{t+1}$ and $y_1=\cdots =y_t>y_{t+1}$. It is now easy to check that $x^{\otimes k}\nprec y^{\otimes k}$ since when taking $l=t^k+kt(d-t)$, we have $$\begin{aligned} \sum\limits_{i=1}^l (x^{\otimes k})^{\downarrow}_i &=& (tx_1)^k + ktx_1^{k-1}\sum\limits_{i=t+1}^d x_i \\ &>& (ty_1)^k + kty_1^{k-1}\sum\limits_{i=t+1}^d y_i\\ &=& \sum\limits_{i=1}^l (y^{\otimes k})^{\downarrow}_i.\end{aligned}$$ So $x\not\in M(y)$ by the arbitrariness of $k$.$\Box$ If we take $x$ and $y$ as in Lemma 2 but $x_n=y_n$ instead of $x_1=y_1$. Let $$d=\max\{l : 1\leq l \leq n,\ \sum_{i=l}^n x_i < \sum_{i=l}^n y_i\}.$$ Denote by $t_1$ the number of components in $x$ which are equal to $x_n$ while $t_2$ the number of components in $y$ which are equal to $y_n$. If $t_1=t_2=t$ , $x_nx_d\leq x_{n-t}^2$, and $y_ny_d\leq y_{n-t}^2$, then we can also deduce that $x\not\in M(y)$. Using the lemmas above, we can now prove that $T(y)\not =M(y)$ for some probability vector $y$ by deriving a sufficient condition under which $T(y)\not\subseteq M(y)$, as the following theorem states. \[thm:mrdeterm\] Suppose $y$ is a nonincreasingly arranged $n$-dimensional probability vector. Denote by $t$ and $m$ the numbers of components which are equal to $y_1$ and which are equal to $y_n$, respectively. Let $d$ be the minimal index of the components which are less than $y_{t+1}$. That is, $$y_1=\cdots=y_{t}>y_{t+1}=\cdots=y_{d-1}>y_{d},$$ and $$y_{n-m}>y_{n-m+1}=\cdots=y_n.$$ If $d<n-m$ and $y_1y_{d}\geq y_{t+1}^2$, then $T(y)\not\subseteq M(y)$. [*Proof.*]{} Take a positive number $\epsilon$ such that $$\epsilon< \min\{\frac{d-t-1}{d-t}(y_{d-1}-y_{d}), \ \frac{m}{m+1}(y_{n-m}-y_{n-m+1})\}.$$ Define two $(n-t)$-dimensional nonnegative vectors $\bar{x}$ and $\bar{y}$ as follows $$\begin{array}{rcl} \displaystyle\bar{x}&=&(y_{t+1}-\displaystyle\frac{\epsilon}{\triangle}, \ \cdots,\ y_{d-1}-\frac{\epsilon}{\triangle},\ y_{d}+\epsilon,\ y_{d+1},\ \cdots, \\ \\ && y_{n-m-1},\ y_{n-m}-\epsilon,\ \displaystyle y_{n-m+1}+\frac{\epsilon}{m}, \ \cdots,\ y_n+\frac{\epsilon}{m}) \end{array}$$ and $$\bar{y}= (y_{t+1},\ y_{t+2},\ \cdots,\ y_n).$$ Here $\triangle=d-t-1$. It is easy to check that $\bar{x}$ and $\bar{y}$ are both nonincreasingly arranged, and $\bar{x}\prec \bar{y}$. Furthermore, $\bar{x}$ is in the interior of $T(\bar{y})$ by Lemma 1 in [@DK01] since $\bar{x}_1<\bar{y}_1$ and $\bar{x}_{n-t}>\bar{y}_{n-t}$. So there exists a sufficiently small but positive $\delta$ such that $\bar{x}'\in T(\bar{y})$ where $$\begin{array}{l} \displaystyle\bar{x}'^{\da}=(y_{t+1}-\displaystyle\frac{\epsilon}{\triangle}, \ \cdots,\ y_{d-1}-\frac{\epsilon}{\triangle},\ y_{d}+\epsilon+\delta,\ y_{d+1},\ \cdots, \\ \\ y_{n-m-1},\ y_{n-m}-\epsilon-\delta,\ \displaystyle y_{n-m+1}+\frac{\epsilon}{m}, \ \cdots,\ y_n+\frac{\epsilon}{m}). \end{array}$$ Now define $x$ as the direct sum of the vectors $(y_1,\dots,y_{t})$ and $\bar{x}'^{\da}$, that is $$x= (y_1,\cdots, y_{t}, \bar{x}'^{\da}).$$ By Lemma 1 we have $x\in T(y)$. On the other hand, $$\sum_{i=1}^{d} x_i = \sum_{i=1}^{d} y_i + \delta> \sum_{i=1}^{d} y_i,$$ $$\sum_{i=1}^{l} x_i \leq \sum_{i=1}^{l} y_i {\ \ \ \rm for \ \ \ \ }1\leq l<d,$$ and $$\begin{array}{rcl} x_1x_{d}&=&\displaystyle y_1(y_{d}+\epsilon+\delta)\\ \\ &>&y_1y_{d}\geq y_{t+1}^2\\ \\&>&\displaystyle (y_{t+1}-\frac{\epsilon}{\triangle})^2=x_{t+1}^2, \end{array}$$ so we have $x\not \in M(y)$ from Lemma 2. That completes our proof.$\Box$ Suppose $t$ and $m$ denote the numbers of the components which are equal to $y_n$ and which are equal to $y_1$, respectively. Let $d$ be the maximal index of the components which are greater than $y_{n-t}$. If $d>m+1$ and $y_ny_{d}\leq y_{n-t}^2$ then we can also deduce that $T(y)\not\subseteq M(y)$. An interesting special case of Theorem \[thm:mrdeterm\] is when $n>4$, if $y_1>y_2>y_3>y_{n-1}>y_n$ and $y_1y_3\geq y_2^2$ then $T(y)\not\subseteq M(y)$. Theorem \[thm:mrdeterm\] in fact gives us a method to construct a vector $y$ for which $T(y)\not\subseteq M(y)$. To be more specific, given a vector $\bar{y}$ such that $T(\bar{y})\neq S(\bar{y})$, we can derive a desired $y$ by the following two steps. First, add a sufficiently large component to $\bar{y}$ such that the conditions presented in Theorem \[thm:mrdeterm\] are satisfied for the new vector (notice that from Theorem 6 of [@DK01], when $T(\bar{y})\neq S(\bar{y})$, the condition $d<n-m$ in Theorem \[thm:mrdeterm\] holds automatically); second, normalize the vector to $y$ such that it is a probability vector. For example, given $\bar{y}=(0.5,0.25,0.25,0)$, we can derive $y=(3,0.5,0.25,0.25,0)/4$ and $T(y)\not\subseteq M(y)$. Furthermore, since the proof of Theorem \[thm:mrdeterm\] is constructive, the states which can be transformed into $y$ by catalyst-assisted transformation while cannot by multiple-copy transformation can also be constructed. We have proved that $T(y)\not = M(y)$ in some cases. Moreover, witness vectors which are in $T(y)$ but not in $M(y)$ are also constructed explicitly. It should be pointed out, however, that the witness vectors we constructed lie on the boundary of $T(y)$ without any exception, that is, they all satisfy the property that $x^\downarrow_1 = y^\downarrow_1$ or $x^\downarrow_n = y^\downarrow_n$. These witness vectors can be involved if we consider the closure of $M(y)$ instead. In fact, we will see in the following section that in probabilistic setting, the two sets $M^{\lambda}(y)$ and $T^{\lambda}(y)$ defined in Eqs.(\[eq:ty\]) and (\[eq:my\]) have exactly the same closure for $0\leq \lambda< 1$. So the question remained is to show whether or not ${M(y)}$ and ${T(y)}$ also have the same closure. Probabilistic case ================== We considered deterministic entanglement transformations in the previous section. In this section, let us turn to examine transformations with maximal probability strictly less than $1$. Given a nonnegative number $\lambda <1$, let $$\label{eq:sy}S^{\lambda}(y)=\{x\in V^n\ |\ P(x\ra y)\geq \lambda \}$$ be the set of states that can be transformed into $y$ by LOCC with the maximal probability not less than $\lambda$, $$\label{eq:ty}T^{\lambda}(y)=\{x\in V^n\ |\ \exists c, P(x\otimes c\ra y\otimes c)\geq \lambda \}$$ be the set of states that can be transformed into $y$ by catalyst-assisted LOCC with the maximal probability not less than $\lambda$, and $$\label{eq:my}M^{\lambda}(y)=\{x\in V^n\ |\ \exists k, P(x^{\otimes{k}}\ra y^{{\otimes{k}}})^{1/k} \geq \lambda\}$$ the set of states which, when some appropriate number of copies are provided, can be transformed into the same number of $y$ by multiple-copy LOCC with the maximal geometric average probability not less than $\lambda$. We have proved in [@DF05a] that $M^{\lambda}(y)\subseteq T^{\lambda}(y)$. In the following we further show that the reverse is not always true. For this purpose, two lemmas which corresponding to Lemma 1 and Lemma 2 in the previous section are useful. Suppose $x,y\in V^n$ and $z\in V^m$ are nonincreasingly arranged nonnegative vectors. If $x\in T^{\lambda}(y)$ then $x'\oplus \lambda z\in T^{\lambda}(y\oplus z)$, where $x'=(x_1',x_2,\dots,x_n)$ with $x_1'=x_1+(1-\lambda)\sum_{i=1}^m z_i$. [*Proof.*]{} From $x\in T^{\lambda}(y)$, there exists $c\in V^r$ such that $P(x\otimes c\ra y\otimes c)\geq \lambda$. For any arbitrarily $1\leq l\leq (n+m)r$, we have $$\label{eq:28} \sum_{i=l}^{(n+m)r} \left((x'\oplus \lambda z)\otimes c\right)_i^\da = \sum_{i=l_1}^{nr}(x'\otimes c)_i^\da + \lambda\sum_{i=l_2}^{mr}(z\otimes c)_i^\da$$ for some $1\leq l_1\leq nr$ and $1\leq l_2\leq mr$. On the other hand, by definition $$\sum_{i=l}^{(n+m)r} \left((y\oplus z)\otimes c\right)_i^\da \leq \sum_{i=l_1}^{nr}(y\otimes c)_i^\da + \sum_{i=l_2}^{mr}(z\otimes c)_i^\da.$$ Notice that $x_i'\geq x_i$ for any $i=1,\dots,n$, and that from $P(x\otimes c\ra y\otimes c)\geq \lambda$ we know $$\label{eq:30} \sum_{i=l_1}^{nr}(x\otimes c)_i^\da \geq \lambda\sum_{i=l_1}^{nr}(y\otimes c)_i^\da.$$ It follows from Eqs.(\[eq:28\])-(\[eq:30\]) that $$\sum_{i=l}^{(n+m)r} \left((x'\oplus \lambda z)\otimes c\right)_i^\da \geq \lambda\sum_{i=l}^{(n+m)r} \left((y\oplus z)\otimes c\right)_i^\da,$$ and $P((x'\oplus \lambda z)\otimes c\ra (y\oplus z)\otimes c)\geq \lambda$ from the arbitrariness of $l$. So we have $x'\oplus \lambda z\in T^{\lambda}(y\oplus z)$. $\Box$ Suppose $x$ and $y$ are two nonincreasingly arranged $n$-dimensional probability vectors, $x_n=\lambda y_n$ but $x\not\in S^{\lambda}(y)$ for $\lambda\in (0,1)$. Let $$d=\max\{l : 1\leq l \leq n,\ \sum_{i=l}^n x_i < \lambda \sum_{i=l}^n y_i\}.$$ Denote by $t_1$ the number of components in $x$ which are equal to $x_n$, while $t_2$ the number of components in $y$ which are equal to $y_n$. If $t_1=t_2=t$ and $$x_nx_d\leq x_{n-t}^2 \mbox{\ \ \ \ and\ \ \ \ } y_ny_d\leq y_{n-t}^2,$$ then $x\not\in M^{\lambda}(y)$. [*Proof.*]{} Similar to Lemma 2. But the last $t^k+kt(n-t-d)$ components of $(x^{\otimes k})^\da$ and $(y^{\otimes k})^\da$ are considered for any $k$ at this time. $\Box$ Let $\lambda\in (0,1)$. There exists $y\in V^n$ such that $T^{\lambda}(y)\nsubseteq M^{\lambda}(y)$. [*Proof.*]{} The proof is similar to but more complicated than that of Theorem \[thm:mrdeterm\]. Beside, due to the asymmetry of roles of the largest and the smallest components in determining the maximal transforming probability presented in Eq.(\[eq:Vidal\]), components at the tail but not at the head of $y$ should be examined. We outline the main steps of the proof here. Let $y\in V^n$ such that $$y_1=\cdots=y_{m}>y_{m+1},$$ and $$y_{d}>y_{d+1}=\cdots=y_{n-t}>y_{n-t+1}=\cdots=y_n.$$ where $d>m+1$ and $y_ny_{d}\leq y_{n-t}^2$. Take a positive number $\epsilon$ such that $$\epsilon< \min\{\frac{\lambda m}{m+1}(y_{m}-y_{m+1}),\frac{\lambda\triangle}{\triangle+1}(y_{d}-y_{d+1})\},$$ where $\triangle=n-t-d$. Define $$\begin{array}{l} \displaystyle\bar{x}=(\widetilde{y_1},\lambda y_{2}-\displaystyle\frac{\epsilon}{m}, \ \cdots,\ \lambda y_{m}-\frac{\epsilon}{m},\ \lambda y_{m+1}+\epsilon,\ \lambda y_{m+2}, \\ \\ \ \cdots,\lambda y_{d-1},\ \lambda y_{d}-\epsilon,\ \displaystyle \lambda y_{d+1}+\frac{\epsilon}{\triangle}, \ \cdots,\ \lambda y_{n-t}+\frac{\epsilon}{\triangle}) \end{array}$$ and $$\bar{y}= (y_{1},\ y_{2},\ \cdots,\ y_{n-t}).$$ Here $\widetilde{y_1}=y_1+(1-\lambda)\sum_{i=2}^{n-t} y_i-\epsilon/m$. Then $\bar{x}$ and $\bar{y}$ are both nonincreasingly arranged, and $\bar{x}\in S^{\lambda}(\bar{y})$. Furthermore, $\bar{x}$ is in the interior of $T^{\lambda}(\bar{y})$ by Theorem 9 in [@FD05a] since $\bar{x}_{n-t}>\lambda\bar{y}_{n-t}$. So there exists a sufficiently small but positive $\delta$ such that $\bar{x}'\in T^{\lambda}(\bar{y})$ where $$\begin{array}{rcl} \bar{x}'^\da&=&(\widetilde{y_1},\lambda y_{2}-\displaystyle\frac{\epsilon}{m}, \ \cdots,\ \lambda y_{m}-\frac{\epsilon}{m},\ \lambda y_{m+1}+\epsilon+\delta,\\ \\ &&\lambda y_{m+2}, \ \cdots,\lambda y_{d-1},\ \lambda y_{d}-\epsilon-\delta,\ \displaystyle \lambda y_{d+1}+\frac{\epsilon}{\triangle}, \\ \\ && \cdots,\ \lambda y_{n-t}+\displaystyle\frac{\epsilon}{\triangle}). \end{array}$$ Now define $x= (x', \lambda y_{n-t+1},\cdots, \lambda y_{n})$, where $x'=(x'_1,\bar{x}'^\da_2,\dots,\bar{x}'^\da_{n-t})$ and $x'_1=y_1+(1-\lambda)\sum_{i=2}^{n} y_i-\epsilon/m$. By Lemma 4 and 5 we can similarly prove that $x \in T^{\lambda}(y)$ but $x\not\in M^{\lambda}(y)$. $\Box$ To draw a clearer picture of the relation between catalyst-assisted transformation and multiple-copy transformation in purely probabilistic setting, we investigate the limit properties of $T^{\lambda}(y)$ and $M^{\lambda}(y)$ about $\lambda$. Since $T^{\lambda'}(y)\subseteq T^{\lambda}(y)$ for any $\lambda'>\lambda$, we can define $$T^{\lambda-}(y)=\bigcap_{\lambda'<\lambda} T^{\lambda'}(y),\ \ \ \ \ \ T^{\lambda+}(y)=\bigcup_{\lambda'>\lambda} T^{\lambda'}(y)$$ which denote respectively the left limit and right limit of the set-valued function $T^{\lambda}(y)$ at the point $\lambda$. Similar notions can be defined for $M^{\lambda+}(y)$ and $M^{\lambda-}(y)$. It is direct from the definition that $$\begin{aligned} T^{\lambda-}(y)&=&\{\ x\ |\ \sup_c P(x\otimes c\ra y\otimes c)\geq \lambda \},\\ M^{\lambda-}(y)&=&\{\ x\ |\ \sup_k P(x^{\otimes{k}}\ra y^{{\otimes{k}}})^{1/k}\geq \lambda \},\end{aligned}$$ and we have shown in [@DF05a] that $T^{\lambda-}(y)=M^{\lambda-}(y)$ for any $\lambda\in [0,1]$. The following theorem tells us that generally, the function $T^{\lambda}(y)$ is neither left continuous nor right continuous at any point $\lambda\in (0,1)$, although it is ‘almost’ right continuous in the sense that the right limit at $\lambda$ shares the same interior points with $T^{\lambda}(y)$. \[lem:tplus\] For any $y\in V^n$ and $0<\lambda<1$, 1\) $T^{\lambda+}(y)$ is open while $T^{\lambda-}(y)$ is closed, 2\) $T^{\lambda+}(y)\varsubsetneq T^{\lambda}(y)\subseteq T^{\lambda-}(y)$, and when $y^{\downarrow}_n>0$, $T^{\lambda}(y)= T^{\lambda-}(y)$ if and only if $y^{\downarrow}_2=y^{\downarrow}_n$, 3\) $T^{\lambda}(y)^{\circ}= T^{\lambda+}(y)$. [*Proof.*]{} 1). For any $x\in T^{\lambda+}(y)$, there exist $\mu>\lambda$ and $c$ such that $P(x\otimes c\ra y\otimes c)\geq \mu$. Let $\nu$ be a number such that $\mu>\nu>\lambda$. From the continuity of ${{P({x}\otimes c\ra y\otimes c)}}$ about $x$ for fixed $c$ and $y$, we have some $\epsilon>0$, such that for any $x'\in B(x,\epsilon)$, $$|{{P({x'}\otimes c\ra y\otimes c)}}-{{P({x}\otimes c\ra y\otimes c)}}|<\mu-\nu.$$ We then derive that $x'\in T^{\lambda+}(y)$ since $$\begin{array}{rl} {{P({x'}\otimes c\ra y\otimes c)}}&\geq {{P({x}\otimes c\ra y\otimes c)}}-A \\ \\ &> \mu-(\mu-\nu)=\nu \end{array}$$ where $A=|{{P({x'}\otimes c\ra y\otimes c)}}-{{P({x}\otimes c\ra y\otimes c)}}|$. That completes the proof that $T^{\lambda+}(y)$ is open. To prove the closeness of $T^{\lambda-}(y)$, we take any sequence $x_i\in T^{\lambda-}(y)$ such that $\lim_i x_i= x$. By definition, for any $x_i$ we have $\sup_c {{P({x_i}\otimes c\ra y\otimes c)}}\geq \lambda$. To realize the transformation from $x\otimes c$ to $y\otimes c$, a possible protocol is first transforming $x$ to $x_i$ and then transforming $x_i\otimes c$ to $y\otimes c$. So we have the following relation $${{P({x}\otimes c\ra y\otimes c)}}\geq P(x\ra x_i){{P({x_i}\otimes c\ra y\otimes c)}}.$$ Thus $$\begin{array}{l} \displaystyle\sup_c {{P({x}\otimes c\ra y\otimes c)}}\\ \\ \geq P(x\ra x_i)\displaystyle\sup_c{{P({x_i}\otimes c\ra y\otimes c)}}\\ \\\geq \lambda P(x\ra x_i). \end{array}$$ The desired result that $x\in T^{\lambda-}(y)$ follows from the above equation by letting $i$ tend to infinity. 2\) is obvious from 1) and the fact that $T^{\lambda}(y)$ is neither closed nor open when $y^{\downarrow}_2>y^{\downarrow}_n$ (see the first two lines of the proof of Theorem 11 of [@FD05a]). When $y^{\downarrow}_2=y^{\downarrow}_n$, we have $T^{\lambda}(y)=S^{\lambda}(y)$ (Theorem 10 of [@FD05a]), then $T^{\lambda}(y)= T^{\lambda-}(y)$ follows from the continuity of $S^{\lambda}(y)$ for any $0<\lambda<1$ (Theorem 3 of [@FD05a]). Now we prove 3). The relation $T^{\lambda+}(y)\subseteq T^{\lambda}(y)^{\circ}$ is easy from 1) and 2). To prove the reverse relation, we take any $x\in T^{\lambda}(y)^{\circ}$. Then ${{P({x}\otimes c\ra y\otimes c)}}\geq \lambda$ for some $c$. There are two cases to consider. Case 1. ${{P({x}\otimes c\ra y\otimes c)}}>\lambda$. In this case, we know immediately that $x\in T^{\lambda'}(y)\subseteq T^{\lambda+}(y)$ for $\lambda'={{P({x}\otimes c\ra y\otimes c)}}$. Case 2. ${{P({x}\otimes c\ra y\otimes c)}}=\lambda$. Since $x$ is an interior point of $T^{\lambda}(y)$, from Theorem 9 of [@FD05a] we have $x^{\downarrow}_n/y^{\downarrow}_n>\lambda$. Then ${{P({x}\otimes c\ra y\otimes c)}}<\min\{1,x^{\downarrow}_n/y^{\downarrow}_n\}$. By Theorem 2 of [@FD04], there exists a catalyst $c'$ such that $$P(x\otimes c\otimes c'\ra y\otimes c\otimes c')>{{P({x}\otimes c\ra y\otimes c)}}=\lambda.$$ So we also have $x\in T^{\lambda'}(y)\subseteq T^{\lambda+}(y)$ for $\lambda'=P(x\otimes c\otimes c'\ra y\otimes c\otimes c')$. $\Box$ Notice that we assume $y^\da_n>0$ in 2) of the above theorem. When $y^\da_n=0$, it is not clear till now whether or not the result still holds. With similar techniques, we can prove a corresponding result of Theorem \[lem:tplus\] for probabilistic multiple-copy transformation. \[lem:Mplus\] For any $y\in V^n$ and $0<\lambda<1$, 1\) $M^{\lambda+}(y)$ is open while $M^{\lambda-}(y)$ is closed, 2\) $M^{\lambda+}(y)\varsubsetneq M^{\lambda}(y)\subseteq M^{\lambda-}(y)$, and when $y^{\downarrow}_n>0$, $M^{\lambda}(y)= M^{\lambda-}(y)$ if and only if $y^{\downarrow}_2=y^{\downarrow}_n$, 3\) $M^{\lambda}(y)^{\circ}= M^{\lambda+}(y)$. [*Proof.*]{} Similar to the proof of Theorem \[lem:tplus\]. $\Box$ Now we can show our main result of this section. Rather surprisingly, when the probability threshold $\lambda$ is strictly less than 1, probabilistic catalyst-assisted transformation and probabilistic multiple-copy transformation are geometrically equivalent in the sense that the two sets $T^{\lambda}(y)$ and $M^{\lambda}(y)$ in fact share the same interior points (or equivalently, the same closure). \[lem:TleqM\] If $0\leq \lambda<\lambda'\leq 1$, then $T^{\lambda'}(y)\subseteq M^{\lambda}(y)$. [*Proof.*]{} By definition, for any $x\in T^{\lambda'}(y)$, there exists $c$ such that $P(x\otimes c\ra y\otimes c)\geq \lambda'$. Then from Theorem 1 of [@DF05a], we have $$\sup_k P(x^{\otimes{k}}\ra y^{{\otimes{k}}})^{1/k}=\sup_c P(x\otimes c\ra y\otimes c)\geq \lambda'>\lambda.$$ Thus there exists $k_0$ such that $P(x^{\otimes{k_0}}\ra y^{{\otimes{k_0}}})^{1/k_0}>\lambda$. So $x\in M^{\lambda}(y)$. $\Box$ For any $y\in V^n$ and $0\leq \lambda <1$, we have $\overline{T^{\lambda}(y)}=\overline{M^{\lambda}(y)}$. [*Proof.*]{} From Theorems \[lem:tplus\] and \[lem:Mplus\], to prove this theorem we need only show that $T^{\lambda+}(y)= M^{\lambda+}(y)$, or $T^{\lambda+}(y)\subseteq M^{\lambda+}(y)$ since the reverse is obvious. For any $\lambda'>\lambda$, take $\mu$ such that $\lambda'>\mu>\lambda$. Then from Lemma \[lem:TleqM\] we have $T^{\lambda'}(y)\subseteq M^{\mu}(y)\subseteq M^{\lambda+}(y)$. So $T^{\lambda+}(y)\subseteq M^{\lambda+}(y)$ by definition. $\Box$ We are now in the appropriate position to elaborate the difference between the geometrical equivalence shown in this paper and the asymptotical equivalence proven in [@DF05a]. The latter can be expressed with our notations as $T^{\lambda-}(y)= M^{\lambda-}(y)$ while our result here indicates that $\overline{T^{\lambda}(y)}=\overline{M^{\lambda}(y)}$. Since the question whether or not $T^{\lambda-}(y)=\overline{T^{\lambda}(y)}$ (or equivalently, $M^{\lambda-}(y)=\overline{M^{\lambda}(y)}$) remains open, our result cannot be derived directly from the one in [@DF05a]. Conclusion ========== To summarize, we show that in some cases catalyst-assisted entanglement transformation is strictly more powerful than multiple-copy one in either deterministic or probabilistic setting. For purely probabilistic setting, however, we can prove that these two kinds of transformations are geometrically equivalent in the sense that the two sets $T^{\lambda}(y)$ and $M^{\lambda}(y)$, denoting the sets of bipartite pure states which can be converted into a given state with Schmidt coefficient vector $y$ with maximal probabilities not less than $\lambda$ by catalyst-assisted transformation and by multiple-copy transformation, respectively, have the same closure. The limit properties of $T^{\lambda}(y)$ and $M^{\lambda}(y)$ as set-valued functions about $\lambda$ are also discussed. The results about the relation between catalyst-assisted transformation and multiple-copy transformation shown in this paper and our previous works can be described by the following diagrams: $$\begin{array}{ccccccccc} M^{\lambda+}(y)&=&M^{\lambda}(y)^\circ&\varsubsetneq&M^{\lambda}(y)&\varsubsetneq&\overline{M^{\lambda}(y)}&{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\subset}}}& M^{\lambda-}(y)\\ \rotatebox{90}{=}&&\rotatebox{90}{=}&&\rotatebox{90}{$\varsupsetneq$}& &\rotatebox{90}{=}& & \rotatebox{90}{=}\\ T^{\lambda+}(y)&=&T^{\lambda}(y)^\circ&\varsubsetneq&T^{\lambda}(y)&\varsubsetneq&\overline{T^{\lambda}(y)}&{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\subset}}}& T^{\lambda-}(y) \end{array}$$ for purely probabilistic case ($\lambda<1$) and $$\begin{array}{ccccccccc} M(y)^\circ&\varsubsetneq&M(y)&\varsubsetneq&\overline{M(y)}& {\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\subset}}}& M^{-}(y)\\ \rotatebox{90}{{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\supset}}}}&&\rotatebox{90}{$\varsupsetneq$}& &\rotatebox{90}{{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\supset}}}}& & \rotatebox{90}{=}\\ T(y)^\circ&\varsubsetneq&T(y)&\varsubsetneq&\overline{T(y)}&{\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\subset}}}& T^{-}(y) \end{array}$$ for deterministic case. Where we write $A(y)\varsubsetneq B(y)$ if $A(y)\subseteq B(y)$ holds for all $y$ but there exists some $y$ such that $A(y)\neq B(y)$; while by $A(y){\ensuremath {\underset{\mbox{\sout{\tiny{\,?\,}}}}{\subset}}}B(y)$ we indicate that whether or not there exists $y$ such that $A(y)\neq B(y)$ is still open, although $A(y)\subseteq B(y)$ always holds for all $y$. From the above two diagrams, the remaining questions for further study are: 1). Whether or not $\overline{T^{\lambda}(y)}= T^{\lambda-}(y)$ (or equivalently, $\overline{M^{\lambda}(y)}= M^{\lambda-}(y)$ ) for any $y$ and $\lambda\leq 1$. In other words, whether or not the function $T^{\lambda}(y)$ (or $M^{\lambda}(y)$) is ‘almost’ left continuous at any $\lambda\leq 1$. 2). Whether or not $\overline{T(y)}= \overline{M(y)}$ (or equivalently, $T(y)^\circ= M(y)^\circ$) for any $y$. That is, whether or not catalyst-assisted transformation and multiple-copy transformation are also geometrically equivalent in deterministic setting. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== The authors thank the colleagues in the Quantum Computation and Quantum Information Research Group for useful discussion. This work was partly supported by the Natural Science Foundation of China (Grant Nos. 60503001, 60321002, and 60305005), and by Tsinghua Basic Research Foundation (Grant No. 052220204). R. Duan acknowledges the financial support of Tsinghua University (Grant No. 052420003).

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--- author: - | [^1]\ Laboratoire de l’Accélérateur Linéaire, Univ. Paris-Sud 11 et IN2P3/CNRS, France\ E-mail: title: Top and EW Physics at the LHeC --- Introduction ============ Deep inelastic scattering (DIS) of a point-like lepton beam over a hadron has played a central role in establishing the quark-parton model and QCD starting with fixed target experiments in the late 1960s at SLAC. Later the Gargamelle neutrino-nucleon experiment at CERN has discovered weak neutral currents. HERA, operated at DESY from 1992 to 2007, was the only $ep$ collider of the world. It has extended the study of the proton structure and quark-gluon interaction dynamics up to a centre-of-mass energy ($\sqrt{s}$) of 320GeV corresponding to an extension by two orders of magnitude towards both higher negative four-momentum transfer squared $Q^2$ and lower Bjorken $x$ in comparison with the kinematic region covered by the fixed target experiments. The LHeC, if realised by adding to the LHC a separate 9km racetrack-shaped recirculating superconducting energy recovery linac providing a polarised electron (possibly also positron) beam of 60GeV, will be a new $ep$ collider of 1.3TeV, running in parallel with the high luminosity phase of the LHC. It has a rich and complementary physics programme to the LHC [@cdr; @1211.5102]. It would enable new precision studies of QCD in general and the precision determination of parton distributions functions (PDFs) in a largely extended kinematic region in particular. It has the potential to reveal new QCD dynamics in an unexplored low $x$ regime where the DGLAP evolution equations may no longer be valid as the latest QCD analysis of the newly combined inclusive neutral and charged current (NC and CC) cross sections at HERA may indicate [@herapdf2]. It would also provide additional and sometimes unique ways for studying top and electroweak (EW) physics as well as Higgs and physics beyond the Standard Model (BSM). This talk focuses on some of the selected topics on top and EW physics at the LHeC and the writeup is organised as follows. In Sec. \[sec:top\], expected limits on anomalous $Wtb$ couplings from the single top production are presented as an exemple. In Sec. \[sec:ew\], the expected precision determination of light quark couplings to the $Z$ boson and the scale dependence of the weak mixing angle $\sin^2\!\theta_W$ based either on the inclusive NC cross section measurements or on polarisation asymmetries of the NC interactions are shown, followed by a summary in Sec. \[sec:summary\]. Top physics {#sec:top} =========== The top quark is the heaviest particle in the SM, which is believed to be most sensitive to BSM physics. It has not been studied so far by any DIS experiments because of the kinematic limit or too small cross section. Therefore the LHeC will be the first DIS experiment capable to study the directly produced single top quark and top pairs in CC and NC interactions, respectively. In the five flavour scheme, the single top-quark production cross section of the $2\to 2$ $t$-channel process $e^-p\to \bar{t}\nu_e+X$ with $\bar{t}\to W^-b$ at $\sqrt{s}=1.3$TeV is predicted to be around 2pb for un polarised electron beam and increases by a factor of $1+P_e$ with $P_e$ being the degree of the longitudinal polarisation of the beam [@dgkm13]. This cross section value is comparable with that of the Tevatron and smaller by about two orders of magnitude than the LHC at 14TeV [@nk15]. The LHeC has however a much cleaner environment due to the absence of pile-up and underlying events. Therefore this process can be used for many precision measurements within the SM, such as the bottom-quark distribution of the proton, the CKM matrix element $V_{tb}$, the $t$-quark polarisation and the $W$ boson helicity. It can also be used to study deviations from the SM such as the anomalous couplings $Wtb$. In addition, the single top production in the NC protoproduction can be used to study top quark flavour changing neutral current couplings $tq\gamma$ with $q$ being a light quark [@cdr]. The top pair events are also produced at the LHeC in NC interactions. Even though the rate is lower than at the LHC, the potential for a better measurement of $tt\gamma$ than LHC is good [@bl13] as in the $t\bar{t}$ photoproduction at the LHeC, the highly energetic incoming photon couples only to the $t$ quark so that the cross section depends directly on the $tt\gamma$ vertex, whereas at the LHC the vertex is probed through $t\bar{t}\gamma$ production, where the outing going photon could come from other charged sources such as the top decays products. The DIS regime of $t\bar{t}$ production will also be able to probe the $ttZ$ coupling though with less sensitivity. A detailed study was performed in [@dgkm13] to evaluate the expected accuracy of measuring the anomalous $Wtb$ couplings at the LHeC based on the single anti-top quark production in $e^-p$ collisions in a model independent way by means of the following effective CP conserving Lagrangian [@dgkm13] $${\cal L}_{Wtb}=\frac{g}{\sqrt{2}}\left[W_\mu\bar{t}\gamma^\mu\left(V_{tb}f^L_1P_L+f^R_1P_R\right)b-\frac{1}{2m_W}W_{\mu\nu}\bar{t}\sigma^{\mu\nu}\left(f^L_2P_L+f^R_2P_R\right)b\right] + h.c.$$ where $f^L_1(\equiv 1+\Delta f^L_1)$ and $f^R_1$ are left- and right-handed vector couplings, $f^{L,R}_2$ are left- and right-handed tansor couplings, $W_{\mu\nu}=\partial_\mu W_\nu-\partial_\nu W_\mu$, $P_{L,R}=\frac{1}{2}(1\mp\gamma_5)$ are left- and right-handed projection operators, $\sigma^{\mu\nu}=i/2(\gamma^\mu\gamma^\nu-\gamma^\nu\gamma^\mu)$ and $g=2/\sin\theta_W$. In the SM, $f^L_1\equiv 1$ and $\Delta f^L_1=f^R_1=f^{L,R}_2\equiv 0$. Several analyses were performed using a simulated event sample corresponding to an integrated luminosity of 100fb$^{-1}$ for three different systematic uncertainties of 1%, 5% and 10%. One of them was based on a $\chi^2$ analysis using differential distributions of a few relevant kinematic variables in the leptonic and hadronic decay modes, respectively. Contours at 68% and 95% confidence level (CL) on two dimensional plane for any coupling combination were presented. One example is shown in Fig. \[fig:wtb\]. The corresponding results in comparison with other results from Tevatron, LHC and indirect one from $B$ decays are shown in Table \[tab:wtb\]. The conservative LHeC limits are thus competitive with or better than similar results from other determinations. ![Contours at (left) 68% and (right) 95% CL on the plane of $|V_{tb}|\Delta^L_1$ and $f^R_2$ for a systematic error of 1%, 5% and 10% on a sample with an integrated luminosity of 100fb$^{-1}$ (figures taken from Ref. [@dgkm13]).[]{data-label="fig:wtb"}](f1l_f2r68_had "fig:"){width=".495\textwidth"} ![Contours at (left) 68% and (right) 95% CL on the plane of $|V_{tb}|\Delta^L_1$ and $f^R_2$ for a systematic error of 1%, 5% and 10% on a sample with an integrated luminosity of 100fb$^{-1}$ (figures taken from Ref. [@dgkm13]).[]{data-label="fig:wtb"}](f1l_f2r95_had "fig:"){width=".495\textwidth"} Upper limit [(95% CL)]{} $|\Delta f^L_1|$ $|f^R_1|$ $|f^L_2|$ $|f^R_2|$ -------------------------- ------------------ --------------------- --------------------- ----------------- LHeC [@dgkm13] $0.005-0.03$ $0.01-0.1$ $0.01-0.1$ $0.01-0.1$ D0 [@d0:wtb] 0.548 0.324 0.347 LHC [@lhc:wtb] $0.03-0.06$ $0.22-0.34$ $0.06-008$ $0.06-0.08$ $B$ decays [@b:wtb] $[-0.13, 0.03]$ $[-0.0007, 0.0025]$ $[-0.0013, 0.0004]$ $[-0.15, 0.57]$ : Comparison of expected upper limits at 95% CL at the LHeC (100fb$^{-1}$, hadronic modes, $\delta_{\rm sy}=0.01-0.1$) [@dgkm13] with the actual limits from D0 (5.4fb$^{-1}$, $W$-helicity, single top) [@d0:wtb] and expected limits at the LHC (100fb$^{-1}$, $\gamma p\to WtX$) [@lhc:wtb] and $B$ decays (indirect) [@b:wtb].[]{data-label="tab:wtb"} EW physics {#sec:ew} ========== Inclusive NC and CC DIS interactions are two main processes which can be measured at the LHeC with high precision providing primary source not only for precision QCD studies but also for EW physics. Three examples are briefly presented in this section. The first example concerns a precision measurement of vector and axial-vector weak NC couplings of the $Z$ boson to light quarks $v_q$ and $a_q$. They were determined together with PDFs in a combined EW and QCD analysis of simulated inclusive NC and CC cross section data samples following Ref. [@h1ew]. This is possible since the double differential NC cross section $\frac{{\rm d}^2\sigma_{\rm NC}}{{\rm d}x{\rm d}Q^2}$ in e.g. $e^-p$ collisions may be expressed in terms of three structure functions as $\frac{2\pi\alpha^2}{xQ^2}\left[Y_+\tilde{F}_2+Y_-\tilde{F}_3-y^2\tilde{F}_L\right]$, where $\alpha$ is the electromagnetic fine structure constant and $Y_\pm=1\pm (1-y)^2$ with $y=Q^2/(xs)$ being the electron inelasticity. The generalised structure $\tilde{F}_2$ can be decomposed as $F_2+P_ea_e\kappa_ZF_2^{\gamma Z}+a^2_e\kappa^2_ZF^Z_2$ corresponding to $\gamma$ exchange, $\gamma Z$ interference and $Z$ exchange contributions. In this expression, $\kappa_Z^{-1}=\frac{2\sqrt{2}\pi\alpha}{G_FM^2_Z}\frac{Q^2+M^2_Z}{Q^2}$ and $a_e$ is the axial-vector coupling of the electron (due to the smallness of the vector coupling $v_e$, terms proportional to $v_e$ have been omitted). Similarly $x\tilde{F}_3=-a_e\kappa_ZxF_3^{\gamma Z}-P_ea^2_e\kappa^2_ZxF_3^{\gamma Z}$. These different structure functions can be further expressed in terms of PDFs $q, \bar{q}$ and the light quark couplings $v_q$ and $a_q$ as $\left[ F_2, F_2^{\gamma Z}, F_2^Z\right]=x\sum_q\left[e^2_q, 2e_qv_q, v^2_q+a^2_q\right]\{q+\bar{q}\}$ and $\left[xF_3^{\gamma Z}, xF_3^Z\right]=2x\sum_q\left[e_qa_q, v_aa_q\right]\{q-\bar{q}\}$, where $e_q$ being the electronic charge of quark $q$. The longitudinal structure function $\tilde{F}_L$ does not contribute at LO. The CC cross section is independent of these couplings but its inclusion in the fit helps to constrain the PDFs. In Ref. [@cdr], different scenarios were considered. The results of one of these, corresponding to $e^\pm$ beams of 50GeV with a longitudinal polarisation of 40% colliding with a proton beam of 7TeV for an integrated luminosity of 1fb$^{-1}$ per beam, are shown in Fig. \[fig:couplings\] in comparison with similar determinations from other experiments. The expected precision at the LHeC is indeed much better and any significant deviation from the SM expectations can thus be observed with this analysis. ![Determination of the vector and axial-vector weak neutral current couplings of the light quarks by LEP [@lep_couplings], D0 [@d0_couplings], H1 [@h1_couplings] and ZEUS [@zeus_couplings], compared with the simulated prospects for the LHeC [@cdr].[]{data-label="fig:couplings"}](couplings_u "fig:"){width=".495\textwidth"} ![Determination of the vector and axial-vector weak neutral current couplings of the light quarks by LEP [@lep_couplings], D0 [@d0_couplings], H1 [@h1_couplings] and ZEUS [@zeus_couplings], compared with the simulated prospects for the LHeC [@cdr].[]{data-label="fig:couplings"}](couplings_d "fig:"){width=".495\textwidth"} The second example is on the scale dependence of the weak mixing angle $\sin^2\!\theta_W$ obtained from a projected measurement of the polarisation asymmetry $A^-=\frac{\sigma^-_{\rm NC}(P_R)-\sigma^-_{\rm NC}(P_L)}{\sigma^-_{\rm NC}(P_R)+\sigma^-_{\rm NC}(P_L)}\simeq \frac{\kappa_Za_e(P_L-P_R)}{2}\frac{F_2^{\gamma Z}}{F_2}$ assuming a left-handed ($P_L$) or right-handed ($P_R$) polarisation of 80% and an integrated luminosity of 10fb$^{-1}$ per polarisation state [@cdr]. The results are compared in Fig. \[fig:wma\_cctot\] (left) with other determinations at different energies. The LHeC measurements are precise and cover a large energy range. It should be mentioned that the NC and CC cross section ratio is also sensitive to $\sin^2\!\theta_W$, provided that the PDFs related uncertainty is under control [@cdr]. ![Left: dependence of the weak mixing angle on the energy scale $\mu$ from [@cdr]. Right: energy dependence of the $\nu N$ cross section. The open points up to 50TeV correspond to the expected precision of the HERA measurements and the solid point corresponds to an expected measurement at the LHeC. The full line represents the predicted cross section including the $W$ propagator while the dashed line is a linear extrapolation from low energy measurements.[]{data-label="fig:wma_cctot"}](sintmu "fig:"){width=".575\textwidth"} ![Left: dependence of the weak mixing angle on the energy scale $\mu$ from [@cdr]. Right: energy dependence of the $\nu N$ cross section. The open points up to 50TeV correspond to the expected precision of the HERA measurements and the solid point corresponds to an expected measurement at the LHeC. The full line represents the predicted cross section including the $W$ propagator while the dashed line is a linear extrapolation from low energy measurements.[]{data-label="fig:wma_cctot"}](cctot_lhec "fig:"){width=".43\textwidth"} The third example is on a measurement of the CC total cross section (Fig. \[fig:wma\_cctot\] (right)). The LHeC measurement together with those from HERA illustrates in a spectacular way the impact of the propagator mass of the $W$ boson on the CC cross section. The dependence on the polarisation of the CC total cross section can further be used to set a stringent lower mass limit on a right-handed $W$ boson following Ref. [@h1cc]. Summary {#sec:summary} ======= A few selected examples of the LHeC measurements from top and EW sectors clearly demonstrate that the realisation of the LHeC can greatly enhance the physics programme and discovery potential of the LHC in a complementary manner. Indeed the LHeC is also considered as the next machine for studying the Higgs boson and a luminosity upgrade by a factor of 10 reaching $10^{34}\,{\rm cm}^{-2}s^{-1}$ is under study [@max], compared to the Conceptional Design Report [@cdr]. More studies are desirable from both theoretical and experimental communities, which will certainly reveal even larger potential of the LHeC. [99]{} J. L. Abelleira Fernandez [*et al.*]{} \[LHeC Study Group\], “A Large Hadron Electron Collider at CERN", J. Phys. G. [**39**]{} (2012) 075001, arXiv:1206.2913 \[physics.acc-ph\]. J. L. Abelleira Fernandez [*et al.*]{} \[LHeC Study Group Collaboration\], “On the Relation of the LHeC and the LHC", arXiv:1211.5102 \[hep-ex\]. F. D. Aaron [*et al.*]{} \[H1 and ZEUS Collaborations\], JHEP [**1001**]{} (2010) 109, arXiv:0911.0884 \[hep-ex\]. S. Dutta, A. Goyal, M. Kumar and B. Mellado, arXiv:1307.1688 \[hep-ph\]. N. Kidonakis, arXiv:1509.02528 \[hep-ph\]. A. O. Bouzas and F. Larios, Phys. Rev. D [**88**]{} (2013) 9, 094007 \[arXiv:1308.5634 \[hep-ph\]\]. V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Lett. B [**713**]{} (2012) 165 \[arXiv:1204.2332 \[hep-ex\]\]. B. Sahin and A. A. Billur, Phys. Rev. D [**86**]{} (2012) 074026 \[arXiv:1210.3235 \[hep-ph\]\]. B. Grzadkowski and M. Misiak, Phys. Rev. D [**78**]{} (2008) 077501 \[Phys. Rev. D [**84**]{} (2011) 059903\] \[arXiv:0802.1413 \[hep-ph\]\]. A. Aktas [*et al.*]{} \[H1 Collaboration\], Phys. Lett. B [**632**]{} (2006) 35 \[hep-ex/0507080\]. S. Schael [*et al.*]{} \[ALEPH and DELPHI and L3 and OPAL and SLD and LEP Electroweak Working Group and SLD Electroweak Group and SLD Heavy Flavour Group Collaborations\], Phys. Rept.  [**427**]{} (2006) 257 \[hep-ex/0509008\]. V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Rev. D [**84**]{} (2011) 012007 \[arXiv:1104.4590 \[hep-ex\]\]. Z. Zhang (for the H1 Collaboration), Combined electroweak and QCD fits including NC and CC data with polarised electron beam at HERA-2, PoS DIS2010 (2010) 056. ZEUS Collaboration, ZEUS-prel-07-027. A. Aktas [*et al.*]{} \[H1 Collaboration\], Phys. Lett. B [**634**]{} (2006) 173 \[hep-ex/0512060\]. O. Bruening and M. Klein, Mod. Phys. Lett. A [**28**]{} (2013) 16, 1330011 \[arXiv:1305.2090 \[physics.acc-ph\]\]. [^1]: for the LHeC Study Group

ArXiv

‘=11 makefntext\#1[ to 3.2pt [-.9pt $^{{\ninerm\@thefnmark}}$]{}\#1]{} makefnmark[to 0pt[$^{\@thefnmark}$]{}]{} PS. @myheadings[mkbothgobbletwo oddhead[ ]{} oddfootevenheadevenfoot \#\#1\#\#1]{} \[appendixc\] \[subappendixc\] \#1 =1.5pc citex\[\#1\]\#2[@fileswauxout citeacite[forciteb:=\#2]{}[\#1]{}]{} @cghi cite\#1\#2[[$\null^{#1}$@tempswa ]{}]{} =cmbx10 scaled1 =cmr10 scaled1 =cmti10 scaled1 =cmbxti10 scaled=cmbx10 scaled=cmr10 scaled=cmti10 scaled=cmbxti10 =cmbx10 =cmr10 =cmti10 =cmbx9 =cmr9 =cmti9 =cmbx8 =cmr8 =cmti8 6.0in 8.6in -0.25truein 0.30truein 0.30truein =1.5pc YUMS 97-018, DO-TH 97-15, SNUTP 97-089\ hep-ph/9706451 (modified 27 June 1997)\ **FLAVOR DEMOCRACY AND QUARK MASS MATRICES [^1]** C. S.  Kim *Department of Physics, Yonsei University, Seoul 120-749, Korea* E-mail: [email protected] and G.  Cvetič *Department of Physics, University of Dortmund, Dortmund, Germany* E-mail: [email protected] Flavor Democracy at Low Energy ============================== In the standard electroweak theory, the hierarchical pattern of the quark masses and their mixing remains an outstanding issue. While a gauge interaction is characterized by its universal coupling constant, the Yukawa interactions have as many coupling constants as there are fields coupled to the Higgs boson. There is no apparent underlying principle which governs the hierarchy of the various Yukawa couplings, and as a result, the Standard Model of strong and electroweak interactions can predict neither the quark (or lepton) masses nor their mixing. This situation can be improved by assuming a universal Yukawa interaction – the resulting spectrum consists then of one massive and two massless quarks in each (up and down) sector in the three generation Standard Model. Flavor–democratic (FD) quark mass matrices, and a perturbed form of such FD matrices, were introduced already in 1978 by Harari, Haut and Weyers[@0)] in a left-right symmetric framework. Flavor democracy has recently been suggested by Koide, Fritzsch and Plankl[@1)], as well as Nambu[@[3]] and many other authors[@[3]] as an analogy with the BCS theory of superconductivity. In this Section we will discuss how this flavor symmetry can be broken by a slight perturbation at low energies, in order to reproduce the quark masses and the CKM matrix[@3)]. As a result, predictions for the top quark mass and for the CP violation parameter $J_{CP}$ are obtained. This Section is based on a work by Cuypers and Kim[@11)]. Considering only quark fields, the gauge invariant Yukawa Lagrangian is $${\cal L}_{\rm Y} = - \sum_{i,j} (\bar Q'_{iL}~\Gamma^D_{ij}~d'_{jR}~\phi~+~ \bar Q'_{iL}~\Gamma^U_{ij}~u'_{jR}~\tilde \phi~+~\mbox{h.c.}) \ . \label{eq1}$$ Here, the primed quark fields are in a flavor \[$SU(2)$\] basis of the $SU(2) \times U(1)$ electroweak gauge group – the left-handed quarks form doublets under the $SU(2)$ transformation, $\bar Q'_L=(\bar u'_L,~\bar d'_L)$, and the right-handed quarks are singlets. The indices $i$ and $j$ run over the number of fermion generations. The Yukawa coupling matrices $\Gamma^{U,D}$ are arbitrary and not necessarily diagonal. After spontaneous symmetry breaking, the Higgs field $\phi$ acquires a nonvanishing vacuum expectation value (VEV) $v$ which yields quark mass terms in the original Lagrangian $${\cal L}_{\rm mass} = - \sum_{i,j} (\bar d'_{iL}~M^D_{ij}~ d'_{jR}~+~\bar u'_{iL}~M^U_{ij}~u'_{jR}~+~\mbox{h.c.}) \ , \label{eq2}$$ and the quark mass matrices are defined as $$M^{U,D}_{ij} \equiv {v \over \sqrt{2}}~\Gamma^{U,D}_{ij} \ . \label{eq3}$$ Mass matrices $M^{U,D}$ are diagonalized by biunitary transformations involving unitary matrices $U^{U,D}_L$ and $U^{U,D}_R$, and the flavor eigenstates are tranformed to physical mass eigenstates by the same unitary transformations, $$U^{U,D}_L~M^{U,D}~(U^{U,D}_R)^{\dagger} = M^{U,D}_{\rm diag}~~{\rm and}~~ U^U_{L,R}~u'_{L,R} = u_{L,R},~~U^D_{L,R}~d'_{L,R} = d_{L,R}~~. \label{eq4}$$ Using the recent CDF data[@4)] of the physical top mass $m_t^{\rm phys.} \approx 175$ GeV, the diagonalized mass matrices $M^{U,D}_{\rm diag}$ at a mass scale of 1 GeV are $$M_{\rm diag}^U \approx m_t \left[ \begin{array}{ccc} 2.5\times10^{-5} & & \\ & 0.006 & \\ & & 1 \end{array} \right] \quad {\rm and} \quad M_{\rm diag}^D \approx m_b \left[ \begin{array}{ccc} 1.7\times10^{-3} & & \\ & 0.03 & \\ & & 1 \end{array} \right]. \label{eq5}$$ The first two eigenvalues in both matrices are almost zero (almost degenerate) when compared to the eigenvalue of the third generation. In order to account for this large mass gap, one can use mass matrices which have in a flavor basis the flavor–democratic (FD) form $$M^U_0 = \frac{m_t}{3} \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right] ~~{\rm and}~~ M^D_0 = \frac{m_b}{3} \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right] ~~. \label{eq6}$$ Diagonalization leads to a pattern similar to the experimental spectrum (5) $$M_{\rm diag}^U = m_t \left[ \begin{array}{ccc} 0 & & \\ & 0 & \\ & & 1 \end{array} \right] \qquad {\rm and} \qquad M_{\rm diag}^D = m_b \left[ \begin{array}{ccc} 0 & & \\ & 0 & \\ & & 1 \end{array} \right] \ . \label{eq7}$$ Arbitrariness in the choice of the Yukawa Lagrangian has been substantially reduced with this symmetric choice. Each (up or down) quark sector is determined in this pure FD approximation by a single universal Yukawa coupling. To induce nonzero masses for the lighter quarks and to reproduce the experimental CKM matrix, small perturbations have to be added to the universal Yukawa interactions. One possibility is to analyze effects of the following two kinds of independent perturbation matrices $$P_1 = \left[ \begin{array}{ccc} \alpha & 0 & 0 \\ 0 & \beta & 0 \\ 0 & 0 & 0 \end{array} \right] \qquad {\rm and} \qquad P_2 = \left[ \begin{array}{ccc} 0 & a & 0 \\ a & 0 & b \\ 0 & b & 0 \end{array} \right] \ , \label{eq8}$$ $\alpha,~\beta,~a$ and $b$ being real parameters to be determined from the quark masses. For simplicity, these perturbations can be applied separately. Quark mass matrices (in a flavor basis) are then sums of the dominant universal FD matrices (6) plus one kind of the perturbation matrices (8). One then has to solve the eigenvalue problem $$\det~|M^{U,D}~-~\lambda|=0,~~{\rm where}~~M^{U,D}=M^{U,D}_0~+~P_i~~ {\rm and}~~\lambda = m_1,~-m_2,~m_3~~ \ , \label{eq9}$$ and $m_1=m_d~{\rm or}~m_u,~m_2=m_s~{\rm or}~m_c$ and $m_3=m_b$ or $m_t$. The six parameters of the perturbed matrices $M^{U,D}$ (e.g., $m_t,~{\alpha}^{(u)},~{\beta}^{(u)};~m_b,~a^{(d)},~b^{(d)}$) are uniquely determined from the experimental input of the five light (current) quark masses and the choice of a particular mass for the top quark. CKM matrix is then constructed as $$V = U^U_L~ \left[ \begin{array}{ccc} 1 & & \\ & e^{i\sigma} & \\ & & e^{i\tau} \end{array} \right]~ U^{D\dagger}_L \ , \label{eq10}$$ where phase angles $\sigma$ and $\tau$ are introduced phenomenologically to generate possible CP violation in the framework of the three generation standard CKM model. The CKM matrix is then uniquely determined by the arbitrary input of the two angles $\sigma$ and $\tau$ in (10). To determine these eight perturbation parameters, a $\chi^2$ analysis was used. For the first five quarks, the masses obtained by Gasser and Leutwyler[@5)] can be used. No constraints on the top quark mass were imposed. Additional constraints were used – for four degrees of freedom of the CKM matrix coming from two sources. Information on the quark mixing angles comes from the measurements of the three absolute values[@6)]: $$|V_{us}| = \sin \theta_{C} = 0.221 \pm 0.002,~~ |V_{cb}| = 0.040 \pm 0.004,~~ \left| V_{ub}/V_{cb} \right| = 0.08 \pm 0.02~~. \label{eq11}$$ Information on the CP violating phase was taken from the experimental value of $\varepsilon$ parameter of K decay $$\varepsilon = (2.26 \pm 0.02)~10^{-3} = B_K\cdot f(m_c,m_t,V) \ , \label{eq12}$$ where $f$ is a complicated function of the charmed and top quark masses and of CKM matrix elements, and $B_K$ is the parameter connecting a free quark estimate to the actual value of $\Delta S =2$ matrix element describing $K - \bar K$ mixing. Following Ref. [@8)], we used the value of $B_K \approx 2/3~\pm~1/3~~.$ Analysis showed that only the combination of perturbations $P_U=P_1$ and $P_D=P_2$ resulted in an acceptable value of $\chi^2/d.o.f. \approx 0.6/1$. The best fit was obtained for $$m_s = 183~{\rm MeV},~~ m_t = 100~{\rm GeV},~~ \sigma = 0.6^{\circ},~~{\rm and}~~ \tau = 5.7^{\circ}~~, \label{eq13}$$ the other quark masses being close to their central values. The three other combinations gave much larger values $\chi^2 > 4$. It appears thus that the prediction for the top quark mass from the low energy FD mass matrices cannot satisfy the TEVATRON[@4)] value of $m_t^{\rm phys.} \approx 175$ GeV. This model’s prediction for $$J_{CP} = {{\rm Im}}(V_{ub}V_{td}V^*_{ud}V^*_{tb}) \ , \label{eq14}$$ as a function of $m_t$ can also be obtained – the approximate value $J_{CP} = (0.3 \pm 0.2)~10^{-4}$ is predicted, which corresponds to $\sin \delta_{13} \approx (0.56 \pm 0.37)$. This result is used to predict $${\varepsilon'/\varepsilon} = (290)\cdot J_{CP}\cdot H(m_t) \ , \label{eq15}$$ where $H(m_t)$ is a decreasing function of the top quark mass[@8))]. The predicted value in the model is ${\varepsilon'/\varepsilon} = (0.6 \pm 0.5)~10^{-3}$, with a weak dependence on the top quark mass. This prediction seems to favor the data from E731[@9)] over the data from NA31[@10)]. To conclude this Section, we described a new set of quark mass matrices based on a perturbation of a universal (FD) Yukawa interaction at [*low energy*]{}. The model contains eight parameters, which have been fitted to reproduce the five known quark masses (except $m_t$), moduli of three known elements of the CKM matrix, and the $K$-physics parameter ${\varepsilon}$. As a result, the physical top quark mass is predicted to be not much heavier than $\approx 100$ GeV, and the direct CP violation parameters are predicted to be $J_{CP} = (0.3 \pm 0.2)~10^{-4}$ and ${\varepsilon'/\varepsilon} = (0.6 \pm 0.5)~10^{-3}$. The analysis will be improved substantially with a better theoretical knowledge of $B_K$, a more precise determination of the light quark masses as well as by taking into account the more accurate measurement of $|V_{cb}|$ and the ratio $|V_{ub}/V_{cb}|$. This [*low energy*]{} model, based on a simple perturbation of a universal FD Yukawa interaction at low energies, has been invalidated by the discovery of the top quark much heavier than 100 GeV. Flavor Democracy at High Energies ================================= Many attempts to unify the gauge interactions of the Standard Model (SM) have been made in the past – within the framework of the Grand Unified Theories (GUT’s). These theories give a unification energy $E_{\rm GUT} \stackrel{>}{\sim} 10^{16}$ GeV, i.e., the energy where the SM gauge couplings would coincide: $5 {\alpha}_1/3 =$ $\alpha_2 =$ $\alpha_3$. Here, $\alpha_j = g_j^2 / 4 \pi$ ($j=1,2,3$) are the gauge couplings of $U(1)_Y,~SU(2)_L$, $SU(3)_C$, respectively. For the unification condition to be satisfied at a single point $\mu (=E_{\rm GUT})$ exactly, supersymmetric theories (SUSY) were used,[@[1]] replacing the SM above the energies $\mu \approx M_{\rm SUSY} \approx 1$ TeV. This changed the slopes of $\alpha_j=\alpha_j (\mu)$ at $\mu \geq M_{\rm SUSY}$, and for certain values of parameters of SUSY the three lines met at a single point. There are several deficiencies in such an approach. The unification energy is exceedingly large ($E_{\rm GUT} \stackrel{>}{\sim} 10^{16}$ GeV) since the proton decay time is large ($\tau_{\rm proton} \geq 5.5 \times 10^{32}$ yr). This implies a large desert between $M_{\rm SUSY}$ and $E_{\rm GUT}$. While eliminating several of the previously free parameters of the SM, SUSY introduces several new parameters and new elementary particles which haven’t been observed. It is our belief that it is more reasonable to attempt first to reduce the number of degrees of freedom (d.o.f.’s) in the Yukawa sector, since this sector seems to be at least as problematic as the gauge boson sector. Any such attempt should be required to lead to an overall reduction of the seemingly independent d.o.f.’s, unlike the GUT–SUSY approach. The symmetry responsible for this reduction of the number of parameters can be “flavor democracy” (FD), valid possibly in certain separate sectors of fermions (e.g., up-type sector, down-type sector). This symmetry could be realized in a flavor gauge theory (FGT)[@[2]] – this is a theory blind to fermionic flavors at high energies $E > {\Lambda}_{\rm FGT}$ and leading at “lower” energies $E \sim {\Lambda}_{\rm FGT}$ to flavor–democratic (FD) Yukawa interactions. Requirement of reduction of as many d.o.f.’s as possible would make it natural for FGT’s to be without elementary Higgs. The scalars of the SM are then tightly bound states of fermion pairs ${\bar f} f$, with ${\bar f}f$ condensation taking place at energies ${\Lambda}$: $E_{\rm ew} \ll {\Lambda} \stackrel{<}{\sim} {\Lambda}_{\rm FGT}$. The idea of FD, and deviations from the exact FD, at [*low energies*]{} ($E \sim 1-10^2$ GeV) have been investigated by several authors[@1); @[3]; @11)]. On the other hand, in this Section we discuss FD and deviations from it at [*higher energies*]{} $E \gg E_{\rm ew}$, and possible connection with FGT’s. This discussion is motivated and partly based on works of Ref. [@[2]]. Let us illustrate first these concepts with a simple scheme of an FGT. Assume that at energies $E \stackrel{>}{\sim} \Lambda_{\rm FGT}$ we have no SM scalars, but new gauge bosons $B_\mu$, i.e., the symmetry group of the gauge theory is extended to a group $G_{\rm SM} \times G_{\rm FGT}$. Furthermore, we assume that the new gauge bosons obtain a heavy mass $M_B~( \sim \Lambda_{\rm FGT})$ by an unspecified mechanism (e.g., dynamically, or via a mechanism mediated by an elementary Higgs). At thus high energies, the SM–part $G_{\rm SM} \equiv$ $SU(3)_c \times SU(2)_L \times U(1)_Y$ is without Higgses, and hence with (as yet) massless gauge bosons and fermions. The FGT–part of Lagrangian in the fermionic sector is written schematically as $${\cal{L}}^{\rm FGT}_{g.b.-f} = -g \Psi \gamma^\mu B_\mu \Psi ~~~({\rm for}~~E \stackrel{>}{\sim} \Lambda_{\rm FGT})~, \label{eqq2}$$ where $\Psi$ is the column of all fermions and $B_\mu=B_\mu^j T_j$. $T_j$’s are the generator matrices of the new symmetry group $G_{\rm FGT}$. Furthermore, we assume that the $T_j$’s corresponding to the electrically neutral $B_\mu^j$’s do not mix flavors (i.e., no FCNC’s at tree level) and are proportional to identity matrices in the flavor space (“flavor blindness”). We will argue in the following lines that the FGT Lagrangian (16) can imply creation of composite Higgs particles through condensation of fermion pairs, and can subsequently lead at lower energies to Yukawa couplings with a flavor democracy. The effective current–current interaction, corresponding to exchanges of neutral gauge bosons $B$ at “low” cutoff energies $E$ ($E \sim \Lambda_{\rm FGT} \sim M_B$), is $${\cal{L}}^{\rm FGT}_{4f} \approx -{g^2 \over 2 M_B^2} \sum_{i,j} (\bar f_i \gamma^\mu f_i)(\bar f_j \gamma_\mu f_j)~~~({\rm for}~~ E \sim \Lambda_{\rm FGT} \sim M_B)~. \label{eqq3}$$ Since we are interested in the possibility of Yukawa interactions of SM originating from (17), and since such interactions connect left–handed to right–handed fermions, we have to deal only with the left–to–right (and right–to–left) part of (17). Applying a Fierz transformation[@[4]] to this part, we obtain four-fermion interactions without $\gamma_\mu$’s $${\cal{L}}^{\rm FGT}_{4f} \approx {2 g^2 \over M_B^2} \sum_{i,j} (\bar f_{iL} f_{jR})(\bar f_{jR} f_{iL})~~~({\rm for}~~E \sim \Lambda_{\rm FGT} \sim M_B)~. \label{eqq4}$$ These interactions can be rewritten in a formally equivalent (Yukawa) form with auxiliary (i.e., as yet nondynamical) scalar fields. One possibility is to introduce only one $SU(2)$ doublet auxiliary scalar $H$ with (as yet arbitrary) bare mass $M_H$, by employing a familiar mathematical trick[@[5]] $$\begin{aligned} {\cal L}^{(E)}_{\rm Y}& \approx & - M_H {\sqrt{2} g \over M_B} \sum_{i,j=1}^{3} {\Bigg \{} \left[ (\bar\psi^q_{iL} \tilde H) u^q_{jR} + (\bar\psi^l_{iL} \tilde H) u^l_{jR} + \mbox{h.c.} \right] \nonumber\\ &&+ \left[ (\bar\psi^q_{iL} H) d^q_{jR} + (\bar\psi^l_{iL} H) d^l_{jR} + \mbox{h.c.} \right] {\Bigg \}} - M_H^2 H^{\dagger} H \ , \label{eqq5a}\end{aligned}$$ where $M_H$ is an unspecified bare mass of the auxiliary $H$, and we use the notations $$H = {H^{+} \choose H^0} \ , \qquad \tilde H = i \tau_2 H^{\ast} \ ; \qquad \psi^q_i = {u^q_i \choose d^q_i} \ , \psi^l_i = {u^l_i \choose d^l_i} \ ,$$ where $u^q_1 = u$, $u_1^l = \nu_e$, $u^q_2=c$, etc. Another possibility is to introduce two auxiliary scalar isodoublets $H^{(U)},~H^{(D)}$, with (as yet) arbitrary bare masses $M_H^{(U)},~ M_H^{(D)}$, and express (18) in the two-Higgs ‘Yukawa’ form $$\begin{aligned} {\cal{L}}^{(E)}_{\rm Y} \approx &- M_H^{(U)} {\sqrt{2} g \over M_B} \sum_{i,j=1}^{3}\left[ (\bar\psi^q_{iL} \tilde H^{(U)}) u^q_{jR} + (\bar\psi^l_{iL} \tilde H^{(U)}) u^l_{jR} + \mbox{h.c.} \right] \nonumber\\ &- M_H^{(D)} {\sqrt{2} g \over M_B} \sum_{i,j=1}^{3} \left[ (\bar\psi^q_{iL} H^{(D)}) d^q_{jR} + (\bar\psi^l_{iL} H^{(D)}) d^l_{jR} + \mbox{h.c.} \right] \\ &- {M_H^{(U)}}^2 ({H^{(U)}}^\dagger H^{(U)}) - {M_H^{(D)}}^2 ({H^{(D)}}^\dagger H^{(D)})~~. \nonumber \label{eqq6}\end{aligned}$$ The cutoff superscript $E$ ($\sim \Lambda_{\rm FGT}$) at the “bare” parameters and fields in (19) and (20) is suppressed for simplicity of notation. Yukawa terms there involve nondynamical scalar fields and are formally equivalent to (18). Equations of motion show that the (yet) nondynamical scalars $H$, $H^{(U)}$, $H^{(D)}$ are proportional to condensates involving fermions and antifermions – i.e., they are composite. When further decreasing the energy cutoff $E$ in the sense of the renormalization group, the composite scalars in (19) and (20) obtain kinetic energy terms and vacuum expectation values (VEV’s) through quantum effects if the FGT gauge coupling $g$ is strong enough – i.e., they become dynamical in an effective SM (or: two-Higgs-doublet SM) framework and they induce dynamically electroweak symmetry breaking (DEWSB). The neutral physical components of these composite Higgs doublets are scalar condensates[@[6]] of fermion pairs $H^0 \sim {\bar f} f$. The low energy effective theory is the minimal SM (MSM) in the case (19) and the SM with two Higgs doublets – type II \[2HDM(II)\] in the case (20). Hence, although (19) and (20) are formally equivalent to four-fermion interactions (18), they lead to two physically different low energy theories[@[2]]. The condensation scenario with the smaller vacuum energy density would physically materialize. We emphasize that the central ingredient distinguishing the described scheme from most of the other scenarios of DEWSB is the flavor democracy in the Yukawa sector near the transition energies, as expressed in (19) and (20). We note that (19) implies that the MSM, if it is to be replaced by an FGT at high energies, should show up a trend of the Yukawa coupling matrix (or equivalently: of the mass matrix) in a flavor basis toward a complete flavor democracy for [**all**]{} fermions, with a common overall factor, as the cutoff energy is increased within the effective MSM toward a transition energy $E_0 (\sim {\Lambda}_{\rm FGT})$ $$M^{(U)}~~{\rm and}~~M^{(D)} \rightarrow {1 \over 3} m_t^0 \pmatrix{N_{FD}^q & 0 \cr 0 & N_{FD}^l \cr}~~~{\rm as}~~ E \uparrow E_0~~, \label{eqq7a}$$ where $m_t^0=m_t(\mu=E_0)$ and $N_{FD}$ is the $3 \times 3$ flavor–democratic matrix $$N_{FD}^f = \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array} \right]~~, \label{eqq7b}$$ with the superscript $f=q$ for the quark sector and $f=l$ for the leptonic sector. On the other hand, if the SM with two Higgses (type II) is to experience such a transition, then (20) implies [**separate**]{} trends toward FD for the up–type and down–type fermions $$M^{(U)}~(M^{(D)}) \rightarrow {1 \over 3}~ m_t^0~(m_b^0)~ \left[ \begin{array}{cc} N_{FD}^q & 0 \\ 0 & N_{FD}^l \end{array} \right]~~~ {\rm as}~~ E \uparrow E_0~~, \label{eqq8}$$ where $m_t^0$ and $m_b^0$ can in general be different. Note that $N_{FD}$, when written in the diagonal form in the mass basis, has the form $$N_{FD}^{\rm mass~basis} = 3 \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right]~~. \label{eqq9}$$ Hence, FD (and FGT) implies in the mass basis as $E$ increases to $E_0 \sim {\Lambda}_{\rm FGT}$: $$\begin{aligned} {m_u \over m_t},~{m_c \over m_t},~{m_{\nu_e} \over m_{\nu_\tau}},~ {m_{\nu_\mu} \over m_{\nu_\tau}} &\rightarrow 0~~, \nonumber\\ {m_d \over m_b},~{m_s \over m_b},~~{m_e \over m_\tau},~~ {m_\mu \over m_\tau} &\rightarrow 0~~, \\ {m_{\nu_\tau} \over m_t},~~{m_\tau \over m_b} &\rightarrow 1~~, \nonumber \label{eqq10}\end{aligned}$$ and in the case of the minimal SM [**in addition**]{} $${m_b \over m_t},~{m_\tau \over m_{\nu_\tau}} \rightarrow 1~~. \label{eqq11}$$ In our previous papers[@[2]] we showed, by considering the quark sector, that the minimal SM does not have the required trend toward FD, but that SM with two Higgs doublets (type II) does. We also checked that these conclusions remain true when we include the leptonic sector. When including also leptons (Ref. [@[2]], first entry), we can neglect for simplicity masses of the first two families of fermions, i.e., only $(t,~b)$ and $(\nu_\tau,~\tau)$ are dealt with (here $\nu_\tau$ is the Dirac tau–neutrino), and then investigate evolution of their Yukawa coupling parameters (or: their masses) with energy. In the case of the effective 2HDM(II) with only the third fermion family, the FD conditions read as (25) (last line). The one–loop renormalization group equations (RGE’s) for the Yukawa coupling parameters $g_t,~g_b~,g_\nu,~g_\tau$ of the third family fermions in any fixed flavor basis for various Standard Models with two Higgs doublets are available, for example, in Ref.[@[7]]. The running masses (at evolution, or cutoff, energies $E$), are proportional to these parameters and to the (running) VEV’s of the two Higgs doublets: $$\left[ \begin{array}{c} m_t(E) \\ m_{\nu_\tau}(E) \end{array} \right] = \frac{ v_{_U}(E) }{\sqrt{2}} \left[ \begin{array}{c} g_t(E) \\ g_{\nu_\tau}(E) \end{array} \right] \ , \qquad \left[ \begin{array}{c} m_b(E) \\ m_{\tau}(E) \end{array} \right] = \frac{ v_{_D}(E) }{\sqrt{2}} \left[ \begin{array}{c} g_b(E) \\ g_{\tau}(E) \end{array} \right] \ , \label{eqq12a}$$ where $$\begin{aligned} \langle H^{(U)(E)} \rangle_0 & = & {1 \over \sqrt{2}} {0 \choose v_{_U}(E) }~,~~ \langle H^{(D)(E)} \rangle_0 = {1 \over \sqrt{2}} {0 \choose v_{_D}(E) }~ \nonumber\\ {\rm and}~~ v_{_U}^2(E) + v_{_D}^2(E)& = & v^2(E) \ (\approx 246^2~{\rm GeV}^2 \ \mbox{ for} \ E \sim E_{\rm ew}) \ . \label{eqq12b}\end{aligned}$$ We recall that the transition energy $E_0$, appearing in FD conditions (25) and (26), is the energy above which SM starts being replaced by an FGT and the composite scalars start “de-condensing.” In Ref.[@[2]], we argued that this $E_0$ lies near the pole of the running fermion masses ($E_0 \stackrel{<}{\approx} \Lambda_{\rm pole}$). We then simply approximate: $E_0 = \Lambda_{\rm FGT} = \Lambda_{\rm pole}$. Hence, the high energy boundary conditions (25) are then $${g_{\nu_\tau} \over g_t}=1,~~{g_\tau \over g_b}=1~~~ {\rm at}~~E \approx \Lambda_{\rm pole}~. \label{eqq13}$$ These conditions are taken into account in numerical calculations, together with the low energy boundary conditions $$\begin{aligned} &m_\tau = 1.78~{\rm GeV},~~m_b(\mu=1~{\rm GeV}) = 5.3~{\rm GeV}, \nonumber\\ &m_t(\mu=m_t) \approx 167~{\rm GeV}~, \label{eqq14}\end{aligned}$$ where $m_\tau$ and $m_b$ are based on the available data of the measured masses[@[8]; @[9]]. The above value of mass $m_t(m_t) \approx m_t^{\rm phys.} [1 + 4 {\alpha}_3(m_t)/(3 \pi) ]^{-1}$ $\approx m_t^{\rm phys.}/1.047$ is based on the experimental value of $m_t^{\rm phys.} \approx 175$ GeV measured at Tevatron[@4)]. For chosen values of VEV’s ratio $v_{_U}/v_{_D}$, we found the masses of Dirac tau–neutrino $m_{\nu_\tau}$, which satisfy the above boundary conditions (29,30), by using numerical integration of RGE’s from $\mu=1$ GeV to $\Lambda_{\rm pole}$. The calculated Dirac neutrino masses are too large to be compatible with results of the available experimental predictions[@[11]]. Therefore, we invoke the usual “see–saw mechanism”[@[12]] of the mixing of the Dirac neutrino masses and the much larger right–handed Majorana neutrino masses $M_R$, in order to obtain a small physical neutrino mass $$m_\nu^{\rm phys.} \approx {m_\nu^{\rm Dirac} \over 4~ M_R}~~. \label{eqq15a}$$ Majorana mass term breaks the lepton number conservation. Therefore, Majorana masses $M_R$ are expected to be of the order of some new (unification) scale $\Lambda~\gg~E_{\rm ew}$. We assume: $M_R \approx \Lambda$. Within our context, the simplest choice of this new unification scale would be the energy $\Lambda_{\rm FGT} = \Lambda_{\rm pole}$ where SM is replaced by FGT. $$m_\nu^{\rm phys.} \approx {m_\nu^{\rm Dirac} \over 4~ \Lambda_{\rm FGT}}~~. \label{eqq15b}$$ The physical tau–neutrino masses $m_{\nu_\tau}^{\rm phys.}$ predicted in this way are very small for the most cases of chosen values of $v_{_U}/v_{_D}$ and $m_t^{\rm phys.}$, i.e., in most cases they are acceptable since being below the experimentally predicted upper bounds[@[11]]. The see–saw scenario leading to our predictions of $m_{\nu_\tau}^{\rm phys.}$ implicitly assumes that: (a) FGT contains in addition Majorana neutrinos, and its energy range of validity also provides the scale for the heavy Majorana masses \[i.e., $M_R \sim \Lambda_{\rm FGT}$\]. (b) At low (SM) energies, Majorana neutrinos remain decoupled from (or very weakly coupled to) the Dirac neutrinos, which is a very plausible assumption in view of assumption (a). In general, it could be assumed $M_R \sim \Lambda_{\rm new-scale} \geq \Lambda_{\rm FGT}$, leading thus to even smaller $m_{\nu_\tau}^{\rm phys.}$ than those in (32). When increasing $m_t^{\rm phys.}$ at a fixed $v_{_U}/v_{_D}$, $m_{\nu_\tau}^{\rm Dirac}$ increases and $\Lambda_{\rm FGT}$ decreases, and hence $m_{\nu_\tau}^{\rm phys.}$ increases. This provides us, at a given ratio $v_{_U}/v_{_D}$, with: (a) [*upper*]{} bounds on $m_t^{\rm phys.}$ for (various) specific upper bounds imposed on $m_{\nu_\tau}^{\rm phys.}$ (e.g., $\leq 31$ MeV[@[11]], or $\leq 1$ MeV, or $\leq 17$ KeV[@[13]]); (b) [*lower*]{} bounds on $m_t^{\rm phys.}$ for (various) specific upper bounds imposed on $\Lambda_{\rm FGT}$ (e.g., $\leq \Lambda_{\rm Planck}$, or $\leq 10^{10}$ GeV, or $\leq 10^5$ GeV). Even with the largest possible upper bounds on $m_{\nu_\tau}^{\rm phys.} \leq 31$ MeV and $\Lambda_{\rm FGT} \leq \Lambda_{\rm Planck}$, we can still get rather narrow bands on the values of $m_t^{\rm phys.}$ at any given $v_{_U}/v_{_D}$. E.g., if $v_{_U}/v_{_D}=1$, then 155 GeV $\stackrel{<}{\approx} m_t^{\rm phys.} \stackrel{<}{\approx} 225$ GeV. Inversely, if $m_t^{\rm phys.} = 175$ GeV \[$m_t(m_t) = 167$ GeV\], $m_{\nu_\tau}^{\rm phys.} \leq 31$ MeV and $\Lambda_{\rm FGT} \leq \Lambda_{\rm Planck}$, then we obtain rather stringent bounds on the VEV ratio: $0.64 \stackrel{<}{\approx} v_{_U}/v_{_D} \stackrel{<}{\approx} 1.35$. To conclude this Section, we stress that we can estimate the masses of top and tau–neutrino within SM with two Higgs doublets, assuming solely that the complete flavor democracy should set in at energies where SM starts breaking down. The gauge theories (FGT’s) which presumably replace SM at such energies remain to be further investigated. For related detailed information, see Ref.[@[2]]. Discussions and Conclusion ========================== We discussed on the one hand flavor–democratic (FD) mass matrices at [*low energies*]{}, and on the other hand conditions under which mass matrices show a trend to flavor–democratic forms at [*high energies*]{} (in a flavor basis) – a behavior possibly related to flavor gauge theories (FGT’s) at high energies. However, we found that the model based on our simple perturbation of a universal FD Yukawa interaction at [*low energies*]{} has been invalidated, because of the discovery of a top quark much heavier than 100 GeV. On the contrary, at [*high energies*]{}, assuming solely that the complete flavor democracy should set in at energies where an effective perturbative two-Higgs-doublet SM (type II) starts breaking down, we can estimate the masses of top and tau–neutrino, which are compatible with the present experimental results. Therefore, the gauge theories (FGT’s) which presumably replace SM at such energies remain to be further investigated. In our forthcoming work[@chk], we would like to investigate further the simple FD mass matrices ansatz which had been applied earlier[@11)] at low energies and had given experimentally unacceptable $m_t$. We would like to apply this ansatz at a high energy scale $E \sim \Lambda_{\rm pole}$, employing RGE evolution within a two-Higgs-doublet SM model (type II). Furthermore, the compositeness nature of the scalars in this framework should be further investigated, particularly in view of the fact that, for cases when VEV ratio is $v_{_U}/v_{_D} \sim 1$, the usual RGE compositeness conditions at ${\Lambda}_{\rm pole}$ suggest that only $H^{(U)}$ can be fully composite, but not $H^{(D)}$ (cf. Ref. [@rev]). Acknowledgements ================ CSK would like to thank Prof. Y. Koide for his kind invitation to the Workshop of MMQL97. 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ArXiv

--- abstract: 'We investigate the influence of quantum fluctuations and weak disorder on the vortex dynamics in a two-dimensional superconducting Berezinskii-Kosterlitz-Thouless system. The temperature below which quantum fluctuations dominate the vortex creep is determined, and the transport in this quantum regime is described. The crossover from quantum to classical regime is discussed and the quantum correction to the classical current-voltage relation is determined. It is found that weak disorder can effectively reduce the critical current as compared to that in the clean system.' author: - 'Aleksandra Petkovi'' c$^{1,2}$, Valerii M. Vinokur$^{2}$, and Thomas Nattermann$^{1}$' date: ',' title: '<span style="font-variant:small-caps;">Transport properties of clean and disordered Josephson junction arrays</span>' --- The physics of the Berezinskii-Kosterlitz-Thouless (BKT) transition in thin superconducting films is re-emerging as one of the mainstreams in the current condensed matter physics. The interest is motivated by recent advances in studies of layered high-temperature superconductors [@Ruef-2006; @Matthey-2007], the discovery of the superconductivity at the interface between the insulating oxides [@Reyren-2007; @Caviglia-2008], new studies in thin superconducting films uncovering the role of the two-dimensional (2D) superconducting fluctuations [@Crane-2007; @Pourret-2006], and the intense developments in the physics of the superconductor-insulator transition where the BKT transition may play a major role [@Vinokur-Nature]. The predicted benchmark of the transition that serves to detect it experimentally is the change of the shape of the $I$-$V$ characteristics, $V\propto I^{1+\alpha}$ from $\alpha=0$ above the transition, $T>T_{BKT}$, to $\alpha=2(T_{BKT}/T)$ at $T\leq T_{BKT}$ [@HN; @Minnhagen]. However, experimental data on superconducting films show appreciable deviations from the theoretical predictions and are still inconclusive [@exp]. Among the possible sources of the deviation from the classic predictions, one can consider the finite size effect, [@Kogan; @GV] and effects of disorder [@Na+_95; @KoNa_96; @Giam]. Another important issue is the role of quantum effects which become crucial when the BKT transition occurs at low enough temperatures. In this Brief Report we will analyze the role of quantum effects in the BKT transition paying a special attention to the intermediate region of the interplay between thermal and quantum contributions. We will discuss the effect of disorder-generated vortices on the BKT transition, neglecting quantum fluctuations, namely the effective reduction of the critical current as compared to that in clean samples. *Model.* We choose a disordered Josephson junction array as a convenient discrete model for the 2D disordered superconducting film [@EcSc_89]. The Hamiltonian describing the system is: $$\label{eq:H} {\cal H} = \frac{1}{2} \sum_{i,j} ( C^{-1})_{i,j}\hat n_i\hat n_j -J\sum_{\langle i,j\rangle}\cos (\hat \varphi_i-\hat \varphi_{j}-A_{ij})$$ where $[\hat n_j, \hat \varphi_k]=-2e i\delta_{j,k}$. We ignore single electron tunneling and other sources of dissipation. The only non-vanishing elements of the capacitance matrix $C_{ij}$ are its diagonal elements $C_{jj} =4C$ (no summation over the repeated index) and $C_{ij}=-C$ for the nearest neighbors $i,j$, i.e., the capacitance to the ground is assumed negligible as compared to the mutual capacitances of the superconducting islands. The second sum in (\[eq:H\]) is taken over all nearest-neighbor pairs on a square lattice. Random phase shifts $A_{ij}$ result from the deviations of the flux in a distorted plaquette from an integer multiple of the flux quantum $\Phi_0=\hbar c/2e$ [@Granato+86]. In the clean classical case, i.e. for $A_{ij}=0$ and in the limit $C\to \infty$, the physics of the system can be most adequately described in terms of vortices that experience the superconducting BKT transition at the temperature $T_{BKT}\simeq \pi \widetilde{J}/2$, where $\widetilde{J}$ denotes the renormalized coupling constant. It is convenient to decompose the phase at the site $i$, as $\varphi_i = \varphi_i^{(v)}+\varphi_{i}^{(sw)}$ where $(v)$ and $(sw)$ stand for the vortex and the spin wave part, respectively. Then, the vortex Hamiltonian can be written as $$\begin{aligned} \label{eq:H_v} \mathcal{H}_{v}=&-{J\pi} \sum_{i\neq j} m_{i} m_j \ln{ \frac{| {\bf r}_{i}-{\bf r}_{j}|}{ \xi}} +\sum_i E_c m_i^2,\end{aligned}$$ where $E_c$ denotes the core energy of a vortex. The sums are taken over the sites ${\bf r}_i$ of a dual lattice; $m_i$ is the vorticity of the $i$th vortex, and we assumed that $\sum_i m_i=0$, where $\xi$ denotes the superconductor coherence length. Next we want to include quantum fluctuations. After going over to the path integral representation of the partition function and integrating out the charge degrees of freedom, the action of the Josephson junction array in the limit $E_c=e^2/2C \ll J$ assumes the form [@EcSc_89; @fazio91] $$\begin{aligned} S=\int d\tau \left(\frac{M}{2}\sum_{i}(\partial_{\tau}\mathbf r_{i})^2 +{\cal H}_v\right).\end{aligned}$$ The vectors ${\bf r}_i(\tau)$ are the world lines of the vortices and $M=h^2 C /(8 e^2 \xi^2)$. *Clean case*. We begin with the discussion of a clean case. If we apply an external transport current, it will exert the force ${\bf f}\sim {\bf j}$ on the vortices, where $\bf{j}$ is the current density [@Tinkham]. This generates an additional term $-\sum_i m_i{\bf f}\cdot{\bf r}_i$ in (\[eq:H\_v\]). In order to describe the effect of vortices on the current-voltage relation quantitatively, we consider the effect of vortices crossing the system transversely to the transport current. This motion dissipates energy. The Bardeen-Stephen flux flow resistance [@Bardeen] gives for the current-voltage ($V-I$) relation $$V=2\pi\xi^2\rho_nn_vI$$ where $n_v$ is the vortex density and $\rho_n$ is the normal state resistivity. The rate equation for the vortex density is $$\begin{aligned} \label{eq:decayrate} \partial_t n_v=\Gamma -\frac{\xi^2}{\tau_{rec}}n_v^2.\end{aligned}$$ Here $\Gamma$ denotes the rate of generation of free vortices, while the second term on the rhs of (\[eq:decayrate\]) describes their recombination, $\tau_{rec}$ denotes the recombination parameter. The steady state value $n_v=(\tau_{rec}\Gamma)^{1/2}/\xi$ of the vortex density determines the current-voltage relation. In order to determine $\Gamma$, we consider the appearance of a vortex-antivortex pair and its subsequent separation via tunneling or thermal activation under the influence of the external force $\mathbf{f}$. In the clean case this process is symmetric, i.e., the coordinates of the vortex ${{\mathbf r}}_1$ and the antivortex ${{\mathbf r}}_2$ satisfy ${\bf r}_1=-{\bf r}_2={{\mathbf r}}$ with ${\mathbf {f}\cdot \bf{r}}=f r$. The action of the vortex pair can be rewritten as $$\begin{aligned} \label{ch2_eq:action} &S=\int d\tau \left[M(\partial_{\tau}r)^2+U(r)\right ],\end{aligned}$$ where $U(r)=2 \pi J \ln{\left( \frac{2r}{\xi} \right)}-2 f r+2 E_c$. The problem effectively reduces to a single particle motion through one-dimensional potential barrier $U(r)$ [@footnote1]. The rate $\Gamma$ is given by [@Affleck81] $$\label{eq:Gamma-T} \Gamma\sim \int_{0}^{\infty} dE \;\Gamma(E)e^{-E/T},$$ where $\Gamma(E)$ denotes the zero temperature tunneling rate of a particle in the potential $U(r)$ having an energy $E$. For low temperatures and hence $E$ smaller than the barrier height $U_0=2\pi J\left[\ln(\frac{2\pi J}{f\xi})-1\right]+2E_c$, $\Gamma(E)$ in the WKB approximation is $$\begin{aligned} \Gamma(E)= e^{-4\sqrt{M}\int_{r_a(E)}^{r_b(E)}d r\sqrt{U(r)-E}/\hbar},\end{aligned}$$ where $r_{a/b}(E)$ satisfy $U(r_{a/b})=E$ (see Fig. \[fig:potential\]). ![Potential barrier for the separation of the vortex-antivortex pair.[]{data-label="fig:potential"}](potential.eps){width="0.7\columnwidth"} In the following different regimes will be considered.\ (i)At zero temperature the only contribution in Eq. (\[eq:Gamma-T\]) comes from $E=0$. The generated voltage for small currents ($f\xi/J\ll 1$) is $$\begin{aligned} \label{eq:quantum} &V\sim\Gamma^{1/2}\sim e^{-S(0,0)/2\hbar}\notag\\ &\frac{S(0,0)}{\hbar}\approx c_1\frac{\sqrt{M}(2J\pi)^{3/2}}{\hbar f}{\left(\ln{\frac{2J\pi}{f\xi}}\right)}^{3/2}.\end{aligned}$$ $c_1$ is a positive constant of the order of unity and $$\begin{aligned} \frac{S(E,T)}{\hbar}=\frac{E}{T}+4\sqrt{M}\int_{r_a(E)}^{r_b(E)}d r\frac{\sqrt{U(r)-E}}{\hbar}\end{aligned}$$ is the action of the classical path of the particle in the potential $-U(r)$ with the energy $E$ and mass $2M$. The result (\[eq:quantum\]) is in an agreement with that of Ref. [@Iengo+96] where it is obtained using the different technique [@footnote2]. We find that the result (\[eq:quantum\]) holds also at finite temperatures as long as $$\begin{aligned} \label{eq:To} T\leq T_0=\frac{1}{c_2}\frac{\hbar f}{\sqrt{\pi 2J M}}\frac{1}{\sqrt{\ln{\frac{\pi 2J}{f\xi}}}},\end{aligned}$$ where $c_2$ is positive constant of the order of unity.\ (ii) At intermediate temperatures $T_0<T<T^*$, where $$\begin{aligned} \label{eq:T^*} T^*=\frac{\hbar}{2\pi}\sqrt{\frac{-U''(r_c)}{2M}}=\frac{\hbar f}{2\pi}\sqrt{\frac{1}{M J \pi}},\end{aligned}$$ the main contribution in Eq. (\[eq:Gamma-T\]) comes from the stationary point $E_T$. Therefore, $V\sim \exp \left [ -S(E_T,T)/2\hbar \right ]$. $E_T$ depends on the temperature and is implicitly given by the equation $$\begin{aligned} \label{eq:finiteT} \frac{\hbar}{T}=2\sqrt{M}\int_{r_a(E_T)}^{r_b(E_T)}d r \frac{1}{\sqrt{U(r)-E_T}}=\tau(E_T),\end{aligned}$$ where $\tau(E)$ can be interpreted as the period of the classical motion of a particle with the mass $2M$ and energy $E$, in the potential $-U(r)$. Since $\tau(E)$ is the monotonically decreasing function of $E$ for small currents, Eq. (\[eq:finiteT\]) has the unique solution $E_T$ for every $T$ in a range $T_0\leq T\leq T^*$. We come back to the discussion of the voltage characteristic in this regime below.\ (iii)At even higher temperatures $T^*< T \leq T_{\mathrm{BKT}}$, the decay rate is dominated by $E>U_0$ [@Goldanskii; @Affleck81] and the thermally activated breaking of vortex pairs dominates the dynamics. Then, the decay rate is given by the Arrhenius law $\Gamma\sim \exp[-S_{class}/ \hbar]$ where $S_{class}=\hbar U_0/T$. The voltage-current relation reads [@HN; @Doniah+79] $$\begin{aligned} \label{ch2_eq:VIclassical} V\sim f e^{-U_0/(2T)} \sim j^{\delta(T)},\quad\quad \delta(T)=1+{\pi J}/{T}.\end{aligned}$$ Taking into account the presence of other vortices by replacing $J\to \widetilde{J}$, the coefficient assumes a universal value $\delta(T_{BKT})=3$.\ (iv) At $T>T_{\mathrm{BKT}}$ a finite density of free vortices appears in an equilibrium, and the system is characterized by a linear current-voltage relation for small enough currents. ![Dynamic phase diagram in current-temperature coordinates showing different types $V(j,T)$ dependencies for $T<T_{BKT}$. The dashed and the solid lines sketch $T_0(j)$ and $T^*(j)$, respectively. In the domain $T<T_0$ quantum tunneling of vortices dominates the vortex dynamics, while at $T>T^*$ the voltage-current characteristics is determined by the thermally activated motion. In the shaded region the quantum correction to the classical result, given by Eq. (\[eq:correction\]), applies.[]{data-label="Fig:regions"}](regimes.eps){width="0.8\linewidth"} Next, we consider crossover from the quantum- ($T\leq T_0$) to the classical regime ($T>T^*$) in more detail. Within the semiclassical approximation the decay rate is given, with the exponential accuracy, by $\Gamma\sim \exp[-S_{\mathrm{min}}/\hbar]$, where $S_{\mathrm{min}}$ is the action of the trajectory minimizing the Euclidean action of Eq. (\[ch2\_eq:action\]). For temperatures below $T_0$ the extremal action is $S_{\mathrm{min}}=S(0,0)$, in the intermediate region ($T_0<T<T^*$) the minimal action is $S_{\mathrm{min}}=S(E_T,T)$, and in the high temperature regime the trajectory extremizing the action is time independent, and therefore $S_{\mathrm{min}}=\hbar U_0/T$. We find that $S_{\mathrm{min}}$ at $T^*$ has a continuous first derivative with respect to temperature, while the second derivative has a jump: $$\begin{aligned} \label{ch2_eq:secondorder} &\frac{{{\mathrm d}}S(E_T,T)}{{{\mathrm d}}T}\Big|_{T^*}=\frac{{{\mathrm d}}S_{\mathrm{class}}}{{{\mathrm d}}T}\Big|_{T^*}\notag\\ &\frac{{{\mathrm d}}^2 S(E_T,T)}{{{\mathrm d}}T^2}\Big|_{T^*} < \frac{{{\mathrm d}}^2 S_{\mathrm{class}}}{{{\mathrm d}}T^2}\Big|_{T^*}.\end{aligned}$$ Following Ref. [@Larkin+83] we call this situation a second-order transition at the crossover point [@footnote3]. The result of Eqs. (\[ch2\_eq:secondorder\]) is a general property of a massive particle trapped in a metastable state formed by a potential $U(r)$, provided $\tau(E)$ is a monotonously decreasing function of energy [@Chudnovsky92]. Generally, in the case of a second-order transition the trajectory extremizing the action can be written as [@Larkin+83] $$\begin{aligned} r(\tau)=r_c+\sum_{n=1}^{\infty}a_n \cos{\left( \frac{2\pi T}{\hbar}n \tau\right)},\end{aligned}$$ where the coefficients $|a_n|\ll |a_1|$ ($n>1$) are small near the transition temperature $T^*$. Substituting $r(\tau)$ in Eq. (\[ch2\_eq:action\]), the action can be expanded in powers of $a_n$, yielding an effective action $ S\approx {U_0\hbar}/{T}+\alpha a_1^2+\beta a_1^4, $ where the coefficient $\alpha$ is negative below $T^*$ and vanishes at the transition temperature $T^*$ [@Larkin+83]. Then the coefficient $a_1$ can be found from the minimization of the action $S$ and the minimal action is $ S_{\mathrm{min}}=U_0\hbar/T-\alpha^2/(4\beta). $ Following Refs. [@Larkin+83], we determine the coefficients $\alpha$ and $\beta$ and find a quantum correction to the classical result of Eq. (\[ch2\_eq:VIclassical\]) $$\begin{aligned} \label{eq:correction} V&\sim j^{\delta(T)} e^{\Delta},\notag\\ \Delta & = \frac{(T^2-T^{*2})^2}{T T^{*^3}} \frac{\sqrt{M J^3} }{\hbar f} \frac{\pi^{5/2}}{1+2(1-4(T/T^*)^2)^{-1}} .\end{aligned}$$ This result is valid near the transition, for temperatures approaching $T^*$ from below, see Fig. \[Fig:regions\]. We conclude that quantum effects significantly enhance the decay rate in comparison to the classical rate for the asymptotically small currents. It would be interesting to probe the result of Eq. (\[eq:correction\]) in experiments.\ *Disordered case*. Next we include disorder into the consideration, in the limit $C\to \infty$. The phase shifts are assumed to be uncorrelated from bond to bond, and each is Gaussian distributed with the mean value and the variance $$\label{eq:A} \overline {A_{ij}} = 0,\quad\quad\overline {A_{ij}^2} =\sigma,$$ respectively. Then, an additional term $\sum _i m_iV({\bf r}_i)$ is generated in (\[eq:H\_v\]), where $V({\bf r}_i)=2\pi J\sum_j Q_j\ln(| {\bf r}_{i}-{\bf r}_{j}| / \xi)$. $Q_i=(1/2\pi)\sum_{<plaq>} A_{ij}$ are the frozen charges sitting on the dual lattice in the center of a plaquette whereas the sum is over a plaquette formed by 4 bonds. From (\[eq:A\]) follows $\overline{(V({\bf r}_i)-V({\bf r}_j))^2}=4\pi\sigma J^2\ln (| {\bf r}_{i}-{\bf r}_{j}|/ \xi)$. It was shown in Ref. [@Na+_95], that the system in the classical case, at $T = 0$ undergoes a disorder driven transition from the ‘ordered’ BKT state to a disordered phase at the critical disorder strength $\sigma_c=\pi/8$. In the ordered BKT phase vortices appear, on average, only in a form of the bound pairs. Indeed, the energy of a vortex pair with the separation $R$ and $m_1=-m_2=1$ in a clean sample is given by $2 \pi J \ln (R/ \xi)$. Since $V({\bf r}_i)$ is Gaussian distributed, the typical energy gain is $-2J \sqrt {\pi \sigma \ln(R/ \xi)}$ which is smaller by a factor $\sim (\ln (R/\xi))^{-1/2}$ than the energy cost of a pair. However, the maximum energy gain of a vortex dipole in a region of linear size $L>R$ is larger by a factor $\sqrt{2\ln N}$ than the typical energy gain, which arises from the $N$ independent realizations of the vortex positions [@KoNa_96]. The disorder potential, that one vortex-antivortex pair of size $R+dR$ feels, is uncorrelated when the pair is translated over a distance larger than $R$ [@LHT96]. Therefore, we introduce a lattice with a lattice constant $R$. Since also the correlations of disorder potential inside the cell give only subleading-order corrections [@LHT96], we estimate $N\approx (L/R)^2 (R/\xi)^2 (2\pi R/\xi) dR/\xi$. For $dR\approx R$, we get the free energy of the pair at $T=0$ $$\begin{aligned} \label{eq:pairenergy} E\approx 2\pi J \ln{\frac{R}{\xi}}\left[1-\sqrt{\frac{4\sigma}{\pi} \frac{\ln{(L R/ \xi^2)}}{\ln{(R/\xi)}}}\right].\end{aligned}$$ Thus, if $R\approx L$, the total energy of the coresponding vortex pair becomes negative and free vortices are favored by disorder provided $\sigma >\sigma_c=\pi/8$, in an agreement with the renormalization group result in Ref. [@Na+_95]. Note that strictly speaking these vortices are “pseudo-free" since despite the fact that their attraction is overruled by disorder, they remain pinned by the same disorder-induced forces. It follows from the above reasoning that even for $\sigma<\sigma_c$ some rare vortex pairs of the negative energy can appear. From (\[eq:pairenergy\]) we get that their maximal size is $R_c\approx \xi (L/\xi)^{\frac{1}{2\sigma_c/\sigma-1}}$, which reaches the size of the system for $\sigma\to \sigma_c-0$, as expected. Typically there is a single dipole of the size $R_c$ in the system. If we divide the system into $M^2$ subsystems, each part will contain a dipole of the maximum size $R_{M} \approx R_c M^{-{\frac{1}{2\sigma_c/\sigma-1}}}$. The density of dipoles of the size $R_M$ is $\xi^{-2}(R_M/ \xi)^{2(1-2\sigma_c/\sigma)}$ at $T=0$, in agreement with Ref. [@LHT96]. We further determine the critical current. If the transport current is strong enough, it will depin vortices such that the dissipation sets in. A crude estimate for the critical depinning force at $T=0$ and $\sigma<\sigma_c$ is given by $$\label{eq:criticalforce} f_c\sim \frac{J}{R_c} \sim \frac{J}{\xi}\left(\frac{L}{\xi}\right)^{\frac{-1}{2\sigma_c/\sigma-1}},$$ since smaller dipoles are depinned at larger forces. The influence of disorder on the voltage-current relation is left for further studies. *Conclusion.* We have investigated transport properties of Josephson junction arrays taking into account the influence of quantum fluctuations on the unbinding of vortex pairs for $E_c\ll J$. At sufficiently low temperatures the quantum tunneling of vortices turns out to be more probable than their thermal activation. We have derived the $V$-$I$ relation corresponding to the quantum creep of the BKT-vortices and found the range of temperatures, $0\leq T\leq T_0$, where this law is applicable. We have determined the temperature $T^*$ above which the thermally activated breaking of vortex pairs dominates the vortex nucleation. We have discussed the region of intermediate temperatures $T_0<T<T^*$ where a crossover from classical to quantum behavior occurs, and found the quantum correction to the classical result, see Eq. (\[eq:correction\]). 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ArXiv

--- abstract: 'We study the effect of algebraically localized impurities on striped phases in one space-dimension. We therefore develop a functional-analytic framework which allows us to cast the perturbation problem as a regular Fredholm problem despite the presence of essential spectrum, caused by the soft translational mode. Our results establish the selection of jumps in wavenumber and phase, depending on the location of the impurity and the average wavenumber in the system. We also show that, for select locations, the jump in the wavenumber vanishes.' --- \ Gabriela Jaramillo$\,^1$, Arnd Scheel$\,^2$, and Qiliang Wu$\,^3$\ *$\,^1$The University of Arizona, Department of Mathematics, 617 N. Santa Rita Ave, Tucson, AZ 85721, USA\ $\,^2$University of Minnesota, School of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA\ $\,^3$Michigan State University, Department of Mathematics, 619 Red Cedar RD, East Lansing, MI 48824, USA* [**Keywords:**]{} Turing patterns, inhomogeneities, Fredholm, essential spectrum Introduction ============ We are interested in the effect of localized impurities on self-organized, spatially periodic patterns, in particular in the idealized situation of an unbounded domain. Our goal is to quantify the effect of the impurity on phases and wavenumbers in the far field. A prototypical example for the formation of self-organized periodic patterns is the Swift-Hohenberg equation $$u_t=-(\Delta +1)^2 u + \mu u - u^3,$$ where, for $0<\mu\ll 1$, periodic patterns of the form $u_*(kx;k)$, $u_*(\xi;k)=u_*(\xi+2\pi;k)$, exist for a band of admissible wavenumbers $k\in (k_-(\mu),k_+(\mu))$. Our results are concerned with this system in one-dimensional space, $x\in{\mathbb{R}}$, including an impurity, $$\label{e:sh} u_t=-(\partial_x^2 +1)^2 u + \mu u - u^3+\varepsilon g(x,u),$$ where $|g(x,u)|{\leqslant}C(u)(1+|x|)^{-\gamma_*}$, for some $\gamma_*$ sufficiently large. We find such perturbation problems interesting for a variety of reasons. First, small impurities are simple examples of defects in spatially extended systems, and a systematic description of such defects is essential to various multi-scale descriptions of extended systems. In particular, defects can be responsible for the selection of wavenumbers $k$ in extended systems. Second, perturbations of periodic patterns pose challenging technical problems since the linearization at such periodic structures is generally not Fredholm when considered as an operator on translation-invariant (or algebraically weighted) function spaces. The difficulty stems from the presence of a non-localized neutral (or soft) mode, in this case the derivative $\partial_xu_*$ of the periodic pattern, which induces a branch of essential spectrum near the origin. In this regard, our results can be viewed as a continuation of a variety of results on perturbation and bifurcation in the presence of essential spectrum. Third, one can interpret the effect of inhomogeneities in relation to the notorious question of asymptotic stability of periodic patterns, where the pattern is perturbed at time $t=0$, whereas in our case the perturbation is constant in time. It would be quite interesting to bring those two view points together and study spatio-temporal perturbations of striped phases; see, for instance, [@dsss; @gallay; @zumbrunwith; @johnsonzumbrun; @uecker; @scheelwu; @schneider]. The effect of inhomogeneities on patterns with soft modes, that is, with eigenmodes of the linearization that exhibit neutral or weak temporal decay, has been studied in detail when periodic patterns are oscillatory in time [@kollar; @SSdef]. In this case, inhomogeneities may create wave-sources such as target patterns, or act as weak sinks. In fact, in this case, the effects are quite similar to the effect of boundary conditions on oscillatory media, or, more generally, the effect of self-organized coherent structures on waves in the far-field. In the case of stationary periodic patterns, with vanishing group velocities, as they arise in the Swift-Hohenberg equation, the literature on defects and their characterization is quite extensive [@defectsSH], albeit arguably not at the level of detail as we are striving for, here. In the direction of the present work, the characterization of boundary conditions on striped phases in [@morrissey] is closest. Results there show how to identify and compute strain-displacement relations, that is, relations between wavenumbers and phases (translations) of periodic patterns in the far field, induced by the presence of the boundary. Our present work can be viewed as matching such relations at $+\infty$ and $-\infty$. Technically, our work is following up on recent studies of inhomogeneities in a variety of contexts [@jara3; @jara1; @jara2], where Kondratiev spaces were used to study perturbations of spatio-temporally periodic patterns by inhomogeneities. The present work goes however significantly past those techniques by treating non-normal form, actual periodic patterns, where in [@jara3; @jara1; @jara2] the periodic patterns were, after appropriate transformations, constant in space. Our results are concerned with the spatially one-dimensional situation, only, but we hope that our approach will allow us to tackle higher-dimensional problems, as well. From a phenomenological point of view, the one-dimensional case is most difficult since effective diffusion of the neutral mode is weakest in one space-dimension, such that the effect of the inhomogeneity on the far-field is the most significant. This phenomenon is well understood in the case of diffusive stability, where decay of localized data is faster in $n$ space-dimensions $t^{-n/2}$, or in the case of impurities in oscillatory media, where small impurities can generate wave sources only in dimensions $n{\leqslant}2$ [@jara3; @jara2; @kollar]. From a technical point of view, the one-dimensional case is easiest since the problem of finding stationary solutions can be cast as an ordinary differential equation; see for instance [@morrissey; @SSdef] for this point of view. Our approach is different and in some sense more direct. We will however comment on how to implement a proof using such “spatial dynamics” methods in the discussion. #### Notations We collect some useful notation. Let $\mathbb{P}_j({\mathbb{R}})$ and $\mathbb{P}_j({\mathbb{Z}})$ denote the set of complex-coefficient polynomials of degree less than $j\in{\mathbb{Z}}^+$ defined on the real line and on the set of integers, respectively. The inner product in a Hilbert space $H$ is denoted as $\langle\cdot, \cdot\rangle$ and the linear subspace spanned by $u\in H$ is denoted as $\langle u \rangle$. The Fourier transform on $L^2({\mathbb{R}}, H)$ and $L^2({\mathbb{Z}}, H)$ are denoted respectively as $\mathcal{F}$ and $\mathcal{F}_{\rm d}$. Moreover, for a Banach space $B$, the notation ${\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}u^*, u {\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}$ represents the action of a linear functional $u^*\in B^*$ on $u\in B$. Throughout, the Lie bracket, $[L_1, L_2]$, of two operators $L_1$ and $L_2$ is the operator $$[L_1, L_2]:=L_1\circ L_2-L_2\circ L_1.$$ We will use Banach spaces of functions on ${\mathbb{R}}$ and ${\mathbb{Z}}$. Given $s\in{\mathbb{Z}}^+\cup\{0\}$, $p \in (1, \infty)$, $\gamma\in{\mathbb{R}}$, and denoting $\lfloor x \rfloor = \sqrt{ 1 +|x|^2}$, the weighted Sobolev space $W^{s,p}_\gamma$ is defined as $$W^{s,p}_\gamma:=\left\{u \in L^1_{\mathrm{loc}}({\mathbb{R}}, H)\middle| \lfloor x \rfloor ^{\gamma}\partial_x^\alpha u \in L^p({\mathbb{R}}, H), \text{for all }\alpha\in[0, s]\cap{\mathbb{Z}}\right\},$$ with norm $\sum_{\alpha=0}^s\|\lfloor x \rfloor^{\gamma}\partial_x^\alpha u \|_{L^p}$, while the Kondratiev space $M^{s,p}_\gamma$ on ${\mathbb{R}}$ is defined as $$M^{s,p}_\gamma:=\left\{u \in L^1_{\mathrm{loc}}({\mathbb{R}}, H)\middle| \lfloor x \rfloor^{\gamma+\alpha}\partial_x^\alpha u \in L^p({\mathbb{R}}, H), \text{for all }\alpha\in[0, s]\cap{\mathbb{Z}}\right\},$$ with norm $\sum_{\alpha=0}^s\| \lfloor x \rfloor^{\gamma+\alpha}\partial_x^\alpha u \|_{L^p}$. Their dual spaces are defined in the standard way and we write $$W^{-s,q}_{-\gamma}:=(W^{s,p}_\gamma)^*,\quad M^{-s,q}_{-\gamma}:=(M^{s,p}_\gamma)^*, \text{ where }1/p+1/q=1.$$ For $s=0$, both spaces are simply weighted $L^p$-space, denoted as $L^p_\gamma$. For $p=2$, we denote $W^{s,2}_\gamma$ as $H^s_\gamma$. Additionally, one can allow different weights on ${\mathbb{R}}^\pm$ to obtain an anisotropic version of these spaces. More specifically, letting $\chi_\pm$ be a smooth partition of unity, with $\mathrm{supp}(\chi_+)\subset (-1,\infty)$, $\chi_-(x)=\chi_+(-x)$, we define $$W^{s,p}_{\gamma_-, \gamma_+}:=\left\{u \in L^1_{\mathrm{loc}}({\mathbb{R}}, H)\middle| \chi_\pm u\in W^{s,p}_{\gamma_\pm}\right\}, \quad M^{s,p}_{\gamma_-, \gamma_+}:=\left\{u \in L^1_{\mathrm{loc}}({\mathbb{R}}, H)\middle| \chi_\pm u\in M^{s,p}_{\gamma_\pm}\right\},$$ which are Banach spaces respectively with norms $$\|u\|_{W^{s,p}_{\gamma_-, \gamma_+}}:=\|\chi_+u\|_{W^{s,p}_{\gamma_+}} + \|\chi_-u\|_{W^{s,p}_{\gamma_-}},\quad \|u\|_{M^{s,p}_{\gamma_-, \gamma_+}}:=\|\chi_+u\|_{M^{s,p}_{\gamma_+}} + \|\chi_-u\|_{M^{s,p}_{\gamma_-}}.$$ Replacing ${\mathbb{R}}$ with ${\mathbb{Z}}$ and $\partial_x$ with the discrete derivative $\delta_+(\{u_j\}_{j\in{\mathbb{Z}}}):=\{u_{j+1}-u_j\}_{j\in{\mathbb{Z}}}$, the discrete counterparts of $L^p_{\gamma_-,\gamma_+}$ and $M^{s,p}_{\gamma_-, \gamma_+}$ are denoted respectively as $\ell^p_{\gamma_-,\gamma_+}$, and ${\mathscr{M}}^{s,p}_{\gamma_-, \gamma_+}$. We point out that the discrete counterparts of $W^{s,p}_{\gamma_-,\gamma_+}$ are isomorphic to $\ell^p_{\gamma_-,\gamma_+}$ due to the fact that $\delta_+$ is a bounded linear operator on $\ell^p_{\gamma_-,\gamma_+}$. #### Outline. The remainder of the paper is organized as follows. In Section \[s:2\], we present our main results. Section \[s:3\] establishes Fredholm properties of one-dimensional differential operators with periodic coefficients in suitable algebraically weighted spaces. Section \[s:4\] exploits these weighted spaces to treat impurities via an implicit function theorem and establishes expansions for solutions. We conclude with a discussion in Section \[s:5\]. #### Acknowledgment. The authors acknowledge partial support through the National Science Foundation through grants NSF-DMS-1311740 (AS) and NSF DMS-1503115 (GJ). Main Result {#s:2} =========== We state assumptions and main results. \[h:0\] We consider with smooth inhomogeneity $g(x,u)$ that is algebraically localized, $$\label{:gloc} |\partial^{j_1}_x\partial^{j_2}_ug(x,u)|{\leqslant}(1+|x|)^{-\gamma_*}, j_1+j_2{\leqslant}3,$$ where $\gamma_*>6$. We next assume the existence of a periodic pattern. \[h:1\] We assume that there exists an even, periodic solution ${u_\mathrm{p}}$ with wavenumber $k_*>0$, ${u_\mathrm{p}}(\xi;k_*)={u_\mathrm{p}}(\xi+2\pi;k_*)={u_\mathrm{p}}(-\xi;k_*)$, to $$\label{e:ssh} -(k_*^2\partial_\xi^2 +1)^2 u + \mu u - u^3=0,$$ for some $\mu>0$, fixed. Note that this assumption is satisfied for $0<\mu\ll 1$, $|k_*-1|\ll 1$. The next assumption requires in particular that ${u_\mathrm{p}}$ is Eckhaus-stable. In order to state this assumption precisely, we introduce the family of Bloch-wave operators $$\label{e:bl} L_\mathrm{B}(\sigma):=-\left(1+(\partial_x+{\mathrm{i}}\sigma)^2\right)^2+\mu-3{u_\mathrm{p}}^2(x), \quad \sigma\in [0,k_*),$$ defined on $\mathcal{D}(L_\mathrm{B}(\sigma))=H^4_\mathrm{per}(0,2\pi/k_*)\subset L^2_\mathrm{per}(0,2\pi/k_*)$. Note that all $L_\mathrm{B}(\sigma)$ have compact resolvent and depend analytically on $\sigma$ as closed operators with Fredholm index 0. \[h:2\] We assume that the periodic solution ${u_\mathrm{p}}$ is spectrally stable, that is, $0\in \mathrm{spec}(L_\mathrm{B}(\sigma))$ precisely for $\sigma=0$, when the eigenvalue $\lambda=0$ is algebraically simple, with eigenfunction ${u_\mathrm{p}}'$. For $\sigma \sim 0$, the expansion of the zero eigenvalue in $\sigma$ does not vanish at second order, $\lambda(\sigma)=\lambda_2\sigma^2+{\mathrm{O}}(\sigma^3)$, for some $\lambda_2\neq 0$. We note that for $\mu\ll 1$, Eckhaus-stable patterns satisfy this hypothesis with $\lambda_2<0$ [@mielke], and Eckhaus-unstable patterns do not, due to a kernel of $L_B(\sigma)$ for some $\sigma\neq 0$. On the other hand, long-wavelength unstable patterns may satisfy this assumption with $\lambda_2>0$; see for instance [@sslong]. We will give an expression for $\lambda_2$ in . \[l:family\] There exists a smooth family of stripe solutions, ${u_\mathrm{p}}(kx-\varphi;k)$, to , parameterized by wavenumber $k\sim k_*$ and phase $\varphi\in {\mathbb{R}}/2\pi{\mathbb{Z}}$. We solve $$-(1+k^2\partial_\xi^2)^2 u + \mu u - u^3=0,$$ as an equation $H^4_\mathrm{per,even}\to L^2_\mathrm{even}$ using the implicit function theorem near ${u_\mathrm{p}}(\xi;k_*)$. The assumption that the kernel of $L_\mathrm{B}(0)$ is simple, spanned by ${u_\mathrm{p}}'$, odd, guarantees invertibility of the linearization. Our main result is as follows. \[t:1\] Assume Hypotheses \[h:0\]–\[h:2\]. Then there exists $\varepsilon_0$ and a two-parameter family of stationary solutions to of the form $$u(x;\varepsilon)=\sum_\pm \chi_\pm(x){u_\mathrm{p}}((k_*+k_0\pm k_1)x-\varphi_0\mp\varphi_1; k_*+k_0\pm k_1)+w(x),$$ where $w\in H^4_{\gamma_*},$ $\gamma_*>6$, and $\varphi_1,k_1$ are $C^1$-functions of $\varepsilon,k_0\in (-\varepsilon_0,\varepsilon_0)$, $\varphi_0\in {\mathbb{R}}$. Moreover, $k_1$ and $\varphi_1$ have the leading-order expansions $$\begin{aligned} \label{e:k1} k_1&=M_k(\varphi_0,0)\varepsilon+{\mathrm{O}}(\varepsilon^2),\\ \varphi_1&=M_\varphi(\varphi_0,0)\varepsilon+{\mathrm{O}}(\varepsilon^2),\end{aligned}$$ where for the case $k_0=0$, $$\begin{aligned} M_k(\varphi_0,0)&=\frac{\pi \displaystyle \int_{\mathbb{R}}g(x, {u_\mathrm{p}}(k_*x-\varphi_0; k_*)) \cdot\partial_\xi {u_\mathrm{p}}(k_*x-\varphi_0; k_*)\,{\mathrm{d}}x }{\lambda_2 k_*\int_0^{2\pi/k_*} (\partial_\xi {u_\mathrm{p}}(k_*x; k_*))^2 {\mathrm{d}}x},\\ M_\varphi(\varphi_0,0)&=\frac{\pi \displaystyle \int_{\mathbb{R}}g(x, {u_\mathrm{p}}(k_*x-\varphi_0; k_*)) \cdot [(x-\varphi_0/k_*)\partial_\xi {u_\mathrm{p}}(k_*x-\varphi_0;k_*)+\partial_k{u_\mathrm{p}}(k_*x-\varphi_0; k_*)]\,{\mathrm{d}}x }{\lambda_2k_* \int_0^{2\pi/k_*} (\partial_\xi {u_\mathrm{p}}(k_*x; k_*))^2 {\mathrm{d}}x}.\end{aligned}$$ We note that when the inhomogeneity is a gradient field, i.e. $g = \partial_uG(x,u)$, then $$\dashint M_k\,{\mathrm{d}}\varphi_0:=\frac{1}{2\pi}\int_0^{2\pi}M_k(\varphi_0,0)\,{\mathrm{d}}\varphi_0=0,$$ and $M_k$ necessarily vanishes for certain relative phase shifts $\varphi_0$. We can therefore find relative phase shifts for which $k_1=0$. Assume that $g \in H^1_{\gamma_*} $, $\gamma_*>6$, $M_k(\varphi_*,0)=0$, and $M_k'(\varphi_*,0)\neq 0$. Then there exists $\bar{\varepsilon}, \bar{k}_0>0$ and a function $\phi_0(\varepsilon,k_0): [0,\bar{\varepsilon}]\times[0,\bar{k}_0]\to {\mathbb{R}}$ with $\phi_0(0,0)=\varphi_*$ such that the wavenumber difference $k_1$ from Theorem \[t:1\] vanishes for $\varphi_0=\phi_0(\varepsilon,k_0)$. Scaling the equation by $\varepsilon$ we may write $k_1 = \varepsilon \bar{k}$ where $$\bar{k}(\varepsilon; \varphi_0, k_0)= M_k(\varphi_0,k_0) + O(\varepsilon).$$ Our assumptions $M_k(\varphi_*,0)=0$, $M_k'(\varphi_*,0)\neq 0$ imply that $\bar{k}=0$ satisfies the conditions for the implicit function theorem, guaranteeing the results of the corollary. The conditions on $g$ allow us to obtain a well defined value for $M'_k(\varphi,0)$ . Fredholm properties in weighted spaces near the essential spectrum {#s:3} ================================================================== The results in this section can be viewed independently of the remainder of the paper. The difficulty of perturbing a striped pattern is due to the fact that the linearization is not Fredholm, which in turn can be attributed to the presence of essential spectrum at the origin, which in turn is induced by the non-localized eigenfunction ${u_\mathrm{p}}'$. It is well known that the linearization “behaves” in many ways like an effective diffusion. We therefore expect that the linearization at a periodic pattern possesses properties similar to the Laplacian $\partial_{xx}$. The Laplacian, on the other hand, while not Fredholm when posed as a closed, densely defined operator mapping $\mathcal{D}(\partial_{xx})\subset L^2\to L^2$, is Fredholm when posed as a closed, densely defined operator mapping $\mathcal{D}(\partial_{xx})\subset L^2_{\gamma-2}\to L^2_\gamma$, for $\gamma\not\in \{\frac{1}{2},\frac{3}{2}\}$. The goal of this section is to generally describe Fredholm properties of operators with translation symmetry in ${\mathbb{R}}$ or ${\mathbb{Z}}$ near points of the essential spectrum. The main restrictions are to one unbounded spatial direction, to “algebraically simple” points of the essential spectrum, and to non-critical weights $\gamma$. Throughout, we consider bounded operators, only. We will point out how these results imply Fredholm properties for more general operators. The outline for this section is as follows. We first consider operators with unbounded variable $x\in{\mathbb{R}}$ in Section \[s:3.1\], then show how to adapt in a straight-forward fashion to operators with unbounded direction $\ell\in{\mathbb{Z}}$ in Section \[s:3.2\]. We finally show how to relate those results to Floquet-Bloch theory for operators on $x\in{\mathbb{R}}$ with periodic coefficients. We establish Fredholm properties for those operators in Section \[s:3.3\]. For convenience, we recall Fredholm properties of $\partial_{xx}$ and of its discrete analogue in the appendix. Operators with continuous translation symmetry {#s:3.1} ---------------------------------------------- #### Setup — operator symbols and essential spectrum. We consider bounded operators $\mathcal{L}$ on $L^2({\mathbb{R}},Y)$, where $Y$ is a complex separable Hilbert space, that possess a translation symmetry, that is, they commute with the action of translations on $L^2({\mathbb{R}},Y)$. The Fourier transform is an isomorphism of $L^2({\mathbb{R}},Y)$, and, due to translation symmetry, the induced operator $\hat{\mathcal{L}}$ on the Fourier space is a direct integral of multiplication operators with Fourier symbol $\hat{\mathcal{L}}=\int_{k\in{\mathbb{R}}}L(k){\mathrm{d}}k$, that is, $$\begin{matrix} \hat{\mathcal{L}}: & \mathcal{D}(\hat{\mathcal{L}})\subset L^2({\mathbb{R}},Y) & \longrightarrow & L^2({\mathbb{R}},Y) \\ & u(k) & \longmapsto & L(k)u(k), \end{matrix}$$ with $L(k)$ linear and bounded on $Y$ for all $k\in{\mathbb{R}}$, see [@amann1]. Formally, we have $\mathcal{L}=L(-{\mathrm{i}}\partial_x)$. We denote the Banach space of bounded operators on $Y$ by $B(Y)$, \[h:3.1\] We assume that $L(k)$ is analytic, uniformly bounded, with values in $B(Y)$, in a strip $k\in \Omega_0:={\mathbb{R}}\times (-{\mathrm{i}}k_{\mathrm{i}},{\mathrm{i}}k_{\mathrm{i}})$ for some $k_{\mathrm{i}}>0$. Moreover, we require that $L(k)$ is Fredholm for all $k\in{\mathbb{R}}$ and invertible with uniform bounds for $|\Re k|{\geqslant}k_0>0$ for some $k_0$ sufficiently large. We mainly think of $L(k)$ rational, $L(k)=P(k)Q(k)^{-1}$, with matrix-valued polynomials $P$ and $Q$, where the values of $k$ such that $Q(k)$ is singular lie off the real axis. On the other hand, our results allow to include convolution operators with exponentially localized kernels. Specific examples are $\partial_{xx}(1-\partial_{xx})^{-1}$, $\partial_x(1+\partial_x)^{-1}$, $(-\mathrm{id}+K*)$, $K$ an exponentially localized kernel, or $(1+\partial_x^2)^2 (1-\partial_x^2)^{-2}$. Note that the spectrum of $\mathcal{L}$ is bounded, given through $$\mathrm{spec}_{L^2({\mathbb{R}},Y)}\mathcal{L}=\{\lambda \mid L(k)-\lambda \text{ not bounded invertible for some }k\in{\mathbb{R}}\}.$$ In the case $Y={\mathbb{R}}^n$, this can be more explicitly characterized through $$\mathrm{spec}_{L^2({\mathbb{R}},{\mathbb{R}}^n)}\mathcal{L}=\{\lambda\mid\mathrm{det}\,(L(k)-\lambda)=0\}.$$ Since $L(k)$ is invertible for large $k$ and Fredholm for all $k\in{\mathbb{R}}$, $L(k)$ is Fredholm of index $0$ for all $k\in{\mathbb{R}}$ and the set of $k\in{\mathbb{R}}$ where $L(k)$ is not invertible is discrete. We are interested in the case where $\mathcal{L}$ is not invertible. \[h:3.2\] There exists a unique $k_*$ and a unique (up to scalar multiples) $e_0\neq 0$ such that $L(k_*)e_0=0$. We then scale $\langle e_0, e_0 \rangle=1$. In particular, $\lambda=0$ belongs to the essential spectrum of $\mathcal{L}$. We can assume without loss of generality that $k_*=0$, possibly conjugating $\mathcal{L}$ with the multiplication operator ${\mathrm{e}}^{{\mathrm{i}}k_* x}$. We write $e_0^*$ for the kernel of the adjoint $L^*(0)$ with $\langle e_0^*, e_0^*\rangle=1$. #### Spatial multiplicities in the essential spectrum. We are interested in the unfolding of the zero-eigenvalue at $k=0$ for the family $L(k)$. We therefore view $L(k)$ as an analytic operator pencil and define the *spatial multiplicity* as the multiplicity of $k=0$ as an eigenvalue of the operator pencil. Since such constructions are possibly not widely known, and the use here is less standard, we include the relevant constructions. Recall that, according to Hypothesis \[h:3.2\], the kernel of $L(0)$ is one-dimensional. \[c:eigenexp\] There exists $m>0$, maximal, and $e(k)=\sum_{j=0}^{m}e_jk^j$ such that $$\label{e:LE2} L(k)e(k)=\lambda_mk^me_0^*+{\mathrm{O}}(k^{m+1}),$$ or, equivalently, $$\sum_{j=0}^k L_je_{k-j}=0,\quad k=0,\ldots,m-1; \qquad \lambda_m:=\left\langle \sum_{j=0}^{m-1} L_{m-j}e_j,e_0^*\right\rangle\neq 0.$$ Here, we expanded $L(k)=\sum_{j=0}^{m}L_j k^j+{\mathrm{O}}(k^{m+1})$. We refer to $m$ as the *spatial multiplicity* of $\lambda=0$. Write $Q_0$ for the orthogonal projection onto $\mathrm{span} \{e_0^*\}$. We solve $L(k)(e_0+v)=z$ by decomposing $$\begin{aligned} \langle L(k)(e_0+v),e_0^*\rangle&=z_1\label{e:gLS1}\\ ({\mathrm{\,id}\,}-Q_0)L(k)(e_0+v)&=z_2,\label{e:gLSeig}\end{aligned}$$ where $z=z_1e_0^*+z_2$, $z_1\in{\mathbb{R}}$ and $z_2\in{\mathrm{Rg}}({\mathrm{\,id}\,}-Q_0)$. Since $L(0)$ is Fredholm of index 0, $L(0):e_0^\perp\to (e_0^*)^\perp$ is an isomorphism, and the second equation can be solved using the implicit function theorem, with solution $v=v_*(k, z_2)$, where $|k|, |z_2|$ small. We then plug $v_*(k, z_2)$ into , yielding $$f(k, z_1, z_2):=\langle L(k)(e_0+v_*(k,z_2)),e_0^*\rangle-z_1=0.$$ Due to the fact that $L(k)$ is invertible for all $k\neq 0\in\Omega_0$, the reduced analytic function $f(k,0,0)$ has non-trivial Taylor jet, that is, there exists $m\in{\mathbb{Z}}^+$ and $\lambda_m\neq0\in{\mathbb{C}}$ so that $f(k,0,0)=\lambda_mk^m+{\mathrm{O}}(k^{m+1})$. Taking $v=v_*(k,0)$, we have $$L(k)(e_0+v_*(k,0))=f(k,0,0)e^*_0=\lambda_mk^m e^*_0+{\mathrm{O}}(k^{m+1}).$$ Letting $e(k)$ be the Taylor expansion up to order ${\mathrm{O}}(k^m)$ of $e_0+v_*(k,0)$, the claims follow quickly. \[r:mult\] In the case where $\lambda$ is an algebraically simple eigenvalue of $L(0)$, one can slightly modify the construction in the proof of Lemma \[c:eigenexp\] and solve $L(k)e(k)=\lambda(k)e(k)$ together with $\langle e(k)-e_0,e_0\rangle =0$ using Lyapunov-Schmidt reduction in much the same way. The linearization with respect to $(e,\lambda)$ is onto and one finds the function $\lambda(k)$ which is of course the expansion of the “temporal eigenvalue ” $\lambda$ in the Fourier parameter $k$. From this construction, one finds $\lambda(k)=\tilde{\lambda}_mk^m+{\mathrm{O}}(k^{m+1})$, for some $\tilde{\lambda}_m\neq 0$, with $m$ as in Lemma \[c:eigenexp\]. Since expansions typically do not converge globally, we introduce localized expansions as follows. Define the pseudo-derivative symbols $$\begin{aligned} D(k)&={\mathrm{i}}k(1+{\mathrm{i}}k)^{-1},\nonumber\\ D_{C,m}(k)&= k\left(1+C{\mathrm{i}}k^m\right)^{-1},\label{e:pd} $$ with associated operators $D(-{\mathrm{i}}\partial_x), D_{C,m}(-{\mathrm{i}}\partial_x)$. Here $C>0$ will eventually be chosen sufficiently large so that the norm of the bounded multiplier $D_{C,m}$ is arbitrarily small. Restricting to the strip $$\Omega_0(C,m):=\{k\in\Omega_0\mid |\Im k|{\leqslant}k_1:= \frac{1}{\sqrt[m]{2C}}\sin(\frac{\pi}{2m})\},$$ $D_{C,m}(k)$ is in fact analytic and uniformly bounded, that is, there exists a constant $C(m)$ such that $$\|D_{C,m}(k)\|{\leqslant}\frac{C(m)}{\sqrt[m]{C}},\quad \text{ for all }k\in\Omega_0(C,m).$$ On the enlarged strip, $\{k\in{\mathbb{C}}\mid |\Im k|< \frac{1}{\sqrt[m]{C}}\sin(\frac{\pi}{2m})\}$, the pseudo-derivative $D_{C,m}$ is analytic but not bounded. To obtain boundedness, we can restrict ourselves to any narrower strip, $\{k\in{\mathbb{C}}\mid |\Im k|< \frac{1}{\sqrt[m]{N C}}\sin(\frac{\pi}{2m})\}$, for any $N>1$. For convenience, we simply chose $N=2$ and $\Omega_0(C,m)\subset \Omega_0$, where the strip $\Omega_0$ is introduced in Hypothesis \[h:3.1\]. Note that replacing $k$ by $D_{C,m}(k)$ in the expansion of $e(k)$ does not alter its Taylor expansion up to order $m$. We therefore may define, for all $k\in\Omega_0(C,m)$, $$\tilde{e}(k):=\sum_{j=0}^{m} \left[D_{C,m}(k)\right]^je_j,$$ such that $$\label{e:expmod} L(k)\tilde{e}(k)= \lambda_me_0^*k^m+{\mathrm{O}}(k^{m+1}).$$ Repeating these considerations for the adjoint, we also find $e^*(k)=\sum_{j=0}^{m}e_j^*\bar{k}^j$ and define $$\tilde{e}^*(k):=\sum_{j=0}^{m} \left[\overline{D_{C,m}(k)}\right]^je_j^*,$$ so that $$\label{e:expmodad} L^*(k)\tilde{e}^*(k)= \bar{\lambda}_me_0 k^m+{\mathrm{O}}(k^{m+1}).$$ Since $L^*(k)$ is anti-analytic, $e^*(k)$ is anti-analytic, and we use the complex conjugate $\overline{D_{C,m}(k)}$ to guarantee that $\tilde{e}^*(k)$ is anti-analytic. #### Fredholm properties of $\mathcal{L}$. The main results on Fredholm properties of $\mathcal{L}$ are stated in the following theorem. \[p:f1\] Suppose the operator $\mathcal{L}$ satisfies Hypothesis \[h:3.1\] and \[h:3.2\], with $k^*=0$. Let $m$ be the spatial multiplicity according to Lemma \[c:eigenexp\]. Then, for $\gamma_-,\gamma_+ \not\in \{1/2,3/2, \cdots, m-1/2\}$, the operator $$\label{e:superL} \mathcal{L}:\mathcal{D}(\mathcal{L})\subset L^2_{\gamma_--m,\gamma_+-m}({\mathbb{R}},Y)\to L^2_{\gamma_-,\gamma_+}({\mathbb{R}},Y),$$ is closed, densely defined, and Fredholm. Moreover, setting $\gamma_{\max} = \max \{ \gamma_-, \gamma_+\} , \gamma_{\min} = \min\{ \gamma_-, \gamma_+\}$, we have that - for $\gamma_{\min} \in I_m:=( m-1/2,\infty)$, the operator is one-to-one with cokernel $${\mathrm{Cok}\,}(\mathcal{L}) ={\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta (-{\mathrm{i}})^\alpha (\partial_x^\alpha x^\beta)e_\alpha^* \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, m-1 \right \};$$ - for $\gamma_{\max} \in I_0:=(-\infty, 1/2)$, the operator is onto with kernel $${\mathrm{Ker}\,}(\mathcal{L})= {\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta (-{\mathrm{i}})^\alpha (\partial_x^\alpha x^\beta)e_\alpha \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, m-1 \right\};$$ - for $\gamma_{\min} \in I_i$ and $\gamma_{\max} \in I_j$ with $I_k:=(k-1/2, k+1/2)$ for $0< k\in {\mathbb{Z}}<m$, the kernel of is $${\mathrm{Ker}\,}(\mathcal{L})={\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta (-{\mathrm{i}})^\alpha (\partial_x^\alpha x^\beta)e_\alpha \hspace{2mm}\bigg|\hspace{2mm} \beta=0,1,\cdots,m-j-1 \right\};$$ and its cokernel is $${\mathrm{Cok}\,}(\mathcal{L})= {\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta (-{\mathrm{i}})^\alpha (\partial_x^\alpha x^\beta)e_\alpha^* \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, i-1 \right\}.$$ On the other hand, the operator does not have closed range for $\gamma_-,\gamma_+ \in \{ 1/2, 3/2, \cdots, m-1/2\}$. The proof of the proposition will occupy the remainder of this section. The key ingredient is the construction of a normal form representation of the operator $L$, through which we conclude that Fredholm properties of the operator $\mathcal{L}$ are equivalent to those of the regularized derivative $[D(-{\mathrm{i}}\partial_x)]^\ell$. We organize the proof by first establishing Fredholm properties of regularized derivatives defined in the Kondratiev spaces, then Fredholm properties of the normal form of the operator $L$, and eventually concluding the proof by returning to physical space. #### Fredholm properties of regularized derivatives. We employ regularized derivatives as model operators. More specifically, for any $\ell\in{\mathbb{Z}}^+$ and $\gamma_\pm\in{\mathbb{R}}$, we define the regularized derivative, $$\label{e:rell2} \begin{matrix} [D(-{\mathrm{i}}\partial_x)]^\ell: & \mathcal{D}([D(-{\mathrm{i}}\partial_x)]^\ell)\subset L^2_{\gamma_--\ell, \gamma_+-\ell} & \longrightarrow & L^2_{\gamma_-, \gamma_+} \\ & u & \longmapsto & \partial_x^\ell ( 1 + \partial_x)^{-\ell}u, \end{matrix}$$ with its domain $\mathcal{D}([D(-{\mathrm{i}}\partial_x)]^\ell)=\{u\in L^2_{\gamma_--\ell, \gamma_+-\ell}\mid (1+\partial_x)^{-\ell}u\in M^{\ell,2}_{\gamma_--\ell, \gamma_+-\ell}\}$. Moreover, the Fredholm properties of the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ are summarized in the following proposition. \[p:regdrv2\] For $\gamma_\pm\in \mathbb{R} \setminus \{ 1/2, 3/2, \cdots, \ell-1/2\}$, the regularized derivative $[D(-{\mathrm{i}}\partial_x)]^\ell$ as defined in is Fredholm. Moreover, the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ satisfies the following conditions. - If $\gamma_{max}\in I_0:=(-\infty,1/2)$, the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ is onto with its kernel equal to $\mathbb{P}_{\ell}({\mathbb{R}})$. - If $\gamma_{min}\in I_\ell:=(\ell-1/2, \infty)$, the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ is one-to-one with its cokernel equal to $\mathbb{P}_{\ell}({\mathbb{R}})$. - If $\gamma_{min}\in I_i$ and $ \gamma_{max}\in I_j$ with $I_k:=(k-1/2,k+1/2)$ for $0<k\in{\mathbb{Z}}<\ell$, the kernel and cokernel of the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ are respectively spanned by $\mathbb{P}_{\ell-j}({\mathbb{R}})$ and $\mathbb{P}_{i}({\mathbb{R}})$. On the other hand, the range of the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ is not closed if $\gamma_-, \gamma_+\in\{1/2, 3/2,...,\ell-1/2\}$. The proof is relegated to Appendix \[ss:a1\], where we prove a more general result. #### Normal form operators. We diagonalize every operator $L(k)$ defined in $Y$ into the direct sum of the Fourier counterpart of a regularized derivative and an isomorphism. To start with, recalling the definitions of the modified kernel and cokernel expansions and , for any $k\in\Omega_0(C,m)$, we define the projections, $$P(k)u:=\langle u, e_0\rangle \tilde{e}(k), \qquad Q(k)v:=\langle v, \tilde{e}^*(k)\rangle e_0^*,$$ from which it is straightforward to conclude the following lemma. \[l:pro\] There exists $C_0>0$ so that, for any $C>C_0$ and $k\in\Omega_0(C,m)$, the operators $${\mathrm{\,id}\,}- P(k): \langle \tilde{e}(k) \rangle^\perp\rightarrow \langle e_0 \rangle^\perp, \quad {\mathrm{\,id}\,}- Q(k): \langle e_0^* \rangle^\perp\rightarrow \langle \tilde{e}^*(k)\rangle^\perp$$ are isomorphisms whose inverses take the form, $$\begin{matrix} ({\mathrm{\,id}\,}-P(k))^{-1}: & \langle e_0 \rangle^\perp & \longrightarrow & \langle \tilde{e}(k) \rangle^\perp \\ & u & \longmapsto & u-\frac{\langle u, \tilde{e}(k) \rangle}{\langle \tilde{e}(k), \tilde{e}(k) \rangle} \tilde{e}(k), \end{matrix} \qquad \begin{matrix} ({\mathrm{\,id}\,}-Q(k))^{-1}: & \langle \tilde{e}^*(k)\rangle^\perp & \longrightarrow & \langle e_0^* \rangle^\perp \\ & u & \longmapsto & u-\langle u, e_0^* \rangle e_0^*. \end{matrix}$$ Moreover, for fixed $C>C_0$, both operators and their inverses admit uniform bounds for $k\in\Omega_0(C,m)$. We also introduce analytic isomorphisms $\iota(k):\langle \tilde{e}(k)\rangle \to \langle e_0^*\rangle$ and $\iota_\perp(k): \langle e_0\rangle^\perp \to \langle \tilde{e}^*(k)\rangle^\perp$. Such isomorphisms can be constructed in many ways and we outline one construction here. Define $$\label{e:pro} \begin{matrix} \iota(k):&\langle \tilde{e}(k)\rangle & \longrightarrow & \langle e_0^*\rangle \\ &\alpha \tilde{e}(k) & \longmapsto & \alpha e_0^*, \end{matrix} \hspace{2cm} \begin{matrix} \iota_\perp(k): &\langle e_0\rangle^\perp & \longrightarrow & \langle \tilde{e}^*(k)\rangle^\perp\\ &u & \longmapsto & ({\mathrm{\,id}\,}-Q(k))\iota_\perp(0)u, \end{matrix}$$ where we define the isomorphism $\iota_\perp(0): \langle e_0\rangle^\perp \to \langle e_0^*\rangle^\perp$ to be a direct sum of the identity map on $\langle e_0\rangle^\perp\cap \langle e_0^* \rangle^\perp$ and a linear length-preserving map from $E_{0,\perp}:={\mathrm{\,span}\,}\{e_0^*-\langle e_0^*, e_0 \rangle e_0\}$ to $E_{0,\perp}^*:={\mathrm{\,span}\,}\{e_0-\langle e_0, e_0^* \rangle e_0^*\}$. More specifically, we choose $$\iota_\perp(0)u:=\begin{cases} u, & u\in \langle e_0\rangle^\perp\cap \langle e_0^* \rangle^\perp,\\ c(e_0-\langle e_0, e_0^* \rangle e_0^*), &u=c(e_0^*-\langle e_0^*, e_0 \rangle e_0)\in E_{0,\perp}. \end{cases}$$ We are now ready to define the normal form operators, $$\begin{matrix} L_{\rm NF}(k) :&\mathcal{D}(L_{\rm NF}(k))\subset Y &\longrightarrow &Y \\ & u & \longmapsto & D^m(k)\iota(k)P(k)u+\iota_\perp(k)({\mathrm{\,id}\,}- P(k))u, \end{matrix}$$ and prove the following lemma. \[l:fact\] For fixed $C>C_0$ and any $k\in\Omega_0(C,m)$, the operator $L(k)$ admits the decomposition, $$L(k)=M_\mathrm{L}(k)L_\mathrm{NF}(k)=L_\mathrm{NF}(k)M_\mathrm{R}(k),$$ where $M_{\rm L\backslash R}:\Omega_0(C, m)\to B(Y)$ are analytic, $L^\infty$-bounded with an $L^\infty$-bounded inverse. For $k\neq 0$, the inverse of $L_{\rm NF}(k)$ is analytic and takes the form, $$L_{\rm NF}^{-1}(k)u=D^{-m}(k)\iota^{-1}(k)Q(k)u+\iota_\perp^{-1}(k)({\mathrm{\,id}\,}-Q(k))u= D^{-m}(k)\langle u, \tilde{e}^*(k) \rangle \tilde{e}(k)+\iota_\perp^{-1}(0)\left( u- \langle u, e_0^*\rangle e_0^*\right).$$ In addition, we have that, based on , $$\begin{aligned} \lim_{k\rightarrow 0}L(k)L_{\rm NF}^{-1}(k)u=& \lim_{k\rightarrow 0}\left[ \frac{(1+{\mathrm{i}}k)^m}{k^m}\langle u, \tilde{e}^*(k)\rangle L(k) \tilde{e}(k)+L(k)\iota^{-1}_\perp(0)\left( u-\langle u, e_0^*\rangle e_0^*\right)\right]\\ =&\lambda_m\langle u, e_0^*\rangle e_0^*+L(0)\iota^{-1}_\perp(0)\left( u-\langle u, e_0^*\rangle e_0^*\right), \end{aligned}$$ is an invertible bounded operator. We now define $$\label{e:ldef} M_{\rm L}(k)u:= \begin{cases} L(k)L_{\rm NF}^{-1}(k)u, & k\neq 0,\\ \lim_{k\rightarrow 0}L(k)L_{\rm NF}^{-1}(k)u, & k=0, \end{cases}$$ which, according to Riemann’s removable singularity theorem and Hypothesis \[h:3.2\], implies $M_{\rm L}(k)$ is analytic and invertible for all $k$ in the strip $\Omega_0$. Furthermore, noting that, according to Hypothesis \[h:3.1\], $L(k)$ is invertible with uniform bounds for $k\in\Omega_0(C,m)$ with $|\Re k|>k_0$ and $$\lim_{\Re k\rightarrow \infty}L_{\rm NF}^{-1}(k)=\langle u, e_0^*\rangle e_0 + \iota_\perp^{-1}(0)\left( u- \langle u, e_0^* \rangle e_0^*\right),$$ is bounded and invertible, we conclude that $M_{\rm L}(k)$ is uniformly bounded with uniformly bounded inverses. We can define and analyze $M_{\rm R}(k)$ in a completely analogous fashion. #### Back to physical space — proof of Proposition \[p:f1\]. We introduce the multiplier operators $$\label{e:multiplier} \begin{matrix} \mathcal{M}_{{\rm L}\backslash {\rm R}}: & \mathcal{S}({\mathbb{R}}, Y) & \longrightarrow & \mathcal{S}({\mathbb{R}}, Y) \\[1mm] & u(x) & \longmapsto & \widecheck{M_{{\rm L}\backslash {\rm R}}\hat{u}}(x). \end{matrix}$$ which, according to the $L^\infty$-boundedness and invertibility of $\partial_k^\alpha M_{{\rm L}}$ and $\partial_k^\alpha M_{ {\rm R}}$ for all $\alpha\in {\mathbb{Z}}^+\cup \{0\}$, are isomorphisms on the Schwartz space $\mathcal{S}({\mathbb{R}}, Y)$. For any given $\gamma_\pm \in{\mathbb{R}}$, it is straightforward to see that $\mathcal{S}({\mathbb{R}}, Y) \subset L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)$ is a continuous embedding. We claim that we can continuously extend the multiplier operators $\mathcal{M}_{{\rm L}\backslash {\rm R}}$ onto $L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)$. In other words, we have the following lemma. \[l:L2iso\] For any given $\gamma_\pm \in{\mathbb{R}}$, the multiplier operators $\mathcal{M}_{{\rm L}\backslash {\rm R}}: L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)\to L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)$ are isomorphisms. We suspect that results analogous to Lemma \[l:L2iso\] hold for general anisotropic weighted spaces $L^p_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)$ with $p\in(1,\infty)$. It appears however that necessary-and-sufficient condition for Fourier multipliers on $L^p_{\gamma_-,\gamma_+}({\mathbb{R}}, {\mathbb{C}})$ with general $p\in(1,\infty)$ are not available, only sufficient conditions such as the Marcinkiewicz and the Hörmander-Mikhlin multiplier theorems, which both can be generalized to certain families of weighted $L^p({\mathbb{R}},{\mathbb{C}})$ spaces; see [@muckenhoupt; @fefferman1974; @kurtz] for details and [@amann1; @girardiweis; @lutz; @bu] for general background on operator-valued Fourier multipliers. We first prove the case of isotropic weights, that is, $\gamma_-=\gamma_+=\gamma$. For $\gamma\in{\mathbb{Z}}_+\cup \{0\}$, we adopt the notation $L^2_{\gamma}({\mathbb{R}}, Y):=L^2_{\gamma,\gamma}({\mathbb{R}}, Y)$ and exploit the Plancherel theorem to derive that $$\|\mathcal{M}_{{\rm L}\backslash {\rm R}} u\|_{L^2_{\gamma}({\mathbb{R}}, Y)}=\|M_{{\rm L}\backslash {\rm R}}\hat{u}\|_{H^\gamma({\mathbb{R}}, Y)} {\leqslant}C(\gamma)\|\hat{u}\|_{H^\gamma({\mathbb{R}}, Y)}=C(\gamma)\|u\|_{L^2_\gamma({\mathbb{R}}, Y)},$$ which, together with a similar inequality for $\mathcal{M}_{{\rm L}\backslash {\rm R}}^{-1}$, shows that $\mathcal{M}_{{\rm L}\backslash {\rm R}}: L^2_{\gamma}({\mathbb{R}}, Y)\to L^2_{\gamma}({\mathbb{R}}, Y)$ are isomorphisms for $\gamma\in{\mathbb{Z}}_+\cup \{0\}$ and thus for $\gamma\in {\mathbb{Z}}_-$ due to duality. By classical interpolation results, see, for example, Theorem 6.4.5 in [@bergh], $H^{n+\theta}({\mathbb{R}}, Y)$ is a complex interpolation space between $H^{n}({\mathbb{R}}, Y)$ and $H^{n+1}({\mathbb{R}}, Y)$ for any given $n\in{\mathbb{Z}}$ and $\theta\in(0,1)$. Therefore, we conclude that $\mathcal{M}_{{\rm L}\backslash {\rm R}}: L^2_{\gamma}({\mathbb{R}}, Y)\to L^2_{\gamma}({\mathbb{R}}, Y)$ are isomorphisms for $\gamma\in{\mathbb{R}}$. To prove the case of anisotropic weights, we start by introducing the exponentially weighted space $$L^2_{\rm exp, \eta}({\mathbb{R}}, Y):=\left\{u \in L^1_{\rm loc}({\mathbb{R}}, Y)\middle| {\mathrm{e}}^{\eta \cdot}u(\cdot)\in L^2({\mathbb{R}}, Y)\right\},$$ with its norm $\|u\|_{L^2_{\rm exp, \eta}({\mathbb{R}}, Y)}:=\|{\mathrm{e}}^{\eta \cdot}u(\cdot)\|_{L^2({\mathbb{R}}, Y)}$ for any given $\eta\in{\mathbb{R}}$. Our strategy is to exploit the fact that the space $L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)$ admits the decomposition, $$\label{e:intadd} L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)=\left(L^2_{\gamma_-}({\mathbb{R}}, Y)\cap L^2_{\rm exp, \eta}({\mathbb{R}}, Y)\right) + \left(L^2_{\gamma_+}({\mathbb{R}}, Y)\cap L^2_{\rm exp, -\eta}({\mathbb{R}}, Y)\right),$$ for any $\eta>0$, where norms on intersections and sums are defined in the usual way; see below. With this in mind, we first study the multipliers on $\mathcal{M}_{{\rm L}\backslash {\rm R}}: L^2_{\rm exp, \eta}({\mathbb{R}}, Y)\to L^2_{\rm exp, \eta}({\mathbb{R}}, Y)$ and claim that they are isomorphisms, for any fixed $|\eta|{\leqslant}k_1$, where $k_1$ is half of the width of the strip $\Omega_0(C,m)$. Note that the multiplier on the Schwartz space can be viewed as a convolution operator. More specifically, denoting the reflection $(\mathcal{R}u)(x):=u(-x)$, we define the distribution $$\begin{matrix} \check{M}_{\rm L\backslash R}: & \mathcal{S}({\mathbb{R}}, Y) & \longrightarrow & {\mathbb{C}}\\ & u & \longmapsto & (\mathcal{M}_{\rm L\backslash R}\mathcal{R}u)(0), \end{matrix}$$ from which we readily derive that, for all $u\in\mathcal{S}({\mathbb{R}}, Y)$, $$(\mathcal{M}_{\rm L\backslash R}u)(x)=(\check{M}_{\rm L\backslash R}\ast u)(x)=\int_{\mathbb{R}}\check{M}_{\rm L\backslash R}(x-y) u(y){\mathrm{d}}y.$$ Since the Fourier transform is given through $\mathcal{F}({\mathrm{e}}^{\eta\cdot}\check{M}_{\rm L\backslash R}(\cdot))(k)=M_{\rm L\backslash R}(k+{\mathrm{i}}\eta)$ for $|\eta|{\leqslant}k_1$, we have that the inequality $$\begin{aligned} \|\mathcal{M}_{\rm L\backslash R}u\|_{L^2_{\rm exp, \eta}({\mathbb{R}}, Y)}= &\|\int_{\mathbb{R}}\big[{\mathrm{e}}^{\eta(x-y)}\check{M}_{\rm L\backslash R}(x-y) \big]\big[{\mathrm{e}}^{\eta y}u(y)\big]{\mathrm{d}}y \|_{L^2({\mathbb{R}}, Y)} \\ =& \|\mathcal{F}\big({\mathrm{e}}^{\eta\cdot}\check{M}_{\rm L\backslash R}(\cdot)\big)\mathcal{F}({\mathrm{e}}^{\eta\cdot}u(\cdot)) \|_{L^2({\mathbb{R}}, Y)}\\ =& \|M_{\rm L\backslash R}(\cdot+{\mathrm{i}}\eta)\mathcal{F}({\mathrm{e}}^{\eta\cdot}u(\cdot)) \|_{L^2({\mathbb{R}}, Y)}\\ {\leqslant}& \|M_{\rm L\backslash R}(\cdot+{\mathrm{i}}\eta)\|_{L^\infty({\mathbb{R}}, B(Y))} \|\mathcal{F}({\mathrm{e}}^{\eta\cdot}u(\cdot)) \|_{L^2({\mathbb{R}}, Y)}\\ {\leqslant}& C \|u \|_{L^2_{\rm exp, \eta}({\mathbb{R}}, Y)}, \end{aligned}$$ holds for any $|\eta|{\leqslant}k_1$ and $u\in \mathcal{S}({\mathbb{R}}, Y)$. Noting that $\mathcal{S}({\mathbb{R}}, Y)\subset L^2_{\rm exp, \eta}({\mathbb{R}}, Y)$ is dense, there are natural extensions of $\mathcal{M}_{\rm L\backslash R}$ as a bounded linear operator on $L^2_{\rm exp, \eta}({\mathbb{R}}, Y)$. Analogous reasoning applied to the inverses of $\mathcal{M}_{\rm L\backslash R}$ lets us conclude that the multipliers $\mathcal{M}_{{\rm L}\backslash {\rm R}}: L^2_{\rm exp, \eta}({\mathbb{R}}, Y)\to L^2_{\rm exp, \eta}({\mathbb{R}}, Y)$ are isomorphisms for any fixed $|\eta|{\leqslant}k_1$. We are now ready to prove the case of anisotropic weights. Given two Banach spaces $E$ and $F$, the linear space $E\cap F$ and $E + F$ are also Banach spaces respectively with norms $$\|u\|_{E\cap F}:= \|u\|_E + \| v \|_F, \quad \| u \|_{E + F} := \inf\{\|v\|_E+\|w\|_F\mid v+w=u, v\in E, w\in F \}.$$ Moreover, for a linear operator $L$ bounded on both $E$ and $F$, it is straightforward to check that $L$ is also bounded on $E\cap F$ and $E+F$. Therefore, given $\gamma\pm\in{\mathbb{R}}$ and $\eta\in[0,k_1]$, due to the fact that $\mathcal{M}_{\rm L\backslash R}$ are isomorphisms on $L^2_{\gamma\pm}$ and $L^2_{\rm exp, \pm\eta}$, we conclude that $\mathcal{M}_{\rm L\backslash R}$ are isomorphisms on the Banach space $$\label{e:bintadd} B(\gamma_-,\gamma_+, \eta, Y):=\left(L^2_{\gamma_-}({\mathbb{R}}, Y)\cap L^2_{\rm exp, \eta}({\mathbb{R}}, Y)\right) + \left(L^2_{\gamma_+}({\mathbb{R}}, Y)\cap L^2_{\rm exp, -\eta}({\mathbb{R}}, Y)\right).$$ Es defined in , the Banach spaces $L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)$ and $B(\gamma_-,\gamma_+, \eta, Y)$ constitute the same linear space. It is therefore sufficient to show that the natural norm on $L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)$ is equivalent to the norm on $B(\gamma_-,\gamma_+, \eta, Y)$ induced by the intersection and sum property. For any $u\in L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)$, we have $$u=\chi_+u+\chi_-u, \quad \chi_\pm u\in L^2_{\gamma_\pm}({\mathbb{R}}, Y)\cap L^2_{\rm exp, \mp\eta}({\mathbb{R}}, Y),$$ and $$\begin{aligned} \|u\|_{B(\gamma_-,\gamma_+, \eta, Y)}{\leqslant}& \| \chi_+u \|_{L^2_{\gamma_+}({\mathbb{R}}, Y)\cap L^2_{\rm exp, -\eta}({\mathbb{R}}, Y)}+ \| \chi_-u \|_{L^2_{\gamma_-}({\mathbb{R}}, Y)\cap L^2_{\rm exp, \eta}({\mathbb{R}}, Y)}\\ =& \| \chi_+u \|_{L^2_{\gamma_+}({\mathbb{R}}, Y)}+\| \chi_+u \|_{L^2_{\rm exp, -\eta}({\mathbb{R}}, Y)}+ \| \chi_-u \|_{L^2_{\gamma_-}({\mathbb{R}}, Y)}+\| \chi_-u \|_{ L^2_{\rm exp, \eta}({\mathbb{R}}, Y)}\\ {\leqslant}&C(\gamma\pm,\eta)\big[ \| \chi_+u \|_{L^2_{\gamma_+}({\mathbb{R}}, Y)}+ \| \chi_-u \|_{L^2_{\gamma_-}({\mathbb{R}}, Y)}\big]\\ =&C(\gamma\pm,\eta)\|u\|_{L^2_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)}, \end{aligned}$$ which implies that the two norms are equivalent, concluding the proof. Denoting the inverse Fourier transform of $L_{\rm NF}$ as $\mathcal{L}_{\rm NF}$, we have $$\mathcal{L}=\mathcal{M}_{\rm L}\mathcal{L}_{\rm NF}, \quad \mathcal{L}^{\rm ad}=\mathcal{M}_{\rm R}^{\rm ad}\mathcal{L}_{\rm NF}^{\rm ad}.$$ The proof of Proposition \[p:f1\] now reduces to establishing Fredholm properties of $\mathcal{L}_{\rm NF}$. Noting that $Y\cong \langle \tilde{e}(k) \rangle \oplus \langle e_0\rangle^{\perp} \cong\langle e^*_0 \rangle \oplus \langle \tilde{e}^*(k) \rangle^{\perp}$, the normal form operator $L_{\rm NF}(k)$ admits an isomorphic diagonal form, $$\begin{matrix} L_{\rm D}(k):& \langle \tilde{e}(k) \rangle \oplus \langle e_0\rangle^{\perp} &\longrightarrow &\langle e^*_0 \rangle \oplus \langle \tilde{e}^*(k) \rangle^{\perp} \\ & \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} & \longmapsto & \begin{pmatrix} D^m(k) \iota(k)& 0 \\ 0 & \iota_{\perp}(k) \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \end{matrix}.$$ According to Lemma \[l:pro\]-\[l:fact\] and definition of projections $\iota(k)$ and $\iota^\perp(k)$, we derive that $$\begin{matrix} \mathcal{L}_{\rm NF}: & \mathcal{D}(\mathcal{L}_{\rm NF})\subset L^p_{\gamma_--m, \gamma_+-m}({\mathbb{R}}, Y) & \longrightarrow & L^p_{\gamma_-, \gamma_+}(R, Y)\\ & u &\longmapsto & \langle D^m(-{\mathrm{i}}\partial_x)u, e_0 \rangle e_0^* +\check{\iota}_\perp ( u-\sum_{j=0}^m\langle D_{C,m}^j(-{\mathrm{i}}\partial_x)u, e_0 \rangle e_j), \end{matrix}$$ where $u(x)-\sum_{j=0}^m\langle D_{C,m}^j(-{\mathrm{i}}\partial_x)u(x), e_0 \rangle e_j\in \langle e_0 \rangle^\perp$ for all $x\in{\mathbb{R}}$ and the mapping $$\begin{matrix} \check{\iota}_\perp: & L^p_{\gamma_-,\gamma_+}({\mathbb{R}}, \langle e_0\rangle^\perp) & \longrightarrow & \{u\in L^p_{\gamma_-,\gamma_+}({\mathbb{R}}, Y)\mid \sum_{j=0}^m\langle D_{C,m}^j(-{\mathrm{i}}\partial_x)u(x), e_i^*\rangle=0, \text{ for all }x\in{\mathbb{R}}\}\\[2mm] & v & \longmapsto &\iota_\perp(0)v-\sum_{j=0}^m\langle D_{C,m}^j(-{\mathrm{i}}\partial_x)[\iota_\perp(0)v], e_i^*\rangle e_0^*, \end{matrix}$$ is an isomorphism. As a result, Fredholm properties of $\mathcal{L}_{\rm NF}$ are encoded in the regularized derivative operator $[D(-{\mathrm{i}}\partial_x)]^m$. More specifically, we note that $$\mathcal{F}^{-1}\left[ D^m(k)\iota(k)\Big(\hat{u}(k) \tilde{e}(k)\Big) \right] =\Big( [D(-{\mathrm{i}}\partial_x)]^m u(x)\Big) e_0^*, \quad \mathcal{F}^{-1}\Big(\hat{u}(k) \tilde{e}(k)\Big)=\sum_{j =0}^m \Big(\left[D_{C,m} (-{\mathrm{i}}\partial_x)\right]^j u(x) \Big)e_j,$$ which implies that the kernel and cokernel of $\mathcal{L}_{\rm NF}$ are given respectively by $$\begin{aligned} &{\mathrm{Ker}\,}(\mathcal{L}_{\rm NF}) = \left \{ \sum_{j =0}^m \Big(\left[D_{C,m} (-{\mathrm{i}}\partial_x)\right]^j u(x) \Big)e_j \hspace{2mm}\bigg|\hspace{2mm} u(x) \in {\mathrm{Ker}\,}\Big( [D(-{\mathrm{i}}\partial_x)]^m \Big) \right\},\\ &{\mathrm{Cok}\,}(\mathcal{L}_{\rm NF}) = \left \{ \sum_{j =0}^m \Big(\left[\overline{D_{C,m} ({\mathrm{i}}\partial_x)}\right]^j u(x) \Big)e_j^* \hspace{2mm}\bigg|\hspace{2mm} u(x) \in {\mathrm{Cok}\,}\Big( [D(-{\mathrm{i}}\partial_x)]^m \Big) \right\} . \end{aligned}$$ The statements in Proposition \[p:f1\] then follow by applying the statement of Proposition \[p:regdrv\] to the above analysis and noting that, for any $u\in\mathbb{P}_m({\mathbb{R}})$, $$\left[D_{C,m} (-{\mathrm{i}}\partial_x)\right]^j u(x)=(-{\mathrm{i}})^\alpha\partial_x^\alpha u(x).$$ Operators with discrete translation symmetry {#s:3.2} -------------------------------------------- The results from Section \[s:3.1\] can be easily adapted to the case of an operator $\mathcal{L}$ on $\ell^2({\mathbb{Z}}, Y)$, that commutes with the discrete translation group ${\mathbb{Z}}$. The discrete Fourier transform takes the form $$\label{e:disft} \begin{matrix} \mathcal{F}_d: & \ell^2({\mathbb{Z}}, Y) & \longrightarrow &L^2(\mathcal{T}_1, Y)\\ &\underline{u}= \{u_j\}_{j\in{\mathbb{Z}}}& \longmapsto & \hat{u}(\sigma)=\sum_{j\in{\mathbb{Z}}}u_j{\mathrm{e}}^{-2\pi{\mathrm{i}}j\sigma}, \end{matrix}$$ where $\mathcal{T}_1:={\mathbb{R}}/{\mathbb{Z}}$ denotes the unit circle. The counterparts of the derivative $\partial_x$ are the discrete derivatives, $$\delta_+ (\{a_j\}_{j\in{\mathbb{Z}}}) := \{a_{j+1}-a_{j}\}_{j\in{\mathbb{Z}}},\hspace{0.6cm} \delta_- (\{a_j\}_{j\in{\mathbb{Z}}}) := \{a_j-a_{j-1}\}_{j\in{\mathbb{Z}}}, \hspace{0.6cm} \delta:=-{\mathrm{i}}(\delta_++\delta_-)/2.$$ The Fourier transform of $\mathcal{L}$, denoted as $\hat{\mathcal{L}}=\int_{\mathcal{T}_1}L(\sigma){\mathrm{d}}\sigma$, is an isomorphism of $L^2(\mathcal{T}_1,Y)$, that is, $$\begin{matrix} \hat{\mathcal{L}}: & \mathcal{D}(\hat{\mathcal{L}})\subset L^2(\mathcal{T}_1,Y) & \longrightarrow & L^2(\mathcal{T}_1,Y) \\ & u(\sigma) & \longmapsto & L(\sigma)u(\sigma), \end{matrix}$$ with $L(\sigma)$ linear and bounded on $Y$ for all $\sigma\in\mathcal{T}_1$. \[h:3.3\] We assume that $L(\sigma)$ is analytic, uniformly bounded, $1$-periodic, with values in the set of bounded operators on $Y$, in a strip $\sigma\in \Omega_1:={\mathbb{R}}\times (-{\mathrm{i}}\sigma_{\mathrm{i}},{\mathrm{i}}\sigma_{\mathrm{i}})$ for some $\sigma_{\mathrm{i}}>0$. Moreover, we require that $L(\sigma)$, restricted to $\sigma\in[-1/2, 1/2]$, is invertible except at $\sigma=0$ and $L(0)$ admits a simple kernel spanned by $e_0$ with $\langle e_0, e_0 \rangle=1$. For convenience, we identify the interval $[-1/2, 1/2]$ with the unit circle $\mathcal{T}_1$, collapsing endpoints $-1/2\sim 1/2$. We adopt all the notations in the continuous case, except for those related to pseudo-derivative symbols. The new pseudo-derivatives take the following forms, $$\label{e:dispd} D_+(\sigma)={\mathrm{e}}^{2\pi{\mathrm{i}}\sigma}-1, \qquad D_-(\sigma)=1-{\mathrm{e}}^{-2\pi{\mathrm{i}}\sigma}, \qquad D_{C,m}(\sigma)=({\mathrm{e}}^{2\pi{\mathrm{i}}\sigma}-1)\left[1+{\mathrm{i}}C\sin^m(2\pi\sigma)\right]^{-1},$$ whose associated physical operator are respectively $\delta_+$, $\delta_-$ and $\delta_+\left[1+{\mathrm{i}}C\delta^m\right]^{-1}$. Here $m\in{\mathbb{Z}}^+$ is the power related to the expansion of the zero eigenvalue, $\lambda(\sigma)=\lambda_m \sigma^m+O(\sigma^{m+1})$, with $\lambda_m\neq0$ for $\sigma\sim 0\in{\mathbb{C}}$. The constant $C>0$ will eventually be chosen sufficiently large so that the norm of the bounded multiplier $D_{C,m}$ is arbitrarily small. As a matter of fact, in the strip $$\Omega_1(C,m):=\left\{\sigma\in\Omega_1\middle| {|\Re \sigma|{\leqslant}1/2,} |\Im \sigma|< \frac{1}{2\pi}\sinh^{-1}\left(\frac{1}{\sqrt[m]{2C}}\sin(\frac{\pi}{2m})\right)\right\},$$ $D_{C,m}(\sigma)$ is analytic and uniformly bounded, that is, there exists a constant C(m) so that $$\|D_{C,m}(\sigma)\|{\leqslant}\frac{C(m)}{\sqrt[m]{C}}, \text{ for all }\sigma\in\Omega_1(C,m).$$ Moreover, we define $e(\sigma)=\sum_{j=0}^{m} e_j\sigma^j$ and $e^*(\sigma)=\sum_{j=0}^{m} e_j^*\bar{\sigma}^j$ so that $$L(\sigma)e(\sigma)={\mathrm{O}}(\sigma^m),\quad L^*(\sigma)e^*(\sigma)={\mathrm{O}}(\sigma^m), \quad \left\langle \sum_{j=0}^{m-1} L_{m-j}e_j,e_0^*\right\rangle\neq0, \quad \sum_{j=0}^k L_je_{k-j}=0,\quad k=0,\ldots,m-1.$$ There exist $\{\tilde{e}_j, \tilde{e}_j^*\}_{j=0}^m\subset Y$, independent of $C$, and $$\tilde{e}(\sigma):=\sum_{j=0}^{m} \left[D_{C,m}(\sigma)\right]^j\tilde{e}_j, \qquad \tilde{e}^*(\sigma):=\sum_{j=0}^{m} \left[\overline{D_{C,m}(\sigma)}\right]^j\tilde{e}_j^*, \qquad \sigma\in\Omega_1(C,m).$$ so that $L(\sigma)\tilde{e}(\sigma)={\mathrm{O}}(\sigma^{m})$ and $L^*(\sigma)\tilde{e}^*(\sigma)={\mathrm{O}}(\sigma^{m})$. \[p:f2\] For $\gamma_\pm \not\in \{1/2,3/2, \cdots, m-1/2\}$, the operator satisfying Hypothesis \[h:3.3\], $$\label{e:dissuperL} \mathcal{L}:\mathcal{D}(\mathcal{L})\subset \ell^2_{\gamma_--m,\gamma_+-m}({\mathbb{Z}},Y)\to \ell^2_{\gamma_-,\gamma_+}({\mathbb{Z}},Y),$$ is closed, densely defined, and Fredholm. Letting $\gamma_{\max} = \max \{ \gamma_-, \gamma_+\} , \gamma_{\min} = \min\{ \gamma_-, \gamma_+\}$ and $\underline{\eta}^\beta:=\{\eta^\beta\}_{\eta\in{\mathbb{Z}}}$, we have that - for $\gamma_{\min} \in I_m:=(m-1/2, \infty)$, the operator is one-to-one with cokernel $${\mathrm{Cok}\,}={\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta (\delta_+^\alpha \underline{\eta}^\beta)\tilde{e}_\alpha^* \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, m-1 \right\};$$ - for $\gamma_{\max} \in I_0:=(-\infty, 1/2)$, the operator is onto with kernel $${\mathrm{Ker}\,}={\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta (\delta_+^\alpha \underline{\eta}^\beta)\tilde{e}_\alpha \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, m-1 \right\};$$ - for $\gamma_{\min} \in I_i$ and $\gamma_{\max} \in I_j$ with $I_k:=(k-1/2, k+1/2)$ for $0<k\in{\mathbb{Z}}<m$, the kernel of is $${\mathrm{Ker}\,}= {\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta (\delta_+^\alpha \underline{\eta}^\beta)\tilde{e}_\alpha \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, m-j-1 \right\};$$ and its cokernel is $${\mathrm{Cok}\,}={\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta (\delta_+^\alpha \underline{\eta}^\beta)\tilde{e}_\alpha^* \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, i-1 \right\}.$$ On the other hand, the operator does not have closed range for $\gamma_-,\gamma_+ \in \{ 1/2, 3/2, \cdots, m-1/2\}$. Just as in the continuous case, the proof reduces to the verification of Fredholm properties of the discrete derivative $\delta_+^{m-j}\delta_-^{j}$, for $j=0,1,\cdots, m$, which is relegated to Appendix \[ss:a2\]. Floquet-Bloch theory and periodic coefficients {#s:3.3} ---------------------------------------------- We are interested in operators posed on the real line, with only a discrete translational symmetry. Examples are of course the linearization at periodic structures, but include more generally operators with periodic coefficients, $\mathcal{P}(\partial_x,x)$, periodic in $x$. One commonly introduces the Bloch-wave transform $$\label{e:bloch} \begin{matrix} \mathcal{B}: &L^2(\mathcal{T}_1, [L^2([0,2\pi])]^n)&\longrightarrow &[L^2({\mathbb{R}})]^n\\ &\mathbf{U}(\sigma,x)&\longmapsto& \int_{\mathcal{T}_1}{\mathrm{e}}^{{\mathrm{i}}\sigma x}\mathbf{U}(\sigma,\cdot){\mathrm{d}}\sigma, \end{matrix}$$ which is an isometric isomorphism with its inverse $$\label{e:bin} \begin{matrix} \mathcal{B}^{-1}: &[L^2({\mathbb{R}})]^n &\longrightarrow&L^2(\mathcal{T}_1, [L^2([0,2\pi])]^n)\\ &\mathbf{u}(x)&\longmapsto&\frac{1}{2\pi}\sum_{\ell\in {\mathbb{Z}}}{\mathrm{e}}^{{\mathrm{i}}\ell x}\widehat{\mathbf{u}}(\sigma+\ell). \end{matrix}$$ We refer to [@reedsimon .16.] for details. Under the Bloch-wave transform, $\mathcal{P}(\partial_x,x)$ defined on $[L^2({\mathbb{R}})]^n$ becomes a direct integral — the Bloch-wave decomposition, $$\label{e:blod} \mathcal{B}^{-1}\circ \mathcal{P} \circ\mathcal{B}=\int_{\mathcal{T}_1} P_{\mathrm{BL}}(\sigma){\mathrm{d}}\sigma,$$ where the Bloch-wave operator $P_{\mathrm{BL}}(\sigma)$ takes the form $$\label{e:blow} \begin{matrix} P_{\mathrm{BL}}(\sigma): & \mathcal{D}(P_{\mathrm{BL}}(\sigma))\subset [L^2([0,2\pi])]^n & \longrightarrow & [L^2([0,2\pi])]^n\\ & u(x) & \longmapsto & P(\partial_x+{\mathrm{i}}\sigma, x)u(x). \end{matrix}$$ We assume that the family of Bloch-wave operators $P_{\rm BL}(\sigma)$ satisfies the following hypothesis. \[h:3.4\] We assume that $P_{\rm BL}(\sigma)$ is analytic and uniformly bounded, 1-periodic, with values in the set of bounded operators on $Y$, in a strip $\sigma\in \Omega_1:={{\mathbb{R}}}\times (-{\mathrm{i}}\sigma_{\mathrm{i}},{\mathrm{i}}\sigma_{\mathrm{i}})$ for some $\sigma_{\mathrm{i}}>0$. Moreover, we require that $P_{\rm BL}(\sigma)$,restricted to $[-1/2,1/2]$, is invertible except at $\sigma=0$ and $P_{\rm BL}(0)$ admits a simple kernel spanned by $e_0$ with $\langle e_0, e_0 \rangle=1$. In order to exploit the results from Section \[s:3.2\], we first define the chopping operator $\mathcal{C}$ that identifies $[L^2({\mathbb{R}})]^n$ with $\ell^2({\mathbb{Z}}, [L^2([0,2\pi])]^n)$, that is, $$\label{e:chop} \begin{matrix} \mathcal{C}: & [L^2({\mathbb{R}})]^n & \longrightarrow & \ell^2({\mathbb{Z}}, [L^2([0,2\pi])]^n)\\ & u& \longmapsto & \{u(2\pi j+x)\}_{j\in{\mathbb{Z}}}, \end{matrix}$$ and the discrete Fourier transform taking the form $$\label{e:fourier} \begin{matrix} \mathcal{F}_d: & \ell^2({\mathbb{Z}}, [L^2([0,2\pi])]^n) & \longrightarrow &L^2(\mathcal{T}_1, [L^2([0,2\pi])]^n)\\ &\underline{u}= \{u_j\}_{j\in{\mathbb{Z}}}& \longmapsto & \sum_{j\in{\mathbb{Z}}}u_j(x){\mathrm{e}}^{-2\pi{\mathrm{i}}j\sigma}. \end{matrix}$$ Under the transformations $\mathcal{C}$ and $\mathcal{F}_{\rm d}$, $\mathcal{P}(\partial_x,x)$ again becomes a direct integral with the notation $$\label{e:abloch} \int_{\mathcal{T}_1} P(\sigma){\mathrm{d}}\sigma:=\mathcal{F}_{\rm d}\circ\mathcal{C}\circ \mathcal{P} \circ \mathcal{C}^{-1}\circ \mathcal{F}_{\rm d}^{-1}.$$ In fact, for any $U\in \mathcal{D}(\int_{\mathcal{T}_1} P(\sigma){\mathrm{d}}\sigma)$, we have that $$\begin{aligned} \Big(\mathcal{F}_{\rm d}\circ\mathcal{C}\circ \mathcal{P} \circ \mathcal{C}^{-1}\circ \mathcal{F}_{\rm d}^{-1}(U)\Big)(\sigma,x) &=\sum_{j\in{\mathbb{Z}}}{\mathrm{e}}^{-2\pi{\mathrm{i}}j\sigma}\left(\mathcal{P}(\partial_x,x)\int_{\mathcal{T}_1} U(\eta, x){\mathrm{e}}^{2\pi{\mathrm{i}}j \eta}{\mathrm{d}}\eta\right)\\ &=\mathcal{P}(\partial_x,x)\int_{\mathcal{T}_1} U(\eta,x)\left(\sum_{j\in{\mathbb{Z}}}{\mathrm{e}}^{2\pi{\mathrm{i}}j (\eta-\sigma)}\right){\mathrm{d}}\eta\\ &=\mathcal{P}(\partial_x,x)\int_{\mathcal{T}_1} U(\eta,x)\delta(\eta-\sigma){\mathrm{d}}\eta\\ &=\mathcal{P}(\partial_x,x)U(\sigma,x), \end{aligned}$$ which shows that, for any $\sigma\in \mathcal{T}_1$, $$\begin{matrix} P(\sigma): & \mathcal{D}(P(\sigma))\subset [L^2([0,2\pi])]^n & \longrightarrow & [L^2([0,2\pi])]^n\\ & u(x) & \longmapsto & \mathcal{P}(\partial_x, x)u(x). \end{matrix}$$ We conclude with a commutative diagram of isomorphisms as follows, dropping the superscript $n$ for ease of notation, $$\label{e:cmd} \begin{matrix} L^2(\mathcal{T}_1, L^2([0,2\pi]))&\overset{\mathcal{B}}{\longrightarrow}& L^2({\mathbb{R}})&\overset{\mathcal{C}}{\longrightarrow}& \ell^2({\mathbb{Z}}, L^2([0,2\pi]))&\overset{\mathcal{F}_{\rm d}}{\longrightarrow}& L^2(\mathcal{T}_1, L^2([0,2\pi]))\\ \downarrow\int_{\mathcal{T}_1} P_{\mathrm{BL}}(\sigma){\mathrm{d}}\sigma&&\downarrow \mathcal{P} &&&&\downarrow \int_{\mathcal{T}_1} P(\sigma){\mathrm{d}}\sigma\\ L^2(\mathcal{T}_1, L^2([0,2\pi]))&\overset{\mathcal{B}}{\longrightarrow}& L^2({\mathbb{R}})&\overset{\mathcal{C}}{\longrightarrow}& \ell^2({\mathbb{Z}}, L^2([0,2\pi]))&\overset{\mathcal{F}_{\rm d}}{\longrightarrow}& L^2(\mathcal{T}_1, [^2([0,2\pi])), \end{matrix}$$ from which it is straightforward to see that $\int_{\mathcal{T}_1} P_{\mathrm{BL}}(\sigma){\mathrm{d}}\sigma$ and $\int_{\mathcal{T}_1} P(\sigma){\mathrm{d}}\sigma$ are isomorphic. Moreover, we have the following lemma. \[l:bceq\] The operators $P(\sigma)$ and $P_\mathrm{BL}(\sigma)$ are canonically isomorphic for all $\sigma\in\mathcal{T}_1$. From (\[e:bin\]-\[e:blod\]) and (\[e:fourier\]-\[e:abloch\]), we summarize that for any $\sigma\in \mathcal{T}_1$, $$\mathcal{D}(P(\sigma))=\{{\mathrm{e}}^{{\mathrm{i}}\sigma x}u(x)\in [L^2([0,2\pi])]^n\mid u(x)\in\mathcal{D}(P_{\mathrm{BL}}(\sigma))\},$$ which directly implies that we have the isomorphism $$\label{e:isoblo} P_{\mathrm{BL}}(\sigma)={\mathrm{e}}^{-{\mathrm{i}}\sigma x}P(\sigma){\mathrm{e}}^{{\mathrm{i}}\sigma x}.$$ According to Hypothesis \[h:3.4\], there exist $m\in{\mathbb{Z}}^+$, $\lambda_m\neq0$, $e(\sigma)=\sum_{j=0}^{m} e_j\sigma^j$ and $e^*(\sigma)=\sum_{j=0}^{m} e_j^*\bar{\sigma}^j$ with $$\label{e:expmod2} P_{\rm BL}(\sigma)e(\sigma)= \lambda_me_0\sigma^m+{\mathrm{O}}(\sigma^{m+1}), $$ and $$\label{e:expmodad2} P_{\rm BL}^*(\sigma)e^*(\sigma)= \bar{\lambda}_me_0^*\sigma^m+{\mathrm{O}}(\sigma^{m+1}),$$ so that $$\left\langle \sum_{j=0}^{m-1} P_{{\rm BL}, m-j}e_j,e_0^*\right\rangle\neq0, \quad \sum_{j=0}^k P_{{\rm BL}, j}e_{k-j}=0,\quad k=0,\ldots,m-1.$$ According to Lemma \[l:bceq\] and Proposition \[p:f2\], we have the following proposition. \[p:f3\] For $\gamma_-,\gamma_+ \not\in \{1/2,3/2, \cdots, m-1/2\}$, the operator satisfying Hypothesis \[h:3.4\], $$\label{e:dissuperL2} \mathcal{P}:\mathcal{D}(\mathcal{P})\subset L^2_{\gamma_--m,\gamma_+-m}\to L^2_{\gamma_-,\gamma_+},$$ is closed, densely defined, and Fredholm. Letting $\gamma_{\max} = \max \{ \gamma_-, \gamma_+\} , \gamma_{\min} = \min\{ \gamma_-, \gamma_+\}$, we have that - for $\gamma_{\min} \in I_m:=(m-1/2, \infty)$, the operator is one-to-one with cokernel $${\mathrm{Cok}\,}={\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta \frac{({\mathrm{i}}x)^\alpha}{\alpha !}e_{\beta-\alpha}^* \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, m-1 \right\};$$ - for $\gamma_{\max} \in I_0:=(-\infty, 1/2)$, the operator is onto with kernel $${\mathrm{Ker}\,}={\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta \frac{({\mathrm{i}}x)^\alpha}{\alpha !}e_{\beta-\alpha} \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, m-1 \right\};$$ - for $\gamma_{\min} \in I_i$ and $\gamma_{\max} \in I_j$ with $I_k:=(k-1/2, k+1/2)$ for $0<k\in{\mathbb{Z}}<m$, the kernel of is $${\mathrm{Ker}\,}= {\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta \frac{({\mathrm{i}}x)^\alpha}{\alpha !}e_{\beta-\alpha} \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, m-j-1 \right\};$$ and its cokernel is $${\mathrm{Cok}\,}={\mathrm{\,span}\,}\left\{ \sum_{\alpha =0}^\beta \frac{({\mathrm{i}}x)^\alpha}{\alpha !}e_{\beta-\alpha}^* \hspace{2mm}\bigg|\hspace{2mm} \beta=0, 1,\cdots, i-1 \right\}.$$ On the other hand, the operator does not have closed range for $\gamma_-,\gamma_+ \in \{ 1/2, 3/2, \cdots, m-1/2\}$. All results in this proposition, except explicit forms of kernels and cokernels, are direct consequences of Proposition \[p:f2\]. From the isomorphism property and the expansion , we have, for $\beta=0, 1, \cdots, m-1$, $$\mathcal{P} \sum_{\alpha =0}^\beta \frac{({\mathrm{i}}x)^\alpha}{\alpha !}e_{\beta-\alpha} =0,$$ which, combined with the domain of $\mathcal{P}$ for given $\gamma_\pm$, concludes the proof. There is an alternative way to obtain the explicit forms of kernels and cokernels. The first step is to obtain explicit forms of $\tilde{e}_j$ and $\tilde{e}_j^*$. Taking $\tilde{e}_j$ for example, we note that the first $m+1$ terms of the Taylor expansion of ${\mathrm{e}}^{{\mathrm{i}}x\sigma}e(\sigma)$ and $\sum_{j=0}^m({\mathrm{e}}^{2\pi{\mathrm{i}}\sigma}-1)^j \tilde{e}_j$ with respect to $\sigma$ are the same. More specifically, we have $$\begin{aligned} {\mathrm{e}}^{{\mathrm{i}}x\sigma}e(\sigma)&=e_0+\sum_{k=1}^m\left(\sum_{j=0}^k\frac{({\mathrm{i}}x)^j}{j!}e_{k-j}\right)\sigma^k+O(\sigma^{m+1}),\\ \sum_{j=0}^m({\mathrm{e}}^{2\pi{\mathrm{i}}\sigma}-1)^j \tilde{e}_j&=\tilde{e}_0+\sum_{k=1}^m\frac{(2\pi{\mathrm{i}})^k}{k!}\left(A(k,j)\tilde{e}_j\right)\sigma^k+O(\sigma^{m+1}), \end{aligned}$$ where $$A(k,j)=\sum_{\ell=1}^j\binom{j}{\ell}\ell^k(-1)^{j-\ell},$$ with $A(k,j)=0$ for $1<k<j$. We can then solve $\{\tilde{e}_j\}_{j=0}^{m}$ in terms of $\{e_j\}_{j=0}^{m}$. In a second step, we plug all these explicit expansions of $\tilde{e}_j$’s into Proposition \[p:f2\] to derive explicit forms of kernels and cokernels. Impurities {#s:4} ========== We now prove Theorem \[t:1\]. Recalling $\chi_\pm$ is a smooth partition of unity with $\mathrm{supp}(\chi_+)\subset (-1,\infty)$, $\chi_-(x)=\chi_+(-x)$, we write $\theta=\chi_+-\chi_-$ and $$\label{e:phi} \begin{aligned} \varphi(x)&=k_0 x-\varphi_0 + k_1 \Theta- \varphi_1 \theta(x),& \varphi'(x)&=k_0 + k_1\theta(x) -\varphi_1\theta'(x),\\ \varphi^\pm(x)&=k_0 x-\varphi_0 \pm(k_1 x-\varphi_1), & (\varphi^\pm)'(x)&=k_0\pm k_1, \end{aligned}$$ where $\Theta(x):=\int_0^x\theta(y){\mathrm{d}}y+c$ with the constant $c>0$ chosen so that $\Theta(x)=|x|$ for $|x|>1$. We think of $\varphi_j$ and $k_j$ as matching variables in the far field and we will consider $\psi_0=(\varphi_0,k_0)$ as free parameters and $\psi_1=(\varphi_1,k_1)$ as variables, and write $\psi=(\psi_0,\psi_1)$, so that $\varphi=\varphi(x;\psi),\varphi^\pm=\varphi^\pm(x;\psi) $. We write $$\label{e:defup} {u_\mathrm{p}}^\psi(x):={u_\mathrm{p}}(k_*x+\varphi(x;\psi);k_*+\varphi'(x;\psi)),\qquad {u_\mathrm{p}}^{\pm,\psi}(x):={u_\mathrm{p}}(k_*+\varphi^\pm(x;\psi);k_*+(\varphi^\pm)'(x;\psi)).$$ We then substitute the ansatz $u(x)={u_\mathrm{p}}^\psi+w$ into the stationary Swift-Hohenberg equation, to obtain $$\label{e:ansatz} L_\mathrm{SH}({u_\mathrm{p}}^\psi+w)+F({u_\mathrm{p}}^\psi+w)+\varepsilon g=0,$$ where $$L_\mathrm{SH}=-(1+\partial_x^2)^2,\qquad F(u)=\mu u - u^3.$$ The phase shifts $\varphi^\pm$ encode simply shifted phases and wavenumbers, so that ${u_\mathrm{p}}^{\pm,\psi}$ are solutions to the Swift-Hohenberg equation and, for both $+$ and $-$, $$\chi_\pm\left(L_\mathrm{SH} {u_\mathrm{p}}^{\pm,\psi}+F({u_\mathrm{p}}^{\pm,\psi})\right)=0.$$ Subtracting these from gives $$\label{e:bif} L_\mathrm{SH} w+F'({u_\mathrm{p}}^\psi)w+N(w, \psi)+K+\varepsilon G=0,$$ where $$N(w,\psi)=F({u_\mathrm{p}}^\psi+w)-F({u_\mathrm{p}}^\psi)-F'({u_\mathrm{p}}^\psi)w={\mathrm{O}}(w^2),\quad G=g(x,{u_\mathrm{p}}^\psi+w),$$ and the commutator $K$ depends on $\psi$, only, $$K=L_\mathrm{SH}{u_\mathrm{p}}^\psi-\sum_\pm\chi_\pm L_\mathrm{SH} {u_\mathrm{p}}^{\pm,\psi}+F({u_\mathrm{p}}^\psi)-\sum\chi_\pm F({u_\mathrm{p}}^{\pm,\psi}).$$ In particular, one readily finds that $K$ is compactly supported and smooth in $\psi$ as an element of $H^k_\gamma$ for any $k,\gamma$. Expanding $$K=K_1\cdot \psi+ K_2, \quad K_2= {\mathrm{O}}(|\psi|^2),$$ gives $$\label{e:bif1} \mathcal{L}^\psi (w,\psi)+\mathcal{N}(w,\psi)+\varepsilon G(w,\psi)=0,$$ where $$\mathcal{L}^\psi (w,\psi)=L_\mathrm{SH} w+F'({u_\mathrm{p}}^\psi)w+K_1\cdot\psi,$$ with the following notation $$K_1:=\partial_\psi K|_{\psi=0}=(K_{\varphi_0},K_{k_0},K_{\varphi_1},K_{k_1}), \quad \mathcal{N}(w,\psi):=N(w,\psi)+K_2={\mathrm{O}}(|w|^2+|\psi|^2).$$ Our goal is to use Lyapunov-Schmidt reduction to solve with variables $w,\psi_1$ and parameters $\varepsilon,\psi_0$, near the trivial solution $k_0=k_1=\varphi_1=\varepsilon=0$, $w=0$, and fixed $\varphi_0\in [0,2\pi)$. \[r:shift\] Without loss of generality, we can also redefine the primary pattern, shifting its location by $\frac{\varphi_0}{k_*}$ in a $\varphi_0$-dependent fashion, and subsequently applying the shift $x'=x-\frac{\varphi_0}{k_*}$ in . As a consequence, in our proof, $\varphi_0\equiv 0$, or, in other words, $\varphi_0$ as a variable does not appear within ${u_\mathrm{p}}^\psi$ and the dependence on $\varphi_0$ is moved to $g=g(x'+\frac{\varphi_0}{k_*},u)$. Making the role of variables versus parameters explicit, we further decompose $$\mathcal{L}^\psi (w,\psi)=\mathcal{L}_1^\psi (w,\psi_1)+\mathcal{L}_0^\psi \psi_0,$$ with $$\mathcal{L}^\psi_1(w,\psi_1) = L_\mathrm{SH}w+ F'({u_\mathrm{p}}^\psi)w + K_{\varphi_1} \varphi_1 + K_{k_1}k_1, \qquad \mathcal{L}_0^{\psi}\psi_0 = K_{\varphi_0} \varphi_0+ K_{k_0}k_0.$$ In order to implement Lyapunov-Schmidt reduction, we proceed as follows. We precondition with $\mathcal{M}(\psi):=(\mathcal{L}_1^\psi)^{-1}$ and consider the resulting equation $$(w,\psi_1)+\mathcal{M}(\psi)\left(\mathcal{L}_0^\psi \psi_0+\mathcal{N}(w,\psi)+\varepsilon G(w,\psi)\right)=0,$$ on $H^4_{\gamma_*-3-\delta}\times {\mathbb{R}}^2$, in a neighborhood of the origin, with parameters $\psi_0,\varepsilon$. The following two ingredients ensure that we can actually apply the implicit function theorem near the trivial solution $w=\psi_1=0$. 1. The inverse $\mathcal{M}(\psi)$ is bounded from $L^2_\gamma$ to $H^4_{\gamma-2}\times {\mathbb{R}}^2$, and $C^1$ in $\psi$ when considered as an operator from $L^2_\gamma$ to $H^4_{\gamma-3-\delta}$, for $\gamma>3/2$. 2. The nonlinearity $\mathcal{N}$ is of class $C^1$ as a map from $H^4_{\gamma}\times {\mathbb{R}}^4$ into $L^2_{2\gamma}$, with vanishing derivatives at the origin. We then choose $\gamma=\gamma_*$ in (i) and $2\gamma=\gamma_*$ in (ii), which gives the restriction $2(\gamma_*-3-\delta)>\gamma_*$, compatible with $\gamma_*>6$. The second part is quite standard, using that $u\mapsto u\cdot u$ maps $H^k_\gamma$ into $H^k_{2\gamma}$ for $k>1/2$, and we will focus on the first part in the next two sections. We therefore proceed in several steps. We first show bounded invertibility for $\psi=0$ in section \[s:4.1\] , in particular computing the derivatives of $K$ and their projection on the cokernel of $ \mathcal{L}^0_1= L_\textrm{SH} + F'({u_\mathrm{p}})$, where ${u_\mathrm{p}}$ simply stands for ${u_\mathrm{p}}(\xi; k_*)$. We then show bounded invertibility and continuity of $\mathcal{L}^{\psi}_1$ for $\psi\neq 0$ using a decomposition argument in Section \[s:4.2\]. Finally, we compute expansions in Section \[s:4.3\]. Invertibility at $\psi \equiv 0$ {#s:4.1} -------------------------------- In this subsection we drop the subscripts from $\mathcal{L}^0_1$. We first show that $$\label{e:L0} \mathcal{L}^0=L_\textrm{SH} + F'({u_\mathrm{p}}),$$ is Fredholm and identify the cokernel, then compute projections of the partial derivatives of $K_1$ on the cokernel, and finally identify projection coefficients with effective diffusivity. Recall that ${u_\mathrm{p}}(\xi;k_*)$, with $\xi = k_* x$, denotes a periodic solution to the unperturbed Swift-Hohenberg equation. Throughout this section we will write ${u_\mathrm{p}}':=\partial_x {u_\mathrm{p}}=k_*\partial_\xi {u_\mathrm{p}}(\xi; k_*)$, $\partial_\xi {u_\mathrm{p}}:=\partial_\xi {u_\mathrm{p}}(\xi; k_*)$ and $\partial_k{u_\mathrm{p}}:=\partial_k{u_\mathrm{p}}(\xi; k_*)$. #### Fredholm properties of $\mathcal{L}^0$. We start by putting the results from Section \[s:3\] to work. \[prop:fredholm\] Assume Hypotheses \[h:0\]–\[h:2\]. For all $\gamma>3/2$, the linear operator $\mathcal{L}^0:\mathcal{D}(\mathcal{L}^0)\subset H^4_{\gamma-2}\to L^2_\gamma$ is Fredholm of index -2, with trivial kernel and cokernel spanned by ${u_\mathrm{p}}'$ and ${u_{\mathrm{p},k_*}}=x\partial_\xi{u_\mathrm{p}}+\partial_k{u_\mathrm{p}}$. According to Proposition \[p:f3\] and the fact that $m=2$, there exists $e_0$ and $e_1$ so that the operator $\tilde{\mathcal{L}}^0:=-[1+(k_*\partial_\xi)^2]^2+\mu-3{u_\mathrm{p}}^2(\xi;k_*)$, which is the counterpart of the operator $\mathcal{P}$, satisfies $$\tilde{\mathcal{L}}^0 e_0=0, \quad \tilde{\mathcal{L}}^0( e_1 + {\mathrm{i}}\xi e_0)=0.$$ By definition, $\tilde{\mathcal{L}}^0$ is a rescaling of $\mathcal{L}^0$ and thus $e_0$ is the normalized version of $u'_p=k_*\partial_\xi {u_\mathrm{p}}$. According to the dependence on parameter $k$ of ${u_\mathrm{p}}(\xi; k)$, we readily derive $$\tilde{\mathcal{L}}^0 (\partial_k{u_\mathrm{p}}+x\partial_\xi {u_\mathrm{p}})=0,$$ which, combined with the invertibility of $\tilde{\mathcal{L}}^0$ restricted to the subspace of even, $2\pi$-periodic functions, shows that $\partial_k{u_\mathrm{p}}+x\partial_\xi {u_\mathrm{p}}$ is a rescaling of $e_1 + {\mathrm{i}}\xi e_0$. As a result, we now conclude that the results in this proposition follows naturally from the self-adjointness of $\mathcal{L}^0$. #### Spanning the cokernel. As a next step, we compute scalar products between $$K_1:=\partial_\psi K|_{\psi=0}=(K_{\varphi_0},K_{k_0},K_{\varphi_1},K_{k_1}),$$ and the elements in the cokernel. More precisely, we show that $K_{\varphi_0} = K_{k_0} =0$ and that $K_{\varphi_1}$ and $K_{k_1}$ span ${u_{\mathrm{p},k_*}}$ and ${u_\mathrm{p}}'$ in the sense of $$\label{e:spandet} \det\begin{pmatrix} \langle {u_\mathrm{p}}',K_{\varphi_1}\rangle & \langle {u_{\mathrm{p},k_*}},K_{\varphi_1}\rangle \\ \langle {u_\mathrm{p}}',K_{k_1}\rangle & \langle {u_{\mathrm{p},k_*}},K_{k_1}\rangle \end{pmatrix}\neq0.$$ where $\langle\cdot, \cdot\rangle$ denotes the standard inner product in $L^2({\mathbb{R}})$. To start with, a straight forward calculation shows that the total derivative of $K$ is $$\label{e:K1} \partial_\psi K|_{\psi =0} =\mathcal{L}^0 (\partial_\xi {u_\mathrm{p}}\partial_\psi \varphi |_{\psi =0}+\partial_k {u_\mathrm{p}}\partial_\psi \varphi' |_{\psi =0})-\sum_{\pm}\chi_\pm \mathcal{L}^0(\partial_\xi {u_\mathrm{p}}\partial_\psi \varphi^\pm |_{\psi =0}+\partial_k {u_\mathrm{p}}\partial_\psi (\varphi^\pm)' |_{\psi =0})$$ where $\mathcal{L}_0=L_\mathrm{SH}+F'({u_\mathrm{p}})$ as defined in and $$\begin{aligned} \partial_\psi \varphi &=(-1,x, -\theta,\Theta), & \partial_\psi \varphi' &=(0, 1, -\theta', \theta), \\ \partial_\psi \varphi^\pm &=(-1,x,\mp 1, \pm x), & \partial_\psi (\varphi^\pm)' &=(0, 1, 0, \pm 1). \end{aligned}$$ We then exploit the fact that $\chi_\pm$ is a partition of unity and $\theta = \chi_+ - \chi_-$ to obtain expressions for each partial derivative in , $$\begin{aligned} K_{\varphi_0} &= K_{k_0}=0,\\ K_{\varphi_1} & = [\theta, \mathcal{L}^0] \partial_\xi {u_\mathrm{p}}- \mathcal{L}^0( \theta' \partial_k {u_\mathrm{p}}),\\ K_{k_1} &= \mathcal{L}^0 \left( \Theta \partial_\xi {u_\mathrm{p}}+ \theta \partial_k {u_\mathrm{p}}\right) - \theta \mathcal{L}^0( x \partial_\xi {u_\mathrm{p}}+ \partial_k {u_\mathrm{p}}).\end{aligned}$$ Recalling that ${u_{\mathrm{p},k_*}}= x \partial_\xi {u_\mathrm{p}}+ \partial_k {u_\mathrm{p}}$, we can further simplify the formula for $K_{k_1}$ into the following form, $$K_{k_1} = [\mathcal{L}^0, \theta] {u_{\mathrm{p},k_*}}+ \mathcal{L}^0 \left( \Theta \partial_\xi {u_\mathrm{p}}- \theta x \partial_\xi {u_\mathrm{p}}\right).$$ We now proceed to show that is true. Noting that $\mathcal{L}^0$ is self-adjoint, $\theta'$ and $\Theta-\theta x$ are compactly supported, ${u_\mathrm{p}}'= k_* \partial_\xi {u_\mathrm{p}}$ and $$[\mathcal{L}^0,w]v=L_\mathrm{SH}(w v) - w L_\mathrm{SH}v=[-\partial_x^4-2\partial_x^2,w]v,$$ we derive the expressions of projections of $K_{\varphi_1}$ and $K_{k_1}$ on the cokernel, $$\begin{aligned} \langle {u_\mathrm{p}}',K_{\varphi_1}\rangle &= k_*^{-1}\langle {u_\mathrm{p}}',[\theta,\mathcal{L}^0]{u_\mathrm{p}}'\rangle =k_*^{-1}\langle {u_\mathrm{p}}',[\partial_x^4+2\partial_x^2,\theta]{u_\mathrm{p}}'\rangle, \label{e:cok1}\\ \langle {u_{\mathrm{p},k_*}},K_{\varphi_1}\rangle &= k_*^{-1}\langle {u_{\mathrm{p},k_*}},[\theta,\mathcal{L}^0]{u_\mathrm{p}}'\rangle = k_*^{-1}\langle {u_{\mathrm{p},k_*}},[\partial_x^4+2\partial_x^2,\theta]{u_\mathrm{p}}'\rangle,\label{e:cok2}\\ \langle {u_\mathrm{p}}',K_{k_1}\rangle &= \langle {u_\mathrm{p}}',[\mathcal{L}^0,\theta]{u_{\mathrm{p},k_*}}\rangle = -\langle {u_\mathrm{p}}',[\partial_x^4+2\partial_x^2,\theta]{u_{\mathrm{p},k_*}}\rangle,\label{e:cok3}\\ \langle {u_{\mathrm{p},k_*}},K_{k_1}\rangle &= \langle {u_{\mathrm{p},k_*}},[\mathcal{L}^0,\theta]{u_{\mathrm{p},k_*}}\rangle = -\langle {u_{\mathrm{p},k_*}},[\partial_x^4+2\partial_x^2,\theta]{u_{\mathrm{p},k_*}}\rangle.\label{e:cok4} \end{aligned}$$ A straightforward computation gives $$\label{e:liebra} \int_{\mathbb{R}}u[\partial_x^{2m},w]v\,{\mathrm{d}}x=\int_{\mathbb{R}}w'\sum_{j=0}^{2m-1}(-1)^{j}u^{(j)}v^{(2m-1-j)}\,{\mathrm{d}}x,$$ which has the following two consequences related to . - Applying to equation and , we conclude that the off-diagonal elements in coincide, taking the form $$\label{e:offd} \langle {u_\mathrm{p}}',K_{k_1}\rangle=k_*\langle {u_{\mathrm{p},k_*}},K_{\varphi_1}\rangle=\int_{\mathbb{R}}\theta'\left[\sum_{j=0}^{3}(-1)^{j}{u_{\mathrm{p},k_*}}^{(j)}{u_\mathrm{p}}^{(4-j)}+2\sum_{j=0}^{1}(-1)^{j}{u_{\mathrm{p},k_*}}^{(j)}{u_\mathrm{p}}^{(2-j)}\right]\,{\mathrm{d}}x.$$ - The expression is zero if $u\cdot v\cdot w$ is odd and each of $u, v, w$ is either even or odd. Noting that $ {u_\mathrm{p}}'$ and $\theta$ are odd and ${u_{\mathrm{p},k_*}}$ is even, we conclude that the diagonal elements in vanish, that is, $$\label{e:Kd} \langle {u_\mathrm{p}}',K_{\varphi_1}\rangle = \langle {u_{\mathrm{p},k_*}},K_{k_1}\rangle=0.$$ To further simplify the expression of off-diagonal elements , we notice that the projections on the cokernel are independent of the choice of $\theta$. More specifically, suppose $\theta_1$ and $\theta_2$ differ by a compactly supported term, $\delta\theta$, we can evaluate the contribution of $\delta\theta$ to our projections, $$\int_{\mathbb{R}}{u_\mathrm{p}}' [\mathcal{L}^0,\delta\theta] {u_{\mathrm{p},k_*}}\,{\mathrm{d}}x= \int_{\mathbb{R}}{u_\mathrm{p}}' \mathcal{L}^0(\delta\theta {u_{\mathrm{p},k_*}})-{u_\mathrm{p}}' \delta\theta \mathcal{L}^0 {u_{\mathrm{p},k_*}}\,{\mathrm{d}}x=0.$$ As a result, the expression in converges, as $\theta'\to 2\delta_{x_0}$, to $$\begin{aligned} \langle {u_\mathrm{p}}',K_{k_1}\rangle=k_*\langle {u_{\mathrm{p},k_*}},K_{\varphi_1}\rangle=\left.2\left[\sum_{j=0}^{3}(-1)^{j}{u_{\mathrm{p},k_*}}^{(j)}{u_\mathrm{p}}^{(4-j)}+2\sum_{j=0}^{1}(-1)^{j}{u_{\mathrm{p},k_*}}^{(j)}{u_\mathrm{p}}^{(2-j)}\right]\right|_{x=x_0},\label{e:kk}\end{aligned}$$ where $x_0\in{\mathbb{R}}$ is arbitrary. Now, using ${u_{\mathrm{p},k_*}}=\frac{x}{k_*}{u_\mathrm{p}}'+\partial_k{u_\mathrm{p}}$ and ${u_\mathrm{p}}'(0)={u_\mathrm{p}}'(2\pi/k_*)=0$, averaging the constant expression in over a period $x_0\in [0,2\pi/k_*]$ and integrating by parts, we find, $$\label{e:coker} \begin{aligned} \langle {u_\mathrm{p}}',K_{k_1}\rangle=k_*\langle {u_{\mathrm{p},k_*}},K_{\varphi_1}\rangle &= \frac{2}{\pi }\int_0^{2\pi/k_*} \left[ k_*\partial_k\left(({u_\mathrm{p}}'')^2-({u_\mathrm{p}}')^2\right) + \left( 3({u_\mathrm{p}}'')^2-({u_\mathrm{p}}')^2 \right) \right] \,{\mathrm{d}}x. \end{aligned}$$ We will see how this expression relates to the effective diffusivity, next, and hence conclude that it does not vanish. As a consequence, $\mathcal{L}^0$ is bounded invertible. #### Computing the effective diffusivity. We first recall the definition of $L_\mathrm{B}(\sigma)$ from , and consider the eigenvalue equation $$\label{e:eigen} L_\mathrm{B}(\sigma)e(\sigma)=\lambda(\sigma)e(\sigma),$$ for $\lambda(0)=0$ and $\sigma \sim 0$. Expanding $$L_\mathrm{B}(\sigma)=L_0+L_1\sigma+L_2\sigma^2+{\mathrm{O}}(\sigma^3), \quad e(\sigma)=e_0+e_1\sigma+e_2\sigma^2+{\mathrm{O}}(\sigma^3), \quad \lambda(\sigma)=\lambda_2\sigma^2 + {\mathrm{O}}(3),$$ and setting $e_0={u_\mathrm{p}}'$ and $\langle e_0, e(\sigma)-e_0\rangle_{L^2(0,2\pi/k_*)}=0$, we find explicitly $$L_0=-(1+\partial_x^2)^2+\mu -3{u_\mathrm{p}}^2(x),\quad L_1=-4{\mathrm{i}}(1+\partial_x^2)\partial_x, \quad L_2=2+6\partial_x^2,$$ which, plugged in the eigenvalue equation , solve $$L_0e_0=0,\quad L_1e_0+L_0e_1=0, \quad L_0e_2+L_1e_1+L_2e_0=\lambda_2e_0.$$ Noting $\langle e_1,e_0\rangle_{L^2(0,2\pi/k_*)}=0$, we project the equation for $\lambda_2$ onto $e_1$, that is, $$\label{e:lambda2} \lambda_2\langle e_0,e_0\rangle_{L^2(0,2\pi/k_*)}=\langle L_1e_1+L_2e_0,e_0\rangle_{L^2(0,2\pi/k_*)}.$$ In order to determine $e_1$, we recall Lemma \[l:family\] and notice that the derivative $\partial_k{u_\mathrm{p}}(kx;k)$ at $k=k_*$ satisfies $$-4k_*(1+k_*^2\partial_\xi^2)\partial_\xi^2{u_\mathrm{p}}+\left(-(1+k_*^2\partial_\xi^2)^2+\mu-3{u_\mathrm{p}}^2\right)\partial_k{u_\mathrm{p}}=0,$$ or equivalently, $L_1 e_0+L_0({\mathrm{i}}k_*\partial_k{u_\mathrm{p}})=0$, which gives $$e_1={\mathrm{i}}k \partial_k {u_\mathrm{p}}.$$ Inserting the expansion for $L_1$, $L_2$ and $e_1$ into equation gives $$\label{e:d} \lambda_2\int_0^{2\pi/k_*}({u_\mathrm{p}}')^2\,{\mathrm{d}}x=-2\int_0^{2\pi/k_*} \left[ k_*\partial_k\left(({u_\mathrm{p}}'')^2-({u_\mathrm{p}}')^2\right) + \left( 3({u_\mathrm{p}}'')^2-({u_\mathrm{p}}')^2 \right) \right] \,{\mathrm{d}}x.$$ Therefore, combining and , we conclude $$\label{e:diffu} \langle {u_\mathrm{p}}',K_{k_1}\rangle=k_*\langle {u_{\mathrm{p},k_*}},K_{\varphi_1}\rangle=-\frac{\lambda_2}{\pi}\int_0^{2\pi/k_*} ({u_\mathrm{p}}')^2\,{\mathrm{d}}x.$$ Notice that a similar reasoning to the proof of Proposition \[prop:fredholm\] shows that for $\gamma>3/2$ the operators $\mathcal{L}^{\pm,\psi} = L_\textrm{SH} + F'({u_\mathrm{p}}^{\pm, \psi})$, with ${u_\mathrm{p}}^{\pm ,\psi}$ as in equation , are also Fredholm operators from $H^4_{\gamma-2}$ to $L^2_{\gamma}$. Moreover, because the inner products ,,, and depend continuously on the parameter $\psi$, the terms $K_{\phi_1}$ and $K_{k_1}$ span the cokernel of these operators as well. Invertibility of $ \mathcal{L}_1^{\psi} $ {#s:4.2} ------------------------------------------ The invertibility of $ \mathcal{L}_1^{\psi} $ for $\psi=(0,\varphi_0, 0,0)$ can be derived in a straightforward fashion from the invertibility of $\mathcal{L}_1^0$ due to the simple fact that $ \mathcal{L}_1^{\psi} $ for $\psi=(0,\varphi_0, 0,0)$ is conjugate to $\mathcal{L}_1^0$ via a spatial translation. As a result, we only need to deal with the operator $\mathcal{L}_1^{\psi}$ for $\psi\sim0$. The operators $\mathcal{L}_1^\psi $ are close to $\mathcal{L}^{0}_1$, but the difference is in general not relatively bounded. The difficulty stems from the fact that $ \mathcal{L}^0_1$ “gains localization” in certain components, whereas the difference $ \mathcal{L}_1^\psi-\mathcal{L}^0_1$, a bounded multiplication operator, does not affect localization. Therefore, a simple Neumann series perturbation argument will not suffice to establish invertibility of $\mathcal{L}_1^\psi$. We establish somewhat weaker bounds on an inverse of $ \mathcal{L}^\psi_1$ as follows. First, using the results from subsection \[s:4.1\] and changing notation in oder to make the distinction between variables and parameters explicit, we write a more complete definition of $\mathcal{L}_1^{\vartheta}$, that is, $$\label{e:lphiinv} \mathcal{L}^{\vartheta}_1(w,\psi_1):= -(1+ \partial_x^2)^2 w + \mu w - 3( {u_\mathrm{p}}^{\vartheta})^2 w+ K_{\varphi_1} \alpha_0+K_{k_1} \alpha_1 = h$$ where $\vartheta = (\vartheta_1,\vartheta_2,\vartheta_3,\vartheta_4)$ denotes the parameter, and $w,\psi_1=(\alpha_0,\alpha_1)$ are variables. The following proposition then shows the invertibility of this operator and its differentiability with respect to $\vartheta$. For $\gamma>3/2$, equation possesses a solution $(w,\psi_1)$ such that $$\|w\|_{H^4_{\gamma-2}}+|\psi_1|{\leqslant}C \|h\|_{L^2_\gamma},$$ with constant $C$ independent of $\vartheta$, sufficiently small. Moreover, the solution depends continuously on $\vartheta$ in $H^4_{\gamma-2-\delta}$, and is differentiable in $\vartheta$, when considered in spaces with weaker localization, $$\|\partial_{\vartheta}w\|_{H^4_{\gamma-3-\delta}}+|\partial_{\vartheta}\psi_1|{\leqslant}C \|h\|_{L^2_\gamma}.$$ For ease of notation we let $m_0 = K_{\varphi_1}, m_1 = K_{k_1}$, and look for solutions to $$\label{e:Lvarphi} \mathcal{L}_1^\vartheta(w,\psi_1) = \mathcal{L}^\vartheta w + \alpha_0m_0+ \alpha_1 m_1=h,$$ where $w\in H^4_{\gamma-2}$, $\alpha_0,\alpha_1\in{\mathbb{R}}$ are variables, $h \in L^2_{\gamma-2}$, and $$\mathcal{L}^{\vartheta}w = -(1+ \partial_x^2)^2 w + \mu w - 3( {u_\mathrm{p}}^{\vartheta})^2 w.$$ We recall as well that $m_0$ and $m_1$ span the cokernel of $ \mathcal{L}^{\pm, \vartheta}= -(1+ \partial_x^2)^2 + \mu - 3( {u_\mathrm{p}}^{\pm, \vartheta})^2,$ where ${u_\mathrm{p}}^{\pm, \vartheta}$ follows the same definition as in equation . We decompose using the partition of unity, $w=w_++w_-$, $h=h_++h_-$, $w_\pm=\chi_\pm w, h_\pm=\chi_\pm h$, and obtain $$\begin{aligned} \mathcal{L}^{ +,\vartheta}w_+ +\sum_{j=0}^1(\alpha_j-\beta_j)m_j+\left(\mathcal{L}^\vartheta- \mathcal{L}^{-,\vartheta} \right)w_--h_+&=0,\label{e:+}\\ \mathcal{L}^{-, \vartheta } w_-+\sum_{j=0}^1 \beta_j m_j+\left(\mathcal{L}^\vartheta- \mathcal{L}^{+,\vartheta}\right)w_+-h_-&=0.\label{e:-}\end{aligned}$$ To solve and for $w_\pm$, $\alpha_j, \beta_j$, $j \in\{0,1\}$, we will consider the cross-coupling terms $\left(\mathcal{L}^\vartheta- \mathcal{L}^{\pm, \vartheta} \right)w_\pm$ as small perturbations. Note that, given $h\in L^2_\gamma$, the system $$\begin{aligned} \mathcal{L}^{+, \vartheta} w_++\sum(\alpha_j-\beta_j)m_j-h_+&=0\\ \mathcal{L}^{-, \vartheta} w_-+\sum \beta_j m_j-h_-&=0,\end{aligned}$$ possesses a unique solution, $(w_+, w_-, \alpha_1, \alpha_2, \beta_1, \beta_2)$, where $w_-\in H^4_{\gamma-2,\gamma'},w_+\in H^4_{\gamma',\gamma-2}$, with $\gamma'$ arbitrarily large since $h_\pm$ are supported on $\pm x>-1$. Given $|\vartheta|$ small, the cross terms are small, bounded operators when considered on these spaces since, for instance, $\mathrm{supp}(\mathcal{L}^\vartheta-\mathcal{L}^{-,\vartheta} )\subset {\mathbb{R}}^+$, and $w_-|_{{\mathbb{R}}^+}\in H^4_{\gamma'}$. This establishes the existence of a bounded inverse, with $w=w_++w_-\in H^4_{\gamma-2}$. It remains to establish the desired smooth dependence of the solution $\underline{w}=(w,\alpha_0,\alpha_1)$ on $\vartheta$. Writing $\mathcal{L}_1^{\vartheta} \underline{w} = h$ briefly as $\mathcal{L}(\vartheta)(\underline{w}(\vartheta))=h$, we find $$\underline{w}(\vartheta+\zeta\varrho)-\underline{w}(\vartheta)=-\mathcal{L}(\vartheta)^{-1}\left(\mathcal{L}(\vartheta+\zeta\varrho)-\mathcal{L}(\vartheta)\right)\underline{w}(\vartheta+\zeta \varrho),$$ where $0<\zeta\ll 1$, $\vartheta, \varrho\in{\mathbb{R}}^4$ with $|\varrho|=1$ and $|\vartheta|$ sufficiently small. Now $\mathcal{L}(\vartheta)^{-1}(\mathcal{L}(\vartheta+\zeta\varrho)-\mathcal{L}(\vartheta) )$ converges to zero when considered as an operator from $H^4_{\gamma-2}\to H^4_{\gamma-2-\delta}$, for any $\delta>0$, which, using uniform bounds for $\underline{w}(\vartheta+\zeta\varrho)$, establishes continuity. Difference quotients and therefore continuity of partial derivatives can be established in a similar fashion. Notice however that the dependence of the operator $\mathcal{L}^{\vartheta}$ on the parameter comes from the coefficient $$3({u_\mathrm{p}}^{\vartheta})^2= 3[{u_\mathrm{p}}( k_*x+ \varphi;k_* + \varphi')]^2,$$ via $$\varphi(x) = \vartheta_1 x + \vartheta_2 + \vartheta_3\Theta(x) - \vartheta_4 \theta(x).$$ Therefore, derivatives of $\underline{w}(\vartheta)$ with respect to $\vartheta_j$, $j=1,3$ induce linear growth and involve loss of one degree of localization. Reduced equations and expansions {#s:4.3} -------------------------------- In order to obtain approximations for the variables $(w,\varphi_1, k_1)$, we assume expansions of the form $$\begin{aligned} w&= w_1(\varphi_0,k_0) \varepsilon + {\mathrm{O}}( \varepsilon^2),\\ \varphi_1& = M_\varphi(\varphi_0,k_0) \varepsilon +{\mathrm{O}}(\varepsilon^2),\\ k_1 & = M_k(\varphi_0,k_0) \varepsilon + {\mathrm{O}}(\varepsilon^2),\end{aligned}$$ and we observe that the first order approximations of $(w_1,M_\varphi,M_k)$ satisfy the following equation $$\mathcal{L}^0 w_1 + K_{\varphi_1} M_{\varphi}+ K_{k_1} M_k + G_1=0,$$ where by Remark \[r:shift\] we have that $$G_1 = g(x'+\frac{\varphi_0}{k_*}, {u_\mathrm{p}}((k_*+k_0)x'; k_*+k_0)).$$ We then proceed to use Lyapunov-Schmidt reduction and obtain the following reduced equations by projecting on the cokernel of $\mathcal{L}^0$, $$\begin{aligned} 0&= \langle {u_{\mathrm{p},k_*}}, K_{\varphi_1} \rangle M_\varphi + \langle {u_{\mathrm{p},k_*}}, G_1\rangle\\ 0&= \langle {u_\mathrm{p}}' , K_{k_1}\rangle M_k + \langle {u_\mathrm{p}}' , G_1\rangle,\end{aligned}$$ where the variables $M_\varphi$ and $M_k$ depend on $k_0$ and $\varphi_0$. Then, combining these results with and , and in the particular case of $k_0=0$, we obtain formulas for $M_\varphi(\varphi_0,0)$ and $M_k(\varphi_0,0)$, that is, $$\begin{aligned} M_{\varphi}(\varphi_0,0) &= \frac{\pi k_* \displaystyle \int_{\mathbb{R}}g(x'+\frac{ \varphi_0}{k_*}, {u_\mathrm{p}}) {u_{\mathrm{p},k_*}}\,{\mathrm{d}}x'}{\lambda_2 \int_0^{2\pi/k_*} ({u_\mathrm{p}}')^2 \,{\mathrm{d}}x}, \\[2ex] M_{k}(\varphi_0,0)&= \frac{\pi \displaystyle \int_{\mathbb{R}}g(x'+\frac{ \varphi_0}{k_*}, {u_\mathrm{p}}) {u_\mathrm{p}}' \,{\mathrm{d}}x'}{\lambda_2 \int_0^{2\pi/k_*} (u'_p)^2 \,{\mathrm{d}}x} .\end{aligned}$$ It is useful to consider again the change of variables $x' = x - \frac{\varphi_0}{k_*}$, and write $$\int_{\mathbb{R}}g(x'+\frac{ \varphi_0}{k_*}, {u_\mathrm{p}}) {u_\mathrm{p}}' \,{\mathrm{d}}x' = \int_{\mathbb{R}}g(x, {u_\mathrm{p}}(k_*x - \varphi_0; k_*)) {u_\mathrm{p}}' (k_*x - \varphi_0; k_*)\,{\mathrm{d}}x,$$ which, in the case of $g = \partial_u H(x,u)$ for some function $H$, implies that $$\dashint M_k\,{\mathrm{d}}\varphi_0:=\frac{1}{2\pi}\int_0^{2\pi}M_k(\varphi_0,0)\,{\mathrm{d}}\varphi_0=0.$$ Discussion {#s:5} ========== In this paper, we developed a functional-analytic framework for perturbation theory in the presence of essential spectrum, induced by non-compact translation symmetry. The key ingredient are algebraically weighted spaces, including loss of localization by the inverse according to the spatial multiplicity of the essential spectrum. We restricted to “simple” branches of essential spectrum for notational simplicity but the methods generalize to more complicated situations. The framework included problems on infinite lattices and cylinders. A crucial assumption is that there is precisely one unbounded direction. We showed how such results can be used to study defects, here impurities, in striped phases. The framework of algebraically localized spaces here allows for algebraic decay of impurities. One naturally encounters negative Fredholm indices in the linearization, which one compensates for by adjusting parameters in the far field. In fact, the spatial multiplicity is related in a direct way to the fact that periodic patterns come in two-parameter families. Technically, the decomposition into core deformations (algebraically localized functions) and far field deformations (wavenumber and phase corrections) can be employed in a variety of different contexts. In particular, our approach lays the basis for the continuation of localized deformations such as defects in parameters using more classical algorithms of numerical continuation [@lloyd; @morrissey]. We emphasize that our results do not depend on the particular equation, studied, as long as one is able to determine the existence of periodic patterns and establish properties of the linearization. It is worth noting that both, existence and stability properties, can be established in very reliable ways solving simple periodic boundary-value problems. In particular, one can treat reaction-diffusion systems without much adaptation. Technically more interesting would be systems with conserved quantities such as Cahn-Hilliard, Phase-Field, or DiBlock Copolymer models, since mass conservation induces an additional multiplicity in the essential spectrum, thus violating Hypothesis \[h:3.2\] on simple kernels of $L(0)$. One could also study problems in channels or infinite cylinders, in particular deformations of hexagonal spot arrays with periodicity of inhomogeneities in one direction. There are at least two alternative approaches. First, one could work in exponentially weighted spaces, resorting to stronger assumptions on the inhomogeneity. Fredholm properties of differential operators on the real line in exponentially weighted spaces are well known [@palmer; @ssmorse] and have been used in the context of perturbation and bifurcation theory in the presence of essential spectrum [@ssmorse; @goharma]. In a similar vein, one could cast the existence problem as a non-autonomous differential equation in space $x$, and use dynamical systems tools to investigate the effect of inhomogeneities. From this point of view, the periodic patterns form a two-dimensional normally hyperbolic manifold of equilibria. One can then readily calculate the effect of inhomogeneities on the periodic flow on this center manifold, using traditional methods of averaging. A major drawback of these more subtle methods is the reliance on a phase space and exponential behavior in normal directions. In particular, there is no clear path towards perturbation of two-dimensional patterns. Algebraic weights, however, allow for finite-dimensional reductions in the presence of essential spectrum also in higher dimensions [@jara1; @jara2]. Appendix ======== Fredholm properties of pseudo-derivatives $[D(-{\mathrm{i}}\partial_x)]^{-\ell}$ {#ss:a1} -------------------------------------------------------------------------------- In this section we prove a more general version of Proposition \[p:regdrv\]. More specifically, for any $\ell\in{\mathbb{Z}}^+$, $p \in (1, \infty)$ and $\gamma_\pm\in{\mathbb{R}}$, we define the regularized derivative, $$\label{e:rell} \begin{matrix} [D(-{\mathrm{i}}\partial_x)]^\ell: & \mathcal{D}([D(-{\mathrm{i}}\partial_x)]^\ell)\subset L^p_{\gamma_--\ell, \gamma_+-\ell} & \longrightarrow & L^p_{\gamma_-, \gamma_+} \\ & u & \longmapsto & \partial_x^\ell ( 1 + \partial_x)^{-\ell}u, \end{matrix}$$ with its domain $\mathcal{D}([D(-{\mathrm{i}}\partial_x)]^\ell)=\{u\in L^p_{\gamma_--\ell, \gamma_+-\ell}\mid (1+\partial_x)^{-\ell}u\in M^{\ell,p}_{\gamma_--\ell, \gamma_+-\ell}\}$. From Lemma \[p:1ppx\] it is straightforward to see that $\mathcal{D}([D(-{\mathrm{i}}\partial_x)]^\ell)$ is a Banach space under the norm $$\| u \|:=\| u \|_{L^p_{\gamma_--\ell, \gamma_+-\ell}}+ \| (1+\partial_x)^{-\ell}u\|_{M^{\ell,p}_{\gamma_--\ell, \gamma_+-\ell}}.$$ Moreover, the Fredholm properties of the bounded operator $[D(-{\mathrm{i}}\partial_x)]^\ell: \mathcal{D}([D(-{\mathrm{i}}\partial_x)]^\ell) \to L^p_{\gamma_-, \gamma_+}$ are summarized in the following proposition. \[p:regdrv\] For $\gamma_\pm\in \mathbb{R} / \{ 1-1/p, 2-1/p, \cdots, \ell-1/p\}$, the regularized derivative $[D(-{\mathrm{i}}\partial_x)]^\ell$ as defined in is Fredholm. Moreover, the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ satisfies the following conditions. - If $\gamma_{max}\in I_0:=(-\infty,1-1/p)$, the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ is onto with its kernel equal to $\mathbb{P}_{\ell}({\mathbb{R}})$. - If $\gamma_{min}\in I_\ell:=(\ell-1/p, \infty)$, the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ is one-to-one with its cokernel equal to $\mathbb{P}_{\ell}({\mathbb{R}})$. - If $\gamma_{min}\in I_i$, $ \gamma_{max}\in I_j$ with $I_k:=(k-1/p, k+1/p)$ for $0< k\in {\mathbb{Z}}< \ell$, the kernel and cokernel of the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ are respectively spanned by $\mathbb{P}_{\ell-j}({\mathbb{R}})$ and $\mathbb{P}_{i}({\mathbb{R}})$. On the other hand, the range of the operator $[D(-{\mathrm{i}}\partial_x)]^\ell$ is not closed if $\gamma_-, \gamma_+\in\{1-1/p, 2-1/p,...,\ell-1/p\}$. We will only prove the result in the isotropic case, that is for $\gamma_-= \gamma_+ = \gamma$, since the proof for the anisotropic case follows the same arguments with straightforward modifications. We start by showing in Lemma \[p:1ppx\] that the operator $(1 \pm \partial_x) : W^{\ell,p}_{\gamma} \rightarrow W^{\ell-1,p}_{\gamma}$ is an isomorphism and then establish the Fredholm properties of $\partial_x^\ell:M^{k+\ell,p}_{\gamma-\ell} \rightarrow M^{k,p}_{\gamma}$ in Lemma \[p:px\]. By combining these two results one arrives at Proposition \[p:regdrv\]. \[p:1ppx\] Given $\ell\in {\mathbb{Z}}^+$, $p\in(1,\infty)$, $\gamma\in{\mathbb{R}}$, the operator $1\pm \partial_x: W^{\ell,p}_{\gamma} \longrightarrow W^{\ell-1,p}_{\gamma}$ is an isomorphism. We have the following commutative diagram $$\begin{matrix} W^{\ell,p}_\gamma & & \overset{1\pm\partial_x}{\longrightarrow} & &W^{\ell-1,p}_\gamma \\[2mm] \lfloor x \rfloor^\gamma\downarrow & & & &\lfloor x \rfloor^\gamma\downarrow\\ W^{\ell,p} & & \overset{\mathcal{M}_\pm}{\longrightarrow} & &W^{\ell-1,p}. \end{matrix}$$ As a result, we have $(\mathcal{M}_\pm u)(x)= \lfloor x \rfloor^\gamma (1\pm\partial_x)(\lfloor x \rfloor^{-\gamma}u(x))= (1\pm\partial_x)u(x)-\gamma x\lfloor x \rfloor^{-2}u(x)$, that is, according to the Kondrachov embedding theorem, the operator $\mathcal{M}_\pm$ is equal to a compact perturbation of the invertible operator $(1\pm \partial_x): W^{\ell,p} \rightarrow W^{\ell-1,p}$. Noting that ${\mathrm{Ker}\,}{\mathcal{M}_\pm}=\{0\}$, we conclude that $\mathcal{M}_\pm$ is invertible. To obtain the Fredholm properties of $\partial_x^\ell$, we first generalize the canonical definition of $\partial_x: M^{k+1,p}_{\gamma-1} \rightarrow M^{k,p}_{\gamma}$ where $k{\geqslant}0$ to the $k<0$ regime: given $k\in{\mathbb{Z}}^-$, the operator $\partial_x :M^{k+1,p}_{\gamma-1} \rightarrow M^{k,p}_{\gamma}$ is defined as $$\label{e:gpx} \partial_x u (v) = - {\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}u, \partial_x v {\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}, \quad \forall u \in M^{k+1,p}_{\gamma-1}, v \in M^{-k,q}_{-\gamma},$$ where $1/p+1/q=1$. The generalized operator $\partial_x :L^p_{\gamma-1} \rightarrow M^{-1,p}_{\gamma}$ is an extension of the canonical operator $\partial_x: M^{1,p}_{\gamma-1} \rightarrow L^p_{\gamma}$ in the sense that $\partial_xu(v) = {\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}\partial_x u , v {\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}$, for any $ u \in M^{1,p}_{\gamma-1}$ and $ v \in M^{1,q}_{-\gamma}$. For this generalized operator, we have the following lemma whose proof will occupy the rest of this section. \[p:px\] Given $k\in\mathbb{Z}$, $\ell\in\mathbb{Z}^+$, $p\in(1,\infty)$, and $\gamma\in \mathbb{R}\setminus\{1-1/p, 2-1/p,...,\ell-1/p\}$, the operator $$\label{e:partialxm} \partial_x^\ell: M^{k+\ell,p}_{\gamma-\ell}\longrightarrow M^{k,p}_{\gamma},$$ is Fredholm. Moreover, - if $\gamma<1-1/p$, the operator is onto with its kernel equal to $\mathbb{P}_\ell({\mathbb{R}})$; - if $\gamma>\ell-1/p$, the operator is one-to-one with its cokernel equal to $\mathbb{P}_\ell({\mathbb{R}})$; - if $j-1/p<\gamma<j+1-1/p$, where $j \in {\mathbb{Z}}^+\cap[1,\ell-1]$, the kernel and cokernel of the operator are respectively $\mathbb{P}_{\ell-j}({\mathbb{R}})$ and $\mathbb{P}_j({\mathbb{R}})$. On the other hand, the operator does not have a closed range if $\gamma\in\{1-1/p, 2-1/p,...,\ell-1/p\}$. We focus on the proof of the two primary cases when $\ell=1$ and $k=0,-1$, which can be readily generalized to the case when $\ell=1$ and $k=n, -n-1$ for $n\in{\mathbb{Z}}^+$, and then the case $\ell>1$. The proof is given in various steps written as lemmas. We first establish Fredholm properties of the operator $\partial_x: M^{1,p}_{\gamma-1} \rightarrow L^p_{\gamma}$ when $ \gamma>1-1/p$ in Lemma \[l:k1\]. We then establish Fredholm properties of the operator $\partial_x: L^p_{\gamma-1} \rightarrow M^{-1,p}_{\gamma}$ when $\gamma\neq 1-1/p$ in Lemma \[l:k0\]-\[l:k01\], where Fredholm properties of the operator $\partial_x: M^{1,p}_{\gamma-1} \rightarrow L^p_{\gamma}$ when $ \gamma<1-1/p$ follow. Finally, we show in Lemma \[l:1-1/p\] that for $\gamma =1-1/p$ both operators do not have closed range. \[l:k1\] Given $p \in (1,\infty)$ and $\gamma > 1-1/p$, the operator, $ \partial_x: M^{1,p}_{\gamma-1} \rightarrow L^p_{\gamma}$, is Fredholm and one-to-one with its cokernel spanned by $\mathbb{P}_1({\mathbb{R}})$. We can readily apply the techniques from the following proof to show that, given $p \in (1,\infty)$ and $[\gamma_+ - (1-1/p)][\gamma_--(1-1/p)]<0$, the operator, $ \partial_x: M^{1,p}_{\gamma_--1,\gamma_+-1} \rightarrow L^p_{\gamma_-,\gamma_+}$, is bounded and invertible. Given $\gamma>1-1/p$, we write $$L^p_{\gamma,\perp}:=\{f\in L^p_\gamma\mid \int_{\mathbb{R}}f=0\},$$ which is closed in $L^p_\gamma$ since $1$ is a bounded linear functional on $L^p_\gamma$. It is not hard to see that, for any $u\in M^{1,p}_{\gamma-1}$, its derivative $\partial_x u\in L^1$. We then consider $v(x):=\int_{\infty}^x\partial_x u(y){\mathrm{d}}y$ and take $C_1=\lim_{x\to-\infty}v(x)$. It is clear that there exists some $C_2\in \mathbb{R}$ such that $u(x)-v(x)=C_2,$ which leads to $$\lim_{x\to\infty}u(x)=C_2,\quad \lim_{x\to-\infty}u(x)=C_2+C_1.$$ The fact that $u\in L^p_{\gamma-1}$ implies that if the $\lim_{x\to \pm\infty}u(x)$ exists, it must be zero. Thus, we have $C_1=C_2=0$, that is, $\int_{\mathbb{R}}\partial_x u {\mathrm{d}}x=0$, and consequently $${\mathrm{Rg}}(\partial_x )\subseteq L^p_{\gamma,\perp}.$$ We now claim that the inverse of $\partial_x$ can be defined as $$\label{e:inv} \begin{aligned} \partial_x^{-1}:&\quad L^p_{\gamma,\perp} &\longrightarrow & \quad M^{1,p}_{\gamma-1}\\ \quad &\quad\quad f & \longmapsto & \quad \int_{\infty}^x f(y){\mathrm{d}}y. \end{aligned}$$ The fact that $\partial_x^{-1}$ is well defined reduces to verifying that $u(x)=\int_{\infty}^x f(y){\mathrm{d}}y\in L^p_{\gamma-1}$. To do that, we let $\tilde{\gamma}:=\gamma-(1-1/p)>0$ and split ${\mathbb{R}}$ into three intervals, that is, ${\mathbb{R}}=(-\infty,-1)\cup [-1,1]\cup (1,\infty)$. First, it is not hard to see that $$\label{e:ineq1} \|u(x)\|_{L^p_{\tilde{\gamma}-1/p}([-1,1])}{\leqslant}C(\gamma,p)\max_{|x|{\leqslant}1}|u(x)| {\leqslant}C(\gamma,p)\|f\|_{L^1({\mathbb{R}})}{\leqslant}C(\gamma,p)\|f\|_{L^p_\gamma({\mathbb{R}})},$$ where $C(\gamma)$ is a constant varying with $\gamma$ and $p$. For the interval $(1,\infty)$, we use a logarithmic scaling, that is, $$\tau:=\ln (x),\quad w(\tau):={\mathrm{e}}^{\tilde{\gamma}\tau}u({\mathrm{e}}^\tau),\quad g(\tau):={\mathrm{e}}^{(\tilde{\gamma}+1)\tau}f({\mathrm{e}}^\tau),$$ so that the ODE $w_\tau-\tilde{\gamma}w=g$ admits a solution $w(\tau)=\int_\infty^\tau {\mathrm{e}}^{\tilde{\gamma}(\tau-s)}g(s){\mathrm{d}}s$. Applying Young’s inequality to the above integral equation, we obtain $$\label{e:ineq2} \sqrt{2}^{(1/p-\tilde{\gamma})}\|u(x)\|_{L^p_{\tilde{\gamma}-1/p}((1,\infty))}{\leqslant}\|w(\tau)\|_{L^p((0,\infty))}{\leqslant}\frac{1}{\tilde{\gamma}}\|g(\tau)\|_{L^p((0,\infty))} {\leqslant}\frac{1}{\tilde{\gamma}}\|f(x)\|_{L^p_{\tilde{\gamma}+1-1/p}((1,\infty))}.$$ For the interval $(-\infty,1)$, a similar argument can be applied and leads to the inequality, $$\label{e:ineq3} \|u(x)\|_{L^p_{\tilde{\gamma}-1/p}((-\infty,-1))}{\leqslant}C(\gamma, p)\|f(x)\|_{L^p_{\tilde{\gamma}+1-1/p}((-\infty,-1))}.$$ Combining the inequalities (\[e:ineq1\])–(\[e:ineq3\]), we conclude that the operator is well defined and we have $$\|\partial_x^{-1}f\|_{M^{1,p}_{\gamma-1}}=\|u\|_{L^p_{\gamma-1}}+\|f\|_{L^p_{\gamma}}{\leqslant}C(\gamma)\|f\|_{L^p_{\gamma}},$$ which implies that $\partial_x^{-1}$ is also a bounded linear operator. \[l:k0\] Given $p \in (1, \infty)$, we have that, - for $\gamma >1-1/p$ , the operator $\partial_x : L^p_{\gamma-1} \rightarrow M^{-1,p}_{\gamma}$ is one-to-one; - for $\gamma < 1-1/p$, the operator $\partial_x : L^p_{\gamma-1} \rightarrow M^{-1,p}_{\gamma}$ is Fredholm, onto with its kernel equal to $\mathbb{P}_1({\mathbb{R}})$. For $\gamma> 1-1/p$, consider $u \in L^p_{\gamma-1}$ with $\partial_xu =0$. We let $\{ u_n\}_{n \in \mathbb{N}} \subset C^{\infty}_0$ such that $u_n \rightarrow u$ in $L^p_{\gamma-1}$ and then have that, for any $v\in M^{1,q}_{-\gamma}$, $$\partial_x u (v) =- {\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}u, \partial_x v {\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}= \lim_{n \rightarrow \infty} {\savebox{\@brx}{\(\m@th{\langle}\)} \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}\partial_x u_n, v {\savebox{\@brx}{\(\m@th{\rangle}\)} \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}=0,$$ which implies $\partial_x u_n \rightarrow 0$ in $L^p_{\gamma}$. We therefore have $u=0$, proving the first statement of the lemma. For $\gamma< 1-1/p$, the operator $\partial_x: M^{1,q}_{-\gamma} \rightarrow L^q_{1-\gamma}$, according to Lemma \[l:k1\], is a Fredholm operator with index $-1$ and cokernel equal to $\mathbb{P}_1({\mathbb{R}})$. Therefore, the operator $\partial_x: L^p_{\gamma-1} \rightarrow M^{-1,p}_{\gamma}$, as the adjoint operator of $\partial_x: M^{1,q}_{-\gamma} \rightarrow L^q_{1-\gamma}$ with an extra negative sign, is Fredholm with index $1$ and kernel equal to $\mathbb{P}_1({\mathbb{R}})$. \[l:k01\] Given $p \in (1, \infty)$, we have - for $\gamma< 1-1/p$, the Fredholm operator $\partial_x: M^{1,p}_{\gamma-1} \rightarrow L^p_{\gamma}$ is onto with its kernel equal to $ \mathbb{P}_1({\mathbb{R}})$. - for $\gamma>1-1/p$, the Fredholm operator $\partial_x: L^p_{\gamma-1} \rightarrow M^{-1,p}_{\gamma}$ is one-to-one with its cokernel equal to $ \mathbb{P}_1({\mathbb{R}})$. To prove the lemma we just need to show that each operator has a closed range. We restrict our attention to the first operator, the second being analogous. By way of contradiction, suppose that $\partial_x: M^{1,p}_{\gamma-1} \rightarrow L^p_{\gamma}$ does not have a closed range for $\gamma<1-1/q$, then there exists a sequence $\{ u_n \}_{n \in \mathbb{N}} \subset M^{1,p}_{\gamma-1}$ such that dist$(u_n , \mathbb{P}_1({\mathbb{R}})) =1$ and $\| \partial_x u_n \|_{L^p_{\gamma}} \rightarrow 0$. The norm inequality $\| \partial_x u_n \|_{M^{-1,p}_{\gamma}} {\leqslant}\|\partial_x u_n \|_{L^p_{\gamma}}$, together with the fact that the operator $\partial_x: L^p_{\gamma-1} \rightarrow M^{-1,p}_{\gamma}$ has closed range show that we can find a subsequence $\{ v_n\} \subset {\mathrm{Ker}\,}(\partial_x) \subset M^{1,p}_{\gamma-1}$ such that $\|u_n -v_n\|_{L^p_{\gamma-1}} \rightarrow 0$. Therefore, we have $$\|u_n -v_n\|_{M^{1,p}_{\gamma-1}} {\leqslant}\|u_n -v_n\|_{L^p_{\gamma-1}} + \|\partial_x u_n - \partial_x v_n\|_{L^p_{\gamma}}\rightarrow 0, \text{ as }n\to\infty,$$ that is, $\mathrm{dist}(u_n , \mathbb{P}_1({\mathbb{R}})) \rightarrow 0$, which is a contradiction and concludes the proof. \[l:1-1/p\] Given $p \in (1,\infty)$ and $\gamma=1-1/p$, the operators $\partial_x : M^{1,p}_{\gamma-1} \rightarrow L^p_{\gamma}$ and $\partial_x: L^p_{\gamma-1} \rightarrow M^{-1,p}_{\gamma}$ do not have closed range. Let $\phi \in C^{\infty}_0$ with $0 {\leqslant}\phi {\leqslant}1$ and $\mathrm{supp}(\phi)=[-1,1]$. Let $u_n(x) = \phi(x/n)$, then $\{\partial_x u_n\}_{n\in{\mathbb{Z}}^+}$ is a bounded sequence in $M^{-1,p}_{\gamma}$ (also, in $L^p_\gamma$). However, if $\gamma=1-1/p$ the sequence $\{ u_n \}_{n \in \mathbb{N}}$ is unbounded in $L^p_{\gamma-1}$ (also, in $W^{1,p}_{\gamma-1}$). Therefore, both operators do not have closed range. Fredholm properties of operators $\delta_+^{\ell-i}\delta_-^i$ {#ss:a2} -------------------------------------------------------------- \[p:delta\] Given $k\in\mathbb{Z}$, $\ell\in\mathbb{Z}^+$, $p\in(1,\infty)$, and $\gamma\in \mathbb{R}\setminus\{1-1/p, 2-1/p,...,\ell-1/p\}$, the operator $$\label{e:deltafred} \delta_+^{\ell-i}\delta_-^i: \mathcal{M}^{k+\ell,p}_{\gamma-\ell}\longrightarrow \mathcal{M}^{k,p}_{\gamma},$$ is Fredholm for $i\in[0,\ell]\cap{\mathbb{Z}}$. Moreover, - if $\gamma<1-1/p$, the operator in is onto with its kernel equal to $\mathbb{P}_\ell({\mathbb{Z}})$; - if $\gamma>\ell-1/p$, the operator in is one-to-one with its cokernel equal to $\mathbb{P}_\ell({\mathbb{Z}})$; - if $j-1/p<\gamma<j+1-1/p$, where $j \in {\mathbb{Z}}^+\cap[1,\ell-1]$, the kernel and cokernel of the operator in are respectively $\mathbb{P}_{\ell-j}({\mathbb{Z}})$ and $\mathbb{P}_j({\mathbb{Z}})$. On the other hand, the operator in does not have a closed range if $\gamma\in\{1-1/p, 2-1/p,...,\ell-1/p\}$. The proof of Proposition \[p:delta\] is essentially the same as in the continuous case, that is, the proof of Lemma \[p:px\]. The main technical difference lies in the proof of the the discrete version of Lemma \[l:k1\], which we shall establish now. For $\gamma>1-1/p$ and $p\in[1,\infty]$, discrete derivative operators, $\delta_\pm: \mathscr{M}^{1,p}_{\gamma-1} \mapsto \ell^p_\gamma$, are one-to-one Fredholm operators with both cokernels spanned by ${\mathbb{P}}_1({\mathbb{Z}})$. It is straightforward to see that $\delta_\pm$ are isomorphic and we only need to prove the results for $\delta_+$. Just like the continuous, the essential part is to prove that $$\begin{matrix} \delta_+^{-1}: & \ell^p_{\gamma,\perp} & \longrightarrow & \ell^p_{\gamma-1} \\ & \{b_j\}_{j\in{\mathbb{Z}}} & \longmapsto & \{-\sum_{i=j}^\infty b_{i}\}_{j\in{\mathbb{Z}}}, \end{matrix}$$ where $\ell^p_{\gamma,\perp}=\{\{b_j\}_{j\in{\mathbb{Z}}}\in\ell^p_\gamma\mid \sum_{j\in{\mathbb{Z}}}b_j=0\}$, is the bounded inverse of $\delta_+$. To do that, we instead consider the following operator $$\begin{matrix} \widetilde{\delta}_+^{-1}: & \ell^p_{\gamma,\perp}({\mathbb{N}}) & \longrightarrow & \ell^p_{\gamma-1}({\mathbb{N}}) \\ & \{b_j\}_{j\in{\mathbb{N}}} & \longmapsto & \{-\sum_{i=j}^\infty |b_{i}|\}_{j\in{\mathbb{N}}}, \end{matrix}$$ We denote $a_j=-\sum_{i=j}^\infty b_{i}$ for all $j\in {\mathbb{Z}}$ and $\widetilde{a}_j=-\sum_{i=j}^\infty |b_{i}|$ for all $j\in{\mathbb{N}}$. It is then not hard to conclude that - $a_{j+1}-a_j=b_j$, for all $j\in{\mathbb{Z}}$; - $\widetilde{a}_{j+1}-\widetilde{a}_j=|b_j|$, for all $j\in{\mathbb{N}}$; - $\{\widetilde{a}_j\}_{j\in {\mathbb{N}}}$ is an increasing sequence with non-negative entries; - $|\widetilde{a}_j|{\geqslant}|a_j|$, for all $j\in{\mathbb{N}}$. For any $\widetilde{\gamma}>0$ and $j\in{\mathbb{N}}$, we introduce $$A_j=2^{j\widetilde{\gamma}}\widetilde{a}_{2^j},\quad B_j=2^{j\widetilde{\gamma}}\sum_{i=2^j}^{2^{j+1}-1}|b_j|,$$ and have $2^{-\widetilde{\gamma}}A_{j+1}-A_{j}=B_j$, or equivalently, $A_j=-\sum_{i=j}^\infty 2^{(j-i)\widetilde{\gamma}}B_i$, which, according to Young’s inequality, leads to that $$\label{e:ABI} \|\{A_j\}_{j\in{\mathbb{N}}}\|_{\ell^p({\mathbb{N}})}{\leqslant}\|\{2^{-\widetilde{\gamma}j}\}_{j\in{\mathbb{N}}}\|_{\ell^1}\|\{B_j\}_{j\in{\mathbb{N}}}\|_{\ell^p({\mathbb{N}})}{\leqslant}\frac{2^{\widetilde{\gamma}}}{2^{\widetilde{\gamma}}-1}\|\{B_j\}_{j\in{\mathbb{N}}}\|_{\ell^p({\mathbb{N}})}.$$ Moreover, on the one hand, we have $$\label{e:AI} \begin{aligned} \|\{A_j\}_{j\in{\mathbb{N}}}\|_{\ell^p({\mathbb{N}})}^p = \sum_{j=0}^\infty 2^{\widetilde{\gamma}pj-j}\left(2^j|\widetilde{a}_{2^j}|^p\right) &{\geqslant}\sum_{j=0}^\infty 2^{(\widetilde{\gamma}p-1)j}\left(\sum_{i=2^j}^{2^{j+1}-1}|\widetilde{a}_{i}|^p\right)\\ &{\geqslant}\min\{4^{1-\widetilde{\gamma}p},1\}\sum_{j=0}^\infty \left(\sum_{i=2^j}^{2^{j+1}-1}\lfloor i\rfloor^{\widetilde{\gamma}p-1}|\widetilde{a}_{i}|^p\right)\\ &=\min\{4^{1-\widetilde{\gamma}p},1\}\|\{\widetilde{a}_j\}_{j\in{\mathbb{Z}}^+}\|_{\ell^p_{\widetilde{\gamma}-1/p}({\mathbb{Z}}^+)}^p\\ &{\geqslant}\min\{4^{1-\widetilde{\gamma}p},1\}\|\{a_j\}_{j\in{\mathbb{Z}}^+}\|_{\ell^p_{\widetilde{\gamma}-1/p}({\mathbb{Z}}^+)}^p. \end{aligned}$$ On the other hand, we have $$\label{e:BI} \begin{aligned} \|\{B_j\}_{j\in{\mathbb{N}}}\|_{\ell^p({\mathbb{N}})}^p =\sum_{j=0}^\infty 2^{(\widetilde{\gamma}+1)pj}\left(\frac{1}{2^j}\sum_{i=2^j}^{2^{j+1}-1}|b_{i}|\right)^p &{\leqslant}\sum_{j=0}^\infty 2^{[(\widetilde{\gamma}+1)p-1]j}\left(\sum_{i=2^j}^{2^{j+1}-1}|b_{i}|^p\right)\\ &{\leqslant}\max\{4^{1-(\widetilde{\gamma}+1)p},1\}\sum_{j=0}^\infty \left(\sum_{i=2^j}^{2^{j+1}-1}i^{(\widetilde{\gamma}+1)p-1}|b_{i}|^p\right)\\ &=\max\{4^{1-(\widetilde{\gamma}+1)p},1\}\|\{b_j\}_{j\in{\mathbb{Z}}^+}\|_{\ell^p_{\widetilde{\gamma}+1-1/p}({\mathbb{Z}}^+)}^p. \end{aligned}$$ Combining these inequalities , and , we conclude that, there exists $C(\tilde{\gamma},p)>0$ so that $$\|\{a_j\}_{j\in{\mathbb{Z}}^+}\|_{\ell^p_{\widetilde{\gamma}-1/p}({\mathbb{Z}}^+)}{\leqslant}C(\tilde{\gamma},p) \|\{b_j\}_{j\in{\mathbb{Z}}^+}\|_{\ell^p_{\widetilde{\gamma}+1-1/p}({\mathbb{Z}}^+)}{\leqslant}C(\tilde{\gamma},p) \|\{b_j\}_{j\in{\mathbb{Z}}}\|_{\ell^p_{\widetilde{\gamma}+1-1/p}({\mathbb{Z}})}.$$ By shifting and letting $j\rightarrow -j$, we can also show that $$\|\{a_j\}_{j\in{\mathbb{Z}}^-\cup\{0\}}\|_{\ell^p_{\widetilde{\gamma}-1/p}({\mathbb{Z}}^-\cup\{0\})}{\leqslant}C(\tilde{\gamma},p) \|\{b_j\}_{j\in{\mathbb{Z}}}\|_{\ell^p_{\widetilde{\gamma}+1-1/p}({\mathbb{Z}})}.$$ In conclusion, letting $\tilde{\gamma}=\gamma-1-1/p>0$, there exists $C(\gamma,p)>0$ such that $$\|\{a_j\}_{j\in{\mathbb{Z}}}\|_{\ell^p_{\gamma-1}}{\leqslant}C(\gamma, p) \|\{b_j\}_{j\in{\mathbb{Z}}}\|_{\ell^p_{\gamma}},$$ which concludes the proof. 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ArXiv

--- abstract: 'We propose an optical read-out scheme allowing a demonstration of principle of information extraction below the diffraction limit. This technique, which could lead to improvement in data read-out density onto optical discs, is independent from the wavelength and numerical aperture of the reading apparatus, and involves a multi-pixel array detector. Furthermore, we show how to use non classical light in order to perform bit discrimination beyond the quantum noise limit.' author: - 'V. Delaubert' - 'N. Treps' - 'G. Bo' - 'C. Fabre' title: 'Optical storage of high density information beyond the diffraction limit : a quantum study' --- Introduction {#introduction .unnumbered} ============ The reconstruction of an object from its image beyond the diffraction limit, typically of the order of the wavelength, is a hot field of research, though a very old one, as Bethe already dealt with the theory of diffraction by sub-wavelength holes in 1944 [@Bethe]. More recently, theory has been developed to be applied to the optical storage problem, in order to study the influence of very small variations of pit width or depth relative to the wavelength [@Bethe; @Marx1; @Marx2; @Wang; @Liu; @Brok]. To date, only a few super-resolution techniques [@Kolobov] include a quantum treatment of the noise in the measurement, but to our knowledge, none has been applied to the optical data storage problem. Optical discs are now reaching their third generation, and have improved their data capacity from 0.65 GB for compact discs (using a wavelength of 780 nm), to 4.7 GB for DVDs ($\lambda$ = 650 nm), and eventually to 25 GB for the Blu-Ray discs (using a wavelength of 405 nm). In addition to new coding techniques, this has been achieved by reducing the spot size of the diffraction-limited focused laser beam onto the disc, involving higher numerical apertures and shorter wavelengths. Several further developments are now in progress, such as the use of volume holography, 266 nm reading lasers, immersion lenses, near field systems, multi-depths pits [@Hsu], or information encoding on angle positions of asymmetrical pits [@TorokCD]. These new techniques rely on bit discrimination using small variations of the measured signals. Therefore, the noise is an important issue, and ultimately, quantum noise will be the limiting factor. In this paper, we investigate an alternative and complementary way to increase the capacity of optical storage, involving the retrieval of information encoded on a scale smaller than the wavelength of the optical reading device. We investigate a way to optimize the detection of sub-wavelength structures using multi-pixel array. An attempt to a full treatment of the optical disc problem being far too complex, we have chosen to illustrate our proposal on a very simple example, leaving aside most technical constraints and complications, but still involving all the essence of the overall problem. We first explain how the use of an array detector can lead to an improvement of the detection and distinction of sub-wavelength structures present in the focal spot of a laser beam. We then focus on information extraction from an optical disc with a simple but illustrative example, considering that only a few bits are burnt on the dimensions of the focal spot of the reading laser, and show how the information is encoded from the disc to the light beam, propagated to the detector, and finally detected. We explain the gain configuration of the array detector that has to be chosen in order to improve the signal-to-noise ratio (SNR) of the detection. Moreover, as quantum noise is experimentally accessible, and will be a limiting factor for further improvements, we perform a quantum calculation of the noise in the detection process. Indeed, we present how this detection can be optimized to perform simultaneous measurements below the quantum noise limit, using non classical light. Proposed scheme for bit sequence recognition in optical discs ============================================================= ![Color online [**Scheme for information extraction from optical disc, using an array detector.**]{}[]{data-label="general_scheme"}](Fig1-general_scheme){width="7cm"} We propose a novel optical read-out scheme shown on figure (\[general\_scheme\]) allowing information extraction from optical discs beyond the diffraction limit, based on multi-pixel detection. Bits, coded as pits and holes on the optical disc, induce phase flips in the electric field transverse profile of the incident beam at reflection. The reflected beam is imaged in the far field of the disc plane, where the detector stands. In the far field, the phase profile induced by the disc is converted into an intensity profile, that the multi-pixel detectors can, at least partly, reconstruct. Taking into account that a lot of a priori information is available - i.e. only a finite number of intensity profiles is possible - we propose to use a detector with a limited number of pixels $D_{k}$ whose gains can independently be varied depending on which bit sequence one wants to detect. The signal is then given by $$\begin{aligned} \label{signal} S = \sum_{k}\sigma_{k}N_{k}\end{aligned}$$ where $N_{k}$ is the mean photon number detected on pixel $D_{k}$, and $\sigma_{k}$ is the electronic gain of the same pixel. Ideally, to each bit sequence present on the disc corresponds a set of gains chosen so that the value of the measurement is zero, thus cancelling noise from the mean field. Measuring the signal for a given time interval $T$ around the centered position of a bit sequence in the focal spot, and testing, in parallel, all the pre-define sets of gain in the remaining time, allows to deduce which bit sequence is present on the disc. We will first show that this improvement in density of information encoded on an optical disc is already possible using classical resources. Moreover, as the measurement is made around a zero mean value, the classical noise is mostly cancelled. Hence, we reach regimes where the quantum noise can be the limiting factor. We will demonstrate how to perform measurements beyond the quantum noise limit, using previous results on quantum noise analysis in multi-pixel detection developed in reference [@Treps]. Encoding information from a disc onto a light beam ================================================== We have explained the general principle of reading-out sub-wavelength bit sequences encoded on an optical disc, and now focus on the information transfer from the optical disc to the laser beam, through an illustrative example. Let us recall that bits are encoded by pits and holes on the disc surface: a step change from hole to pit (or either from pit to hole) encodes bit $1$, whereas no depth change on the surface encodes bit $0$, as represented on figure (\[bits\]). A hole depth of $\lambda/4$ insures a $\pi$ phase shift between the fields reflected by a pit and a hole. In this section, we compute the incident field distribution on the optical disc affected by the presence of a bit sequence in the focal spot, and finally analyze the intensity back reflected in the far field, in the detection plane, as sketched on figure (\[general\_scheme\]). ![Color online [**Example of bit sequence on an optical disc. The spacing between the bits is smaller than the wavelength, the minimum waist of the incident laser beam being of the order of $\lambda$. A hole depth of $\lambda/4$ insures a $\pi$ phase shift between fields reflected on a pit and a hole.**]{}[]{data-label="bits"}](Fig2-bits){width="7cm"} Beam focalization ----------------- Current optical disc read-out devices involve a linearly polarized beam strongly focused on the disc surface to point out details whose size is of the order of the laser wavelength. The numerical aperture ($NA$) of the focusing lens can be large ($0.47$ for CDs, $0.6$ for DVDs, and $0.85$ for BLU RAY discs), and the exact calculation of the field cannot be done in the paraxial and scalar approximation. Thus, the vectorial theory of diffraction has to be taken into account. The structure of the electromagnetic field in the focal plane of a strongly focused beam has been investigated for decades now [@VanNie], as its applications include areas such as microscopy, laser micro-fabrication, micromanipulation, and optical storage [@Landesman; @Rodriguez; @Ulanowski; @Lax; @Seshadri; @Ciattoni; @Cao; @Nieminen]. In our case of interest, we can restrict the field calculation to the focal plane, which is the disc plane. Thus Richards and Wolf integrals [@Richards], which are not suitable for a general propagation of the field, but which can provide the field profile in the focal plane for any type of polarization of the incoming beam as long as the focusing length is much larger than the wavelength, can be used to achieve this calculation. These integrals have already been used in many publications dealing with tight focusing processes [@Quabis1; @Novotny; @Dorn; @Quabis2; @Sheppard; @Torok; @Youngworth; @Zhan]. As highlighted in these references, the importance of the vectorial aspect of the field can easily be understood when a linearly polarized beam is strongly focused, as the polarization of the wave after the lens is not perpendicular to the propagation axis anymore and has thus components along this axis. In order to estimate the limit of validity of the paraxial approximation, we computed focused spot sizes of linearly polarized beam in the focal plane for different numerical apertures, first in the paraxial approximation, and then calculated with Richards and Wolf integrals. The results are compared on figure (\[waistcomparison\]) for an incident plane wave in air medium with $\lambda=780nm$, where the spot size is defined as the diameter which contains $86\%$ of the focused energy, as in reference [@Siegman]. ![Color online[**Evolution of the focused spot size of an incident plane wave with the numerical aperture (for $\lambda$ = 780 nm in air medium). The spot size is limited to the order of the wavelength in the non-paraxial case (o), whereas it goes to zero for very high numerical apertures in the paraxial case($\ast$).**]{}[]{data-label="waistcomparison"}](Fig3-waist_NA){width="9cm"} We see that when the numerical aperture exceeds $0.6$, a good prediction requires a non-paraxial treatment. Moreover, whereas there is no theoretical limit to focalization in the paraxial case, we see that non paraxial effects prevent us to reach a waist smaller than the order of the wavelength. Note that this limit is not fundamental and can be overcome by modifying the polarization of the incoming beam. Quabis [*et al.*]{} have indeed managed to reduce the spot area to about $0.1~\lambda^{2}$ using an incident radially polarized doughnut beam [@Quabis1; @Quabis2]. As our aim is to present a demonstration of principle and not a full treatment of the optical disc problem, the following calculations will be done using the physical parameters of the actual Compact Discs ($\lambda=780nm$ and $NA=0.47$, corresponding to a focalization angle of $27$ degrees in air medium). In this case, the paraxial and scalar approximations are still valid. Indeed, figure (\[fieldcomponents\]), giving the transverse profile of the three field components and the resultant intensity in the focal plane using the former parameters, shows that although the field is not strictly linearly polarized as foreseen before, $E_{y}\ll E_{z}\ll E_{x}$, and we can thus consider that only $E_{x}$ is different from zero with a good approximation. Note that the exact expression would not intrinsically change the problem, as our scheme can be adapted to any field profile discrimination. ![Color online[**Norm of the different field components and resultant intensity in the focal plane with a linearly polarized incident field along the $x$ axis, focused with a $0.47$ numerical aperture.**]{}[]{data-label="fieldcomponents"}](Fig4-components){width="7cm"} Reflection onto the disc ------------------------ In order to compute the reflected field, we simply assume that bumps and holes are generated in such a way that they induce a $\pi$ phase shift between them at reflection on the field profile. Note that the holes depth is usually $\lambda/4$, but precise calculations would be required to give the exact shape of the pits, as they are supposed to be burnt below the wavelength size, and as the field penetration in those holes is not trivial [@Wang; @Brok; @Liu]. As we have shown that only one vectorial component of the field was relevant in the focal plane, we can directly apply this phase shift to the amplitude profile of this component. We first envision a scheme with only three bits in the focal spot, which means that $2^{3}$ different bit sequences, i.e. a byte, have to be distinguished from each other, using the information extracted from the reflected field. Note that we neglect the influence of other bits in the neighborhood. A more complete calculation involving this effect with more bits will be considered in a further approach. The amplitude profiles obtained when the incident beam is centered on a bit of the CD are presented on figure (\[summary\]), for a particular bit sequence. Note that we have chosen the space between two bits on the disc equal to the waist size of the reading beam. The first three curves respectively show the field amplitude profile incident on the disc, an example of a bit sequence, and the corresponding profile just after reflection onto the disc. We see that binary information is encoded from bumps and holes on the CD to phase flips in the reflected field. ![Color online[**Modifications of the transverse amplitude field profile trough propagation, in the case of a $111$ bit sequence in the focal spot : a) incoming beam profile, b) 111 bit sequence, c) corresponding reflected field in the disc plane, d) far field profile in the detector plane .**]{}[]{data-label="summary"}](Fig5-summary){width="9cm"} Back propagation to the detector plane -------------------------------------- In order to extract the information encoded in the transverse amplitude profile of the beam, the field has to be back propagated to the detector plane. A circulator, composed of a polarizing beam splitter and a Faraday rotator, ensures that the linearly polarized reflected beam reaches the array detector, as shown on figure (\[general\_scheme\]). Assuming that the detector is positioned just behind the lens plane, the expression of the detected field is given by the far field of the disc plane, apertured by the diameter of the focusing lens. As the focal length and the diameter of the lens are large compared to the wavelength, we use Rayleigh Sommerfeld integral to compute the field in the lens plane [@Born]. As an example, the calculated far field profile when the bit sequence $111$ is present in the focal spot is shown on the fourth graph of figure (\[summary\]). The presence of the lens provides a limited aperture for the beam and cuts the high spatial frequencies of the field, which can be a source of information loss, as the difference between each bit sequence can rely on those high frequencies. However, we will see that enough information remains in the low frequency part of the spatial spectrum, so that the $8$ bits can be distinguished. This is due to the fact that we have in this problem a lot of a priori information on the possible configurations to distinguish. We see on figure (\[far\_field\]) that, with the physical parameters used in compact disc read out devices, 6 over 8 profiles in the detector plane are still different enough to be distinguished. ![Color online[**Field profiles in the array detector plane, for each of the $8$ bit sequence configuration. Note that they are clearly distinguishable, except for the bit sequences $100$ and $001$, and $011$ and $110$, which have the same profile because of the symmetry of the bit sequence relative to the position of the incident laser beam.** ]{}[]{data-label="far_field"}](Fig6-far_field.eps){width="9cm"} At this stage, we are nevertheless unable to discriminate between symmetric configurations, because they give rise to the same far field profile. Therefore, 100 and 001, and 110 and 011, cannot be distinguished. Note that this problem can be solved thanks to the rotation of the disc. Indeed, an asymmetry is created when the position of the disc relative to the laser beam is shifted, thus modifying differently the two previously indistinguishable profiles. As shown on figure (\[non\_centered\]), where the far field profiles are represented after a shift of $w_{0}/6$ in the position of the disc, the degeneracy has been removed. Moreover, it is important to notice that the other profiles experience a small shape modification. This redundant information is very useful in order to remove ambiguities while the disc is rotating. ![Color online[**Field profiles in the array detector plane, for each of the $8$ bit sequence configuration, when the position of the disc has been shifted of $w_{0}/6$ relative to the incident beam. The profile degeneracy for $100$ and $001$, and $011$ and $110$ is raised. Note that the other profiles have experienced a much smaller shape modification between the two positions of the disc.** ]{}[]{data-label="non_centered"}](Fig7-FF_non_centered.eps){width="9cm"} Information extraction for bit sequence recognition =================================================== In this section, we describe the detection,present some illustrative results, and the way they can be used to increase the read-out precision of information encoded on optical discs. We show here that a pixellised detector with a very small number of pixels is enough to distinguish between the $8$ bit sequences. Note that for technical and computing time reasons, it is not realistic to use a CCD camera to record the reflected images, as such cameras cannot yet combine good quantum efficiency and high speed. Detected profiles ----------------- For simplicity reason, we limit our calculation to a $5$ pixels array detector $D_{1}..D_{5}$, each of whom has an electronic gain $\sigma_{1}..\sigma_{5}$, as shown on figure (\[FF\_detection\]). The size of each detector has been chosen without a systematic optimization, which will be done in a further approach. Gain values are adapted to detect a mean signal equal to zero for each bit configuration present in the focal spot, in order to cancel the common mode classical noise present in the mean field [@Treps]. It means that for each bit sequence $i$, gains are chosen to satisfy the following relation $$\label{gaindef} \sum_{k=1}^{5} \sigma_{k}(i)N_{k}(i) =0$$ where $N_{k}(i)$ is the mean photon number detected on pixel $D_{k}$ when bit $i$ is present in the focal spot on the disc $$\label{Idef} N_{k}(i) = \int_{D_{k}}n_{i}(x)dx$$ where $n_{i}(x)$ is the number of photon incident on the array detector, at position $x$, when bit sequence $i$ is present in the focal spot. As all profiles are symmetrical when the incident beam is centered on a bit, we have set $\sigma_{1}=\sigma_{5}$ and $\sigma_{2}=\sigma_{4}$. In addition, we have chosen $\sigma_{3}=-\frac{\sigma_{1}}{2}$. Using these relations and equation (\[gaindef\]), we compute gain values adapted to the recognition of each bit sequence. ![Color online[**Far field profiles in detection the plane for each bit configuration, and array detector geometry. The 5 detectors $D_{1}..D_{5}$ have electronic gains $\sigma_{1}(i)..\sigma_{5}(i)$ according to the bit sequence $i$ which is present in the focal spot.**]{}[]{data-label="FF_detection"}](Fig8-FF_detection.eps){width="8cm"} Note that the calculation of each gain configuration requires a priori information on the far field profiles, or at least an experimental calibration using a well-known sample. Now that these gain configurations are set, we can investigate for a bit sequence on the optical disc. Classical results ----------------- The expression of the detected signal $S_{i}(j)$ is given by $$\begin{aligned} \label{signal} S_{i}(j) = \sum_{k=1}^{5}\sigma_{k}(j)N_{k}(i)\end{aligned}$$ where $i$ refers to the bit sequence effectively present in the focal spot, and $j$ to the gain set adapted to the detection of the bit sequence $j$. It merely corresponds to the intensity weighted by the electronic gains. Note that for $i=j$ - and only in this case if the detector is well chosen - the mean value of the signal $S_{i}(i)$ is equal to zero, according to equation (\[gaindef\]). All possible values of $S_{i}(j)$ are presented for a total number of incident photons $N_{inc}=25$, in table (\[table\]) where $i$ is read vertically, and corresponds to the bit sequence on the disc, whereas $j$ is read horizontally and refers to the gain set adapted to the detection of bit $j$. In order not to have redundant information, we have gathered results corresponding to identical far field profiles. A zero value is obtained for only one gain configuration, allowing an identification of the bit sequence present in the focal spot. 000 001/100 010 011/110 101 111 --------- ----- --------- ------ --------- ----- ------ 000 0 -34 -204 -254 -77 -303 001/100 15 0 -76 -99 -19 -121 010 23 20 0 -6 16 -13 011/110 24 22 5 0 19 -5 101 19 11 -36 -50 0 -63 111 24 23 9 5 20 0 : [**Detected signals $S_{i}(j)$ where $i$ is read vertically and corresponds to the bit sequence on the disc, whereas $j$ is read horizontally and refers to the gain set adapted to the detection of bit $j$. A zero value means that the tested gain configuration is adapted to the bit sequence.**]{}[]{data-label="table"} The reading process to determine which bit sequence is lit on the disc follows these few steps : - the time dependent intensity is first measured on each of the five detectors with all electronic gains set to one. - these intensities are integrated for a time $T$. - the signal is then calculated, using the different gain configurations $j$ - the bit sequence effectively present in the focal spot is determined by the only signal yielding a zero value. Note that the second step just corresponds to the $N_{k}$ measurements. The integration time $T$ is chosen as the time interval during which the signal leads to the determination of a unique bit sequence. The third step corresponds to the simple calculation of a line in table \[table\]. This can be done in parallel thanks to the speed of data processing on dedicated processors, and the reading rate will thus not be affected compared to current devices. Finally, note that the last step requires a good choice of the parameters in order to be able to distinguish all bit sequences. It means that the noise level has to be smaller than the difference between the two closest values from $0$, in order to get a zero mean value for only one bit sequence. Indeed, there must be no overlap between the expectation values when we take into account the noise and thus the uncertainty relative to each measurement. Note that using the zero value as the discriminating factor could be combined with the use of all the calculated values, as each line of table (\[table\]) is distinct. We just need to know how to weight each data point according to the noise related to its obtention. Noise calculation ================= The shot noise limit -------------------- To include the noise in our calculation, we separate classical and quantum noise contributions. The classical noise comprises residual noise of the laser diode, mechanical and thermal vibrations. The major part of this noise is directly proportional to the signal, i.e. to the number of detected photons. For a detection of the total number of photons $N_{inc}$ in the whole beam during the integration time of the detector, the classical noise contribution $\sqrt{\langle\delta N_{inc}^{2}\rangle}$ would thus be written as $$\label{noise1} \sqrt{\langle\delta N_{inc}^{2}\rangle}=\beta N_{inc}$$ where $\beta$ is a constant factor. And the individual noise variable $\delta N_{i}(k)$ arising from detection on pixel $D_{k}$ is given by $$\label{noise2} \delta N_{i}(k)= \frac{N_{i}(k)}{N_{inc}}\delta N_{inc}$$ Using equations \[signal\], \[noise1\] and \[noise2\], a simple calculation yields the variance of the signal arising from the classical noise $$\langle \delta \hat{S}^{2}_{i}(j)\rangle_{Cl}=\frac{B {S}^{2}_{i}(j)}{N_{inc}}$$ where the constant $B=N_{inc}\beta^{2}$ is the classical noise factor, and is chosen so that, when $B=1$ and when all the intensity is detected by one detector, the classical noise term is equal to the shot noise term. Note that classical noise does not deteriorate measurements having a zero mean value. For this reason, we have chosen to discriminate bit sequences by choosing gains such as $S_{i}(i)=0$, as mentioned earlier. ![Color online[**Classical noise ($10~dB$ of excess noise) represented as error bars, for $\lambda=0.78 \mu m$, $NA=0.47$, and $25$ detected photons. Each inset corresponds to the $6$ signals obtained for the different gain configurations, when one particular bit sequence is present in the focal spot. Each bit sequence present in the focal spot can be clearly identified as only one gain configuration can give a zero value for each inset.**]{}[]{data-label="classical"}](Fig9-noise_classical.eps){width="8cm"} The calculation of the quantum contribution requires the use of quantum field operators, describing the quantum fluctuations in all transverse modes of the field. By changing the gain configuration of the array detector, not only the signal $S_{i}(j)$ is modified, but also the related quantum noise denoted $\langle \delta\hat{S}^{2}_{i}(j)\rangle_{Qu}$, as different gain configurations are sensitive to noise in different modes of the field. We have shown in reference [@Treps] that for a multi-pixel detection of an optical image, the measurement noise arises from only one mode component of the field, referred to as the [*detection mode*]{}, or [*noise-mode*]{} [@Del]. The expression of the quantum noise is then : $$\begin{aligned} \label{qnoise} \langle \delta \hat{S}^{2}_{i}(j)\rangle_{Qu} = f_{i,j}^{2} N_{inc}\langle\delta \hat{X}^{2}_{w_{i,j}}\rangle\end{aligned}$$ where $\delta \hat{X}_{w_{i,j}}$ is the quantum noise contribution of the noise-mode $w_{i,j}(x)$ which is defined for one set of gain $j$, when the bit sequence $i$ is present in the focal spot, as $$\label{wdef} \forall x \in D_{k} \qquad w_{i,j}(x) = \frac{\sigma_{k}(j)n_{i}(x)}{f_{i,j}}$$ and where $f_{i,j}$ is a normalization factor, which expression is $$\label{fdef} f^{2}_{i,j} = \frac{\sum_{k=1}^{5}\sigma^{2}_{k}(j)N_{k}(i)dx}{N_{inc}}$$ The noise-mode corresponds in fact to the incident field profile weighted by the gains. The shot noise level corresponds to $\langle\delta \hat{X}^{2}_{w_{i,j}}\rangle=1$. The variance of the signal can eventually be written as: $$\begin{aligned} \label{noisedef} \langle \delta \hat{S}^{2}_{i}(j)\rangle = f_{i,j}^{2} N_{inc}\langle\delta \hat{X}^{2}_{w_{i,j}}\rangle + \frac{B {S}^{2}_{i}(j)}{N_{inc}}\end{aligned}$$ ![Color online[**Shot noise represented as error bars, for $\lambda=0.78 \mu m$, $NA=0.47$, $25$ detected photons. Some bit sequences cannot be determined without ambiguity because of the noise level.**]{}[]{data-label="shot"}](Fig10-noise_shot.eps){width="8cm"} We have first represented the classical noise with an excess noise of $10~dB$, as error bars for each result $S_{i}(j)$, on figures (\[classical\]). We have chosen a representation with a number of detected photons of only $25$. Each of the $6$ insets refers to the measurement obtained for a particular bit sequence in the focal spot. The $6$ data points and associated error bars refer to the results obtained when the $6$ gain configurations are tested. One inset thus corresponds to one line in table (\[table\]). We can see that with this choice of parameters, the bit sequence effectively present in the focal spot can be determined without ambiguity by the only zero value. The sequence corresponds to the one for which the gains were optimized. We see that bit sequence discrimination can be achieved even with a very low number of photons. The relative immunity to classical noise of our scheme arises from the fact that measurements are performed around a zero mean value. Thus, given this limit in the minimum necessary photon number and the flux of photons one can calculate the maximum data rate, which is found to be $2.10^{7} Mbits/s$ (this estimation takes into account an integration time $T$ corresponding to $1/10$ of the delay between the read-out process of two adjacent bits with a $1mW$ laser). This very high value shows that classical noise should not be a limit for data rate in such a scheme. The effect of quantum noise is very small, but becomes a limiting factor for such a small number of detected photons, or for a large number of bits encoded on the disc in the wavelength size. In order to see independently the effect of each contribution to the noise, we have thus represented on figure (\[shot\]) the shot noise also for $25$ detected photons, appearing as the threshold under which it is impossible to distinguish bit sequences because of the quantum noise. Note that for the represented case, the shot noise is the most important contribution, and that it prevents a bit sequence discrimination, as a zero value for the signal can be obtained for several gain configurations in the same inset. Beyond the shot noise limit --------------------------- When the shot noise is the limiting factor, non classical light can be used to perform measurements beyond the quantum noise limit. We have shown in reference [@Treps] that squeezing the noise-mode of the incident field was a necessary and sufficient condition to a perfect measurement. What we are interested in is improving the measurements that yield a zero value, which are obtained when the gain configuration matches the bit sequence in the focal spot, as $S_{i}(i)=0$. Using equation (\[noisedef\]), we see that $w_{i,i}$ has to be squeezed. As no information on the bit present in the focal spot is available before the measurement, in order to improve simultaneously all the bit sequences detections, the $6$ noise-modes have to be squeezed at the same time in the incident field. These $6$ transverse modes are not necessarily orthogonal, but one can show that squeezing the subspace that can generate all of them is enough to induce the same amount of squeezing. ![Color online[**Quantum detection noise represented as error bars, for $\lambda=0.78 \mu m$, $NA=0.47$, $25$ detected photons and $-10~dB$ of simultaneous squeezing for all the flipped modes. The ambiguity in presence of shot noise has been removed and each bit sequence can be identified.**]{}[]{data-label="SQZ"}](Fig11-noise_sqz.eps){width="8cm"} The quantum noise with $10~dB$ of squeezing on the sub space generated by the $w_{i,i}$ is represented as error bars on figure (\[SQZ\]). The noise of each noise-mode $w_{i,j}$ is computed using its overlap integrals with the generator modes of the squeezed sub space, assuming that all modes orthogonal to the squeezed subspace are filled with coherent noise. In this case, the effect of squeezing, reducing the quantum noise on the measurements, and especially on the measurement for which the gains have been optimized, is enough to discriminate bit sequences that were masked by quantum noise. Conclusion ========== We have proposed a novel way of information extraction from optical discs, based on multi-pixel detection. We have first demonstrated, using only classical resources, that this detection could allow large data storage capacity, by burning several bits in the spot size of the reading laser. We have presented a demonstration of principle through a simple example which will be refined in further studies. We have also shown that in shot noise limited measurements, using squeezed light in appropriate modes of the incident laser beam can lead to improvement in bit sequence discrimination. The next steps are to study in details how to extract the redundant information when the disc is spinning, and to systematically optimize the number of bits in the focal spot, the number and size of pixels in the array detector. Such a regime involving a large number of bits in the focal spot will ultimately be limited by the shot noise, and will require the quantum noise calculations presented in this paper. Acknowledgment ============== We thank Magnus Hsu, Ping Koy Lam and Hans Bachor for fruitful discussions. Laboratoire Kastler Brossel, of the Ecole Normale Superieure and University Pierre et Marie Curie, is associated to CNRS. This work has been supported by the European Union in the frame of the QUANTIM network (contract IST 2000-26019). H. A. Bethe, Phys. Rev. 66 [**7 and 8**]{}, 163 (1944) D. S. Marx, and P. Psaltis, J. Opt. Soc. Am. A 14 [**6**]{}, 1268 (1997) D. S. Marx, and P. Psaltis, Appl. Opt. 36 [**25**]{}, 6434 (1997) X. Wang, J. Mason, M. Latta, T. C. Strand, D. S. 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A 5, 1610 (1988). G. Rodriguez-Morales and S. Chavez-Cerda, Opt. Let. 5, 430 (2004). Z.Ulanowski and I. K. Ludlow, Opt. Let. 25 1792 (2000). T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, submitted to Elsevier. B. Richards, E.Wolf, Electromagnetic diffraction in optical systems, Proc. R. Soc. A 253, 359 (1959). S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Appl. Phys. B 72, 109 (2001). L. Novotny, R. D. Grober, and K. Karrai, Ot. Let. 26, 789 (2001). R. Dorn, S. Quabis, and G. Leuchs, preprint : arxiv/physics/0310007. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Com. 179, 1 (2000). C. J. R. Sheppard, J. Opt. Soc. Am. A 18, 1579 (2001). P. Török, P. Varga, Z. Laczik, and G. R. Booker, J. Opt. Soc. Am. A 12, 325 (1995). K. S. Youngworth, and T. G. Brown, Opt. Expr. 7, 77 (2000). Q. Zhan, and J. R. Leger, Opt. Expr. 10, 324 (2002). 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ArXiv

--- abstract: 'The problem of the $\Xi_{c}^{+}$ lifetime is considered in the framework of [*Heavy-Quark Expansion*]{} and $SU(3)_{flavor}$ symmetry. The lifetime of $\Xi_{c}^{+}$ is expressed in terms of measurable inclusive quantities of the other two charmed baryons belonging to the same $SU(3)_{flavor}$ multiplet in a model-independent way. In such a treatment, inclusive decay rates of singly Cabibbo suppressed decay modes have a prominent role. An analogous approach is applied to the multiplet of charmed mesons yielding interesting predictions on $D_{s}^{+}$ properties. The results obtained indicate that a more precise measurement of inclusive decay quantities of some charmed hadrons (such as $\Lambda_{c}^{+}$) that are more amenable to experiment can contribute significantly to our understanding of decay properties of other charmed hadrons (such as $\Xi_{c}^{+}$) where discrepancies or ambiguities exist.' author: - 'B. Guberina' - 'H. Štefančić' title: 'Cabibbo suppressed decays and the $\Xi_{c}^{+}$ lifetime' --- The investigation of inclusive decays and lifetimes of hadrons containing heavy quarks [@review] is already a mature subject with many fruitful applications and numerous significant achievements. The fusion of the [*Operator Product Expansion (OPE)*]{} techniques developed in the nineties [@90] with the phenomenological insights of the eighties [@80] has created a consistent framework known as [*Heavy-Quark Expansion (HQE)*]{}, within which one can systematically treat inclusive decays of heavy quarks and hadrons containing them. A host of experimental data, first on $c$ hadron decays and then, with the advent of $B$ factories, on $b$ decays, have made possible a comparison of experimental and theoretical results and revealed broad agreement with several notable exceptions [^1]. Addressing these discrepancies has become one of the most important tasks in heavy-quark physics, given that data extracted from inclusive weak decays represent an essential input in research of fundamental questions of the Standard Model (such as [*CP*]{} violation) or its extensions. Increasing quantity and quality of experimental data opens new directions in treating inclusive weak decays which may contribute to the resolution of existing problems. Consideration of inclusive weak decay rates of Cabibbo suppressed modes as individual objects (not only as a small correction to inclusive weak decay rates of Cabibbo dominant modes) supported by the application of standard symmetries (such as $SU(3)_{flavor}$ or [*Heavy-Quark Symmetry (HQS)*]{}) traces along one of these directions. As the [*HQE*]{} depends crucially on the heaviness of the decaying heavy quark, the predictions are more reliable in the sector of $b$ hadrons than in the sector of $c$ hadrons. Nevertheless, rather acceptable predictions of lifetime hierarchies and lifetime ratios were obtained in the sector of charmed hadrons too. Furthermore, very reasonable agreement was achieved in the sector of singly-charmed baryons [@GM; @PDG2000]. However, recent measurements of the $\Xi_{c}^{+}$ lifetime by [*FOCUS*]{} [@focus] and [*CLEO*]{} [@cleo] collaborations indicate substantial discrepancy between new experimental data and the presently available theoretical result [@GM; @Bigi]: $$\begin{aligned} \label{eq:exp} \left ( \tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+}) \right )_{FOCUS} & = & 2.29 \pm 0.14 \, ,\nonumber \\ \left ( \tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+}) \right )_{CLEO} & = & 2.8 \pm 0.3 \, , \nonumber \\ \left ( \tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+}) \right )_{th} & \sim & 1.3 \, .\end{aligned}$$ The results displayed above show that there is a difference by a factor of $\sim 2$ between experiment and theory. It is reasonable to pose a question whether the [*HQE*]{} can correctly describe lifetimes of singly-charmed baryons. The new experimental data on the lifetime of $\Xi_{c}^{+}$ are certainly out of reach of the calculations performed so far. However, experimental data for other singly-charmed, weakly-decaying baryons ($\Lambda_{c}^{+}$, $\Xi_{c}^{0}$, and $\Omega_{c}^{0}$) are consistent with theoretical calculations of [@GM]. However, as the data on the lifetimes of $\Xi_{c}^{0}$ and $\Omega_{c}^{0}$ are presently of marginal quality, it is not excluded that future updates of these lifetimes might disturb the agreement in the case of these baryons, too. The theoretical procedure is based on some assumptions (e.g., calculation of four-quark operator matrix elements in a nonrelativistic quark model) that may limit its explanatory power in the case of $\Xi_{c}^{+}$. Therefore, it is justified to investigate if a theoretical procedure based on the [*HQE*]{} with relaxed assumptions of analysis [@GM] can be formulated so that it might explain new experimental results. To this end, one must invoke Cabibbo suppressed modes of decay as a new source of information. Let us begin our analysis with a brief discussion about the inclusive weak decay rate. The principal result of the [*HQE*]{} is the expression for any inclusive weak decay rate of a heavy hadron given as a series of matrix elements of local operators with the inverse mass of the decaying heavy quark as an expansion parameter: $$\begin{aligned} \label{eq:master} \Gamma(H_{Q} \rightarrow f) & = & \frac{G_{F}^{2} m_{Q}^{5}}{192 \pi^{3}} \mid V \mid^{2} \frac{1}{2 M_{H_{Q}}} \\ \nonumber & & \times \left [ \sum_{D=3}^{\infty} c_{D}^{f} \frac{\langle H_{Q} \mid O_{D} \mid H_{Q} \rangle}{m_{Q}^{D-3}} \right ] \, ,\end{aligned}$$ where $D$ denotes the canonical dimension of the local operator $O_{D}$. The coefficients $c_{D}$ are calculated perturbatively (therefore given as a series in $\alpha_{S}$). $V$ stands for a product of [*CKM*]{} matrix elements appearing in a given weak decay mode. For the sake of practical calculations, one has to truncate the series at some point in the series hoping that the disregarded remainder of the series does not contribute significantly to the final result. The quality of such an approximation depends on the magnitude of the expansion parameter, i.e., on the speed of convergence of the series. The underlying hypothesis is that the inclusive hadron decay rates can be described by calculating the inclusive quark decay rates – the [*ansatz*]{} known as quark-hadron duality. The [*ansatz*]{} is not trivially obvious as one can see by inspection of the leading term in (\[eq:master\]). The decay rate is given by $\Gamma^{dec} \sim m_{Q}^{5}$ and this expression has, [*prima facie*]{}, nothing to do with the hadrons in the final states. This is, however, misleading since the summation of hadronic widths of different channels agrees with the widths computed at the quark level [^2]. Another problem stems from the matrix elements appearing in the expansion. They are dominated by nonperturbative dynamics and therefore so far there has been no systematic way of calculating them. The matrix elements of several operators of the lowest dimensions can be determined by applying [*Heavy-Quark Effective Theory (HQET)*]{}, lattice [*QCD*]{}, or, in some cases, extracted from the lepton energy spectra, but the matrix elements of some operators essential for understanding lifetime differences of heavy hadrons (e.g., four-quark operators) are still not generally calculable in such a manner, but one must recourse to quark models, which introduces the undesirable feature of model dependence. A further source of uncertainty is the heavy-quark mass $m_{Q}$. Since in the leading order the inclusive weak decay rate depends on the fifth power of $m_{Q}$, very small uncertainties in the determination of this mass parameter can lead to a significant uncertainty in the inclusive weak decay rate. Finally, using a truncated expression instead of the entire series raises the possibility of violation of quark-hadron duality [@Shifmandual; @BUdual], which emerges as another possible source of contributions beyond the present theoretical control. The [*OPE*]{} was originally formulated for deep Euclidean kinematics and its net effect was to factorize perturbative short-distance physics (Wilson coefficients) from soft, nonperturbative one (matrix elements). On the other hand, the quark-hadron duality is the concept dealing exclusively with Minkowskian dynamics [^3]. It appears that the small corrections that one safely neglects in the Euclidean regime often turn out to be enhanced in the Minkowski regime [@Shifmandual; @BUdual]. The Wilson coefficients themselves are generally not free of nonperturbative (nonlogarithmic) terms. They are generated, e.g., by small-size instantons [@Shifmandual]. Similarly, perturbative corrections appear in the soft physics of matrix elements. Generally, the truncation of the series (\[eq:master\]) in $\alpha_{s}$ and condensate terms is known to be necessary since both series are factorially divergent [@Mueller]. Therefore, a “practical” calculation at any given order $\alpha_{s}^{m}$ and $m_{Q}^{-n}$ will have a “natural uncertainty” coming from the higher-order terms $\alpha_{s}^{m+1}$ and $m_{Q}^{-(n+1)}$. The “natural uncertainty” also includes the ordinary uncertainties like the uncertainties in quark masses, $\Lambda_{QCD}$, etc. The uncertainties beyond this “natural uncertainty” are considered to be violations of quark-hadron duality. Resolutions of the problems stated above presumably lead to the explanation of discrepancies between present experimental and theoretical results. Since the contributions of higher-dimensional operators, uncertainties in matrix elements and $m_{Q}$ as well as effects of duality violation are all intertwined in the full expression for the weak decay rate, it is very difficult to distinguish precisely which of these factors causes the problem and should be improved accordingly. One possible strategy is to eliminate or reduce the importance of all (in practice as many as possible) factors but one in order to test the influence of the remaining factor. In this paper we adopt this strategy and implement it using symmetries in multiplets of heavy hadrons. Investigations along similar lines (connecting the charmed with the beauty sector) were performed in [@Vol; @GMS1; @GMS2]. The standard procedure of truncating the series (\[eq:master\]) is to keep operators of dimensions 3 (decay operator) and 5 (chromomagnetic operator) [^4], which are insensitive to the light-quark content of the heavy hadron (at least on the quark-gluon operator level). Operators of dimension 6, which are sensitive to flavors of light quarks (four-quark operators), also have to be kept in order to describe lifetime differences within multiplets of heavy hadrons. The effects of four-quark operators (clearly presented in [@GM]) are traditionally referred to as W-exchange, positive and negative Pauli interference in baryons, and W-annihilation, W-exchange, and negative Pauli interference in mesons. We shall adopt this procedure along with the assumption of $SU(3)_{flavor}$ symmetry at the level of matrix elements. The validity of this assumption and its influence on the final result will be discussed below. We start by expressing decay rates of individual Cabibbo modes for singly-charmed baryons within the framework that we have set. As already mentioned, operators of dimension 3 and 5 are insensitive to the light-(anti)quark content of a heavy hadron. Nevertheless, their coefficients comprise a phase-space correction coming from the fact that some of the resulting quarks in the decay of a heavy quark have a nonnegligible mass compared with the heavy-quark mass. Thus, contributions of operators of dimensions 3 and 5 have slightly different values in the treatment of various Cabibbo modes of the decay of the heavy quark. In the case of $c$ quark decays, these corrections are generally not large and we shall neglect them in our initial treatment. Their effect will be taken into account in the discussion of our results. The assumption of $SU(3)_{flavor}$ symmetry guarantees that the matrix elements of operators of dimension 3 and 5 are the same for all hadrons in any $SU(3)_{flavor}$ multiplet of heavy hadrons. These approximations allow us to describe the contribution of the aforementioned operators with a single quantity $\Gamma_{35}$ in all Cabibbo modes, for all members of the multiplet, up to the product of the [*CKM*]{} matrix elements specific for every individual Cabibbo mode. The coefficients of four-quark operators also include phase-space corrections owing to the massive particles in the final state of the decay of the heavy quark. In this case, however, these corrections are at the percent level and can be safely disregarded in $c$ quark decays. The contributions of these operators of dimension 6 for the case of baryons can then be expressed in terms of several parameters (under the assumption of $SU(3)_{flavor}$ symmetry) related to the aforementioned types of four-quark effects: W-exchange ($\Gamma_{exch}$), negative Pauli interference ($\Gamma_{negint}$), and positive Pauli interference ($\Gamma_{posint}$), again up to the [*CKM*]{} matrix elements. Analogous claims are valid in the case of charmed meson decays. We should emphasize that $\Gamma$’s are conveniently chosen combinations of products of coefficients and operator matrix elements which appear in expressions for the inclusive weak decay rates of all Cabibbo modes. As we do not engage in a direct calculation of matrix elements, all these matrix elements can be taken as determined at the same scale $\mu$, i.e., there is no need for the hybrid renormalization in the case of four-quark operators. One needs to know nothing else about the matrix elements of the operators. In such a suitably defined theoretical environment one can express inclusive decay rates in a straightforward manner. The decay rates for nonleptonic modes are $$\begin{aligned} \label{eq:nllambda} \frac{\Gamma^{c \rightarrow s \overline{d} u} (\Lambda_{c}^{+})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{exch} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(\Lambda_{c}^{+})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(\Lambda_{c}^{+})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{exch} + \Gamma_{negint} + \Gamma_{posint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(\Lambda_{c}^{+})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{negint} \end{aligned}$$ for $\Lambda_{c}^{+}$, $$\begin{aligned} \label{eq:nlxiplus} \frac{\Gamma^{c \rightarrow s \overline{d} u}(\Xi_{c}^{+})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(\Xi_{c}^{+})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{negint} + \Gamma_{posint} + \Gamma_{exch} \nonumber \, , \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(\Xi_{c}^{+})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(\Xi_{c}^{+})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{exch} + \Gamma_{negint}\end{aligned}$$ for $\Xi_{c}^{+}$, and $$\begin{aligned} \label{eq:nlxi0} \frac{\Gamma^{c \rightarrow s \overline{d} u}(\Xi_{c}^{0})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(\Xi_{c}^{0})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{exch} \nonumber \, , \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(\Xi_{c}^{0})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(\Xi_{c}^{0})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{posint} + \Gamma_{exch} \end{aligned}$$ for $\Xi_{c}^{0}$. For the decay rates of the semileptonic modes one obtains ($l = e, \mu$) $$\begin{aligned} \label{eq:sllambda} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(\Lambda_{c}^{+})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(\Lambda_{c}^{+})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL} + \Gamma_{posint}^{SL}\end{aligned}$$ for $\Lambda_{c}^{+}$, $$\begin{aligned} \label{eq:slxiplus} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(\Xi_{c}^{+})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} +\Gamma_{posint}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(\Xi_{c}^{+})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL}\end{aligned}$$ for $\Xi_{c}^{+}$, and $$\begin{aligned} \label{eq:slxi0} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(\Xi_{c}^{0})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} +\Gamma_{posint}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(\Xi_{c}^{0})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL} + \Gamma_{posint}^{SL}\end{aligned}$$ for $\Xi_{c}^{0}$. One can introduce the following notation for the [*CKM*]{} matrix elements: $|V_{cs}|^2 = |V_{ud}|^2 = (\cos \theta_{c})^2 \equiv c^2$ and $|V_{cd}|^2 = |V_{us}|^2 = (\sin \theta_{c})^2 \equiv s^2$. Combining relations from (\[eq:nllambda\]) and (\[eq:nlxiplus\]), one obtains $$\begin{aligned} \label{eq:nllamxi} \Gamma^{c \rightarrow s \overline{d} u} (\Xi_{c}^{+}) & = & \frac{c^2}{s^2} \left ( \Gamma^{c \rightarrow s \overline{s} u} (\Lambda_{c}^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (\Lambda_{c}^{+}) \right ) \nonumber \\ & - & \Gamma^{c \rightarrow s \overline{d} u} (\Lambda_{c}^{+}) \, , \nonumber \\ \Gamma^{c \rightarrow s \overline{s} u} (\Xi_{c}^{+}) & + & \Gamma^{c \rightarrow d \overline{d} u} (\Xi_{c}^{+}) = \nonumber \\ & & \Gamma^{c \rightarrow s \overline{s} u} (\Lambda_{c}^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (\Lambda_{c}^{+}) \, , \nonumber \\ \Gamma^{c \rightarrow d \overline{s} u} (\Xi_{c}^{+}) & = & \frac{s^4}{c^4} \Gamma^{c \rightarrow s \overline{d} u} (\Lambda_{c}^{+})\end{aligned}$$ for the nonleptonic decay rates and from (\[eq:sllambda\]), (\[eq:slxiplus\]), and (\[eq:slxi0\]) we have $$\label{eq:slbar} \Gamma_{SL}(\Xi_{c}^{+}) = \Gamma_{SL}(\Xi_{c}^{0}) + \frac{s^2}{c^2} ( \Gamma_{SL}(\Lambda_{c}^{+}) - \Gamma_{SL}(\Xi_{c}^{0}))$$ for the semileptonic decay rates, where $\Gamma_{SL} (X) = \Gamma^{c \rightarrow s \overline{l} \nu_{l}} (X) + \Gamma^{c \rightarrow d \overline{l} \nu_{l}} (X)$, $X = \Xi_{c}^{+}, \Xi_{c}^{0}, \Lambda_{c}^{+}$. Expressions (\[eq:nllamxi\]) and (\[eq:slbar\]) show that all contributions to the total inclusive weak decay rate of $\Xi_{c}^{+}$ are expressed in terms of some of the analogous contributions of $\Lambda_{c}^{+}$ and $\Xi_{c}^{0}$. In this way, we have succeeded in expressing the lifetime of a “problematic” baryon $\Xi_{c}^{+}$ in terms of quantities of “nonproblematic” baryons $\Lambda_{c}^{+}$ and $\Xi_{c}^{0}$. If we introduce the notation $$\label{eq:brlambda} BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+}) = \frac{\left ( \Gamma^{c \rightarrow s \overline{s} u} (\Lambda_{c}^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (\Lambda_{c}^{+}) \right )} {\Gamma_{TOT}(\Lambda_{c}^{+})} \, ,$$ the final expression (after neglecting all terms $\sim s^{4}$) for the ratio specified in (\[eq:exp\]) becomes $$\begin{aligned} \label{eq:ratioxilam} \frac{\tau(\Xi_{c}^{+})}{\tau(\Lambda_{c}^{+})} & = & \left[ -1 + \left( 2 + \frac{c^2}{s^2} \right) BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+}) \right. \nonumber \\ & & + \left. 2 \left( 1 - \frac{s^2}{c^2} \right) \frac{\tau(\Lambda_{c}^{+})} {\tau(\Xi_{c}^{0})} BR_{SL} (\Xi_{c}^{0}) + 2 \left( 1 + \frac{s^2}{c^2} \right) BR_{SL} (\Lambda_{c}^{+}) \right] ^{-1} \, .\end{aligned}$$ This type of analysis can be extended to the sector of charmed mesons. The hierarchy of charmed meson lifetimes is in general well understood in the framework of the [*HQE*]{} [@Bigi95], although some disrepancies exist that motivate alternative approaches [@Nussinov] and raise corresponding controversies [@Bigi2001]. We shall pursue our analysis in the framework of [*HQS*]{} and perform a model-independent analysis. This analysis, apart from its intrinsic value as a contribution to the understanding of charmed meson lifetimes, can also be a testing ground of our approach because of a higher quality of available experimental data for charmed mesons. Therefore, we conduct our analysis on a $SU(3)_{flavor}$ antitriplet of charmed mesons. The inclusive weak decay rates for individual Cabibbo nonleptonic decay modes are ($\Gamma$’s used in the mesonic case are different from those used in the baryonic case although the notation is very similar) $$\begin{aligned} \label{eq:nlDplus} \frac{\Gamma^{c \rightarrow s \overline{d} u} (D^{+})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(D^{+})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(D^{+})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{ann} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(D^{+})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{ann} \end{aligned}$$ for $D^{+}$, $$\begin{aligned} \label{eq:nlD0} \frac{\Gamma^{c \rightarrow s \overline{d} u}(D^{0})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(D^{0})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(D^{0})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{exch} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(D^{0})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{exch} \end{aligned}$$ for $D^{0}$, and $$\begin{aligned} \label{eq:nlDsplus} \frac{\Gamma^{c \rightarrow s \overline{d} u}(D_{s}^{+})} {|V_{cs}|^2 |V_{ud}|^2} & = & \Gamma_{35} + \Gamma_{ann} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow s \overline{s} u}(D_{s}^{+})} {|V_{cs}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{ann} + \Gamma_{negint} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{d} u}(D_{s}^{+})} {|V_{cd}|^2 |V_{ud}|^2} & = & \Gamma_{35} \, , \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{s} u}(D_{s}^{+})} {|V_{cd}|^2 |V_{us}|^2} & = & \Gamma_{35} + \Gamma_{negint}\end{aligned}$$ for $D_{s}^{+}$. For the decay rates of the semileptonic modes one obtains ($l = e, \mu$) $$\begin{aligned} \label{eq:slDplus} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(D^{+})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(D^{+})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL} + \Gamma_{ann}^{SL}\end{aligned}$$ for $D^{+}$, $$\begin{aligned} \label{eq:slD0} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(D^{0})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(D^{0})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL}\end{aligned}$$ for $D^{0}$, and $$\begin{aligned} \label{eq:slDsplus} \frac{\Gamma^{c \rightarrow s \overline{l} \nu_{l}}(D_{s}^{+})} {|V_{cs}|^2} & = & \Gamma_{35}^{SL} +\Gamma_{ann}^{SL} \nonumber \\ \frac{\Gamma^{c \rightarrow d \overline{l} \nu_{l}}(D_{s}^{+})} {|V_{cd}|^2} & = & \Gamma_{35}^{SL}\end{aligned}$$ for $D_{s}^{0}$. Combining relations from (\[eq:nlDplus\]) and (\[eq:nlDsplus\]), one obtains $$\begin{aligned} \label{eq:nlDplusDsplus} \Gamma^{c \rightarrow s \overline{d} u} (D_{s}^{+}) & = & \frac{c^2}{s^2} \left ( \Gamma^{c \rightarrow s \overline{s} u} (D^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (D^{+}) \right ) \nonumber \\ & & - \Gamma^{c \rightarrow s \overline{d} u} (D^{+}) \, , \nonumber \\ \Gamma^{c \rightarrow s \overline{s} u} (D_{s}^{+}) & + & \Gamma^{c \rightarrow d \overline{d} u} (D_{s}^{+}) = \nonumber \\ & & \Gamma^{c \rightarrow s \overline{s} u} (D^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (D^{+}) \, , \nonumber \\ \Gamma^{c \rightarrow d \overline{s} u} (D_{s}^{+}) & = & \frac{s^4}{c^4} \Gamma^{c \rightarrow s \overline{d} u} (D^{+})\end{aligned}$$ for the nonleptonic decay rates and from (\[eq:slDplus\]), (\[eq:slD0\]) and (\[eq:slDsplus\]) we have $$\label{eq:slmes} \Gamma_{SL}(D_{s}^{+}) = \Gamma_{SL}(D^{0}) + \frac{c^2}{s^2} (\Gamma_{SL}(D^{+}) - \Gamma_{SL}(D^{0}))$$ for the semileptonic decay rates, where $\Gamma_{SL} (X) = \Gamma^{c \rightarrow s \overline{l} \nu_{l}} (X) + \Gamma^{c \rightarrow d \overline{l} \nu_{l}} (X)$, $X = D^{+}, D^{0}, D_{s}^{+}$. Expressions (\[eq:nlDplusDsplus\]) and (\[eq:slmes\]) show that all contributions to the total inclusive weak decay rate of $D_{s}^{+}$ are expressed in terms of some of the analogous contributions of $D^{+}$ and $D^{0}$. Let us comment briefly on the findings of [@slCher; @slShif; @slVol] which indicate that the [*HQE*]{} could not reproduce semileptonic inclusive widths of charmed mesons. Let us point out that although the expressions for semileptonic inclusive decay widths are calculated using the [*HQE*]{}, the relations among them (such as (\[eq:slmes\])) simply state that inclusive semileptonic widths for all three charmed mesons are very close, which is satisfied very well experimentally [@PDG2000]. Therefore, the possibility that the [*HQE*]{} does not describe semileptonic inclusive widths ideally (although contributions of higher dimensional operators should be investigated before making this statement definite) does not bare a consequence on our final results which depend only on the relations among semileptonic decay widths. If we introduce the notation $$\label{eq:brDplus} BR_{\Delta C =-1, \Delta S = 0} (D^{+}) = \frac{\left ( \Gamma^{c \rightarrow s \overline{s} u} (D^{+}) + \Gamma^{c \rightarrow d \overline{d} u} (D^{+}) \right )} {\Gamma_{TOT}(D^{+})} \, ,$$ we obtain the following final expression (after neglecting terms $\sim s^{4}$) for the ratio of lifetimes of $D^{+}$ and $D^{0}$ mesons $$\begin{aligned} \label{eq:ratioDplusDsplus} \frac{\tau(D^{+})}{\tau(D_{s}^{+})} (1 - BR_{\tau}(D_{s}^{+} )) & = & -1 + \left( 2 + \frac{c^2}{s^2} \right) BR_{\Delta C =-1, \Delta S = 0} (D^{+}) \nonumber \\ & + & 2 \left( 1 - \frac{c^2}{s^2} \right) \frac{\tau(D^{+})} {\tau(D^{0})} BR_{SL} (D^{0}) + 2 \left( 1 + \frac{c^2}{s^2} \right) BR_{SL} (D^{+}) \, , $$ where $BR_{\tau}(D_{s}^{+} )$ denotes the branching ratio of the leptonic $D_{s}^{+} \rightarrow \tau^{+} \nu_{\tau}$ decay [^5]. Once we have obtained the results (\[eq:ratioxilam\]) and (\[eq:ratioDplusDsplus\]), we can clearly see their theoretical and experimental appeal. These relations have an intrinsic value since they express the lifetime of one charmed hadron in terms of measurable quantities of other two charmed hadrons belonging to the same $SU(3)_{flavor}$ multiplet. This result represents exploitation of advantages of the [*HQE*]{} at a new deeper level. The approach that leads to (\[eq:ratioxilam\]) and (\[eq:ratioDplusDsplus\]) also suppresses some of the problems referred to in the introduction. Let us briefly discuss these problems in the light of our approach. The problem of convergence seems rather important in charmed baryon decays. The operators of the lowest dimension in (\[eq:master\]), which are neglected in our approach, are some operators of dimension 6 (which are insensitive to the light content of the heavy hadron) followed by the operators of dimension 7 and higher. In our approach, all operators that are insensitive to the light content of heavy hadrons give the same contribution to the inclusive weak decay rate of each Cabibbo mode (up to the [*CKM*]{} matrix elements) and for every hadron within a given $SU(3)_{flavor}$ multiplet. If we look at the relations (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]) as relations between exact inclusive weak decay rates for individual Cabibbo modes (and not only as approximations with several lowest dimensional operators), we can see that contributions of all light-flavor insensitive operators (of any dimension) get cancelled. Thus, these relations are correct up to the contributions of higher light-flavor sensitive operators. Since apart from four-quark operators there are other operators of dimension 6 in (\[eq:master\]) but they are all light-flavor insensitive, the aforementioned relations get the first correction from those operators of dimension 7 (or higher) which are light-flavor sensitive. Therefore, relations (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]) are in the form that ameliorates the convergence issue. The phase-space corrections represent corrections which are different in various Cabibbo modes, depending on the number of massive quarks in the final state. Still, relations (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]) are in such a form that the effect of phase space is significantly reduced. Let us consider the first equation of (\[eq:nllamxi\]): the sum of decay rates of two modes with one $s$ quark in the final state equals (up to the [*CKM*]{} matrix elements) the sum of decay rates of modes with two and zero $s$ quarks in the final state. Numerical values of the phase-space corrections to operators of dimensions 3 and 5 [@Bigi95] indicate that the sum of corrections for two $s$ quarks and zero $s$ quarks in the final state is very close to the double correction for one $s$ quark in the final state. The effects of phase-space corrections largely cancel. A similar situation appears in all other relations in (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]). Therefore, inclusion of phase-space corrections does not notably worsen the accuracy of the aforementioned relations. The problem of calculating matrix elements is in our approach completely avoided. From the span of lifetimes of charmed hadrons [@PDG2000] it is clear that four-quark operators must play a very prominent role. Since, in contradistinction to operators of dimension 3 and 5, the matrix elements of four-quark operators cannot be calculated in a model-independent way, it is clear that even a modest inaccuracy in their determination may lead to significant deviations from the correct result. Moreover, a recent analysis [@Voloshin] indicates that there might exist serious deviations from some standard approximations, such as the valence quark approximation. Evading these pitfalls is one of the greatest advantages of our approach. Another advantage is that all crucial relations in this paper do not depend on the heavy quark mass $m_{Q}$ in the case when the assumed symmetries apply. In the realistic case, the form of relations (\[eq:nllamxi\]), (\[eq:slbar\]), (\[eq:nlDplusDsplus\]), and (\[eq:slmes\]) reduces the dependence of results on $m_{Q}$ significantly (to the level of breaking of underlying symmetries). Finally, there remains the assumption on $SU(3)_{flavor}$ symmetry. The effects of breaking of this symmetry were analyzed in [@GM]. From that analysis one can conclude that the effects of $SU(3)_{flavor}$ breaking are generally less than $30\%$ and probably significantly smaller. Therefore, we expect the same level of accuracy in our treatment, too. After the discussion of theoretical advantages and limitations of our approach there remains an important problem of confrontation of theoretical findings with experimental values. From the final relation for baryons (\[eq:ratioxilam\]) and mesons (\[eq:ratioDplusDsplus\]) it is evident that theoretical predictions depend on the branching ratios of the singly Cabibbo suppressed nonleptonic modes $BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+})$ and $BR_{\Delta C =-1, \Delta S = 0} (D^{+})$, respectively. These values are not available from experiment and their determination represents a crucial step in numerical analysis. An estimate of these quantities can be obtained indirectly from exclusive modes and depends on the quality of data for these modes. From the flavor quantum numbers of the final decay products in heavy-hadron decays one can determine which Cabibbo mode governed that particular decay at the quark level. The only exceptions are the modes $c \rightarrow s \overline{s} u$ and $c \rightarrow d \overline{d} u$ which lead to the final hadronic state with the same flavor quantum numbers. However, this fact does not pose a problem since in all expressions the decay rates of these two modes appear in the form of sum and therefore there is no need to make difference between them. From the flavor quantum numbers of the final states of any particular exclusive mode one can determine whether it was governed by the Cabibbo dominant, singly Cabibbo suppressed, or doubly Cabibbo suppressed nonleptonic modes at the quark level. An analogous conclusion follows for semileptonic decays. It is, therefore, possible to obtain a decay rate for any Cabibbo inclusive mode (all decay channels coming from the same Cabibbo mode at the quark level) by summing the decay rates of associated exclusive modes. In performing this procedure one encounters the effect of quantum interference. Namely, different final states originating from the same quark decay mode can mix owing to final state strong interactions. The most notable manifestation of this effect is that summing of the branching ratios of all exclusive modes taken from [@PDG2000] can lead to a result well over $100 \%$ (e.g., for $D^{0}$ or $D^{+}$). To minimize this effect, we invoke the following procedure: we calculate the inclusive decay rate of singly Cabibbo suppressed modes by summing the decay rates of all appropriate exclusive decay modes; then we calculate the [*total decay rate*]{} by summing decay rates of [*all*]{} exclusive modes and then divide the two numbers to obtain the wanted ratio. Using the sum of all exclusive modes instead of the measured lifetime for the total decay rate insures the same treatment of interference effects in both quantities in the ratio. Other quantities appearing in the expressions (\[eq:ratioxilam\]) and (\[eq:ratioDplusDsplus\]) are lifetimes and semileptonic branching ratios, which are a standard part of information on any weakly decaying particle. In general, they are well measured and available in [@PDG2000]. Let us first consider the presently very interesting question of the $\tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+})$ ratio. The sum of branching ratios of all measured exclusive decay modes is approximately $50 \%$ which shows that the set of available decay modes is not complete. The branching ratio $BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+})$ is obtained at the level of $0.0295 \pm 0.0115$, which is probably an underestimated result since only a few exclusive modes corresponding to singly Cabibbo suppressed modes are available [@PDG2000]. Another problem is the lack of data on the semileptonic branching ratio of the $\Xi_{c}^{0}$ baryon. This value can be taken from [@GM] to be $BR_{SL}(\Xi_{c}^{0}) = (0.092 \pm 0.006)$. As the contribution coming from $BR_{SL}(\Xi_{c}^{0})$ is the nonleading one (the leading one coming from the $BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+})$), this mixing of theoretical and experimental results does not introduce a significant model dependence. Still, only the future arrival of experimental data on $BR_{SL}(\Xi_{c}^{0})$ will complete the set of experimental values needed for a fully consistent analysis. The rest of the data is taken to be [@PDG2000]: $BR_{SL}(\Lambda_{c}^{+}) = (0.045 \pm 0.017)$, $\tau(\Lambda_{c}^{+}) = (0.206 \pm 0.012) \, \rm ps$, and $\tau(\Xi_{c}^{0}) = (0.098 \pm 0.019) \, \rm ps$. The analysis using the set of parameters specified above yields a result for the $\tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+})$ ratio which is far above the new experimental results and has a very large error. The principal reason for such a result can be seen from (\[eq:ratioxilam\]). The value of $BR_{\Delta C =-1, \Delta S = 0} (\Lambda_{c}^{+})$ is multiplied by a large factor $c^2/s^2$, which makes the final result very sensitive to the value of this branching ratio. The conclusion stemming from this analysis is that the presently available data on $\Lambda_{c}^{+}$ exclusive modes are insufficiently accurate and abundant to insure a reliable result. A more extensive and precise measurement of exclusive decay modes of $\Lambda_{c}^{+}$ (especially Cabibbo suppressed ones) can however lead to interesting new predictions on $\Xi_{c}^{+}$. Numerical analysis in the sector of charmed meson decays is more promising. Addition of the branching ratios of all exclusive modes of $D^{+}$ gives a value of $110 \%$, which shows that the data on exclusive decay modes of $D^{+}$ can be considered complete. The branching ratio $BR_{\Delta C =-1, \Delta S = 0} (D^{+})$ attains the value $0.140 \pm 0.026$. Other values taken from [@PDG2000] are $BR_{SL}(D^{+}) = 0.172 \pm 0.019$, $BR_{SL}(D^{0}) = 0.0675 \pm 0.0029$, $\tau(D^{+}) = (1.051 \pm 0.013) \, \rm ps$, and $\tau(D^{0}) = (0.4126 \pm 0.0028) \, \rm ps$. Using the expression (\[eq:ratioDplusDsplus\]) one obtains the value $(\tau(D^{+})/\tau(D_{s}^{+}))(1-BR_{\tau}(D^{+}_{s}))_{th} = 2.63 \pm 0.98$. This result obtained from theoretical considerations can be compared with the value for the same quantity following from the experiment. To this end, we use the experimental values [@PDG2000]: $\tau(D_{s}^{+}) = (0.496 \pm 0.0095) \rm ps$ and $BR_{\tau}(D_{s}^{+}) = 0.07 \pm 0.04$. This leads to a value $(\tau(D^{+})/\tau(D_{s}^{+}))(1-BR_{\tau}(D^{+}_{s}))_{exp} = 1.971 \pm 0.096$. Comparison of these two results shows that they are consistent within their errors. A relatively large error of the result obtained through relation (\[eq:ratioDplusDsplus\]) originates to a great extent from the expression (\[eq:slmes\]) where the inclusive semileptonic decay rate of $D_{s}^{+}$ is expressed in terms of respective quantities for the other two charmed mesons. In this relation a large factor $c^{2}/s^{2}$ multiplies a small quantity $\Gamma_{SL}(D^{+}) - \Gamma_{SL}(D^{0})$ (the inclusive decay rates for these two charmed mesons are numerically very close). In the final expression, this fact contributes very little to the central value, but gives a significant contribution to the error since $\Gamma_{SL}(D^{+})$ and $\Gamma_{SL}(D^{0})$ are treated as independent quantities and their individual errors are significantly larger than their difference. The consequences of these facts can be better observed if one performs the following analysis. For the sake of error analysis, we take that $\Gamma_{SL}(D^{+})$ and $\Gamma_{SL}(D^{0})$ are identically equal (while in reality they differ by the small Cabibbo suppressed correction). This approximation removes the problematic term of a large factor multiplying a small quantity. This procedure changes the central value at the permille level while the error is almost halved (even with this reduced errors our two results are in a 2$\sigma$ interval). The procedures presented so far are by no means restricted to the calculation of the lifetimes of $\Xi_{c}^{+}$ and $D_{s}^{+}$. Any inclusive quantity (such as semileptonic branching ratios) for these hadrons can be expressed by means of inclusive quantities of the other two charmed hadrons belonging to the same multiplet. Similar relations can also be established in multiplets of $b$ hadrons bearing in mind that, e.g., phase-space corrections in the $b$ case can be substantial. Nevertheless, the full success of this approach is dependent on accumulation of experimental data [^6] and measurement of inclusive decay rates of suppressed decay modes. Considerations displayed in this paper are motivated by recent experimental results on charmed baryon lifetimes and the need to establish whether a standard existing formalism can be brought into agreement with these results by eliminating or reducing some of its uncertainties. Our formalism procures model-inedependent results with the assumption of some symmetries. Apart from these desirable properties, the theoretical appeal of our approach consists in expressing some measurable quantity of a heavy hadron in terms of measurable quantities of other heavy hadrons from the same $SU(3)_{flavor}$ multiplet. This feature enables us to set a new course in testing the formalism of inclusive weak decays. Using relations such as (\[eq:ratioxilam\]) and (\[eq:ratioDplusDsplus\]) one can use the data for those hadrons the decays of which are more amenable to experimental determination to produce predictions for hadrons where experimental data lack or need theoretical verification (like in the $\Xi_{c}^{+}$ case). As any advantage, this one has its price, too. One has to introduce inclusive decay rates of singly Cabibbo suppressed modes which so far have not been measured (as inclusive modes). Use of data on exclusive decay modes can give a reasonable estimate of necessary decay rates. Nevertheless, the full strength of our approach would manifest itself if direct measurements of inclusive decay rates of singly Cabibbo suppressed modes of $\Lambda_{c}^{+}$ should be possible in the near future. Even better and more detailed data on exclusive decay modes of $\Lambda_{c}^{+}$ could improve our understanding of new experimental data on the $\Xi_{c}^{+}$ lifetime. The real challenge now faces the experimental community. There is a clear indication that by measuring the parameters of one heavy hadron ( $\Lambda_{c}^{+}$) we can draw definite conclusions on the other heavy hadron ($\Xi_{c}^{+}$). These conclusions may clarify the question of applicability of the [*HQE*]{} in charmed decays or at least decide whether $\Xi_{c}^{+}$ really fits into the, so far successful, description of charmed baryon lifetime hierarchy. [**Acknowledgments**]{} The authors would like to thank the referee for kind suggestions which have improved the comprehensibility of the paper. This work was supported by the Ministry of Science and Technology of the Republic of Croatia under the contract No. 0098002. [88]{} I. Bigi, The Lifetimes of Heavy Flavor Hadrons - a Case Study in Quark-Hadron Duality, in: Proc. $\rm 3^{rd}$ Intern. Conf. on B Physics and CP Violation, Taipei, Taiwan, December 3-7, 1999, to appear, hep-ph/0001003;\ M. Neubert, B Decays and the Heavy-Quark Expansion, in: A.J. Buras, M. Lindner (Eds.), Heavy Flavors II, p. 239, hep-ph/9702375;\ Bellini, Bigi, Dornan, Phys. Rep. 289 (1997) 1. B. Blok, M. Shifman: Lifetimes of Charmed Hadrons Revised - Facts and Fancy, in: J. Kirkby, R. Kirkby (Eds.), Proc. Workshop on the Tau-Charm Factory, p. 247, Marbella, Spain, 1993, Editions Frontieres, Gif-sur-Yvette, 1994, hep-ph/9311331; N. Uraltsev in Heavy Flavor Physics: A Probe of Nature’s Great Design, Proceedings of the International School of Physics “Enrico Fermi”, Course CXXXVII, Varenna, July 7-18, 1997, I. Bigi, L. Moroni (Eds.), IOS Press, Amsterdam, 1998) p. 329, hep-ph/9804275. B. Guberina, S. Nussinov, R.D. Peccei, R. Rückl, Phys. Lett. B 89 (1979) 111;\ N. Bilić, B. Guberina, J. Trampetić, Nucl. Phys. B 248 (1984) 261;\ M. Shifman, M.B. Voloshin, Yad. Fiz. 41 (1985) 463 \[Sov. J. Nucl. Phys. 41 (1985) 120\];\ M. Shifman, M.B. Voloshin, Zhetf 91 (1986) 1180 \[JETP 64 (1986) 698\];\ B. Guberina, R. Rückl, J. Trampetić, Z. Phys. C 33 (1986) 297. B. Guberina, B. Melić, Eur. Phys. J. C 2 (1998) 697. D. E. Groom et al.(Particle Data Group), Eur. Phys. J. C 15 (2000) 1. J.M. Link et al. (FOCUS collaboration), Phys. Lett. B 523 (2001) 53. A.H. Mahmood et al. (CLEO collaboration), Phys. Rev. D 65 (2002) 031102. I. Bigi, hep-ph/0111182 I. Bigi, N. Uraltsev, Phys. Lett. B 457 (1999) 163; Phys. Rev. D 60 (1999) 114034. R. Lebed, N. Uraltsev, Phys. Rev. D 62 (2000) 094011. M. Shifman,to be published in the Boris Ioffe Festschrift “At the Frontier of Particle Physics/Handbook of QCD”, Ed. M. Shifman, World Scientific, Singapore, 2001, hep-ph/0009131. I. Bigi, N. Uraltsev, Int. J. Mod. Phys. A 16 (2001) 5201. A.H. Mueller, in “QCD: 20 Years Later”, Eds. P.M. Zerwas and H.A. Kastrup, World Scientific, Singapore, 1993, p. 162. M.B. Voloshin, Phys. Rep. 320 (1999) 275. B. Guberina, B. Melić, H. Štefančić, Phys. Lett. B 469 (1999) 253. B. Guberina, B. Melić, H. Štefančić, Phys. Lett. B 484 (2000) 43. I. Bigi, UND-HEP-95-BIG02, hep-ph/9508408. S. Nussinov, M.V. Purohit, Phys. Rev. D 65 (2002) 034018. I. Bigi, UND-HEP-01-BIG08, hep-ph/0112155. V. Chernyak, Nucl. Phys. B 457 (1995) 96. B. Blok, R.D. Dikeman, M.A. Shifman, Phys. Rev. D 51 (1995) 6167. M.B. Voloshin, TPI-MINN-02/01-T, UMN-TH-2041-02, hep-ph/0202028. M.B. Voloshin, Phys. Rev. D 61 (2000) 074026. V. Eiges (BELLE collaboration), private communication with B. Guberina. [^1]: Like the still present problem of the $\tau(\Lambda_{b}^{0})/\tau(B_{d}^{0})$ ratio or the recently escalating problem of the $\tau(\Xi_{c}^{+})/\tau(\Lambda_{c}^{+})$ ratio. [^2]: As nicely demonstrated in $(1 + 1)$-dimensional [*QCD*]{} [@11QCD1; @11QCD2; @Shifmandual]. [^3]: Therefore, it cannot be studied in lattice QCD, which is essentially a numerical Euclidean approach. [^4]: there are no operators of dimension 4 owing to color-gauge invariance [^5]: This mode contributes significantly only to the decays of the $D_{s}^{+}$ meson and therefore cannot be related to the analogous decay rates of other members of the $SU(3)_{flavor}$ multiplet. [^6]: The upcoming high-statistics measurements, especially for charmed baryons [@private], are in this respect very encouraging.

ArXiv

--- author: - William Sacks - Alain Mauger - Yves Noat bibliography: - 'library.bib' title: | From Cooper-pair glass to unconventional superconductivity:\ a unified approach to cuprates and pnictides --- Despite its wide applications, the BCS theory  [@PR_BCS1957] fails to account for the physical properties of a large variety of high-$T_c$ superconductors (SC), the cuprate family, but also the more recent iron-based superconductors. A striking feature of these materials is the proximity to an insulating phase, whether anti-ferromagnetic (cuprates), spin density wave (iron based SC, Bechgard salts) or localization (ultra-thin films). Just beyond the insulating phase, the SC dome appears in the phase diagram as a function of carrier concentration between two critical points. Understanding the transition from such an insulating to SC state is still a major challenge. Microscopic measurements reveal an unconventional quasiparticle (QP) dispersion, the ‘peak-dip-hump’ structure [@RevModPhys_Fisher2007], often attributed to the coupling to a collective mode[@AdvPhys_Eschrig2006; @PRB_Berthod2013; @PRL_Ahmadi2011; @Chilett_Jing2015; @PRL_Chi2012; @PRL_Fasano2010]. Although the peak to dip energy follows both the neutron resonance and $T_c$ as a function of doping [@Trends_Zaza2002; @CurrOp_Song2013; @Chilett_Jing2015], the finer shape of the QP spectra and their temperature dependence remain a challenge. Moreover, in the temperature range \[$T_c$,$T^*$\] a pseudogap (PG) state persists, having a Fermi-level gap $\Delta_p$ much larger than the critical energy scale $k_B\,T_c$ in cuprates (see [@Nat_Hashimoto2014] and ref. therein) and also in iron-based SC [@Nat_Xu2011; @NJPhys_Kwon2012; @PRB_Shimojima2014]. to 5.0 cm[ ![Illustration of the boson-fermion PPI model [@SSciTech_Sacks2015]. The SC ground state (at $\Delta_p$) has two distinct types of excitations: a distribution $\Delta_k^i$ of pair (boson) excitations, left panel, and quasiparticle (fermion) excitations. We demonstrate the strong coupling of the condensate QP of energy $E^c_k$ to pre-existing excited pair states of equal energy $\Delta_k^j$. The composite object is called a ‘super-quasiparticle’.[]{data-label="Fig1"}](condensate.png "fig:"){width="8.6"}]{} In this letter, these questions are addressed within the pair-pair interaction model (PPI). We show that the main unconventional features of high-$T_c$ SC can be understood in a microscopic theory wherein incoherent pairs in the [*Cooper-glass state*]{} interact to form the coherent superconducting state. As a result of this novel PPI, the quasiparticles become coupled to the excited pair states (see Fig.\[Fig1\]). These ‘super-quasiparticles’ give rise to an unconventional excitation spectrum wherein the gap function in the SC state is energy dependent but non-retarded. The theory is in full agreement with the experimental spectra on cuprates and pnictides, despite the order of magnitude variation in the energy gap. The results point to a universal mechanism in high-$T_c$ driven by the interaction between pairs, giving key physical quantities such as the condensation energy and elementary excitations, as a function of temperature and doping. In particular, the ‘peak-dip-hump’ originates from instantaneous electron interactions, thus discarding a bosonic mode as its origin in these materials. [*Microscopic model*]{}. The hamiltonian describes normal electrons coexisting with interacting preformed pairs: $$\label{ham1} H = H_{0} + H_{pair}+H_{int}$$ where the first term $H_{0}$ describes the normal metal phase, and the second term is the pairing hamiltonian: $$\label{hpair} H_{pair} = -\sum_i\,\sum_k\ (\Delta^i_k\,\bkidag + {\Delta^i_k}^*\,\bki)$$ Here $\bkidag$ creates the $i$th pair state as composites of two fermions: $\bki = \akidown\,\akiup$, and of binding energy $\Delta^i_k$. The first two terms $H_{PG} = H_{0} + H_{pair}$ describe a non-superconducting state, a Cooper-pair glass having no global phase, formed by the superposition of pairs in random states. SC coherence is achieved due to the pair-pair interaction term giving rise to the characteristic DOS (Fig.\[Fig2\], red curve): $$\label{Hint} H_{int} = \frac{1}{2}\, \sum_{i\neq j} \sum_{k,k'}\, \beta^{i,j}_{k,k'}\ \bkpj\,\bkidag + h.c.$$ where $\beta^{i,j}_{k,k'}$ are the coupling coefficients, which we later tie to $\beta^c$, the SC order parameter. [*Cooper-pair glass state*]{}. to 5.8 cm[![DOS in the Cooper-pair glass (CPG) state, blue curve, showing a broad pseudogap of width $2\,<\Delta_i> = 2\,\Delta_0$ and no coherence peaks. DOS in the SC state, red curve, with pronounced dips due to the QP excited-pair coupling.[]{data-label="Fig2"}](DOS3.png "fig:"){width="8"}]{} The accumulated results of photoemission [@PRL_Kanigel2008], local tunneling experiments [@PRL_Renner1998_Temp; @PRL_Renner1998_B; @Nat_Gomes2007] and normal coherence length [@SciRep_Kirzhne2014] imply a scenario in which, contrary to BCS theory, the pseudogap is linked to some form of precursor pairing [@Nat_Emery1995]. The existence of Fermi-surface arcs just above $T_c$, as seen using ARPES [@PRL_Kanigel2008], is further evidence. Without the PPI ($H_{int} = 0$) we consider that the system consists of incoherent preformed pairs with an energy distribution: $$\label{P0} P_0(\Delta^i) \propto \frac{\sigma^2}{(\Delta^i-\Delta_0)^2 + \sigma_0^2}$$ where $\Delta_0$ and $\sigma_0$ are the average gap and the half-width, respectively. In the spinor notation: $\tilde{a}^i_k = (\akiup,\akidowndag)$, the equation of motion is: $ i\hbar\,\frac{d\,\tilde{a}^i_k}{dt} = [\tilde{a}^i_k, H_{PG}] = H^i_{PG}\, \tilde{a}^i_k$, where $H^i_{PG}$ is the effective matrix: $$\label{matrix} H^i_{PG} = \left( \begin{array}{cc} \epsilon_k & -\Delta^i_k \\ -\Delta^i_k & -\epsilon_k \\ \end{array} \right)$$ The latter is diagonal in the quasiparticle basis: $\tilde{\gamma}^i_k = \Lambda^{i}_{k}\ \tilde{a}^i_k$ with eigenvalues, $E^{i\pm}_k = \pm \sqrt{\epsilon_k^2 + {\Delta^i_k}^2}$, leading to: $ H_{PG} = \sum_{i}\,\sum_{k}\ {\tilde{\gamma}^{i\,\,\dag}_k}\, (E^i_k\ \sigma_z)\,\tilde{\gamma}^i_k $, where $\sigma_z$ is the standard Pauli matrix. In the continuum limit the spectral function ${A}_{PG}(k,E)$ acquires a significant width [@PhysicaC_Sacks2014] and the $T = 0$ DOS becomes a convolution: $$\label{dos2} N_{PG}(E) = N_n(E_F)\,\int_0^\infty \,d\Delta^i\,P_0(\Delta^i)\,\frac{E}{\sqrt{E^2-\Delta^{i\,\,2}}}$$ As a result of the pair distribution, the coherence peaks in the DOS are absent (blue curve, Fig.\[Fig2\]), a key feature of the incoherent [*Cooper-pair glass*]{}. This state is intimately related to the pseudogap observed once SC coherence is lost, i.e. at $T_c$ or within a vortex core. 0.2cm [*Equations of motion with $H_{int}$*]{}. Adding the term $[\tilde{a}^i_k, H_{int}]$ to the equation of motion, we obtain: $$\begin{aligned} \label{eq_motion} i\hbar\, \dot a^i_{k\uparrow} &=& \epsilon_k\,\akiup - \Delta^i_k\,\akidowndag + \sum_{j,k'}\null^{'}\, \beta^{i,j}_{k,k'}\ \bkpj\,\akidowndag \ \ \ \null \nonumber \\ i\hbar\, \dot a^{i\,\,\dag}_{-k\downarrow} &=& - \epsilon_k\, \akidowndag - \Delta^i_k\,\akiup + \sum_{j,k'}\null^{'}\, \beta^{i,j}_{k,k'}\ \bkpjdag\,\akiup \ \ \ \null\end{aligned}$$ Obviously, without pairing ($\Delta^i = 0$), electrons are uncoupled from holes, reflecting the normal state. To the contrary, the second (anomalous) terms in (\[eq\_motion\]) are generated by the removal of an electron-pair by a hole or a hole-pair by an electron (Fig.\[Fig3\], middle panel). The third term is new: the final state now contains a fermion triplet, which we call ‘super-anomalous’. For a fixed ($j,k$), a [*quadron*]{} of zero spin and charge is annihilated leaving a pair plus a fermion (Fig.\[Fig3\], lower panel). Since the $i$th electron (hole) is also coupled to all $j \neq i$, the hamiltonian cannot be simply diagonalized in terms of a set of quasiparticle operators {$\tilde{\gamma}^i_k $}. However, the fermion operator triplet can be decoupled by the quantum average of pair permutations: $$\begin{aligned} \label{approx1 bis} \bkpj\,\akidowndag &\simeq& <\akpjdown \akpjup>\,\akidowndag \\ \null &+& <\akidowndag \akpjdown> \akpjup + <\akpjup \akidowndag> \akpjdown \nonumber\end{aligned}$$ resulting in the equation of motion: $$\begin{aligned} \label{mfeq4} i\hbar\, \frac{d\,\tilde{a}^i_k}{dt} &=& (H^i_{PG}\, + \,\delta\Delta^i_k\,\mathcal{J})\,\tilde{a}^i_k \\ &+& \sum_{j,k'}\null^{'}\, \beta^{i,j}_{k,k'}\,\left[\Gamma^{i,j}_{k,k'}(\uparrow \uparrow)\, \tilde{a}^j_{k'} + \Gamma^{i,j}_{k,k'}(\uparrow \downarrow)\,\tilde{a}^{j\,\,\dag}_{k'} \right] \nonumber\end{aligned}$$ in which the two $\Gamma$-matrix coefficients are: $$\Gamma^{i,j}_{k,k'}(\uparrow \uparrow) = \left( \begin{array}{cc} <\akidowndag \akpjdown> & 0 \\ 0 & <\akiup \akpjupdag> \\ \end{array} \right)$$ $$\Gamma^{i,j}_{k,k'}(\uparrow \downarrow) = \left( \begin{array}{cc} 0 & <\akpjup \akidowndag> \\ <\akpjdowndag \akiup> & 0 \\ \end{array} \right)$$ and $\mathcal{J} = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right) $. These equations display coupling coefficients depending on different states $(k,k')$ and different pairs $(i,j)$, where two are [*spin aligned*]{}, $\Gamma(\uparrow \uparrow)$, and two are [*spin reversed*]{}, $\Gamma(\uparrow \downarrow)$. The vertex amplitudes imply instantaneous electron/hole interactions, all inherent to the super-anomalous term of Fig.\[Fig3\]. The new correction to the gap function is: $$\label{gapcorrection} \delta\Delta^i_k = \sum_{j,k'}\null^{'}\, \beta^{i,j}_{k,k'} <\bkpj>$$ which follows directly from (\[eq\_motion\]) with: $\bkj \rightarrow \,<\bkj>$. to 7 cm[ ![Summary of the various fermion/hole terms encountered in the ‘exact’ equation of motion (\[eq\_motion\]). Upper panel: normal state electron/hole of energy $\epsilon_k$. Middle panel: (anomalous term) standard BCS-type processes. Lower panel: (super-anomalous term) an electron (hole) annihilates a [*quadron*]{}, having zero spin and zero charge, leaving a [*super-quasiparticle*]{} consisting of a pair plus a fermion. The associated coupling energy is $\beta^{i,j}$.[]{data-label="Fig3"}](process3.png "fig:"){width="8.8"}]{} 0.2cm [*Quasiparticle coupling*]{}. In order to define Bogoliubov quasiparticles in such a system, we apply the basis transformation which diagonalizes the first two terms on the r.h.s. of equation (\[mfeq4\]): $$\overline \mathcal{O} = \overline \Lambda^{i}_{k}\,\mathcal{O}\,\overline \Lambda^{i\,\,{-1}}_{k}$$ for a general operator $\mathcal{O}$. Writing the quasiparticle basis as $\tilde \gamma^i_k = \overline \Lambda^{i}_{k}\ \tilde{a}^i_k$, yields: $$\label{mfterm2} i\hbar\, \frac{d\,\tilde \gamma^i_k}{dt} = \overline \Lambda^{i}_{k}\,(H^i_{PG} + \,\delta\Delta^i_k\,\mathcal{J})\,\overline \Lambda^{i\,\,{-1}}_{k} \,\tilde{\gamma}^{i}_k = \overline E^i_k\,\sigma_z\,\tilde \gamma^i_k$$ It is important to stress that the eigenvalues ($\overline E^i_k$) depend on the [*modified gap function*]{}; their dispersion is: $ %\begin{equation}\label{Ekibar} \overline E^i_k = \sqrt{\epsilon_k^2 + (\overline{\Delta}^i_k)^2} %\end{equation} $, where $\overline{\Delta}^i_k = \Delta^i_k - \delta\Delta^i_k$, which we note is first order in $\beta$. Using the $\overline \Lambda^{i}_{k}$ transformation, the equations of motion (\[mfeq4\]) can now be written in terms of QP operators: $$\begin{aligned} \label{mfeq5} i\hbar\, \frac{d\,\tilde{\gamma}^i_k}{dt} &=& \overline E^i_k\,\sigma_z\,\tilde \gamma^i_k \\ &+& \sum_{j,k'}\null^{'}\, \beta^{i,j}_{k,k'}\,\left[\overline \Gamma^{i,j}_{k,k'}(\uparrow \uparrow)\, \tilde{\gamma}^j_{k'} + \overline \Gamma^{i,j}_{k,k'}(\uparrow \downarrow)\,\tilde{\gamma}^{j\,\,\dag}_{k'} \right] \nonumber\end{aligned}$$ As a result of the PPI, the second term of the full equation of motion (\[mfeq5\]) contains the coupling of the $i$th quasiparticle to all other quasiparticles $j\neq i$, with the QP-QP coupling proportional to the $\overline\Gamma$ coefficients. It implies that quasiparticles interact via pair states and conversely that pair states interact via quasiparticles: a novel QP-pair vertex is thus revealed. To illustrate the effect of the coupling we focus on the case where, for wave vectors $k$ and $k'$, only two quasiparticles of energy $E^i_k$ and $E^j_{k'}$ become degenerate (higher degeneracies are possible) and the exact operators satisfy: $i\hbar\, \frac{d\,\tilde \gamma^i_k}{dt} = i\hbar\, \frac{d\,\tilde \gamma^j_{k'}}{dt} = E^{ex}_{kk'}\, \tilde \gamma^i_{k}$. In the bi-spinor basis $ \tilde{\tilde{\gamma}}_k^j = ( \tilde{\gamma}_k^j, \tilde{\gamma}_k^{j\,\,\dag})$, a new object of dimension 4 in the $a_\mu$ fermions, the secular equation is obtained: $$\label{Secmatrix} \left( \begin{array}{cc} (E^{ex}_{kk'}{\bf 1} - \overline E^i_k\,\sigma_z)\cdot {\bf 1} & - \beta\ L^{i,j}_{k,k'} \\ - \beta\ L^{j,i}_{k'k} & (E^{ex}_{kk'}{\bf 1} - \overline E^j_{k'}\,\sigma_z)\cdot {\bf 1} \\ \end{array} \right) \times \left( \begin{array}{l} \tilde{\tilde{\gamma}}_k^i \\ \tilde{\tilde{\gamma}}_k^{j\,\,\dag} \\ \end{array} \right) = 0$$ with $L^{i,j}$ a matrix of dimension 4: $$\label{Lmatrix} L^{i,j}_{k,k'} = \left( \begin{array}{cc} \overline \Gamma^{i,j}_{k,k'}(\uparrow \uparrow) & \overline \Gamma^{i,j}_{k,k'}(\uparrow \downarrow) \\ \overline \Gamma^{i,j}_{k,k'}(\uparrow \downarrow) & \overline \Gamma^{i,j}_{k,k'}(\uparrow \uparrow) \\ \end{array} \right)$$ The analogy with the lowest order pairing matrix $H^i_{PG}$, Eq.(\[matrix\]), is striking. In the conventional BCS theory, electron ($\epsilon_k$) and hole ($-\epsilon_k$) states are coupled via the pair potential $\Delta$; here the PPI ($\sim \beta$) leads to the coupling of the QP states ($\overline E^i_k$, $\pm \overline E^j_{k'}$). Since the determinant of the secular matrix must vanish, we obtain: $$\label{secequation} (E^{ex\,\,2}_{kk'} - \overline E^{i}_k\null^2) (E^{ex\,\,2}_{kk'} - \overline E^{j}_{k'}\null^2) = \beta^4\ {\rm \bf det} \left( L^{j,i}\cdot L^{i,j} \right)$$ where the explicit QP-QP coupling $\sim\,\beta^4 \Gamma^4$, appearing on the r.h.s., is assumed to be small but finite. The exact eigenstates $E^{ex}_{kk'}$ correspond to a new super-quasiparticle, $(\tilde \gamma^i_k,\tilde \gamma^j_{k'})$, and thus to the quadron ($\overline{\Delta}_k^i,\overline{\Delta}_{k'}^j$). While the $\overline E^i_k$ are to first order in the PPI $\propto \beta$, the coupling involved in the super-quasiparticle is to higher order in $\beta\,\Gamma$. Since the latter is small, the coupling of the quasiparticles $\tilde \gamma^i_k$ and $\tilde \gamma^j_{k'}$ need only be considered at the [*degeneracy point*]{}: $$\label{degen} \overline E^{i}_k = \overline E^{j}_{k'}$$ while otherwise, $\overline E^i_k$ and $\overline E^j_{k'}$ are uncoupled. This degeneracy condition thus plays a central role in the theory. 0.2cm [*Superconducting gap function*]{}. The SC ground state can be derived from the mean-field expression (\[gapcorrection\]) wherein all pairs are assumed to be degenerate. The final-state gap function is thus written: $$\label{gapbarSC} \overline{\Delta}_k = \Delta_{k,0} - \delta\Delta^{i=c}_k$$ where $i=c$ indicates pairs of the condensate and $\Delta_{k,0}\,=$ $<\Delta^i_k>$. As in our previous work Refs.[@SSciTech_Sacks2015], we take the interaction to be proportional to the DOS of preformed pairs: $ \beta^{i,j}_{k,k'} = g_k \,g_{k'}\,P_0(\Delta_k^i)\,P_0(\Delta_{k'}^j) $, where $g_k$ takes into account the $d$-wave pairing. The crucial point is that all the pairs $\Delta_{k'}^j$ are degenerate in the condensate. to 10.6 cm[ ![Fits to the tunneling DOS of 4 very different SC materials using the same gap function (\[gapequation5\]). We compare the cuprate with iron-based materials: BiSrCaCuO (slightly overdoped with $\Delta_p = 27$meV), TlBaCaCuO ($\Delta_p = 35$meV), LiFeAs ($\Delta_p = 6$meV), FeSe ($\Delta_p = 2$meV) taken from Refs.([@Sci_McElroy2005; @JphysJap_Sekine2016; @PRL_Chi2012; @Sci_Song2011]) respectively. The other numerical values used for the fits ($\beta^c, \Delta_0, \sigma_0$) are summarized in TableI. The dip position, indicated by the arrow in each case, follows approximately: $E_{dip} \simeq \Delta_p + 2\,\beta^c$. []{data-label="Fig5"}](spec4.png "fig:"){width="8.8"}]{} The gap equation (\[gapbarSC\]) must be self-consistent for zero kinetic energy, wherein the QP states are at the Fermi level. Thus, setting $\epsilon_k = 0$ and $\overline{\Delta}_k = \Delta_{k}^c$, yields: $$\label{gapbarSC2} \Delta^{c}_k = \Delta_{k,0} - 2\,\beta_k^c\,P_0(\Delta^{c}_k)$$ where $\beta_k^{c} = \frac{N_{oc}}{2}\,g_{k}\, \sum_{k'}\,g_{k'}\,P_0(\Delta_{k'}^c)\,<b^c_{k'}>$ is the mean-field [*condensate*]{} pair-pair interaction and $N_{oc}$, the number of pairs ($N_{oc}\gg 1$). Since the mean-field parameter $\beta^c$ is proportional to $N_{oc}(T)$, as a result of the quasi-Bose transition, the second term represents the [*condensation energy*]{}. As the temperature rises, it gradually decreases and finally vanishes at $T_c$, contrary to the spectral gap[@EPJB_Sacks2016] – a clear departure from conventional SC. A key aspect of the problem is that the gap function $\Delta^{c}_k$ in equation (\[gapbarSC2\]) must be modified for non-vanishing kinetic energy, where a quasiparticle [*becomes degenerate with an excited pair state*]{}(see Fig.\[Fig1\]). The latter coupling energy $\sim \beta^2\,\Gamma^2$ is to second order while the renormalized gap function, proportional to $\beta^c$, remains large. Thus, for excited states, $\epsilon_k > 0$, the gap equation (\[gapbarSC\]) is: $$\label{gapbarSC4} \overline{\Delta}^{i=ex}_k = \Delta_{k,0} - 2\,\beta_k^c\,P_0(\Delta^{i=ex}_k)$$ where both $\Delta_{k,0}$ and $\beta_k$ are assumed independent of $\epsilon_k$. Recalling equation (\[degen\]), the [*correct degeneracy point*]{} is $\Delta^{i=ex}_k = \overline E_k$ where we identify $i \rightarrow \Delta^{i=ex}_k$ as the excited pair, degenerate with the state $j \rightarrow \overline E_k= \sqrt{\epsilon_k^2 + (\overline \Delta_k)^2}$ of the condensate. Dropping the overbar, the full gap equation for excited states, reads: $$\label{gapequation5} {\Delta}_k(E_k)=\Delta_{k,0} - 2\,\beta_k^c\,P_0(\sqrt{\epsilon_k^2 + \Delta_k(E_k)^2})$$ We thus have an energy dependent and self-consistent equation for the gap function which leads to a strictly non-hyperbolic QP dispersion and, most significantly, gives rise to the dip in the spectral function (Fig.\[Fig2\]). One can now identify its physical origin: the strong coupling of the SC quasiparticle with excited pair states. 0.2cm [*Comparaison with experiments*]{}. The instantaneous interactions in the hamiltonian imply that the DOS can be calculated with [*no retardation effects*]{} in the Green’s function. If no quasiparticle lifetime effect is invoked, at $T = 0$, the DOS for the $d$-wave condensate can be calculated by the standard formulae using $\frac{\partial \epsilon_k}{\partial E_k}$: $$\begin{aligned} \label{DOS} &\null&{N}_{SC}^{d}(E) = \mathcal{N}_n(E_F) \int_{0}^{2\,\pi}\frac{d\theta}{2\,\pi}\,\int_0^\infty d\epsilon_k\ \delta(E_k - E) \nonumber\\ & \null &=\mathcal{N}_n(E_F) \int_{0}^{2\,\pi}\frac{d\theta}{2\,\pi}\, \left[\frac{E_k - \Delta_k(E_k,\theta) \frac{\partial\Delta_k}{\partial E_k}}{\sqrt{E_k^2 - \Delta_k(E_k)^2}}\right]_{E_k=E}\end{aligned}$$ where $\mathcal{N}_n(E_F)$ is the normal DOS at the Fermi energy. The SC DOS is thus proportional to the derivative of the gap function (\[gapequation5\]) wherein the peak-dip-hump is due to the interaction term: $ \frac{\partial\,\Delta_k(E_k)}{\partial\, E_k} = 2\,\beta^c_k\, \frac{d\,P_0(E_k)}{dE_k}$. The controlling parameters are thus the pair-pair interaction $\beta^c$ and the condensate pair number, $N_{oc}$, but the distribution $P_0(E_k)$ plays an essential role. Since the derivative has two extrema, the first one reinforces the QP coherence peaks, giving them an unconventional wide shape, while the second extremum produces the dip [@EPL_Cren2000]. As in our previous work[@SSciTech_Sacks2015], the DOS (\[DOS\]) can be used to fit a wide variety of tunneling spectra of high T$_c$ superconductors with remarkably few parameters: $\beta^c$, the mean pair-pair interaction, $\Delta_0$ and $\sigma_0$ which characterize the distribution of pair states. Among the cuprates, the tunneling characteristics of BiSrCaCuO (2212) or BiSrCaCuO (2203) have been the most clearly established (see [@RevModPhys_Fisher2007] and references therein). Much success has recently been done on iron-based SC (see [@CurrOp_Song2013] and references therein), such as BaKFeAs, doped Fe(Se,Te) [@Chilett_Jing2015], as well as LiFeAs [@PRL_Chi2012] or FeSe [@Sci_Song2011] where typical spectra are shown in Fig.\[Fig5\]. Since we focus on the SC aspects of the DOS, the background density is removed and the spectra symmetrized, without affecting adversely the SC gap and peak-dip-hump features. Along with a slightly overdoped BiSrCaCuO [@Sci_McElroy2005], Fig.\[Fig5\] shows a recent high-quality spectrum on a 3-layer TlBaCaCuO [@JphysJap_Sekine2016], indicating the universality of the peak-dip-hump features. In the same figure, the spectra are fitted using the PPI model. The parameters of the fits are given in TableI. First, we note the relatively sharp peaks at $e\,V = \pm \Delta_p$ in the iron-based SC as compared to BiSrCaCuO and TlBaCaCuO. Indeed, higher peaks are quite rare, partly due to thermal smearing at 4.2K but also due to a finite quasiparticle lifetime, which we estimate to be $\sim 1.5$ meV in the case of BiSrCaCuO and an order of magnitude less for FeSe. The detailed shape of all the spectra are accurately fitted using the same gap function(\[gapequation5\]) in the DOS, from energies within the gap, to the wide QP peaks and the pronounced dip. The parameters thus have the same meaning despite the range of values, and the very different composition and structure of the materials. We find that $E_{dip}-\Delta_p\simeq 2\beta^c$, where $\beta^c\simeq 2k_BT_c$ follows the SC dome but without the collective mode scenario. Rather, it emerges from the novel QP-pair vertex inherent to the super-anomalous term of the equation of motion (\[eq\_motion\]). These super-quasiparticles cause the dip in the spectrum and signal the long range SC order. The condensate PPI energy $\beta^c$ depends on the product of the pairing amplitude $\Delta_p$ with the carrier density $p$ : $\beta^c(p) \propto p \times \Delta_p$, providing a simple explanation for the SC dome. The mechanism is thus the interplay between the pair binding energy, decreasing with $p$, and the number of pairs increasing with $p$. 0.2cm [*Conclusion*]{}. We propose a scenario for high-$T_c$ superconductors wherein the initial incoherent state is the Cooper-pair glass, whose properties explain the observed pseudogap and Fermi-arc phenomena in agreement with both tunneling [@PRL_Renner1998_Temp; @JphysJap_Sekine2016; @PhysicaC_Kawashima2010] and ARPES [@Nat_Hashimoto2014] experiments. SC coherence results from the novel pair-pair interaction, which adds a [*quadron*]{} term to the hamiltonian giving rise to a new type of fundamental excitation, the [*super-quasiparticle*]{}. The important effect is the renormalized gap function, which is energy-dependent, but non retarded. The theory gives for the first time the correct temperature and doping dependence of the quasiparticles in the SC to PG transition. It reproduces quantitatively the experimental spectra of both pnictides and cuprates, including the peak-dip-hump structure, and attributes a common meaning to the fundamental parameters. In conclusion, these features are not due to the coupling to a bosonic mode, but rather emerge from instantaneous all-electron interactions.

ArXiv

--- abstract: 'A measurement of the top quark pair production cross section in proton anti-proton collisions at an interaction energy of $\sqrt{s}=1.96~{\rm TeV}$ is presented. This analysis uses 405 pb$^{-1}$ of data collected with the DØ detector at the Fermilab Tevatron Collider. Fully hadronic $t\bar{t}$ decays with final states of six or more jets are separated from the multijet background using secondary vertex tagging and a neural network. The $t\bar{t}$ cross section is measured as $\sigma_{t\bar{t}}=4.5_{-1.9}^{+2.0}({\rm stat}) _{-1.1}^{+1.4}({\rm syst}) \pm 0.3 ({\rm lumi})~{\rm pb }$ for a top quark mass of $m_{t} = 175~{\rm GeV/c^2}$.' date: 'December 13, 2006' title: 'Measurement of the $p\bar{p} \to t\bar{t}$ production cross section at $\sqrt{s}=1.96$ TeV in the fully hadronic decay channel ' --- list\_of\_authors\_r2.tex The standard model (SM) predicts that the top quark decays primarily into a $W$ boson and a $b$ quark. The measurement presented here tests the prediction of the SM in the dominant decay mode of the $t\bar{t}$ system: when both $W$ bosons decay to quarks, the so-called fully hadronic decay channel. This topology occurs in 46% of $t\bar{t}$ events. The theoretical signature for fully hadronic $t\bar{t}$ events is six or more jets originating from the hadronization of the six quarks. Of the six jets, two originate from $b$ quark decays. Fully hadronic $t\bar{t}$ events are difficult to identify at hadron colliders because the background rate is many orders of magnitude larger than that of the $t\bar{t}$ signal. We report a measurement of the production cross-section of top quark pairs, $\sigma_{t\bar{t}}$, using data collected with DØ in the fully hadronic channel, that exploits the long lifetime of the $b$ hadrons in identifying $b$ jets. To increase the sensitivity for $t\bar{t}$ events, we used a neural network to distinguish signal from the overwhelming background of multijet production through Quantum Chromodynamic processes (QCD). The DØ detector [@d0det] has a central tracking system consisting of a silicon micro strip tracker (SMT) and a central fiber tracker (CFT), both located within a 2 T superconducting solenoidal magnet, with designs optimized for tracking and vertexing at pseudorapidities $|\eta|<3$ and $|\eta|<2.5$, respectively. Rapidity $y$ and pseudorapidity $\eta$ are defined as functions of the polar angle $\theta$ and parameter $\beta$ as $y(\theta,\beta)= \frac{1}{2} \ln [ (1+\beta \cos \theta)/(1-\beta \cos \theta )]$ and $\eta(\theta)=y(\theta,1)$, where $\beta$ is the ratio of the particle’s momentum to its energy. The liquid-argon and uranium calorimeter has a central section (CC) covering pseudorapidities $|\eta|$ up to $\approx 1.1$ and two end calorimeters (EC) that extend coverage to $|\eta| \approx 4.2$, with all three housed in separate cryostats. Each calorimeter cryostat contains a multilayer electromagnetic calorimeter, a finely segmented hadronic calorimeter and a third hadronic calorimeter that is more coarsely segmented, providing both segmentation in depth and in projective towers of size $0.1 \times 0.1$ in $\eta$-$\phi$ space, where $\phi$ is the azimuthal angle in radians. An outer muon system, covering $|\eta|<2$, consists of a layer of tracking detectors and scintillation trigger counters in front of 1.8 T iron toroids, followed by two similar layers after the toroids. The luminosity is measured using plastic scintillator arrays placed in front of the EC cryostats. The data set was collected between 2002 and 2004, and corresponds to an integrated luminosity $\mathcal{L}=405~\pm~25~{\rm pb}^{-1}$ [@newlumi]. To isolate events with six jets, we used a dedicated multijet trigger. The requirements on the trigger, particularly on jet and trigger tower energy thresholds, were tightened during the collection of the data set to manage the increasing instantaneous luminosities delivered by the Fermilab Tevatron Collider. The change in trigger requirements had little effect on the efficiency for signal, while removing an increasing number of background events [@footnote1]. The trigger was tuned for the fully hadronic $t\bar{t}$ channel and was optimized to remain as efficient possible while using limited bandwidth. The collection rate after all trigger levels was fixed to a few Hz, which was completely dominated by QCD multijet events as the hadronic $t\bar{t}$ event production rate is expected to be a few events per day. We required three or four trigger towers above an energy threshold of 5 GeV at the first trigger level, three reconstructed jets with transverse energies ($E_T$) above 8 GeV at the second trigger level, combined with a requirement on the sum of the transverse momenta ($p_{T}$) of the jets, and four or five reconstructed jets at transverse energy thresholds between 10 and 30 GeV at the highest trigger level [@d0det]. We simulated $t\bar{t}$ production using [alpgen 1.3]{} to generate the parton-level processes, and [pythia 6.2]{} to model hadronization [@alpgen; @pythia]. We used a top quark invariant mass of $m_{t}=175~{\rm GeV/c^2}$. The decay of hadrons carrying bottom quarks was modeled using [evtgen]{} [@evtgen]. The simulated $t\bar{t}$ events were processed with the full [geant]{}-based DØ detector simulation, after which the Monte Carlo (MC) events were passed through the same reconstruction program as was used for data. The small differences between the MC model and the data were corrected by matching the properties of the reconstructed objects. The residual differences were very small and were corrected using factors derived from detailed comparisons between the MC model and the data for well understood SM processes such as the jets in $Z$ boson and QCD dijet production. In the offline analysis, jets were defined with an iterative cone algorithm [@jetsdef]. Before the jet algorithm was applied, calorimeter noise was suppressed by removing isolated cells whose measured energy was lower than four standard deviations above cell pedestal. In the case that a cell above this threshold was found to be adjacent to one with an energy less than four standard deviations above pedestal, the latter was retained if its signal exceeded 2.5 standard deviations above pedestal. Cells that were reconstructed with negative energies were always removed. The elements for cone jet reconstruction consisted of projective towers of calorimeter cells. First, seeds were defined using a preclustering algorithm, using calorimeter towers above an energy threshold of 0.5 GeV. The cone jet reconstruction, an iterative clustering process where the jet axis was required to match the axis of a projective cone, was then run using all preclusters above 1.0 GeV as seeds. As jets from $t\bar{t}$ production are relatively narrow due to relatively high jet $p_{T}$, the jets were defined using a cone with radius $R_{{\rm cone}}=0.5$, where $\Delta R = \sqrt{(\Delta y)^2+(\Delta \phi)^2}$ . The resulting jets (proto-jets) took into account all energy deposits contained in the jet cone. If two proto-jets were within $1<\Delta R / R_{{\rm cone}} <2$, an additional midpoint clustering was applied, where the combination of the two proto-jets was used as a seed for a possible additional proto-jet. At this stage, the proto-jets that share transverse momentum were examined with a splitting and merging algorithm, after which each calorimeter tower was assigned to one proto-jet at most. The proto-jets were merged if the shared $p_T$ exceeded 50% of the $p_T$ of the proto-jet with the lowest transverse momentum and the towers were added to the most energetic proto-jet while the other candidate was rejected. If the proto-jets shared less than half of their $p_{T}$, the shared towers were assigned to the proto-jet which was closest in $\Delta R$ space. The collection of stable proto-jets remaining was then referred to as the [*reconstructed*]{} jets in the event. The minimal $p_{T}$ of a reconstructed jet was required to be 8 GeV/$c$ before any energy corrections were applied. We removed jets caused by electromagnetic particles and jets resulting from noise in hadronic sections of the calorimeter by requiring that the fraction of the jet energy deposited in the calorimeter ($EMF$) was $0.05 < EMF < 0.95$ and the fraction of energy in the coarse hadronic calorimeter was less than 0.4. Jets formed from clusters of calorimeter cells known to be affected by noise were also rejected. The remaining noise contribution was removed by requiring that the jet also fired the first level trigger. To correct the calorimeter jet energies back to the level of particle jets, a jet energy scale (JES) correction $C^{JES}$ was applied. The same procedure has to be applied to Monte Carlo jets to ensure an identical calorimeter response in data and simulation. The particle level or true jet energy $E^{true}$ was obtained from the measured jet energy $E^m$ and the detector pseudorapidity, measured from the center of the detector ($\eta_{det}$), using the relation $$E^{true}= \frac{E^m - E_0 ( \eta_{det}, \mathcal{L})}{\mathcal{R}(\eta_{det}, E^m) S(\eta_{det},E^m)} = C^{JES} (E^m,\eta_{det},\mathcal{L}) \cdot E^m .$$ In data and MC the total correction was applied to the measured energy $E^m$ as a multiplicative factor $C^{JES}$. $E_{0}(\eta_{det},\mathcal{L})$ was the offset energy created by electronics noise and noise signal caused by the uranium in the calorimeter, pile-up energy from previous collisions and the additional energy from the underlying physics event. The dependence on the luminosity $\mathcal{L}$ was caused by the fact that the number of additional interactions was dependent on the instantaneous luminosity, while the dependence on $y$ was caused by variations in the calorimeter occupancy as a function of the jet rapidity. $\mathcal{R}(\eta_{det},E^m)$ parameterized the energy response of the calorimeter, while $S(\eta_{det},E^m)$ represents the fraction of the true partonic jet energy that was deposited inside the jet cone. This out-of-cone showering correction depended on the energy of the jet and its location in the calorimeter. The JES was measured directly using $p_T$ conservation in photon + jet events. The method was identical for data and simulation and used transverse momentum balancing between the jet and the photon. As the energy scale of the photon was directly and precisely measured (the electromagnetic calorimeter response was derived from measurements of resonances in the $e^+ e^-$ spectrum like the $Z$ boson), the true jet energy could be derived from the difference between the photon and jet energy. $E_0$, $\mathcal{R}$ and $S$ were fit as a function of jet rapidity and measured energy, which lead to uncertainties coming from the fit (statistical) and the method (systematic). The total correction $C^{JES}$ was approximately 1.4 for data jets in the energy range expected for jets associated with top quark events. The uncertainties on $C^{JES}$, which were dominated by the systematic uncertainty of the out-of-cone showering correction $S(\eta_{det},E^m)$, were a few percent and were dependent on the jet energy and rapidity. The jet energy resolution was measured in photon + jet data for low jet energies and dijet data for higher jet energy values. Fits to the transverse energy asymmetry $[p_T(1) - p_T(2)]/[p_T(1)+p_T(2)]$ between the transverse momenta of the back-to-back jets and/or photon ($p_T(1)$ and $p_T(2)$) were then used to obtain the jet energy resolution as a function of jet rapidity and transverse energy. The uncertainties on the jet energy resolution were dominated by limited statistics in the samples used. In this analysis, we considered a data set consisting of events with four or more reconstructed jets, in which the scalar sum of the uncorrected transverse momenta $H_T^{uncorr}$ of all the jets in the event was greater than 90 GeV/$c$. The final analysis sample was a subset of this sample, where at least six jets with corrected transverse momentum greater than 15 GeV/$c$ and $|y|<2.5$ were required. Events with isolated high transverse momentum electron or muon candidates were vetoed to ensure that the all-hadronic and leptonic $t\bar{t}$ samples were disjoint [@ttbarlepjetsrun2; @ttbarlepjetsvtxtag]. In addition, we rejected events where two distinct $p\bar{p}$ interactions with separate primary vertices were observed and the jets in the event were not assigned to only one of the two primary vertices. The primary vertex requirement did not affect minimum bias interactions or $t\bar{t}$ events. Table \[effmostcuts\] lists the efficiencies after the first set of selection cuts, commonly referred to as preselection, which includes the requirements on the primary vertex, the number of reconstructed jets and the presence of isolated leptons, and the efficiency after preselection and after preselection and the trigger. Besides selecting all hadronic $t\bar{t}$ events, the analysis was also expected to accept a small contribution from the semi-leptonic (lepton+jets) $t\bar{t}$ decay channel. The combined efficiency included the fully hadronic and semi-leptonic $W$-boson branching fractions of $0.4619\pm0.0048$ and $0.4349\pm0.0027$ respectively [@pdg]. We used a secondary vertex tagging algorithm (SVT) to identify $b$-quark jets. The algorithm was the same as used in previously published DØ $t\bar{t}$ production cross section measurements [@ttbarlepjetsrun2; @ttbarlepjetsvtxtag]. Secondary vertex candidates were reconstructed from two or more tracks in the jet, removing vertices consistent with originating from long-lived light hadrons as for example $K_S^0$ and $\Lambda$. Two configurations of the secondary vertex algorithm were used; these were labeled “loose” and “tight” respectively. If a reconstructed secondary vertex in the jet had a transverse decay length $L_{xy}$ significance ($L_{xy}/\sigma_{L_{xy}}$) $>$ 5 (7), the jet was tagged as a loose (tight) $b$-quark jet. The loose SVT was chosen to efficiently identify $b$-quark jets, while the tight SVT was configured to accept only very few light quark jets while sacrificing a small reduction in the efficiency for $b$-quark jets. Events with two or more loosely tagged jets were called double-tag events. The sample that did not contain two loosely tagged jets was inspected for events with one tight tag. Events thus isolated were labeled single-tag events. The fully exclusive samples of single-tag and double-tag events were treated separately because of their different signal-to-background ratios. The use of the tight SVT selection for single tagged events optimized the rejection of mistags, the main background in the single-tag analysis. When two tags were required, the background sample started to be dominated by direct $b\bar{b}$ production. The choice to use the loose SVT optimized the double-tag analysis for signal efficiency instead of background rejection. cut $t\bar{t} \to {\rm hadrons}$ $t\bar{t} \to \ell + {\rm jets}$ any $t\bar{t}$ ------------------ ------------------------------ ---------------------------------- ------------------------- [preselection]{} [$0.2706 \pm 0.0016$]{} [$0.0311\pm0.0008$]{} [$0.1385\pm 0.0011$]{} [trigger]{} [$0.2527\pm 0.0015$]{} [$0.0268 \pm 0.0007$]{} [$0.1284 \pm 0.0010$]{} : \[effmostcuts\] Efficiency for selection criteria applied before $b$-jet identification. Efficiencies listed include the efficiency for all previous selection criteria. The trigger efficiency is quoted for events that have passed the preselection. The uncertainties are due to Monte Carlo statistics. Listed are the selection efficiencies as determined for $t\bar{t}$ in the hadronic decay channel, the lepton+jets decay channel and the efficiency for all different decay channels corrected for $W$ boson branching fractions. Compared to light-quark QCD multijet events, $t\bar{t}$ events on average have more jets of higher energy and with less boost in the beam direction, resulting in events with many central jets that all have similar and relatively high energies. Moreover, the fully hadronic decay makes it possible to reconstruct the $W$ boson and $t$ quark four-momenta. To distinguish between signal and background, we used the following event characteristics [@run1alljetsxsec1]: ![\[fig1\] The $H_T$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig1.eps){width="\linewidth"} \(1) $H_T$: The scalar sum of the corrected transverse momenta of the jets (Fig. \[fig1\]). ![\[fig2\] The $E_{T}^{56}$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig2.eps){width="\linewidth"} \(2) $E_{T}^{56}$: The square root of the product of the transverse momenta of the fifth and sixth leading jet (Fig. \[fig2\]). ![\[fig3\] The $\mathcal{A}$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig3.eps){width="\linewidth"} \(3) $\mathcal{A}$: The aplanarity as calculated from the normalized momentum tensor (Fig. \[fig3\]) [@ttbarlepjetsrun2; @ttbarlepjetsvtxtag; @run1alljetsxsec1]. ![\[fig4\] The $\langle \eta^2 \rangle$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig4.eps){width="\linewidth"} \(4) $\langle \eta^2 \rangle$: The $p_T$-weighted mean square of the $y$ of the jets in an event (Fig. \[fig4\]), see also Ref.  [@run1alljetsxsec1]. ![\[fig5\] The $\mathcal{M}$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig5.eps){width="\linewidth"} \(5) $\mathcal{M}$: The mass-$\chi^2$ variable, which was defined as $\mathcal{M} = (M_{W_1}- M_W)^2/\sigma_{M_W}^2 +(M_{W_2}- M_W)^2/\sigma_{M_W}^2 +(m_{t_1}-m_{t_2})^2/\sigma_{m_t}^2$, where the parameters $M_W$, $\sigma_{M_W}$ and $\sigma_{m_t}$ were the invariant mass and mass resolution from the jet four-momenta calculated as observed in all-hadronic $t\bar{t}$ MC, respectively 79, 11 and 21 GeV/$c^2$ after all corrections and resolutions were included [@footnote2]. $M_{W_i}$ and $m_{t_i}$ were calculated for every possible permutation of the jets in the event. We did not distinguish between tagged and untagged jets. The combination of jets that yielded the lowest value of $\mathcal{M}$ is used (Fig. \[fig5\]). ![\[fig6\] The $M_{min}^{34}$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig6.eps){width="\linewidth"} \(6) $M_{min}^{34}$: The second-smallest dijet mass in the event. First, all possible dijet masses were considered and the jets that yield the smallest mass were rejected. $M_{min}^{34}$ was the smallest dijet mass as found from the remaining jets (Fig. \[fig6\]). ![\[fig7\] The output discriminant of an artificial neural network ($NN$) with six input nodes. All distributions are normalized to area. $NN$ is optimized to distinguish between fully hadronic $t\bar{t}$ Monte Carlo events (signal) and the background from multijet production (background) as predicted by the tag rate functions. ](./fig7.eps){width="\linewidth"} The top quark production cross section was calculated from the output of [*NN*]{}, an artificial neural network trained to force its output near 1 for $t\bar{t}$ events and near $-1$ for QCD multijet events, using the multilayer perceptron in the [root]{} analysis program [@root]. The six parameters illustrated in Figs. \[fig1\]-\[fig6\] were used as input for the neural net. The very large background-to-signal ratio in the untagged data allowed us to use untagged data as background input for the training of [*NN*]{}, while $t\bar{t}$ MC was used for the signal. Fig. \[fig7\] shows the [*NN*]{} discriminant for $t\bar{t}$ signal and multijet background. Although the distributions for single- and double-tag events were different due to increased heavy flavor content in the double-tag sample, both samples showed a clear discrimination between signal and background. The overwhelming background also made it possible to use the entire (tagged and untagged) sample to estimate the background. For the loose and tight SVT, we derived a tag rate function ([trf]{} — the probability for any individual jet to have a secondary vertex tag ) from the data with $N_{tags} \leq 1$. The [trf]{} was parameterized in terms of the $p_T$, $\phi$ and $y$ of the jet and the coordinate along the beam axis ($z$) of the primary vertex of the event, $z_{PV}$, in four different $H_T$ bins. To predict the number of tagged jets in the event, it was necessary to correct for a possible correlation between tagged jets. In the single-tag analysis the correlation factor was negligible, unlike in the double-tag analysis, where the presence of $b\bar{b}$+jets events in the sample enhanced the correlation correction. We corrected for correlations caused by $b\bar{b}$ background by applying a correlation factor $C_{ij}$, that was parameterized as a function of the cone distance between the tagged jets, $\Delta R$. Figure \[fig8\] shows the number of double-tagged events versus $\Delta R$ as observed in data, and the distribution as modeled by the [trf]{} with and without including $C_{ij}$. We considered significantly different functional forms for the parameterization of $C_{ij}$ and found that the choice of parameterization had little effect on the shape of the modeled background distribution. ![\[fig8\] The performance of the [trf]{} prediction on double-tag events (points), without including the correlation factor $C_{ij}$ (dashed histogram), and including $C_{ij}$ for two different functional parameterizations (solid histograms).](./fig8.eps){width="\linewidth"} The probabilities $p_i$ were used to assign a weight, the probability that the event could have a given number of tags, to every tagged and untagged event in the sample. To ensure the [trf]{} prediction was accurate in the region of phase space outside the “background” peak of the neural network, we used the region $-0.7<NN<0.5$ to determine a normalization. In this region of phase space, the $t\bar{t}$ content was negligible. A possible dependence on $t\bar{t}$ content was studied by the addition and/or subtraction of simulated $t\bar{t}$ events, as was the variation of the interval used for the normalization. Outside the background peak, the [trf]{} predictions were corrected by: [*SF*]{}$_{1} = 1.000 \pm 0.009$ for the single-tag analysis, and [*SF*]{}$_{2} = 0.969 \pm 0.014 $ for the double-tag analysis. The errors on the normalization were taken into account as a systematic uncertainty on the number of background events. ![\[fig9\] The distribution of the $NN$ output variable for single-tag events. Shown are the data (points), background (hashed band), signal (filled histogram) and signal+background (dashed histogram). The vertical line represents the used cut of $NN>0.81$.](./fig9.eps){width="\linewidth"} Both the single-tag and the double-tag analysis were expected to be dominated by background, even at large values of $NN$. Figures \[fig9\] and \[fig10\] show the distribution for data (points), the Monte Carlo simulation prediction for $\sigma_{t\bar{t}}=6.5~{\rm pb}$ (filled histogram), the background prediction (line histogram) and the signal+background distribution (dashed histogram) [@ttbarlepjetsvtxtag; @theoryxsec]. The cross section was calculated from the number of $t\bar{t}$ and background candidates above a cut value of the $NN$ discriminant. The cut value was chosen to maximize the expected statistical significance $s/\sqrt{s+b}$, where $s$ and $b$ were the number of expected signal and background events. The signal and background distributions were estimated using the [trf]{} prediction and $t\bar{t}$ Monte Carlo events [@footnote3]. For both analyses, the expected statistical significance was about two standard deviations. The optimal cut for the single (double)-tag analysis was $NN\geq 0.81~(0.78)$ shown by a vertical line in Figs. \[fig9\] and \[fig10\]. Table \[tabres1\] gives the observed numbers of events ($N_{obs}^i$), the background prediction ($N_{bg}^i$) and the efficiency for signal ($\varepsilon_{t\bar{t}}$) that can be used to calculate the $t\bar{t}$ production cross section via: $$\sigma_{t\bar{t}}=\frac{N_{obs}^i - N_{bg}^i}{ \varepsilon_{t\bar{t}}^i \mathcal{L} (1 - \varepsilon_{TRF}^i) } ,$$ where $i$ was “$=1$” for the single-tag analysis and “$\geq 2$” for the double-tag analysis. The number of background events is predicted using the [trf]{} method. It was likely that at values of $NN$ close to unity a certain fraction of the sample used to predict the background actually consists of tagged or untagged $t\bar{t}$ events, resulting in an increased background prediction. The expected $t\bar{t}$ contamination of the background sample was corrected by a factor $\varepsilon_{TRF}^i$. In the higher value bins of $NN$, the contribution from untagged $t\bar{t}$ events was significant. $\varepsilon_{TRF}^i$ was estimated by applying the [trf]{} on $t\bar{t}$ MC, and comparing the predicted tagging probability for signal to what was expected from background. The size of the Monte Carlo sample dominates the uncertainty on $\varepsilon_{TRF}^i$. Table \[tabres1\] lists the systematic uncertainties on the estimate of the number of background events, the selection efficiency and the background contamination. The first was uncorrelated between the two analyses, while the latter two were correlated as they were derived from the same Monte Carlo samples. For the single-tag analysis, the systematic uncertainty on the selection efficiency was dominated by the uncertainty in the jet calibration and identification, which were estimated by varying the parameterizations used by one standard deviation. The uncertainty on the background prediction was dominated by the uncertainty on the [trf]{} method and the uncertainty on $\varepsilon_{TRF}$ was due to limited Monte Carlo statistics. The uncertainty of the [trf]{} prediction was comprised from the uncertainties coming from the fits of the probability density functions at the jet level, the statistics of the background sample and the uncertainty on the normalization and correlation factors $SF$ and $C_{ij}$. For the double-tag analysis, the contribution from the uncertainties due to calibration of the $b$ quark jet identification efficiency was an additional systematic uncertainty on $\varepsilon_{t\bar{t}}$. These uncertainties were derived by varying the parameterizations used within their known uncertainties.   symbol value -------------------------- ----------------------------------- --------------------------------------------- -- observed events $N_{obs}^{=1}$ 495 background events $N_{bg}^{=1}$ $464.3 \pm 4.6 ({\rm syst})$ $t\bar{t}$ efficiency $\varepsilon_{t\bar{t}}^{=1}$ $0.0242 _{-0.0058}^{+0.0049}({\rm syst})$ $t\bar{t}$ contamination $\varepsilon_{TRF}^{=1}$ $0.245 \pm 0.031 ({\rm syst})$ observed events $N_{obs}^{\geq 2}$ 439 background events $N_{bg}^{\geq 2}$ $400.2_{-6.2}^{+7.3} ({\rm syst})$ $t\bar{t}$ efficiency $\varepsilon_{t\bar{t}}^{\geq 2}$ $0.0254 _{- 0.0070}^{+0.0065} ({\rm syst})$ $t\bar{t}$ contamination $\varepsilon_{TRF}^{\geq 2}$ $0.194 \pm 0.048({\rm syst})$ : \[tabres1\]Overview of observed events, background predictions and efficiencies. ![\[fig10\] The distribution of the $NN$ output variable for double-tag events. Shown are the data (points), background (hashed band), signal (filled histogram) and signal+background (dashed histogram). The vertical line represents the used cut of $NN>0.78$. ](./fig10.eps){width="\linewidth"} The single-tag analysis yielded a cross section of $$\sigma_{t\bar{t}}=4.1_{-3.0}^{+3.0}({\rm stat}) _{-0.9}^{+1.3}({\rm syst}) \pm 0.3 ({\rm lumi})~{\rm pb}.$$ For the double-tag analysis the measured cross section was $$\sigma_{t\bar{t}}=4.7_{-2.5}^{+2.6}({\rm stat}) _{-1.4}^{+1.7}({\rm syst}) \pm 0.3 ({\rm lumi})~{\rm pb}.$$ As the single-tag and double-tag analysis were measured on independent samples, the statistical uncertainties were uncorrelated. The uncertainties on the selection efficiency were completely correlated. Taking all uncertainties into account, a combined cross section measurement of $$\sigma_{t\bar{t}}=4.5_{-1.9}^{+2.0}({\rm stat}) _{-1.1}^{+1.4}({\rm syst}) \pm 0.3 ({\rm lumi})~{\rm pb}$$ was obtained, for a top quark mass of $m_t=175~{\rm GeV}/c^2$. For a top quark mass of $m_t=165~{\rm GeV}/c^2$, the cross section is $\sigma_{t\bar{t}}(165)=6.2_{-2.7}^{+2.8}({\rm stat}) _{-1.5}^{+2.0}({\rm syst}) \pm 0.4 ({\rm lumi})$ pb, while for a top quark mass of $m_t=185~{\rm GeV}/c^2$ the value shifted down to $\sigma_{t\bar{t}}(185)=4.3_{-1.8}^{+1.9}({\rm stat}) _{-1.0}^{+1.4}({\rm syst}) \pm 0.3 ({\rm lumi})$ pb. In summary, we have measured the $t\bar{t}$ production cross section in $p\bar{p}$ interactions at $\sqrt{s}=1.96$ TeV in the fully hadronic decay channel. We used lifetime $b$-tagging and an artificial neural network to distinguish $t\bar{t}$ from background. Our measurement yields a value consistent with SM predictions and previous measurements. [99]{} list\_of\_visitor\_addresses\_r2.tex DØ Collaboration, V.M. Abazov [*et al.*]{}, Nucl. Instrum. Methods Phys. Res. A [**565**]{}, 463 (2006). T. Andeen [*et. al.*]{}, FERMILAB-TM-2365-E (2006), in preparation. The efficiency for signal remained between 85 and 90% throughout the data collection period. Efficiencies were was measured both on $t\bar{t}$ Monte Carlo and derived from parameterizations determined from data. M.L. Mangano [*et al.*]{}, J. High Energy Phys. [**07**]{}, 001 (2003). T. Sjöstrand [*et al.*]{}, Comput. Phys. Commun. [**135**]{}, 238 (2001). D. Lange, Nucl. Instrum. Methods Phys. Res. A [**462**]{}, 152 (2001). G.C. Blazey [*et al.*]{}, in [*Proceedings of the Workshop: QCD and Weak Boson Physics in Run II*]{}, U. Baur, R.K. Ellis and D. Zeppenfeld (ed.), Fermilab, Batavia, IL (2000). DØ Collaboration, V.M. Abazov [*et al.*]{}, Phys. Lett. B [**626**]{}, [ 35]{} (2005). DØ Collaboration, V.M. Abazov [*et al.*]{}, Submitted to Phys. Rev. D, FERMILAB-PUB-06-386-E (2006). W.-M. Yao [*et al.*]{}, Journal of Physics G [**33**]{}, 1 (2006). DØ Collaboration, B. Abbott [*et al.*]{}, Phys. Rev. Lett. [**83**]{} 1908 (1999). The possibility that the wrong permutations of jets could be chosen was taken into account in the determination of the values of the values of 79, 11, and 21 GeV/$c^2$ for $M_W$, $\sigma_{M_W}$ and $\sigma_{m_t}$. R. Brun and F. Rademakers, Nucl. Inst. Meth. in Phys. Res. A [**389**]{} (1997) 81-86. See also http://root.cern.ch/. N. Kidonakis and R. Vogt, Phys. Rev. D [**68**]{}, 114014 (2003). The expected $t\bar{t}$ content used to optimize the $NN$ cut was equivalent with a hypothetical cross section of $\sigma_{t\bar{t}}=6.5~{\rm pb}$. The chosen cuts are stable under variation of the value assumed for the optimization.

ArXiv

--- abstract: 'We propose driven dissipative Majorana platforms for the stabilization and manipulation of robust quantum states. For Majorana box setups, in the presence of environmental electromagnetic noise and with tunnel couplings to quantum dots, we show that the time evolution of the Majorana sector is governed by a Lindblad master equation over a wide parameter regime. For the single-box case, arbitrary pure states (‘dark states’) can be stabilized by adjusting suitable gate voltages. For devices with two tunnel-coupled boxes, we outline how to engineer dark spaces, i.e., manifolds of degenerate dark states, and how to stabilize fault-tolerant Bell states. The proposed Majorana-based dark space platforms rely on the constructive interplay of topological protection mechanisms and the autonomous quantum error correction capabilities of engineered driven dissipative systems. Once a working Majorana platform becomes available, only standard hardware requirements are needed to implement our ideas.' author: - 'Matthias Gau,$^{1,2}$ Reinhold Egger,$^{1}$ Alex Zazunov,$^{1}$ and Yuval Gefen$^{2}$' title: Towards dark space stabilization and manipulation in driven dissipative Majorana platforms --- \#1\#2[\#1|\#2]{}\#1[|\#1|]{}\#1[\#1]{}\#1[|\#1]{}\#1[\#1|]{} Introduction {#sec1} ============ It has been known for a long time that the dynamics of open quantum systems subject to external driving forces and coupled to environmental modes (‘heat bath’) can be described by master equations [@Weiss2007; @Breuer2006; @Gardiner2004]. For a Markovian bath, the memory time of the bath represents the shortest time scale of the problem. The master equation is then of Lindblad type [@Lindblad1976; @Lindblad1983], where a Hamiltonian describes the coherent time evolution of the system’s density matrix and a Lindbladian captures the dissipative dynamics. (We here use ‘Lindbladian’ for the dissipator terms in the master equations below.) The Lindblad equation is the most general Markovian master equation which preserves the trace and positive semi-definiteness of the density matrix. A major development over the past two decades has come from the realization that driven dissipative (DD) quantum systems can be stabilized in a pure quantum state by appropriate engineering of the driving fields and of the coupling to the dissipative environment [@Plenio1999; @Beige2000; @Plenio2002; @Diehl2008; @Kraus2008; @Diehl2010; @Diehl2011; @Bardyn2013; @Zanardi2014; @Albert2014; @Jacobs2014; @Albert2016; @Goldman2016; @Wiseman2010]. Such states are eigenstates of the corresponding Lindbladian with zero eigenvalue, i.e., the operation of the Lindbladian leaves them inert. We therefore will refer to these DD stabilized states as *dark states* in what follows. Rather than viewing the coupling to a dissipative environment as foe (e.g., leading to decoherence of quantum states and undermining the utilization of similar platforms for quantum information processing), the combined effect of drive and dissipation can thus be harnessed to engineer quantum-coherent pure states. Going beyond dark states, the stabilization of a *dark space* [@Iemini2015; @Iemini2016; @Santos2020] — a manifold spanned by multiple degenerate dark states — raises the prospects of employing such systems as viable platform for quantum information processing. Reference [@Touzard2018] reports on recent experimental results in this direction. Using trapped ions or superconducting qubits, the above ideas have already allowed for first qubit stabilization experiments [@Geerlings2013; @Lu2017; @Touzard2018], for the implementation of quantum simulators [@Barreiro2011; @Schindler2013], and for the generation of selected highly entangled multi-particle states [@Shankar2013; @Leghtas2013; @Reiter2016; @Liu2016]. Systems composed of many coupled qubits stabilized by DD mechanisms could eventually result in universal quantum computation platforms [@Verstraete2009; @Fujii2014], where fault tolerance is the consequence of autonomous error correction [@Terhal2015] due to the engineered dissipative environment, without the need for active feedback [@Wiseman2010; @Kerckhoff2010; @Murch2012; @Kapit2015; @Kapit2016]. Recent experimental progress on autonomous error correction in DD qubit systems has been described in Refs. [@Leghtas2013; @Liu2016; @Reiter2017; @Puri2019]. At present, reported fidelities in DD qubit setups (which by construction are stable in time) are typically below 90$\%$ for state stabilization, with significantly lower fidelities for single- or two-qubit gate operations. Another important and at first glance unrelated development towards the (so far elusive) goal of fault-tolerant universal quantum computation comes from the field of topological quantum computation [@Nayak2008]. By using topological quasiparticles [@Wen2017] for encoding and processing quantum information, the latter is nonlocally distributed in space and thereby protected against local environmental fluctuations. In general terms, for practically useful and scalable DD systems with multiple degenerate dark states, the coupling to the environment has to be carefully engineered such that it is blind to all system operators acting within the targeted dark space manifold [@Facchi2000]. It will thus be imperative to avoid residual (uncontrolled and unwanted) noise sources. In that regard, platforms harboring topological quasiparticles may offer a key advantage since they should come with a strongly reduced intrinsic sensitivity to residual environmental fluctuations as compared to conventional systems. The simplest candidate for topological quasiparticles is given by Majorana bound states (MBSs), which are localized zero-energy states in topological superconductors. For Majorana reviews, see Refs. [@Alicea2012; @Leijnse2012; @Beenakker2013; @Sarma2015; @Aguado2017; @Lutchyn2018; @Zhang2019a]. Topological codes relying on MBSs have so far been discussed in the context of active error correction [@Alicea2011; @Terhal2012; @Hyart2013; @Vijay2015; @Aasen2016; @Landau2016; @Plugge2016; @Plugge2017; @Karzig2017; @Litinski2017; @Wille2019], where periodically repeated stabilizer measurements are needed for fault tolerance. It remains an important challenge to devise feasible and scalable Majorana platforms exploiting passive error correction strategies, where DD mechanisms serve to continuously measure the system in a way that the desired highly entangled many-body quantum state becomes stabilized automatically, see, e.g., Ref. [@Herold2017]. While this ambitious goal is beyond the scope of our work, we here analyze related questions for DD systems with up to eight MBSs. For a mesoscopic floating (not grounded) topological superconductor harboring four MBSs, strong charging effects [@Fu2010] imply that the ground state is doubly degenerate under Coulomb valley conditions (see Sec. \[sec2a\] for details). Such a superconducting island is therefore a good candidate for a topologically protected Majorana qubit, named Majorana box qubit [@Plugge2017] or tetron [@Karzig2017]. Thanks to the nonlocal Majorana encoding of quantum information, such a qubit allows for unique addressability options via electron cotunneling when quantum dots (QDs) or normal leads are attached to the island by tunneling contacts, see also Refs. [@Gau2018; @Munk2019]. Majorana qubits have not yet been experimentally realized. However, the recent emergence of new Majorana platforms (see, e.g., Refs. [@Liu2018b; @Zhang2018b; @Wang2018b; @Sajadi2018; @Ghatak2018; @Murani2019]) in addition to the semiconductor nanowire platform mainly explored so far [@Lutchyn2018; @Zhang2019a] indicates that they may be available in the foreseeable future. We note that alternative Majorana qubit designs have been put forward, e.g., in Refs. [@Terhal2012; @Hyart2013; @Aasen2016]. Many of the ideas discussed below can be adapted to those setups as well. Motivation and goals of this work --------------------------------- We here show that once available, Majorana box devices yield highly attractive platforms for implementing DD protocols aimed at the realization of dark states and/or dark spaces. The driving field is applied to the tunnel link connecting a pair of QDs, and dissipation is due to environmental electromagnetic noise. To the best of our knowledge, apart from a distantly related proposal for the DD stabilization of Majorana-based quantum memories [@Bardyn2016], no studies of DD Majorana systems have appeared in the literature so far. We note that the DD engineering of MBSs in cold-atom based Kitaev chains [@Diehl2011; @Bardyn2013; @Goldman2016] differs from our ideas: We consider topological superconductors harboring native MBSs, and then subject the resulting Majorana systems to DD stabilization and manipulation protocols targeting dark states and/or dark spaces. Our unique platform enables us to employ QDs as external knobs to be used not only for state engineering but also for state manipulation. Our motivation for designing and studying novel DD stabilization and manipulation schemes using Majorana platforms rests on several arguments and expectations: 1. Since uncontrolled environmental effects are largely suppressed by topological protection mechanisms, one may reach higher fidelities than those reported so far for DD dark state or dark space implementions using conventional (topologically trivial) platforms. This point should be especially important for high-dimensional dark spaces, where residual noise effects could break the degeneracy of the dark states spanning the dark space manifold [@Facchi2000]. Such spaces are highly attractive candidates for implementing fault-tolerant quantum computing platforms. These topological protection elements are especially important for platforms where the Lindblad spectrum is not gapped. 2. It is known that for large-scale Majorana surface codes, where active feedback is needed for code stabilization, the fault-tolerance error threshold is much more benign than for conventional bosonic surface codes, see Refs. [@Vijay2015; @Plugge2016; @Fowler2012] and references therein. In particular, in Majorana surface codes no ancilla qubits are needed for stabilizer readout at all. We expect that our dark space constructions using MBS systems can allow for similar fault tolerance advantages over conventional dark space realizations. However, more work is needed to reach a quantitative conclusion on this point. 3. The DD stabilization and manipulation of Majorana-based dark states or dark spaces offers several practical advantages. In particular, the robustness of such states as quantified by the dissipative gap is expected to be superior to quantum states that are encoded without DD mechanisms in native Majorana devices, see Sec. \[sec3\]. Moreover, a small overlap between MBSs is often tolerable, without causing dephasing of dark states, cf. Sec. \[sec3e\]. 4. When steering a state into the dark space or manipulating a state within the dark space, one may need to maximize its purity, having in mind quantum information manipulation protocols. For this purpose, we may adiabatically switch on a suitable perturbation either to the Lindbladian dissipator or to the accompanying Hamiltonian, thereby breaking the degeneracy of the dark space. In this manner, one can revert to a specific pure dark state, manipulate this state, and subsequently adiabatically switch off this perturbation again. The DD Majorana platforms discussed below offer convenient tools to switch on and off such degeneracy-breaking perturbations. ![Schematic sketch of a driven dissipative Majorana box setup. The superconducting island harbors four Majorana operators $\gamma_\nu$, three of which are tunnel-coupled to two single-level quantum dots (QDs, in blue). The Majoranas could be realized as end states of two parallel topological superconductor nanowires (green) which are electrically connected by a superconducting bridge (orange) [@Plugge2017]. The tunnel links connecting QDs to MBSs are shown as dashed lines. The phases $\beta_j$ in Eq.  are also indicated. Due to the large box charging energy, transport between different QDs through the Majorana island proceeds only via cotunneling processes. These cotunneling processes can be inelastic, involving the emission or absorption of photons from the dissipative electromagnetic environment. In addition, a driving field can pump electrons via a tunnel link between the QDs (solid line). []{data-label="fig1"}](f1){width="0.9\columnwidth"} The dynamics of the Majorana degrees of freedom in a device such as the one depicted in Fig. \[fig1\] will here be discussed on several conceptual levels. We show that our DD protocols indeed give rise to master equations of Lindblad type. These equations contain both a Hamiltonian (governing the unitary part of the time evolution) and a Lindbladian (causing dissipative dynamics). By choosing suitable parameter values as discussed in Sec. \[sec3\], we demonstrate that an arbitrary dark state can be stabilized. In more complex two-box devices, see Sec. \[sec4\], the Lindbladian can be engineered to support a multi-dimensional dark space. As a generic initial state is driven towards the dark space, we show (see also Ref. [@ourprl]) how to optimize the purity, the fidelity (i.e., the overlap of the state with the target dark space), and the speed of approach. In our accompanying short paper [@ourprl], we provide a summary of our key ideas and apply them to show that in a two-box setup, one can stabilize and manipulate ‘dark qubit’ states. In effect, the topologically protected native Majorana qubit discussed in Refs. [@Plugge2017; @Karzig2017] (which exists in a single box) is thereby stabilized by adding another protection layer due to DD mechanisms (at the prize of adding a second box). The main benefit of applying DD strategies to a topologically nontrivial system comes from the insight that in the latter class of systems, one can implement unidirectional cotunneling processes in an elementary and practically useful manner. These cotunneling processes in turn directly determine the structure of the jump operators in the Lindblad equation. Overview -------- In order to guide the focused reader through this long article, we here provide a short overview summarizing the content of the subsequent sections. In addition, Table \[table1\] summarizes the key symbols and notations used throughout this paper. - In Sec. \[sec2\], we introduce the theoretical concepts and physical ingredients needed for the DD stabilization and manipulation of dark states using a single Majorana box, see Fig. \[fig1\], and we derive the dynamical equations. Our model is introduced in Sec. \[sec2a\], where the dissipation arises from environmental electromagnetic fluctuations and the drive is applied to a pair of QDs. We subsequently derive the Lindblad equation [@Weiss2007; @Breuer2006; @Gardiner2004; @Lindblad1976; @Lindblad1983] governing the time evolution of the combined QD-Majorana system in Sec. \[sec2b\], where we also present numerical results for the dynamics obtained from this Lindblad master equation. Remarkably, up to initial transient behaviors, one can describe the dynamics in the Majorana sector in terms of a reduced Lindblad equation, where the QD degrees of freedom have been traced out. We describe this step in Sec. \[sec2c\], along with a discussion of the conditions under which this reduced Lindblad equation applies. All of our subsequent results are obtained by employing this reduced Lindblad equation. - In Sec. \[sec3\] we then describe dark state stabilization protocols for the single-box device in Fig. \[fig1\]. We begin in Sec. \[sec3a\] with the case of Pauli operator eigenstates, followed by the stabilization of the so-called magic state in Sec. \[sec3b\]. In Sec. \[sec3c\], the role of increasing temperature on our stabilization protocols is examined. Interestingly, as shown in Sec. \[sec3d\], we find that for certain parameter settings, dark states can be stabilized even in the absence of any drive. Finally, in Sec. \[sec3e\], we discuss additional points, e.g., concerning the role of Majorana state overlaps or how to perform a parity readout of the stabilized states. - In Sec. \[sec4\], we turn to a setup with two coupled boxes and present our DD stabilization and manipulation protocols for quantum states that belong to a dark space manifold. The Lindblad equation for this setting is derived in Sec. \[sec4a\]. We explain how one can engineer a degenerate dark space in Sec. \[sec4b\]. This topic is the main focus of Ref. [@ourprl], and the discussion is therefore kept rather short here. Finally, in Sec. \[sec4c\], we show how to stabilize Bell states in the two-box setting. - The paper concludes with a summary and an outlook in Sec. \[secConc\]. Technical details and additional information can be found in three Appendices. Let us also remark that we often use units with $\hbar=k_B=1$. Symbol Meaning First appearance --------------------------------------------------------------- ------------------------------------------------------------------------------- ------------------ *Model parameters:* $A$ drive amplitude $\alpha$ dimensionless system-bath coupling for Ohmic bath $\beta_j$ phases of the tunnel couplings $\lambda_{j\nu}$ $E_C$ charging energy of the Majorana box $\epsilon_{j}$ level energy of the respective quantum dot $g_0$ cotunneling scale for single-box setup, $g_0=t_0^2/E_C$ $\tilde g_0^{}$ cotunneling scale for double-box setup $\lambda_{j\nu}$ tunnel coupling between QD fermion $d_j$ and Majorana operator $\gamma_{\nu}$ (‘state design parameters’) $M$ number of MBSs on Majorana box $\omega_0$ drive frequency $\omega_c$ cut-off frequency for Ohmic bath $T$ temperature $t_0$ overall scale of tunnel couplings between QDs and Majorana box $t_{LR}$ tunnel coupling connecting both Majorana boxes, see Sec. \[sec4\] *Dynamical quantities:* $D$ dark space dimension Sec. \[sec3e4\] $\Delta_{z,x,y,m}$ dissipative gap for the respective dark state e.g., see $h_{\pm}, \tilde h^{}_\pm$ Lamb shift parameters for full and reduced Lindblad eq., respectively , $J_\pm, \Gamma_{\pm}, H_{\rm L}$ jump operators, transition rates, and Hamiltonian for full Lindblad eq. , , $\tilde J^{}_\pm, \tilde\Gamma^{}_{\pm}, \tilde H^{}_{\rm L}$ jump operators, transition rates, and Hamiltonian for reduced Lindblad eq. , , $K_{j=1,\ldots,6}, \tilde \Gamma^{}_{j}$ jump operators and transition rates for two-box setup , $p$ occupation probability of high-lying QD $\rho(t)$ reduced density matrix for combined QD-Majorana system $\rho_{\rm M}(t)$ reduced density matrix for the Majorana sector $(\tau_x,\tau_y,\tau_z)$ Pauli operators for QD pair in single-occupancy regime $N_{\rm d}=1$ $\theta_{j\nu},\theta$ fluctuating electromagnetic phases , $\hat W_{jk}, \hat W_{x,y,z}$ fluctuating cotunneling operators , $W_{jk}, W_{x,y,z}$ cotunneling operators for $\theta_{j,\nu}=0$ $(X,Y,Z)$ Pauli operators of Majorana box Driven dissipative Majorana dynamics {#sec2} ==================================== We start this section by discussing the Majorana box [@Plugge2017; @Karzig2017]. Our DD model as well as the physical assumptions behind it are explained in Sec. \[sec2a\]. We then derive the Lindblad master equation governing the dynamics of the reduced density matrix of the Majorana sector. To that end, we first trace over the environmental degrees of freedom in Sec. \[sec2b\], and then over the QD fermions in Sec. \[sec2c\]. Model and low-energy theory {#sec2a} --------------------------- In this subsection, we introduce the model for the DD Majorana setup illustrated in Fig. \[fig1\]. We also outline the hardware ingredients needed for implementing our dark state stabilization and manipulation protocols. For concreteness, we refer to a possible realization using proximitized semiconductor nanowires [@Plugge2017; @Karzig2017]. In addition, we describe the effective low-energy Hamiltonian obtained after the high-energy charge states on the Majorana island are projected away. ### Majorana box Consider the setup depicted in Fig. \[fig1\], where a floating topological superconductor island harbors $M$ zero-energy MBSs. For this case we have $M=4$, but for generality, we shall allow for general (even) values of $M$. The MBSs correspond to the Majorana operators $\gamma^{}_\nu=\gamma_\nu^\dagger$, with anticommutator $\{\gamma_\nu,\gamma_{\nu'}\}=2\delta_{\nu\nu'}$ and $\nu=1,\ldots,M$. As indicated in Fig. \[fig1\], they could be realized as end states of two parallel InAs/Al nanowires [@Lutchyn2018]. We consider class-$D$ topological superconductor wires, where time reversal symmetry is broken by a magnetic field [@Alicea2012]. Both nanowires are electrically connected by a superconducting bridge such that the entire island has a common charging energy, $E_C=e^2/(2C)$, with typical values of the order $E_C\approx 1$ meV [@Lutchyn2018]. The isolated island (‘box’) has the Hamiltonian (we work in the Schrödinger picture for now) $$\label{Hbox} H_{\rm box}=E_C( \hat N-N_g)^2.$$ The operator $\hat N$ refers to the total electron number on the box, and $N_g$ is a tunable backgate parameter. In Eq.  we have neglected hybridization energies resulting from a finite overlap between different MBS pairs. These energy scales are exponentially small in the respective MBS-MBS distance. As will be discussed in Sec. \[sec3e\], a small hybridization between MBSs is often tolerable for DD-generated dark states or dark spaces. For the native Majorana qubit, such effects cause dephasing. Our theory requires several conditions to be satisfied. First, we assume that our DD protocols only involve energy scales well below both $E_C$ and the superconducting (proximity) gap $\Delta$. This assumption implies that the ambient temperature satisfies $T\ll {\rm min}\{E_C,\Delta\}$. We can then neglect the effects of above-gap continuum quasiparticles, as has tacitly been assumed in Eq. , which otherwise constitute an intrinsic source of dissipation in the Majorana sector. In practice, one also needs to ensure that accidental low-energy Andreev states are not accessible, see Ref. [@Manousakis2019] for a recent discussion. Second, we consider Coulomb valley conditions [@Nazarov; @AltlandBook], i.e., $N_g$ is tuned close to an integer value and the box is only weakly coupled to the QDs in Fig. \[fig1\]. In that case, $H_{\rm box}$ leads to charge quantization, which dictates the fermion number parity of the island. At temperatures well below the superconducting gap, only the Majorana sector of the full Hilbert space of the box has to be kept [@Fu2010]. For $M=4$, we arrive at a parity constraint in the Majorana sector, $\gamma_1\gamma_2\gamma_3\gamma_4=\pm 1$ [@Beri2012], and the lowest-energy island state is then doubly degenerate. The corresponding Pauli operators associated with the resulting Majorana qubit are represented by Majorana bilinears [@Beri2012; @Landau2016; @Plugge2016], $$\label{PauliOp} X=i\gamma_1\gamma_3,\quad Y=i\gamma_3\gamma_2,\quad Z=i\gamma_1\gamma_2.$$ The fact that Pauli operators correspond to spatially separated pairs of Majorana operators allows for unusually versatile qubit access options. ### Quantum dots We next turn to the Hamiltonian describing the two QDs, $H_{\rm d}$, in Fig. \[fig1\]. We start from a general single-dot Hamiltonian, $H_{\rm QD}=\sum_{\alpha} h_{\alpha} d_{\alpha}^{\dagger}d^{}_{\alpha}+\epsilon_C\left(\hat n-n_g\right)^2$, where $\alpha$ labels electron spin and orbital degrees of freedom, $d_{\alpha}$ are fermion operators with $\hat n=\sum_{\alpha} d_{\alpha}^{\dagger}d^{}_{\alpha}$, $h_{\alpha}$ describes a single-particle energy, and $\epsilon_C$ is the (large) dot charging energy [@Karzig2017; @Flensberg2011; @Nazarov; @AltlandBook]. On low energy scales, the dot can then effectively be described by a single spinless fermion level. Denoting the corresponding level energy by $\epsilon_j$ for QD $j=1,2$, one arrives at $$\label{HDots} H_{\rm d}=\sum_{j=1,2} \epsilon_j d_j^{\dagger} d^{}_j,$$ see Ref. [@Karzig2017] for details. The energies $\epsilon_j$ can be controlled by variation of the gate voltage parameter $n_g$. Without loss of generality, we take $\epsilon_2>\epsilon_1$ throughout, where both energies should satisfy $|\epsilon_j|\ll {\rm min}\{E_C,\Delta\}$. In addition, we employ a time-dependent electromagnetic driving field which can pump single electrons between the two QDs via a tunnel link. To that end, a suitable AC voltage can be applied to a gate electrode located near this link. The respective Hamiltonian contribution is given by [@Platero2004] $$\label{Hdriv} H_{{\rm drive}}(t) = w(t) d_1^{\dagger} d_2^{}+{\rm h.c.},\quad w(t) = t_{12}+2A \cos\left(\omega_0 t\right),$$ where $\omega_0$ denotes the drive frequency and $A$ the drive amplitude. In what follows, we assume that the static contribution vanishes, $t_{12}=0$, because a small coupling $t_{12}\ne 0$ will not affect the dissipator in the Lindblad equation, see Eq.  below, and thus does not change the physics in a qualitative manner. In this work, we consider the Coulomb valley regime where the total charge on the box is fixed by the charging term in Eq.  on time scales $\delta t>1/E_C$ [@Romito2014]. The total particle number on the QDs, $N_{\rm d}=\sum_j d_j^\dagger d_j^{}$, is therefore also conserved on such time scales. For even $N_{\rm d}\in\{0,2\}$, the inter-QD dynamics is effectively frozen out. We here mainly focus on the case $N_{\rm d}=1$, where the pair of QDs forms a spin-1/2 degree of freedom corresponding to Pauli operators $\tau_{x,y,z}$ with $\tau_\pm=(\tau_x\pm i \tau_y)/2$, $$\tau_+ = \tau_-^\dagger = d_1^{\dagger}d_2^{}, \quad \tau_z = d_1^{\dagger}d_1^{} - d_2^\dagger d_2^{} = 2\tau_+\tau_--1.\label{taudef}$$ We next turn to the tunnel couplings connecting the QDs to the island. ### Tunnel couplings and electromagnetic environment In the above parameter regime, tunneling to the box has to proceed via MBSs since no other low-energy island states are available. Such processes can be inelastic due to the coupling to a bosonic environment. We here consider the case of a dissipative electromagnetic environment, which can be modeled by including fluctuating phases $\theta_{j\nu}$ in the tunneling matrix elements [@Nazarov; @Devoret1990; @Girvin1990], $$\label{hatlambda} \hat\lambda_{j\nu}=\lambda_{j\nu}e^{i\theta_{j\nu}},$$ with dimensionless complex-valued parameters $\lambda_{j\nu}$ subject to ${\rm max}\{|\lambda_{j\nu}|\}=1$. Here $\lambda_{j\nu}$ determines the transparency of the tunnel link between the QD fermion $d_j$ and the Majorana state $\gamma_\nu$ in the absence of electromagnetic noise [@Zazunov2016]. With the overall hybridization energy $t_0$ characterizing the QD-MBS couplings, the tunneling Hamiltonian is given by [@Nazarov; @Devoret1990; @Girvin1990] $$\label{Htun} H_{{\rm tun}} =t_0 e^{-i\hat\phi}\sum_{j,\nu} \hat \lambda_{j\nu} d^{\dagger}_j \gamma_\nu + {\rm h.c.}$$ The phase operator $\hat\phi$ of the island has the commutator $[\hat N,\hat\phi]=-i$ with the number operator $\hat N$ in Eq. . The $e^{i\hat\phi}$ ($e^{-i\hat \phi}$) factor in Eq.  thus ensures that an electron charge is added to (subtracted from) the island in a tunneling process. It is well known that the electromagnetic potential fluctuations predominantly couple to the phase of the wave function [@Devoret1990; @Girvin1990]. This fact is expressed by the appearance of the fluctuating tunnel couplings $\hat\lambda_{j\nu}$, see Eq. , in the tunneling Hamiltonian . For concreteness, we assume that the electromagnetic environment can be modeled by a single bosonic bath, see also Ref. [@Munk2019]. Representing the bath by an infinite set of harmonic oscillators [@Weiss2007; @Breuer2006], the environmental Hamiltonian is $H_{\rm env}=\sum_m E_m b_m^{\dagger} b_m^{}$, with the energy $E_m>0$ of the photon mode described by the boson annihilation operator $b_m$. In practice, the relevant bath energies $E_m$ are strongly suppressed above a cutoff frequency $\omega_c$. With dimensionless real-valued couplings $g_{j\nu,m}$, the stochastic phase operators $\theta_{j\nu}$ are written as $$\label{phasedef} \theta_{j\nu}=\sum_{m} g_{j\nu,m} \left(b^{}_m+b_m^\dagger\right).$$ Clearly, they commute with each other, $[\theta_{j\nu},\theta_{j'\nu'}]=0$. ### Low-energy theory {#spsec1} We are interested in the parameter regime defined by the conditions $$\label{condit} {\rm max}\{T,A,t_0,\omega_0,\omega_c,|\epsilon_j|\}\ll {\rm min}\{E_C,\Delta\}.$$ The parameters on the left side of Eq.  affect the dissipative transition rates in the Lindblad equation below. These rates in turn set the time scale on which dark states are approached. We will adopt a concise description, whereby for engineering a stabilization protocol targeting a specific dark state, it suffices to adjust the complex-valued tunnel link parameters $\lambda_{j\nu}$, see Sec. \[sec3\]. In practice, those *state design parameters* can be changed via gate voltages. We also note that under the conditions in Eq. (\[condit\]), boson-assisted processes can neither excite above-gap quasi-particles nor higher-energy charge states on the island. The full Hamiltonian can then be projected onto the doubly degenerate ground-state space of the box, $H(t)\to H_{\rm eff}(t)$. Using a Schrieffer-Wolff transformation to implement this projection, and noting that $H_{\rm box}$ then reduces to an irrelevant constant energy shift, we arrive at the effective low-energy Hamiltonian $$\label{heff0} H_{\rm eff}(t)=H_{\rm d}+ H_{\rm env} + H_{\rm drive}(t)+H_{\rm cot},$$ with the drive term in Eq.  and the cotunneling contribution $$H_{\rm cot} = g_0\sum_{j,k=1,2}\hat W_{jk} \left( 2 d_j^\dagger d_k^{}-\delta_{jk} \right), \quad g_0\equiv \frac{t_0^2}{E_C}.\label{Heff}$$ We here use the operators $$\label{Wopdef} \hat W_{jk}= \sum_{1\le\mu<\nu\le M} \left( \hat \lambda_{j\nu} \hat \lambda_{k\mu}^\dagger - \hat \lambda_{j\mu}\hat \lambda_{k\nu}^\dagger \right)\gamma_\mu\gamma_\nu.$$ Equation describes cotunneling paths through the box, where the energy of the intermediate virtual state has been approximated by $E_C$, cf. Eq. , and photon emission and absorption processes are encoded by the $\hat \lambda$ factors in Eq. . For even QD occupation number $N_{\rm d}$, Eq.  reduces to $$\label{heffeven} H_{\rm cot}^{(N_{\rm d}=0,2)}=g_0 \ {\rm sgn}(N_{\rm d}-1) \ \sum_j \hat W_{jj}.$$ For $N_{\rm d}=1$, using the notation $$\begin{aligned} \nonumber \hat W_+&\equiv& \hat W_{12}, \quad \hat W_-=\hat W_+^\dagger,\quad \hat W_x=\hat W_++\hat W_-,\\ \label{wdef1} \hat W_y&=&i(\hat W_+ -\hat W_-),\quad \hat W_z=\hat W_{11}-\hat W_{22},\end{aligned}$$ we find that Eq.  can instead be expressed in the form $$\label{cot1} H_{\rm cot}^{(N_{\rm d}=1)} = g_0 \sum_{a=x,y,z} \hat W_a \tau_a,$$ with the QD Pauli operators $\tau_a$ in Eq. . We emphasize that like the $\hat W_{jk}$ operators in Eq. , also the $\hat W_a$ still contain the phase fluctuation operators due to the electromagnetic environment. In order to realize the most general qubit-qubit exchange coupling between the QD spin $\{\tau_a\}$ and the $M=4$ Majorana box spin $(X,Y,Z)$ in the cotunneling regime, one has to specify nine independent (tunable) real-valued coupling constants. For the $M=4$ case in Fig. \[fig1\], taking into account gauge invariance — which allows us to set one of the $\lambda_{j\nu}$ to a real value —, the five different complex-valued hopping parameters $\lambda_{j \nu}$ are sufficient. On top of that, the direct tunnel amplitude between the QDs is assumed to be real-valued after setting $t_{12} = 0$ in Eq. . To simplify the subsequent analysis, we assume that the dominant contribution to the environmental electromagnetic noise comes from the long wavelength part. In effect, such contributions will cause dephasing of the QDs, e.g., due to the presence of a backgate electrode. This assumption is also consistent with the picture of a single bath. To good accuracy, the couplings $g_{j\nu,m}$ in Eq.  then do not depend on the Majorana index $\nu$, i.e., $g_{j\nu,m}=g_{j,m}$. As a consequence, also the fluctuating phases become $\nu$-independent, $\theta_{j\nu}=\theta_j$. In that case, the diagonal entries $\hat W_{jj}$ are insensitive to electromagnetic noise and the bath completely decouples for even $N_{\rm d}$, see Eq. . From now on, we therefore focus on the case of a single electron shared by the QDs, $N_{\rm d}=1$. Defining the phase operator $$\theta\equiv \theta_1-\theta_2 = \sum_m (g_{1,m}-g_{2,m}) \left(b_m^{}+b_m^\dagger\right),$$ Eq.  then yields $$\label{cotfinal} H_{\rm cot}= 2g_0 \left(e^{i\theta} W_+\tau_+ + {\rm h.c.}\right)+ g_0 W_z \tau_z.$$ The operators $W_+$ and $W_z$ correspond to ‘undressed’ ($\theta_{j\nu}\to 0$) versions of $\hat W_+$ and $\hat W_z$, respectively. These operators act only on the Hilbert space sector describing Majorana states. Comparing to Eq. , we have $$\label{wdef2} W_{jk} = \sum_{\mu<\nu}^M \left( \lambda_{j\nu} \lambda_{k\mu}^\ast - \lambda_{j\mu} \lambda_{k\nu}^\ast \right)\gamma_\mu\gamma_\nu.$$ For the device in Fig. \[fig1\], the $W_{jk}$ operators can be expressed in terms of the Pauli operators $(X,Y,Z)$ in Eq. , see below. ### Bath correlation functions The equilibrium density matrix of the thermal environment is given by $$\label{envtrace} \rho_{\rm env}=Z^{-1}_{\rm env} e^{-H_{\rm env}/T}\quad {\rm with} \quad Z_{\rm env}={\rm tr}_{\rm env} \ e^{-H_{\rm env}/T},$$ with ‘tr$_{\rm env}$’ denoting a trace operation over the environmental bosons. Using $\braket{\hat O}_{\rm env}\equiv {\rm tr}_{\rm env} (\hat O\rho_{\rm env})$, we define the correlation function [@Weiss2007] $$\begin{aligned} \nonumber && J_{\rm env}(t)=\braket{[\theta(t)-\theta(0)] \theta(0)}_{\rm env} = \int_0^\infty \frac{d\omega}{\pi} \frac{{\cal J}(\omega)}{\omega^2}\times \\ &&\quad \times \left\{ [\cos(\omega t)-1] \coth\left(\frac{\omega}{2T}\right) - i \sin(\omega t) \right\},\label{BathCorr}\end{aligned}$$ with the spectral density $$\label{spectraldensity} {\cal J}(\omega)=\pi\sum_m (g_{1,m}-g_{2,m})^2 E_m^2 \delta(\omega-E_m).$$ Switching to the continuum limit in bath frequency space, we focus on the practically most important Ohmic case with ${\cal J}(\omega)\propto \omega$ in the low-frequency limit. In concrete calculations, we use the model spectral density [@Weiss2007] $$\label{Ohmic} {\cal J}(\omega) = \alpha \omega e^{-\omega/\omega_c},$$ where $\alpha$ is a dimensionless system-bath coupling and frequencies above $\omega_c$ are exponentially suppressed. For a related discussion in the context of Majorana qubits, see Ref. [@Munk2019]. The parameter $\alpha$ is related to the environmental impedance $Z(\omega)$ [@Devoret1990], $$\label{alphadef} \alpha= \frac{e^2}{2h} {\rm Re} Z(\omega=0).$$ We consider the case $\alpha<1$ below. For the subsequent discussion, we rewrite $H_{\rm cot}$ in normal-ordered form relative to the phase fluctuations, $$\label{normalorder1} H_{\rm cot} = H_{\rm cot}^{(0)} + V,$$ where $H_{\rm cot}^{(0)}$ is the expectation value of $H_{\rm cot}$ with respect to phase fluctuations and $V$ represents the coupling of the combined QD-MBS system to phase fluctuations. Since $\langle\theta^2\rangle_{\rm env}$ diverges in the Ohmic case, we have $\langle e^{i\theta}\rangle_{\rm env}=0$, resulting in $$\label{normalorder2} H_{\rm cot}^{(0)} \equiv \braket{H_{\rm cot}}_{\rm env}= g_0 W_z\tau_z.$$ The interaction term in Eq.  is then given by $$\label{normalorder3} V= 2g_0 \left( e^{ i\theta} W_+ \tau_+ + {\rm h.c.}\right) .$$ By construction, $\braket{V}_{\rm env}=0$. Correlation functions of exponentiated phase fluctuations are given by ($s=\pm 1$) $$\label{xicor} \braket{e^{is\theta (t)} e^{-is\theta(0)}}_{\rm env} = e^{J_{\rm env}(t)}$$ with $J_{\rm env}(t)$ in Eq. . ### Interaction picture and Rotating Wave Approximation From now on, we shall switch to the interaction picture with respect to $H_{\rm d}+H_{\rm env}$. The Hamiltonian then takes the form, see Eqs.  and , $$\begin{aligned} \label{Hfull} H_{{\rm eff},I}(t) &=& H_{0,I}(t)+ V_I(t),\\ \nonumber H_{0,I}(t) &=& H_{{\rm drive},I}(t) + H^{(0)}_{{\rm cot},I}(t).\end{aligned}$$ For simplicity, we drop the ‘$I$’ index (for interaction picture) in what follows and focus on resonant drive conditions, $$\label{omegadef} \omega_0=\epsilon_2-\epsilon_1.$$ In the regime $\omega_0\gg T$ considered below, see Eq. , we can then apply the rotating wave approximation (RWA) [@Gardiner2004]. As a consequence, $H_{\rm drive}(t)\to \tilde H_{\rm drive}$ with $$\label{Hdriv2} \tilde H_{\rm drive}=A\left( d_1^{\dagger} d^{}_2+ d_2^{\dagger} d^{}_1\right) = A \tau_x,$$ resulting in a time-independent drive Hamiltonian in the interaction picture. If the drive frequency is slightly detuned, $\omega_0=\epsilon_2-\epsilon_1+ \delta\omega_0$, a residual time dependence remains, $H_{\rm drive}(t)= e^{-i\delta\omega_0 t} A d_1^{\dagger} d^{}_2 +$ h.c., after applying the RWA. However, we find that the final Lindblad equation for the dynamics of the Majorana sector in Sec. \[sec2c\] is not affected to leading order in $\delta\omega_0$. A small mismatch in the resonance condition will therefore not obstruct our findings. We then put $\delta\omega_0=0$ from now on. Master equation {#sec2b} --------------- In this subsection, we consider the time evolution of the reduced density matrix, $\rho(t)$, describing the coupled system defined by the MBSs and the pair of QD fermions. After tracing over the environmental bosons, we arrive at a Lindblad master equation for the dynamics of $\rho(t)$. In Sec. \[sec2c\], we will subsequently trace over the QD fermions to obtain a Lindblad equation for the Majorana sector only. With $\omega_0=\epsilon_2-\epsilon_1$ and $g_0=t_0^2/E_C$, we consider the regime $$\label{basiccond} g_0\ll T\ll \omega_0, \quad A\alt g_0.$$ In particular, $T\ll \omega_0$ is needed to justify the RWA, while $g_0\ll T$ is required for the Born-Markov approximation. In addition, the regime $g_0\ll T$ allows us to neglect emission and absorption processes taking place only in the Majorana sector since the bath is then unable to resolve such transitions. Of course, we will account for boson-assisted inter-QD transitions resulting from cotunneling processes. Finally, Eq.  states that we study a weakly driven system with drive amplitude $A\alt g_0$. The opposite strongly driven case is briefly discussed in App. \[appA\] and will be studied in detail elsewhere. We note that inelastic corrections to the drive Hamiltonian due to electromagnetic phase fluctuations, see Eq. , can be neglected by the secular approximation, cf. Sec. II.B of Ref. [@Shavit2019]. ### Lindblad master equation for $\rho(t)$ The master equation governing the dynamics of the density matrix $\rho(t)$ for the combined system (QDs and Majorana sector) is obtained by following the standard derivation of Born-Markov master equations [@Weiss2007; @Breuer2006; @Gardiner2004]. We assume a factorized initial (time $t=0$) density matrix of the total system, $\rho_{\rm tot}(0)=\rho(0) \otimes \rho_{\rm env}$, with $\rho_{\rm env}$ in Eq. . Starting from the von-Neumann equation for $\rho_{\rm tot}(t)$ subject to $H_{\rm eff}(t)$ in Eq. , we trace over the environmental modes and apply the Born-Markov approximation [@Weiss2007; @Breuer2006; @Gardiner2004]. As a result, $\rho(t)$ obeys the master equation $$\begin{aligned} \label{UnwantedTerms} &&\partial_t\rho(t)=-i\left[ H_{0}(t),\rho(t)\right] \\ \nonumber && -\, {\rm tr}_{\rm env} \int_0^{\infty} d\tau\left[ V(t-\tau),\left[ V(t)+H_0(t),\rho(t)\otimes\rho_{\rm env}\right]\right],\end{aligned}$$ where we have used that, by construction, ${\rm tr}_{\rm env}\left[ V(t),\rho(0)\otimes\rho_{\rm env}\right]=0$. Similarly, the mixed term involving $V(t-\tau)$ and $H_{0}(t)$ vanishes identically. We are then left with the coherent evolution term due to $H_{0}(t)$, and the double commutator containing two $V$ terms. Unfolding the double commutator, we arrive at a master equation of Lindblad [@Lindblad1976; @Lindblad1983] type, $$\label{GeneralLindblad} \partial_t\rho(t) = -i\left[ H_{\rm L},\rho(t)\right]+ \sum_{\pm} \Gamma_\pm \mathcal{L}[J_\pm]\rho(t) .$$ The subscript ‘L’ in $H_{\rm L}$ is meant to clarify that this Hamiltonian appears in a Lindblad equation. The dissipator ${\cal L}$ acts as superoperator on $\rho$ [@Breuer2006], $$\label{Dissipator} \mathcal{L}[J]\rho=J\rho J^\dagger -\frac{1}{2}\lbrace J^{\dagger} J,\rho \rbrace.$$ The two *jump operators* in Eq.  are given by $$\label{jumpops} J_\pm = 2W_\pm \tau_\pm = J_\mp^\dagger,$$ with the corresponding dissipative transition rates, $$\label{dissrate} \Gamma_{\pm}= 2g_0^2\, {\rm Re} \Lambda_\pm .$$ Here, we define the quantities $$\label{lambdadef} \Lambda_\pm= \int_0^\infty dt \, e^{\pm i\omega_0 t} e^{J_{\rm env}(t)},$$ with the bath correlation function . Their imaginary parts give Lamb shift parameters, $$\label{shifts2} h_\pm = g_0^2 \, {\rm Im}\Lambda_\pm ,$$ which appear in the Hamiltonian governing the coherent time evolution in Eq. , $$\label{Hq} H_{\rm L} = A \tau_x + g_0 W_z\tau_z + \sum_{\pm} h_\pm J_{\pm}^{\dagger} J_\pm^{}.$$ The first two terms in $H_{\rm L}$ originate from $H_0$ in Eq. , while the third term contains the Lamb shifts . Next we observe that Eq.  implies the general relation $$\label{detbal} J_{\rm env}\left(-t-i/T\right)=J_{\rm env}(t)$$ in the complex-time plane. Using Eq.  in Eq.  then results in a detailed balance relation, $\Lambda_-=e^{-\omega_0/T} \Lambda_+$. As a consequence, for arbitrary parameters, we find $$\Gamma_-=e^{-\omega_0/T}\Gamma_+, \quad h_-=e^{-\omega_0/T} h_+.$$ In particular, for $T\ll \omega_0$, the dissipative rate $\Gamma_-$ associated with the jump operator $J_-$ will be exponentially suppressed against the rate $\Gamma_+$. The dissipative part of the Lindblad equation is therefore completely dominated by the jump operator $J_+$. It is a distinguishing feature of our DD platform that jump operators can be directly implemented by designing *unidirectional* inelastic cotunneling paths connecting pairs of QDs via the box, with the overall energy scale $g_0$. The QDs are also directly coupled by a driven tunnel link $w(t)$, see Eq. , with overall energy scale $A$. For $T\ll \omega_0$, as far as inter-dot transitions via the box are concerned, only photon emission processes are relevant. As a consequence, only transitions from the energetically high-lying QD 2 to QD 1 may take place, corresponding to the jump operator $J_+\propto \tau_+$, see Eqs.  and . Such transitions act on the Majorana state according to the operator $W_+$. As we show below, this operator can be engineered at will by adjusting the tunneling parameters $\lambda_{j\nu}$, which in turn is possible by changing suitable gate voltages. The driving field pumps the dot electron in the opposite direction, i.e., from QD $1\to 2$, and for a small pumping rate, $A\alt g_0$, we obtain a steady state circulation $1\to 2\to 1$ by alternating pumping and cotunneling processes. On the other hand, for $A>g_0$, pumping processes will dominate and the cotunneling channel is effectively suppressed, see App. \[appA\]. To facilitate analytical progress, we consider the case $\omega_0\ll \omega_c$. (Otherwise Eq.  can be solved numerically in a straightforward manner.) One then finds [@Weiss2007] $$J_{\rm env}(t) \simeq -2\alpha \ln\left( \frac{\omega_c}{\pi T} \sinh(\pi T t)\right) - i\pi\alpha \,{\rm sgn}(t),$$ and with the Gamma function $\Gamma(z)$, we arrive at $$\begin{aligned} \label{rate2} \Gamma_+ &\simeq& \Gamma(1-2\alpha)\sin(2\pi\alpha) \left(\frac{\omega_0}{\omega_c}\right)^{2\alpha} \frac{2 g_0^2 }{\omega_0} ,\\ \nonumber h_+ &\simeq& \frac12 \cot(2\pi \alpha) \Gamma_+.\end{aligned}$$ For the device in Fig. \[fig1\], using the Pauli operators , the jump operators $J_\pm^{}=J_\mp^\dagger$ follow from Eq.  in the general form $$\begin{aligned} \nonumber J_{+}&=& \tilde J_+ \tau_+ , \\ \tilde J_+ &=& 2ie^{i\beta_2}|\lambda_{23}|\left(e^{-i\beta_3}|\lambda_{11}|X-e^{-i\beta_1}|\lambda_{12}|Y\right) \nonumber \\ &-& 2i\left[e^{-i\beta_1}|\lambda_{12}\lambda_{21}|-e^{i\beta_4}|\lambda_{11}\lambda_{22}|\right]Z, \label{jumpN}\end{aligned}$$ where the phases $\beta_{1,2,3,4}$ are indicated in Fig. \[fig1\]. They are connected to the phases $\chi_{j\nu}$ in the tunneling parameters, $\lambda_{j\nu}=|\lambda_{j\nu}|e^{-i\chi_{j\nu}}$, by the relations $$\label{betadef} \beta_1 = \chi_{12}, \quad \beta_2 = \chi_{23},\quad \beta_3= \chi_{11}, \quad \beta_4=\chi_{22},$$ with the gauge choice $\chi_{21}=0$. In particular, $\beta_1-\beta_3$ ($\beta_2$) is the loop phase accumulated along the shortest closed tunneling trajectory involving only QD 1 (QD 2), cf. Eq. . These phases, as well as the absolute values $|\lambda_{j\nu}|$, can be experimentally varied, e.g., by changing the voltages on nearby gates. We emphasize that $\tilde J_+$ is fully determined by selecting the state design parameters $\lambda_{j\nu}$. The Hamiltonian $H_{\rm L}$ then follows as $$\begin{aligned} H_{\rm L}&=& A\tau_x + 2g_0 \tilde J_z \tau_z + \sum_\pm h_\pm J_\pm^\dagger J_\pm^{},\nonumber \\ \label{jzdef} \tilde J_z &=& \frac12\bar\lambda^2+\sin\beta_2|\lambda_{21}\lambda_{23}| X+\\ \nonumber &+& \sin\left(\beta_4-\beta_2\right)|\lambda_{22}\lambda_{23}|Y+\\ \nonumber &+&\left[\sin\beta_4|\lambda_{21}\lambda_{22}|-\sin\left(\beta_1-\beta_3\right)|\lambda_{11}\lambda_{12}|\right]Z ,\end{aligned}$$ where $\bar \lambda^2\equiv |\lambda_{11}|^2+|\lambda_{12}|^2+|\lambda_{21}|^2+|\lambda_{22}|^2+|\lambda_{23}|^2$. It is worth mentioning that the operators $\tilde J_\pm$ and $\tilde J_z$ act only on the Majorana subsector. To illustrate the above general expressions, let us consider a simple example. We take stabilization parameters subject to the conditions $$\begin{aligned} \label{parameterchoice1} |\lambda_{11}| &=& |\lambda_{12}|, \quad \lambda_{22}=0, \\ \nonumber \beta_1 &=&- \beta_2 =\pi/2, \quad \beta_3 = \beta_4 =0.\end{aligned}$$ Using Eq. , the dominant jump operator contributing to the Lindbladian is then given by $$J_+=2|\lambda_{11}| \left(2|\lambda_{23}|\sigma_+ + |\lambda_{21}|Z\right)\tau_+, \label{NumericJumpOperator}$$ where $\sigma_\pm=(X\pm iY)/2$. For $|\lambda_{23}|\gg |\lambda_{21}|$, the Lindbladian will then automatically drive an arbitrary Majorana state $\rho_{\rm M}$ towards $|0\rangle\langle 0|$, with the $Z$-eigenstate $|0\rangle$ to eigenvalue $+1$, i.e., $Z|0\rangle=|0\rangle$. Here, the reduced density matrix $\rho_{\rm M}(t)$ describes the Majorana sector only, see Sec. \[sec2c\]. However, the operator $\tilde J_z$ appearing in the Hamiltonian $H_{\rm L}$ still contains a small $X$ component, see Eq. , which could potentially disrupt the action of the dissipator. Nonetheless, we find below that for small $|\lambda_{21}|$, the desired state $|0\rangle$ is approached with high fidelity, regardless of the initial system state $\rho(0)$. An optimized parameter choice for stabilizing $|0\rangle$ will be discussed in Sec. \[sec3\]. ### Numerical results {#specialsec} ![ Driven dissipative dynamics for the setup in Fig. \[fig1\], illustrating the time-dependent expectation values of the Pauli operators $\tau_{x,y,z}$ describing the QDs, see Eq. . We also show the purity, $P_s(t)$, of the system state, see Eq. . All results were obtained by numerical integration of the Lindblad equation for the density matrix $\rho$ describing the QDs and the Majorana sector, with $H_{\rm L}$ in Eq. . We used the parameters in Eq. , with $T/g_0=4$, $\omega_0/g_0=40$, $\omega_c/g_0=200$, $A/g_0=0.1$, $\alpha=1/4$, $|\lambda_{11}|=|\lambda_{12}|=|\lambda_{23}|= 1$, and $|\lambda_{21}|=0.1$. Fast transient oscillations in $\langle\tau_a(t)\rangle$ are not resolved on the shown time scale, corresponding to shaded regions. The respective dynamics in the Majorana sector is depicted in Fig. \[fig3\]. []{data-label="fig2"}](f2){width="\columnwidth"} ![Time evolution of the Bloch vector, $(\langle X\rangle,\langle Y\rangle,\langle Z\rangle)(t)$, describing the Majorana state $\rho_{\rm M}(t)$ for the same parameters as in Fig. \[fig2\]. The expectation value is computed by numerically integrating the Lindblad equation. Starting from the initial $X$-eigenstate $|+\rangle$, the DD protocol stabilizes the dark state $|0\rangle$ at long times, corresponding to the north pole of the Bloch sphere. The intermediate states (with alternating colors) were obtained at times $g_0t\in\lbrace 5\times 10^{3}, 10\times 10^3,\ldots,15\times10^4\rbrace$. \[fig3\]](f3){width="0.7\columnwidth"} We next turn to a numerical integration of Eq.  using the approach of Refs. [@Johansson2012; @Johansson2013]. Numerical results for the above parameters are shown in Figs. \[fig2\] and \[fig3\]. While the goal of the DD protocol is to stabilize a selected state in the Majorana sector, it is useful to also study the dynamics in the QD sector, see Fig. \[fig2\]. We start from a pure initial state, $\rho(0)=|\Psi(0)\rangle\langle\Psi(0)|$, with $|\Psi(0)\rangle= |+\rangle \otimes |0\rangle_{\rm d}$, where the $\tau_z=+1$ QD eigenstate, $|0\rangle_{\rm d}$, describes an electron located in the energetically lower QD 1, with QD 2 left empty, see Eq. . The initial Majorana state has been chosen as the $X$-eigenstate $|+\rangle$ with eigenvalue $+1$. However, we have checked that the same long-time limit of $\rho(t)$ is reached for other initial states. We define the purity of the system state as $$\label{puritydef} P_s(t)= {\rm tr} \rho^2(t).$$ The upper left panel of Fig. \[fig2\] shows that the purity approaches a value close to the largest possible value ($P_s=1$) at long times. Moreover, as observed from Fig. \[fig3\], the DD protocol steers the Majorana state towards the pure state $|0\rangle$, i.e., towards the north pole of the corresponding Bloch sphere. For the shown example, the QD state $\rho_{\rm d}$ has most probability weight in the energetically lower QD 1. Indeed, Fig. \[fig2\] shows that at long times, the electron shared by the two QDs will predominantly relax to QD 1, corresponding to the state $|0\rangle_{\rm d}$. Nonetheless, it is of crucial importance that the occupation probability $p$ for encountering the electron in the energetically higher QD 2 (corresponding to the state $|1\rangle_{\rm d}$) remains finite at long times. We find $p\approx 0.001$ for the parameters in Fig. \[fig2\]. We conclude that the system state factorizes at long times, $\rho(t)\simeq \rho_{\rm M}\otimes \rho_{\rm d}$ with $\rho_{\rm M}=|0\rangle \langle 0|$. The approach of the Majorana state towards $|0\rangle$ takes place on a time scale given by the inverse of the dissipative gap of the reduced Lindbladian describing the Majorana sector only, see Sec. \[sec3\] below. The relaxation time scales for the QD subsystem can be longer, cp. Figs. \[fig2\] and \[fig3\]. Finally, we remark that for the special case $\lambda_{21}=0$, the electron shared by the two QDs will *not* predominantly relax to the energetically lower QD $1$. One here has only two cotunneling paths between both QDs, namely the constituents forming the operator $4|\lambda_{11}\lambda_{23}|\sigma_+$ in Eq. . Both paths interfere destructively once the Majorana island is stabilized in the state $\ket{0}$. An arbitrarily weak drive can then overcome all dissipative effects in the long-time limit. In contrast to what happens for $\lambda_{21}\ne 0$, the QDs will thus realize an equal-weight mixture of $|0\rangle_{\rm d}$ and $|1\rangle_{\rm d}$. Nonetheless, the reduced Lindblad equation below still applies, with $p\to 1/2$ and $p_\perp\to 0$ in Eq. . We note that those parameters are also appropriate in the strongly driven case, cf. App. \[appA\]. Lindblad equation for the Majorana sector {#sec2c} ----------------------------------------- The above observations allow us to derive a reduced Lindblad equation, which directly describes the dynamics of $\rho_{\rm M}(t)$ in the Majorana sector alone. To that end, we now trace also over the QD subspace. At long times, our numerical simulations generically show that $\rho(t)$ factorizes into a Majorana part, $\rho_{\rm M}(t)$, and a QD contribution, $\rho_{\rm d}(t)$, $$\label{factorize} \rho(t \to \infty) \simeq \rho_{\rm M}(t) \otimes \rho_{\rm d}(t).$$ For tracing over the QD part, we can effectively use a time-independent *Ansatz*, $$\label{ssform} \rho_{\rm d}=\left( \begin{array}{cc} 1-p & p_\perp \\ p^\ast_\perp & p \end{array}\right),$$ written in the basis $\{ |0\rangle_{\rm d},|1\rangle_{\rm d}\}$ selected by the coupling to the QDs. Here, $p\ne 0$ refers to the occupation probability of the energetically higher QD 2. This probability can be determined by numerically solving Eq. , cf. Sec. \[sec2b\], or it may be treated as phenomenological parameter. A simple estimate predicts $p\approx {\rm max}(A,g_0)/\omega_0$. Noting that a small but finite expectation value $\langle\tau_x\rangle\ne 0$ is observed in Fig. \[fig2\] at long times, we have also included an off-diagonal term $(p_\perp)$ in Eq. . Inserting Eq.  into Eq.  and tracing over the QD subsystem, we arrive at a Lindblad equation for the $2\times 2$ density matrix $\rho_{\rm M}(t)$ only, $$\label{LindbladMBQ} \partial_t\rho_{\rm M}(t) = -i[\tilde H_{\rm L},\rho_{\rm M} ]+ \sum_{s=\pm} \tilde\Gamma_s \mathcal{L}[\tilde J_s]\rho_{\rm M}(t),$$ where the jump operators $\tilde J_\pm$ have been defined in Eq. . The dissipative transition rates $\tilde \Gamma_\pm$ in Eq.  are given by $$\label{mod1} \tilde \Gamma_+ = p \Gamma_+,\quad \tilde \Gamma_- = (1-p)\Gamma_-,$$ cf. Eqs.  and . The coherent time evolution in Eq.  is governed by the Hamiltonian $$\label{mod22} \tilde H_{\rm L} = 2(1-2p)g_0 \tilde J_z + \sum_\pm \tilde h_\pm \tilde J_\pm^\dagger \tilde J_\pm^{},$$ where $\tilde J_z$ has been specified in Eq.  and the Lamb shifts $\tilde h_\pm$ are given by $$\label{mod2} \tilde h_+ = p h_+ ,\quad \tilde h_- = (1-p) h_-.$$ The drive amplitude $A$ then appears only implicitly through the dependence $p=p(A)$. We note that within the RWA, no contributions $\propto p_\perp$ appear in Eq. . Indeed, the RWA allows one to neglect terms $\propto \tau_+\rho\tau_+$ which stem from $p_\perp\ne 0$. Importantly, apart from the initial transient behavior, all of our numerical results for the Majorana dynamics obtained from the full Lindblad equation for the combined QD-MBS system, Eq. , are quantitatively reproduced by using the simpler Lindblad equation . This statement is valid for arbitrary model parameters subject to Eqs.  and . We emphasize that the integration over the QD degrees of freedom as carried out above relies on the facts that (i) the convergence towards the target state is dictated by the Majorana sector, and that (ii) the QD and MBS degrees of freedom always decouple in the long-time limit, see Eq. . The latter feature has been established by extensive numerical simulations of Eq. . The reduced Lindblad equation is applicable as long as transient behaviors are not of interest. In particular, when studying, e.g., the dynamics of $\rho_{\rm M}(t)$ in the presence of time-dependent QD level energies $\epsilon_j(t)$, Eq.  should only be used for very slow (adiabatic) time dependences. For rapidly varying QD level energies, one has go back to the full Lindblad equation for the combined QD-MBS system in Eq. . Dark state stabilization {#sec3} ========================= Using the Lindblad master equation and the Choi isomorphism [@Albert2014] summarized in App. \[appB\], we now turn to a detailed analysis of our stabilization protocols for the single-box device in Fig. \[fig1\]. The parameter values for stabilizing a specific dark state can be determined by solving the zero-eigenvalue condition of the Lindbladian, cf. App. \[appB\]. We recall that the key state design parameters of our DD protocol are given by the complex-valued tunneling amplitude parameters $\lambda_{j\nu}$, which also define the phases $\beta_j$ in Fig. \[fig1\]. In Sec. \[sec3a\], we show how to stabilize Pauli operator eigenstates. In Sec. \[sec3b\], we discuss magic state stabilization protocols, followed by a study of temperature effects in Sec. \[sec3c\]. We show in Sec. \[sec3d\] that in certain cases, a dark state can be stabilized even in the absence of any driving field. Finally, we conclude in Sec. \[sec3e\] with several remarks. Pauli operator eigenstates {#sec3a} -------------------------- ![Dark-state stabilization protocols for Pauli operator eigenstates. Left side panels (blue curves): Stabilization of $|0\rangle$. Right side panels (red curves): Stabilization of $|+\rangle$, where $X|+\rangle=|+\rangle$. In both cases, the Majorana island has initially been prepared in the $Y$-eigenstate with eigenvalue $+1$. We use the parameters in Eq.  with $p=1/2$, all other parameters are as in Fig. \[fig2\]. With $E_C=1$ meV and $g_0/E_C=2.5\times 10^{-3}$, the time units follow as shown. As explained in the main text, for the chosen parameter set, Rabi oscillations are absent. []{data-label="fig4"}](f4){width="\columnwidth"} We start by discussing DD protocols targeting Pauli operator eigenstates. Typical numerical results obtained by solving Eq.  are illustrated in Fig. \[fig4\]. Following the method in App. \[appB\], the $Z=\pm 1$ eigenstates can be realized by choosing $$\label{sigmazcond} |\lambda_{11}|=|\lambda_{12}|,\quad \lambda_{21}=\lambda_{22}=0, \quad \beta_1-\beta_3=\pm\pi/2,$$ with arbitrary $\lambda_{23}$ and $\beta_{2,4}$, see Eq. . (We note that for $\lambda_{23}=0$, the phases $\beta_{2,4}$ are not defined.) At this point, we use the concept of a *dissipative map* $\hat E$ [@Breuer2006], which is defined in terms of a jump operator mapping the system onto a specific state when acting inside the Lindblad dissipator. For example, the dissipative maps targeting the $Z=\pm 1$ eigenstates are $$\hat E_{\pm}=\sigma_\pm=(X\pm iY)/2.$$ For the stabilization parameters in Eq. , the jump operator $\tilde J_+\propto \hat E_\pm$, with the $\pm$ sign determined by Eq. , completely dominates the Lindbladian part of Eq.  at low temperatures, $T\ll\omega_0$. The dissipative dynamics then maps every input state to $|0\rangle$ (for the $+$ sign) or $|1\rangle$ (for the $-$ sign). At the same time, the Hamiltonian evolution in Eq.  comes from $\tilde H_{\rm L}\propto Z$, see Eq. . Evidently, this Hamiltonian commutes with the targeted state $\rho_{\rm M}(\infty)$, and therefore does not affect the dynamics towards the steady state generated by the dissipative map $\hat E_\pm$. The Majorana state $\rho_{\rm M}(t)$ is thus automatically steered towards the corresponding $Z$-eigenstate by the Lindbladian, with no obstruction from the Hamiltonian dynamics. For the above protocol, the *dissipative gap* is given by, cf. App. \[appB\], $$\label{deltaz} \Delta_z = |4\lambda_{11}\lambda_{23}|^2\sum_{s=\pm}\tilde\Gamma_{s}.$$ In general terms, the dissipative gap is defined as the real part of the smallest non-vanishing eigenvalue of the Lindbladian (the dark state itself has eigenvalue zero) [@Breuer2006]. The time scale on which the dark state will be approached is therefore given by $\Delta_z^{-1}$. Moreover, the approach of the Bloch vector towards the dark state $|0\rangle$ is in general accompanied by damped oscillations in the $(X,Y)$ components, where $\Delta_z$ is the damping rate and the Rabi frequency follows from Eq.  as $$\label{rabiz} \Omega_z \simeq \left|2g_0(1-2p)|\lambda_{11}|^2-8|\lambda_{11}\lambda_{23}|^2\tilde h_+\right|.$$ For the special case $\lambda_{21}=0$ with $p=1/2$, cf. Sec. \[sec2b\], and noting that $\tilde h_+=0$ for $\alpha=1/4$, cf. Eq. , we obtain $\Omega_z=0$ in Eq. . The left panels in Fig. \[fig4\] therefore exhibit only damping in the $(X,Y)$ components, without Rabi oscillations. Next, $X=\pm 1$ eigenstates are realized by choosing $$|\lambda_{21}|=|\lambda_{23}|,\quad \lambda_{11}=\lambda_{22}=0,\quad \beta_2=\mp\pi/2,$$ with the dissipative gap $\Delta_x=|4\lambda_{12}\lambda_{21}|^2\sum_s\tilde\Gamma_s.$ As shown in the right panels of Fig. \[fig4\], $X$-eigenstates, e.g., the state $|+\rangle$ for eigenvalue $+1$, can be stabilized using the setup in Fig. \[fig1\]. As for the $Z$-stabilization shown in the left panels, there are no Rabi oscillations for this parameter set. Finally, for stabilizing the $Y$-eigenstates with eigenvalue $\pm 1$, one requires $$|\lambda_{22}|=|\lambda_{23}|, \quad \lambda_{12}= \lambda_{21}=0,\quad \beta_2-\beta_3-\beta_4=\pm\pi/2,$$ with the dissipative gap $\Delta_y = |4\lambda_{11}\lambda_{22}|^2\sum_s\tilde\Gamma_s$. In all these examples, the target axis (say, $\hat e_z$ for $Z$-eigenstates) is controlled by selecting appropriate tunneling amplitude parameters $\lambda_{j\nu}$. Two links are switched off, and two are matched in amplitude such that the desired jump operator $\tilde J_+$ is implemented. For $T\ll \omega_0$, dissipative transitions are fully governed by this jump operator which is due to inelastic cotunneling transitions from QD $2\to 1$. Under these conditions, we find that $\tilde H_{\rm L}$ commutes with the Pauli operator $\hat\sigma$ corresponding to the target axis (e.g., $\hat\sigma=Z$ for $Z$-states). Finally, by adjusting the phases $\beta_j$, one can select the stabilized state, say, $|0\rangle$ or $|1\rangle$. It is a remarkable feature of our Majorana-based DD setup that the Hamiltonian $\tilde H_{\rm L}$ can be engineered to only generate $\hat\sigma$. As a consequence, the Lindbladian dissipator already drives the system to the desired dark state. Magic states {#sec3b} ------------ ![ Fidelity for a stabilization protocol targeting the magic state $|m\rangle$. Here the Majorana state follows by numerical integration of Eq.  using the parameters in Eq.  with $|\lambda_{23}|=1$. Other parameters are $E_C = 1$ meV, $g_0/E_C=2.5\times 10^{-3}, T/g_0=4$, $\omega_0/g_0=40$, $\omega_c/g_0 =200$, $\alpha=1/4$, and $p = 0.01$. Main panel: Time dependence of the fidelity for ideal parameters \[Eq. \] (red curve), with a mismatch of order $10\%$ in all state design parameters \[$|\lambda_{11}|=-0.1+1/\sqrt2,|\lambda_{21}|=+0.1+1/\sqrt{2},|\lambda_{12}|=|\lambda_{22}|=0.9,\beta_3=-\beta_2=11 \pi/20$\] (blue), and a mismatch of order $20\%$ in the same parameters (orange). Inset: Steady-state fidelity vs deviation $\Delta\beta_2$ with otherwise ideal parameters, where $\beta_2=-\frac{\pi}{2}(1+\Delta\beta_2)$. \[fig5\]](f5){width="\columnwidth"} In order to highlight the power of our DD stabilization protocols, we next consider the magic state [@Nielsen] $$\label{magicstate} |m\rangle = e^{-i\frac{\pi}{8}Y} |0\rangle.$$ The practical importance of this state comes from the fact that a large number of ancilla qubits approximately prepared in the state $|m\rangle$ are needed for the magic state distillation protocol. The latter is an essential ingredient for implementing the $T$-gate required for universal surface code quantum computation [@Fowler2012; @Vijay2015; @Landau2016; @Plugge2016; @Nielsen]. Targeting $|m\rangle$, the stabilization conditions now involve all tunnel links in Fig. \[fig1\] and are given by $$\begin{aligned} \label{magiccond} |\lambda_{12}|&=&|\lambda_{23}|, \quad |\lambda_{21}|=|\lambda_{11}|=|\lambda_{23}|/\sqrt2,\\ \nonumber \lambda_{22}&=&0,\quad \beta_3=\beta_1+\beta_2, \quad \beta_2 = -\pi/2.\end{aligned}$$ We here define the *fidelity* of the state $\rho_{\rm M}(t)$ with respect to a specific pure state, $\rho_{\rm M}^{(0)}=|\psi\rangle\langle \psi|$, as $$\label{fidelity} F(t)={\rm tr}\left[ |\psi\rangle\langle \psi|\rho_{\rm M}(t)\right].$$ We show numerical results for the magic state fidelity with $|\psi\rangle=|m\rangle$ in Fig. \[fig5\], using the parameters in Eq. . We find $F=1$ at long times for the ideal parameter choice in Eq. . Figure \[fig5\] also illustrates the long-time fidelity when allowing for small deviations from Eq.  which are inevitable in practical implementations. Remarkably, even for sizeable deviations from the ideal parameter set, the fidelity remains close to unity. By determining the spectrum of the Lindbladian, we obtain the dissipative gap as $$\label{dissgapm} \Delta_m = |4\lambda_{11}\lambda_{23}|^2\sum_s\tilde \Gamma_s.$$ Using the parameters in Fig. \[fig5\], we find $\Delta_m^{-1}\simeq 80$ ns. Even though our magic state stabilization protocol requires more parameter fine tuning than the stabilization of $|0\rangle$, the dark state $|m\rangle$ is reached on essentially the same time scale. Effect of temperature {#sec3c} --------------------- ![ Steady-state fidelity, $F(\infty)$, and purity, $P(\infty)$, vs temperature (in Kelvin) for the state $|0\rangle$ and for the magic state $|m\rangle$. We use ideal state design parameters, see Eqs.  and , with all other parameters as in Figs. \[fig4\] and \[fig5\], respectively. The numerical results for both states cannot be distinguished for these parameter choices on the shown scales. The frequency $\omega_0$ corresponds to a temperature of $\approx 2.5$ K. []{data-label="fig6"}](f6){width="0.95\columnwidth"} We next address the effect of raising temperature within the conditions set by Eq. , in particular $T\ll \omega_0$. Figure \[fig6\] shows numerical results for the $T$-dependent steady state fidelity $F(\infty)$ with respect to the states $|0\rangle$ and $|m\rangle$, choosing ideal parameters as in Eqs.  and , respectively. At very low temperatures, the fidelity stays very close to the ideal value ($F=1$) since here only the rate $\tilde \Gamma_+$, see Eqs.  and , is significant. In this limit, corrections to $F=1$ are exponentially small and appear to be governed by the dissipative gap, $1-F\propto \exp(-\Delta_{z/m}/T)$. The same scaling behavior also applies to the purity. As temperature increases, the thermal excitation rate $\tilde\Gamma_-=e^{-\omega_0/T}\tilde \Gamma_+$ cannot be neglected anymore. Focusing on the stabilization of the state $|0\rangle$, we have $\tilde J_-\propto \sigma_-$. The Lindblad dissipator $\tilde \Gamma_- {\cal L}[\tilde J_-]$ will then target the ‘wrong’ $Z$-eigenstate $|1\rangle$. The competition between ${\cal L}[\tilde J_+]$ and ${\cal L}[\tilde J_-]$ implies that the fidelity will deteriorate as temperature increases. This expectation is confirmed by our numerical results. For the parameters in Fig. \[fig6\], the fidelity noticeably drops once $T$ exceeds the crossover temperature $T_c\approx 250$ mK. Figure \[fig6\] also shows the temperature dependent purity of the steady state, $P(\infty) ={\rm tr}\rho_{\rm M}^2(t\to \infty)$. For $T\ll T_c$, we find $P(\infty) \simeq 1$. As $T$ increases, however, the maximally mixed state $\rho_{\rm M}(\infty)=\frac12 \mathbb{1}$ with $F(\infty)= P(\infty)=1/2$ is approached, and consequently the purity also becomes smaller. Stabilization without driving field {#sec3d} ----------------------------------- In certain cases, it is possible to stabilize dark states even without drive Hamiltonian, $H_{\rm drive}=0$. In this subsection, we demonstrate the feasibility of this idea for special choices of the state design parameters. We are not aware of other DD systems allowing for dark states in the absence of driving. In our setup, we will see that the dissipative dynamics can also generate terms that mimic the effects of a weak driving field. To be specific, we apply the Lindblad equation to setups where $\lambda_{j\nu}\ne 0$ only for $(j \nu) \in \{ 11,12,23 \}$. In particular, since $\lambda_{21}=0$, this case corresponds to the special parameter regime discussed in Sec. \[specialsec\]. For simplicity, below we drop the exponentially small contribution to the dissipator due to $\tilde J_-$. From Eq. , the only relevant jump operator is then given by $$\label{jplus1} \tilde J_{+} = 2i \lambda_{23}^\ast \left( \lambda_{11} X - \lambda_{12} Y \right).$$ In addition, we keep Lamb shift effects implicit. In particular, they can be taken into account by renormalizing $B_z$ in Eq.  below. The operator $\tilde J_z$ entering $\tilde H_{\rm L}$, see Eqs.  and , has the form $$\label{jz1} \tilde J_z =-\sin\beta_1\left| \lambda_{11} \lambda_{12} \right| \, Z .$$ We now study the undriven ($A=0$) scenario for two parameter sets allowing for analytical progress. The stabilization of pure dark states may then be possible because the Hamiltonian $\tilde H_L$ can effectively take over the role of the drive. The frequency $\omega_0$ now simply represents the (positive) energy difference $\epsilon_2-\epsilon_1$, see Eq. , instead of a drive frequency. Moreover, we assume $p_\perp = 0$ while the probability $p$ in Eq.  is estimated by $p\approx g_0/\omega_0$. We note in passing a finite static contribution to the inter-QD tunnel coupling, $t_{12}\ne 0$ in Eq. , can be taken into account here. This coupling will modify $p$ according to $p \approx{\rm max}(t_{12}, g_0)/\omega_0$. We also recall that for $A\ne 0$, one instead finds $p=1/2$ since we have $\lambda_{21}=0$, cf. Sec. \[specialsec\]. ### Case 1: $\lambda_{11}=\pm i \lambda_{12}$ {#case-1-lambda_11pm-i-lambda_12 .unnumbered} The first case is defined by $\lambda_{11}=is \lambda_{12}$, with $s=\pm 1$. We observe that the dot fermion operator $d_1$ corresponding to QD 1 is then tunnel-coupled to a nonlocal fermion formed from the Majorana operators, $c=(\gamma_1-is \gamma_2)/2$. With $\sigma_\pm=(X\pm iY)/2$, Eqs.  and simplify to $$\label{case1} \tilde J_+=4i\lambda_{23}^\ast \lambda_{11}\sigma_{-s}, \quad\tilde J_z=-s |\lambda_{11}|^2 \, Z.$$ The Lindblad equation is then given by $$\label{nodrive:LBMcase1} \partial_t \rho_{\rm M}(t) = - i [ \tilde H_{\rm L}, \rho_{\rm M}(t) ] + \Gamma_1 {\cal L} \left[ \sigma_{-s} \right] \rho_{\rm M}(t),$$ where the Hamiltonian follows from Eq.  as $$\label{HM1} \tilde H_{\rm L} = -2s (1-2p) g_0 |\lambda_{11}|^2 Z=s B_z Z.$$ We note that the Lamb shift $\tilde h_+$ can be taken into account by redefining $B_z$. Furthermore, the rate $\Gamma_1$ in Eq.  is proportional to $\tilde \Gamma_+$ in Eq. . The only zero eigenstate of the Lindbladian is the $Z$-eigenstate $|0\rangle$ (for $s=-1$) \[or $|1\rangle$ (for $s=+1$)\], e.g., ${\cal L} \left[ \sigma_+ \right] |0 \rangle \langle 0 | = 0$. The same $Z$-eigenstate is also the lowest energy eigenstate of $\tilde H_{\rm L}$ in Eq. . Using the $Z$-eigenstate basis $\{ |0\rangle , |1\rangle \}$ for $s=-1$ \[and $\{|1\rangle,|0\rangle \}$ for $s=+1$\], we can parametrize the time-dependent density matrix $\rho_{\rm M}(t)$ solving Eq.  with real-valued $x(t)$ subject to $0\le x\le 1$ and complex-valued $y(t)$ as $$\rho_{\rm M}(t) = \left( \begin{array}{cc} 1 - x(t) & y(t) \\ y^\ast(t) &x(t)\end{array}\right).$$ The quantities $x(t)$ and $y(t)$ represent the diagonal and off-diagonal density matrix deviations, respectively, from the steady-state density matrix corresponding to the stabilized $Z$-eigenstate. Using Eq. , these deviations obey the equations of motion $$\label{relax1} \partial_t x = - \Gamma_1 x,\quad \partial_t y = - 2 i B_z y - \frac{\Gamma_1}{2} y,$$ which explicitly shows the relaxation and decoherence dynamics of $\rho_{\rm M}(t)$ towards the stabilized pure state. The above example demonstrates that the dissipative stabilization of a dark state can be achieved even in the absence of a driving field in our Majorana box setup. ### Case 2: $\beta_1=0$ {#case-2-beta_10 .unnumbered} Putting the phase $\beta_1$ to zero, $d_1$ is effectively coupled to a single Majorana operator, $\gamma_{\rm eff}=\gamma_1 \cos \delta + \gamma_2\sin \delta$, with $\delta= \tan^{-1} \left| \lambda_{12} / \lambda_{11} \right|$. One then obtains $\tilde J_z=0$. The jump operator $\tilde J_+$ is now given by $$\label{case2} \tilde J_+= B_{\perp}\sigma_+ e^{i\delta}+{\rm h.c.},\quad B_\perp= 2i \lambda_{23}^\ast \lambda_{11}/|\cos\delta|.$$ Noting that the Lamb shifts in $\tilde H_{\rm L}$ only give an irrelevant constant, we arrive at the Lindblad equation $$\label{nodrive:LBMcase2} \partial_t \rho_{\rm M}(t) = \frac{\Gamma_2}{4} {\cal L} \left[ \sigma_{\bf n} \right] \rho_{\rm M}(t),$$ where we define $$\sigma_{\bf n} = {\bf n \,\cdot} {\bm \sigma} = \sigma_+ e^{i \delta} + \sigma_-e^{-i\delta},$$ with the unit vector ${\bf n} = (\cos \delta, - \sin \delta, 0)$. Again, the rate $\Gamma_2$ is proportional to the respective rate $\tilde \Gamma_+$ in Eq. . For the case in Eq. , the Lindbladian has two zero eigenstates, ${\cal L} \left[ \sigma_{\bf n} \right] | s \rangle \langle s | = {\cal L} \left[ \sigma_{\bf n} \right] | a \rangle \langle a | = 0$, corresponding to the eigenstates of ${\bf \sigma}_{\bf n}$, i.e., $\sigma_{\bf n}| s \rangle = | s \rangle$ and $\sigma_{\bf n}| a \rangle =- | a \rangle$. Using the $X$-eigenstates $|\pm\rangle$, one finds $$| s / a \rangle =\frac{1}{ \sqrt{2}} \left( e^{i \delta} |+\rangle \pm e^{-i \delta} |-\rangle \right).$$ In the $\{ |s\rangle,|a\rangle\}$ basis, $\rho_{\rm M}(t)$ can be parametrized as $$\rho_{\rm M}(t) = \left( \begin{array}{cc} \frac12 + x(t) & y(t) \\ y^\ast(t) & \frac12-x(t) \end{array} \right),$$ where the real-valued parameter $x(t)$ has to satisfy $|x|\le 1/2$. Equation  then yields $$\partial_t x = 0,\quad \partial_t y = - \frac{\Gamma_2}{2} y.$$ Clearly, there is no relaxation in the basis selected by the environment via the QDs, i.e., $x(t)$ remains constant. Only the off-diagonal elements of the density matrix are subject to decay with the rate $\Gamma_2/2.$ One can therefore prepare an arbitrary mixed state as steady state. Discussion {#sec3e} ---------- We conclude this section with several additional points. ### Mixed states As pointed out in Sec. \[sec3d\], one can also use our protocols for stabilizing mixed states, see also Ref. [@Kumar2020]. To give another example, now for $A\ne 0$, we consider changing the above phase conditions such that a mixture of Pauli eigenstates can be prepared as dark state. For instance, by choosing the state design parameters as in Eq.  but keeping $\bar\beta=\beta_1-\beta_3$ arbitrary, one obtains the dark state $$\rho_{\rm M}(\infty)=\frac{1+\sin\bar\beta}{2}|0\rangle\langle 0|+\frac{1-\sin\bar\beta}{2}|1\rangle\langle 1|.$$ The relative weight of the two components can then be altered by adjusting the phase difference $\bar\beta$. ### Majorana overlaps So far we have assumed that the overlap between different MBSs is negligibly small. What are the effects of a finite (but small) hybridization between different MBS pairs on the above stabilization protocols? Such terms could arise, e.g., due to the finite nanowire length [@Alicea2012]. They are described by a Hamiltonian term $ H'=\sum_{\nu<\nu'}i\epsilon_{\nu\nu'}\gamma_\nu\gamma_{\nu'}$, with hybridization energies $\epsilon_{\nu\nu'}$. By construction, such a term survives the RWA and the Schrieffer-Wolff projection in Sec. \[sec2\] and thus contributes to the Hamiltonian $\tilde H_{\rm L}$ in the Lindblad equation without affecting the Lindbladian dissipator. In the Pauli operator language, such terms act like a weak magnetic Zeeman field. If the corresponding field is parallel to the target axis of the dark state, it does not cause any dephasing. For instance, for the stabilization of the $Z$-eigenstate $|0\rangle$, the hybridization parameters $\epsilon_{12}$ and $\epsilon_{34}$ can be tolerated as they only couple to the Pauli operator $Z$ in Eq. . Clearly, such couplings have no detrimental effects on our stabilization protocols. ### Readout dynamics For reading out a stabilized dark state, it is possible to use the same techniques suggested previously for the native Majorana qubit [@Plugge2017; @Karzig2017; @Munk2019]. In particular, one can perform capacitance spectroscopy using additional single-level QDs that are tunnel-coupled to MBS pairs. These QDs are used for measurements only, where the spectroscopic signal contains an interference term which depends on the respective Pauli matrix in Eq. . This projective readout yields the Pauli eigenvalue $\pm 1$ with a state-dependent probability [@Karzig2017]. Of course, this method can also be used to prepare the Majorana island in a Pauli eigenstate before the DD protocol is started. In order for the readout not to interfere with the DD stabilization protocol, one has to make sure that the characteristic projective measurement time scale (see Refs. [@Plugge2017; @Karzig2017] for detailed expressions) is much longer than the typical inelastic cotunneling time $\tilde\Gamma_+^{-1}$. Similarly, single-electron pumping protocols via a pair of QDs attached to different MBSs allow one to apply a Pauli operator to the tetron state in a topologically protected manner [@Plugge2017]. ### Beyond the horizon of a dark state {#sec3e4} So far we have discussed DD stabilization protocols targeting a desired dark state. The dark space dimension for those protocols is $D=1$, see App. \[appC\]. Since there is a unique dark state for a given choice of the state design parameters, one could utilize a DD single-box device as a self-correcting quantum memory. By means of adiabatic changes of the state design parameters, one can in principle steer the Majorana state on its Bloch sphere. However, for general state manipulation protocols, it is advantageous to have access to a dark space manifold with $D>1$, which may be engineered in systems with more than four MBSs. We address this case in the next section. Dark space engineering {#sec4} ======================= We continue with DD protocols targeting quantum states within a dark space manifold. A degenerate manifold of dark states may be engineered by employing a device with at least two Majorana boxes as depicted in Fig. \[fig7\]. After introducing our model and the corresponding Lindblad equation in Sec. \[sec4a\], we show in Sec. \[sec4b\] how a dark space can be created and classified. In Sec. \[sec4c\], we then describe how to stabilize Bell states. In Ref. [@ourprl], we describe external perturbations for moving the dark state to another state within the protected dark space manifold, and we show how to create a dark space manifold realizing a ‘dark Majorana qubit’. In such a system, topological and DD mechanisms reinforce each other and thereby can provide exceptionally high levels of fault tolerance. Moreover, we remark that the stabilization of Bell states can also be implemented in a hexon device (i.e., a Majorana box with six MBSs [@Karzig2017]), see Ref. [@GauThesis]. ![Schematic two-box layout for DD dark space stabilization and manipulation protocols, cp. Fig. \[fig1\] for the single-box case. The left (right) box harbors four MBSs described by $\gamma_{\nu}^L$ ($\gamma^R_\nu$). The tunneling bridge with amplitude $t_{LR}$ connects $\gamma_4^L$ and $\gamma_2^R$. QD 3 has independently driven tunneling bridges to QD 1 and to QD 2 (solid lines). The three QDs are operated in the single-electron regime, $N_{\rm d}=1$. The electromagnetic environment affects the phases of the tunnel links betweens QDs and MBSs (dashed lines). The phases $\beta_{j}$ for this geometry are also indicated.[]{data-label="fig7"}](f7){width="\columnwidth"} Lindblad equation for two coupled boxes {#sec4a} --------------------------------------- ### Model Following the discussion in Sec. \[sec2a\], we describe the two islands in Fig. \[fig7\] by $H_{\rm box}=H_{{\rm box},L}+H_{{\rm box},R}$, with $H_{{\rm box},L/R}$ as in Eq. . Here, the four MBSs on the left (right) box correspond to Majorana operators $\gamma_\nu^L$ ($\gamma_\nu^R$). Both islands are separately operated under Coulomb valley conditions. For notational simplicity, we assume that they have the same charging energy, $E_{C,L}=E_{C,R}=E_C$. Focusing on the long-wavelength components of the electromagnetic environment, we again work with a single bosonic bath, $H_{\rm env}=\sum_m E_m b_m^\dagger b_m^{}$, where photons couple to the QDs and MBSs via fluctuating phases, $\theta_j$, in the tunneling Hamiltonian, see Sec. \[sec2a\]. The setup in Fig. \[fig7\] requires up to three single-level QDs, $H_{\rm d}=\sum_{j=1}^3 \epsilon_j d_j^\dagger d_j^{}$, where QD 3 couples to both other QDs via independently driven tunnel links. We consider the regime $N_{\rm d}=1$, where on time scales $\delta t>1/E_C$, the three QDs share a single electron. Using the interaction picture with respect to the dot Hamiltonian $H_{\rm d}$, the full Hamiltonian is then given by $$\label{Hfull2MBQ} H(t) = H_{\rm box} + H_{\rm env}+ H_{LR} + H_{\rm drive}(t) + H_{\rm tun}(t),$$ where a phase-coherent tunnel link couples the boxes. Without loss of generality, we assume a real-valued tunneling amplitude $t_{LR}>0$, $$\label{LRCoup} H_{LR} = i t_{LR} \gamma_4^L \gamma_2^R.$$ The drive Hamiltonian now has the form $$\label{Hdrive2} H_{\rm drive}(t)=\sum_{j=1,2} 2A_j\cos\left(\omega_jt\right)e^{i\left(\epsilon_j-\epsilon_3\right)}d_j^{\dagger}d^{}_3 + {\rm h.c.},$$ where the two driving fields have the respective amplitude $A_{1,2}$ and frequency $\omega_{1,2}$. In analogy to Eq. , the QD-MBS tunnel links are described by $$\label{Htun2} H_{\rm tun}(t)=t_0 \sum_{j\nu, \kappa=L/R}\lambda_{j,\nu\kappa}^{} e^{-i\phi_\kappa} e^{i\theta_j} e^{i\epsilon_jt} d^\dagger_j \gamma_{\nu}^\kappa +{\rm h.c.},$$ with the phase operators $\phi_{L/R}$ for the left/right Majorana island. Using the same approximations as in Sec. \[spsec1\], the electromagnetic environment enters Eq.  through the fluctuating phases $\theta_j$. With the overall energy scale $t_0$, the complex-valued parameters $\lambda_{j,\nu\kappa}$ parametrize the transparency of the tunnel contact between $d_j$ and $\gamma_\nu^{\kappa=L/R}$. Similar to Eq. , the phases $\beta_j$ in Fig. \[fig7\] follow from the phases of these parameters. Since $\beta_4$ can be absorbed by a renormalization of $\beta_3$ for the purposes below, we put $\beta_4=0$. To simplify the presentation, we next assume that QDs 1 and 2 have the same energy level, $\epsilon_1=\epsilon_2$. Moreover, we consider the case of equal drive frequencies, $\omega_1=\omega_2\equiv \omega_0$, and identical drive amplitudes, $A_1=A_2\equiv A$, and again impose a resonance condition, $\omega_0=\epsilon_3-\epsilon_1$. However, in analogy to our discussion in Sec. \[sec2\], we expect that overly precise fine tuning with respect to those conditions is not necessary. We now proceed in analogy to Sec. \[sec2a\] with the construction of an effective low-energy model by means of a Schrieffer-Wolff transformation to the lowest-energy charge state in each box. We can then define Pauli operators $(X_\kappa,Y_\kappa,Z_\kappa)$ with $\kappa=L,R$ referring to the left and right box, respectively, using the Majorana representation in Eq. . In the present case, it is crucial to keep all terms up to third order in the expansion parameters when accounting for cotunneling trajectories connecting pairs of QDs, cf. Fig. \[fig7\]. (For the single-box case in Sec. \[sec2a\], it is sufficient to go to second order only.) The electromagnetic environment then enters the low-energy theory via the three phase differences $\theta_j-\theta_k$ with $j<k$. This fact implies that, in general, we have six different spectral densities ${\cal J}_{jk;j'k'}(\omega)$. We model these spectral densities by the Ohmic form in Eq. (\[Ohmic\]), with system-bath couplings $\alpha_{jk;j'k'}$. For simplicity, we employ an average value $\alpha$ for these couplings below. The bath is then described by a single spectral density ${\cal J}(\omega)$ again. Importantly, the physics is not changed in an essential manner by this approximation. In particular, no additional jump operators appear when allowing for different $\alpha_{jk;j'k'}$. ### Lindblad equation We consider again the weak driving regime with $T\ll \omega_0$. Under these conditions, proceeding along similar steps as in Sec. \[sec2b\], one obtains a Lindblad master equation for the density matrix, $\rho(t)$, describing both the Majorana sector and the QD degrees of freedom. In order to arrive at a Lindblad equation for the reduced density matrix, $\rho_{\rm M}(t)$, which refers only to the Majorana sector of both boxes, we next trace over the QD subsector, see Sec. \[sec2c\]. For the QD steady-state density matrix, $\rho_{\rm d}$, we use the *Ansatz* $$\label{ansatz2} \rho_{\rm d}= {\rm diag}\left(\frac{1-p}{2}, \frac{1-p}{2},p\right),$$ expressed in the basis $\{ |100\rangle,|010\rangle,|001\rangle \}$ with QD occupation states $|n_1,n_2,n_3\rangle$ for $N_{\rm d}=1$. Note that since we assumed $\epsilon_1=\epsilon_2$, the occupation probabilities of QDs 1 and 2 are equal. The occupation probability $0<p\ll 1$ refers to the energetically highest QD 3. Equation is consistent with our numerical analysis of the Lindblad equation for $\rho(t)$, where we again find a factorized density matrix at long times, $\rho(t)\simeq \rho_{\rm M}(t)\otimes \rho_d$. We note that the dark space turns out to be independent of the concrete value of $p$. Going through the corresponding steps in Sec. \[sec2c\], we arrive at a Lindblad equation for $\rho_{\rm M}(t)$, $$\label{LindbladTwoMBQ} \partial_t\rho_{\rm M}(t)=-i[\tilde H_{\rm L},\rho_{\rm M}(t)]+ \sum_{a=1}^6 \tilde \Gamma_a \mathcal{L}[K_a] \rho_{\rm M}(t).$$ The six jump operators are denoted by $K_a$, with the respective dissipative transition rates $\tilde\Gamma_a$. With $\lambda_{LR}\equiv t_{LR}/E_C \ll 1$, we obtain $$\begin{aligned} \nonumber K_1^{} &=& K_4^\dagger = ie^{i(\beta_3-\beta_1)}\frac{|\lambda_{1,1R}\lambda_{3,3R}|}{\lambda_{LR}} X_R \\ &&\qquad - \,e^{i\beta_3} |\lambda_{1,3L}\lambda_{3,3R}| Z_L Y_R,\nonumber\\ \label{jumpK} K_2^{} &=& K_5^\dagger =-ie^{i(\beta_3-\beta_2)}\frac{|\lambda_{2,4R}\lambda_{3,3R}|}{\lambda_{LR}}Z_R\\ &&\qquad + \, e^{i\beta_3}|\lambda_{2,2L}\lambda_{3,3R}| X_LY_R,\nonumber\\ K_3^{} &=& K_6^\dagger = i\frac{|\lambda_{1,3L}\lambda_{2,2L}|}{\lambda_{LR}}Y_L- ie^{i(\beta_2-\beta_1)} \frac{|\lambda_{1,1R}\lambda_{2,4R}|}{\lambda_{LR}} Y_R\nonumber\\ &&\qquad+ \,e^{-i\beta_1} |\lambda_{1,1R}\lambda_{2,2L}| \,X_LZ_R \nonumber \\ \nonumber && \qquad -\,e^{i\beta_2}|\lambda_{1,3L}\lambda_{2,4R}| \,Z_LX_R. \nonumber\end{aligned}$$ The coherent evolution in Eq.  is governed by the Hamiltonian $$\label{h2qdef} \tilde H_{\rm L} = 2p\tilde g_0 K_z + \sum_{a=1}^6 \tilde h_a K_a^\dagger K_a,$$ with the operator $$\label{kz} K_z^{} =\sin\beta_1|\lambda_{1,1R}\lambda_{1,3L}| Z_LZ_R+ \sin\beta_2|\lambda_{2,2L}\lambda_{2,4R}| X_L X_R.$$ We here used the energy scale $$\label{tildeg0} \tilde g_0= \lambda_{LR} g_0 = \frac{t_0^2 t_{LR}}{E_C^2},$$ which characterizes the relevant inelastic cotunneling processes in the double-box setup. The transition rates $\tilde\Gamma_a$ follow in the form $$\begin{aligned} \tilde\Gamma_1 &=& \tilde \Gamma_2 = 2p\tilde g_0^2 \,{\rm Re} \int_0^\infty dt e^{i\omega_0 t} e^{J_{\rm env}(t)},\nonumber\\ \tilde\Gamma_3 &=& \tilde \Gamma_6 =(1-p)\tilde g_0^2\, {\rm Re} \int_0^\infty dt e^{J_{\rm env}(t)} ,\label{trans22}\\ \tilde\Gamma_4 &=&\tilde\Gamma_5= \frac{(1-p)}{2p} e^{-\omega_0/T}\tilde \Gamma_1,\nonumber\end{aligned}$$ and the Lamb shifts $\tilde h_a$ are given by $$\begin{aligned} \tilde h_1 &=& \tilde h_2 = p\tilde g_0^2 \,{\rm Im} \int_0^\infty dt e^{i\omega_0 t} e^{J_{\rm env}(t)},\nonumber\\ \tilde h_3 &=& \tilde h_6 = \frac12 (1-p)\tilde g_0^2 \,{\rm Im} \int_0^\infty dt e^{J_{\rm env}(t)} ,\label{lamb22}\\ \tilde h_4 &=&\tilde h_5= \frac{(1-p)}{2p} e^{-\omega_0/T}\tilde h_1.\nonumber\end{aligned}$$ For $\omega_0\ll \omega_c$, we can then make further analytical progress. Explicit expressions for $\tilde \Gamma_{1,2}$ and $\tilde h_{1,2}$ follow by comparison with Eq. . In addition, we find $$\begin{aligned} \nonumber \tilde \Gamma_{3,6} &\simeq& (1-p) \frac{\cos(\pi \alpha)\Gamma(\alpha)\Gamma(1-2\alpha)}{2^{1-2\alpha} \Gamma(1-\alpha)} \left( \frac{\pi T}{\omega_c}\right)^{2\alpha-1} \frac{2g_0^2}{\omega_c} ,\\ \tilde h_{3,6}&=& -\frac{1}{2}\tan(\pi \alpha) \tilde \Gamma_{3,6}.\end{aligned}$$ By following the derivation of the reduced master equation , we observe that the operator $K_1$ ($K_2$) comes from unidirectional transitions transferring an electron from the energetically high-lying QD 3 to QD 1 (QD 2) via the double-box setup, collecting all possible cotunneling trajectories allowed by third-order perturbation theory. Likewise, the jump operator $K_4$ ($K_5$) describes the reversed process, with a cotunneling transition from QD 1 (QD 2) to QD 3. For $T\ll \omega_0$, the transition rates $\tilde \Gamma_{4,5}$ and Lamb shifts $\tilde h_{4,5}$ are exponentially suppressed, $\propto e^{-\omega_0/T}$, against the respective contributions from $K_{1,2}$. Moreover, the jump operators $K_3$ and $K_6$ in Eq.  describe cotunneling transitions between QDs 1 and 2. Since these QDs are not directly connected by a driven tunnel link and have the same energy, $\epsilon_1=\epsilon_2$, the corresponding rates and Lamb shifts coincide, $\tilde\Gamma_3=\tilde\Gamma_6$ and $\tilde h_3=\tilde h_6$. Importantly, for $1/2< \alpha < 1$, these quantities are reduced by a factor $(T/\omega_0)^{2\alpha-1}\ll 1$ against $\tilde\Gamma_{1,2}$ and $\tilde h_{1,2}$, respectively. In the remainder of this section, we shall study this parameter regime where the most important jump operators in Eq.  are given by $K_1$ and $K_2$. Nonetheless, we retain the other jump operators in our numerical analysis as well. Finally, we note that all terms without the factor $\lambda^{-1}_{LR}\gg 1$ in Eqs.  and stem from third-order processes. While one *a priori* expects that the corresponding dissipative terms in Eq.  are suppressed against second-order contributions, by careful tuning of the link transparencies $\lambda_{j,\nu\kappa}$, they can become of comparable magnitude. As a consequence, all relevant cotunneling paths will then have amplitudes corresponding to third-order processes. This means that for the present two-box setup, the energy scale $g_0=t_0^2/E_C$ appearing in Eq.  has to be replaced by $\tilde g_0$ in Eq. . The Lindblad equation describing the weak driving limit is therefore valid under the conditions $$\label{basiccond2} \tilde g_0\ll T\ll \omega_0,\quad A \alt \tilde g_0.$$ ### Dissipative maps Before entering our discussion of stabilization protocols for the layout in Fig. \[fig7\], it is convenient to introduce the dissipative maps [@Barreiro2011] $$\label{bellmap} \hat E_{1,\pm} = (\mathbb{1}\pm Z_L Z_R) X_R,\quad \hat E_{2,\pm} = (\mathbb{1}\pm X_L X_R) Z_R.$$ These maps can be used to target the four Bell states, $$\label{bellstates} |\psi_{\pm} \rangle =\frac{1}{\sqrt2}(|00\rangle\pm|11\rangle), \quad|\phi_{\pm}\rangle=\frac{1}{\sqrt2}(|01\rangle\pm|10\rangle),$$ which are eigenstates of both $Z_LZ_R=\pm 1$ and $X_LX_R=\pm 1$. We observe that $\hat E_{1,-}$ maps even-parity onto the respective odd-parity states, $\hat E_{1,-}|\psi_{\pm}\rangle=|\phi_{\pm}\rangle$, while odd-parity states do not evolve in time, $\hat E_{1,-}|\phi_{\pm}\rangle=0$. Under this dissipative map, the system will thus be driven into the degenerate odd-parity subsector spanned by the $|\phi_\pm\rangle$ states. Similarly, $\hat E_{2,-}$ can drive the system into the antisymmetric subsector spanned by $|\phi_-\rangle$ and $|\psi_-\rangle$. The key idea in our DD protocols below is to identify state design parameters such that the jump operators effectively realize the needed dissipative map(s) in Eq. . Recalling that a dissipative map breaks a number of conserved quantities (and therefore symmetries) in our system, see Refs. [@Albert2014; @Albert2016] and App. \[appC\], we here employ this insight to either stabilize a dark space, see Sec. \[sec4b\] and Ref. [@ourprl], or to target protected and maximally entangled two-qubit dark states, see Sec. \[sec4c\]. Stabilization of a dark space {#sec4b} ----------------------------- In this subsection, we briefly outline how one can stabilize a dark space in the setup of Fig. \[fig7\], see also Ref. [@ourprl]. For convenience, we decouple QD 2 from the system by using the parameter choice $$\label{decouple2} \lambda_{2,2L}=\lambda_{2,4R}=0,\quad \beta_2=0.$$ We note that this is not the only possible parameter set for constructing a dark space. As a consequence of Eq. , many of the jump operators in Eq.  vanish identically, $K_2=K_3=K_5=K_6=0$. The jump operator $K^{}_1=K_4^\dagger$ then yields the dissipative map $\hat E_{1,-}$ in Eq.  upon choosing $$\beta_1=-\pi, \quad \beta_3 = -\pi/2, \quad |\lambda_{1,1R}|=\lambda_{LR} |\lambda_{1,3L}|. \label{spacecond}$$ Noting that $\hat E_{1,-}= X_R-i Z_L Y_R$, see Eq. , we indeed arrive at $K_1\propto \hat E_{1,-}$ from Eq. . In addition, Eq.  shows that under the above conditions, $\tilde H_{\rm L}$ only generates terms $\propto Z_L Z_R$ which do not obstruct the dissipative dynamics. For $T\ll \omega_0$, we next observe that to exponential accuracy, $K_1$ is the only jump operator contributing to the Lindbladian in Eq.  for the parameters in Eqs.  and . The DD protocol therefore will stabilize the system in the odd-parity ($Z_L Z_R=-1$) Bell state manifold spanned by $\{ |\phi_+\rangle, |\phi_-\rangle \}$. We show in Ref. [@ourprl] that this degenerate manifold has the dark space dimension $D=4$, see also App. \[appC\], which is equivalent to a degenerate qubit space [@Albert2014]. It is possible to manipulate dark states within a dark space by following different strategies [@ourprl]. For instance, one can adiabatically switch on a perturbation that breaks at least one conservation law. An alternative possibility is to employ single-electron pumping protocols, in analogy to previous proposals for native Majorana qubits [@Plugge2017; @Karzig2017]. Stabilizing Bell states {#sec4c} ----------------------- We next turn to the stabilization of Bell states in the setup of Fig. \[fig7\], where the couplings between QD 2 and the Majorana islands are now assumed finite again. In that case, the jump operator $K_2$ in Eq.  does not vanish anymore. In the low temperature regime, the corresponding Lindbladian term in Eq.  contributes with the same transition rate, $\tilde\Gamma_2=\tilde\Gamma_1$, as for $K_1$, see Eq. . Importantly, $K_2$ breaks additional conservation laws and thereby allows one to engineer stabilization protocols targeting maximally entangled two-qubit states. We again study the regime $1/2< \alpha<1$, where the jump operators $K_{3,6}$ give only subleading contributions. Let us start with the Bell singlet state $|\phi_-\rangle$ in Eq. , where $Z_LZ_R=-1$ and $X_LX_R=-1$. By choosing the state design parameters as $$\begin{aligned} \label{bellcond} \beta_1 &=& -\pi,\quad \beta_2 = 0,\quad \beta_3 = -\pi/2,\\ |\lambda_{1,1R}|&=&\lambda_{LR}|\lambda_{1,3L}|,\quad |\lambda_{2,4R}|=\lambda_{LR}|\lambda_{2,2L}|, \nonumber\end{aligned}$$ we observe from Eq.  that $K_1\propto \hat E_{1,-}$ and $K_2\propto \hat E_{2,-}$ are directly expressed in terms of the corresponding dissipative maps, see Eq. . The Lindbladian will therefore drive the system to the dark state $|\phi_-\rangle$. The dark space dimension is thus given by $D=1$. ![Fidelity for stabilizing the Bell singlet state $|\phi_-\rangle$ in the setup of Fig. \[fig7\]. We show numerical results obtained from Eq.  with the parameters in Eq.  and $|\lambda_{1,3L}|=|\lambda_{2,2L}|=|\lambda_{3,3R}|=1$, using the initial state $\rho_{\rm M}(0)=|00\rangle\langle00|$. Other parameters are $E_C=1$ meV, $\tilde g_0/E_C=10^{-5}$, $T/\tilde g_0=2, \omega_0/\tilde g_0=2\times 10^3, \omega_c/\tilde g_0=10^4, \alpha=0.99$, and $p=0.01$. Main panel: Time dependence of $F(t)$ for ideal parameters \[Eq. \] (red curve), and for a mismatch of order $10\%$ in all state design parameters \[$|\lambda_{1,1R}|=1.1\lambda_{LR}|\lambda_{1,3L}|, \, |\lambda_{2,4R}|=0.9\lambda_{LR}|\lambda_{2,2L}|, \, \beta_1=-1.1\pi, \, \beta_3=-9\pi/20$\] (blue). Inset: Steady-state fidelity vs deviation $\Delta\beta_1$ from the ideal value, i.e., $\beta_1=-\pi(1+\Delta\beta_1)$, with otherwise ideal parameters.[]{data-label="fig8"}](f8){width="\columnwidth"} As is shown in Fig. \[fig8\], the numerical solution of Eq.  confirms this expectation. For the stabilization parameters in Eq. , the Bell singlet state is reached with nearly perfect fidelity when taking ideal parameter values. One can rationalize the almost perfect fidelity by noting that the coherent evolution due to $\tilde H_{\rm L}$, see Eq. , involves only the operators $Z_L Z_R$ and $X_L X_R$. As a consequence, the dynamics induced by the dissipative maps $K_{1,2}\propto \hat E_{1/2,-}$ will not be disturbed. Note that the parameters in Fig. \[fig8\] were chosen such that $\tilde\Gamma_1\gg\tilde\Gamma_3$ while staying in the regime specified in Eq. . Indeed, the observed small deviations from the ideal value $F=1$, see Fig. \[fig8\], can be traced back to the jump operators $K_3$ and $K_6$, which give nominally subleading but practically important contributions to the Lindblad equation. Figure \[fig8\] shows that the stabilization protocol is rather robust against deviations of state design parameters from their ideal values in Eq. , see Sec. \[sec3\]. Following the approach in App. \[appB\], we find that the dissipative gap for stabilizing $|\phi_-\rangle$ is given by $$\label{bellgap} \Delta_{\rm Bell}=|2\lambda_{3,3R}|^2\left(|\lambda_{1,3L}|^2+|\lambda_{2,2L}|^2\right)\sum_{a=1,2,4,5}\tilde\Gamma_a.$$ Due to the importance of third-order inelastic cotunneling processes, this dissipative gap is several orders of magnitude below the corresponding gaps in the single-box case, cf. Sec. \[sec3\]. For the parameters in Fig. \[fig8\], we obtain the time scale $\Delta_{\rm Bell}^{-1}\approx 0.3$ ms. The other Bell states in Eq.  can be targeted by changing the phases $\beta_j$ in Eq. . The jump operators $K_1$ and $K_2$ will then directly implement the desired dissipative maps, with the dissipative gap still given by Eq. . For stabilization of the Bell state $|\psi_+\rangle$ ($|\psi_-\rangle$), one has to put $\beta_1=0, \, \beta_2=\pi \, (\beta_2=0)$, and $\beta_3=\pi/2$. Similarly, $|\phi_+\rangle$ is stabilized for $\beta_1=-\pi, \beta_2 = \pi,$ and $\beta_3= -\pi/2$. We thus always have $\beta_3-\beta_1=\pi/2$, and the remaining two phases select the targeted Bell state. In particular, $\beta_1$ selects the parity of the target state while $\beta_2$ determines the symmetric vs antisymmetric state. Summary and prospects {#secConc} ===================== In this paper, we have described DD protocols in Majorana-based layouts for stabilizing as well as manipulating dark states and dark spaces. For devices with one or two Majorana boxes coupled to driven QDs and subject to electromagnetic noise, we have shown that in a wide parameter regime the dynamics in the Majorana sector is accurately described by Lindblad master equations. The underlying topological nature of the Majorana states significantly boosts the power of DD schemes in several directions. First, the role of uncontrolled environmental noise sources should be suppressed compared to topologically trivial realizations, which is a key advantage for high-dimensional dark space constructions. Second, the fact that Pauli operators describing native Majorana qubits correspond to products of Majorana operators (pertaining to spatially separated MBSs), see Eq. , allows for unique addressability options. Only through this feature, which is rooted in topology, it is possible to design the special unidirectional cotunneling paths which directly implement the jump operators appearing in the Lindblad equation. In the simplest single-box case, see Fig. \[fig1\], the basic pumping-cotunneling cycle involves (i) pumping the dot electron from QD 1 to the high-lying QD 2 by means of a weak driving field, and (ii) the back transfer of the electron from QD 2 to QD 1 by cotunneling through the box. In general, competing transfer mechanisms may also contribute to both steps, and the parameter regime has to be carefully adjusted to minimize their impact. Taking step (ii) as example, the drive Hamiltonian in Eq. , possibly together with photon emission processes, may provide such a competing rate. By choosing both a sufficiently small drive amplitude, $A<g_0$, and a very small direct tunnel coupling $t_{12}$ between both QDs, these competing rates can be systematically suppressed against the cotunneling rates through the box. We also note that in most cases of interest, the Lindbladian dissipator alone is responsible for driving the system into the desired dark state or dark space, i.e., the Hamiltonian appearing in coherent part of the Lindblad equation does not obstruct the dissipative dynamics. For a single-box architecture, we have shown how to stabilize arbitary pure dark states, i.e., states that are fault tolerant and stable on arbitrary time scales. For multiple-box devices, one can also stabilize dark spaces, i.e., manifolds of degenerate dark states, as well as protected two-qubit Bell states. In our accompanying short paper [@ourprl], we show that a two-box device allows one to implement a dark Majorana qubit, which in turn could serve as basic ingredient for dark space quantum computation schemes. Our stabilization and manipulation protocols can be implemented with available hardware elements once a working Majorana platform becomes available. The above concepts and ideas raise many interesting perspectives for future research. First, we expect that one can devise robust Majorana braiding protocols [@Alicea2012; @Leijnse2012; @Beenakker2013] that are stabilized by working within a dark space manifold. Second, for chains of many boxes, our DD stabilization protocols may allow for interesting quantum simulation applications, e.g., a realization of the topologically nontrivial ground state of spin ladders [@Ebisu2019] or of the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin chain [@Kraus2008; @Affleck1987]. For clarifying the feasibility of such ideas, one needs to analyze the spectrum of the Lindbladian for DD multiple-box networks. We leave this endeavor to future work. We thank A. Altland, S. Diehl, and K. Snizhko for discussions. This project has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant No.  277101999, TRR 183 (project C01), under Germany’s Excellence Strategy - Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 - 390534769, and under Grant No. EG 96/13-1. In addition, we acknowledge funding by the Israel Science Foundation. On the strong driving limit {#appA} =========================== We here briefly discuss the strong driving limit for the single-box device in Fig. \[fig1\], with total QD occupancy $N_{\rm d}=1$ and under resonant driving conditions, $\omega_0=\epsilon_2-\epsilon_1$. We consider the regime $$g_0 \ll T<A\ll \omega_0,$$ with otherwise identical conditions as in Sec. \[sec2\]. After imposing the RWA, the steady-state density matrix of the QDs is given by Eq.  with $p=1/2$ and $p_\perp=0$. Starting from the effective Hamiltonian $H_{\rm eff}(t)$ in Eq. , we then arrive at a Lindblad equation for the density matrix $\rho(t)$ describing the combined system of MBSs and QDs, $$\label{AAA} \partial_t \rho(t) = -i [ H_{\rm L}, \rho(t) ] + 2 g_0^2 \sum_{a = 1}^3 \sum_{s = \pm} {\rm Re}\,\Lambda_{a,s} \, {\cal L} [ J_{a,s} ] \rho(t),$$ with the effective Hamiltonian $$H_{\rm L} = A\tau_x + g_0^2 \sum_{a = 1}^3 \sum_{s = \pm} {\rm Im}\,\Lambda_{a,s} \, J_{a,s}^\dagger J_{a,s}^{}.$$ We here encounter *six* jump operators ($s=\pm$), $$\label{rwa:Jaq} J_{1,s} = \tilde J_s \tau_x,\quad J_{2,s} = J_{3,-s}^\dagger = \tilde J_s (\tau_z + i \tau_y )/2,$$ with the operators $\tilde J_\pm$ in Eq. . The dissipative transition rates as well as the Lamb shifts follow from $\Lambda_{1,\pm}\equiv \Lambda_\pm$, see Eq. , and $$\label{rwa:Lambdaq} \Lambda_{2/3,s} = \int_0^\infty dt \, e^{i s \omega_0 t \pm i A t} e^{J_{\rm env}(t)},$$ with the bath correlation function . Comparing to the weakly driven case in Sec. \[sec2b\], the strong driving field $A$ splits the two jump operators $J_s$ in Sec. \[sec2b\] into the six jump operators in Eq. . Tracing over the QD degrees of freedom, we arrive at a Lindblad equation for the density matrix $\rho_{\rm M}(t)$, cf. Sec. \[sec2c\], $$\label{rwa:LBM} \partial_t \rho_{\rm M}(t) = - i [ \tilde H_{\rm L}, \rho_{\rm M}(t) ] + \sum_{s = \pm} \tilde\Gamma_s \, {\cal L }[ \tilde J_s ] \rho_{\rm M}(t),$$ with $\tilde H_{\rm L} = {\rm Tr}_{\rm d} \left\{ \rho_{\rm d} H_{\rm L} \right\}$. In this expression, $\rho_{\rm d}$ follows from Eq.  with $p\to 1/2$ and $p_\perp \to 0$. Only the two jump operators $\tilde J_\pm$ appear in the reduced Lindblad equation , with the dissipative transition rates $$\label{rwa:Gammaq} \tilde\Gamma_s =2 g_0^2 \, {\rm Re}\left[\Lambda_{1,s} +\frac12 \left( \Lambda_{2,s} + \Lambda_{3,s} \right) \right].$$ Finally, we note that for $T>A\gg g_0$, the Lindblad equation holds with $p \to 1/2$. On the dissipative gap {#appB} ====================== An elegant way to study the spectrum of a general Lindbladian uses the so-called Choi isomorphism in order to map the $N\times N$ system density matrix, $\rho(t)$, to an $N^2\times 1$ vector, $|\rho(t)\rangle$, and the Liouvillian, $\mathcal{\hat L}$, to an $N^2\times N^2$ superoperator ${\bm L}$ [@Albert2014]. We here include the Hamiltonian part in $\mathcal{\hat L}$. Let us consider a general Lindblad master equation, cf. Eq. , $$\label{genB1} \partial_t\rho(t)=\mathcal{\hat L}\rho(t)=-i[ H,\rho(t)]+\sum_{a}\Gamma_{a}{\cal L} [ J_{a}] \rho(t),$$ with jump operators $J_a$ and the corresponding transition rates $\Gamma_a$. Using the isomorphism, we have the correspondence $J\rho J^\dagger \leftrightarrow ( J\otimes J^*)|\rho\rangle$, and Eq.  takes the equivalent form $\partial_t|\rho(t)\rangle={\bm L}|\rho(t)\rangle$ with $$\begin{aligned} \label{genB2} {\bm L}&=&-i\left(H\otimes \mathbb{1}- \mathbb{1}\otimes H^\ast\right)+ \sum_{a}\frac{\Gamma_{a}}{2} \times \\ \nonumber &\times& \left(2 J_{a}^{}\otimes J^{\ast}_{a}-\mathbb{1}\otimes \bigl(J_a^{\dagger} J_{a}^{}\bigr)^\ast -J_{a}^{\dagger} J_{a}\otimes\mathbb{1}\right).\end{aligned}$$ In this language, the steady state, $\rho_{\rm ss}$, follows as right eigenvector of ${\bm L}$ with eigenvalue zero, $${\bm L}\left|\rho_{\rm ss}\right\rangle=0. \label{Criterion}$$ Equation allows one to systematically search for stabilization conditions targeting a desired dark state. Moreover, the spectrum of the Lindbladian coincides with the eigenvalues of the superoperator $\bm L$. In particular, the number of zero eigenvalues defines the dark space dimension, $D$, and the dissipative gap equals the real part of the smallest non-zero eigenvalue [@Albert2014]. On conserved quantities {#appC} ======================= For an open quantum system described by a Lindbladian as in Eq. , where we assume that $\mathcal{\hat L}$ has no purely imaginary eigenvalues, it is known that all conserved quantities are linked to the basis states spanning the dark space [@Albert2014]. For a Lindbladian with $D$ conserved quantities $C_{\mu=1,\ldots,D}$, we have the commutation relations $$[H, C_\mu]=[J_a, C_\mu] = 0.$$ Using an orthonormal basis, $\lbrace M_\mu\rbrace_{\mu=1}^D$, to span the resulting $D$-dimensional dark space, the steady state can be written as $$\rho_{\rm ss} = \underset{t\to \infty}{\rm lim} e^{\mathcal{\hat L}t} \rho(0)=\sum_{\mu=1}^D c_\mu M_{\mu}, \label{generaldarkspace}$$ where $\rho(0)$ is the initial density matrix and the $c_\mu={\rm tr}[C_\mu^{\dagger} \rho(0)]$ are weights determining in which of the degenerate steady states the system ends up. As first illustration, let us consider the stabilization of the dark state $|0\rangle$ for a single-box device, cf. Sec. \[sec3a\] and Eq. . The jump operators are then given by $\tilde J_\pm\propto \sigma_\pm$. The only operator commuting with both $\tilde J_+$ and $\tilde J_-$ is the identity, $C_\mu = \mathbb{1}$, and thus the dark space dimension is $D=1$. For this example, we also have $H=\tilde H_{\rm L}\propto Z$, see Eq. . We conclude that $M_1=|0\rangle\langle 0|$ spans the corresponding space. As second example, we discuss the dark space stabilization for a two-box device in Sec. \[sec4b\]. Using the Lindblad equation and assuming that QD $2$ remains decoupled from the system, see Eq. , the four conserved quantities $C_{\mu}$ listed in Ref. [@ourprl] are readily identified. Given these quantities, a basis spanning the dark space can be constructed from Eq. . One may view the basis elements, $M_{\mu}$, as linearly independent ‘vectors’ with the orthogonality relation ${\rm tr}( M_{\mu}^{\dagger}M_{\nu}^{})=\delta_{\mu\nu}$. The existence of four conserved operators $C_{\mu}$ now implies that we have four basis vectors spanning the dark space, see Ref. [@ourprl] for explicit expressions. 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ArXiv

--- abstract: 'We settle the question of tight thresholds for offline cuckoo hashing. The problem can be stated as follows: we have $n$ keys to be hashed into $m$ buckets each capable of holding a single key. Each key has $k \geq 3 $ (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. We seek thresholds $c_k$ such that, as $n$ goes to infinity, if $n/m \leq c$ for some $ c < c_k$ then a hash table can be constructed successfully with high probability, and if $n/m \geq c$ for some $c > c_k$ a hash table cannot be constructed successfully with high probability. Here we are considering the offline version of the problem, where all keys and hash values are given, so the problem is equivalent to previous models of multiple-choice hashing. We find the thresholds for all values of $k > 2$ by showing that they are in fact the same as the previously known thresholds for the random $k$-XORSAT problem. We then extend these results to the setting where keys can have differing number of choices, and provide evidence in the form of an algorithm for a conjecture extending this result to cuckoo hash tables that store multiple keys in a bucket.' author: - 'Martin Dietzfelbinger[^1]' - 'Andreas Goerdt[^2]' - 'Michael Mitzenmacher[^3]' - | \ Andrea Montanari[^4] - 'Rasmus Pagh[^5]' - 'Michael Rink${}^\star$' title: Tight Thresholds for Cuckoo Hashing via XORSAT --- =1 =1 =1 Introduction ============ Consider a hashing scheme with $n$ keys to be hashed into $m$ buckets each capable of holding a single key. Each key has $k \geq 3$ (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. This setting describes the offline load balancing problem corresponding to multiple choice hashing [@ABKU] and cuckoo hashing [@FPSS:2005; @cuckoo1] with $k \geq 3$ choices. An open question in the literature (see, for example, the discussion in [@MMsurvey]) is to determine a tight threshold $c_k$ such that if $n/m \leq c$ for some $ c < c_k$ then a hash table can be constructed successfully with high probability, and if $n/m \geq c$ for some $c > c_k$ a hash table cannot be constructed successfully with high probability. In this paper, we provide these thresholds. We note that, in parallel with this work, two other papers have similarly provided means for determining the thresholds [@Fountoulakis:2009; @Frieze:2009]. Our work differs from these works in substantial ways. Perhaps the most substantial is our argument that, somewhat surprisingly, the thresholds we seek were actually essentially already known. We show that tight thresholds follow from known results in the literature, and in fact correspond exactly to the known thresholds for the random $k$-XORSAT problem. We describe the $k$-XORSAT problem and the means for computing its thresholds in more detail in the following sections. Our argument is somewhat indirect, although all of the arguments appear to rely intrinsically on the analysis of corresponding random hypergraphs, and hence the alternative arguments of [@Fountoulakis:2009; @Frieze:2009] provide additional insight that may prove useful in further explorations. With this starting point, we extend our study of the cuckoo hashing problem in two ways. First, we consider [*irregular cuckoo hashing*]{}, where the number of choices corresponding to a key is not a fixed constant $k$ but itself a random variable depending on the key. Our motivations for studying this variant include past work on irregular low-density parity-check codes [@ldpc] and recent work on alternative hashing schemes that have been said to behave like cuckoo hashing with “3.5 choices” [@cuckoo35]. Beyond finding thresholds, we show how to optimize irregular cuckoo hashing schemes with a specified average number of choices per key; for example, with an average of 3.5 choices per key, the optimal scheme is the natural one where half of the keys obtain 3 choices, and the other half obtain 4. Second, we consider the generalization to the setting where a bucket can hold more than one key. We provide a conjecture regarding the appropriate threshold behavior for this setting, and provide a simple algorithm that, experimentally, appears to perform remarkably close to the thresholds predicted by our conjecture. =1 Paper overview -------------- Section \[sect:facts:cores\] presents an exposition of known results on cores of random hypergraphs. Readers familiar with this material may want to skip directly to Section \[sec:main\], which provides our proof that the thresholds for $k$-XORSAT and $k$-ary cuckoo hashing are identical. In Section \[sec:non-integer\] we extend the discussion of thresholds to the case where $k$ is any real number greater than 2. Finally, Section \[subsec:selfless:algorithm\] presents our simple algorithm to construct hash tables and presents experimental evidence that it is able to achieve load factors close to the thresholds. =1 Further details appear in the appendices. Technical background on cores {#sect:facts:cores} ============================= The key to our analysis will be the behavior of cores in random hypergraphs. We therefore begin by providing a review of this subject. To be clear, the results of this section are not new; the reader is encouraged to see [@MezMontBook:2009 Ch. 18], as well as references [@Coo:2004; @DubMan:2002; @Molloy:RSA:2005] for more background. We consider the set of all $k$-uniform hypergraphs with $m$ nodes and $n$ hyperedges ${\mathcal{G}^k_{m,n}}$. More precisely, each hypergraph $G$ from ${\mathcal{G}^k_{m,n}}$ consists of $n$ (labeled) hyperedges of a fixed size $k\ge2$, chosen independently at random, with repetition, from the $\binom{m}{k}$ subsets of $\{1,\ldots,m\}$ of size $k$. This model will be regarded as a probability space. We always assume $k$ is fixed, $m$ is sufficiently large, and $n=c m$ for a constant $c$. For $\ell\ge2$, the $\ell$[*-core*]{} of a hypergraph $G$ is defined as the largest induced sub-hypergraph that has minimum degree $\ell$ or larger. It is well known that the $\ell$-core can be obtained by the following iterative “peeling process”: While there are nodes with degree smaller than $\ell$, delete them and their incident hyperedges. By pursuing this process backwards one sees that the $\ell$-core, conditioned on the number of nodes and hyperedges it contains, is a uniform random hypergraph that satisfies the degree constraint. The fate of a fixed node $a$ after a fixed number of $h$ iterations of the peeling procedure is determined by the $h$-neighborhood of $a$, where the $h$-neighborhood of $a$ is the sub-hypergraph induced on the nodes at distance at most $h$ from $a$. For example, the $1$-neighborhood contains all hyperedges containing $a$. In our setting where $n$ is linear in $m$ the $h$-neighborhood of node $a$ is a hypertree of low degree (at most $\log \log m$) with high probability. We assume this in the discussion to come. We can see whether a node $a$ is removed from the hypergraph in the course of $h$ iterations of the peeling process in the following way. Consider the hypertree rooted from $a$ (so the children are nodes that share a hyperedge with $a$, and similarly the children of a node share a hyperedge with that node down the tree). First, consider the nodes at distance $h-1$ from $a$ and delete them if they have at most $\ell-2$ child hyperedges; that is, their degree is at most $\ell-1.$ Second, treat the nodes at distance $h-2 $ in the same way, and so on, down to distance $1$, the children of $a$. Finally, $a$ is deleted if its degree is at most $\ell-1.$ The analysis of such random processes on trees has been well-studied in the literature. (See, for example, [@BroderFriezeUpfal; @AndOrTrees] for similar analyses.) We wish to determine the probability $q_h$ that node $a$ is deleted after $h$ rounds of the peeling process. For $j < h$ let $p_j$ be the probability that a node at distance $h-j$ from $a$ is deleted after $j$ rounds of the peeling process. The discussion becomes easier for the binomial random hypergraph with an expected number of $cm$ hyperedges: Each hyperedge is present with probability $k! \cdot c /m^{k-1}$ independently. It is well known that ${\mathcal{G}^k_{m,n}}$ and the binomial hypergraph are equivalent as far as asymptotic behavior of cores are concerned when $c$ is a constant. Let ${\mathrm{Bin}}(N,p)$ denote a random variable with a binomial distribution, and ${\mathrm{Po}}(\beta)$ a random variable with a Poisson distribution. Below we make use of the Poisson approximation of the binomial distribution and the fact that the number of child hyperedges of a node in the hypertree asymptotically follows the binomial distribution. This results in additive terms that tend to zero as $m$ goes to infinity. We have $p_0 = 0$, $$\begin{aligned} p_1 & =& \Pr\bigg[{\mathrm{Bin}}\left( \binom{m-1}{k-1} \,\,,\,\,k! \cdot \frac{c}{m^{k-1}} \right) \le \ell-2 \bigg]\, \\ & & \\ & = & \Pr[{\mathrm{Po}}(k c ) \le \ell-2] \pm o(1), \\ \mbox{ } & \\ p_{j+1} &= & \Pr\bigg[{\mathrm{Bin}}\left(\binom{m-1}{k-1} \,\, , \,\, k! \cdot \frac{c}{m^{k-1}} \cdot (1- p_j)^{k-1}\right)\, \le \ell-2\bigg]\, \\ & & \\ &=& \Pr[ {\mathrm{Po}}(kc(1-p_j)^{k-1}) \le \ell-2] \pm o(1), \mbox{ for } j=1,\dots,h-2.\end{aligned}$$ The probability $q_h$ that $a$ itself is deleted is given by the following different formula: $$\label{del} q_h = \Pr[{\mathrm{Po}}(kc(1-p_{h-1})^{k-1} )\le \ell-1] \pm o(1).$$ The $p_j$ are monotonically increasing and $0 \le p_j \le 1$, so $p= \lim p_j$ is well-defined. The probability that $a$ is deleted approaches $p$ from below as $h$ grows. Continuity of the functions involved implies that $p$ is the smallest non-negative solution of $$\begin{aligned} p & = & \Pr[ {\mathrm{Po}}(kc(1-p)^{k-1}) \le \ell-2].\end{aligned}$$ Observe that $1$ is always a solution. Equivalently, applying the monotone function $t\mapsto kc(1-t)^{k-1}$ to both sides of the equation, $p$ is the smallest solution of $$\label{eq:def:p} kc(1-p)^{k-1} = kc \left( 1 - \Pr [ {\mathrm{Po}}(kc(1-p)^{k-1}) \le \ell-2] \right)^{k-1}.$$ Let $\beta = kc (1-p)^{k-1}$. It is helpful to think of $\beta$ with the following interpretation: Given a node in the hypertree, the number of child hyperedges (before deletion) follows the distribution ${\mathrm{Po}}(kc)$. Asymptotically, a given child hyperedge is not deleted with probability $(1-p)^{k-1}$, independently for all children. Hence the number of child hyperedges after deletion follows the distribution ${\mathrm{Po}}(kc(1-p)^{k-1}).$ And $\beta$ is the key parameter for the node giving the expected number of hyperedges containing it that could contribute to keeping it in the core. Note that (\[eq:def:p\]) is equivalent to $$c = \frac{1}{k}\cdot\frac{\beta}{\left (\Pr[{\mathrm{Po}}(\beta ) \,\ge \ell-1]\right)^{k-1}}.$$ This motivates considering the function $$g_{k,\ell}(\beta) =\frac{1}{k}\cdot\frac{\beta}{(\Pr[{\mathrm{Po}}(\beta) \ge \ell-1])^{k-1}}, \label{eq:definition:g}$$ which has the following properties in the range $(0,\infty)$: It tends to infinity for $\beta\to0$, as well as for $\beta\to\infty$. Since it is convex there is exactly one global minimum. Let $\beta^*_{k,\ell}=\arg\min_{\beta}g_{k,\ell}(\beta)$ and $c^*_{k,\ell}=\min g_{k,\ell}(\beta)$. For $\beta > \beta^*_{k,\ell}$ the function $g_{k,\ell}$ is monotonically increasing. For each $c>c^*_{k,\ell}$ let $\beta(c) \,= \, \beta_{k,\ell}(c)$ denote the unique $\beta>\beta^*_{k,\ell}$ such that $g_{k,\ell}(\beta)=c$. Coming back to the fate of $a$ under the peeling process, Equation (\[del\]) shows that $a$ is deleted with probability approaching $\Pr[{\mathrm{Po}}(\beta(c))\le \ell-1].$ This probability is smaller than $1$ if and only if $c > c^*_{k, \ell}$, which implies that the expected number of nodes that are [*not*]{} deleted is linear in $n$. As the $h$-neighborhoods of two nodes $a$ and $b$ are disjoint with high probability, by making use of the second moment we can show that in this case a linear number of nodes survive with high probability. (The sophisticated reader would use Azuma’s inequality to obtain concentration bounds.) Following this line of reasoning, we obtain the following results, the full proof of which is in [@Molloy:RSA:2005]. (See also the related argument of [@MezMontBook:2009 Ch. 18].) Note the restriction to the case $k + \ell > 4$, which means that the result does not apply to $2$-cores in standard graphs; since the analysis of standard cuckoo hashing is simple, using direct arguments, this case is ignored in the analysis henceforth. \[prop:one\] Let $k+\ell >4$ and $G$ be a random hypergraph from ${\mathcal{G}^k_{m,n}}$. Then $c^*_{k,\ell}$ is the threshold for the *appearance* of an $\ell$-core in $G$. That is, for constant $c$ and $m\to \infty$, - if $n/m = c < c^*_{k,\ell}$, then $G$ has an empty $\ell$-core with probability $1-o(1)$. - if $n/m = c > c^*_{k,\ell}$, then $G$ has an $\ell$-core of linear size with probability $1-o(1)$. In the following we assume $c > c^*_{k,\ell}$. Therefore $\beta(c)>\beta^*_{k,\ell}$ exists. Let $\hat m$ be the number of nodes in the $\ell$-core and $\hat n$ be the number of hyperedges in the $\ell$-core. We will find it useful in what follows to consider the [*edge density*]{} of the $\ell$-core, which is simply the ratio of the number of hyperedges to the number of nodes. =1 \[prop:two\] Let $c>c^*_{k,\ell}$ and $n/m = c\, (1\pm o(1))$. Then with high probability in ${\mathcal{G}^k_{m,n}}$ $$\hat m = \Pr[{\mathrm{Po}}({\beta(c))}\ge \ell]\cdot m \pm o(m)$$ and $$\hat n = (\Pr[{\mathrm{Po}}({\beta(c))}\ge \ell-1])^k \cdot n \pm o(m).$$ =1 \[prop:two\] Let $c>c^*_{k,\ell}$ and $n/m = c\, (1\pm o(1))$. Then with high probability in ${\mathcal{G}^k_{m,n}}$ $$\hat m = \Pr[{\mathrm{Po}}({\beta(c))}\ge \ell]\cdot m \pm o(m) \mbox{ and } \hat n = (\Pr[{\mathrm{Po}}({\beta(c))}\ge \ell-1])^k \cdot n \pm o(m).$$ The bound for $\hat m$ follows from the concentration of the expected number of nodes surviving when we plug in the limit $p$ for $p_h$ in equation (\[del\]). The result for $\hat n$ follows similar lines: Consider a fixed hyperedge $e$ that we assume is present in the random hypergraph. For each node of this hyperedge we consider its $h$-neighborhood modified in that $e$ itself does not belong to this $h$-neighborhood. We have $k$ disjoint trees with high probability. Therefore each of the $k$ nodes of $e$ survives $h$ iterations of the peeling procedure independently with probability $\Pr[{\mathrm{Po}}(\beta(c)) \ge \ell-1]$. Note that we use $\ell-1$ here (instead of $\ell$) because the nodes belong to $e.$ Then $e$ itself survives with $ (\Pr[{\mathrm{Po}}(\beta(c)) \ge \ell-1])^k.$ Concentration of the number of surviving hyperedges again follows from second moment calculations or Azuma’s inequality. With this we have the information needed regarding the edge density of the $\ell$-core. \[prop:three\] If $c>c^*_{k,\ell}$ and $n/m = c\, (1\pm o(1))$ then with high probability the edge density of the $\ell$-core of a random hypergraph from ${\mathcal{G}^k_{m,n}}$ is $$\frac{\beta(c)\cdot \Pr[{\mathrm{Po}}(\beta(c))\ge \ell-1]}{k\cdot\Pr[{\mathrm{Po}}(\beta(c))\ge \ell]}\pm o(1).$$ This follows directly from Proposition \[prop:two\], where we have also used equation (\[eq:definition:g\]) to simplify the expression for $\hat n$. We define $c_{k,\ell}$ as the unique $c$ that satisfies $$\label{ckll+1} \frac{\beta(c)\cdot \Pr[{\mathrm{Po}}(\beta(c))\ge \ell-1]} {k\cdot\Pr[{\mathrm{Po}}(\beta(c))\ge \ell]}\,= \, \ell-1.$$ The values $c_{k,\ell}$ will prove important in the work to come; in particular, we next show that $c_{k, 2}$ is the threshold for $k$-ary cuckoo hashing for $k > 2$. We also conjecture that $c_{k,\ell+1}$ is the threshold for $k$-ary cuckoo hashing when a bucket can hold $\ell$ keys instead of a single key. The following table contains numerical values of $c_{k,\ell}$ for $\ell=2, \ldots, 7$ and $k=2,\ldots,7$ (rounded to 10 decimal places). Some of these numbers are found or referred to in other works, such as [@Coo:2004 Sect. 5], [@MezRicZec:2003 Sect.4.4], [@MezMontBook:2009 p.423], [@FerRam:2007], and [@CaiSanWor:2007]. $\ell \backslash k$ 2 3 4 5 6 7 --------------------- -------------- -------------- -------------- -------------- -------------- -------------- 2 $-$ 0.9179352767 0.9767701649 0.9924383913 0.9973795528 0.9990637588 3 1.7940237365 1.9764028279 1.9964829679 1.9994487201 1.9999137473 1.9999866878 4 2.8774628058 2.9918572178 2.9993854302 2.9999554360 2.9999969384 2.9999997987 5 3.9214790971 3.9970126256 3.9998882644 3.9999962949 3.9999998884 3.9999999969 6 4.9477568093 4.9988732941 4.9999793407 4.9999996871 4.9999999959 5.0000000000 7 5.9644362395 5.9995688805 5.9999961417 5.9999999733 5.9999999998 6.0000000000 =1 Equality of thresholds for random $k$-XORSAT and $k$-ary cuckoo hashing {#sec:main} ======================================================================= We now recall the random $k$-XORSAT problem and describe its relationship to cores of random hypergraphs and cuckoo hashing. The $k$-XORSAT problem is a variant of the satisfiability problem in which every clause has $k$ literals and the clause is satisfied if the XOR of values of the literals is 1. Equivalently, since XORs correspond to addition modulo 2, and the negation of $X_i$ is just $1$ XOR $X_i$, an instance of the $k$-XORSAT problem corresponds to a system of linear equations modulo 2, with each equation having $k$ variables (none of which is negated), and randomly chosen right hand sides. (In what follows we simply use the addition operator where it is understood we are working modulo 2 from context.) For a random $k$-XORSAT problem, let ${{\Phi}^k_{m,n}}$ be the set of all sequences of $n$ linear equations over $m$ variables $x_1,\ldots,x_m,$ where an equation is $$x_{j_1}+\cdots+x_{j_k}=b_j,$$ where $b_j\in\{0,1\}$ and $\{j_1,\ldots,j_k\}$ is a subset of $\{1,\ldots,m\}$ with $k$ elements. We consider ${{\Phi}^k_{m,n}}$ as a probability space with the uniform distribution. Given a $k$-XORSAT formula $F$, it is clear that $F$ is satisfiable if and only if the formula obtained from $F$ by repeatedly deleting variables that occur only once (and equations containing them) is satisfiable. Now consider the $k$-XORSAT formula as a hypergraph, with nodes representing variables and hyperedges representing equations. (The values $b_j$ of the equations are not represented.) The process of repeatedly deleting all variables that occur only once, and the corresponding equations, is exactly equivalent to the peeling process on the hypergraph. Hence, after the peeling process, we obtain the 2-core of the hypergraph. This motivates the following definition. Let $\Psi^k_{m,n}$ be the set of all sequences of $n$ equations such that each variable appears at least twice. We consider $\Psi^k_{m,n}$ as a probability space with the uniform distribution. Recall that if we start with a uniformly chosen random $k$-XORSAT formula, and perform the peeling process, then conditioned on the remaining number of equations and variables ($\hat n$ and $\hat m$), we are in fact left with a uniform random formula from $\Psi^k_{\hat m,\hat n}$. Hence, the imperative question is when a random formula from $\Psi^k_{\hat m,\hat n}$ will be satisfiable. In [@DubMan:2002], it was shown that this depends entirely on the edge density of the corresponding hypergraph. If the edge density is smaller than 1, so that there are more variables than equations, the formula is likely to be satisfiable, and naturally, if there are more equations than variables, the formula is likely to be unsatisfiable. Specifically, we have the following theorem from [@DubMan:2002]. \[thresh\] \[DM:Theorem\] Let $k > 2$ be fixed. For $n/m = \gamma$ and $m\to\infty$, - if $\gamma>1$ then a random formula from $\Psi^k_{m,n}$ is unsatisfiable with high probability. - if $\gamma<1$ then a random formula from $\Psi^k_{m,n}$ is satisfiable with high probability. The proof of Theorem \[thresh\] in Section 3 of [@DubMan:2002] uses a first moment method argument for the simple direction (part (a)). Part (b) is significantly more complicated, and is based on the second moment method. Essentially the same problem has also arisen in coding theoretic settings; analysis and techniques can be found in for example [@MMU]. =1 It has been suggested by various readers of earlier drafts of this paper that previous proofs of Theorem \[thresh\] have been insufficiently complete, particularly for $k > 3$. We therefore provide a detailed proof in Appendix C for completeness. We have shown that the edge density is concentrated around a specific value depending on the initial ratio $c$ of hyperedges (equations) to nodes (variables). Let $c_{k,2}$ be the value of $c$ such that the resulting edge density is concentrated around 1. Then Proposition \[prop:three\] and Theorem \[thresh\] together with the preceding consideration implies: \[corthresh\] Let $k > 2$ and consider ${{\Phi}^k_{m,n}}.$ The satisfiability threshold with respect to the edge density $c=n/m$ is $c_{k, 2}$. Again, up to this point, everything we have stated was known from previous work. We now provide the connection to cuckoo hashing, to show that we obtain the same threshold values for the success of cuckoo hashing. That is, we argue the following: \[thm:equivalence:variant\] For $k > 2$, $c_{k,2}$ is the threshold for $k$-ary cuckoo hashing to work. That is, and with $n$ keys to be stored and $m$ buckets, with $c=n/m$ fixed and $m\to\infty$, - if $c > c_{k,2}$, then $k$-ary cuckoo hashing does not work with high probability. - if $c < c_{k,2}$, then $k$-ary cuckoo hashing works with high probability. Assume a set of $n$ keys $S$ is given, and for each $x \in S$ a random set $A_x\subseteq\{1,\ldots,m \}$ of size $k$ of possible buckets is chosen. To prove part (a), note that the sets $A_x$ for $x\in S$ can be represented by a random hypergraph from ${\mathcal{G}^k_{m,n}}$. If $n/m = c > c_{k,2}$ and $m\to\infty$, then with high probability the edge density in the 2-core is greater than 1. The hyperedges in the 2-core correspond to a set of keys, and the nodes in the 2-core to the buckets available for these keys. Obviously, then, cuckoo hashing does not work. To prove part (b), consider the case where $n/m = c < c_{k,2}$ and $m \to\infty$. Picking for each $x$ a random $b_x \in \{0, 1\}$, the sets $A_x$, $x\in S$, induce a random system of equations from ${{\Phi}^k_{m,n}}.$ Specifically, $A_x = \{ j_1 , \dots, j_k\}$ induces the equation $x_{j_1} + \dots + x_{j_k} = b_x.$ By Corollary \[corthresh\] a random system of equations from ${{\Phi}^k_{m,n}}$ is satisfiable with high probability. This implies that the the matrix $M$ made up from the left-hand sides of these equations consists of linearly independent rows with high probability. This is because a given set of left-hand sides with dependent rows is only satisfiable with probability at most $1/2$ when we pick the $b_x$ at random. Therefore we have an $n \times n$-submatrix in $M$ with a nonzero determinant. The expansion of the determinant of this submatrix as a sum of products by the Leibniz formula must contain a product with all factors being variables $x_{i_j}$ (as opposed to 0). This product term corresponds to a permutation mapping keys to buckets, showing that cuckoo hashing is indeed possible. We make some additional remarks. We note that the idea of using the rank of the key-bucket matrix to obtain lower bounds on the cuckoo hashing threshold is not new either; it appears in [@DP:2008]. There the authors use a result bounding the rank by Calkin [@Calkin] to obtain a lower bound on the threshold, but this bound is not tight in this context. More details can be found by reviewing [@Calkin Theorem 1.2] and [@MezMontBook:2009 Exercise 18.6]. Also, Batu et al. [@BBC] note that 2-core thresholds provide an upper bound on the threshold for cuckoo hashing, but fail to note the connection to work on the $k$-XORSAT problems. Non-integer choices {#sec:non-integer} =================== The analysis of $k$-cores in Section \[sec:main\] and the correspondence to $k$-XORSAT problems extends nicely to the setting where the number of choices for a key is not necessarily a fixed number $k$. This can be naturally accomplished in the following way: when a key $x$ is to be inserted in the cuckoo hash table, the number of choices of location for the key is itself determined by some hash function; then the appropriate number of choices for each key $x$ can also be found when performing a lookup. Hence, it is possible to ask about for example cuckoo hashing with 3.5 choices, by which we would mean an average of 3.5 choices. Similarly, even if we decide to have an average of $k$ choices per key, for an integer $k$, it is not immediately obvious whether the success probability in $k$-ary cuckoo hashing could be improved if we do not fix the number of possible positions for a key but rather choose it at random from a cleverly selected distribution. Let us consider a more general setting where for each $x\in U$ the set $A_x$ is chosen uniformly at random from the set of all ${\ensuremath{k}}_x$-element subsets of $[{\ensuremath{m}}]$, where ${\ensuremath{k}}_x$ follows some probability mass function ${\ensuremath{{\rho}}}_x$ on $\{2,\ldots,{\ensuremath{m}}\}$.[^6] Let ${\ensuremath{\kappa}}_x=E({\ensuremath{k}}_x)$ and ${\ensuremath{\kappa}}^*=\frac{1}{{\ensuremath{n}}}\sum_{x\in S}{\ensuremath{\kappa}}_x$. Note that ${\ensuremath{\kappa}}^*$ is the average (over all $x \in S$) worst case lookup time for successful searches. We keep ${\ensuremath{\kappa}}^*$ fixed and study which sequence $({\ensuremath{{\rho}}}_x)_{x\in S}$ maximizes the probability that cuckoo hashing is successful. We fix the sequence of the expected number of choices per key $({\ensuremath{\kappa}}_x)_{x\in S}$ and therefore ${\ensuremath{\kappa}}^*$. Furthermore we assume ${\ensuremath{\kappa}}_x\leq n-2$, for all $x \in S$; obviously this does not exclude interesting cases. For compactness reasons, there is a system of probability mass functions ${\ensuremath{{\rho}}}_x$ that maximizes the success probability. We will show the following: \[prop:constant\_expected\_degree\] Let $({\ensuremath{{\rho}}}_x)_{x \in S}$ be an optimal sequence. Then we have, for all $x \in S$: $${\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor )=1-({\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor), \text{ and }{\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor+1 )={\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor.$$ That is, the success probability is maximized if for each $x\in S$ the number of choices ${\ensuremath{k}}_x$ is concentrated on $\lfloor {\ensuremath{\kappa}}_x\rfloor$ and $\lfloor {\ensuremath{\kappa}}_x\rfloor +1$ (when the number of choices is non-integral). Further, in the natural case where all keys $x$ have the same expected number ${\ensuremath{\kappa}}^*$ of choices, the optimal assignment is concentrated on $\lfloor {\ensuremath{\kappa}}^* \rfloor$ and $\lfloor {\ensuremath{\kappa}}^*\rfloor +1$. Also, if ${\ensuremath{\kappa}}_x$ is an integer, then a fixed degree ${\ensuremath{k}}_x={\ensuremath{\kappa}}_x$ is optimal. This is very different from other similar scenarios, such as erasure- and error-correcting codes, where irregular distributions have proven beneficial [@ldpc]. =1 The proof is given in Appendix A. =1 We consider a random bipartite graph $G_S$ with left node set $S$, right node set $[{\ensuremath{m}}]$ and an edge between two nodes $x\in S$ and $a\in [{\ensuremath{m}}]$ if and only if $a\in A_x$. Let the sequence $({\ensuremath{\kappa}}_x)_{x\in S}$ be fixed. For each $x \in S$ we want to obtain a distribution ${\ensuremath{{\rho}}}_x$ for the degree ${\ensuremath{k}}_x$ (or, equivalently, the cardinality of $A_x$), such that we have $E({\ensuremath{k}}_x)={\ensuremath{\kappa}}_x$ and the following quantity is maximized: $$\label{eq:success_prob} \Pr(\text{``success''}):=\Pr( (A_x)_{x\in S} \text{ admits a left-perfect matching}\footnote{In the following ``matching'' and ``left-perfect matching'' are used synonymously.}\text{ in $G_S$}){\text{ }}.$$ We study the sequence $({\ensuremath{{\rho}}}_x)_{x\in S}$ that realizes the maximum. Let $z$ be an arbitrary but fixed element of $S$ with probability mass function ${\ensuremath{{\rho}}}_z$. To prove Proposition \[prop:constant\_expected\_degree\] it is sufficient to show that if there exist two numbers ${\ensuremath{l}}$ and ${\ensuremath{k}}$ with ${\ensuremath{l}}<{\ensuremath{\kappa}}_z < {\ensuremath{k}}$ and ${\ensuremath{k}}-{\ensuremath{l}}\geq2$ as well as ${\ensuremath{{\rho}}}_z({\ensuremath{l}})>0$ and ${\ensuremath{{\rho}}}_z({\ensuremath{k}})>0$ then cannot be maximal. We start by fixing ${\ensuremath{k}}_x$ and $A_x$ for each $x\in S-\{z\}$ and consider the corresponding bipartite graph $G_{S-\{z\}}$. Let $B\subseteq [{\ensuremath{m}}]$ be the set of right nodes in $G_{S-\{z\}}$ that are matched in every matching. Then there is a matching for the whole key set $S$ in $G_S$ if and only if $A_z \not \subseteq B$. Note that $0\leq |B|<{\ensuremath{m}}$, i.e., there must be at least one right node that is not matched. Let $p=\min\{ {\ensuremath{{\rho}}}_z({\ensuremath{l}}), {\ensuremath{{\rho}}}_z({\ensuremath{k}}) \}>0$ and $|B|=b$. We will show that changing ${\ensuremath{{\rho}}}_z$ to $$\begin{aligned} {\ensuremath{{\rho}}}_z'({\ensuremath{l}})&:={\ensuremath{{\rho}}}_z({\ensuremath{l}})-p &{\ensuremath{{\rho}}}_z'({\ensuremath{k}})&:={\ensuremath{{\rho}}}_z({\ensuremath{k}})-p \\ {\ensuremath{{\rho}}}_z'({\ensuremath{l}}+1)&:={\ensuremath{{\rho}}}_z({\ensuremath{l}}+1)+p &{\ensuremath{{\rho}}}_z'({\ensuremath{k}}-1)&:={\ensuremath{{\rho}}}_z({\ensuremath{k}}-1)+p,\end{aligned}$$ with ${\ensuremath{{\rho}}}_z'(j)={\ensuremath{{\rho}}}_z(j)$ for $j\notin\{{\ensuremath{l}},{\ensuremath{k}}\}$, increases , while leaving ${\ensuremath{\kappa}}_z$ unchanged. This is the case if and only if $$\label{eq:first_ineq} p\cdot \frac{\binom{b}{{\ensuremath{l}}}}{\binom{{\ensuremath{m}}}{{\ensuremath{l}}}} + p\cdot \frac{\binom{b}{{\ensuremath{k}}}}{\binom{{\ensuremath{m}}}{{\ensuremath{k}}}} \geq p\cdot \frac{\binom{b}{{\ensuremath{l}}+1}}{\binom{{\ensuremath{m}}}{{\ensuremath{l}}+1}} + p\cdot \frac{\binom{b}{{\ensuremath{k}}-1}}{\binom{{\ensuremath{m}}}{{\ensuremath{k}}-1}}, $$ and the strict inequality holds for at least one value $b$ that occurs with positive probability. The left sum of is the $2\cdot p$ fraction of the failure probability (by ${\ensuremath{{\rho}}}_z({\ensuremath{l}})$ and ${\ensuremath{{\rho}}}_z({\ensuremath{k}})$) before the change of ${\ensuremath{{\rho}}}_z$ under the condition that $B$ has cardinality $b$; the right sum is the corresponding fraction of the failure probability after the change. Depending on $b$ we have to distinguish several cases. $b={\ensuremath{m}}-1$. In this case both sides of are equal, i.e., the modification we do to ${\ensuremath{{\rho}}}_z$ will not change the success probability. ${\ensuremath{k}}\leq b < {\ensuremath{m}}-1$. Canceling $p$ and subtracting ${\binom{b}{{\ensuremath{k}}}}/{\binom{{\ensuremath{m}}}{{\ensuremath{k}}}}$ and ${\binom{b}{{\ensuremath{l}}+1}}/{\binom{{\ensuremath{m}}}{{\ensuremath{l}}+1}}$ from both sides of shows that the strict inequality holds if and only if $$\label{eq:intermediate} \begin{split} \frac{b\cdots(b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}}+1)}-\frac{b\cdots(b-{\ensuremath{l}})}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots(b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+2)}-\frac{b\cdots(b-{\ensuremath{k}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)} {\text{ }}. \end{split}$$ Factoring out $\frac{b\cdots(b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}}+1)}$ on the left side and $\frac{b\cdots(b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+2)}$ on the right side gives $$\frac{b\cdots (b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots ({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots (b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)} \Leftrightarrow \frac{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)}{{\ensuremath{m}}\cdots ({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots (b-{\ensuremath{k}}+2)}{b\cdots (b-{\ensuremath{l}}+1)}{\text{ }}.$$ Since ${\ensuremath{l}}\leq {\ensuremath{k}}-2$, this is equivalent to $$({\ensuremath{m}}-{\ensuremath{l}}+1)\cdot({\ensuremath{m}}-{\ensuremath{l}})\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)>(b-{\ensuremath{l}})\cdot(b-{\ensuremath{l}}-1)\cdots(b-{\ensuremath{k}}+2){\text{ }},$$ which is true for $m-1>b$. ${\ensuremath{l}}\leq b<{\ensuremath{k}}$. Calculations along the lines of case 2 show that the strict inequality of also holds in this case. Note that $\binom{b}{k}$, $\binom{b}{k-1}$ and $\binom{b}{l+1}$ can be zero. $0\leq b<{\ensuremath{l}}$. In this case both sides of are zero, i.e., the modifications we do to ${\ensuremath{{\rho}}}_z$ will not change the success probability. Since in cases 1 and 4 above there was no change in the success probability, to show that cannot be maximal when ${\ensuremath{k}}-{\ensuremath{l}}\geq2$ as we are considering, it remains to show that at least one of the Cases 2 and 3 occurs with positive probability. We construct a situation in which one of these cases applies, and which occurs with positive probability. Choose degrees ${\ensuremath{k}}_x$ for all elements $x \in S-\{z\}$ such that ${\ensuremath{k}}_x\leq {\ensuremath{\kappa}}_x$ and ${\ensuremath{{\rho}}}_x({\ensuremath{k}}_x)>0$. Consider a permutation of the elements $x \in S-\{z\}$ such that these degrees are ordered, i.e., ${\ensuremath{k}}_{x_1}\leq {\ensuremath{k}}_{x_2}\leq \ldots \leq {\ensuremath{k}}_{x_{{\ensuremath{n}}-1}}$. Choose the first element $x_i$ with $i \geq {\ensuremath{l}}$ and ${\ensuremath{k}}_{x_i}\leq i$. Such an element must exist, since we assume ${\ensuremath{k}}_x\leq {\ensuremath{\kappa}}_x\leq {\ensuremath{n}}-2$, in particular we have $l<{\ensuremath{n}}-2$. Arrange that $A_{x_j}\subseteq [i],1\leq j \leq i,$ such that there is a matching in $G_{\{x_1,\ldots,x_i\}}$. This implies $b\geq {\ensuremath{l}}$. Then arrange that $|A_{x_j}-\bigcup_{1\leq {j'}<j} A_{x_{j'}}|=1$, for all $i<j\leq n-2$, as well as $|A_{x_{n-1}}-\bigcup_{1\leq {j'}<n-1} A_{x_{j'}}|=2$, which implies $b<m-1$. This finishes the proof of Proposition \[prop:constant\_expected\_degree\]. Thresholds for non-integral degree distributions ------------------------------------------------ We now describe how to extend our previous analysis to derive thresholds for the case of a non-integral number of choices per key; equivalently, we are making use of thresholds for XORSAT problems with an irregular number of literals per clause. Following notation that is frequently used in the coding literature, we let $\Lambda_k$ be the probability that a key obtains $k$ choices, and define $\Lambda(x) = \sum_k \Lambda_k x^k$. Clearly, then, $\Lambda'(x) = \sum_k \Lambda_k k x^{k-1}$, and $\Lambda'(1)=\kappa^*$. (We assume henceforth that $\Lambda_0 = \Lambda_1 = 0$ and $\Lambda_k = 0$ for all $k$ sufficiently large for technical convenience.) We now follow our previous analysis from Section \[sect:facts:cores\]; to see if a node $a$ is deleted after $h$ rounds of the peeling process, we let $p_j$ be the probability that a node at distance $h-j$ from $a$ is deleted after $j$ rounds. We must now account for the differing degrees of hyperedges. Here, the appropriate asymptotics is given by a mixture of binomial hypergraphs, with each hyperedge of degree $k$ present with probability $k! \cdot c \Lambda_k /m^{k-1}$ independently. The corresponding equations are then given by $p_0 = 0$, $$\begin{aligned} p_1 & =& \Pr\bigg[\sum_k {\mathrm{Bin}}\left( \binom{m-1}{k-1} \,\,,\,\,k! \cdot \frac{c\Lambda_k}{m^{k-1}} \right) \le \ell-2 \bigg]\, \\ & = & \Pr\bigg[\sum_k {\mathrm{Po}}(k c \Lambda_k) \le \ell-2\bigg] \pm o(1), \\ & = & \Pr[{\mathrm{Po}}(c \Lambda'(1)) \le \ell-2] \pm o(1), \\ \mbox{ } & \\ p_{j+1} &= & \Pr\bigg[\sum_k {\mathrm{Bin}}\left(\binom{m-1}{k-1} \,\, , \,\, k! \cdot \frac{c \Lambda_k}{m^{k-1}} \cdot (1- p_j)^{k-1}\right)\, \le \ell-2\bigg]\, \\ &=& \Pr\bigg[ \sum_k {\mathrm{Po}}(kc \Lambda_k (1-p_j)^{k-1}) \le \ell-2\bigg] \pm o(1), \mbox{ for } j=1,\dots,h-2, \\ &=& \Pr[ {\mathrm{Po}}(c \Lambda'(1-p_j)) \le \ell-2] \pm o(1), \mbox{ for } j=1,\dots,h-2.\end{aligned}$$ Note that we have used the standard fact that the sum of Poisson random variables is itself Poisson, which allows us to conveniently express everything in terms of the generating function $\Lambda(x)$ and its derivative. As before we find $p= \lim p_j$, which is now given by the smallest non-negative solution of $$\begin{aligned} p & = & \Pr[ {\mathrm{Po}}(c\Lambda'(1-p)) \le \ell-2].\end{aligned}$$ When given a degree distribution $(\Lambda_k)_k$, we can proceed as before to find the threshold load that allows that the edge density of the 2-core remains greater than 1; using =1 a second moment argument, =1 the approach of Appendix C, this can again be shown to be the required property for the corresponding XORSAT problem to have a solution, and hence for there to be a permutation successfully mapping keys to buckets. Notice that this argument works for all degree distributions (subject to the restrictions given above), but in particular we have already shown that the optimal thresholds are to be found by the simple degree distributions that have all weight on two values, $\lfloor{\ensuremath{\kappa}}^*\rfloor$ and $\lfloor{\ensuremath{\kappa}}^*\rfloor + 1$. Abusing notation slightly, let $c_{{\ensuremath{\kappa}}^*,2}$ be the unique $c$ such that the edge density of the 2-core of the corresponding mixture is equal to 1, following the same form as in Proposition \[prop:three\] and equation (\[ckll+1\]). The corresponding extension to Theorem \[thm:equivalence:variant\] is the following: \[thm:equivalence:variant2\] For ${\ensuremath{\kappa}}^* > 2$, $c_{{\ensuremath{\kappa}}^*,2}$ is the threshold for cuckoo hashing with an average of ${\ensuremath{\kappa}}^*$ choices per key to work. That is, with $n$ keys to be stored and $m$ buckets, with $c=n/m$ fixed and $m\to\infty$, - if $c > c_{{\ensuremath{\kappa}}^*,2}$, for any distribution on the number of choices per key with mean ${\ensuremath{\kappa}}^*$, cuckoo hashing does not work with high probability. - if $c < c_{{\ensuremath{\kappa}}^*,2}$, then cuckoo hashing works with high probability when the distribution on the number of choices per key is given by ${\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor )=1-({\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor)$ and ${\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor+1 )={\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor$, for all $x \in S$. ![Thresholds for non-integral ${\ensuremath{\kappa}}^*$-ary cuckoo hashing, with optimal degree distribution. The values in the tables are rounded to the nearest multiple of $10^{-10}$.[]{data-label="fig:non-integral"}](\figPath/threshold-nonintegral.pdf){width="\linewidth"} ${\ensuremath{\kappa}}^*$ $c_{{\ensuremath{\kappa}}^*,2}$ --------------------------- --------------------------------- 2.25 0.6666666667 2.50 0.8103423635 2.75 0.8788457372 3.00 0.9179352767 3.25 0.9408047937 3.50 0.9570796377 3.75 0.9685811888 4.00 0.9767701649 ${\ensuremath{\kappa}}^*$ $c_{{\ensuremath{\kappa}}^*,2}$ --------------------------- --------------------------------- 4.25 0.9825693463 4.50 0.9868637629 4.75 0.9900548807 5.00 0.9924383913 5.25 0.9942189481 5.50 0.9955692011 5.75 0.9965961383 6.00 0.9973795528 We have determined the thresholds numerically for a range of values of ${\ensuremath{\kappa}}^*$. The results are shown in Figure \[fig:non-integral\]. One somewhat surprising finding is that the threshold for ${\ensuremath{\kappa}}^*\leq 2.25$ appears to simply be given by $c = 0.5/(3-{\ensuremath{\kappa}}^*)$. Consequently, in place of using 2 hash functions per key, simply by using a mix of 2 or 3 hash functions for a key, we can increase the space utilization by adding 33% more keys with the same (asymptotic) amount of memory. Algorithm for computing a placement {#subsec:selfless:algorithm} =================================== In this section, we describe an algorithm for finding a placement for the keys using $k$-ary cuckoo hashing when the set $S$ of keys is given an advance. The algorithm is an adaptation of the “selfless algorithm” proposed by Sanders [@Sanders:SOFSEM:2004], for the case $k=2$, and analyzed in [@CaiSanWor:2007], for orienting standard undirected random graphs so that all edges are directed and the maximum indegree of all nodes is at most $\ell$, for some fixed $\ell \geq 2$. We generalize this algorithm to hypergraphs, including hypergraphs where hyperedges can have varying degrees. Of course, maximum matching algorithms can solve this problem perfectly. However, there are multiple motivations for considering our algorithms. First, it seems in preliminary experiments that the running times of standard matching algorithms like the Hopcroft-Karp algorithm [@HK:1973] will tend to increase significantly as the edge density approaches the threshold (the details of this effect are not yet understood), while our algorithm has linear running time which does not change in the neighborhood of the threshold. This proves useful in our experimental evaluation of thresholds. Second, we believe that algorithms of this form may prove easier to analyze for some variations of the problem. We first describe the generalized selfless algorithm for bucket size $\ell=1$. A description in pseudocode is given as Algorithm \[algo:GeneralizedSelfless\]. The algorithm can deal with arbitrary hypergraphs, uniform or not. The aim is to “orient” the hyperedges of the hypergraph $G$, i.e., associate a node $v\in e$ to each hyperedge $e$ so that at most one hyperedge is directed towards any one node $v$. Initially, all hyperedges are unoriented. Nodes that have an hyperedge directed towards them are saturated and are not considered further, and similarly hyperedges once oriented are fixed. At each step, if there is a node $v$ that is incident to only one undirected hyperedge $e$, we direct $v$ to $e$, breaking ties arbitrarily. (In the pseudocode, this is realized by giving such nodes the highest *priority*, which is 0. Note that this rule entails that the algorithm starts by carrying out the peeling process for the $2$-core. But the rule is also applied when hyperedges from the 2-core have already been treated.) If there are no such nodes, every unoriented hyperedge is assigned as its *weight* the number of unsaturated nodes it contains. (Intuitively, a smaller weight means a higher need to direct the hyperedge.) The priority of a node $v$ then is the sum of the inverses of the weights of the hyperedges that contain $v$. This corresponds to the expected number of hyperedges $v$ would have directed toward it if all its unoriented hyperedges were directed to one of their nodes at random. Now a vertex $v$ of smallest (highest) priority is chosen, again breaking ties at random. If this priority is larger than 1, then the algorithm stops and reports “failure”. This is because the sum of all priorities is the number of undirected hyperedges, so if the smallest priority is bigger than 1, the number of undirected hyperedges is larger than the number of unsaturated nodes, and it is impossible to complete the process of directing the hyperedges. Otherwise the algorithm directs the minimum weight incident hyperedge of $v$ toward $v$, breaking ties randomly. (Intuitively, this means that the algorithm tries to continue the peeling process “on average”.) This step is repeated until all hyperedges have been oriented or failure occurs. =1 We ran the generalized selfless algorithm for hypergraphs with $10^5$ and $10^6$ nodes and tabulated the failure rate around the theoretical threshold values $c_{k,2}$ for $k=3,4,5$. Results demonstrate that the generalized selfless algorithm achieves results quite near the threshold; more details and figures are given in Appendix B. =1 We ran the generalized selfless algorithm for hypergraphs with $10^5$ and $10^6$ nodes and tabulated the failure rate around the theoretical threshold values $c_{k,2}$ for $k=3,4,5$. For each pair $({\ensuremath{m}},k)$ we considered $81$ edge densities $c=\frac{{\ensuremath{n}}}{{\ensuremath{m}}}$, spaced apart by $0.0001$, thus covering an interval of length $0.008$, which encloses the theoretical threshold value for the particular parameter pair $({\ensuremath{m}},k)$. The hyperedges of the hypergraphs were randomly chosen via pseudo random number generator MT19937 “Mersenne Twister” of the GNU Scientific Library [@GNU_Scientific]. We measured the average failure rate of the algorithm over $100$ random hypergraphs for each combination $({\ensuremath{m}},{\ensuremath{n}},k)$ within the parameter space. To get an estimation of the threshold, i.e., the rate $c$ where the algorithm switches from success to failure, we fit the sigmoid function $$\label{eq:fit_function} \sigma(c;a,b)=\frac{1}{1+e^{-(c-a)/b}}$$ to the measured failure rate (via gnuplot[^7]), using the method of least squares. We determined the parameters $a,b$ that lead to a (local) minimum of the sum of squares of the $81$ residuals, denoted by $\sum_{res}$. The parameter $a$ is the inflection point of and therefore the approximation of the threshold of the generalized selfless algorithm. Figures \[fig:gen\_selfless\_edge\_size\_3\], \[fig:gen\_selfless\_edge\_size\_4\] and \[fig:gen\_selfless\_edge\_size\_5\] show the results of the experiments. One observes that this simple algorithm is able to construct the placements for edge densities quite close to the calculated thresholds $c_{k,2}$. The slope of the sigmoid curve increases and $\sum_{res}$ decreases with growing ${\ensuremath{m}}$ and $k$, leading to a sharp transition from total success to total failure. Clearly the algorithm can fail on hypergraphs that admit a matching. Experimental comparisons with a perfect matching algorithm [@HK:1973] showed that this is very unlikely for random hypergraphs. An example is given in Figure \[fig:selfless\_vs\_perfect\_matching\], which shows the failure rate of perfect matching in comparison to the generalized selfless algorithm. Note that the plot shows an interval of size $0.004$, i.e., 41 data points instead of $81$. The differences in the failure rates of the algorithms become very small as ${\ensuremath{m}}$ grows. A conjecture, with evidence from a generalized selfless algorithm {#subsec:conjecture} ------------------------------------------------------------------ Now consider a situation in which buckets have a capacity of $\ell>1$ keys. There is as yet no rigorous analysis of the appropriate thresholds for cuckoo hashing for the cases $k > 2$ and $\ell>1$. However, our results of Section \[sect:facts:cores\] suggest a natural conjecture: \[conj:threshold\] For $k$-ary cuckoo hashing with bucket size $\ell$, it is *conjectured* that cuckoo hashing works with high probability if $n/m = c > c_{k,\ell+1}$, and does not work if $n/m = c < c_{k,\ell+1}$, i.e., that the threshold is at the point where the $(\ell+1)$-core of the cuckoo hypergraph starts having edge density larger than $\ell$. In order to provide evidence for this conjecture, we generalize our algorithm further so that it can deal with bucket size $\ell>1$. The pseudocode is given as Algorithm \[algo:GeneralizedSelfless2\]. In hypergraph language, we are now looking for an orientation of the hyperedges of $G$ so that every node has at most $\ell$ hyperedges directed toward it. Now a node is saturated if it has $\ell$ edges pointing to it. As long as there are nodes $v$ such that the number of hyperedges directed toward $v$ and the number of undirected hyperedges containing $v$ taken together does not exceed $\ell$, one such node is chosen and its undirected edges are directed toward it. Again, the effect of this rule is that the algorithm starts by carrying out the peeling process that finds the $(\ell+1)$-core. Otherwise, the algorithm assigned weights and priorities as before, and if all priorities exceed $\ell$, the algorithm stops and reports failure. If the smallest (highest) priority is at most $\ell$, a vertex of smallest priority is chosen and one of the incident undirected hyperedges of minimum weight is directed toward it. The process is carried out until all hyperedges have been directed or failure occurs. Experiments with Algorithm \[algo:GeneralizedSelfless2\] corroborate Conjecture \[conj:threshold\], in that they show that the failure rate of the algorithm changes from 0 to 1 very close to the possible threshold values suggested in Conjecture \[conj:threshold\]. =1 Again, numerical results are given in Appendix B. =1 (For an example see Figure \[fig:gen\_selfless\_edge\_size\_3\_bucket\_size=2\].) Conclusion ========== We have found tight thresholds for cuckoo hashing with 1 key per bucket, by showing that the thresholds are in fact the same for the previous studied $k$-XORSAT problem. 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Specifically, we show that if $({\ensuremath{{\rho}}}_x)_{x \in S}$ is an optimal sequence, then for all $x \in S$: $${\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor )=1-({\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor), \text{ and }{\ensuremath{{\rho}}}_x( \lfloor {\ensuremath{\kappa}}_x \rfloor+1 )={\ensuremath{\kappa}}_x-\lfloor {\ensuremath{\kappa}}_x\rfloor.$$ We consider a random bipartite graph $G_S$ with left node set $S$, right node set $[{\ensuremath{m}}]$ and an edge between two nodes $x\in S$ and $a\in [{\ensuremath{m}}]$ if and only if $a\in A_x$. Let the sequence $({\ensuremath{\kappa}}_x)_{x\in S}$ be fixed. For each $x \in S$ we want to obtain a distribution ${\ensuremath{{\rho}}}_x$ for the degree ${\ensuremath{k}}_x$ (or, equivalently, the cardinality of $A_x$), such that we have $E({\ensuremath{k}}_x)={\ensuremath{\kappa}}_x$ and the following quantity is maximized: $$\label{eq:success_prob} \Pr(\text{``success''}):=\Pr( (A_x)_{x\in S} \text{ admits a left-perfect matching}\footnote{In the following ``matching'' and ``left-perfect matching'' are used synonymously.}\text{ in $G_S$}){\text{ }}.$$ We study the sequence $({\ensuremath{{\rho}}}_x)_{x\in S}$ that realizes the maximum. Let $z$ be an arbitrary but fixed element of $S$ with probability mass function ${\ensuremath{{\rho}}}_z$. To prove Proposition \[prop:constant\_expected\_degree\] it is sufficient to show that if there exist two numbers ${\ensuremath{l}}$ and ${\ensuremath{k}}$ with ${\ensuremath{l}}<{\ensuremath{\kappa}}_z < {\ensuremath{k}}$ and ${\ensuremath{k}}-{\ensuremath{l}}\geq2$ as well as ${\ensuremath{{\rho}}}_z({\ensuremath{l}})>0$ and ${\ensuremath{{\rho}}}_z({\ensuremath{k}})>0$ then cannot be maximal. We start by fixing ${\ensuremath{k}}_x$ and $A_x$ for each $x\in S-\{z\}$ and consider the corresponding bipartite graph $G_{S-\{z\}}$. Let $B\subseteq [{\ensuremath{m}}]$ be the set of right nodes in $G_{S-\{z\}}$ that are matched in every matching. Then there is a matching for the whole key set $S$ in $G_S$ if and only if $A_z \not \subseteq B$. Note that $0\leq |B|<{\ensuremath{m}}$, i.e., there must be at least one right node that is not matched. Let $p=\min\{ {\ensuremath{{\rho}}}_z({\ensuremath{l}}), {\ensuremath{{\rho}}}_z({\ensuremath{k}}) \}>0$ and $|B|=b$. We will show that changing ${\ensuremath{{\rho}}}_z$ to $$\begin{aligned} {\ensuremath{{\rho}}}_z'({\ensuremath{l}})&:={\ensuremath{{\rho}}}_z({\ensuremath{l}})-p &{\ensuremath{{\rho}}}_z'({\ensuremath{k}})&:={\ensuremath{{\rho}}}_z({\ensuremath{k}})-p \\ {\ensuremath{{\rho}}}_z'({\ensuremath{l}}+1)&:={\ensuremath{{\rho}}}_z({\ensuremath{l}}+1)+p &{\ensuremath{{\rho}}}_z'({\ensuremath{k}}-1)&:={\ensuremath{{\rho}}}_z({\ensuremath{k}}-1)+p,\end{aligned}$$ with ${\ensuremath{{\rho}}}_z'(j)={\ensuremath{{\rho}}}_z(j)$ for $j\notin\{{\ensuremath{l}},{\ensuremath{k}}\}$, increases , while leaving ${\ensuremath{\kappa}}_z$ unchanged. This is the case if and only if $$\label{eq:first_ineq} p\cdot \frac{\binom{b}{{\ensuremath{l}}}}{\binom{{\ensuremath{m}}}{{\ensuremath{l}}}} + p\cdot \frac{\binom{b}{{\ensuremath{k}}}}{\binom{{\ensuremath{m}}}{{\ensuremath{k}}}} \geq p\cdot \frac{\binom{b}{{\ensuremath{l}}+1}}{\binom{{\ensuremath{m}}}{{\ensuremath{l}}+1}} + p\cdot \frac{\binom{b}{{\ensuremath{k}}-1}}{\binom{{\ensuremath{m}}}{{\ensuremath{k}}-1}}, $$ and the strict inequality holds for at least one value $b$ that occurs with positive probability. The left sum of is the $2\cdot p$ fraction of the failure probability (by ${\ensuremath{{\rho}}}_z({\ensuremath{l}})$ and ${\ensuremath{{\rho}}}_z({\ensuremath{k}})$) before the change of ${\ensuremath{{\rho}}}_z$ under the condition that $B$ has cardinality $b$; the right sum is the corresponding fraction of the failure probability after the change. Depending on $b$ we have to distinguish several cases. $b={\ensuremath{m}}-1$. In this case both sides of are equal, i.e., the modification we do to ${\ensuremath{{\rho}}}_z$ will not change the success probability. ${\ensuremath{k}}\leq b < {\ensuremath{m}}-1$. Canceling $p$ and subtracting ${\binom{b}{{\ensuremath{k}}}}/{\binom{{\ensuremath{m}}}{{\ensuremath{k}}}}$ and ${\binom{b}{{\ensuremath{l}}+1}}/{\binom{{\ensuremath{m}}}{{\ensuremath{l}}+1}}$ from both sides of shows that the strict inequality holds if and only if $$\label{eq:intermediate} \begin{split} \frac{b\cdots(b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}}+1)}-\frac{b\cdots(b-{\ensuremath{l}})}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots(b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+2)}-\frac{b\cdots(b-{\ensuremath{k}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)} {\text{ }}. \end{split}$$ Factoring out $\frac{b\cdots(b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{l}}+1)}$ on the left side and $\frac{b\cdots(b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+2)}$ on the right side gives $$\frac{b\cdots (b-{\ensuremath{l}}+1)}{{\ensuremath{m}}\cdots ({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots (b-{\ensuremath{k}}+2)}{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)} \Leftrightarrow \frac{{\ensuremath{m}}\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)}{{\ensuremath{m}}\cdots ({\ensuremath{m}}-{\ensuremath{l}})}>\frac{b\cdots (b-{\ensuremath{k}}+2)}{b\cdots (b-{\ensuremath{l}}+1)}{\text{ }}.$$ Since ${\ensuremath{l}}\leq {\ensuremath{k}}-2$, this is equivalent to $$({\ensuremath{m}}-{\ensuremath{l}}+1)\cdot({\ensuremath{m}}-{\ensuremath{l}})\cdots({\ensuremath{m}}-{\ensuremath{k}}+1)>(b-{\ensuremath{l}})\cdot(b-{\ensuremath{l}}-1)\cdots(b-{\ensuremath{k}}+2){\text{ }},$$ which is true for $m-1>b$. ${\ensuremath{l}}\leq b<{\ensuremath{k}}$. Calculations along the lines of case 2 show that the strict inequality of also holds in this case. Note that $\binom{b}{k}$, $\binom{b}{k-1}$ and $\binom{b}{l+1}$ can be zero. $0\leq b<{\ensuremath{l}}$. In this case both sides of are zero, i.e., the modifications we do to ${\ensuremath{{\rho}}}_z$ will not change the success probability. Since in cases 1 and 4 above there was no change in the success probability, to show that cannot be maximal when ${\ensuremath{k}}-{\ensuremath{l}}\geq2$ as we are considering, it remains to show that at least one of the Cases 2 and 3 occurs with positive probability. We construct a situation in which one of these cases applies, and which occurs with positive probability. Choose degrees ${\ensuremath{k}}_x$ for all elements $x \in S-\{z\}$ such that ${\ensuremath{k}}_x\leq {\ensuremath{\kappa}}_x$ and ${\ensuremath{{\rho}}}_x({\ensuremath{k}}_x)>0$. Consider a permutation of the elements $x \in S-\{z\}$ such that these degrees are ordered, i.e., ${\ensuremath{k}}_{x_1}\leq {\ensuremath{k}}_{x_2}\leq \ldots \leq {\ensuremath{k}}_{x_{{\ensuremath{n}}-1}}$. Choose the first element $x_i$ with $i \geq {\ensuremath{l}}$ and ${\ensuremath{k}}_{x_i}\leq i$. Such an element must exist, since we assume ${\ensuremath{k}}_x\leq {\ensuremath{\kappa}}_x\leq {\ensuremath{n}}-2$, in particular we have $l<{\ensuremath{n}}-2$. Arrange that $A_{x_j}\subseteq [i],1\leq j \leq i,$ such that there is a matching in $G_{\{x_1,\ldots,x_i\}}$. This implies $b\geq {\ensuremath{l}}$. Then arrange that $|A_{x_j}-\bigcup_{1\leq {j'}<j} A_{x_{j'}}|=1$, for all $i<j\leq n-2$, as well as $|A_{x_{n-1}}-\bigcup_{1\leq {j'}<n-1} A_{x_{j'}}|=2$, which implies $b<m-1$. This finishes the proof of Proposition \[prop:constant\_expected\_degree\]. Performance results for the generalized selfless algorithm ========================================================== We present some performance results for the generalized selfless algorithm We ran the generalized selfless algorithm for hypergraphs with $10^5$ and $10^6$ nodes and tabulated the failure rate around the theoretical threshold values $c_{k,2}$ for $k=3,4,5$. For each pair $({\ensuremath{m}},k)$ we considered $81$ edge densities $c=\frac{{\ensuremath{n}}}{{\ensuremath{m}}}$, spaced apart by $0.0001$, thus covering an interval of length $0.008$, which encloses the theoretical threshold value for the particular parameter pair $({\ensuremath{m}},k)$. The hyperedges of the hypergraphs were randomly chosen via pseudo random number generator MT19937 “Mersenne Twister” of the GNU Scientific Library [@GNU_Scientific]. We measured the average failure rate of the algorithm over $100$ random hypergraphs for each combination $({\ensuremath{m}},{\ensuremath{n}},k)$ within the parameter space. To get an estimation of the threshold, i.e., the rate $c$ where the algorithm switches from success to failure, we fit the sigmoid function $$\label{eq:fit_function} \sigma(c;a,b)=\frac{1}{1+e^{-(c-a)/b}}$$ to the measured failure rate (via gnuplot[^8]), using the method of least squares. We determined the parameters $a,b$ that lead to a (local) minimum of the sum of squares of the $81$ residuals, denoted by $\sum_{res}$. The parameter $a$ is the inflection point of and therefore the approximation of the threshold of the generalized selfless algorithm. Figures \[fig:gen\_selfless\_edge\_size\_3\], \[fig:gen\_selfless\_edge\_size\_4\] and \[fig:gen\_selfless\_edge\_size\_5\] show the results of the experiments. One observes that this simple algorithm is able to construct the placements for edge densities quite close to the calculated thresholds $c_{k,2}$. The slope of the sigmoid curve increases and $\sum_{res}$ decreases with growing ${\ensuremath{m}}$ and $k$, leading to a sharp transition from total success to total failure. Clearly the algorithm can fail on hypergraphs that admit a matching. Experimental comparisons with a perfect matching algorithm [@HK:1973] showed that this is very unlikely for random hypergraphs. An example is given in Figure \[fig:selfless\_vs\_perfect\_matching\], which shows the failure rate of perfect matching in comparison to the generalized selfless algorithm. Note that the plot shows an interval of size $0.004$, i.e., 41 data points instead of $81$. The differences in the failure rates of the algorithms become very small as ${\ensuremath{m}}$ grows. Similarly, we find our generalized algorithm for the case where the bucket size $\ell$ is greater than 1 has similar behavior. For an example see Figure \[fig:gen\_selfless\_edge\_size\_3\_bucket\_size=2\]. Proof of the threshold for $k$-XORSAT ===================================== In this section we give a full proof of the threshold for $k$-XORSAT (Corollary \[corthresh\]). The proof employs the notation and facts developed in Sections \[sect:facts:cores\] and \[sec:main\], especially Propositions \[prop:two\] and \[prop:three\], and the following fact (known as “Friedgut’s Theorem” for $k$-XORSAT [@CD03; @CD09]). \[fact:Friedgut\] For every $k\geq 3$ there exists a function $c_k(m)\leq 1$ such that, for every ${\ensuremath{\varepsilon}}>0$ and a random formula $F$ (system $Ax=b$ of equations) from $\Phi^k_{m,n}$ we have the following: $$\lim_{m\to\infty} \Pr[\text{$F$ is satisfiable}]= \begin{cases} 1, \text{ if } n=c_k(m)(1-{\ensuremath{\varepsilon}})m\\ 0, \text{ if } n=c_k(m)(1+{\ensuremath{\varepsilon}})m {\text{ }}.\\ \end{cases}$$ Recall from Section \[sec:main\] that $\Phi^k_{m,n}$ can be regarded as a probability space whose elements are pairs $(A,b)$ where $A$ is an $n\times m$ matrix with entries in $\{0,1\}$, each row containing $k$ 1’s, and $b\in\{0,1\}^n$. Alternatively, $A$ can be regarded as a node-edge incidence matrix $A_G$ of a $k$-uniform hypergraph $G\in {\mathcal{G}^k_{m,n}}$. Via the obvious correspondence we identify ${\mathcal{G}^k_{m,n}}$ with the set of bipartite graphs $G$ with $n$ left nodes (“check nodes”) and $m$ right nodes (“variable nodes”) and degree $k$ at each left node. Similarly, $\Psi^k_{\hat{m},\hat{n}}$ is the probability space whose elements are pairs $(\hat{A},\hat{b})$, $\hat{b}\in\{0,1\}^{\hat{n}}$, where $\hat{A}$ is either the incidence matrix of a $k$-uniform hypergraph $H$ with $\hat{m}$ nodes, $\hat{n}$ edges, and minimum degree 2 or the adjacency matrix $\hat{A}_H$ of a bipartite graph $H$ with $\hat{n}$ left nodes and $\hat{m}$ right nodes, with degree $k$ at each left node and minimum degree 2 at each right node. We use the same notation for both and let ${\mathcal{H}^k_{\hat{m},\hat{n}}}$ be the set of all these graphs.[^9] The following lemma is central. \[lemma:Z2\] For any $\delta>0$ there exists ${\ensuremath{\varepsilon}}= {\ensuremath{\varepsilon}}(\delta)>0$ such that the following happens. Let $H\in{\mathcal{H}^k_{\hat{m},\hat{n}}}$ be uniformly random with $\hat{n}<\hat{m}(1-\delta)$ and denote by $Z_{H}$ the number of solutions of the linear system $\hat{A}_Hx = 0$ (over $\mathrm{GF}[2]$). Then $$\begin{aligned} \Pr[Z_{H} = 2^{\hat{m}-\hat{n}}]\ge {\ensuremath{\varepsilon}}(\delta)>0 \, . $$ We note that a full proof of this lemma for the special case $k=3$, with $\Pr[Z_{H} = 2^{\hat{m}-\hat{n}}]=1-o(1)$, was given in [@DubMan:2002]. Consider the following two cases. $c^*_{k,2}<c<c_{k,2}$. Let a system $A_Gx=b$ be chosen at random from $\Phi^k_{m,n}$. Reducing $G$ to its 2-core $H$ leads to a system $\hat{A}_Hx=\hat{b}$ with $\hat{m}$ variables, $\hat{n}$ equations, and $\operatorname{rank}(\hat{A}_H)=\operatorname{rank}(A_G)-(m-\hat{m})$. The graph $H$ is random in ${\mathcal{H}^k_{\hat{m},\hat{n}}}$. By Propositions \[prop:two\] and \[prop:three\], with high probability $\hat{n}\leq (1-\delta)\hat{m}$ for some $\delta=\delta(c)$, and $\hat{m}=\Theta(m)$. By Lemma \[lemma:Z2\], for $m$ large enough, we get $\Pr[Z_{H} = 2^{\hat{m}-\hat{n}}]\geq {\ensuremath{\varepsilon}}(\delta)>0$. This implies $\Pr[A_Gx=b\text{ is satisfiable}] \geq \Pr[A_G \text{ has full row rank}]=\Pr[Z_G=2^{m-n}]\geq {\ensuremath{\varepsilon}}(\delta)$. $c>c_{k,2}$. Let $A_Gx=b$ and its reduced version $\hat{A}_Hx=\hat{b}$ be as in (i). By Propositions \[prop:two\] and \[prop:three\], with high probability $\hat{n}\geq (1+\delta)\hat{m}$ for some $\delta=\delta(c)$, and $\hat{m}=\Theta(m)$. We have $\operatorname{rank}(\hat{A})\leq \hat{m}$, and by the randomness of $\hat{b}$ we have $\Pr[\hat{A}_Hx=\hat{b}\text{ is satisfiable}]\leq2^{\hat{m}-\hat{n}}\leq2^{-\delta\hat{m}}$. Combining parts (i) and (ii) with Friedgut’s Theorem (Fact \[fact:Friedgut\]) shows that $\lim_{m\to\infty} c_k(m)=c_{k,2}$, which implies Corollary \[corthresh\]. We now move to the proof of Lemma \[lemma:Z2\], which focuses on the 2-core $H$ of the graph $G$, and we condition on its number of nodes. With a slight abuse of notation we will drop the “hat” from our notations. In other words, we now let $H$ be a uniformly random graph from ${\mathcal{H}^k_{m,n}}$ and let $\gamma = n / m$ (see Theorem \[thresh\]). It is convenient to introduce some additional notation. Given a formal series $p(z)$, $\operatorname{coeff}[p(z),z^r]$ denotes the coefficient of $z^r$ in $p(z)$. We further introduce the notations $$\begin{aligned} &q(z) = (e^{z}-1-z)\, ,\;\;\;\;\;\;\;\;\;\;\; Q(z) = \frac{z q'(z)}{q(z)}\, ,\\ &p_k(z) = \frac{1}{2}(1+z)^k+ \frac{1}{2}(1-z)^k\, ,\;\;\;\;\;\;\; P_k(z) = \frac{z p_k'(z)}{p_k(z)}\, . $$ It is easy to see that $z\mapsto Q(z)$ is a strictly increasing function with $\lim_{z\to 0}Q(z)=2$, and $\lim_{z\to\infty}Q(z) = \infty$. Further $z\mapsto P_k(z)$ is strictly increasing with $\lim_{z\to 0}P_k(z)=0$, and $\lim_{z\to\infty}P_k(z) = k$ for $k$ even, and $k-1$ otherwise. Further we define the domain sets $$\begin{aligned} {\mathcal{D}_{m,n}}&= &\big\{(w,l)\in{\mathbb{Z}}^2\,:\, 0\le w\le m\, , 2w\le l\le kn-2(m-w)\, , l\mbox{ even }\big\}\, , \\ {\mathcal{D}_{\gamma}}({\ensuremath{\varepsilon}})&= &\big\{(\omega,\lambda)\in{\mathbb{R}}^2\,:\, {\ensuremath{\varepsilon}}\le \omega\le 1-{\ensuremath{\varepsilon}}\, , \tfrac{2\omega}{k\gamma}+{\ensuremath{\varepsilon}}\le \lambda\le 1-\tfrac{2(1-\omega)}{k\gamma}-{\ensuremath{\varepsilon}}\big\}\, ,\\ {\mathcal{D}_{m,n}}({\ensuremath{\varepsilon}})&= &\big\{(w,l)\in{\mathcal{D}_{m,n}}\,:\, (\tfrac{w}{m},\tfrac{l}{kn})\in {\mathcal{D}_{\gamma}}({\ensuremath{\varepsilon}})\big\}\, ,\\ {\overline{\mathcal{D}}_{m,n}}({\ensuremath{\varepsilon}})&= & {\mathcal{D}_{m,n}}\setminus {\mathcal{D}_{m,n}}({\ensuremath{\varepsilon}})\, . $$ The assertion of Lemma \[lemma:Z2\] now follows from the following sequence of lemmas, to be proven in the subsections below. \[lemma:Comb\] Let $Z_{H}$ be the number of solutions of the linear system $A_Hx=0$. Then $$\begin{aligned} \operatorname{E}[Z_H] & = \frac{1}{N_0}\sum_{(w,l)\in{\mathcal{D}_{m,n}}} N(w,l)\, , $$ where we define $$\begin{aligned} N_0 & = (kn)!\,\operatorname{coeff}[(e^z-1-z)^m,z^{kn}]\, ,\\ N(w,l) & = \binom{m}{w}\, l!(kn-l)!\, \operatorname{coeff}[(e^z-1-z)^w,z^l]\,\operatorname{coeff}[(e^z-1-z)^{m-w},z^{kn-l}]\, \operatorname{coeff}[p_k(z)^n,z^l]\, . $$ \[lemma:Boundary\] For any $\delta>0$ there exists ${\ensuremath{\varepsilon}}>0$ such that, if $n\le m(1-\delta)$, then $$\begin{aligned} \frac{1}{N_0}\cdot\sum_{(w,l)\in{\overline{\mathcal{D}}_{m,n}}({\ensuremath{\varepsilon}})}\! \! N(w,l)\le 2^{m\delta}\, . $$ \[lemma:Exp\] For any $\delta>0, {\ensuremath{\varepsilon}}>0$ there exists $C= C(\delta,{\ensuremath{\varepsilon}})$ such that, if $m\delta\le n\le m(1-\delta)$ and $(w,l)\in{\mathcal{D}_{m,n}}({\ensuremath{\varepsilon}})$, then $$\begin{aligned} \frac{N(w,l)}{N_0}\le \frac{C}{m}\, \exp\Big(m\,\psi\big(\tfrac{w}{m},\tfrac{l}{kn}\big) \Big)\, , $$ where, letting $h(z) = -z\log z-(1-z)\log(1-z)$,[^10] we define $$\begin{aligned} \psi(\omega,\lambda) &= h(\omega)-k\gamma\, h(\lambda)- \log q(s)+ k\gamma\log s\label{eq:Phi}\\ &+\omega \log q(a)-k\gamma\lambda \log a+(1-\omega)\log q(b)-k\gamma(1-\lambda) \log b\nonumber\\ & +\gamma\log p_k(c)-k\gamma\lambda \log c\, .\nonumber $$ Finally, $a=a(\omega,\lambda)$, $b = b(\omega,\lambda)$, $c = c(\omega,\lambda)$, and $s$ are the unique non-negative solutions of $$\begin{aligned} Q(s) = k\gamma\, , \;\;\;\;\; Q(a) = \frac{k\gamma\lambda}{\omega}\, , \;\;\;\;\;Q(b) = \frac{k\gamma(1-\lambda)}{(1-\omega)}\, , \label{Qeq}\\ P_k(c) = k\lambda\, .\label{Peq} $$ \[lemma:Calculus\] For any $\gamma<1$, the function $\psi: {\mathcal{D}_{\gamma}}(0)\to{\mathbb{R}}$ achieves its unique global maximum at $(\omega,\lambda) =(1/2,1/2)$, with $\psi(1/2,1/2) = (1-\gamma)\log 2$. Further, there exists $\xi>0$ such that $-\operatorname{Hess}_{\psi}(1/2,1/2)\succeq\, \xi\, I_2$.[^11] Finally, let us recall a well known fact about lattice sums (see for instance [@BR]). \[lemma:Sum\] Let $D$ be an open domain in ${\mathbb{R}}^d$, and $F: D\to{\mathbb{R}}$ be continuously differentiable, achieving its unique maximum in $z_*\in D$, with $\operatorname{Hess}_F(z_*)\succeq \xi I_{d}$ for some $\xi>0$. Then there exists $C>0$ such that, for any $\delta\ge 0$ $$\begin{aligned} \sum_{x\in {\mathbb{Z}}^d\, :\; x\,\delta \in D} \exp\Big( \tfrac{1}{\delta}F(\delta x)\Big) \le \frac{C}{\delta^{d/2}}\, \exp\Big(\tfrac{1}{\delta}F(z_*)\Big)\, . $$ The proof is simply obtained by putting together Lemmas \[lemma:Comb\], \[lemma:Boundary\], \[lemma:Exp\], \[lemma:Calculus\] and using Lemma \[lemma:Sum\] (with $F(x_1,x_2)= \psi(x_1,2x_2/(k\gamma))$, $d=2$ and $\delta = 1/n$) to bound the sum. Proof of Lemma \[lemma:Comb\] ----------------------------- Clearly $N_0$ is the number of graphs in ${\mathcal{H}^k_{m,n}}$. Indeed it is the number of way of putting $nk$ distinct balls in $m$ bins in such a way that each bin contains at least $2$ balls. The claim follows by proving that, for each $(w,l)\in{\mathcal{D}_{m,n}}$, $N(w,l)$ is the number of couples $(H,x)$ where $H\in {\mathcal{H}^k_{m,n}}$ and $x\in\{0,1\}^m$ with $A_Hx = 0 \mod 2$, such that $x$ has $w$ ones and $H$ has $l$ edges incident on variable (right) nodes $i$ such that $x_i=1$. Indeed, $\binom{m}{w}$ gives the number of ways of choosing the ones. Paint by red the $l$ edges incident on these nodes, and by blue the other $(kn-l)$ edges. The coefficient factors give the number of ways of attributing red/blue edges to nodes on the two sides. The factorials give the number of ways of matching edges of the same color on the two sides. [$\square$]{} Proof of Lemma \[lemma:Exp\] ---------------------------- Let us start by proving a lower bound on $N_0$. For any $s>0$, we have $$\begin{aligned} N_0 = (kn)! \frac{q(s)^m}{s^{kn}}\, \Pr_{s}\Big[\sum_{i=1}^mX_i =kn\Big] \, , $$ where $X_1,\dots, X_m$ are i.i.d. Poisson random variable (with parameter $s$) conditioned to $X_i\ge 2$, i.e., for any $q\ge 2$, $$\begin{aligned} \Pr_s[X_i = q] = \frac{1}{e^s-1-s}\, \frac{s^q}{q!}\, . $$ By assumption $s$ is chosen such that $\operatorname{E}_s[X_i]=Q(s)=k\gamma\in (k\delta,k(1-\delta))$. By the local central limit theorem for lattice random variables of [@BR Corollary 22.3], we have $\Pr_{s}\big[\sum_{i=1}^mX_i =kn\big] \ge C'/\sqrt{m}$ for sone constant $C'(\delta)$, whence, using Stirling’s formula $$\begin{aligned} N_0 \ge C_1(\delta) \left(\frac{kn}{e}\right)^{kn} \frac{q(s)^m}{s^{kn}}\, .\label{eq:BoundN0} $$ Consider now $N(w,l)$. By the central limit theorem for the sum of Bernoulli random variables, for any $m\delta\le w\le m(1-\delta)$, we have $$\begin{aligned} \binom{m}{w}\le \frac{C_2(\delta)}{\sqrt{m}}\, e^{mh(w/m)}\, . $$ Treating the coefficient terms as above, and using Stirling’s formula for $l,(kn-l)=\Theta(m)$, we get $$\begin{aligned} N(w,l) \le \frac{C_3(\delta)}{m}\,e^{mh(w/m)}\, \left(\frac{l}{e}\right)^l \left(\frac{knl}{e}\right)^{(kn-l)}\frac{q(a)^w}{a^l}\, \frac{q(b)^{m-w}}{b^{kn-l}}\, \frac{p_k(c)^n}{c^l}\, .\label{eq:BoundNwl} $$ The claim is proved by taking the ratio of the bounds (\[eq:BoundNwl\]) and (\[eq:BoundN0\]). [$\square$]{} Proof of Lemma \[lemma:Calculus\], outline ------------------------------------------ We now present an outline of the proof of Lemma \[lemma:Calculus\]. Appendix \[app:fullproof5\] contains the additional details for a complete proof. For $(\omega,\lambda)=(1/2,1/2)$, Eqs. (\[Qeq\]), (\[Peq\]) admit the unique solution $a=b=s$ and $c=1$. A straightforward calculation yields $\psi(1/2,1/2) = (1-\gamma)\log 2$. Call $\Psi(\omega,\lambda;a,b,c)$ the right hand side of Eq. (\[eq:Phi\]). Notice that the derivatives of $\Psi$ with respect to $a,b,c$ vanish by Eqs. (\[Qeq\]), (\[Peq\]). Therefore it is easy to compute the partial derivatives $$\begin{aligned} \frac{\partial \psi}{\partial \omega} & = & \log \frac{1-\omega}{\omega} +\log\frac{q(a)}{q(b)}\, ,\\ \frac{\partial \psi}{\partial \lambda} & = & -k\gamma \log \frac{1-\lambda}{\lambda}-k\gamma\log\frac{a}{b}-k\gamma\log c\, . $$ Using the fact that $a=b=s$ and $c=1$ at $(\omega,\lambda)=(1/2,1/2)$, we get that the gradient of $\psi$ vanishes at $(1/2,1/2)$, and again, $\psi(1/2,1/2) = (1-\gamma)\log 2$. By a somewhat longer calculation, we obtain the following second derivatives $$\begin{aligned} \left.\frac{\partial^2 \psi}{\partial \omega^2} \right|_{1/2,1/2} & = & -4\,\Big(1+\frac{(k\gamma)^2}{s^2C}\Big)\, ,\\ \left. \frac{\partial^2 \psi}{\partial \lambda\partial\omega}\right|_{1/2,1/2} & = & 4\, \frac{(k\gamma)^2}{s^2C}\, , \\ \left.\frac{\partial^2 \psi}{\partial \omega^2} \right|_{1/2,1/2} & = & -4\, \frac{(k\gamma)^2}{s^2C}\, , $$ with $$\begin{aligned} C= \frac{q''(s)}{q(s)}-\frac{q'(s)^2}{q(s)^2}+\frac{k\gamma}{s^2}>0\, . $$ It is easy to deduce that $-\operatorname{Hess}_\psi(1/2,1/2)$ is positive definite. The function $\psi:{\mathcal{D}_{\gamma}}(0)\to{\mathbb{R}}$ is continuous in ${\mathcal{D}_{\gamma}}(0)$ and differentiable in its interior. Further, we have the following asymptotic behaviors (first two at fixed $\lambda$, second two at fixed $\omega$): $$\begin{aligned} \lim_{\omega\to 0}&\frac{\partial\psi}{\partial\omega}=+ \infty\, , \;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;\; \lim_{\omega\to 1}\frac{\partial\psi}{\partial\omega}=- \infty\, ,\\ \lim_{\lambda\to 2\omega/(k\gamma)} &\frac{\partial\psi}{\partial\lambda}=+ \infty\, , \;\;\;\;\;\;\;\;\;\; \lim_{\lambda\to 1-2(1-\omega)/(k\gamma)} \frac{\partial\psi}{\partial\lambda}=- \infty\, . $$ Therefore any global maximum of $\psi$ must be a stationary point in the interior of ${\mathcal{D}_{\gamma}}(0)$. We next will prove that $(1/2,1/2)$ is the only such point. Notice that $\Psi(\omega,\lambda;a,b,c)$ is convex with respect to $a,b,c$. As a consequence $$\begin{aligned} \psi(\omega,\lambda)=\min_{a,b,c}\Psi(\omega,\lambda;a,b,c)\, . \label{eq:Variational} $$ We will construct an upper bound on $\psi$ by choosing $a,b,c$ appropriately. The first remark is that $$\begin{aligned} \Psi(1-\omega,1-\lambda;b,a,1/c) = \Psi(\omega,\lambda;a,b,c) -\gamma\, \log \frac{p_k(c)}{c^{k}p_k(1/c)} \, . $$ Since, for $c\in[0,1]$ (which is guaranteed by Eq. (\[Peq\]) for $\lambda\in [0,1/2]$) we have $p_k(c)\ge c^{k}p_k(1/c)$, we can restrict without loss of generality to $\lambda\le 1/2$ (whence $c\in [0,1]$). Next notice that, maximizing $\Psi$ over $\omega$, we get $\Psi(\omega,\lambda;a,b,c)\le \Psi_1(\lambda;a,b,c)$, where $$\begin{aligned} \Psi_1(\lambda;a,b,c) = & \log\big(q(a)+q(b)\big)-k\gamma\, h(\lambda)- \log q(s)+ k\gamma\log s\\ &-k\gamma\lambda \log a-k\gamma(1-\lambda) \log b +\gamma\log p_k(c)-k\gamma\lambda \log c\, .\nonumber $$ Next fix $c = c(\lambda) = b\lambda/(a-a\lambda)$. Since this transformation is invertible, we can as well keep $c$ as a free parameter, and let $\lambda =ac/(ac+b)$. If we let $\Psi_2(a,b,c) = \Psi_1(ac/(ac+b);a,b,c)$, we get $$\begin{aligned} \Psi_2(a,b,c) = & \log\big(q(a)+q(b)\big)- \log q(s)+\gamma\log p_k(c)\\ &-k\gamma\log(ac+b)+k\gamma\lambda \log s\, .\nonumber\end{aligned}$$ Also, without loss of generality, we can rescale $a$ by a factor $s$, and set $b=s$, therefore defining $\Psi_3(a,c)=\Psi_2(sa,s,c)$. If we introduce the notation $$\begin{aligned} \Lambda_s(x)=\frac{q(sx)}{q(s)} = \frac{e^{sx}-1-sx}{e^{s}-1-s}\,, $$ we get the expression $$\begin{aligned} \Psi_3(a,c) = -k\gamma \log(1+ac) +\log\big(1+\Lambda_s(a)\big)+\gamma\log p_k(c)\, . $$ By the above derivation we have the following relation with $\psi(\omega,\lambda)$: $$\begin{aligned} &\psi(\omega,\lambda)\le \left.\Psi_4(c)\right|_{c=\lambda/a_*(c)(1-\lambda)} \, ,\\ &\Psi_4(c) = \Psi_3(a_*(c),c)\, ,\;\;\;\;\;\;\; a_*(c) = \arg\min_{a\ge 0}\Psi_3(a,c)\, . $$ A direct calculation shows that $a_*(1) = 1$ and $\Psi_3(1,1) = (1-\gamma)\log 2$. This point corresponds to $(\omega,\lambda)=(1/2,1/2)$ through the above derivation. We will show that $c=1$ is indeed the global maximum of $\Psi_4(c)$ for $c\in [0,1]$, which implies the assertion. Maximizing $\Psi_3(a,c)$ with respect to $c$ implies $a_*(c)$ to be the unique non-negative solution of the stationarity condition $$\begin{aligned} a = \frac{(1+c)^{k-1}-(1-c)^{k-1}}{(1+c)^{k-1}+(1-c)^{k-1}}\, . \label{Beq} $$ On the other hand, the stationarity condition with respect to $a$ yields $$\begin{aligned} c = \frac{\lambda_s(a)}{1+\Lambda_s(a)-a\lambda_s(a)}\, . \label{Ceq} $$ where we used the fact that $\Lambda_s'(1) = k\gamma$ and defined $\lambda_s(x) = \Lambda_s'(x)/\Lambda_s'(1)$. Equations (\[Beq\]) and (\[Ceq\]) admit the solutions $a=c=0$ and $a=c=1$, and is easy to check that these are both local maxima of $\Psi_4$. We will show that they admit only one more solution with $c\in(0,1)$, that necessarily is a local minimum of $\Psi_4$. Indeed, if we let $a=\tanh x$, $c= \tanh y$, Eq. (\[Beq\]) becomes $$\begin{aligned} x = (k-1) y\, . $$ Our claim is therefore implied by Lemma \[lemma:Elementary\] below. [$\square$]{} \[lemma:Elementary\] For $s> 0$, let $$\begin{aligned} \Lambda_s(t) = \frac{e^{st}-1-st}{e^{s}-1-s}\, , \;\;\;\;\;\;\; \lambda_s(t) = \frac{e^{st}-1}{e^{s}-1}\, . $$ Define $F_s:{\mathbb{R}}\to{\mathbb{R}}$ by $$\begin{aligned} F_s(x) = \operatorname{atanh}\big(f(\tanh x)\big)\, ,\;\;\;\;\;\;\;\;\; f_s(t)= \frac{\lambda_s(t)}{1+\Lambda_s(t)-t\lambda_s(t)}\, . $$ Then $F_s$ is convex on $[0,\infty)$. This can be seen simply by graphing $F_s(x)$, or by some calculus which we omit. Proof of Lemma \[lemma:Boundary\] --------------------------------- The proof is analogous to the one of Lemma \[lemma:Exp\]. We have just to be careful to the values of $w,l$ near the boundary of the domain ${\mathcal{D}_{m,n}}$. Luckily we only need a loose upper bound. Equation (\[eq:BoundN0\]) remains true in the present case (as it only hinges on $n=\Theta(m)$). On the other hand using $\binom{m}{w}\le \exp(mh(w/m))$, $\operatorname{coeff}[f(x)^k,x^l]\le f(a)^k/a^l$ and $m!\le \sqrt{2\pi}\, (m/e)^{m+1/2}$, we get $$\begin{aligned} N(w,l) \le 2\pi\, e^{-kn-1} e^{mh(w/m)}\, l^{l+1/2} (kn-l)^{(kn-l+1/2)}\frac{q(a)^w}{a^l}\, \frac{q(b)^{m-w}}{b^{kn-l}}\, \frac{p_k(c)^n}{c^l}\, . $$ for any $a,b,c>0$. Taking the ratio, and bounding polynomial factors $\sqrt{l(kn-l)}\le C m$ we get $$\begin{aligned} \frac{N(w,l)}{N_0}\le C\, m\, \exp\big(m\psi(w/m,\lambda/kn)\big)\, , $$ whence $$\begin{aligned} \frac{1}{N_0}\sum_{(w,l)\in{\overline{\mathcal{D}}_{m,n}}({\ensuremath{\varepsilon}})} N(w,l)\le Cm^3\, \exp\big(m\sup\{ \psi(\omega,\lambda):\, (\omega,\lambda)\in {\mathcal{D}_{\gamma}}(0)\setminus {\mathcal{D}_{\gamma}}({\ensuremath{\varepsilon}})\} \big) $$ with $\psi(\omega,\lambda)$ defined as in Eq. (\[eq:Phi\]). Notice that $\psi:{\mathcal{D}_{\gamma}}(\delta)\to {\mathbb{R}}$ is a continuous function. It is therefore sufficient to show that it is strictly smaller than $(1-\gamma)\log 2$ on the boundaries of its domain. This indeed follows from Lemma \[lemma:Calculus\]. [$\square$]{} [^1]: Fakultät für Informatik und Automatisierung, Technische Universität Ilmenau. Research supported by DFG grant DI 412/10-1. [{martin.dietzfelbinger,michael.rink}@tu-ilmenau.de]{} [^2]: Fakultät für Informatik, Technische Universität Chemnitz. [[email protected]]{} [^3]: Harvard University, School of Engineering and Applied Sciences. Part of this work was done while visiting Microsoft Research New England. [[email protected]]{} [^4]: Department of Electrical Engineering and Department of Statistics, Stanford University. Part of this work was done while visiting Microsoft Research New England. [[email protected]]{} [^5]: Efficient Computation group, IT University of Copenhagen. [[email protected]]{} [^6]: We could in principle also consider the possibility of keys having only a single choice. However, this is generally not very interesting since even a small number of keys with a single choice would make an assignment impossible whp., by the birthday paradox. Hence, we restrict our attention to at least two choices. [^7]: gnuplot, an interactive plotting program, version 4.2, <http://www.gnuplot.info> [^8]: gnuplot, an interactive plotting program, version 4.2, <http://www.gnuplot.info> [^9]: For simplicity we assume that for each left node a sequence of $k$ right nodes is chosen at random, allowing and ignoring repetitions. The difference from $k$-uniform hypergraphs is negligible. [^10]: $\log$ means logarithm to the base $e$ [^11]: $\operatorname{Hess}_\psi$ denotes the Hessian matrix of $\psi$ and $I_2$ the $2\times 2$ unit matrix

ArXiv

--- abstract: | We report the discovery of a very cool d/sdL7+T7.5p common proper motion binary system, SDSS J1416+13AB, found by cross-matching the UKIDSS Large Area Survey Data Release 5 against the Sloan Digital Sky Survey Data Release 7. The d/sdL7 is blue in J-H and H-K and has other features suggestive of low-metallicity and/or high gravity. The T7.5p displays spectral peculiarity seen before in earlier type dwarfs discovered in UKIDSS LAS DR4, and referred to as CH$_4$-J-early peculiarity, where the CH$_4$-J index, based on the absorption to the red side of the $J$-band peak, suggests an earlier spectral type than the H$_2$O-J index, based on the blue side of the $J$-band peak, by $\sim 2$ subtypes. We suggest that CH$_4$-J-early peculiarity arises from low-metallicity and/or high-gravity, and speculate as to its use for classifying T dwarfs. UKIDSS and follow-up UKIRT/WFCAM photometry shows the T dwarf to have the bluest near-infrared colours yet seen for such an object with $H-K = -1.31 \pm 0.17$. Warm [*Spitzer*]{} IRAC photometry shows the T dwarf to have extremely red $H - [4.5]~=~4.86 \pm 0.04$, which is the reddest yet seen for a substellar object. The lack of parallax measurement for the pair limits our ability to estimate parameters for the system. However, applying a conservative distance estimate of 5–15 pc suggests a projected separation in range 45–135 AU. By comparing $H - K:H - [4.5]$ colours of the T dwarf to spectral models we estimate that $T_{\rm eff} = 500$ K and \[M/H\]$\sim -0.30$, with $\log g \sim 5.0$. This suggests a mass of $\sim$30 M$_{Jupiter}$ for the T dwarf and an age of $\sim$10 Gyr for the system. The primary would then be a 75 M$_{Jupiter}$ object with $\log~g \sim 5.5$ and a relatively dust-free $T_{\rm eff} \sim 1500$K atmosphere. Given the unusual properties of the system we caution that these estimates are uncertain. We eagerly await parallax measurements and high-resolution imaging which will constrain the parameters further. author: - | Ben Burningham$^{1}$[^1], S. K. Leggett$^{2}$, P.W. Lucas$^{1}$, D.J. Pinfield$^{1}$, R.L. Smart$^{3}$, A.C. Day-Jones$^{4}$, H.R.A. Jones$^1$, D.Murray$^1$ E. Nickson$^{5,1}$, M. Tamura$^{6}$, Z. Zhang$^{1}$, N. Lodieu$^{7}$, C.G. Tinney$^{8}$, M. R. Zapatero Osorio$^{9}$\ $^{1}$ Centre for Astrophysics Research, Science and Technology Research Institute, University of Hertfordshire, Hatfield AL10 9AB\ $^{2}$ Gemini Observatory, 670 N. A’ohoku Place, Hilo, HI 96720, USA\ $^{3}$ Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Torino, Strada Osservatrio 20, 10025 Pino Torinese, Italy\ $^{4}$ Universidad de Chile,Camino el Observatorio \# 1515, Santiago, Chile, Casilla 36-D\ $^{5}$ University of Southampton, Southampton, UK\ $^{6}$ National Astronomical Observatory, Mitaka, Tokyo 181-8588\ $^{7}$ Instituto de Astrofísica de Canarias, 38200 La Laguna, Spain\ $^{8}$ School of Physics, University of New South Wales, 2052. Australia\ $^{9}$ Centro de Astrobiología (CSIC-INTA), E-28850 Torrejón de Ardoz, Madrid, Spain\ bibliography: - 'refs.bib' title: The discovery of a very cool binary system --- surveys - stars: low-mass, brown dwarfs Introduction {#sec:intro} ============ The current generation of wide-field surveys [e.g. UKIRT Infrared Deep Sky Survey, UKIDSS; Canada-France Brown Dwarf Survey, CFBDS; @ukidss; @cfbds] is significantly expanding the sample of late type T dwarfs [e.g. @delorme08; @lod07; @pinfield08; @ben10]. Recent discoveries of extremely cool T dwarfs probe new low-temperature extremes, with $T_{\rm eff}$ as low as 500K [@ben08; @delorme08; @ben09; @sandy09]. In addition to probing new $T_{\rm eff}$ regimes, we can expect the expanded sample to populate other hitherto unexplored regions of T dwarf parameter space. Of particular interest is the growing diversity seen in metallicity and gravity for late-T dwarfs [e.g. @sandy10], and the potential for extending the low-metallicity subdwarf sequence to very low temperatures. To date, the sample of ultracool subdwarfs (UCSDs) consists of just one proposed T subdwarf, 2MASS J09373487+2931409 [@burgasser02; @burgasser06], along with a small number of L subdwarfs [e.g. 2MASS J1626+3925 - sdL4; SDSS J1256–0224 - sdL4; 2MASS J0616–6407 - sdL5; ULAS J1350+0815 - sdL5; 2MASS J0532+8246 - sdL7; @burgasser04a; @sivarani09; @cushing09; @lodieu2009; @burgasser03 respectively]. Recent parallax determinations and model comparisons by @schilbach09 suggest that of these, only the earliest type objects (2MASS J1626+39 and SDSS J1256-02) have metallicities consistent with subdwarf classification on the scheme that @gizis97 defined for M subdwarfs. Based on this @schilbach09 suggest that an intermediate d/sd classification should be applied to the two coolest objects (2MASS J0532+82 and 2MASS J0937+29). It is important to remember, however, that the subdwarf classification scheme is empirically based, and metallicities are associated with specific subdwarf classes only by model comparisons. That the model comparisons for the latest type UCSDs suggest higher metallicities than seen for earlier type objects should not be a sole basis for reclassification. The higher metallicity inferred from the colours of the coldest objects may actually highlight problems with the models to which they are compared. The spectral classification of subdwarfs should be based on observed spectral features that distinguish these objects from “normal” ultracool dwarfs (UCDs). As such, in this paper we adopt the position that the sdL objects described above are subdwarfs, since their spectra are clearly distinct from those of the bulk population of L dwarfs in a manner broadly consistent with subdwarf status. The more limited sample of T dwarfs, however, precludes such classification at this time, and we adopt the “peculiar” description for possible subdwarfs of this type [e.g. @burgasser06; @ben10]. However, in both cases the limited sample of “subdwarf” objects means that the current classification system may require significant revision as the true diversity of the spectra of low-metallicity UCDs becomes apparent in the era of larger, deeper surveys such as VISTA and WISE. We report here the discovery of a nearby d/sdL7+T7.5p common proper motion binary. The rest of the paper is laid out as follows. In Section \[sec:ident\] we describe the identification, photometric follow-up, spectral classification and proper motion determination for the two objects. In Section \[sec:binary\] we demonstrate their association as a common proper motion binary pair, and we provide initial estimates for some of their properties in Section \[sec:properties\]. Our results and conclusions are summarised in Section \[sec:summ\]. Two new ultracool dwarfs {#sec:ident} ======================== Our searches of the UKIDSS Large Area Survey [LAS; see @ukidss] have been successful at identifying late-type T dwarfs [e.g. @lod07; @warren07; @pinfield08; @ben08; @ben09; @ben10]. Using the same search methodology as previously described in detail in @pinfield08 and @ben10, we identified ULAS J141623.94+134836.30 (hereafter ULAS J1416+13) as a candidate late-T dwarf in Data Release 5 of the LAS with unusually blue $H-K =-1.35$. The subsequent photometric and spectroscopic follow-up, which resulted in its classification as a T7.5p dwarf, are described in the following sub-sections. Inspection of the surrounding field in SDSS, required to establish the red nature of ULAS J1416+13, revealed the presence of a nearby, very red object at a separation of 9. Interrogation of SDSS DR7 revealed this object, SDSS J141624.08+134826.7 (hereafter SDSS J1416+13), to have an SDSS spectrum with L dwarf spectral morphology (see also Table \[tab:optmags\] for SDSS photometry of this object). Since our initial identification of this L dwarf, its discovery has been published by @schmidt10 and @bowler10, who have classified it as a blue L5 and L6pec $\pm 2$ dwarf respectively. In the following sub-sections, we also describe our follow-up photometry of this target, and describe our analysis of this source that was carried out independently prior to the @schmidt10 and @bowler10 publications. Figure \[fig:finder\] shows a UKIDSS $J$ band finding chart for both the L and T dwarf. ![A 1’$\times$1’ $J$ band finding chart for ULAS J1416+13 and SDSS J1416+13 taken from the UKIDSS database. []{data-label="fig:finder"}](Jfinder_11.ps){height="200pt"} Near-infrared photometry {#subsec:photo} ------------------------ Near-infrared follow-up photometry was obtained using the Wide Field CAMera [WFCAM; @wfcam] on UKIRT on the night of 17$^{th}$ June 2009, and the data were processed using the WFCAM science pipeline by the Cambridge Astronomical Surveys Unit (CASU) [@irwin04], and archived at the WFCAM Science Archive [WSA; @wsa]. Observations consisted of a three point jitter pattern in the $Y$ and $J$ bands, and five point jitter patterns in the $H$ and $K$ bands repeated twice, all with 2x2 microstepping and individual exposures of 10 seconds resulting in total integration times of 120 seconds in $Y$ and $J$ and 400 seconds in $H$ and $K$. The resulting photometry for both our targets is given in Table \[tab:nirmags\]. The WFCAM filters are on the Mauna Kea Observatories (MKO) photometric system [@mko] Object $u'$ $g'$ $r'$ $i'$ $z'$ $g'-r'$ $r'-i'$ $i'-z'$ --------------- ------------------ ------------------ ------------------ ------------------ ------------------ ----------------- ----------------- ----------------- SDSS J1416+13 $23.55 \pm 0.57$ $23.08 \pm 0.18$ $20.69 \pm 0.04$ $18.38 \pm 0.01$ $15.92 \pm 0.01$ $2.39 \pm 0.19$ $2.31 \pm 0.04$ $2.46 \pm 0.01$ Object $Y$ $J$ $H$ $K$ $Y-J$ $J-H$ $H-K$ --------------- ------------------ ------------------ ------------------ ------------------ ----------------- ------------------ ------------------ -- -- -- -- ULAS J1416+13 $18.16 \pm 0.02$ $17.26 \pm 0.02$ $17.58 \pm 0.03$ $18.93 \pm 0.24$ $0.90 \pm 0.03$ $-0.32 \pm 0.03$ $-1.35 \pm 0.25$ $18.13 \pm 0.02$ $17.35 \pm 0.02$ $17.62 \pm 0.02$ $18.93 \pm 0.17$ $0.78 \pm 0.03$ $-0.27 \pm 0.03$ $-1.31 \pm 0.17$ SDSS J1416+13 $14.25 \pm 0.01$ $12.99 \pm 0.01$ $12.47 \pm 0.01$ $12.05 \pm 0.01$ $1.26 \pm 0.01$ $0.52 \pm 0.01$ $0.42 \pm 0.01$ $14.28 \pm 0.01$ $13.04 \pm 0.01$ $12.49 \pm 0.01$ $12.08 \pm 0.01$ $1.24 \pm 0.01$ $0.55 \pm 0.01$ $0.41 \pm 0.01$ Warm-Spitzer IRAC photometry {#subsec:irac} ---------------------------- The [*Spitzer*]{} General Observer program 60093 allowed us to obtain IRAC photometry of apparently very late-type T dwarfs discovered in the UKIDSS data. This Cycle 6 warm mission program provides only photometry at the shortest two wavelengths, \[3.6\] and \[4.5\]. Note that \[3.6\] and \[4.5\] are nominal filter wavelengths and, as the photometry is not colour-corrected for the dwarfs’ spectral shapes, the results cannot be translated to a flux at the nominal wavelength [e.g. @cushing08; @reach05]. Data were obtained for SDSS J1416+13 and ULAS J1416+13 on 23$^{rd}$ August 2009. The telescope was pointed mid-way between the L and T dwarf; with a separation of 9  both dwarfs were near the centre of the 5.2 arcminute field of view. Individual frame times were 30 seconds, repeated three times, with a 16 position spiral dither pattern, for a total integration time of 24 minutes in each band. The post-basic-calibrated-data (pbcd) mosaics generated by the [ *Spitzer*]{} pipeline were used to obtain aperture photometry. The photometry was derived using a 0.6-arcsecond pixel aperture radius, with separate (i.e. not annular) skies chosen to avoid the flaring due to the bright primary. The aperture correction was taken from the IRAC handbook[^2]. The error is estimated by the variation with sky aperture, which is larger than that implied by the uncertainty images (noise pixel maps) provided by the [*Spitzer*]{} pipeline, and is much less than 1% for the A component in both bands, and 4% and 0.7% for the B component at \[3.6\] and \[4.5\] respectively. The description of the primary issues with early release warm IRAC data[^3] indicates that the only significant concern is the uncertainty in the linearity correction for SDSS J1416+13; the total uncertainty due to this correction is estimated to be 5–7% at \[3.6\] and 4% at \[4.5\] for bright sources. Otherwise the photometry for both sources is uncertain by the usual 3% due to uncertainties in the absolute calibration and pipeline processing. Table \[tab:irac\] gives the photometry and the total uncertainties for both dwarfs. Near-infrared spectroscopy {#subsec:spectra} -------------------------- We used $JH$ and $HK$ grisms in the InfraRed Camera and Spectrograph [IRCS; @IRCS2000] on the Subaru telescope on Mauna Kea to obtain a R$\sim 100$ $JH$ and $HK$ spectra for ULAS J1416+13 on 7$^{th}$ May 2009 and 31$^{st}$ December 2009 respectively. The observations were made up of a set of eight 300s sub-exposures for the $JH$ spectrum and eighteen 200s sub-exposures in an ABBA jitter pattern to facilitate effective background subtraction, with a slit width of 1 arcsec. The length of the A-B jitter was 10 arcsecs. The spectrum was extracted using standard IRAF packages. The AB pairs were subtracted using generic IRAF tools, and median stacked. The data were found to be sufficiently uniform in the spatial axis for flat-fielding to be neglected. We used a comparison argon arc frame to obtain the dispersion solution, which was then applied to the pixel coordinates in the dispersion direction on the images. The resulting wavelength-calibrated subtracted pairs had a low-level of residual sky emission removed by fitting and subtracting this emission with a set of polynomial functions fit to each pixel row perpendicular to the dispersion direction, and considering pixel data on either side of the target spectrum only. The spectra were then extracted using a linear aperture, and cosmic rays and bad pixels removed using a sigma-clipping algorithm. Telluric correction was achieved by dividing each extracted target spectrum by that of the F4V star HIP 72303, which was observed just after the target and at a similar airmass. Prior to division, hydrogen lines were removed from the standard star spectrum by interpolating the stellar continuum. Relative flux calibration was then achieved by multiplying through by a blackbody spectrum of the appropriate $T_{\rm eff}$. The spectra were then normalised using the measured near-infrared photometry to place the spectra on an absolute flux scale, and rebinned by a factor of three to increase the signal-to-noise, whilst avoiding under-sampling of the spectral resolution. Spectral types {#subsec:sptypes} -------------- As noted in the Section \[sec:intro\], the discovery SDSS J1416+13 has recently been published by @schmidt10 and @bowler10. @schmidt10 find an optical spectral type of L5 and an infrared type of L5–6 [using the @geballe02 indices]. @bowler10 similarly find an optical type of L6$\pm$0.5 and an infrared type of L7–7.5. The template fits carried out in both papers show some discrepancies beyond 9000Å however, and here we use the SDSS spectrum of the source to produce an alternative classification as follows. The top two panels of Figure \[fig:sdssspec\] show the SDSS DR7 spectrum of SDSS J1416+13 along with the optical spectra of the L6 and L7 spectral templates 2MASS J0103+19 and DENIS J0205–11. Whilst the SDSS J1416+13 is good match over much of the range to the L6 template, they disagree significantly beyond 9000Å. On the other hand, the slope of the pseudo-continuum is very similar to that of an L7 across the entire 6000–9200Å range, although the prominent TiO, FeH and CrH features are considerably stronger in the spectrum of SDSS J1416+13. This behaviour is more typical of low-metallicity objects, where it has been speculated that that the low-metallicity atmosphere inhibits the formation of the condensate dust clouds, allowing the opacity due to alkali and hydride species to become more apparent [e.g. @burgasser03; @reiners06]. Hence, we do not classify this object following the system for L dwarfs defined by @kirkpatrick99, and instead rely on comparison to other metal-poor L dwarfs. The lower panel of Figure \[fig:sdssspec\] shows the close similarity between the spectrum of SDSS J1416+13 and that of the metal-poor L dwarf 2MASS J0532+8246. @burgasser03 demonstrated that this object not only displays features characteristic of a low-metallicity atmosphere, but also has kinematics consistent with halo membership, and classify it as sdL7. Whilst the general agreement between the spectrum of SDSS J1416+13 and the sdL7 spectrum is good across the entire range considered, there are specific areas of disagreement that should be noted. In particular the Cs[I]{} and Na[I]{} absorption features are somewhat deeper than in the sdL7 template, and more suggestive of dwarf classification than that of a subdwarf. This suggests that SDSS J1416+13 may be less metal poor than 2MASS J0532+82. Given the apparent intermediate nature of SDSS J1416+13 between the L7 and sdL7 spectra, we classify it as d/sdL7 (optical). We note that @bowler10 suggest that SDSS J1416+13 is unlikely to have significantly reduced metallicity based on the optical TiO and CaH features. @burgasser08b and @stephens09 discuss various mechanisms which may lead to unusually blue L dwarfs including low metallicity, high gravity and thin condensate cloud decks. We explore the physical properties of the L dwarf further in Section \[sec:properties\]. The IRCS spectrum of the T dwarf, ULAS J1416+13, is shown in Figure \[fig:JHKspec\], along with spectra of the T7 and T8 spectral standards [@burgasser06]. With the exception of the poor match to both templates on the red side of the J-band peak and the heavily suppressed $K$ band peak, the spectrum appears intermediate between the two. This is reflected in the spectral typing ratios (see Table \[tab:indices\]), and we classify this object as T7.5p. The early type suggested by the CH$_{4}$-K index clearly reflects the small amount of flux in the $K$ band peak. The type of peculiarity seen here in the red side of the $J$ band peak, and reflected in the spectral typing ratios, has been described for at least three other T dwarfs in @ben10, and has been suggested as a possible tracer of low-metallicity and/or high-gravity. The significance of this feature is discussed in more detail in Section \[sec:properties\]. Name \[3.6\] \[4.5\] --------------- ------------------ ------------------ SDSS J1416+13 $10.99 \pm 0.07$ $10.98 \pm 0.05$ ULAS J1416+13 $14.69 \pm 0.05$ $12.76 \pm 0.03$ : [*Spitzer*]{} IRAC photometry for the d/sdL7 and T7.5p dwarfs presented here. \[tab:irac\] {height="300pt"} {height="500pt"} [**Index**]{} [**Ratio**]{} [**Value**]{} [**Type**]{} ------------------- ----------------------------------------------------------------------------------------- ----------------- -------------- H$_2$O-J $\frac{\int^{1.165}_{1.14} f(\lambda)d\lambda}{\int^{1.285}_{1.26}f(\lambda)d\lambda }$ $0.07 \pm 0.01$ T7/8 \[+1mm\] CH$_4$-J $\frac{\int^{1.34}_{1.315} f(\lambda)d\lambda}{\int^{1.285}_{1.26}f(\lambda)d\lambda }$ $0.34 \pm 0.01$ T6 $W_J$ $\frac{\int^{1.23}_{1.18} f(\lambda)d\lambda}{2\int^{1.285}_{1.26}f(\lambda)d\lambda }$ $0.34 \pm 0.01$ T7/8 H$_2$O-$H$ $\frac{\int^{1.52}_{1.48} f(\lambda)d\lambda}{\int^{1.60}_{1.56}f(\lambda)d\lambda }$ $0.20 \pm 0.01$ T7/8 CH$_4$-$H$ $\frac{\int^{1.675}_{1.635} f(\lambda)d\lambda}{\int^{1.60}_{1.56}f(\lambda)d\lambda }$ $0.20 \pm 0.01$ T7 NH$_3$-$H$ $\frac{\int^{1.56}_{1.53} f(\lambda)d\lambda}{\int^{1.60}_{1.57}f(\lambda)d\lambda }$ $0.61 \pm 0.01$ ... CH$_4$-K $\frac{\int^{2.255}_{2.215} f(\lambda)d\lambda}{\int^{2.12}_{2.08}f(\lambda)d\lambda }$ $0.29 \pm 0.02$ T4 : The spectral flux ratios for ULAS J1416+13. Those used for spectral typing are indicated on Figure \[fig:JHKspec\].The NH$_3$ index is not used for assigning a type [see @ben08 and Burningham et al 2010 for a discussion of this], but is included for completeness and to permit future comparison with other late T dwarfs. []{data-label="tab:indices"} Proper motions {#subsec:propermotion} -------------- The photometric follow-up observations that were carried out provided a second epoch of imaging data, showing the position of the two sources of interest 1.1 years after the LAS image was measured. We used the [IRAF]{} task [GEOMAP]{} to derive spatial transformations from the WFCAM follow-up $J$-band image into the original UKIDSS LAS $J$-band image based on the positions of 18 reference stars. The transform allowed for linear shifts and rotation, although the rotation that was required was negligible. We then transformed the WFCAM follow-up pixel coordinates of the targets into the LAS images using [GEOXYTRAN]{}, and calculated their change in position (relative to the reference stars) between the two epochs. The uncertainties associated with our proper motion measurement primarily come from the spatial transformations, and the accuracy with which we have been able to measure the position of the targets (by centroiding) in the image data. Centroiding uncertainties for the targets should be small, since the seeing and signal-to-noise of the sources was good in both epochs, so this latter source of uncertainty will be neglected. For the LAS image the seeing was $\sim$0.9 in the $J$-band, whilst for the WFCAM image it was $\sim 1.1$. The root-mean-square (rms) scatter in the difference between the transformed positions of the reference stars and their actual measured positions was $\pm$0.24 pixels in declination and $\pm$0.18 pixels in right ascension, corresponding to 0.048 and 0.036  in the $J$-band LAS image. We thus estimate proper motion uncertainties of $\pm$45 mas/yr and $\pm$33 mas/yr in declination and right ascension respectively. The final, relative, proper motion measurements are $\mu_{\alpha cos\delta}=248 \pm 33$mas/yr, $\mu_{\delta}= 100 \pm 45$mas/yr for SDSS J1416+13 and $\mu_{\alpha cos\delta}=221 \pm 33$mas/yr, $\mu_{\delta}= 115 \pm 45$mas/yr for ULAS J1416+13. It should be noted that the relative proper motions calculated here disagree with the absolute values found for the primary by @schmidt10 and @bowler10 at the 4$\sigma$ level. This discrepancy likely arises as a result of two factors. Firstly, in the first epoch images both targets lie within 30 of the detector edge. As a result, the distribution of reference stars is not even about the targets. Since there is likely to considerable geometric distortion across the field of view, this poor distribution of reference stars will likely result in an unreliable absolute fit to the coordinates. Secondly, they do not take into account the parallax of the targets. The first and second epoch data were taken on 12$^{th}$ May 2008 and 17$^{th}$ June 2009 respectively, which would suggest the influence of parallax should be small. However, given that the distance for both objects may be as low as 5 pc (see Section \[sec:properties\]), we do not rule this out as a significant effect. These concerns should not effect the reliability of these proper motions as relative values, but we caution that they include systematic effects that prevent their use in any absolute manner. A wide low-mass binary {#sec:binary} ====================== The close agreement of the proper-motions for these two objects, and their 9  proximity on the sky suggests that they represent a common proper motion binary pair. To estimate the probability that the proper motions are aligned by chance, rather than because of a bona-fide association, we have considered the proper motions of objects in the SuperCosmos Sky Survey [@supercos1] in the direction of our targets. Since we do not have a parallax for either object, we instead estimate a liberal range of distances based on their spectral types and apparent magnitudes for the purposes of placing broad limits on their shared volume. In Section \[sec:properties\] we refine this distance estimate based on subsequent analysis of these objects. If we apply the $M_J$ vs. spectral type relations of @liu06 we find that an L7 and a T7.5 dwarf with the apparent magnitudes of our objects can be expected to lie at distances ranging from 5 pc to 25 pc. Of the $\sim 50$ SuperCosmos objects with apparent distances (based on colour-magnitude relation for field stars) similar to those of our targets, none shared a common motion to within $2 \sigma$. We thus conclude that the likelihood of a common proper motion occurring by chance in this direction is less than $1/50$. Since the statistics for the properties of the ultracool subdwarf population are not currently known, we will use the space density of “normal” L dwarfs to estimate a conservative probability that this pair are unrelated, and are found in close proximity by chance. Using our liberal distance range of 5–25 pc, and given the separation of 9, we can thus estimate that two objects likely share a volume of $\leq$0.01 pc$^3$. The space density for field L dwarfs was determined by @cruz07 to be 0.0038 pc$^{-3}$. The probability of finding an L dwarf within the same 0.01 pc$^3$ as our T7.5 dwarf is thus $3.8 \times 10^{-5}$. It is reasonable to surmise that the probability of finding two ultracool subdwarfs within this volume would be considerably smaller. These combined arguments suggest the probability of a chance alignment in space and motion for these two objects is less than $10^{-6}$. If we apply these arguments to the total UKIDSS LAS T dwarf sample up to DR4 [@ben10] we find that we would need a sample of approximately 1000 times larger before we would expect to identify one chance alignment such as this. It is worth stressing that our estimate for this probability is somewhat conservative. Given the apparently unusual nature of the objects discussed here, it is likely that true probability for chance alignment is considerably lower. We thus conclude that SDSS J1416+13 and ULAS J1416+13 represent a binary pair, which we shall henceforth refer to as SDSS J1416+13AB. The properties of SDSS J1416+13AB {#sec:properties} ================================= The optical spectral classification of SDSS J1416+13A as a dwarf/subdwarf implies that we could reasonably classify the secondary as a dwarf/subdwarf also, given that most binary systems are expected to be coevally formed in the same cloud core. Figure \[fig:colplot\] shows near-infrared colours as a function of spectral type for L and T dwarfs, with SDSS J1416+13AB indicated with red asterisks. Blue $H-K$ near-infrared colours for mid-to late T dwarfs have typically been interpreted as indicative of low-metallicity and/or high-gravity [e.g. @burgasser02; @knapp04; @liu07], caused by $K$ band suppression by pressure sensitive collisionally induced absorption by hydrogen [CIA H$_2$; @saumon94]. Blue $J-H$ colours in metal poor L dwarfs have also been interpreted in terms of $H$ band suppression by CIA H$_2$ [e.g. @burgasser03]. The blue $J-H$ colour of SDSS J1416+13A, and the blue $H-K$ colour of SDSS J1416+13B, therefore, support the interpretation that both objects have low-metallicity and/or high-gravity, and we interpret the peculiar spectral shape of SDSS J1416+13B in this context. ![$J-H$ and $H-K$ colour as a function of spectral type for L and T dwarfs. Data for L and T  dwarfs on the MKO system are taken from @knapp04 with T spectral types updated to the @burgasser06 system. Additional data for late-T dwarfs taken from @ben10. Known binary systems are shown as green dots, whilst known metal poor objects discussed in the text are shown as red dots, and labelled in the lower plot. The only other known T dwarf with $K$ band photometry that displays CH$_4$-J-early peculiarity is shown as an orange dot, whilst SDSS J1416+13AB are shown as red asterisks. With the exception of SDSS J1416+13A and 2MASS J0532+82, all spectral types are near-infrared types. 2MASS photometry for 2MASS0532+82 has been converted to the MKO system using the @stephens04 relationships, which give consistent results with synthetic colours calculated from the object’s near-infrared spectrum. []{data-label="fig:colplot"}](colplot.ps){height="300pt"} The spectral morphology in the $J$ band peak of SDSS J1416+13B is reminiscent of a number of T dwarfs recently discovered that have been classified as peculiar [@ben10]. These also show a $J$ band peak that appears earlier in type on the red side (as indicated by the CH$_4$-J index) compared to the blue side (as indicated by the H$_2$0-J and $W_J$ indices). This morphology was referred to by @ben10 as CH$_4$-J-early peculiarity, and we continue this convention here. Only one of the objects already found with CH$_4$-J-early peculiarity, ULAS J1233+1219, currently has $K$ band photometry. It also appears very blue, with $H-K = -0.75$ (indicated by an orange filled circle in Figure \[fig:colplot\]), and is as notable an outlier in $H-K$ for its type as SDSS J1416+13AB. It thus seems plausible that CH$_4$-J-early peculiarity is indicative of low-metallicity and/or high gravity. There is some theoretical basis for preferring a low-metallicity interpretation of CH$_4$-J-early peculiarity. Figure \[fig:metmod\] shows comparisons of @bsh2006 model spectra for $\log~g = 5.0$, $T_{\rm eff} = 700$K T dwarfs with solar and \[Fe/H\]=-0.5 metallicity, and also for solar metallicity with $\log g = 5.0$ and $\log g = 5.5$. Enhancement of the red side of the $J$ band peak is apparent in both the low-metallicity and high-gravity cases, but is most pronounced in the former. We speculate that CH$_4$-J-early peculiarity may represent a useful tracer of low-metallicity atmospheres, although its presence in a system with fiducial metallicity and age constraints will need to be observed before a robust interpretation will be possible. It is interesting to note that the spectral shape of SDSS J1416+13B also deviates from that of the spectral templates blueward of 1.1$\mu$m, in a manner similar to that seen in Figure \[fig:metmod\] for the low-metallicity case. The same behaviour is not predicted for the high-gravity case. A spectrum with better coverage in the $Y$ band may provide a useful means of breaking the gravity-metallicity degeneracy. The need for a more complex spectral classification scheme to take account of spectral variations that result from changes in metallicity and gravity in addition to $T_{\rm eff}$ has been highlighted by @kirkpatrick05. As more objects that exhibit CH$_4$-J-early spectral peculiarity are identified, its behaviour may provide a convenient method for more detailed classification of T dwarf spectra. ![@bsh2006 models for $\log~g = 5.0$ 700K T dwarfs with solar and \[Fe/H\]0=-0.5 metallicity, and for solar metallicity combined with $\log g = 5.0$ and $\log g = 5.5$. []{data-label="fig:metmod"}](modcomp.ps){height="250pt"} The lack of a known parallax for this binary pair precludes a detailed assessment of their properties, since spectroscopic distances are not well constrained for ultracool T dwarfs. In the case of the one previously identified sdL7, 2MASS J0532+82, the determined absolute magnitude ($M_J$ = 13.00) is 1-2 mags brighter than might otherwise be expected for a field dwarf of type L7 [@burgasser08]. If we assume that SDSS J1416+13A has $M_J = 13.0$ as was the case for for 2MASS J0532+82 we arrive at a distance estimate of 10 pc. However, SDSS J1416+13A is considerably less blue in $J-H$ than 2MASS J0532+82 ($J-H = 0.55$ vs $J-H = 0.08$ respectively, see Figure \[fig:colplot\]) and, as previously discussed, may have rather different properties. Assuming spectral types L7 and T7.5, however, suggests distances of 5 pc and 20 pc for the objects respectively by applying the $M_J$ vs spectral type relations of @liu06. In the case of the metal poor T dwarf 2MASS J0937+29 this method would overestimate the distance by $\sim 30$%. Using this as a correction suggests a distance for SDSS J1416+13AB of 14 pc. This would represent a significant discrepancy in the distances of the primary and secondary members of SDSS J1416+13AB, implying that the primary could be an unresolved binary. However, the $K$ band suppression in SDSS J1416+13B is greater than in the case of 2MASS J0937+29 and, as discussed below, it appears to be considerably cooler. It thus seems likely that SDSS J1416+13B is fainter still, and a distance as close as 10 pc seems plausible. A distance of $\sim 10$pc is also in broad agreement with that estimated by @schmidt10 and @bowler10 for the primary. We thus conservatively estimate the distance to SDSS J1416+13AB to lie in the 5-15 pc range. The implied projected separation of the binary pair at this range of distances is 45 – 135 AU. It is thus possible that this pair also represents a rare very low-mass wide binary system [e.g. Figure 9 in @lafreniere08]. The longer baseline provided by our [*Spitzer*]{} IRAC photometry offers the opportunity to estimate parameters of the system through comparison to predictions of model spectra. The IRAC colours of SDSS J1416+13A are normal for a late-type L dwarf, although the colours of these objects show significant scatter (see Figure \[fig:mircol\]). All of the low-metallicity late-T dwarfs plotted in Figure \[fig:mircol\] display $H - [4.5]$ that is at least 0.5 magnitudes redder than would otherwise be expected for a “normal” T dwarf of their subtype. However, the $H_{MKO}$ - \[4.5\] colour of SDSS J1416+13B is the reddest yet measured. In addition to apparently indicating low-metallicity atmospheres, this colour is a good indicator of $T_{\rm eff}$ [e.g. @warren07; @stephens09; @sandy10] and so SDSS J1416+13B appears to be very cool. ![Spitzer IRAC colours as a function of spectral type for L and T dwarfs. Data for L and T  dwarfs on the MKO system taken from @knapp04 with T spectral types updated to the @burgasser06 system. Additional data for late-T dwarfs taken from @sandy10. Known binary systems are shown as green dots, whilst known metal poor objects discussed in the text are shown as red dots, and labelled in the upper plot. SDSS J1416+13AB are shown as red asterisks. []{data-label="fig:mircol"}](mircol.ps){height="300pt"} Figure \[fig:h449\] reproduces Figure 11 of @sandy10 with the location of SDSS J1416+13B indicated. It can be seen that this T dwarf forms a sequence with the other known metal-poor (\[m/H\]$\sim$ -0.3) high-gravity ($\log g \sim 5.0 - 5.3$) dwarfs: 2MASS J0937347+293142, 2MASS J12373919+6526148, 2MASS J11145133–2618235, 2MASS J09393548–2448279. The dwarfs have $T_{\rm eff} \sim 950, 825, 750$ and 600 K respectively [@geballe09; @lb07; @sandy07; @sandy09; @burgasser08a]. Extrapolating these values using Figure \[fig:h449\] and the models [@marley02; @sm08] shown in the figure, implies that SDSS J1416+13B has \[m/H\]$\sim$–0.3, ${\rm log}~g \sim$ 5.0 to 5.3 and $T_{\rm eff} \sim 500$ K. This indicates a mass of 30–40 $M_{Jupiter}$ for SDSS J1416+13B and an age around 10 Gyr or older for the system using the evolutionary models of @sm08. The near-infrared indices of @geballe02 for SDSS 1416+13A suggest a near-infrared spectral type of L7-7.5 [@bowler10]. The near-infrared spectral type-$T_{\rm eff}$ relations of @stephens09 suggest that blue L7 dwarfs have $T_{\rm eff} \sim$1500 K. If the system is aged at $\sim 10$Gyr as implied by the secondary then the evolutionary models of @sm08 suggest that the primary is a $\sim$75M$_{Jupiter}$ dwarf with $\log~g \sim 5.5$. Hence the L dwarf is at the stellar/substellar boundary as also suggested by @bowler10. @stephens09 have shown that the atmospheres of blue L dwarfs are less dusty than the bulk population, deriving a high value of $f_{sed} \sim 3$ using the @marley02 models. These authors also find that there is an indication that dust clearing may occur at higher temperatures for higher gravity systems. The blue colours and almost dust-free atmosphere of this relatively warm $\sim$1500 K L dwarf is therefore consistent with a high gravity and relatively old age for the system. Both @schmidt10 and @bowler10 have estimated $(U,V,W)_{\rm LSR}$ for SDSS J1416+13A, finding $(-7.9 \pm 2.1, 10.2 \pm 1.2, -31.4 \pm 4.7)$ kms$^{-1}$ and $(-6 \pm 4, 10.2 \pm 1.4, -27 \pm 9)$ kms$^{-1}$ respectively[^4], and interpret its kinematics as indicative of thin disk membership. This is consistent with the age of  10 Gyr and slight metal-paucity that we find here [@robin03; @haywood97]. It is intriguing that SDSS J1416+13B appears to be $\sim$250 K and $\sim$100 K cooler than 2MASS J1114–26 2MASS J0939–24 respectively, which are of similar spectral type (T7.5 and T8). It is plausible that the near-infrared spectral type vs. $T_{\rm eff}$ relation for very late T dwarfs shows significant dependence on metallicity and gravity, with lower-metallicity dwarfs of a given subtype having lower $T_{\rm eff}$ than similar type objects of higher metallicity. The cool nature of 2MASS J0939–24 compared to other T8 dwarfs spectral type would tend to support this assertion, although interpretation of this object is complicated by its probable binarity [@burgasser08a]. If SDSS J1416+13B had significantly lower metallicity and/or higher gravity than 2MASS J1114–26 2MASS J0939–24, such an effect might account for their similar types but diverse $T_{\rm eff}$s. However, the same model predictions seen in Figure \[fig:h449\] that suggest such different $T_{\rm eff}$ also suggest fairly similar metallicities and gravities for the three objects. Finally, it should be noted that we cannot currently rule out the presence of a cooler unresolved companion to SDSS J1416+13B, which might explain its extremely red $H-[4.5]$ colour coupled with is T7.5p near-infrared morphology. Unfortunately, the lack of parallax and high-resolution imaging for this target prevent us from adequately exploring this issue here. {height="400pt"} Summary {#sec:summ} ======= We have identified what appears to be the coolest binary system yet found. The association of the T7.5p component with the d/sdL7 primary allows us to now extend the high-gravity and low-metallicity sequence to the lowest observed temperatures, and we suggest that CH$_4$-J-early peculiarity [@ben10] may in future prove to be a useful discriminator for this type of object. The likely close proximity of the system to the Sun should facilitate the determination of the trigonometric parallax in the near-future, which will allow a more robust determination of the properties for this exciting system. Acknowledgements {#acknowledgements .unnumbered} ================ SKL is supported by the Gemini Observatory, which is operated by AURA, on behalf of the international Gemini partnership of Argentina, Australia, Brazil, Canada, Chile, the United Kingdom, and the United States of America. EN received support from a Royal Astronomical Society small grant. NL was funded by the Ramón y Cajal fellowship number 08-303-01-02. CGT is supported by ARC grant DP0774000. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and has benefited from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at http://www.browndwarfs.org/spexprism. [^1]: E-mail: [email protected] [^2]: http://ssc.spitzer.caltech.edu/irac/dh/ [^3]: http://ssc.spitzer.caltech.edu/irac/documents/iracwarmdatamemo.txt [^4]: $U$ positive towards the Galactic centre.

ArXiv

Introduction ============ The single band two dimensional Hubbard Hamiltonian[@HUBBARD] has recently received considerable attention due to possible connections with high temperature superconductors. Indeed, evidence is accumulating that this Hamiltonian may describe, at least qualitatively, some of the normal state properties of the cuprates.[@review] Exact Diagonalization (ED) and Quantum Monte Carlo (QMC) have been used to model static properties like the behavior of spin correlations and magnetic susceptibility both at half-filling and with doping.[@review] Comparisons of dynamic quantities like the spectral weight and density of states with angle-resolved photoemission results[@flat-exper; @flat; @bulut; @hanke; @berlin] have also proven quite successful. Significantly, while analytic calculations have pointed towards various low temperature superconducting instabilities, such indications have been absent in numerical work.[@review] Historically, however, the Hubbard model was first proposed to model magnetism and metal-insulator transitions in 3D transition metals and their oxides,[@HUBBARD] rather than superconductivity. Now that the technology of numerical work has developed, it is useful to reconsider some of these original problems. A discussion of possible links between the 3D Hubbard model and photoemission results for ${\rm YTiO_3}$, ${\rm Sr VO_3}$ and others[@fujimori; @inoue; @morikawa] has already recently occurred. In such perovskite ${\rm Ti^{3+}}$ and ${\rm V^{4+}}$ oxides, which are both in a $3d^1$ configuration, the hopping amplitude $t$ between transition-metal ions can be varied by modifying the $d-d$ neighboring overlaps through a tetragonal distortion. Thus, the strength of the electron correlation $U/t$ can be varied by changing the composition. In fact, a metal-insulator transition has been reported in the series ${\rm SrVO_3}-{\rm Ca VO_3}-{\rm La Ti O_3}-{\rm YTiO_3}$. On the metallic side, a quasiparticle band is experimentally observed near the Fermi energy $E_F$, as well as a high energy satellite associated to the lower Hubbard band (LHB).[@fujimori; @rrmp] Spectral weight is transferred from the quasiparticle to the LHB as $U/t$ is increased at half-filling. In this paper, we report the first use of Quantum Monte Carlo, combined with analytic continuation techniques, to evaluate the spectral function and density of states for the 3D Hubbard Hamiltonian. The motivation is twofold. First, we want to compare general properties of the 3D Hubbard Hamiltonian with the extensive studies already reported in the literature[@WHITE; @jarrell; @rrmp; @review] for the 2D and infinite-D cases. Of particular importance is the presence of quasiparticles near the half-filling regime, as well as the evolution of spectral weight with doping. Many of the high-Tc cuprates contain ${\rm CuO_2}$ planes that are at least weakly coupled with each other, and thus the study of the 3D system may help in understanding part of the details of the cuprates. More generally, the Hubbard Hamiltonian is likely to continue being one of the models used to capture the physics of strongly correlated electrons, so we believe it is important to document its properties in as many environments as possible for potential future comparisons against experiments. Secondly, we discuss a particular illustration of such contact between Hubbard Hamiltonian physics and experiments on 3D transition metal oxides. In addition to the studies of half-filled systems with varying correlation energy mentioned above, experiments where the band filling is tuned by changing the chemical composition have also been reported.[@fujimori2; @morikawa; @tokura] One compound that has been carefully investigated in this context is ${\rm Y_{1-x} Ca_x Ti O_3}$. At $x=0$ the system is an antiferromagnetic insulator. As $x$ increases, a metal-insulator transition is observed in PES studies. The lower and upper Hubbard bands (LHB and UHB) are easily identified even with $x$ close to 1, which would naively correspond to small electronic density in the single band Hubbard model, i.e. a regime where $U/t$ is mostly irrelevant. In the experiments, a very small amount of spectral weight is transferred to the Fermi energy, filling the gap observed at half-filling (i.e. generating a “pseudogap”). Analysis of the PES results of these compounds using the paramagnetic solution of the Hubbard Hamiltonian in infinite-D [@metzner], a limit where dynamic mean field theory becomes exact (see section II), has resulted in qualitative agreement [@jarrell; @georges; @rrmp] with the experimental results. At and close to half-filling there is an antiferromagnetic (AF) solution which becomes unstable against a paramagnetic (PM) solution at a critical concentration of holes. In the PM case, weight appears in the original Hubbard gap as reported experimentally. However, this analysis of the spectral weight in terms of the infinite-D Hamiltonian is in contradiction with results for the density of states reported in the 2D Hubbard model[@review] where it is found that upon hole (electron) doping away from half-filling the chemical potential $\mu$ moves to the top (bottom) of the valence (conduction) band. The results at $\langle n \rangle =1$ in 2D already show the presence of a robust quasiparticle peak which is absent in the insulating PM solution of the $D=\infty$ model. That is, in the 2D system the large peak in the density of states observed away from half-filling seems to evolve from a robust peak already present at half-filling. On the other hand, at $D=\infty$ a feature resembling a “Kondo-resonance” is $generated$ upon doping if the paramagnetic solution is used. This peak in the density of states does not have an analog at half-filling unless frustration is included.[@jarrell] Studies in 3D may help in the resolution of this apparent non-continuity of the physics of the Hubbard model when the dimension changes from 2 to $\infty$. The proper way to carry out a comparison between $D=3$ and $\infty$ features is to base the analysis on ground state properties. With this restriction, i.e. using the AF solution at $D=\infty$ and close to half-filling, rather than the PM solution, we found that the $D=3$ and $\infty$ results are in good agreement. In this paper we will consider which of these situations the 3D Hubbard Hamiltonian better corresponds to, and therefore whether the single band Hubbard Hamiltonian provides an adequate description of the density of states of 3D transition-metal oxides. Model and Methods ================= The single band Hubbard Hamiltonian is $$\begin{aligned} H & = & -t \sum_{\bf \langle ij \rangle } ( c^\dagger_{ {\bf i} \sigma} c_{{\bf j} \sigma} + h.c.) - \mu \sum_{{\bf i}\sigma} n_{{\bf i}\sigma} \nonumber \\ & & + U \sum_{\bf i} (n_{{\bf i} \uparrow} - 1/2 ) (n_{{\bf i} \downarrow} - 1/2 ), \label{hubbard}\end{aligned}$$ where the notation is standard. Here ${\bf \langle ij \rangle }$ represents nearest-neighbor links on a 3D cubic lattice. The chemical potential $\mu$ controls the doping. For $\mu=0$ the system is at half filling ($\langle n \rangle=1$) due to particle-hole symmetry. $t\equiv 1$ will set our energy scale. We will study the 3D Hubbard Hamiltonian using a finite temperature, grand canonical Quantum Monte Carlo (QMC) method [@blankenbecler] which is stabilized at low temperatures by the use of orthogonalization techniques [@white]. The algorithm is based on a functional-integral representation of the partition function obtained by discretizing the “imaginary-time” interval $[0,\beta]$ where $\beta$ is the inverse temperature. The Hubbard interaction is decoupled by a two-valued Hubbard-Stratonovich transformation [@hirsch] yielding a bilinear time-dependent fermionic action. The fermionic degrees of freedom can hence be integrated out analytically, and the partition function (as well as observables) can be written as a sum over the auxiliary fields with a weight proportional to the product of two determinants, one for each spin species. At half-filling ($\langle n \rangle=1$), it can be shown by particle-hole transformation of one spin species $(c_{{\bf i}\downarrow} \rightarrow (-1)^{{\bf i}} c_{{\bf i}\downarrow}^\dagger)$ that the two determinants differ only by a positive factor, hence their product is positive definite. At general fillings, however, the product can become negative, and this “minus-sign problem” restricts the application of QMC to relatively high temperature (of order 1/30 of the bandwidth) off half-filling. The QMC algorithm provides a variety of static and dynamic observables. One equal time quantity in which we are interested is the magnetic (spin-spin) correlation function, $$C({\bf l}) = \frac{1}{N} \sum_{{\bf j}} \langle m_{\bf j} m_{{\bf j+l}} \rangle. \label{correl}$$ Here $m_{\bf j}=\sum_\sigma\sigma n_{{\bf j}\sigma}$ is the local spin operator, and $N$ is the total number of lattice sites. Static correlations have also been investigated in earlier studies of the $3D$ Hubbard model [@HIRSCHn; @rts] where the antiferromagnetic phase diagram at half filling was explored. To obtain dynamical quantities in real time or frequency, the QMC results in imaginary time have to be analytically continued to the real time axis. Since we are mostly interested in the one-particle spectrum we measure the one-particle Green function $G({\bf p},\tau)$. The imaginary part of $G({\bf p},\omega)$ (in real frequency) defines the spectral weight function at momentum ${\bf p}$, $A({\bf p},\omega)$, which is related to $G({\bf p},\tau)$ by: $$G({\bf p},\tau) = \int_{-\infty}^\infty d\omega \, A({\bf p},\omega) \, \frac{e^{-\tau\omega}}{1+e^{\beta\omega}}. \label{conti}$$ $A({\bf p},\omega)$ can in principle be calculated by inverting Eq.(\[conti\]), but the exponential behavior of the kernel at large values of $|\omega|$ makes this inversion difficult numerically. $G({\bf p},\tau)$ is quite insensitive to details of $A({\bf p},\omega)$ in particular at large frequencies. Since $G({\bf p},\tau)$ is known only on a finite grid in the interval $[0,\beta]$ and there only within the statistical errors given by the QMC-sampling, solving Eq.(\[conti\]) for $A({\bf p},\omega)$ is an ill-posed problem. A large number of solutions exists, and the problem is to find criteria to select out the correct one. This can be done by employing the Maximum Entropy (ME) method [@gubernatis]. Basically, the ME finds the “most likely” solution $A({\bf p},\omega)$ which is consistent with the data and all information that is known about the solution (like positivity, normalization, etc.). ME avoids “overfitting” to the data by a “smoothing” technique that tries to assimilate the resulting $A({\bf p},\omega)$ to a flat default model. In the absence of any data ($G({\bf p},\tau)$) ME would converge to the default model which is chosen to be a constant within some large frequency interval. There is no adjustable parameter in the ME application. One needs accurate data for $G({\bf p},\tau)$ with a statistical error of $o(10^{-4})$ to get reliable results for $A({\bf p},\omega)$. In principle, one can calculate analytically the first and second (and higher) moments of the spectral weight and include this information in the ME procedure. However, we chose to calculate the moments afterwards from the resulting function $A({\bf p},\omega)$, and to compare them with the analytically known results as a further test. The agreement was in all cases within 10%. Still, the ME methods provides only a rough estimate of the true spectral weight functions. Band gaps and the positions of significant peaks are usually well captured but fine structure which needs a high frequency resolution is hard to detect within the ME approach. Integrating $A({\bf p},\omega)$ over the momenta gives the one-particle density of states (DOS) $N(\omega)$. However, technically, it is preferrable to integrate first $G({\bf p},\tau)$, which reduces the statistical errors, and then perform the analytic continuation. The DOS will be compared to results from the dynamical mean-field theory of infinite dimensions, $D=\infty,$ [@metzner]. In this limit, with the proper scaling of the hopping element ($t=t^*/\sqrt{Z}$, with $Z$ being the coordination number) the one-particle self energy becomes local or, equivalently, momentum independent and the lattice problem is mapped onto a single site problem. The constant $t^*$ is set to $t^*=\sqrt{6}$ to obtain the same energy scale, $t=1$, when compared to the $3D$ case. [@bethe] In contrast to conventional mean-field theories, the self energy remains frequency dependent, preserving important physics. Spatial fluctuations are neglected, an approximation which becomes exact in the limit $Z\to\infty$ ($Z=2D$ for the simple cubic lattice). Even in $D=\infty$, the remaining local interacting problem cannot be solved analytically but will also be treated by a finite temperature QMC [@fye] supplemented by a self-consistency iteration [@georges; @rrmp]. The advantage is that the system can be investigated in the thermodynamic limit with a modest amount of computer time. Due to its local character the $D=\infty$ approach cannot provide information on momentum dependent spectral functions. However, recently a $k$-resolved spectral function has been studied[@rspec] in $D=\infty$. Among other things, the $D=\infty$ limit has been used recently to study the AF phase diagram in the Hubbard model [@jarrell]. The agreement of the Néel temperature with 3D results is good.[@rts] In $D=\infty$ it is further possible to suppress AF long range order artificially by restricting the calculation to the (at low temperatures unstable) paramagnetic solution at half-filling. In this way, one may simulate frustration due to the lattice structure or orbital degeneracy, although in the absence of calculations for hypercubic lattices with nearest and next-nearest uniform hopping amplitudes it is still a conjecture how close this approach is to including these effects fully. Half-filling ============ Quantum Monte Carlo ------------------- We first study the single particle spectral weight $A({\bf p},\omega)$ at relatively strong coupling, $U=8$, and half-filling ($\langle n \rangle=1$) at a low temperature of $T=1/10$. -10mm A gap is clearly present in the spectrum (Fig. 1) which is compatible with the expectation that the half-filled Hubbard model on a bipartite lattice is an antiferromagnetic insulator for all nonzero values of the coupling $U/t$. The spectral weight has four distinct features (two in the LHB and two identical ones in the UHB, as expected from particle-hole symmetry). In the UHB there is weight at a high energy, roughly in the interval between $\sim 5t$ and $8t$. This broad feature likely corresponds to the incoherent part of the spectral function found in previous simulations for the 2D Hubbard and $t-J$ models.[@review] The dominant scale of this incoherent weight is $t$, and since it is located far from the top of the valence band its presence is not important for the low temperature properties of the system. Much more interesting is the sharper peak found close to the gap in the spectrum. This band dispersion starts at a binding energy of approximately $\omega= -4t$ at momenta $(0,0,0)$ and moves up in energy obtaining its maximum value at $\omega\approx -2t$ at momenta $(0,\pi/2,\pi)$ and $(\pi/2,\pi/2,\pi/2)$ in Fig. 1. The width of the peak diminishes as the top of the valence band is reached. Similar structure was discussed before in studies of 2D systems, which had a somewhat higher resolution, as a “quasiparticle” band corresponding to a hole moving coherently in an antiferromagnetic background.[@review; @moreo] This quasiparticle should be visualized as a hole distorting the AF order parameter in its vicinity. In this respect it is like a spin-polaron or spin-bag,[@schrieffer] although “string states” likely influence its dispersion and shape.[@review] The quasiparticle (hole plus spin distortion) movement is regulated by the exchange $J$, rather than $t$. Using the center of the quasiparticle peaks of Fig. 1 as an indication of the actual quasiparticle pole position, we obtain a bandwidth $W$ of about $2$ to $3t$ or, equivalently, $4$ to $6J$ using ${ J=4t^2/U}$ for $U=8$. However, due to the low resolution of the ME procedure, reflected in part in the large width of the peaks of Fig. 1, it is difficult to show more convincingly within QMC/ME that the quasiparticle bandwidth is indeed dominated by $J$. Note that moving from $(0,0,0)$ to $(\pi,\pi,\pi)$ along the main diagonal of the Brillouin Zone (BZ), the PES part of the spectrum (i.e. the weight at $\omega <0$) loses intensity. There is a clear transfer of weight from PES at small $|{\bf p}|$ to IPES at large $|{\bf p}|$, as observed in 2D simulations.[@bulut1] In addition, note that there is PES weight above the (naive) Fermi momentum of this half-filled system. For example, at ${\bf p} = (0,\pi,\pi)$, spectral weight at $\omega <0$ can be clearly observed. Similarly, at ${\bf p} = (0,0,\pi)$ weight in the IPES region is found. This effective doubling of the size of the unit cell in all three directions is a consequence of the presence of AF long range order. The hole energy at ${\bf p}=(0,0,0)$ and $(\pi,\pi,\pi)$ becomes degenerate in the bulk limit and the quasiparticle band, for example along the main diagonal of the BZ, has a reflection symmetry with respect to $(\pi/2,\pi/2,\pi/2)$, as observed in our results (Fig. 1). However, note that the actual intensity of the AF-induced PES weight close to $(\pi,\pi,\pi)$ is a function of the coupling. As $U/t \rightarrow 0$, the intensity of the AF-induced region is also reduced to zero. The presence of this AF-generated feature has recently received attention in the context of the 2D high-Tc cuprates.[@schrieffer; @shadow] While its presence in PES experiments at optimal doping is still under discussion, these features clearly appear in PES experimental studies of half-filled insulators, like ${\rm Sr_2 Cu O_2 Cl_2}$.[@wells] Thus, while this behavior has been primarily discussed in the context of 2D systems angle-resolved PES (ARPES) studies of 3D insulators like ${\rm LaTiO_3}$ might also show such features. In Fig. 2 we show the density of states (DOS) $N(\omega)$ of the $4^3$ lattice. The two features described before, namely quasiparticle and incoherent background, in both PES and IPES are clearly visible. Also shown in Fig. 2 is the temperature effect on $N(\omega)$ which is weak for the given temperatures ($\beta=10$ and 4). The basic features are still retained, only the gap is slightly reduced and the quasiparticle peak less pronounced at the higher temperature. Results corresponding to a larger coupling, $U=12$, at $\beta=4$ are shown in Fig. 3. The gap increases and the quasiparticle band becomes sharper as $U/t$ grows, as expected if its bandwidth is regulated by $J$. $A({\bf p},\omega)$ at $U=12$, $\beta=4$ are similar to the results shown in Fig. 1. -20mm -20mm A characteristic double peak like that seen in Figs. 2,3 has been observed in the X-ray absorption spectrum of ${\rm LaFeO_3}$ [@sarma], which is a strongly correlated, wide gapped antiferromagnetic insulator. The peaks appear at energies of about 2.2 eV and 3.8 eV. When Fe is substituted by Ni this structure vanishes and the system becomes a paramagnetic metal at a Ni concentration of about 80%. If we choose for comparison with the calculated DOS (Fig. 3) a hopping amplitude of $t=0.5$ eV, giving a reasonable d-bandwidth of $W=6$ eV, the positions of the quasiparticle peak and the maximum of the incoherent band for $U=12t$ are at about $\omega_1\approx 4t=2$ eV and $\omega_2\approx 8.4t=4.2$ eV, respectively. The agreement with the experimental values is fairly good considering the crude simplifications of the Hubbard model such as neglecting orbital degeneracy and charge transfer effects. Even the estimated charge gap of Fig. 3, defined by the onset of spectral weight relative to the Fermi energy, $\Delta_{charge}\approx 3t=1.5$ eV is not too far from the experimental value of $\sim 1.1$ eV. The ratio $r=\omega_2/\omega_1$ decreases with $U$ since for $U\gg t$ both energies are expected to converge to $U/2$. While for $U=12t$, $r\approx 2.1$ is comparable to the experimental value ($\sim 1.7$), it is too large for $U=8$ ($r \approx 2.9 $) showing that under the assumption of a single band Hubbard model description for ${\rm LaFeO_3}$, the effective on-site interaction is at least of the size of the d-bandwidth. Another feature which has been attributed to antiferromagnetic ordering was found in a high resolution PES study of $\rm{V_2O_3}$. [@shin; @rozen2] In the AF insulator at $T=100K$ the spectrum shows a shoulder at $\omega_1=-0.8$ eV which is absent in the paramagnetic metal at $T=200K$. This shoulder might be a reminiscent of the quasiparticle peak. The maximum of the lower Hubbard band is at about $\omega_2 \approx -1.3$ eV, giving a ratio $\omega_2/\omega_1\approx 1.6$ similar to that observed in ${\rm LaFeO_3}$. The on-site interaction was estimated to be about 1.5 times the bandwidth.[@shin] -20mm It is interesting to compare the results obtained in our simulations with those found in the $D=\infty$ limit of the Hubbard model. At half-filling for arbitrary coupling strength, the $D=\infty$ model has an AF insulating ground state. Its DOS is shown in Fig. 4, using the same coupling and temperature as in the 3D simulation. $N(\omega)$ for both cases are similar, and they are also similar to results found before in 2D, suggesting that the physics of holes in an antiferromagnetic system is qualitatively the same irrespective of whether a 2D, 3D or ${\rm \infty}$D lattice is used, at least within the accuracy of present QMC/ME simulations. SDW mean-field and Born approximation ------------------------------------- Since the data shown in the previous subsection correspond to holes in a system with AF long-range order, it is natural to compare our results against those found in mean-field approximations to the half-filled 3D Hubbard model that incorporate magnetic order in the ground state. The “spin-density-wave” mean-field approximation has been extensively used in the context of the 2D Hubbard model,[@sdw] and here we will apply it to our 3D problem. For a lattice of $N$ sites, the self-consistent equation for the gap $\Delta$ is $$1={{U}\over{2N}} \sum_{\bf p} {{1}\over{E_{\bf p}}}, \label{sdw}$$ where $E_{\bf p} = \sqrt{ \epsilon_{\bf p}^2 + \Delta^2 }$ is the quasiparticle energy, and $\epsilon_{\bf p} = -2t (\cos p_x + \cos p_y + \cos p_z)$ is the bare electron dispersion. The resulting quasiparticle dispersion is shown in Fig. 5 compared against the results of the QMC/ME simulation. The overall agreement is good if the coupling $U$ in the gap equation Eq.(4) is tuned to a value $U \sim 5.6$. It is reasonable that a reduced $U$ should be required for such a fit, since the SDW MF gap is usually larger than the more accurate QMC result. Similar renormalizations of $U$ in comparing QMC and approximate analytic work has been discussed in the context of fitting the magnetic response,[@renormu] and has also been explicitly calculated.[@vandongen] Fig. 5 shows many of the features observed in the numerical simulation, namely a hole dispersion which is maximized at $(\pi/2,\pi/2,\pi/2)$ for the momenta shown there, an overall bandwidth smaller than the noninteracting one, and the presence of AF-induced features in the dispersion above the naive Fermi momentum. Thus the SDW MF approach qualitatively captures the correct hole quasiparticle bandwidth $J$ at half-filling. However, a spurious degeneracy appears in the hole dispersion in this approximation. Momenta satisfying $\cos p_x + \cos p_y + \cos p_z =0$ have the same energy. This is not induced by symmetry arguments and is an artifact of the SDW approach. In addition, $A({\bf p},\omega)$ in the SDW approximation only has one peak in the PES region for each value of the momentum, missing entirely the incoherent part. While it may not be necessary to fix this problem in this case, it is important in general to be able to go beyond SDW MF. To do this, the self-consistent Born approximation[@born] (SCBA) for one hole in the 3D $t-J$ model, which corresponds to the strong coupling limit of the Hubbard model, can be used. This technique reproduces accurately exact diagonalization results in the 2D case.[@born] -3mm Actually, the dispersion of a dressed hole in an antiferromagnet within the SCBA for a bilayer system, and also for a 3D cubic lattice, has been recently studied.[@sasha] Here, for completeness, we reproduce some of the results of Ref.[@sasha], and compare them against those of the 3D Hubbard model obtained with the SDW MF approximation and QMC calculations (Fig. 6). The comparison is carried out at $J/t \sim 0.3$ which corresponds to $U/t \sim 13$. The maximum of the dispersion in the valence band using the SCBA now lies at $({\pi/2, \pi/2, \pi/2})$, removing the spurious SDW MF degeneracy. In the scale of Fig. 6 the splitting between this momentum and $(\pi,\pi/2,0)$ is difficult to resolve, since it corresponds to about 100K. Note that the bandwidth predicted by the SDW MF technique is approximately a factor two larger than the more accurate prediction of the SCBA. However, for this larger value of $U$ it does not appear possible to fit simultaneously the SDW MF bandwidth and band-gap to the results of QMC by the same renormalization of $U$, something which can be done successfully at weaker coupling, $U=8$. The QMC points at this intermediate coupling value where $U$=bandwidth lie in between the SDW MF and SCBA. Though the uncertainties in the QMC results are rather large, we expect the agreement between SCBA and QMC results to improve as the coupling increases. The best fit of the SCBA data[@sasha] is $\epsilon({\bf p})\,=\,c\,+\,0.082(\cos p_x\cos p_y+ \cos p_y\cos p_z+\cos p_x\cos p_z)\, +\,0.022(\cos 2p_x+\cos 2p_y+\cos 2p_z)$ (eV), if $J=0.125eV$ and $t=0.4eV$ are used. The constant $c$ is defined by the SDW MF gap (Fig. 6). As in the case of the 2D problem, holes tend to move within the same sublattice to avoid distorting the AF background.[@review] Working at small $J/t$, the bandwidth of the 3D $t-J$ hole quasiparticle was found to scale as $J$,[@sasha] as occurs in two dimensions. Finite Hole Density =================== D=3 --- We can also use the QMC approach to study the 3D Hubbard model away from half-filling for temperatures down to about 1/30 of the bandwidth, a value for which $T\sim J$ for the present strong coupling values. First, we study the influence of doping and temperature on the spin-spin correlation function $C({\bf l})$. At half filling $C({\bf l})$ shows strong antiferromagnetic correlations over the whole $4^3$-lattice at $\beta=10$ (Fig. 7). At $\beta=2$ the correlations are significantly weakened, and with additional doping ($\langle n \rangle =0.88$) all correlations are suppressed besides those between nearest neighbors. These appear to be stable against doping. The density of local moments, $\sqrt{C(0)}$ reaches its low temperature limit at an energy scale set by $U$ and hence is unaffected by the change of $\beta$ from $\beta=2$ to $\beta=10$ (note that longer range spin correlations form at a temperature set by the much smaller energy scale $J$). $\sqrt{C(0)}$ is to first order proportional to the electronic density and hence slightly reduced at $\langle n \rangle =0.88$. There has been considerable discussion concerning the relationship between the spin-spin correlations and the presence of a gap in the density of states. In particular, it was observed[@WHITE] that if $N(\omega)$ is evaluated on lattices of increasing size at fixed temperature a well formed gap appearing on small lattices disappears when the spatial extent exceeds the spin-spin correlation length. Decreasing the temperature (and hence increasing the range of the spin correlation) allows the gap to reform. Similar effects are seen here in 3D. -16mm Fig. 8a shows the density of states on a $4^3$ lattice at several densities, $U=8$ and $\beta=2$. At this temperature the charge gap is not fully developed, and the quasiparticle peaks cannot be resolved. The result with doping is similar to that reported on 2D lattices.[@bulut2] The chemical potential $\mu$ moves to the top of the valence band as the density is reduced from half-filling. A large peak is generated which increases in intensity as $\langle n \rangle$ is further reduced. The weight of the upper part of the spectrum (reminiscent of the UHB) decreases with doping due to the reduced effective interaction. Similar results are shown in Fig. 8b but for a $6^3$ lattice. There is not much difference between the two lattices, showing that within the resolution of the ME procedure finite size effects are small. The large peak that appears in Fig. 8a-b at finite hole density is crossed by $\mu$ as the density is reduced. At $\langle n \rangle = 0.94$, the peak is located to the left of $\mu$, at $\langle n \rangle = 0.88$ it has reached the chemical potential, and at $\langle n \rangle = 0.72$, the peak has moved to the right. This is in agreement with the behavior observed in both 2D QMC and ED simulations,[@dos] and it may be of relevance for estimations of superconducting critical temperatures if a source of carrier attraction is identified.[@dos1] The results of the previous section at half-filling obtained at low temperatures $(T\sim 1/10)$ revealed a sharp quasiparticle peak in the DOS at the top of the valence band and bottom of the conduction band. Numerical studies of 2D lattices have shown that the peak intensity at $T=0$ is the $largest$ at half-filling.[@dos] Away from half-filling, the peak is still visible but it is broader than at $\langle n \rangle =1$.[@dos] Thus there is no evidence that the sharp peak in the DOS of the doped system has been generated dynamically and represents a “Kondo-resonance” induced by doping, as has sometimes been suggested [@jarrell], and as Fig. 8 obtained at relatively high temperature, $\beta=2$, seem to imply. -18mm -18mm Another important quantity to study is the quasiparticle residue Z. The SCBA results show that Z is small but finite for the case of one hole in an antiferromagnetic insulator state, and actually the results are very similar in 3D and 2D systems.[@born; @sasha] Numerical results provide a similar picture.[@review] On the other hand, Z vanishes in the $D=\infty$ approach working in the paramagnetic state as the doping $\delta$ tends to zero. Note that in this state there are no AF correlations ($\xi_{AF} =0$). Thus, it is clear that the hole quasiparticle at half-filling observed in the 2D and 3D systems is $not$ related with the quasiparticle-like feature observed in the PM state at $D=\infty$. -12mm -12mm -6mm In Fig. 9, we show $A({\bf p},\omega)$ obtained on the $4^3$ lattice, $U=8$, $T=1/2$ and various densities away from half-filling. The gap is now absent. From the energy location of the maximum of the dominant peak in Fig. 9a-c, the quasiparticle dispersion can be obtained. The results are shown in Fig. 10. It is remarkable that the quasiparticle dispersion resembles that of a noninteracting system i.e. $\epsilon_{\bf p} = -2t^\star (\cos p_x + \cos p_y + \cos p_z)$, with a scale increasing from $t^\star \sim t/4$ to $t/3$ with doping. This dispersion certainly does not exhaust all the spectral weigth but a large incoherent part still remains at this coupling, density and temperature. Similar results were observed in 2D.[@bulut; @moreo; @dos; @ortolani] Only vestiges remain of the AF induced weight in PES near $(\pi,\pi,\pi)$. However, this drastic reduction of the AF induced intensity may be caused by the high temperature of the simulation as observed in the spin-spin correlation function (Fig. 7). -5mm -10mm $D=\infty$ ----------- The previous subsection and the results at half-filling have shown that the DOS of the 3D Hubbard model has a large peak at the top of the valence band. The peak is crossed by the chemical potential as $\langle n \rangle $ decreases. This behavior is in apparent contradiction with results reported at $D=\infty$ where a peak is generated upon doping if the “paramagnetic” solution to the mean-field problem is selected. At $D=\infty$, there are only two very distinct magnetic ground states. One has AF long-range order, and the other is a paramagnet with strictly $zero$ AF correlation length i.e. without short range antiferromagnetic fluctuations. Thus, at $D=\infty$ the transition is abrupt from a regime with $\xi_{AF} = \infty$ to $\xi_{AF} = 0$. This does not occur in finite dimensions where before the long-range order regime is reached, AF correlations start building up smoothly. This qualitative difference is depicted in Fig. 11. $\xi_{AF}$ as small as a couple of lattice spacings can be robust enough to induce important changes in the carrier dispersion, and may even be enough to induce superconductivity as many theories for the 2D high-Tc cuprates conjecture. We believe that the absence of a regime of intermediate size AF correlations at large D is the key ingredient that explains the differences reported here between D=2,3 and ${\rm D=\infty}$. In Fig. 12a, the D=$\infty$ DOS in the AF phase is shown at $\langle n \rangle =1$ and 0.94. For these densities the AF-phase is energetically stable. We observe the tendency of the large peak at the bottom of the valence band to move towards the chemical potential in good agreement with the 3D Quantum Monte Carlo simulations. As found in 2D, the intensity of the peak decreases as we move away from half-filling if the temperature is low enough. In Fig. 12b, the DOS in the $D=\infty$ limit working in the paramagnetic phase is shown at several densities. For the present interaction, $U=8$, the paramagnetic solution remains metallic at all temperatures even at half filling. [@jarrell] The results are qualitatively different from those observed in the AF regime. At $\langle n \rangle =1$ a large peak at the chemical potential is clearly visible. Upon hole doping this peak gradually moves toward higher energies. At sufficiently strong doping the DOS of the PM-phase (Fig. 12b) resembles the results for the $3D$ lattices (Fig. 8), which is not surprising since AF correlations in $3D$ are strongly suppressed at the present temperature. Close to half filling, however, the $3D$ results are closer to the DOS of the AF-phase where a strong peak is observed on the left hand side of the chemical potential $\mu$. This result is gratifying since the proper way to compare $D=3$ and $\infty$ results is by using the actual ground states in each dimension. In $D=\infty$, at low temperatures, the crossing of the peak by $\mu$ is expected at that point where the AF-Phase becomes unstable against doping. -18mm -18mm **Conclusions** =============== In this paper we have calculated the single particle properties of the 3D single band Hubbard model using Quantum Monte Carlo and the SDW mean field and SCBA approximations. Our results have many similarities with those reported previously in 2D systems. At half-filling, peaks at the top of the valence band and bottom of the conduction band are observed in the DOS. Their behavior is associated with spin polarons with a bandwidth of order the exchange $J$. We found similarities to and semi-quantitive agreement with experimentally observed features in the spectra of strongly correlated 3D AF insulators, $\rm{LaFeO_3}$ and $V_2O_3$. As we dope the system, the sharp peak associated with these quasiparticles is crossed by the chemical potential as the density $\langle n \rangle $ changes. The PES weight observed away from half-filling is already present at half-filling. No new states are generated by doping. This result must be contrasted with that observed experimentally in, e.g., ${\rm Y_{1-x} Ca_x Ti O_3}$ using angle-integrated PES. In this case spectral weight which is not present in the insulator appears at $E_F$ in the metallic regime as we dope the system. This behavior does not seem reproduced by the single band Hubbard model Eq.(1) in 3D, whose physics appears to be very close to that of 2D. Indeed for the 2D cuprates it has been shown experimentally that the states found at $E_F$ upon doping are already present at half-filling.[@fujimori3] An exception among 3D materials is $\rm{NiS_{2-x}Se_x}$ which remains antiferromagnetic throughout the metal-insulator transition induced by (homovalent) Se-substitution or temperature. PES spectra at $x=0.5$ for different temperatures [@matsuura] show a strong peak close to the Fermi energy which does $not$ disappear in the insulator. Instead the peak is shifted off the Fermi energy and only very slightly reduced in weight. Since this situation is not described within the paramagnetic $D=\infty$ approach, AF correlations are presumably essential for the low energy electronic excitations of this system. The success of the $D=\infty$ approach to the Hubbard model in describing the physics of ${\rm Y_{1-x} Ca_x Ti O_3}$, ${\rm SrVO_3}$ and ${\rm CaVO_3}$, however, appears crucially to depend upon forcing the paramagnetic solution of the equations.[@rozen1] In this case, states are actually $generated$ in the Hubbard gap after a small hole doping is introduced. Of course, it may be that the “arbitrary” choice of this paramagnetic solution, which is not the actual minimum of the free energy, is well motivated since it mimics the presence of physical effects like frustration which destroy long range order in real materials. More work is needed to show that this scenario is realized for realistic densities and couplings. An alternative explanation for the discrepancy between the PM solution in infinite-D and finite dimensional results may lie in the finite resolution of the combination of Monte Carlo simulations and Maximum Entropy techniques. However, the SCBA and results at half-filling and low T show that it is likely that at $\langle n \rangle =1$ we have quasiparticle states in the DOS. In studies of the single band Hubbard Hamiltonian in 2D, and in the present analysis in 3D, it is clear that short-range AF correlations play an important role close to $\langle n \rangle =1$. In particular, the states created at the top of the valence band are likely to be spin polarons with a finite quasiparticle residue Z. PES states observed at finite hole doping evolve continuously from those present at half-filling. Experiments on the 2D high-Tc cuprates seem to present similar features, while the results for the 3D perovskites are very different in the sense that no remnants of the coherent part of the spectrum away from half-filling are reported at half-filling. Still, strong AF correlations are apparently present in several 3D transition metal oxides and influence the low energy spectrum at least on the insulating side of the transition. The introduction of frustration in the single band Hubbard model in 3D, perhaps through next-nearest neighbor hoppings, will reduce AF correlations and in particular the AF-induced charge gap, and might be sufficient to observe a evolution of spectral weight upon doping closer to the experimental findings. However, it might be that 3D models which explicitly include orbital degeneracy will be necessary to reproduce the physics of the transition metal oxides, as has recently been described for NiO chains[@nio] and Mn-oxides.[@muller]. Indeed, a recent argument presented by Kajueter et al.[@kajueter] to justify the use of the $D=\infty$ model provides a more realistic explanation for the apparent link between theory in this limit, and 3D transition-metal oxide results. 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ArXiv

--- author: - Roman Klokov - Edmond Boyer - Jakob Verbeek title: | Discrete Point Flow Networks\ for Efficient Point Cloud Generation ---

ArXiv

--- abstract: 'Recent work highlights that tens of Galactic double neutron stars are likely to be detectable in the millihertz band of the space-based gravitational-wave observatory, LISA. Kyutoku and Nishino point out that some of these binaries might be detectable as radio pulsars using the Square Kilometer Array (SKA). We point out that the joint LISA+SKA detection of a $f_\text{gw}\gtrsim\unit[1]{mHz}$ binary, corresponding to a binary period of $\lesssim\unit[400]{s}$, would enable precision measurements of ultra-relativistic phenomena. We show that, given plausible assumptions, multi-messenger observations of ultra-relativistic binaries can be used to constrain the neutron star equation of state with remarkable fidelity. It may be possible to measure the mass-radius relation with a precision of [$\approx$0.2%]{} after $\unit[10]{yr}$ of observations with the SKA. Such a measurement would be roughly an order of magnitude more precise than possible with other proposed observations. We summarize some of the other remarkable science made possible with multi-messenger observations of millihertz binaries, and discuss the prospects for the detection of such objects.' author: - Eric Thrane - Stefan Osłowski - Paul Lasky bibliography: - 'bibliography.bib' title: 'Ultra-relativistic astrophysics using multi-messenger observations of double neutron stars with LISA and the SKA' --- Detecting ultra-relativistic Galactic binaries with LISA ======================================================== Kyutoku and Nishino [@Kyutoku] recently pointed out that the Laser Interferometer Space Antenna (LISA) [@LISA] is likely to detect ultra-relativistic, Galactic double neutron stars, some of which could be subsequently detected in radio with follow-up from the Square Kilometer Array (SKA) [@SKA]. These millihertz binaries have the potential to probe a regime of relativistic astrophysics not accessible with currently known binary systems. For example, the Double Pulsar (PSR J0737–3039) has an orbital period of $\unit[2.5]{hr}$ and semi-major axis $a=\unit[6\times10^{-3}]{AU}$ [@DoublePulsar]. In contrast, the double neutron stars observed by LISA with gravitational-wave frequencies $\gtrsim\unit[1]{mHz}$, will have binary periods of $P_B\lesssim\unit[2000]{s}$ and semi-major axes $a\lesssim\unit[1\times10^{-3}]{AU}$ [^1]. The number of double neutron stars emitting gravitational waves above $\unit[1]{mHz}$ can be estimated using the double neutron star merger rate inferred from LIGO/Virgo following the detection of GW170817 [@GW170817]. Following [@Kyutoku; @Kyutoku2], we estimate $$\begin{aligned} N_\text{LISA} = & (47-690) \left(\frac{{\cal M}}{\unit[1.2]{M_\odot}}\right)^{-5/3} \left(\frac{f_\text{gw}}{\unit[1]{mHz}}\right)^{-8/3} .\end{aligned}$$ Here, ${\cal M}$ is chirp mass, which we take throughout to be $1.2 M_\odot$ (corresponding to an equal mass binary with $1.38 M_\odot$ components). The range of values (90% credible interval) comes from uncertainty in the merger rate. While not all of these binaries will be detectable by LISA, many of them will be. We adopt the convention that a double neutron star is detectable if it produces a matched-filter signal-to-noise ratio $\rho>7$; see, for example, [@Robson]. We calculate typical signal-to-noise ratios using [@Seto] $$\begin{aligned} \widehat\rho \equiv & \langle\rho^2\rangle^{1/2} \\ = & \frac{8G^{5/3}T^{1/2}{\cal M}^{5/3}\pi^{2/3}}{5^{1/2}c^4d} \left(\frac{f_\text{gw}^{2/3}}{S_n^{1/2}(f_\text{gw})}\right) ,\end{aligned}$$ which is the square root of the signal-to-noise ratio squared, averaged over binary orientation and sky location [@Robson]. Here, $T$ is the observation time, $G$ is the gravitational constant, $c$ is the speed of light, $d$ is the distance, and $S_n(f_\text{gw})$ is the noise power-spectral density. We model the LISA noise curve (shown below in Fig. \[fig:asd\]) using the $T=\unit[4]{yr}$ prescription from [@Robson], which includes the effect of foreground from white-dwarf binaries. Using this expression, one finds that a $\unit[1]{mHz}$ binary can be detected to distances of $d\approx\unit[9]{kpc}$ (beyond the distance to the Galactic Center), while a $\unit[5]{mHz}$ binary can be detected to distances of $d\approx\unit[590]{kpc}$, 75% the distance to Andromeda. This result is roughly consistent with other estimates of the number of double neutron stars detectable with LISA. Kremer et al. [@NU] examined the population of LISA-band binaries in the globular clusters of the Milky Way. They estimate $22$ globular-cluster double neutron stars will radiate in the LISA band ($\approx\unit[10^{-2}-100]{mHz}$). Of these, two double neutron stars are likely to be detected above the LISA noise floor. Since many binaries in globular clusters form dynamically, many of these systems have significant eccentricity. The number of millihertz binaries in globular clusters is likely to be small compared to the number of millihertz binaries in the field since the prevalence of millihertz binaries is directly related to the double neutron star merger rate, and $N$-body studies predict that globular clusters are relatively inefficient at merging double neutron stars; see, e.g., [@Belczynski]. See also work by Seto [@Seto] and Lau et al. [@Lau], who considered the population of LISA-band double neutron stars in the Local Group. Our best guess for the Galactic rate of double neutron star mergers is $\unit[1.5\times10^{-4}]{MWEG^{-1} yr^{-1}}$ [@GW170817], which implies a typical time between mergers in the Milky Way of $\sim\unit[6700]{yr}$. (Here, MWEG stands for “Milky Way Equivalent Galaxy.”) Therefore, our best guess for the shortest time to merge for a Galactic double neutron star binary is half that: $\unit[3300]{yr}$. Below, we focus our attention on binaries that are, in principle, observable as pulsars. Here, we assume that $\approx10\%$ of the recycled neutron stars in double neutron star systems are detectable as pulsars due to beaming effects. Thus, our best guess for the shortest time to merge for a Galactic double neutron star binary [*with a potentially observable radio pulsar*]{} is [[$\unit[33]{kyr}$]{}]{}. Let us imagine that such a double neutron star binary exists in the Milky Way and give it a fictitious name: PSR J1234–5678 or “J1234” for short. We proceed to investigate the properties of J1234, and then see how our results would change assuming different times to merge. For reasons that will become clear momentarily, we are especially interested in millihertz binaries with non-negligible eccentricity. Therefore, let us further suppose that J1234 was born recently through unstable “case BB” mass transfer. Such systems have been hypothesized as possible progenitors for binary neutron star mergers [@Ivanova; @Belczynski2; @VignaGomez] as well as sources for $r$-process enrichment in ultra-faint dwarf galaxies [@Safarzadeh]. In this scenario, a neutron star - helium star binary undergoes unstable mass transfer, leading to a common envelope event, and eventually—in some cases—a neutron star - helium core binary with an orbital period of ${\cal O}(\unit[1000]{s})$. Since the binary is so tight at this stage of its evolution, it is likely to survive when the helium core undergoes a supernova, leading to a double neutron star binary likely to merge in $\lesssim\unit[10]{Myr}$. A binary with such a short life time can retain significant eccentricity as it passes through the LISA band. For illustrative purposes, we assume that J1234 was born with an eccentricity of [[$e_0=0.75$]{}]{} (typical of population synthesis studies) and a period of [[$P_b^0=\unit[12]{ks}$]{}]{} giving it a lifetime of $\unit[10]{Myr}$ [@Peters64]. We evolve J1234 forward in time using the standard prescription from [@Peters64], so that it is [[$\unit[33]{kyr}$]{}]{} from merger. At this point in its evolution, the orbital period of J1234 is [[$P_b = \unit[490]{s}$]{}]{} and the eccentricity is [[$e=0.11$]{}]{}, which is comparable to the Double Pulsar. Approximating the orbit as quasi-circular, the gravitational-wave frequency is [[$\unit[4.1]{mHz}$]{}]{}. In the remainder of this note, we consider some of the science possible by multi-messenger study of J1234. (In our closing remarks, we discuss how the results would change given different assumptions about J1234.) In Fig. \[fig:asd\], we show the effective gravitational-wave strain from J1234 relative to the LISA noise curve assuming a typical distance of $d=\unit[10]{kpc}$ and an observation time of $T=\unit[4]{yr}$ $$\begin{aligned} h_\text{eff}^2 \equiv \frac{32}{5}f_\text{gw}^2TS_h(f),\end{aligned}$$ where $$\begin{aligned} S_h(f) = \frac{{\cal A}^2(f)}{2T} ,\end{aligned}$$ and $$\begin{aligned} {\cal A}^2(f) = \frac{5}{24}\frac{(G{\cal M}/c^3)^{5/3}f_\text{gw}^{-7/3}}{\pi^{4/3}d^2/c^2} .\end{aligned}$$ As noted above, the noise curve is calculated using the $T=\unit[4]{yr}$ prescription from [@Robson]. We include instrumental noise (red) as well as confusion noise from the white dwarf background (blue). The effective strain from J1234 is indicated with the green star. Approximating the orbit as quasi-circular, the gravitational waves from J1234 would produce a matched filter signal-to-noise ratio of [[$\rho=360$]{}]{}. (This implies a $\rho=7$ detection distance of [[$\unit[510]{kpc}$]{}]{}, about 67% the distance to Andromeda). The dotted green line shows the evolution of the binary from $\unit[1.06]{mHz}$ ([[$\unit[800]{kyr}$]{}]{} away from merger), when the binary would have first become detectable to LISA with $\rho\gtrsim7$. LISA is capable of precise localisation because of the large baseline created from its orbit around the Sun. Monochromatic sources like J1234 can be localised to within a solid angle of [@Kyutoku] $$\begin{aligned} \Delta\Omega = \unit[0.036]{deg^2} \left(\frac{\rho}{200}\right)^{-2} \left(\frac{f_\text{gw}}{\unit[4]{mHz}}\right)^{-2} ,\end{aligned}$$ which works out to [[$\unit[0.011]{deg^2}$]{}]{} for J1234, well within a single SKA beam. The distance is measured to a fractional uncertainty of of [@Kyutoku] $$\begin{aligned} \frac{\Delta d}{d} = 0.01 \left(\frac{\rho}{200}\right)^{-1} ,\end{aligned}$$ which implies that the distance to J1234 will be known from LISA measurements to [0.6%]{}. Knowing the distance so precisely will be useful in order to control syetmatic error from proper motion. ![ The gravitational-wave effective strain from J1234 relative to the LISA amplitude spectral density noise curve. The confusion noise from white dwarf binaries is shown in blue. The instrumental noise is shown in red. The total noise is black. The effective strain from J1234 is shown as a green star assuming a distance of $d=\unit[10]{kpc}$. The dotted green line shows the evolution of J1234 from $\unit[1]{mHz}$ when it would have first been detectable by LISA. []{data-label="fig:asd"}](asd.png){width="49.00000%"} The early phase of SKA is expected to quadruple the population of known pulsars in the Galaxy, and the full SKA will detect all pulsars in our Galaxy beaming towards Earth, thus increasing the sample of known double neutron stars by a factor of 5-10 [@Keane2018; @Levin2018]. While population synthesis studies often adopt the work of Tauris and Manchester [@TM98] to estimate the beaming fraction, this is an optimistic assumption for double neutron star systems as the relation between the period and beaming fraction is a best fit for pulsars with longer periods. Double neutron star systems are likely to have tighter beams. For example, the recycled star in the Double Pulsar system has a beaming fraction of about 10%, and accordingly only one in ten double neutron star systems might be seen by the SKA. A further complication of finding millihertz binaries is that they are highly accelerated. While significant progress has been made in this area [e.g., @Cameron18], this still remains a computationally challenging aspect of pulsar searching. However, if J1234 was discovered first by LISA, a targeted search would largely alleviate this problem. In the discussion that follows we assume that J1234 is detectable by the SKA. Lense-Thirring Precession ========================= Let us suppose that one of the neutron stars in J1234 is rapidly rotating. This rotation will give rise to Lense-Thirring precession. Lense-Thirring precession is one of four sources of periastron advance contributing to the total periastron advance [@Bagchi] of a pulsar $a$ with companion $b$ $$\begin{aligned} \dot\omega_\text{tot}^a = \dot\omega_\text{PN}^a + \dot\omega_\text{secular} + \dot\omega_\text{LT}^a + \dot\omega_\text{LT}^b .\end{aligned}$$ Here, $\dot\omega_\text{PN}^a$ is the post-Newtonian correction at $a$ due to the spacetime curvature from both $a$ and $b$; $\dot\omega_\text{secular}$ is secular variation due to the change in the apparent orientation of the binary with respect to the line of sight because of the proper motion of the binary barycenter; $\dot\omega_\text{LT}^a$ is the Lense-Thirring precession from $a$; and $\dot\omega_\text{LT}^b$ is the Lense-Thirring precession from $b$. We make a number of simplifying assumptions. We assume that: (i) $b$ is slowly rotating so that $\dot\omega_\text{LT}^b=0$; (ii) the neutron stars have essential equal masses; (iii) the binary is viewed edge-on with inclination angle $\iota=90^\circ$; (iv) the spin vector of pulsar $a$ is aligned to the total angular momentum. Given these assumptions, and following the calculation from [@Bagchi], the advance of periastron from Lense-Thirring precession is  $$\begin{aligned} \dot\omega_\text{LT}^a = & 14\pi^3 \frac{I_a}{P_b^2 P_a M c^2} .\end{aligned}$$ Here, $I_a$ is the moment of inertia for pulsar $a$, $P_b$ is the binary period, and $P_a$ is the spin period of $a$. Plugging in plausible values for the parameters of J1234, we obtain $$\begin{aligned} \dot\omega_\text{LT}^\text{J1234} = & {{\color{black}\unit[4.1\times10^{-2}]{deg\,yr^{-1}}}} \Bigg[ \left(\frac{I_a}{\unit[1.26\times10^{45}]{g\,cm^2}}\right) \nonumber\\ & \left(\frac{{{\color{black}\unit[490]{s}}}}{P_b}\right)^2 \left(\frac{\unit[20]{ms}}{P_a}\right) \left(\frac{\unit[2.76]{M_\odot}}{M}\right) \Bigg] .\end{aligned}$$ Previous work has studied how the SKA will be able to measuring Lense-Thirring precession in the Double Pulsar [@Kehl]. According to these authors, the SKA will achieve a Double Pulsar precession sensitivity of $\dot\omega=\unit[10^{-4}]{deg\,yr^{-1}}$ with after four years of SKA1 and $\dot\omega=\unit[2\times10^{-5}]{deg\,yr^{-1}}$ after ten years of SKA (4 years of SKA1 + an additional 6 years of SKA at design sensitivity). While the magnitude of Lense-Thirring precession is greater for short-period binaries, scaling like $\dot\omega_\text{LT}\propto P_b^{-2}$, the pulsar-timing signal scales like the semi-major axis $\tau\propto a\sin\iota$. Taking into account these two effects, and applying Kepler’s third law,the signal-to-noise ratio for Lense-Thirring precession scales as $$\begin{aligned} \text{SNR} \propto & P_b^{-4/3} \sin\iota . $$ Naively applying this scaling law—we discuss caveats momentarily—we estimate that the precession of J1234 can be measured with signal-to-noise ratio of SNR=[[$60$]{}]{} after four years of operation and SNR=[[$300$]{}]{} after 10 years of operation. The precision of timing experiments is predominantly determined by the sensitivity of the telescope relative to the flux of the pulsar and the spin period of the pulsar. The Double Pulsar spin period is quite typical for a double neutron star system [@Oslowski2011]. The measurement of binary period derivative is largely limited due to various effects affecting the observed magnitude, such as the Shklovskii effect and the uncertainty of the Galactic potential. However, the combination of a good estimate of distance to J1234 combined with a model of the Galactic potential to large distances from Earth provided by Gaia and other surveys [@Gaia2016] will mitigate these potential problems. The main special property of the Double Pulsar is its near edge-on alignment and the sweeping of the pulsar A’s beam across the magnetosphere of pulsar B, enabling exciting physical experiments [@Breton2008]. However, neither of these two conditions are prerequisite for measuring precession. Another property of the Double Pulsar, which may not be unique, but which is relevant here, is that J0737-3039A is an aligned rotator and, not affected by geodetic precession. If that was not the case in J1234, the precession period would be about 150 times shorter than for the Double Pulsar [@Stairs2003], which means it would precess about twice a year. In this case, timing measurements of J1234 could be adversely affected by reduced cadence. However, given the expected length of the observing campaign, and a good coverage of the orbit thanks to the short orbital period, the timing would still provide a good measurement if a phase-connected solution can be found between epoch in which the pulsar is beaming towards Earth. Moreover, precession can increase the fraction of binaries beaming towards Earth while helping to constrain other binary parameters of the system. Here, we assume that J1234 is either an aligned rotator, or that the impact of precession on the timing measurement of required parameters is not deleterious. Therefore, our naive scaling argument provides a reasonable approximation for our ability to measure Lense-Thirring precession in J1234. Equation of State ================= Previous work has noted that Lense-Thirring precession in the Double Pulsar can be used to constrain the neutron star equation of state [@Damour; @Kehl; @Bagchi]. The neutron-star moment of inertia depends on the mass, the radius, and the equation of state [@Ravenhall]. Since the mass of J1234 and its companion are precisely determined using pulsar timing, one can compare the measured moment of inertia with the moment of inertia implied by $M$ and $R(M)$. The uncertainty in $R(M)$ can be approximated as $$\begin{aligned} \frac{\delta R(M)}{R(M)} \approx \frac{1}{2} \frac{\delta I}{I} .\end{aligned}$$ Thus, an SNR=[[300]{}]{} measurement of $\dot\omega$ will constrain $R(M)$ for one value of $M$ to a precision of [$\approx$0.2%]{}. We note that there are unlikely to be many such multi-messenger measurements of ultra-relativistic double neutron stars because there are a limited number in the LISA-band. Thus, other measurements of the neutron star equation of state, for example, from LIGO [@GW170817_eos] and/or NICER [@nicer], will be needed to supplement constraints on $R(M)$ from ultra-relativistic binaries. In order to investigate how this result is affected by uncertainty in the double neutron star merger rate, we repeat the calculation above using the extreme values of the 90% rate credible interval. A minimum (maximum) rate of $\unit[320]{Gpc^{-3}yr^{-1}}$ ($\unit[4740]{Gpc^{-3}yr^{-1}}$) implies a J1234 merger time of [[$\unit[3.2\times10^5]{yr}$ ($\unit[2.2\times10^4]{yr}$)]{}]{} [^2] These times to merger imply gravitational-wave frequencies of [[$f_\text{gw}=\unit[1.6]{mHz}$ ($f_\text{gw}=\unit[4.7]{mHz}$)]{}]{}. Even at a reduced frequency of [[$f_\text{gw}^\text{min}=\unit[1.6]{mHz}$]{}]{}, J1234 could be observed out to a distance of [[$d=\unit[42]{kpc}$]{}]{} (most of the Milky Way). The eccentricity in the LISA band would be [[$e=0.25$ ($e=0.09$)]{}]{}. The resulting Lense-Thirring signal would change to [[$\dot\omega_\text{LT}=\unit[0.03]{deg\,yr^{-1}}$ ($\unit[0.28]{deg\,yr^{-1}}$)]{}]{}, which corresponds to a signal-to-noise ratio of [[SNR=46 (SNR=400)]{}]{} after ten years of SKA. Thus, taking into account Poisson uncertainty from the double neutron star merger rate, we calculate that the neutron star $R(M)$ can be constrained with J1234 at the level of approximately [$0.1\%-1\%$]{} (90% credible interval) using ten years of SKA. Discussion ========== Double neutron stars represent an exciting multi-messenger source for LISA and the SKA. LISA is adept at detecting millihertz double neutron stars that would be challenging to detect in the radio owing to their large accelerations. Once the binary has been identified by LISA, the SKA can (1) separate out double neutron stars from binaries with one or more white dwarf and (2) measure the binary evolution with a sensitivity that is many orders of magnitude better than LISA. Such ultra-relativistic binaries provide a unique laboratory for a variety of tests, including tests of general relativity and the composition of neutron stars. While neutron stars have received considerable attention in the SKA literature, they are to some extent overshadowed by black hole science in the LISA literature. In light of the science made possible by multi-messenger observations by LISA and the SKA, it seems that there are exciting new lines of inquiry worth pursuing in the future. We highlight a couple here. First, using the SKA and LISA, it may be possible to study the population properties of double neutron stars in the Milky Way, studying the population changes with binary period. Second, LISA is capable of detecting the most ultra-relativistic double neutron stars like J1234 ($\gtrsim\unit[10]{mHz}$) in nearby galaxies. Using ephemerides provided by LISA, it may be possible to observe these pulsars with the SKA as well; see, for example, [@Kramer]. We thank Ilya Mandel and Simon Stevenson for helpful discussions on millihertz binaries. This work is supported through Australian Research Council (ARC) Future Fellowships FT150100281, FT160100112, FL150100148, Centre of Excellence CE170100004, and Discovery Project DP180103155. [^1]: We note that LISA is sensitive to gravitational waves at frequencies below $\unit[1]{mHz}$, but we focus on $\gtrsim\unit[1]{mHz}$ since binaries in this band are easier for LISA to detect, and they are more relativistic. [^2]: Note, this time to merger includes the aforementioned factor of ten penalty, which takes into account the fact that not all double neutron stars will are likely to include a visible pulsar.

ArXiv

--- abstract: 'We present the results from a monitoring campaign of the Narrow-Line Seyfert 1 galaxy PG 1211+143. The object was monitored with ground-based facilities (UBVRI photometry; from February to July, 2007) and with [[*Swift*]{}]{}  (X-ray photometry/spectroscopy and UV/Optical photometry; between March and May, 2007). We found PG 1211+143 in a historical low X-ray flux state at the beginning of the [[*Swift*]{}]{} monitoring campaign in March 2007. It is seen from the light curves that while violently variable in X-rays, the quasar shows little variations in optical/UV bands. The X-ray spectrum in the low state is similar to other Narrow-Line Seyfert 1 galaxies during their low-states and can be explained by a strong partial covering absorber or by X-ray reflection onto the disk. With the current data set, however, it is not possible to distinguish between both scenarios. The interband cross-correlation functions indicate a possible reprocessing of the X-rays into the longer wavelengths, consistent with the idea of a thin accretion disk, powering the quasar. The time lags between the X-ray and the optical/UV light curves, ranging from $\sim$2 to $\sim$18 days for the different wavebands, scale approximately as $\sim \lambda^{4/3}$, but appear to be somewhat larger than expected for this object, taking into account its accretion disk parameters. Possible implications for the location of the X-ray irradiating source are discussed.' title: 'Studying X-ray reprocessing and continuum variability in quasars: PG 1211+143' --- \[firstpage\] quasars: individual: PG 1211+143; quasars: general; galaxies:active, photometry; accretion, accretion disks Introduction ============ Powered by accretion, supposedly onto a supermassive black hole, quasars (Active Galactic Nuclei, AGN) are long known mostly as highly energetic, exotic objects in the hearts of the galaxies. Not until recently was their key role in galaxy evolution realized, revealed mostly as a strong correlation between the properties of the central black hole and that of the host galaxy (Magorrian et al. 1998, Ferrarese & Merritt 2000). Studying quasars, therefore, is not only important to understand the underlying physics; it can also help to shed some light on the strange interplay between the accreting matter from the host and outflows from the center, which ultimately shape both – the black hole and the galaxy. Although a general picture of the structure of a typical quasar seems to be widely accepted (e.g. Elvis 2000; see also Krolik 1999), there are still many details in this picture that are not fully understood. Many of the problems to be solved concern AGN continuum variability – a rather common property, observed in practically all energy bands. Its universality indicates perhaps that variability should be an intrinsic property of the processes, responsible for continuum generation. The optical/UV to X-ray part of the continuum spectrum, as typically assumed, originates from an accretion disk around the central supermassive black hole. Generally, X-ray variability can be caused by several factors: a change in the accretion rate; variable absorption (e.g. Abrassart & Czerny 2000); variable reflection (e.g. through a change of the height of the irradiating source, Miniutti & Fabian 2004; see also Gallo 2006, Done & Nayakshin 2007); some combination of reflection and absorption (e.g. Chevalier et al. 2006; Turner & Miller 2009); hot spots orbiting the central black hole (Turner et al. 2006; Turner & Miller 2009); local flares (Czerny et al. 2004), etc. The AGN type with the strongest X-ray variability is the class of Narrow-Line Seyfert 1 galaxies (NLS1s, e.g. Osterbrock & Pogge, 1985). In addition, NLS1s show the steepest X-ray spectra seen among all AGN (e.g. Boller et al. 1996, Brandt et al. 1997, Leighly 1999a, b, Grupe et al. 2001). Most of their observed properties, like spectral slopes, FeII and \[OIII\] line ratios, CIV shifts, etc., appear to be driven by the relatively high Eddington ratio $L/L_{\rm Edd}$ in these objects (e.g. Sulentic et al. 2000, Boroson 2002, Grupe 2004, Bachev et al. 2004). What concerns the optical/UV variability, the picture there is even more puzzling. There are many factors that can contribute to the variations of the optical flux, but most of them can account for the long-term (months to years) changes. There are often reported in many objects, however, short-term (day to week) optical/UV variations, simultaneous with or shortly lagging behind the X-ray variations. An interesting idea that can explain such a behaviour is reprocessing of the highly variable X-ray emission into optical/UV bands. In this paper we address the question of the relations between the X-ray and the optical/UV emission by studying the variability from X-rays to I-band of the NLS1 PG 1211+143. This NLS1 has been the target of almost all major X-ray observatories since EINSTEIN (Elvis et al. 1985). The X-ray continuum displays a strong and variable soft X-ray excess (Pounds & Reeves 2007). From XMM-Newton RGS data, Pounds et al. (2003) suggested the presence of high-velocity outflows in PG 1211+143, a result that was questioned by Kaspi & Behar (2006). However, high-velocity outflows seen in X-rays have been repeatedly reported (e.g. Leighly et al. 1997) and new XMM-Newton data of PG 1211+143 (Pounds & Page 2006) seem to confirm the previous claims made by Pounds et al. (2003). Our primary goal is to find out if and how the X-ray variations are transferred into the longer-wavelength continuum. Time delays between the flaring X-ray emission, presumably coming from a compact, central source and the optical/UV light curves are expected, provided the X-rays are reprocessed in the outer, colder part of an accretion disk. Such a study may have implications on two important problems – the radial temperature distribution of an accretion disk (and hence – the type of the disk) and the location of the X-ray source, based on how much the disk “sees” it. This paper is organized as follows: In Section 2 we describe the [[*Swift*]{}]{} and ground-based optical monitoring observations. Section 3 focuses on presenting the results of this study and in Section 4 we discuss these results in the context of the general picture of AGN. Throughout the paper spectral indexes are denoted as energy spectral indexes with $F_{\nu} \propto \nu^{-\alpha}$. Luminosities are calculated assuming a $\Lambda$CDM cosmology with $\Omega_{\rm M}$=0.27, $\Omega_{\Lambda}$=0.73 and a Hubble constant of $H_0$=75 km s$^{-1}$ Mpc$^{-1}$. Observations and reductions =========================== Swift data ---------- The [[*Swift*]{}]{} Gamma-Ray Burst (GRB) explorer mission (Gehrels et al. 2004) monitored PG 1211+143 between 2007 March 08 and May 20. Note, that scheduled observations were twice bumped by detections of Gamma-Ray-Bursts[^1], explaining the absence of segments 15 and 20 (Table A1). After our monitoring campaign in 2007, PG 1211+143 was re-observed by [[*Swift*]{}]{} in February 2008 (segment 24) However, this observation was used to slew between two targets. Therefore, this observation is very short (188s) and no X-ray spectra or UVOT photometry data were obtained. This observation only allows to measure a count rate. A summary of all [[*Swift*]{}]{} observations is given in Table\[obs\_log\]. The [[*Swift*]{}]{} X-Ray Telescope (XRT; Burrows et al. 2005) was operating in photon counting mode (PC mode; Hill et al. 2004) and the data were reduced by the task [*xrtpipeline*]{} version 0.10.4, which is included in the HEASOFT package 6.1. Source photons were selected in a circular region with a radius of 47$^{''}$ and background region of a close by source-free region with $r=188^{''}$. Photons were selected with grades 0–12. The photons were extracted with [*XSELECT*]{} version 2.4. The spectral data were re-binned by using [*grppha*]{} version 3.0.0 having 20 photons per bin. The spectra were analyzed with [*XSPEC*]{} version 12.3.1 (Arnaud 1996). The ancillary response function files (arfs) were created by [*xrtmkarf*]{} and corrected for vignetting and bad columns/pixels using the exposure maps. We used the standard response matrix [*swxpc0to12s0\_20010101v010.rmf*]{}. Especially during the low-state the number of photons during one segment is too small to derive a spectrum with decent signal-to-noise. Therefore we co-added the data of several segments to obtain source and background spectra. In order to examine spectral changes at different flux/count rate levels, we created spectral for the low, intermediate, and high states with count rates CR $<$0.12 counts s$^{-1}$, 0.13$<$CR$<$0.2, and CR $>$0.2 counts s$^{-1}$. This lead to high state source and background spectra co-adding the data from 2007 April 22, May 09 and 14 (segments 018, 021, and 022), 2007 March 26 and April 02 (segments 13 and 14) for the intermediate state, and all other for the low state. As for the arfs, we created an arf for each segment and coadded them by using the ftool [*addarf*]{} weighted by the exposure times. Due to the low number of photons in the February 2008 observation (segment 024) no spectra could be derived. Fluxes in the 0.2–2.0 and 2–10 keV band for this segment were determined from the count rates in these bands by comparing the fluxes during the high state during segments 018, 021, and 022, assuming no spectral changes. All spectral fits were performed in the observed 0.3–10.0 keV energy band. In order to compare the observations from different missions we use the HEASARC tool [*PIMMS*]{} version 3.8. Data were also taken with the UV/Optical Telescope (UVOT; Roming et al. 2005), which operates between 1700–6500 Å using 6 photometry filters. Before analyzing the data, the exposures of each segment were co-added by the UVOT task [*uvotimsum*]{}. Source counts were selected with the standard 5$^{''}$ radius in all filters (Poole et al. 2008) and background counts in a source-free region with a radius r=20$^{''}$. The data were analyzed with the UVOT software tool [*uvotsource*]{} assuming a GRB like power law continuum spectrum. The magnitudes were all corrected for Galactic reddening $E_{\rm B-V}=0.035$ given by Schlegel et al. (1998) using the extinction correction in the UVOT bands given in Roming et al. (2009). [[*XMM-Newton*]{}]{} data analysis ---------------------------------- In order to compare the results derived from the [[*Swift*]{}]{} observations we also analyzed the [[*XMM-Newton*]{}]{} data of PG 1211+143. [[*XMM-Newton*]{}]{} observed PG 1211+143 on 2001 June 15 and 2004 June 21 for 53 and 57 ks, respectively (Pounds & Reeves 2007). Because our paper focuses on the [[*Swift*]{}]{} and ground-based monitoring campaigns in 2007 we reduced only the [[*XMM-Newton*]{}]{} EPIC pn data. A complete analysis of these [[*XMM-Newton*]{}]{} data sets can be found in Pounds et al. (2003), Pounds & Page (2006) and Pounds & Reeves (2007). The [[*XMM-Newton*]{}]{} EPIC pn data were reduced in the standard way as described e.g in Grupe et al. (2004). Ground-based observations ------------------------- Additional broad-band monitoring in UBVRI bands was performed on 3 telescopes: the 2-m RCC and the 50/70-cm Schmidt telescopes of Rozhen National Observatory, Bulgaria and the 60-cm telescope of Belogradchik Observatory, Bulgaria. All telescopes are equipped with CCD cameras: the 2-m telescope with a VersArray CCD, while the smaller telescopes – with SBIG ST-8. Identical (U)BVR$_{\rm c}$I$_{\rm c}$ filters are used in all telescopes. The ground-based monitoring covered a period of about 5 months (February – July, 2007), during which the object was observed in more than 40 epochs in BVRI bands, and occasionally – in U. The photometric errors varied significantly depending on the telescope, the filter, the camera in use and the atmospheric conditions, but were typically 0.02 – 0.03 mag. (rarely up to $\sim$0.1 mag in some filters for the smaller instruments). Results ======= Long-term X-ray Light Curve --------------------------- Figure\[pg1211\_xray\_lt\_lc\] displays the long-term 0.2–2.0 and 2.0–10.0 keV light curves. Most of the data prior 2000 were taken from Janiuk et al. (2001). The ROSAT All-Sky Survey point at 1990.9 was taken from Grupe et al. (2001). The XMM 2001 and 2004 and the [[*Swift*]{}]{} fluxes were from our analysis as presented in this paper. PG 1211+143 has become fainter over the last decades in both soft and hard bands with the strongest changes in the soft band. Historically, in the early 1980s, PG 1211+143 had been in a much brighter X-ray state than over the last decade. During the beginning of the [[*Swift*]{}]{} monitoring campaign in March 2007, PG 1211+143 appeared to be in the lowest state seen so far. At the end of our monitoring campaign in May 2007 PG 1211+143 was back in the high state that was previously known from the XMM observations. The latest data point is from February 2008. The 0.2–2.0 and 2–10 keV fluxes during that observation are comparable with the XMM observations. As for the [[*Swift*]{}]{} data taken in 2007 and 2008 we used the flux values obtained from the low and high state spectra. As for the February 2008 data flux points we converted the count rates into fluxes assuming an X-ray spectrum as seen during the [[*Swift*]{}]{} high states. [[*Swift*]{}]{} XRT and UVOT light curves ----------------------------------------- The [[*Swift*]{}]{} XRT count rates and hardness ratios[^2], and UVOT magnitudes are listed in Table\[xrt\_uvot\_res\]. These values are plotted in Figure\[swift\_lc\]. At the beginning of the [[*Swift*]{}]{} monitoring campaign in March 2007, PG 1211+143 was found in a very low state. Compared to previous ROSAT and [[*XMM-Newton*]{}]{} observations, reported by e.g. Grupe et al. (2001) and Pounds et al. (2003, 2006, 2007), PG 1211+143 appeared to be fainter by a factor of about 10. At the end of the monitoring campaign in May 2007, PG 1211+143 reached a flux level that was expected from the previous ROSAT and [[*XMM-Newton*]{}]{} observations. A later observation by [[*Swift*]{}]{} on 2008 February 17 found it at a level of 0.375[$\pm$]{}0.045 counts s$^{-1}$ and confirmed that it returned back in a high state. The low-state, found in March 2007, seems to be just a short temporary event. A behaviour like this is not unseen in AGN and has been recently reported for the NLS1 Mkn 335 by Grupe et al. (2007b, 2008a), which had been found in a historical low state by [[*Swift*]{}]{} and [[*XMM-Newton*]{}]{}. Besides the X-ray variability, PG 1211+143 also displays some variability at optical/UV wavelengths, although on a much smaller level than in X-rays. Table\[xrt\_uvot\_res\] lists the magnitudes measured in all 6 UVOT filters. All 6 light curves are also plotted in Figure\[swift\_lc\]. The most significant drop occurred in all 6 filters during the 2007 April 17 observation. During the next observation on 2007 April 22, PG 1211+143 not only became brighter again in the optical/UV, but also showed an increase in count rate by a factor of almost 4 in X-rays. Ground-based monitoring ----------------------- ### Secondary standards In order to facilitate future photometric studies of PG 1211+143, we calibrated secondary standards in the field of the object, shown as stars “A” (USNO B1 1039-0200330) and “B” (USNO B1 1040-0199800) in Figure 3. The magnitudes with the errors (due primarily to the errors of the calibration) are given in Table 1. Mostly Landolt standard sequences (Landolt 1992) were used for the calibration (PG 1633+099) and in some occasions – M67 (Chevalier & Ilovaisky 1991). Star B V R I ------ -------------- -------------- -------------- -------------- A 11.80 (0.08) 11.35 (0.05) 10.97 (0.06) 10.71 (0.05) B – 15.34 (0.10) 14.80 (0.07) 14.35 (0.07) : Field standards ### Magnitude adjustments The light curve (LC) of PG 1211+143 (Sect. 3.4) is built by measuring its differential magnitude in respect to the adjacent field stars, none of which showed signs of variability (with the exception of a known RR Lyr type star, CI Com, located very close to the quasar). Since the data are collected on different instruments using different cameras (even with identical filters) it is not unusual for an object with strong emission lines to show a differential magnitude, slightly depending on the instrument. The reason for this complication is mostly related to the nature of the quasar spectrum: if a strong emission line falls in a wavelength region where the cameras have different sensitivities, or the filter transparency is slightly different, one may get a broad-band differential magnitude depending on the instrument. In our case all R-band magnitudes of the quasar had to be adjusted by 0.1 mag for one of the instruments (the 50/70cm Schmidt telescope), probably due to the reasons described above. The adjustment corrections were easily obtained through comparison of the light curves, which cover each other on many occasions. We should note, however, that the exact quasar magnitudes are not of importance for this study, as the variations are only considered. Additionally, the ground-based UBV magnitudes were similarly adjusted to match the corresponding Swift magnitudes. Actually, after the correction for the Galactic reddening, the adjustments for the ground-based B and V magnitudes were very minor, typically less than 0.03 mag, which is smaller than the uncertainties of the calibrated field standards (Sect. 3.3.1). A log of the ground-based observations, including the obtained UBVRI magnitudes of the quasar after the corrections for the Galactic reddening is presented in Table A3. Combined light curves --------------------- The combined continuum light curves of PG 1211+143 for the time of monitoring are presented in Figs. 4 and 5. Fig. 4 compares the X-ray with the optical (UBVRI) variations, all transformed into arbitrary magnitudes for presentation purposes. The optical data are combined from all participating instruments. One sees that the erratic X-ray variations (almost 2.5 magnitudes) hardly influence the optical flux, which shows only minor variations on a generally decaying trend. Figure 5 presents the most intense period of the monitoring, comparing X-ray and V-band magnitudes. The V-band LC for that period stays remarkably stable, with a RMS smaller or comparable to the typical photometric errors. Time delays ----------- In order to study the time delay dependence of the wavelength we performed a linear-interpolation cross-correlation analysis (Gaskell & Sparke 1986) between the X-ray and the other bands’ light curves. The interpolation between the photometric points is needed due to unevenly sampled data and is one of the frequently used methods. Other methods applied in the literature do not seem to obtain significantly different results (e.g. discrete CCF, Edelson & Krolik 1986; z-transformed CCF, Alexander 1997) when compared. Figure 6 shows the interpolation cross-correlation functions, ICCF($\tau$), between the X-ray LC and the other band LCs. A maximum of an ICCF($\tau$) for a positive $\tau$ indicates a time delay behind the X-ray changes and is a signature of a possible reprocessing. Although the ICCFs are mostly negative, due to the different overall trends of the X-ray and optical/UV wave-bands, a clear maximum for a positive $\tau$ is evident for most wave-bands. Since the X-ray points distribution was far from a Gaussian, even on a magnitude scale, a rank correlation was attempted, but the resulting ICCFs appeared to be very similar. Table 2 and Figure 7 show the wavelength dependence of the time lag. The wavelengths are taken from the corresponding transmission curve of the filters used (with uncertainties associated with the band widths) and the time delays are from the ICCF maxima. One sees that for the I-band the highest peak of the ICCF (Figure 6) is at $\tau \simeq -2$ days, indicating a possible short lag of the X-rays behind the near-infrared emission. Another, lower peak at $\tau \simeq +18$ days is also evident. This maximum could probably be associated with reprocessing and is plotted in Figure 7 mostly to demonstrate its consistence with the fit (see below). However, the I-band response to the X-ray variations seems to be more complicated than the simple reprocessing model suggests, as we discuss later in Sect. 4 Uncertainties of the maximum of the ICCFs are difficult to assess. Although there are methods, described in the literature (Gaskell & Peterson 1987), one can hardly rely completely on so computed uncertainties, since the true behaviour of the light curve at the places where it is interpolated is anyway impossible to predict. That is why we accepted the width of ICCF profile at an appropriate level around the peak as an indicative of the uncertainty. This approach is very simple and in addition incorporates into the errors such unknowns as the inclination of the disk in respect to the observer, the spatial size of the irradiating source, etc. See Bachev (2009) for more discussions on these issues. A clear relation between $\tau$ and $\lambda$ is seen and an acceptable (nonlinear) fit to the data is $\tau_{\lambda}\simeq 9\lambda_{\rm 5000}^{4/3}$ \[days\][^3], where $\lambda_{\rm 5000}$ is $\lambda / 5000$Å. Section 4 discusses possible implications of this result and how it fits into the model of reprocessing from a thin accretion disk. Filter $\lambda_{\rm 0}$ (Å) FWHM (Å) $\tau$ (days) $\Delta \tau$ (days) -------- ----------------------- ------------ --------------- ---------------------- UVW2 1928 657 2.5 3 UVM2 2246 498 2.5 3.5 UVW1 2600 693 2 3 U 3465 785 4 3.5 B 4392 975 5 3.5 V 5468 769 13 3 R $\sim$6500 $\sim$700 14 4 I $\sim$8300 $\sim$1000 $-$2 (18) 3 (4) : Wavelength dependence of the time delays with uncertainties. For I-band, the highest and the first positive (in parentheses) ICCF maxima are shown (see the text). X-ray Spectroscopy ------------------ As described in Section 2.1, the data were combined to derive low, intermediate and high-state spectra of PG 1211+143. These data were first fitted by a single absorbed power law model with the absorption column density fixed to the Galactic value (2.47$\times 10^{20}$ cm$^{-2}$; Kalberla et al. 2005). Table\[xrt\_xspec\] lists the spectral fit parameters. Obviously a single power law model does not represent the observed spectrum. Figure\[xrt\_xspec\_plot\] displays this fit simultaneously to the low and high state [[*Swift*]{}]{} spectra. As a comparison, Table\[xrt\_xspec\] also lists the results for the fits to the 2001 and 2004 [[*XMM-Newton*]{}]{} EPIC pn data. As the next step we fitted the spectra with a broken power law model. Although this model significantly improves the fits and describes the spectra quite well, it is not a physical model. Especially in the low-state the hard X-ray spectral slope appears to be very flat with [$\alpha_{\rm X}$]{}=–0.18. This behaviour is typical when the X-ray spectrum is either affected by partial covering absorption or reflection (e.g. Turner & Miller 2009, Grupe et al. 2008a). Next the spectra were fitted with a power law model with a partial covering absorber. These fits suggest a strong partial covering absorber in the low-state spectrum with an absorption column density of the order of 9$\times 10^{22}$ cm$^{-2}$ and a covering fraction of 95%. During the intermediate state the column density of the absorber decreases to 8$\times 10^{22}$ cm$^{-2}$ with a covering fraction of 93% and drops down to $N_{\rm H, pc}=3.5\times 10^{22}$ cm$^{-2}$ and $f_{\rm pc}$=78% during the high state. In order to check whether the data can be self-consistently fit, we fitted the low and high-state [[*Swift*]{}]{} data simultaneously in [*XSPEC*]{}. In this case we tied the column densities of the partial covering absorber and the X-ray spectral slopes of both spectra but left the covering fractions and the normalisations free. Here we found an absorption column density of $N_{\rm H,pc}=8.1\times10^{22}$ cm$^{-2}$ with covering fractions of 91% and 86% for the low and high states, respectively. In all cases, [[*Swift*]{}]{} as well as [[*XMM-Newton*]{}]{}, the X-ray spectral slope remains around [$\alpha_{\rm X}$]{}=2.1 and does not show any significant changes within the errors. The partial covering absorber model can explain the variability seen in X-rays in PG 1211+143. The results obtained from the [[*Swift*]{}]{} data during the high state are consistent with those derived from the 2001 and 2004 [[*XMM-Newton*]{}]{} EPIC pn data. The spectra were also fitted with a blurred reflection model (Ross & Fabian 2005). Such models, where the primary emission (i.e. the power law component) illuminates the accretion disk producing a reflection spectrum that is blurred by Doppler and relativistic effects close to the black hole (e.g. Fabian et al. 1989) have been successfully applied to several NLS1 X-ray spectra (e.g. Fabian et al. 2004; Gallo et al. 2007a, 2007b; Grupe et al. 2008a; Larsson et al. 2008). As shown in Figures\[pg1211\_refl\_mod\] and \[pg1211\_refl\_ratio\], the reflection model broadly describes the high- and low-flux states of PG 1211+143. In the simplest case, the blurring parameters and disk ionisation are linked between the two epochs. The disk inclination ($i$) and outer radius ($R_{\rm out}$) are fixed to 30$^{\circ}$ and $400 R_{\rm g}$, respectively; $R_{\rm g}$ is the Schwarzschild radius. The continuum shape ($\Gamma$) and normalisations of the reflection and power law were free to vary independently at each epoch. The resulting fit is reasonable ($\chi^{2}_{\nu}/dof = 1.30/105$), considering the obvious over-simplification of our model. The inner disk radius and emissivity index were found to be $R_{\rm in}=1.76^{+0.27}_{-0.35} R_{\rm g}$ and $q=5.57^{+0.54}_{-0.71}$, respectively. The disk ionisation was $\xi = 116 \pm 10$. The intrinsic power law shape was significantly harder during the low-flux state, $\alpha_{\rm x,low}=0.55^{+0.09}_{-0.06}$, compared to $\alpha_{\rm x,high}=0.96^{+0.13}_{-0.10}$ during the high-state. The primary difference between the high and low state is the relative contribution of the power law component to the total 0.3–10 keV flux, being approximately 0.42 and 0.12, respectively. During the low-flux state the reflection component dominates the spectrum. [lcccccccc]{} Model & $\alpha_{\rm x,soft}$ & $E_{\rm Break}$ & $\alpha_{\rm x,hard}$ & $N_{\rm H,pc}$ & $f_{\rm pc}$ & $\chi^2/\nu$ & $F_{\rm 0.2-2.0 keV}$ & $F_{\rm 2-10 keV}$\ \ Powl & 1.73[$\pm$]{}0.10 & — & — & — & — & 312/60 & 1.59[$\pm$]{}0.10 & 0.39[$\pm$]{}0.02\ Bknpo & 2.29$^{+0.14}_{-0.13}$ & 1.42$^{+0.13}_{-0.11}$ & –0.18$^{+0.17}_{-0.18}$ & — & — & 68/58 & 1.81[$\pm$]{}0.16 & 2.45[$\pm$]{}0.21\ Zpcfabs \* powl & 2.18$^{+0.10}_{-0.12}$ & — & — & 9.45$^{+1.87}_{-1.62}$ & 0.95[$\pm$]{}0.02 & 82/58 & 1.76[$\pm$]{}0.26 & 1.54[$\pm$]{}0.23\ \ Powl & 1.71[$\pm$]{}0.15 & — & — & — & — & 103/26 & 3.65[$\pm$]{}0.36 & 0.92[$\pm$]{}0.09\ Bknpo & 2.44[$\pm$]{}0.26 & 1.15$^{+0.27}_{-0.14}$ & 0.31$^{+0.26}_{-0.34}$ & — & — & 30/24 & 4.34[$\pm$]{}0.76 & 3.85[$\pm$]{}0.67\ Zpcfabs \* powl & 2.22[$\pm$]{}0.20 & — & — & 7.88$^{+3.21}_{-2.33}$ & 0.93$^{+0.03}_{-0.04}$ & 35/24 & 4.07[$\pm$]{}1.47 & 2.84[$\pm$]{}1.04\ \ Powl & 1.47[$\pm$]{}0.06 & — & — & — & — & 168/62 & 7.04[$\pm$]{}0.29 & 2.76[$\pm$]{}0.12\ Bknpo & 2.05$^{+0.16}_{-0.15}$ & 1.04$^{+0.13}_{-0.12}$ & 0.84[$\pm$]{}0.14 & — & — & 91/60 & 7.79[$\pm$]{}0.78 & 5.06[$\pm$]{}0.51\ Zpcfabs \* powl & 1.98[$\pm$]{}0.14 & — & — & 3.48$^{+1.99}_{-1.02}$ & 0.78$^{+0.05}_{-0.07}$ & 101/60 & 7.50[$\pm$]{}1.76 & 4.02[$\pm$]{}0.94\ \ Powl & 1.56[$\pm$]{}0.05 & — & — & — & — & 488/123 & 1.49/7.24 & 0.50/2.41\ Bknpo & 2.14[$\pm$]{}0.10 & 1.30[$\pm$]{}0.12 & 0.34$^{+0.12}_{-0.12}$ & — & — & 215/121 & 1.79/7.65 & 1.62/6.92\ Zpcfabs \* Powl & 2.00[$\pm$]{}0.09 & — & — & 8.11$^{+1.64}_{-1.33}$ & 0.91[$\pm$]{}0.02/0.86[$\pm$]{}0.03 & 209/120 & 1.65/7.68 & 1.53/4.52\ \ Powl & 1.97[$\pm$]{}0.02 & — & — & — & — & 13092/994 & 5.56[$\pm$]{}0.03 & 0.86[$\pm$]{}0.01\ Bknpo & 2.20[$\pm$]{}0.01 & 1.28[$\pm$]{}0.05 & 0.71[$\pm$]{}0.05 & — & — & 4860/992 & 5.57[$\pm$]{}0.02 & 3.04[$\pm$]{}0.02\ Zpcfabs \* powl & 2.15[$\pm$]{}0.01 & — & — & 6.41[$\pm$]{}0.15 & 0.87[$\pm$]{}0.02 & 5424/992 & 5.50[$\pm$]{}0.10 & 2.64[$\pm$]{}0.05\ \ Powl & 1.62[$\pm$]{}0.01 & — & — & — & — & 4122/940 & 5.63[$\pm$]{}0.05 & 1.06[$\pm$]{}0.02\ Bknpo & 1.75[$\pm$]{}0.01 & 1.58[$\pm$]{}0.05 & 0.85[$\pm$]{}0.03 & — & — & 1828/938 & 5.58[$\pm$]{}0.05 & 3.12[$\pm$]{}0.03\ Zpcfabs \* powl & 1.73[$\pm$]{}0.01 & — & — & 8.73[$\pm$]{}0.34 & 0.70[$\pm$]{}0.02 & 1958/938 & 5.58[$\pm$]{}0.17 & 2.89[$\pm$]{}0.09\ Spectral energy distribution ---------------------------- Figure\[pg1211\_sed\] displays the spectral energy distributions (SED) during the low state observation on 2007 April 17 (blue squares) and the high state observation on April 22 (red triangles). The optical/UV slope [$\alpha_{\rm UV}$]{} slightly changes from $-$0.67[$\pm$]{}0.12 to $-$0.56[$\pm$]{}0.10 between the low and high states. Most significant, however, is the change in the optical-to-X-ray spectral slope [$\alpha_{\rm ox}$]{}[^4] from [$\alpha_{\rm ox}$]{}=1.84 during the low state to [$\alpha_{\rm ox}$]{}=1.48 during the high state. This low-state [$\alpha_{\rm ox}$]{} almost makes it an X-ray weak AGN according to the definition by Brandt et al. (2000), who defines AGN with [$\alpha_{\rm ox}$]{}$>$2.0 as X-ray weak. The luminosities in the Big-Blue-Bump are log $L_{\rm BBB}$=38.42 and 38.53 \[W\] for the low and high states, respectively. These luminosities correspond to Eddington ratios of $L/L_{\rm Edd}$=0.26 and 0.33, respectively, assuming a mass of the central black hole of 9$\times 10^7$[$M_{\odot}$]{} (Vestergaard & Peterson 2006). Discussion ========== Based on the results from this study of the NLS1 PG 1211+143 we found that the short-time (and perhaps – even the long-time) variations of the X-ray and optical/UV continuums do not seem to correlate well (Smith & Vaughan 2007, and the references within). While the X-ray continuum varied rapidly (more than 5 times during [[*Swift*]{}]{} monitoring campaign) with a general trend of brightness increase, the optical/UV continuum showed minimal changes with a general trend of brightness decrease. This is not unusual and is in fact reported for other objects (e.g. NGC 5548, Uttley et al. 2003). Such a behaviour sets constraints on different reprocessing scenarios. We are going to discuss briefly several possibilities, assuming that the optical/UV emission is produced by a standard, thin accretion disk, the central black hole mass is $M_{\rm BH}\simeq 9 \times 10^{7}$ [$M_{\odot}$]{}  and the accretion rate (in Eddington units) is $\dot m \simeq 0.3$ (Sect. 3.7). The accretion rate is also consistent within the errors with the one found by Kaspi et al. (2000) and Loska et al. (2004). Cause of the X-ray weakness --------------------------- As shown in Figure\[longterm\] over the last 20 years PG 1211+143 appears to be fainter in X-rays compared with the X-ray observation during the 1980th. Especially during our [[*Swift*]{}]{} observations during the beginning of our monitoring in March 2007, PG 1211+143 was found to be in an historical X-ray low state especially in the 0.2–2.0 keV band. The X-ray spectrum during the low state is somewhat similar to that found during the historical low state in Mkn 335 (Grupe et al. 2007b, 2008a). Also, here the low state could be explained by a strong partial covering absorber. Later monitoring with [[*Swift*]{}]{} suggests that the absorber has disappeared again and that Mkn 335 is back in a high-state (Grupe et al. 2009, in prep.). Similarly, the low-state here was just a temporary event that lasted for a maximum of about a year. Partial covering absorbers, however, can last significantly longer. One example is the X-ray transient NLS1 WPVS 007 (Grupe et al. 1995), which has developed strong broad absorption line features in the UV (Leighly et al. 2009) and a strong partial covering absorber in X-rays (Grupe et al. 2008b). It has been in an extreme low X-ray state for more than a decade (Grupe et al. 2007a, 2008b). Statistically we cannot distinguish between the partial covering absorber or the reflection models. Both models result in similar $\chi^2/\nu$. Both models can also be fit self-consistently leaving the intrinsic X-ray spectrum fixed and only affected by either partial covering absorption or reflection. In the case of a partial covering absorber we can expect that the observed light is polarized like it has been seen in the NLS1 Mkn 1239 which is highly optically polarized and shows a strong partial covering absorber in X-rays (Grupe et al. 2004). However, both PG 1211+143 and Mkn 335 do not show any sign of optical continuum polarization (Berriman et al. 1999, Smith et al. 2002). Note, however, that all these polarimetry measurements were done when the objects were in their high-state. There are no polarimetry measurements (at least not to our knowledge) that were performed during their low states. Therefore the non polarization in the optical does not exclude the partial covering model. Compact central X-ray source or extended medium? ------------------------------------------------ It is commonly assumed that the most part of the AGN X-ray emission is produced close to the center, within the inner few tens $R_{\rm g}$. The X-rays in radio-quiet objects may come from the inner part of an accretion disk, perhaps operating there in a mode of a very hot, geometrically thick, low-efficient accretion flow (e.g. ADAF) or from an active corona, sandwiching the disk. In either case, when studying the X-ray irradiation of the *optically* emitting outer parts of the disk (at $\sim 100-1000 R_{\rm g}$), the X-ray producing region could be considered as a point source, elevated slightly above the center of the disk and illuminating the periphery. Thus, a part of this highly variable by presumption X-ray emission could be reprocessed into optical/UV emission, which variations will lag behind the X-ray variations. The temperature of a thin (Shakura-Sunyaev) accretion disk scales with the radial distance $r$ (in Schwarzschild radii) as $T(r) \simeq 6~10^{5}(\dot m /M_{\rm 8})^{1/4} r^{-3/4}$ \[K\] (Frank et al. 2002). If a point-like X-ray source, located at some distance $H$ above the disk, close to the center, irradiates the outer parts, it causes a temperature increase by a certain factor, but the radial temperature dependence happens to be the same (at least for $r>>H$). Since most of the visual/UV light is presumably due to viscous heating, not to irradiation, the temperature increase could be considered as a small addition to the usual disk temperature. Nevertheless, the X-ray variations should transform into *some* optical/UV variations with a time delay, due to the light crossing time. If each disk ring emits mostly wavelengths close to the maximum of the *Planck* curve for the corresponding temperature, one expects the time delays to scale with the wavelength as $\tau_{\rm \lambda} \simeq 5 (\dot m M_{\rm 8}^{2})^{1/3} \lambda_{\rm 5000}^{4/3}$ \[days\], which transforms to $\tau_{\rm \lambda} \simeq 5 \lambda_{\rm 5000}^{4/3}$ \[days\] for the accretion parameters accepted above. The delay obtained from the fit in Sect. 3.5, however, appears to be $\sim$2 times longer than expected, yet consistent with the expected dependence of $\lambda^{4/3}$. Although different fits to the data are possible, due to the uncertainties, the general offset seems to indicate a time inconsistency. A possible explanation may be searched in the spatial location of the X-ray source. Even if the very center produces most of the X-ray emission, the outer disk may not “see” much of it. Instead, a large-scale back-scattering matter may be located at significant height above the disk, thus increasing the light-crossing time 2 – 3 times (Loska et al. 2004; Czerny & Janiuk 2007, see also their Fig. 6). In fact, the presence of such a back-scattering matter, in a form of a high-velocity outflow or a warm absorber is suggested by independent studies of this object (Pounds et al. 2003, Pounds & Page 2006). One is to note however, that for a similar otherwise object – Mkn 335, the delays were found to be consistent with a direct irradiation from a compact central source (Czerny & Janiuk 2007). On the other hand, two recent studies by Arévalo et al. (2008) and Breedt et al. (2009) found significant correlation between the X-ray and optical bands at essentially zero time lag (yet, not necessarily inconsistent with the direct reprocessing, considering the uncertainties) for MR 2251–178 and Mkn 79, respectively. Finally, for a broad-line radio galaxy (3C 120), an otherwise different type of object, but with similar to PG 1211+143 black hole mass and accretion rate, Marshall et al. (2009) found a $\sim$28 days delay of R-band behind the X-rays, being longer than expected and similar to our findings. Since the results from these and other studies have not shown a systematic inter-band behaviour, one approach to resolve the problem could be to study separately different types of objects, grouped by their intrinsic characteristics, like central masses, accretion rates, line widths, X-ray and radio properties, etc. (e.g. Bachev, 2009), in order to reveal how the presence of disk reprocessing might be related to overall quasar appearance. On the other hand, instead from around the central black hole, the variable X-ray emission could come from many active regions (flares) in the corona, covering the optically emitting parts of the disk (Czerny et al., 2004). Such active regions (or hot spots) can irradiate the disk locally, producing almost instantaneous optical/UV continuum changes. The later can be significant enough to be observable under some conditions. The observed wavelength-dependent time delay however, seems to make this possibility unlikely. Why is the optical/UV continuum poorly responding to the X-ray changes? ----------------------------------------------------------------------- A few factors can contribute to the apparent lack of significant correlation between the optical/UV and X-ray continua. First, the possibility that the X-ray emission is highly anisotropic (Papadakis et al. 2000), or the disk geometry is far from flat (bumpy surface, Cackett et al. 2007; or warps, Bachev 1999), leading only to a minor optical response to the huge otherwise X-ray changes, cannot entirely be ruled out. Yet, the observed $\sim$2 times longer lags than expected are difficult to explain in terms of such an assumption, since in the presence of a large-scale backscattering medium, needed to account for the extra light travel path, the unevenness of the surface should be of little significance. Another possibility, which might explain the large X-ray variations and the absence of optical/UV such is the presence of absorbing matter along the line of sight. If located close enough to the center, it can partially obscure the compact X-ray source from the observer, but not too much from the larger, optically emitting part of the accretion disk. In such a case, indeed, the large X-ray variations observed would hardly be transferred into the optical bands. Unfortunately, the X-ray spectral fitting cannot distinguish well enough between reflection and absorption models, to be able to determine which one shapes the X-ray continuum most. X-ray emission – leading or trailing? ------------------------------------- Taking into account the position of the highest maximum of the I-band ICCF (Figure 6), the I-band changes appear to lead the X-ray ones. One way to explain this result, if real at all, is invoking the synchrotron-self Compton (SSC) mechanism to account for part of the produced X-rays. SSC assumes that some of the near-IR photons might have a synchrotron origin, and could later be scattered by the same relativistic electrons to produce the X-ray flares, lagging behind the infrared. However, the available data set, based merely on the light curves information, cannot justify undisputedly such an explanation. Furthermore, no strong jet or significant radio emission is present in this object (where SSC is typically assumed to play a significant role). So, if SSC is to account for the delay of the X-ray behind the I-band, this process has to take place in a the base of a possible failed jet (Czerny et al., 2008, and the references therein) or in the central parts of the disk, where the disk could operate as a very hot flow, and where hot electrons and perhaps strong magnetic fields could be present. Summary ======= In this paper we presented the results of a continuum (X-ray – to optical I-band) monitoring campaign of PG 1211+143, performed with [[*Swift*]{}]{} and ground-based observatories. The main results are summarized below: 1. In spite of being in a very low X-ray state, the quasar PG 1211+143 showed significant X-ray variations (up to 5 times) on daily basis, with only minor optical/UV flux changes. This behaviour indicates that a rather small amount of the hard radiation is reprocessed into longer wavelengths. Since both, reflection and absorption models fit equally well the X-ray spectrum, we are unable to determine the exact cause of the X-ray weakness of PG 1211+143 during its 2007 minimum. 2. Interband cross-correlation functions suggest that a wavelength-dependent time delay between the X-ray and the optical/UV bands is present, indicating that at least a part of the X-rays is reprocessed into longer wavelengths. 3. Although the $\tau - \lambda$ dependence followed the general trend expected for a thin accretion disk (i.e. $\tau_{\rm \lambda} \sim \lambda^{4/3}$), the delays are $\sim$2 times longer, implying the possible existence of a large-scale back-scattering matter above the disk (wind/warm absorber), rather than a central point-like X-ray source, directly irradiating the disk. 4. Even if the object is radio-quiet, with no strong jet known, we found indications that the SSC mechanism may play some role in the X-ray production. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to Neil Gehrels for approving our ToO request and the [[*Swift*]{}]{} team for performing the ToO observations of PG 1211+143. We would also like to thank the anonymous referee for his/her helpful comments and suggestions which significantly improved this paper. 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M., et al., 2003, ApJ, 584, 53 Vestergaard, M., & Peterson, B.M., 2006, ApJ, 641, 689 Log of observations =================== Segment $T_{\rm start}$ $T_{\rm end}$ JD-2454000 $T_{\rm XRT}$ $T_{\rm V}$ $T_{\rm B}$ $T_{\rm U}$ $T_{\rm W1}$ $T_{\rm M2}$ $T_{\rm W2}$ --------- ------------------- ------------------- ------------ --------------- ------------- ------------- ------------- -------------- -------------- -------------- 001 2007 Mar 08 00:48 2007 Mar 08 02:35 167 1111 — — — — — — 002 2007 Mar 09 01:03 2007 Mar 09 02:54 168 1781 148 148 148 298 373 596 003 2007 Mar 10 17:20 2007 Mar 10 22:17 169 1695 154 154 154 309 203 621 004 2007 Mar 11 19:03 2007 Mar 11 23:59 170 1711 154 154 154 310 217 621 005 2007 Mar 12 15:55 2007 Mar 12 20:52 171 1666 149 149 149 300 219 601 006 2007 Mar 13 20:50 2007 Mar 13 22:34 172 844 74 74 74 149 117 301 007 2007 Mar 14 09:10 2007 Mar 14 12:34 173 1995 168 168 168 336 403 670 008 2007 Mar 15 11:07 2007 Mar 15 13:02 174 1769 145 145 145 292 399 583 009 2007 Mar 16 00:02 2007 Mar 16 17:54 175 1566 142 145 145 295 82 593 010 2007 Mar 17 08:21 2007 Mar 17 19:12 176 2689 208 231 231 467 478 932 011 2007 Mar 18 10:05 2007 Mar 18 16:39 177 1738 149 149 149 300 292 601 012 2007 Mar 19 10:11 2007 Mar 19 16:45 178 1683 144 144 144 292 282 582 013 2007 Mar 26 07:09 2007 Mar 26 08:28 185 2006 166 166 166 331 444 664 014 2007 Apr 02 20:57 2007 Apr 02 22:36 192 2450 201 201 201 403 559 808 016 2007 Apr 11 09:12 2007 Apr 11 14:09 201 1756 157 157 157 318 233 633 017 2007 Apr 17 00:13 2007 Apr 17 06:46 207 2043 168 168 168 338 417 678 018 2007 Apr 22 11:53 2007 Apr 22 15:18 212 2063 171 171 171 342 434 684 019 2007 Apr 30 04:44 2007 Apr 30 06:32 220 1356 114 114 114 226 289 454 021 2007 May 09 15:13 2007 May 09 17:04 229 1398 70 139 139 277 183 539 022 2007 May 14 04:04 2007 May 14 05:59 234 2257 185 186 186 372 518 744 023 2007 May 20 01:28 2007 May 20 03:12 240 857 — 176 176 354 — 120 024 2008 Feb 17 07:56 2008 Feb 17 07:59 513 188 ... ... ... ... ... ... Segment CR HR V B U UV W1 UV M2 UV W2 --------- --------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- 001 0.050[$\pm$]{}0.007 –0.19[$\pm$]{}0.13 ... ... ... ... ... ... 002 0.057[$\pm$]{}0.006 +0.02[$\pm$]{}0.11 14.32[$\pm$]{}0.02 14.59[$\pm$]{}0.01 13.33[$\pm$]{}0.01 13.31[$\pm$]{}0.01 13.16[$\pm$]{}0.01 13.18[$\pm$]{}0.01 003 0.049[$\pm$]{}0.006 –0.27[$\pm$]{}0.12 14.28[$\pm$]{}0.02 14.56[$\pm$]{}0.01 13.31[$\pm$]{}0.01 13.25[$\pm$]{}0.01 13.17[$\pm$]{}0.01 13.12[$\pm$]{}0.01 004 0.085[$\pm$]{}0.008 –0.14[$\pm$]{}0.09 14.26[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.29[$\pm$]{}0.01 13.24[$\pm$]{}0.01 13.15[$\pm$]{}0.01 13.12[$\pm$]{}0.01 005 0.119[$\pm$]{}0.009 –0.21[$\pm$]{}0.08 14.26[$\pm$]{}0.02 14.56[$\pm$]{}0.01 13.31[$\pm$]{}0.01 13.26[$\pm$]{}0.01 13.08[$\pm$]{}0.01 13.11[$\pm$]{}0.01 006 0.059[$\pm$]{}0.009 –0.17[$\pm$]{}0.15 14.29[$\pm$]{}0.02 14.52[$\pm$]{}0.01 13.33[$\pm$]{}0.01 13.27[$\pm$]{}0.01 13.07[$\pm$]{}0.03 13.11[$\pm$]{}0.01 007 0.071[$\pm$]{}0.006 –0.09[$\pm$]{}0.08 14.26[$\pm$]{}0.02 14.56[$\pm$]{}0.01 13.28[$\pm$]{}0.01 13.23[$\pm$]{}0.01 13.10[$\pm$]{}0.01 13.09[$\pm$]{}0.01 008 0.060[$\pm$]{}0.006 +0.08[$\pm$]{}0.06 14.26[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.37[$\pm$]{}0.01 13.28[$\pm$]{}0.01 13.17[$\pm$]{}0.01 13.18[$\pm$]{}0.01 009 0.071[$\pm$]{}0.007 –0.16[$\pm$]{}0.10 14.29[$\pm$]{}0.02 14.56[$\pm$]{}0.01 13.29[$\pm$]{}0.01 13.25[$\pm$]{}0.01 13.22[$\pm$]{}0.04 13.09[$\pm$]{}0.01 010 0.059[$\pm$]{}0.005 –0.10[$\pm$]{}0.09 14.27[$\pm$]{}0.02 14.54[$\pm$]{}0.01 13.31[$\pm$]{}0.01 13.26[$\pm$]{}0.01 13.12[$\pm$]{}0.01 13.12[$\pm$]{}0.01 011 0.072[$\pm$]{}0.007 –0.15[$\pm$]{}0.09 14.29[$\pm$]{}0.02 14.60[$\pm$]{}0.01 13.33[$\pm$]{}0.01 13.28[$\pm$]{}0.01 13.12[$\pm$]{}0.01 13.15[$\pm$]{}0.01 012 0.045[$\pm$]{}0.006 –0.05[$\pm$]{}0.12 14.29[$\pm$]{}0.02 14.60[$\pm$]{}0.01 13.38[$\pm$]{}0.01 13.27[$\pm$]{}0.01 13.14[$\pm$]{}0.01 13.13[$\pm$]{}0.01 013 0.139[$\pm$]{}0.009 –0.14[$\pm$]{}0.06 14.27[$\pm$]{}0.02 14.55[$\pm$]{}0.01 13.30[$\pm$]{}0.01 13.20[$\pm$]{}0.01 13.08[$\pm$]{}0.01 13.09[$\pm$]{}0.01 014 0.171[$\pm$]{}0.008 –0.22[$\pm$]{}0.05 14.27[$\pm$]{}0.02 14.57[$\pm$]{}0.01 13.30[$\pm$]{}0.01 13.24[$\pm$]{}0.01 13.10[$\pm$]{}0.01 13.11[$\pm$]{}0.01 016 0.037[$\pm$]{}0.005 –0.18[$\pm$]{}0.13 14.28[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.32[$\pm$]{}0.01 13.29[$\pm$]{}0.01 13.15[$\pm$]{}0.01 13.15[$\pm$]{}0.01 017 0.060[$\pm$]{}0.006 –0.17[$\pm$]{}0.09 14.36[$\pm$]{}0.02 14.68[$\pm$]{}0.01 13.43[$\pm$]{}0.01 13.37[$\pm$]{}0.01 13.28[$\pm$]{}0.01 13.32[$\pm$]{}0.01 018 0.332[$\pm$]{}0.013 –0.21[$\pm$]{}0.04 14.26[$\pm$]{}0.02 14.57[$\pm$]{}0.01 13.34[$\pm$]{}0.01 13.24[$\pm$]{}0.01 13.10[$\pm$]{}0.01 13.10[$\pm$]{}0.01 019 0.095[$\pm$]{}0.009 –0.16[$\pm$]{}0.09 14.33[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.30[$\pm$]{}0.01 13.24[$\pm$]{}0.01 13.11[$\pm$]{}0.01 13.16[$\pm$]{}0.01 021 0.293[$\pm$]{}0.015 –0.11[$\pm$]{}0.05 14.26[$\pm$]{}0.02 14.58[$\pm$]{}0.01 13.32[$\pm$]{}0.01 13.26[$\pm$]{}0.01 13.11[$\pm$]{}0.02 13.13[$\pm$]{}0.01 022 0.294[$\pm$]{}0.012 –0.15[$\pm$]{}0.04 14.27[$\pm$]{}0.02 14.60[$\pm$]{}0.01 13.39[$\pm$]{}0.01 13.26[$\pm$]{}0.01 13.17[$\pm$]{}0.01 13.17[$\pm$]{}0.01 023 0.095[$\pm$]{}0.011 –0.18[$\pm$]{}0.11 ... 14.61[$\pm$]{}0.01 13.35[$\pm$]{}0.01 13.30[$\pm$]{}0.01 ... 13.18[$\pm$]{}0.01 024 0.375[$\pm$]{}0.045 –0.10[$\pm$]{}0.12 ... ... ... ... ... ... JD (2454..) U B V R I Instr. ------------- -------------------- -------------------- -------------------- -------------------- --------------------- -------- 116.52 ... 14.79[$\pm$]{}0.10 14.29[$\pm$]{}0.03 13.96[$\pm$]{}0.01 13.75 [$\pm$]{}0.02 AOB 153.52 ... ... 14.22[$\pm$]{}0.04 ... ... AOB 157.39 ... 14.49[$\pm$]{}0.04 14.24[$\pm$]{}0.02 13.91[$\pm$]{}0.02 13.72 [$\pm$]{}0.02 RSh 171.63 13.28[$\pm$]{}0.07 14.50[$\pm$]{}0.03 14.25[$\pm$]{}0.02 13.91[$\pm$]{}0.02 13.73 [$\pm$]{}0.03 RSh 172.42 13.33[$\pm$]{}0.05 14.51[$\pm$]{}0.02 14.27[$\pm$]{}0.02 13.93[$\pm$]{}0.02 13.74 [$\pm$]{}0.02 RSh 174.34 ... 14.58[$\pm$]{}0.07 14.28[$\pm$]{}0.02 13.94[$\pm$]{}0.01 13.73 [$\pm$]{}0.01 AOB 175.35 ... 14.57[$\pm$]{}0.07 14.28[$\pm$]{}0.02 13.93[$\pm$]{}0.01 13.74 [$\pm$]{}0.01 AOB 176.31 ... 14.47[$\pm$]{}0.10 14.28[$\pm$]{}0.02 13.94[$\pm$]{}0.01 13.75 [$\pm$]{}0.01 AOB 176.45 13.32[$\pm$]{}0.03 14.49[$\pm$]{}0.03 14.26[$\pm$]{}0.02 13.93[$\pm$]{}0.02 13.72 [$\pm$]{}0.02 RSh 177.32 ... ... 14.28[$\pm$]{}0.03 13.96[$\pm$]{}0.02 13.73 [$\pm$]{}0.03 AOB 178.32 ... 14.55[$\pm$]{}0.10 14.28[$\pm$]{}0.02 13.92[$\pm$]{}0.02 13.74 [$\pm$]{}0.02 AOB 179.47 ... 14.57[$\pm$]{}0.10 14.26[$\pm$]{}0.02 13.94[$\pm$]{}0.02 13.74 [$\pm$]{}0.02 AOB 199.49 ... 14.50[$\pm$]{}0.01 14.25[$\pm$]{}0.01 13.98[$\pm$]{}0.01 13.78 [$\pm$]{}0.01 R2m 200.43 ... 14.51[$\pm$]{}0.01 14.23[$\pm$]{}0.01 13.97[$\pm$]{}0.01 13.76 [$\pm$]{}0.01 R2m 201.44 ... 14.56[$\pm$]{}0.03 14.22[$\pm$]{}0.03 13.94[$\pm$]{}0.03 ... R2m 201.45 ... 14.57[$\pm$]{}0.02 14.23[$\pm$]{}0.01 13.97[$\pm$]{}0.01 13.75 [$\pm$]{}0.01 RSh 203.32 ... 14.57[$\pm$]{}0.01 14.23[$\pm$]{}0.03 14.00[$\pm$]{}0.01 13.78 [$\pm$]{}0.01 RSh 204.34 ... 14.63[$\pm$]{}0.01 14.23[$\pm$]{}0.01 13.97[$\pm$]{}0.01 13.78 [$\pm$]{}0.01 RSh 208.49 13.39[$\pm$]{}0.03 14.56[$\pm$]{}0.02 14.28[$\pm$]{}0.02 13.97[$\pm$]{}0.02 13.77 [$\pm$]{}0.02 RSh 210.41 13.35[$\pm$]{}0.02 14.56[$\pm$]{}0.02 14.29[$\pm$]{}0.02 13.97[$\pm$]{}0.02 13.76 [$\pm$]{}0.02 RSh 211.38 13.39[$\pm$]{}0.03 14.59[$\pm$]{}0.02 14.31[$\pm$]{}0.01 13.99[$\pm$]{}0.02 13.77 [$\pm$]{}0.02 RSh 213.29 ... 14.60[$\pm$]{}0.05 14.31[$\pm$]{}0.02 13.94[$\pm$]{}0.01 13.76 [$\pm$]{}0.02 AOB 217.29 ... 14.56[$\pm$]{}0.05 14.36[$\pm$]{}0.03 13.99[$\pm$]{}0.02 13.77 [$\pm$]{}0.02 AOB 230.35 ... 14.62[$\pm$]{}0.05 14.28[$\pm$]{}0.02 13.96[$\pm$]{}0.01 13.76 [$\pm$]{}0.01 AOB 231.30 ... 14.61[$\pm$]{}0.05 14.27[$\pm$]{}0.02 13.95[$\pm$]{}0.01 13.74 [$\pm$]{}0.01 AOB 232.31 ... 14.63[$\pm$]{}0.06 14.29[$\pm$]{}0.02 13.95[$\pm$]{}0.02 13.74 [$\pm$]{}0.02 AOB 233.42 ... 14.59[$\pm$]{}0.05 14.38[$\pm$]{}0.02 13.98[$\pm$]{}0.01 13.78 [$\pm$]{}0.01 AOB 234.30 ... 14.66[$\pm$]{}0.06 14.23[$\pm$]{}0.02 13.98[$\pm$]{}0.01 13.77 [$\pm$]{}0.01 AOB 236.32 ... 14.60[$\pm$]{}0.07 14.31[$\pm$]{}0.02 13.98[$\pm$]{}0.01 13.77 [$\pm$]{}0.01 AOB 238.37 ... 14.57[$\pm$]{}0.01 14.23[$\pm$]{}0.01 13.91[$\pm$]{}0.01 ... R2m 247.45 ... 14.56[$\pm$]{}0.02 14.29[$\pm$]{}0.03 13.99[$\pm$]{}0.03 13.81 [$\pm$]{}0.03 RSh 260.39 13.35[$\pm$]{}0.05 14.60[$\pm$]{}0.03 14.31[$\pm$]{}0.02 14.00[$\pm$]{}0.02 13.82 [$\pm$]{}0.02 RSh 261.36 .... 14.62[$\pm$]{}0.15 14.38[$\pm$]{}0.02 14.02[$\pm$]{}0.02 13.83 [$\pm$]{}0.02 AOB 262.40 ... 14.65[$\pm$]{}0.03 ... 14.01[$\pm$]{}0.02 13.79 [$\pm$]{}0.03 RSh 263.36 ... 14.61[$\pm$]{}0.03 14.33[$\pm$]{}0.02 13.97[$\pm$]{}0.02 13.81 [$\pm$]{}0.03 RSh 265.35 ... 14.64[$\pm$]{}0.10 14.39[$\pm$]{}0.02 14.05[$\pm$]{}0.02 13.87 [$\pm$]{}0.02 AOB 265.42 ... 14.68[$\pm$]{}0.02 14.35[$\pm$]{}0.02 14.00[$\pm$]{}0.03 13.84 [$\pm$]{}0.02 RSh 266.40 ... 14.66[$\pm$]{}0.02 14.33[$\pm$]{}0.02 14.01[$\pm$]{}0.02 13.81 [$\pm$]{}0.02 RSh 289.31 ... 14.72[$\pm$]{}0.13 14.36[$\pm$]{}0.02 14.01[$\pm$]{}0.02 13.84 [$\pm$]{}0.02 AOB 290.35 ... 14.76[$\pm$]{}0.13 14.35[$\pm$]{}0.02 14.01[$\pm$]{}0.02 13.85 [$\pm$]{}0.03 AOB 291.34 ... 14.69[$\pm$]{}0.02 14.32[$\pm$]{}0.02 14.06[$\pm$]{}0.01 13.89 [$\pm$]{}0.01 RSh 292.31 ... 14.68[$\pm$]{}0.02 14.28[$\pm$]{}0.02 14.03[$\pm$]{}0.01 13.87 [$\pm$]{}0.01 RSh 292.36 ... ... 14.32[$\pm$]{}0.06 13.95[$\pm$]{}0.06 ... AOB 293.32 ... 14.67[$\pm$]{}0.01 14.38[$\pm$]{}0.02 14.04[$\pm$]{}0.01 13.85 [$\pm$]{}0.01 RSh 296.33 ... 14.73[$\pm$]{}0.03 14.38[$\pm$]{}0.02 14.02[$\pm$]{}0.01 ... RSh 298.30 ... 14.62[$\pm$]{}0.03 14.37[$\pm$]{}0.02 14.03[$\pm$]{}0.03 13.85 [$\pm$]{}0.03 RSh 299.30 13.51[$\pm$]{}0.09 14.66[$\pm$]{}0.03 14.36[$\pm$]{}0.02 14.02[$\pm$]{}0.02 13.86 [$\pm$]{}0.03 RSh 303.30 ... 14.81[$\pm$]{}0.02 14.38[$\pm$]{}0.01 14.07[$\pm$]{}0.01 13.89 [$\pm$]{}0.02 R2m \[lastpage\] [^1]: Although [[*Swift*]{}]{} has turned into a multi-purpose observatory, its main focus is still on observing GRBs and therefore GRBs will supersede scheduled ToO observations. [^2]: The XRT hardness ratio is defines as HR=(H-S)/(H+S) with S and H are the counts in the 0.3–1.0 keV and 1.0–10.0 keV bands, respectively. [^3]: All the calculations here are performed using the observer’s frame measurements. Due to the similar way the times and the wavelengths are affected by the redshift, for the quasar rest frame the delay in the $\tau - \lambda$ dependence [*increases*]{} only by $(1+z)^{-1/3}$, Sect. 4, which is only $\sim 3\%$ and is much less than the expected errors. [^4]: The X-ray loudness is defined by Tananbaum et al. (1979) as [$\alpha_{\rm ox}$]{}=–0.384 log($f_{\rm 2keV}/f_{2500\rm \AA}$).

ArXiv

--- abstract: | The Virtual Observatory is a new technology of the astronomical research allowing the seamless processing and analysis of a heterogeneous data obtained from a number of distributed data archives. It may also provide astronomical community with powerful computational and data processing on-line services replacing the custom scientific code run on user’s computers. Despite its benefits the VO technology has been still little exploited in stellar spectroscopy. As an example of possible evolution in this field we present an experimental web-based service for disentangling of spectra based on code KOREL. This code developed by P. Hadrava enables Fourier disentangling and line-strength photometry, i.e. simultaneous decomposition of spectra of multiple stars and solving for orbital parameters, line-profile variability or other physical parameters of observed objects. We discuss the benefits of the service-oriented approach from the point of view of both developers and users and give examples of possible user-friendly implementation of spectra disentangling methods as a standard tools of Virtual Observatory. author: - Petr Škoda and Petr Hadrava title: Fourier Disentangling Using the Technology of Virtual Observatory --- Introduction ============ The astronomical spectroscopy uses many special techniques to analyse stellar spectra and estimate physical properties of targets studied. Basically they consist in comparison of the observed spectra with theoretical models which, however, may be of very different level of sophistication. For instance, a simple comparison of suitably defined effective centres of spectral lines with their laboratory wavelengths gives Doppler shifts, which in the case of spectroscopic binaries enables one to determine their orbital parameters. Detailed comparison of equivalent widths and shapes of line profiles with synthetic spectra may reveal effective temperatures, gravity acceleration, abundances and other physical parameters of stellar atmospheres. In practice, however, the spectra of components of the binary are blended and the information on orbital and atmospheric parameters are entangled. Several techniques for separation of component spectra from a series of spectra has been proposed which enable also to develop the so called spectra disentangling, i.e. a method of simultaneous separation of the spectra and determination of physical parameters governing their variability. In particular, the method of Fourier disentangling introduced and implemented in program KOREL by @h95 proved to be efficient and viable for a further generalisation. To allow the application of such a powerful method on a number of different objects in a scalable way, we attempted to embed the KOREL in the infrastructure of Virtual Observatory. The Virtual Observatory ======================= Contemporary astronomy faces an enormous amount of data continuously flowing from large telescopes, space missions and supercomputer simulations, that can hardly be analysed (and even previewed) by the traditional scientific methods. Thus the concept of (astronomical) Virtual Observatory (VO) was recently born aiming at federalisation of all astronomical resources (e.g. catalogues, data archives, simulation databases, data processing and analysing tools) using the global infrastructure based on unified data format and set of rigid, yet extensible communication protocols. The development and implementation of these global standards is the role of the International Virtual Observatory Alliance (IVOA). Technically, VO is a collection of inter-operating data archives and software tools which utilise the internet to form a virtual desktop environment in which astronomical research can be conducted in a user friendly manner allowing the astronomer to concentrate on asking the scientific questions instead of spending most of the time with searching in heterogeneous scattered archives, and with homogenisation of data represented by different units in various file formats. Owing to its huge data-mining potential and easy multiwavelength analysis tools, the VO technology allows to tackle problems not feasible by any other means, like the search of very rare astronomical events, candidates of yet unknown classes of objects (e.g. extremely cold brown dwarfs, supermassive stars etc.), statistics of order of tens of millions target or pan-spectral classification as building the spectra energy distributions of radiation from gamma to radio using the archives of all space and ground-based observations together. For the extensive introduction into the VO science see @2006LNEA....2...71S. The Fourier Disentangling ========================= The disentangling of spectra represents nowadays a whole branch of stellar spectroscopy fairly exceeding the scope of our contribution. We thus refer for a detailed explanation of its physical and mathematical principles, astrophysical consequences and for corresponding literature to the review [@h04] or its update [@h09b]. Here we shall only qualitatively characterise the method of Fourier disentangling implemented in code KOREL and we shall list a recent progress. The instantaneous spectra of many variable objects can be in a good approximation expressed as a superposition of their intrinsic (time independent) components convolved with some broadening functions (e.g. Doppler shifted delta-functions) depending on time and some physical parameters of the variability (e.g. the orbital parameters). In the Fourier conjugate space the intrinsic components can thus easily be solved (independently for each Fourier mode) from a more numerous set of observations. Moreover, the values of the free parameters can be fit by the least-square method. To prevent an ill-determination of the problem, a good coverage of the time interval of the characteristic variability is needed. The main task of the development of the method is thus to find a proper theoretical model of the broadening. Already the very simple assumption of line-strength variability with fixed line profiles [@h97] enables many useful applications. To apply the method successfully to real data, the observers should understand the assumptions and properties of the model and to prepare a set of data decisive for the parameters required from the solution. If the solution for the intrinsic component spectra is well over-determined by a great number of observed spectra, their noise can be substantially reduced by the averaging. A recent improvement of the numerical technique [@h09a] enables to retrieve the radial-velocity shifts with an accuracy surpassing the limitations by the step of spectra sampling (this is sometimes called super-resolution). Our recent work [@hss09] opens a disentangling of Cepheid pulsations. The Virtual Observatory Web Services ==================================== As the Fourier disentangling of the large number of spectra may become computation intensive, its full power may be exploited using the modern technology of VO Web Services (WS). The WS is typically complex processing application using the web technology (http protocol and (X)HTML markup) to transfer input data (files, tables, images, spectra etc.) to the main processing back-end (often run in front of queue scheduling and/or parallelising engine on computer clusters or GRIDS) and the results (after intensive number crunching) back to user. All this can be done using only an ordinary web browser (and in principle the science may be done on the fast palmtop or advanced mobile phone). The more detailed analysis about the benefits of GRID technology in stellar spectroscopy is presented by @2009MmSAI..80..484S. This service-oriented approach has many advantages both for the user and developer. Let’s name some of them: - There is the only one, current, well tested version of the code (and documentation), maintained and updated by its author - The user needs not to install anything from the author - The code is optimised for given HW (native compiler), knowing its limits (memory and cache sizes, number of nodes etc.) - The problem is scalable - the more user requirements may be solved by adding more computing nodes and introducing priority queues - The web technology provides the easy way of interaction (forms) and graphics output (in-line images) even produced dynamically (variable refresh rates or event driven - e.g. AJAX) The KOREL Web Service --------------------- The idea of our service is to have an user interface similar to e-shop portal, starting with user registration. Every set of input parameters creates a job, which may be run in parallel with others, the user may stop or remove them, can return to the previous versions etc. Privileged users may even recompile their own version of KOREL code tailored to their needs (e.g. maximum amount and size of spectra). All user communication is encrypted and the user can see only his/her jobs. The service may be accessed from the KOREL portal at Astronomical Institute in Ondřejov[^1]. At the time of preparation of the proceedings the KOREL Web Service requires to upload the files [korel.data]{} and [korel.par]{} in given strict format. Usually, for the preparation of input data the program [PREKOR]{} run at local computer is used, which reads spectra in various formats, rebins them equidistant in radial velocity (logarithmic wavelength) and optionally applies the precisely computed heliocentric correction. In addition to that, its interactive graphics helps to select the proper spectral regions bordered by the clear continuum and allows the removal of bad spectra. In the future, the role of the [PREKOR]{} may be replaced by another set of web services acquiring the spectra directly from VO servers and using proper metadata (e.g. elements of orbits) obtained from proper catalogues published in VO (especially CDS Vizier and Simbad). The interactive capability will be provided by VO spectral tools (e.g. SPLAT or VOSpec). Conclusions =========== The Fourier disentangling is already well-established method of stellar spectra analysis with the wide range of applications. The KOREL web service is probably one of the first attempts to adapt the legacy stellar spectra analysis code for the Virtual Observatory service. The advantages of solution adopted are evident, although some level of user conservatism has to be expected. This work was supported by grants GAČR 202/06/0041, GAČR 202/09/0772 and by projects AV0Z10030501 and LC06014. We thank all the developers of UK’s AstroGrid Virtual Observatory Project for testbeding the new paradigm of scientific research based on joining supercomputing GRID technologies with Virtual Observatory standards. We are greatly indebted to Pavel Škoda and Jan Fuchs for practical implementation of several versions of the KOREL Web Service. Hadrava, P. 1995, , 114, 393 Hadrava, P. 1997, , 122, 581 Hadrava, P. 2004, Publ. Astron. Inst. ASCR, 92, 15 Hadrava, P. 2009, , 494, 399 Hadrava, P. 2009, arXiv:0909.0172 Hadrava, P., Šlechta, M., & Škoda, P. 2009, , in press (arXiv:0909.0610) Solano, E. 2006, Lecture Notes and Essays in Astrophysics, 2, 71 koda, P. 2009, Memorie della Societa Astronomica Italiana, 80, 484 [^1]: http://stelweb.asu.cas.cz/vo-korel

ArXiv

--- abstract: 'This paper describes algorithms for the exact symbolic computation of period integrals on moduli spaces ${\mathcal{M}}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points, and applications to the computation of Feynman integrals.' author: - Christian Bogner and Francis Brown title: Feynman integrals and iterated integrals on moduli spaces of curves of genus zero --- MaPhy-AvH/2014-10 Introduction ============ Let $n\geq 0$ and let ${\mathcal{M}}_{0,n}$ denote the moduli space of Riemann spheres with $n$ ordered marked points. The main examples of periods of ${\mathcal{M}}_{0,n+3}$ consist of integrals [@Bro2; @Bro3; @Ter] $$\label{introM0nint} \int_{0\leq t_1 \leq \ldots \leq t_n \leq 1} {\prod_{i=1}^n t_i^{a_{i}} (1-t_i)^{b_i} \over \prod_{1 \leq i< j \leq n} (t_i-t_j)^{c_{ij}} } dt_1 \ldots dt_n$$ for suitable choices of integers $a_i,b_i,c_{ij} \in {\mathbb{Z}}$ such that the integral converges. These integrals have a variety of applications ranging from superstring theory [@Sch1; @Sch2] to irrationality proofs [@AperyVar; @Fisch1]. In [@Bro2] it was shown that such integrals are linear combinations of multiple zeta values $$\label{introMZVdef} \zeta(n_1,\ldots, n_r) = \sum_{1\leq k_1 < \ldots < k_r} {1 \over k_1^{n_1} \ldots k_r^{n_r}} \qquad \hbox{ where } n_i \in {\mathbb{N}}, n_r \geq 2$$ with rational coefficients. One of the goals of this paper is to provide effective algorithms, based on [@Bro2], for computing such integrals $(\ref{introM0nint})$ symbolically. The idea is to integrate out one variable at a time by working in a suitable algebra of iterated integrals (or rather, their symbols) which is closed under the two operations of taking primitives and taking limits along boundary divisors. The second main application is for the calculation of a large class of Feynman amplitudes, based on the universal property of the spaces ${\mathcal{M}}_{0,n}$. The general idea goes as follows. Suppose that $X \rightarrow S$ is a stable curve of genus zero. Then the universal property of moduli spaces yields an $n\geq3$ and a commutative diagram: $$\label{Square} \begin{array}{ccc} X & \longrightarrow & \overline{{\mathcal{M}}}_{0,n+1} \\ \downarrow & & \downarrow \\ S & \longrightarrow & \overline{{\mathcal{M}}}_{0,n} \end{array}$$ The idea is that, for a specific class of (multi-valued) forms on $X$, we can integrate in the fibers of $X$ over $S$ by passing to the right-hand side of the diagram and computing the integral on the moduli space ${\mathcal{M}}_{0,n+1}$. In this way, it only suffices to describe algorithms to integrate on the universal curve ${\mathcal{M}}_{0,n+1}$ over ${\mathcal{M}}_{0,n}$. In practice, this involves computing a change of variables to pass from $X$ to a set of convenient coordinates on the moduli space ${\mathcal{M}}_{0,n+1}$, applying the algorithm of [@Bro2] to integrate out one of these coordinates, and finally changing variables back to $S$. This process can be repeated for certain varieties which can be fibered in curves of genus $0$ and yields an effective algorithm for computing a large class of integrals. Necessary conditions for such fibrations to exist (‘linear reducibility’) were described in [@Bro4] and apply to many families of Feynman integrals, as we discuss in more detail presently. Feynman integrals ----------------- Any Feynman integral in even-dimensional space-time can always be expressed as an integral in Schwinger parameters $\alpha_j$: $$\label{introFeynI} I = \int_{0\leq \alpha_j \leq \infty } {P(\alpha_j) \over Q(\alpha_j)} \, d\alpha_1 \ldots d{\alpha_N}$$ where $P$ and $Q$ are polynomials with (typically) rational coefficients and which perhaps depend on other parameters such as masses or momenta. Cohomologial considerations tell us that the types of numbers occurring as such integrals only depend on the denominator $Q$, and not on the numerator $P$. A basic idea of [@Bro5] is to compute the integral $(\ref{introFeynI})$ by integrating out the Schwinger parameters $\alpha_i$ one at a time in some well-chosen order. After $i$ integrations, we require that the partial integral $$\label{Ikpartial} I(\alpha_1, \ldots, \alpha_{N-i}) = \int_{0\leq \alpha_j \leq \infty} {P \over Q}\, d\alpha_{N-i+1} \ldots d \alpha_{N}$$ be expressed as a certain kind of generalised polylogarithm function, or iterated integral. Under certain conditions on the singularities of the integrand, the next variable can be integrated out. A ‘linear reduction’ algorithm [@Bro4; @Bro5] yields an upper bound for the set of singularities of $(\ref{Ikpartial})$ and can tell us in advance whether $(\ref{introFeynI})$ can be computed by this method. It takes the form of a sequence of sets of polynomials (or rather, their associated hypersurfaces): $$S_1 \ , \ S_2 \ , \ \ldots$$ where $S_1 = \{Q\}$, and $S_{i+1}$ is derived from $S_{i}$ by taking certain resultants of polynomials in $S_i$ with respect to $\alpha_{N-i+1}$. When $Q$ is linearly reducible, we obtain a sequence of spaces for $i\geq 1$: $$\begin{aligned} X_i & =& ({\mathbb{P}}^1 \backslash \{0,\infty\})^{N-i+1} \ \backslash \ V(S_i) \nonumber \\ & =& \{ (\alpha_{1}, \ldots, \alpha_{N-i+1}): \alpha_k \neq 0, \infty \hbox{ and } P(\alpha_{1},\ldots, \alpha_{N-i+1}) \neq 0 \hbox{ for all } P \in S_i \} \nonumber \end{aligned}$$ and maps $\pi_i: X_i \rightarrow X_{i+1}$ which correspond to projecting out the variable $\alpha_{N-i+1}$. The linear reducibility assumption guarantees that $X_i$ fibers over $X_{i+1}$ in curves of genus $0$. Thus setting $(X,S) = (X_i, X_{i+1})$ in the discussion above, we can explicitly find changes of variables in the $\alpha_i$ to write $(\ref{Ikpartial})$ as an iterated integral on a moduli space ${\mathcal{M}}_{0,n}$ and do the next integration. It is perhaps surprising that such a method should ever work for any non-trivial Feynman integrals. The fundamental reason it does, however, is that the polynomial $Q$ can be expressed in terms of determinants of matrices whose entries are linear in the $\alpha_i$ parameters. In the case when $Q$ is the first Symanzik polynomial, and to a lesser extent when $Q$ also depends on masses and external momenta, it satisfies many ‘resultant identities’, which only break down at a certain loop order. A method of hyperlogarithms versus a method of moduli spaces {#secthypversusmod} ------------------------------------------------------------ There are two possible approaches to implementing the above algorithm: one which is now referred to as the ‘method of hyperlogarithms’ [@Bro5], which stays firmly on the left-hand side of the diagram $(\ref{Square})$; the other, which is the algorithm described here [@Bro2], which makes more systematic use of the geometry of the moduli spaces ${\mathcal{M}}_{0,n}$ and works on the right-hand side of the diagram $(\ref{Square})$. The first involves working directly in Schwinger parameters, and expressing all partial integrals as hyperlogarithms (iterated integrals of one variable) whose arguments are certain rational functions in Schwinger parameters. It has been fully implemented by Panzer [@Pan1; @Pan2; @Pan3] and various parts of the algorithm have found applications in different contexts, as described below. A conceptual disadvantage of this method is that the underlying geometry of every Feynman diagram is different. The second method, espoused here, is to compute all integrals on the moduli spaces ${\mathcal{M}}_{0,n}$ (which, by no accident, are the universal domain of definition for hyperlogarithms). Thus the underlying geometry is always the same and is well-understood; all the information about the particular integral $(\ref{introFeynI})$ is contained in the changes of variables $(\ref{Square})$. Another key difference is the systematic use of generalised symbols of functions in several complex variables, as opposed to functions of a single variable (hyperlogarithms). That these two points of view are equivalent is theorem \[thmVdecomp\] below, but leads, in practice, to a rather different algorithmic approach. We nonetheless provide algorithms (the symbol and unshuffle maps) to pass between both points of view. Applicability ------------- The above method can be applied to a range of Feynman integrals provided that the initial integral $(\ref{introFeynI})$ is convergent. The case of massless, single-scale, primitively overall-divergent Feynman diagrams in a scalar field theory was detailed in [@Bro4]. Since then, the method was applied to the computation of integrals of hexagonal Feynman graphs, arising in $\mathcal{N}=4$ supersymmetric Yang-Mills theory [@Del1; @Del2; @Del3], integrals with operator insertions contributing to massive matrix elements of quantum chromodynamics (QCD) [@Abl1; @Abl2; @Abl3], one- and two-loop triangular Feynman graphs with off-shell legs [@Cha], phase-space contributions [@Ana1; @Ana2] to the cross-section for threshold production of the Higgs boson from gluon-fusion at N3LO QCD [@Higgs], coefficients in the expansion of certain hypergeometric functions, contributing to superstring amplitudes [@Sch1; @Sch2], massless multi-loop propagator-type integrals [@Pan1], and a variety of three- and four-point Feynman integrals depending on several kinematical scales [@Pan2]. These applications arise from very different contexts and the method is combined with various other computational techniques. Focussing on Feynman integrals, we can summarize by stating that the method can be extended to the following situations: - To Feynman graphs with several masses or kinematic scales. - To gauge theories, or more generally, integrals with arbitrary numerator structures. - To graphs with ultra-violet subdivergences. In particular, it is compatible with the renormalisation procedure due to Bogoliubov, Parasiuk, Hepp and Zimmermann (BPHZ) in a momentum scheme [@BroKre]. - Finally, it can also be combined with dimensional regularisation to treat UV and IR divergences by the method of [@Pan2]. The method is suited for automatization on a computer. For the special case of harmonic polylogarithms, the programs [@Mai1; @Mai2] support direct integration using these functions. For the general approach, using hyperlogarithms, a first implementation of the method was presented in ref. [@Pan3]. A program for the numerical evaluation of these functions is given in ref. [@Vol]. There appear to be other classes of integrals which are not strictly Feynman diagrams, but for which the method of iterated fibration in curves of genus zero $(\ref{Square})$ still applies. A basic example are periods of arbitrary hyperplane complements [@Bro2], and as a consequence, various families of integrals occurring in deformation quantization, for example. Plan of the paper ----------------- In section $\ref{Sect2}$ we review some of the mathematics of iterated integrals on moduli spaces ${\mathcal{M}}_{0,n}$, based on [@Bro2]. The geometric ideas behind the main algorithms are outlined here. In §\[sec:Computing-on-the\], these algorithms are spelled out in complete detail together with some illustrative examples. In §\[sec:Feynman-type-integrals\], it is explained how to pass between Feynman integral representations and moduli space representations. In §\[sec:Applications\] we discuss some applications, before presenting the conclusions. Some introductory background can be found in the survey papers [@Bro6; @Bro7].\ The methods of $\S\ref{sec:Computing-on-the}$ should in principle generalise to genus $1$, using multiple elliptic polylogarithms defined in [@BrLe], but there remains a considerable amount of theoretical groundwork to be done. A different direction for generalisation is to introduce roots of unity, by replacing ${\mathbb{P}}^1\backslash \{0,1,\infty\}$ with ${\mathbb{P}}^1 \backslash \{0,\mu_N, \infty\}$ where $\mu_N$ is the group of $N^\mathrm{th}$ roots of unity. This should be rather similar to the framework discussed here.\ *Acknowledgements*: The second named author is a beneficiary of ERC grant 257638. The first named author thanks Erik Panzer for very useful discussions and especially helpful suggestions regarding the contents of section \[sec:Feynman-type-integrals\]. We thank Humboldt University for hospitality and support. Our Feynman graphs were drawn using [@Hah]. Iterated integrals on the moduli spaces ${\mathcal{M}}_{0,n}$ {#Sect2} ============================================================= Coordinates ----------- Let $n\geq 3$ and let ${\mathbb{C}}_{\infty}= {\mathbb{C}}\cup \{\infty\}$ denote the Riemann sphere. The complex moduli space ${\mathcal{M}}_{0,n}({\mathbb{C}})$ is the space of $n$ distinct ordered points on ${\mathbb{C}}_{\infty}$ modulo automorphisms $${\mathcal{M}}_{0,n}({\mathbb{C}}) = \{ (z_1,\ldots, z_n)\in {\mathbb{C}}_{\infty}^n \hbox{ distinct} \} /\mathrm{PGL_2}({\mathbb{C}}) \ .$$ There are two sets of coordinates, called simplicial and cubical, which are useful for the sequel. By applying an element of $\mathrm{PGL_2}({\mathbb{C}})$, we can assume that $z_1 =0, z_{n-1} =1$ and $z_n = \infty$ and define $$t_1 = z_2 \ , \ t_2 = z_3 \ , \ldots\ , \ t_{n-3} = z_{n-2} \ .$$ The $(t_1,\ldots, t_{n-3})$ are called simplicial coordinates and define an isomorphism $${\mathcal{M}}_{0,n}({\mathbb{C}}) \cong \{(t_1,\ldots, t_{n-3}) \in {\mathbb{C}}^{n-3} \hbox { such that the } t_i \hbox{ are distinct and } t_i \neq 0,1\}\ .$$ Cubical coordinates, on the other hand, are defined by $$\label{simplicialtocube} x_1 = {t_1 \over t_2} \ , \ x_2 = {t_2 \over t_3 } \ , \ \ldots\ , \ x_{n-4} = {t_{n-4} \over t_{n-3}} \ , \ x_{n-3}= t_{n-3}$$ Cubical coordinates define an isomorphism $${\mathcal{M}}_{0,n}({\mathbb{C}}) \cong \{(x_1,\ldots, x_{n-3}) \in {\mathbb{C}}^{n-3} \hbox { such that } x_{i} x_{i+1} \ldots x_j \neq \{0,1\} \hbox{ for all } 1 \leq i \leq j \leq n-3\}\ .$$ Note that the divisors above only involve products of cubical coordinates with consecutive indices. The main advantage of cubical coordinates is that the divisors corresponding to $$x_i=0 \quad \hbox{for } i=1,\ldots, n-3$$ are strict normal crossing in a neighbourhood of the origin $(0,\ldots, 0)$. The reason for the nomenclature is that the standard cell (a connected component of the set of real points ${\mathcal{M}}_{0,n}({\mathbb{R}})$) is either a simplex: $$X_n \cong \{(t_1,\ldots, t_{n-3}) \in {\mathbb{R}}^{n-3}: 0 < t_1 < \ldots < t_{n-3} < 1\}$$ or a cube: $$X_n \cong \{(x_1,\ldots, x_{n-3}) \in {\mathbb{R}}^{n-3}: 0 < x_i < 1 \hbox{ for all } 1\leq i \leq n-3 \}\ ,$$ depending on the choice of coordinate system. Differential forms ------------------ Let $\Omega^k({\mathcal{M}}_{0,n})$ denote the space of global regular differential $k$-forms on ${\mathcal{M}}_{0,n}$ which are defined over ${\mathbb{Q}}$. Consider the following elements of $\Omega^1({\mathcal{M}}_{0,n})$: $$\omega_{ij} = { {dt_i - dt_j} \over t_i -t_j } \ \hbox{ for } \ 0\leq i,j\leq n-2$$ where we set $t_0=0$ and $t_{n-2}= 1$. Clearly $\omega_{ij} = \omega_{ji}$ and $\omega_{ii}=0$. There are no other linear relations between the $\omega_{ij}$ besides these. Define $${\mathcal{A}}^1({\mathcal{M}}_{0,n}) = \langle \omega_{ij}: \hbox{ for } i < j \ , \ (i,j) \neq (0,n-2)\rangle_{{\mathbb{Q}}}$$ Thus ${\mathcal{A}}^1({\mathcal{M}}_{0,4})$ has the basis ${dt_1 \over t_1}, {dt_1 \over t_1 -1}$. The $\omega_{ij}$ satisfy the following quadratic relation: $$\label{quadrel} \omega_{ij} \wedge \omega_{jk}+ \omega_{jk} \wedge \omega_{ki} + \omega_{ki} \wedge \omega_{ij} =0$$ for all indices $i,j,k$. Define ${\mathcal{A}}^{\bullet}({\mathcal{M}}_{0,n})$ to be the differential graded algebra which is the quotient of the exterior algebra generated by ${\mathcal{A}}^1({\mathcal{M}}_{0,n})$ by the quadratic relations $(\ref{quadrel})$. A theorem due to Arnold states that $${\mathcal{A}}^{\bullet}({\mathcal{M}}_{0,n}) \longrightarrow H_{dR}^{\bullet}({\mathcal{M}}_{0,n};{\mathbb{Q}})$$ is an isomorphism of algebras. Thus ${\mathcal{A}}^{\bullet}({\mathcal{M}}_{0,n})$ is an explicit model for the de Rham cohomology of ${\mathcal{M}}_{0,n}$. In cubical coordinates, it is convenient to take a different basis for ${\mathcal{A}}^1({\mathcal{M}}_{0,n})$ formed by $${dx_i \over x_i} \quad \hbox{ and } \quad { { d (x_i\ldots x_j) \over x_i x_{i+1} \ldots x_j -1 } } \ \hbox{ for } 1\leq i \leq j \leq n-3 \ .$$ We will consider iterated integrals in these one-forms. Iterated integrals and symbols ------------------------------ Recall the definition of iterated integrals from [@Che]. Let $M$ be a smooth complex manifold and let $\omega_1, \ldots, \omega_n$ denote smooth 1-forms. Let $\gamma:[0,1] \rightarrow M$ be a smooth path. The iterated integral of these forms along $\gamma$ is defined by $$\int_{\gamma} \omega_1 \ldots \omega_n = \int_{0\leq t_1 \leq t_2 \leq \ldots \leq t_n \leq 1} \gamma^{*}(\omega_n) (t_1) \ldots \gamma^{*}(\omega_1) (t_n)\ .$$ There are different conventions for iterated integrals: here we integrate starting from the right. The argument of the left-hand integral is ${\mathbb{C}}$-multilinear in the forms $\omega_i$ and can be viewed as a functional on the tensor product $\Omega^1(M)^{\otimes n}$. Elements of this space are customarily written using the bar notation $[\omega_1 | \ldots | \omega_n]$ to denote a tensor product $\omega_1 \otimes \ldots \otimes \omega_n$. Chen’s theorem states that iterated integration defines an isomorphism from the zeroth cohomology of the reduced bar construction on the $C^{\infty}$ de Rham complex of $M$ to the space of iterated integrals on $M$ which only depend on the homotopy class of $\gamma$ relative to its endpoints. The reduced bar construction on ${\mathcal{M}}_{0,n}$ can be written down explicitly using the model ${\mathcal{A}}$ defined above, in terms of a certain algebra of symbols. For $n\geq 3$, define a graded ${\mathbb{Q}}$ vector space $$V({\mathcal{M}}_{0,n}) \subset \bigoplus_{m\geq 0} {\mathcal{A}}^1({\mathcal{M}}_{0,n})^{\otimes m}$$ by linear combinations of bar elements $$\sum_{I=(i_1,\ldots, i_m)} c_I [\omega_{i_1} | \ldots | \omega_{i_m}]$$ which satisfy the integrability condition $$\label{intcond} \sum_I c_I [\omega_{i_1} | \ldots | \omega_{i_{j-1}} | \omega_{i_j} \wedge \omega_{i_{j+1}} |\omega_{i_{j+2}}| \ldots | \omega_{i_m}] = 0 \qquad \hbox{ for all } 1\leq j \leq m-1\ .$$ Then $V({\mathcal{M}}_{0,n})$ is an algebra for the shuffle product ${\, \hbox{\rus x} \,}$ and is equipped with the deconcatenation coproduct $\Delta$, which is defined by: $$\Delta [\omega_{i_1} | \ldots | \omega_{i_m}] = \sum_{k=0}^m [\omega_{i_1} | \ldots | \omega_{i_k}] \otimes [\omega_{i_{k+1}} | \ldots | \omega_{i_m}]$$ Thus $V({\mathcal{M}}_{0,n})$ is a graded Hopf algebra over ${\mathbb{Q}}$. Iterated integration defines a homomorphism $$\begin{aligned} V({\mathcal{M}}_{0,n}) &\longrightarrow& \{\hbox{Multivalued functions on } {\mathcal{M}}_{0,n}({\mathbb{C}}) \} \label{itintonMod0n} \\ \sum_{I=(i_1,\ldots, i_m)} c_I [ \omega_{i_1} | \ldots | \omega_{i_m}] & \mapsto & \sum_I c_I \int_{\gamma_z} \omega_{i_1} \ldots \omega_{i_m} \nonumber \end{aligned}$$ where $\gamma_z$ is a homotopy equivalence class of paths from a fixed (tangential) base-point to $z \in {\mathcal{M}}_{0,n}({\mathbb{C}})$. By a version of Chen’s theorem, this map gives an isomorphism between homotopy invariant iterated integrals (viewed as multi-valued functions of their endpoint) on ${\mathcal{M}}_{0,n}$ and symbols. Equivalently, this means that the map $(\ref{itintonMod0n})$ is a homomorphism of differential algebras (for a certain differential to be defined in $(\ref{BdRdifferential})$) and the constants of integration are fixed as follows. One can show that, in cubical coordinates $(x_1,\ldots, x_{n-3})$, every iterated integral $(\ref{itintonMod0n})$ admits a finite expansion of the form $$\sum_{I=(i_1,\ldots, i_{n-3})} f_I(x_1,\ldots, x_{n-3}) \log(x_1)^{i_1} \ldots \log(x_{n-3})^{i_{n-3}}$$ where $f_I(x_1,\ldots, x_{n-3})$ is a formal power series in the $x_i$ which converges in the neighbourhood of the origin. The normalisation condition is that the regularised value at zero vanishes: $$f_{0,\ldots, 0} (0,\ldots, 0) = 0\ .$$ This gives a bijection between symbols and certain multivalued functions (whose branch is fixed, for example, on the standard cell $X_n)$, and in this way we can work entirely with symbols. Various operations on functions can be expressed algebraically in terms of $V({\mathcal{M}}_{0,n})$. For example, the monodromy of functions around loops can be expressed in terms of the coproduct $\Delta$. The bar-de Rham complex {#sectbardeRham} ----------------------- Differentiation of iterated integrals with respect to their endpoint corresponds to the following left-truncation operator $$\begin{aligned} \label{BdRdifferential} d: V({\mathcal{M}}_{0,n}) & \longrightarrow & \Omega^1({\mathcal{M}}_{0,n}) \otimes V({\mathcal{M}}_{0,n})\\ \sum_I c_I [ \omega_{i_1} | \ldots | \omega_{i_m}] & \mapsto & \sum_I c_I \omega_{i_1} \otimes [ \omega_{i_2} | \ldots | \omega_{i_m}] \nonumber \end{aligned}$$ where $I= (i_1,\ldots, i_m)$. The bar-de Rham complex is defined to be $$B({\mathcal{M}}_{0,n}) = \Omega^{\bullet}({\mathcal{M}}_{0,n}) \otimes V({\mathcal{M}}_{0,n})$$ equipped with the differential induced by $d$. In [@Bro2] it was shown that The cohomology of the bar-de Rham complex of ${\mathcal{M}}_{0,n}$ is trivial: $$H^i ( B({\mathcal{M}}_{0,n})) = \begin{cases} {\mathbb{Q}}\quad \hbox{ if } i = 0 \nonumber \\ 0 \quad \hbox{ if } i > 0 \end{cases}$$ In particular, $B({\mathcal{M}}_{0,n})$ is closed under the operation of taking primitives, which is one ingredient for computing integrals symbolically. The next ingredient states that one can compute regularised limits along irreducible boundary divisors $D \subset \overline{{\mathcal{M}}}_{0,n} \backslash {\mathcal{M}}_{0,n}$ with respect to certain local canonical sections $v$ of the normal bundle of $D$. Let $\mathcal{Z}$ denote the ${\mathbb{Q}}$-vector space generated by multiple zeta values $(\ref{introMZVdef}).$ \[thmreglimmap\] There exist canonical ‘regularised limit’ maps $$\mathrm{Reg}^v_D : V({\mathcal{M}}_{0,n}) \longrightarrow V({\mathcal{M}}_{0,r}) \otimes V({\mathcal{M}}_{0,n+2-r}) \otimes \mathcal{Z}$$ for every irreducible boundary divisor $D$ of $\overline{{\mathcal{M}}}_{0,n}$ which is isomorphic to $\overline{{\mathcal{M}}}_{0,r} \times \overline{{\mathcal{M}}}_{0,n+2-r}$. This states that the regularised limits of iterated integrals on moduli spaces are products of such iterated integrals with coefficients in the ring $\mathcal{Z}$ of multiple zeta values. By applying these two operations of primitives and limits, one can compute period integrals on ${\mathcal{M}}_{0,n}$. In more detail: ### Total primitives {#sectTotalPrimitives} Taking primitives of differential one-forms is a trivial matter. Let $\eta $ be a $1$-form in $B^1({\mathcal{M}}_{0,n})$ such that $d\eta=0$. We can write it as a finite sum $$\eta = \sum_k \omega^{k}_{0} \otimes [ \omega_1^k| \ldots | \omega_n^k]$$ A primitive is given explicitly by $$\int \eta = \sum_k [ \omega^{k}_{0}| \omega_1^k| \ldots | \omega_n^k] \ .$$ The constant of integration is uniquely (and automatically) determined by the property $$\varepsilon(\int \eta ) =0$$ where $\varepsilon: V({\mathcal{M}}_{0,n}) \rightarrow {\mathbb{Q}}$ is the augmentation map (projection onto terms of weight $0$). The fact that $\int \eta $ satisfies the integrability condition $(\ref{intcond})$ follows from the integrability of $\eta$ and the equation $d\eta=0$. In practice, the algorithm we actually use for taking primitives on the universal curve needs to be more sophisticated and is described below. ### Limits When taking limits, one must bear in mind the fact that the elements of $V({\mathcal{M}}_{0,n})$ represent multivalued functions, and hence depend on the (homotopy class) of the path $\gamma_z$ of analytic continuation $(\ref{itintonMod0n})$. When computing period integrals by the method described above, however, all iterated integrals which occur will be single-valued on the domain of integration ([@Bro4], theorem 58). In cubical coordinates, the domain of integration is the unit cube $X_n = [0,1]^{n-3}$, and so it suffices in this case to define limits along the divisors in $\overline{{\mathcal{M}}}_{0,n}$ defined by $x_i=0$ and $x_i=1$, where $x_i$ are cubical coordinates. Recall that the integration map from $V({\mathcal{M}}_{0,n})$ to multivalued functions is normalised at the point $(0,\ldots, 0)$ with respect to unit tangent vectors in cubical coordinates $x_i$, and it follows that the limits at $x_i=0$ are trivial to compute. Any function $f$ in the image of $(\ref{itintonMod0n})$ is uniquely determined on the simply connected domain $X_n=[0,1]^{n-3}$, and admits a unique expansion for some $N$ $$\label{fexpansion} f(x_1,\ldots, 1-\epsilon_i,\ldots, x_{n-3}) = \sum_{k=0}^N \log(\epsilon)^k p_k(\epsilon) f_k(x_1,\ldots, x_{i-1}, x_{i+1}, \ldots,x_{n-3})$$ where $p_k(\epsilon)$ is holomorphic at $\epsilon=0$ and where $f_k$ is in the image of $V({\mathcal{M}}_{0,i+2}) \otimes V({\mathcal{M}}_{0,n-i})$. The ‘regularised limit’ of $f$ along $x_i=1$ (with respect to the normal vector $-{\partial \over \partial x_i}$) is the function $$\mathrm{Reg}_{x_i=1}\, f = p_0(0) f_0(x_1,\ldots, x_{i-1}, x_{i+1}, \ldots,x_{n-3}) \ .$$ It is the composition of the realisation map $(\ref{itintonMod0n})$ with a certain map (theorem $\ref{thmreglimmap}$) $$V({\mathcal{M}}_{0,n}) \longrightarrow \mathcal{Z} \otimes V({\mathcal{M}}_{0,i+2}) \otimes V({\mathcal{M}}_{0,n-i})$$ where $\mathcal{Z}$ is the ring of multiple zeta values. This map can be computed explicitly as follows. Recall first of all the general formula for the behaviour of iterated integrals with respect to composition of paths, where $\gamma_1 \gamma_2$ denotes the path $\gamma_2$ followed by the path $\gamma_1$: $$\int_{\gamma_1 \gamma_2} \omega_1 \ldots \omega_n = \sum_{i=0}^n \int_{\gamma_1} \omega_1 \ldots \omega_i \int_{\gamma_2} \omega_{i+1} \ldots \omega_n\ .\label{pathconcat}$$ If $E_{\gamma}$ is the function on $V({\mathcal{M}}_{0,n})$ which denotes evaluation of a (regularised) iterated integral along a path $\gamma$, then the previous equation can be interpreted as a convolution product: $$\label{convolution} E_{\gamma_1 \gamma_2} = m ( E_{\gamma_1} \otimes E_{\gamma_2}) \circ \Delta$$ Ignoring, for the time being, issues to do with tangential base points and regularisation, a path from the origin $0$ to a point $z=(x_1,\ldots,x_{i-1}, 1, x_{i+1}, \ldots, x_{n-3})$ which lies inside the cube $X_n=[0,1]^{n-3}$ is homotopic to a composition of paths $\gamma_1 \gamma_2$ (‘up the $i^{\mathrm{th}}$ axis and then along to the point $z$’), where $$\gamma_2 =\hbox{straight line from } \ 0\ \hbox{ to } \ 1_i =( \underbrace{0,\ldots, 0}_{i-1}, 1, \underbrace{0,\ldots 0}_{n-i-4})$$ and $\gamma_1$ is a path from $1_i$ to $z$ which lies inside $x_i=1$. The segment of path $\gamma_2$ can be interpreted as a straight line from $0$ to $1$ in ${\mathcal{M}}_{0,4} = {\mathbb{P}}^1\backslash \{0,1,\infty\}$ (with coordinate $x_i$). Iterated integrals along this path give rise to coefficients of the Drinfeld associator, which are multiple zeta values. Iterated integrals along $\gamma_1$ can be identified with our class of multivalued functions on the boundary divisor $D$ of $\overline{{\mathcal{M}}}_{0,n}$ defined by $x_i=1$, which is canonically isomorphic to $\overline{{\mathcal{M}}}_{0,i+2} \times \overline{{\mathcal{M}}}_{0,n-i}$. One can check that the above argument makes sense for regularised (divergent) iterated integrals, and putting the pieces together yields the regularisation algorithm which is described below. For the computation of period integrals, one needs slightly more. We actually require an expansion of the function $(\ref{fexpansion})$ as a polynomial in $\log(\epsilon)$ and a Taylor expansion of $p_k(\epsilon)$ up to some order $K$ in $\epsilon$. This is because $f$ can occur with a rational prefactor which may have poles in $\epsilon$ of order $K$. This Taylor expansion is straightforward to compute recursively by expanding ${\partial \over \partial x_i} f$ and integrating (we know the constant terms by the previous discussion). The partial derivative ${\partial \over \partial x_i} f$ is simply a component of the total differential $d$ defined in $(\ref{BdRdifferential})$, which decreases the length and hence this gives an algorithm which terminates after finitely many steps, also described below. Note that in order to compute period integrals $(\ref{introM0nint})$, one only requires taking limits with respect to the final cubical variable $x_i$ for $i=n$. ### More general limits It can happen, for example when computing Feynman integrals, that one wants to take limits at more general divisors on $\overline{{\mathcal{M}}}_{0,n}$. The compactification of the standard cell $X_n$ (the closure of $X_n$ in $\overline{{\mathcal{M}}}_{0,n}$ for the analytic topology) is a closed polytope $$\overline{X}_n \subset \overline{{\mathcal{M}}}_{0,n}$$ which has the combinatorial structure of a Stasheff polytope. It can happen that one wants to compute limits at a (tangential) base point on the boundary of $\overline{X}_n$. An example is illustrated in the figure below in the case $n=5$, and where $\overline{X}_5$ is a pentagon. (-330,-10)[$x_1=0$]{} (-230,-10)[$x_1=1$]{} (-380,22)[$x_2=0$]{} (-380,120)[$x_2=1$]{} (-265,150)[$x_1x_2=1$]{} (5,80)[$\mathcal{E}$]{} (-50,128)[$z_1$]{} (-25,102)[$z_2$]{} The case of such limits can be dealt with using explicit local normal crossing coordinates on the boundary of $\overline{X}_n$ such as the dihedral coordinates $u_{ij}$ defined in [@Bro2]. One can show that any such limit is in fact a composition of regularised limits along divisors $x_{i_k} =1$ and $x_{i_k}=0$ in some specified (but not necessarily unique) order. This order can be determined from the combinatorics of the dihedral coordinates, and gives an algorithm to compute limits in this more general sense. For example, in the figure, the point $z_1$ is reached by taking the limit first as $x_2$ goes to $1$ and then $x_1$ goes to $1$; the point $z_2$ corresponds to the opposite order. The regularised limits of iterated integrals (such as $\mathrm{Li}_{1,1}(x,y)$) at $(1,1)$ along each path are different. Note that a path which approaches $(1,1)$ with a gradient which is strictly in between $0$ and $\infty$ corresponds to a limit point which is not equal to either $z_1$ or $z_2$ on $\mathcal{E}$ and could take us outside the realm of multiple zeta values. Finally, it is worth noting that one can imagine situations when one needs to take limits at points ‘at infinity’ corresponding to the case when, for example, some cubical coordinates $x_i$ go to infinity. This will not be discussed here. Fibrations ---------- The space $V({\mathcal{M}}_{0,n})$ is defined by a system of quadratic equations $(\ref{intcond})$ and its structure is hard to understand from this point of view. We will never need to actually solve the integrability equations $(\ref{intcond})$. A different description of $V({\mathcal{M}}_{0,n})$ comes from considering the morphism $$\begin{aligned} {\mathcal{M}}_{0,n} & \longrightarrow & {\mathcal{M}}_{0,n-1} \label{fibration} \\ (x_1,\ldots, x_{n-3}) & \mapsto & (x_1,\ldots, x_{n-4}) \nonumber\end{aligned}$$ which is obtained by forgetting the last cubical coordinate. It is a fibration, whose fiber over the point $(x_1,\ldots, x_{n-4})$ is isomorphic to the punctured projective line $$C_{n} = {\mathbb{P}}^1 \backslash \{ 0, (x_1\ldots x_{n-4})^{-1} , \ldots, x_{n-4}^{-1}, 1, \infty\}$$ with coordinate $x_{n-3}$. Let ${\mathcal{A}}_n={\mathcal{A}}({\mathcal{M}}_{0,n})$ denote the model for the de Rham complex on ${\mathcal{M}}_{0,n}$ defined earlier, and let ${\,{}^F \!\!\bar{\mathcal{A}}}_n= {\mathcal{A}}_{n}/{\mathcal{A}}_{n-1}$ denote the ${\mathbb{Q}}$-vector space of relative differentials. Denote the natural projection by $$\label{projectontoAf} \omega \mapsto \overline{\omega}: {\mathcal{A}}_{n} \rightarrow {\,{}^F \!\!\bar{\mathcal{A}}}_n$$ Using the representation of forms in cubical coordinates, we can choose a splitting $$\label{lambdadefn} \lambda_n \quad : \quad {\,{}^F \!\!\bar{\mathcal{A}}}_n \overset{\sim}{\rightarrow} {\,{}^F \!\!\mathcal{A}}_n \subseteq {\mathcal{A}}_{n}$$ which is defined explicitly in $(\ref{lambdaexplicit})$, and obtain a decomposition of ${\mathcal{A}}_{n-1}$-modules: $$\label{Aosproductdecomp} {\mathcal{A}}_{n} \cong {\mathcal{A}}_{n-1} \otimes {\,{}^F \!\!\bar{\mathcal{A}}}_n\ .$$ Armed with this decomposition, the quadratic relation $(\ref{quadrel})$ can be reinterpreted as a multiplication law on $1$-forms on the fiber: $$\begin{aligned} \label{Wedgedecomp} \mu_n \quad : \quad {\,{}^F \!\!\mathcal{A}}_n^1 \wedge {\,{}^F \!\!\mathcal{A}}_n^1 \longrightarrow {\mathcal{A}}^1_{n-1} \otimes {\,{}^F \!\!\mathcal{A}}^1_n \end{aligned}$$ which is used intensively in all computations. The product of two elements in ${\,{}^F \!\!\mathcal{A}}^1_n$ lies in ${\mathcal{A}}_n^2 \cong {\mathcal{A}}^2_{n-1} \oplus ({\mathcal{A}}^1_{n-1} \otimes {\,{}^F \!\!\bar{\mathcal{A}}}^1_n)$ since ${\,{}^F \!\!\bar{\mathcal{A}}}_n^2=0$. In fact, our choice of splitting $\lambda_n$ is such that the component of the previous isomorphism in ${\mathcal{A}}_{n-1}^2$ vanishes, which defines the map $(\ref{Wedgedecomp})$. [@Bro2] \[thmVdecomp\] The choice of map $\lambda_n$ gives a canonical isomorphism of algebras $$\label{Vdecomp} V({\mathcal{M}}_{0,n}) \cong V({\mathcal{M}}_{0,{n-1}}) \otimes V(C_n)\ ,$$ (which does not respect the coproducts on both sides) where $$V(C_n) = \bigoplus_{k\geq 0} ({\,{}^F \!\!\bar{\mathcal{A}}}^1_n)^{\otimes k}$$ is the ${\mathbb{Q}}$-vector space spanned by all words in ${\,{}^F \!\!\bar{\mathcal{A}}}^1_n$, equipped with the shuffle product. This gives a very precise description of the algebraic structure on $V({\mathcal{M}}_{0,n})$: by applying this theorem iteratively, every element of $V({\mathcal{M}}_{0,n})$ can be uniquely represented by a sum of tensors of words in prescribed alphabets. In order to go back and forth between the two representations on the left and right hand sides of $(\ref{Vdecomp})$ we have the symbol and unshuffle maps, defined as follows. 1. The *symbol map* is a homomorphism, which depends on the choice $(\ref{lambdadefn})$, $$\Psi: V(C_n) \longrightarrow V({\mathcal{M}}_{0,n})$$ which can be thought of as the map which takes a function defined on a fiber of the universal curve $C_n$ and extends it to a function on the entire moduli space ${\mathcal{M}}_{0,n}$. It is constructed as follows. One can define a Gauss-Manin connection, corresponding to ‘differentiation under an iterated integral’ which is a linear map $$\nabla : V(C_n) \longrightarrow {\mathcal{A}}^1_{n-1} \otimes V(C_n)$$ by the following recipe: lift words in ${\,{}^F \!\!\bar{\mathcal{A}}}_n$ to words in ${\mathcal{A}}_{n}$ via the map $\lambda_n$; then apply the usual internal differential of the bar construction in degree $0$ (all signs simplify since the $\omega_i$ are $1$-forms): $$\label{integrability} D [\omega_1 | \ldots | \omega_n] = (-1)^n\big( \sum_{i=1}^n [\omega_1 | \ldots | d\omega_i| \ldots | \omega_n] + \sum_{i=1}^{n-1} [\omega_1 | \ldots | \omega_i \wedge \omega_{i+1} | \ldots | \omega_n] \big)$$ and finally project all one-forms on the right-hand side to ${\,{}^F \!\!\bar{\mathcal{A}}}^1$ via the map $(\ref{projectontoAf})$ and project all two forms (namely, $d \omega_i$ and $\omega_i \wedge \omega_{i+1}$) onto ${\mathcal{A}}^1_{n-1} \otimes {\,{}^F \!\!\bar{\mathcal{A}}}^1$ via the decomposition $(\ref{Wedgedecomp})$. Pulling out all factors in ${\mathcal{A}}^1_{n-1}$ to the left gives the required formula for $\nabla$. The connection $\nabla$ can be promoted to a total connection $$\begin{aligned} \label{TotalConnection} \nabla_T: V(C_n) \longrightarrow {\mathcal{A}}^1_{n} \otimes V(C_n)\end{aligned}$$ by setting $\nabla_T= d- \nabla $, and identifying ${\mathcal{A}}^1_{n-1} \oplus {\,{}^F \!\!\bar{\mathcal{A}}}_n^1 \cong {\mathcal{A}}^1_{n}$ via the decomposition $(\ref{Aosproductdecomp})$. It is straightforward to show that in this context the total connection is flat $(\nabla_T^2=0)$. Finally, the symbol map is the unique linear map (necessarily a homomorphism) $$\label{SymbolMap} \Psi: V(C_n) \longrightarrow V({\mathcal{M}}_{0,n})$$ which satisfies the equation $$(id \otimes \Psi) \circ \nabla_T = d \circ \Psi.$$ This can be viewed as a recursive formula to compute the symbol map $\Psi$ since $\nabla_T$ strictly decreases the length of bar elements. Explicitly, it can be rewritten $$\Psi= \int (id \otimes \Psi) \circ \nabla_T$$ where the total primitive operator $\int$ was defined in §\[sectTotalPrimitives\]. 2. In the other direction, there is the *unshuffle* map which is a homomorphism of graded algebras $$\Phi: V({\mathcal{M}}_{0,n}) \overset{\sim}{\longrightarrow} V({\mathcal{M}}_{0,n-1}) \otimes V(C_n)$$ which is the inverse of the map $m({\mathrm{id}}\otimes \Psi) : V({\mathcal{M}}_{0,n-1}) \otimes V(C_n) \rightarrow V({\mathcal{M}}_{0,n}) $ (which we abusively denote simply by $\Psi$), where $m$ denotes multiplication. It can be computed as follows. Denote the natural map $$\begin{aligned} r: V({\mathcal{M}}_{0,n}) & \longrightarrow & V(C_n) \nonumber \\ {[}\omega_1 | \ldots | \omega_r] & \mapsto& [\overline{\omega}_1| \ldots | \overline{\omega}_r] \nonumber \end{aligned}$$ given by restriction of iterated integrals to the fiber induced by $(\ref{projectontoAf})$ component-wise on bar elements. Note that the map $\Psi$ has the property that $r\circ \Psi$ is the identity on $V(C_n)$. Recall the morphism $(\ref{fibration})$ from ${\mathcal{M}}_{0,n}$ to ${\mathcal{M}}_{0,n-1}$ defined in terms of cubical coordinates. The projection map $\pi : {\mathcal{A}}_n \rightarrow {\mathcal{A}}_{n-1}$ implied by the section $\lambda_n$ is given by sending first $dx_{n-3}$ to zero and then $x_{n-3}$ to zero. One can see that it is a homomorphism by inspection of the explicit equations in §\[sub:Arnold-relations\]: the product of two elements in ${}^F\!{\mathcal{A}}_n^1$ have no component in ${\mathcal{A}}^2_{n-1}$. It defines a homomorphism $$\pi: V({\mathcal{M}}_{0,n})\rightarrow V({\mathcal{M}}_{0,n-1})$$ and one easily verifies that the homomorphism $\Phi$ defined by $$\Phi( \xi) = (r \otimes \pi ) \circ \Delta$$ is an inverse to the symbol map $\Psi$. Alternatively, we can view ${\mathcal{M}}_{0,n-1}$ as being embedded in $\overline{{\mathcal{M}}}_{0,n}$ by identifying it with the divisor defined by $x_{n-3}=0$. An element of $V({\mathcal{M}}_{0,n})$ can be thought of as an iterated integral along a path from the unit tangential base point at the origin $0$ in cubical coordinates to a point $x=(x_1,\ldots, x_{n-3})$. It is the composition of a path from the unit tangential base point at $0$ to $(x_1,\ldots, x_{n-4})$ in the base ${\mathcal{M}}_{0,n-1}$, followed by a path in $C_n$ from the unit tangential base point at $x_{n-3}=0$ to $x$. Since composition of paths is dual to deconcatenation in $V({\mathcal{M}}_{0,n})$, this yields a geometric interpretation of the above formula for $\Phi$. Thus it is possible, via the symbol and unshuffle maps, to pass back and forth between a representation of an iterated integral on ${\mathcal{M}}_{0,n}$ as a symbol in $V({\mathcal{M}}_{0,n})$ or a product of words in $V(C_i)$’s. This gives a precise algorithmic equivalence between the two approaches described in §\[secthypversusmod\]. Representation as functions --------------------------- In order to represent elements of $V({\mathcal{M}}_{0,n})$ as functions (although in principle one never needs to do this) the simplest method is to apply the unshuffle map $\Phi$ defined above, which reduces to the problem of representing elements of $V(C_{k})$, for $4\leq k \leq n$ as functions. This is simply the case of computing iterated integrals in a single variable $x_{n-3}$, i.e. hyperlogarithms. $$\begin{aligned} V(C_n) &\longrightarrow & \hbox{Iterated integrals on } C_n \\ \nonumber [ \omega_1 | \ldots | \omega_k] & \mapsto & \int \omega_1 \ldots \omega_k \nonumber\end{aligned}$$ The iterated integrals on $C_n$ are normalised with respect to the tangential base point ${\partial \over \partial x_{n-3}}$ at $x_{n-3}=0$. They can be written as polynomials in $\log(x_{n-3})$ and explicit power series which were studied in the work of Lappo-Danilevsky [@Lap]. In this way, the iterated use of the unshuffle map reduces the expression of elements of $V({\mathcal{M}}_{0,n})$ as functions to a product of hyperlogarithms. These are well understood, and can be expressed in terms of multiple polylogarithms $$\mathrm{Li}_{n_1,\ldots, n_r}(x_1,\ldots, x_r) = \sum_{0<k_1<\ldots< k_r} {x_1^{k_1} \ldots x_r^{k_r} \over k_1^{n_1} \ldots k_r^{n_r}}$$ which can be computed to arbitrary accuracy by standard techniques [@Vol]. ‘Mixed’ primitives {#subMixedPrimitives} ------------------ Suppose that we have an element $\xi \in V({\mathcal{M}}_{0,n})$, and a one form $\omega \in {\,{}^F \!\!\mathcal{A}}_{n}^1$ which is only defined on the fiber. The mixed primitive is defined to be $$\omega \star \xi := \Psi \big( \int \omega \,\Phi(\xi)\big) \qquad \in V({\mathcal{M}}_{0,n})\ .$$ In other words, $\xi$ is viewed as an element of $V({\mathcal{M}}_{0,n-1})\otimes V(C_n)$ via the unshuffle map, then multiplied by $1\otimes \omega$ before computing its primitive $\int$ in $V(C_n)$ (which is simply given by left concatenation of forms in ${\,{}^F \!\!\mathcal{A}}^1$, as in §\[sectTotalPrimitives\]). Clearly, the map $\star$ is bilinear over ${\mathbb{Q}}$ and satisfies $$\label{starproperty1} \omega_0 \star \Psi([\omega_1 | \ldots | \omega_k]) = \Psi( [\omega_0 | \ldots | \omega_k])$$ for all $\omega_i \in {\,{}^F \!\!\bar{\mathcal{A}}}_{n}^1$. Furthermore, $\star$ is right-linear over $V({\mathcal{M}}_{0,n-1})$: $$\label{starproperty2} \omega \star ( b {\, \hbox{\rus x} \,}\xi) = b {\, \hbox{\rus x} \,}(\omega \star \xi)$$ for all $b\in V({\mathcal{M}}_{0,n-1})$, and $\xi \in V({\mathcal{M}}_{0,n})$, and $\star$ is uniquely determined by $(\ref{starproperty1})$, $(\ref{starproperty2})$ and $(\ref{Vdecomp})$. Evidently, one does not want to have to compute $\star$ by applying the unshuffling and symbol maps $\Phi$ and $\Psi$ which would be highly inefficient (and largely redundant). The approach we have adopted is more direct. Suppose that $\xi =\sum_I c_I [\omega_{i_1} | \ldots | \omega_{i_m}]$. As a first approximation to the mixed primitive $ \omega \star \xi $ take the element $$\xi_0=\sum_{I=(i_1,\ldots, i_m)} c_I [\lambda_n(\omega) | \omega_{i_1} | \ldots | \omega_{i_m}]$$ The projection of $\xi_0$ onto $V(C_n)$ coincides with that of $\omega \star \xi$, but $\xi_0$ does not satisfy the integrability condition $(\ref{intcond})$. The idea is to add correction terms $\xi_1, \ldots, \xi_k$ to $\xi_0$ so that the sum $\xi_0 + \ldots + \xi_k = \sum_J c'_J [ \eta_{j_1}| \ldots | \eta_{j_{m+1}}]$ satisfies the first $k$ integrability equations (with the notation of $(\ref{intcond})$) $$\sum_J c'_J [\eta_{i_1} | \ldots | \eta_{j_r} \wedge \eta_{j_{r+1}} | \ldots | \eta_{j_{m+1}}]=0 \qquad \hbox{ for } 1 \leq r \leq k$$ The correction term $\xi_{k+1}$ is obtained using the quadratic relations $\mu_n$ to expand out each wedge product $\omega_i \wedge \omega_j$ in the $k+1$th integrability equation, applied to $\xi_0+\ldots+ \xi_k$. The mixed primitive $\omega \star \xi$ is equal to the sum $\xi_0+ \ldots + \xi_{m}$ if $\xi$ is of length $m$. The precise details are described below. Feasibility and orders of magnitude ----------------------------------- By iterating theorem $(\ref{thmVdecomp})$ one obtains a formula for the dimension of all symbols on ${\mathcal{M}}_{0,n+3}$ in weight $N$: $$\label{dimformula} \sum_{N\geq 0}\, (\dim_{{\mathbb{Q}}} V({\mathcal{M}}_{0,n+3})_{N}) t^N = {1 \over (1-2t) (1-3t) \ldots (1-(n+1)t)}$$ This gives a coarse upper bound for the possible size of expressions which can occur during the integration process. At the initial step of integration, the integrand is of weight $0$ on a moduli space of high dimension ${\mathcal{M}}_{0,n+3}$, and at the final step, the integrand is of high weight on a moduli space of low dimension ${\mathcal{M}}_{0,4}$. The dimensions $(\ref{dimformula})$ peak somewhere in the middle of the computation. For example, for (the maximal weight part) of a period integral $(\ref{introM0nint})$ in five variables, one works in a sequence of vector spaces of dimension $20,125,285, 211,32$ (these are the dimensions of the spaces of functions after taking each primitive and before taking each limit). In the case of Feynman diagrams, one can estimate in advance (using the linear reduction algorithm) the number of marked points $n$ which will be required at each step of the integration to get a bound on the size of the computation. In practice, it seems that the limit of what is reasonable with current levels of computing power should be adequate to reach the ‘non-polylogarithmic’ boundary where amplitudes which are not periods of mixed Tate motives first start to appear. Computing on the moduli space {#sec:Computing-on-the} ============================= In this section we spell out the details of the above algorithms and present them in a version which is ready for implementation on a computer. As a proof of concept we implemented these algorithms in a Maple-based computer program. With this program we computed all examples below and all applications of section \[sec:Applications\]. For notational convenience let $m=n-3$ denote the number of cubical coordinates $x_i$ on ${\mathcal{M}}_{0,n}$. As bases for ${\mathcal{A}}^1_{n}$, ${\,{}^F \!\!\bar{\mathcal{A}}}^1_n$ and ${\,{}^F \!\!\mathcal{A}}^1_n$ we choose the sets of closed 1-forms $$\begin{aligned} \Omega_{m} & = & \left\{ \frac{dx_{1}}{x_{1}},\,...,\,\frac{dx_{m}}{x_{m}},\,\frac{d\left(\prod_{a\leq i\leq b}x_{i}\right)} {\prod_{a\leq i\leq b}x_{i}-1}\textrm{ for }1\leq a\leq b\leq m\right\} ,\\ \bar{\Omega}^F_{m} & = & \left\{ \frac{dx_{m}}{x_{m}},\,\frac{\left(\prod_{a\leq i\leq m-1}x_{i}\right)dx_{m}} {\prod_{a\leq i\leq m}x_{i}-1}\textrm{ for }1\leq a\leq m\right\} ,\\ \Omega_{m}^{F} & = & \left\{ \frac{dx_{m}}{x_{m}},\,\frac{d\left(\prod_{a\leq i\leq m}x_{i}\right)}{\prod_{a\leq i\leq m}x_{i}-1} \textrm{ for }1\leq a\leq m\right\} ,\\\end{aligned}$$ respectively. The isomorphism $\bar{{\,{}^F \!\!\mathcal{A}}_n}\overset{\lambda_{n}}{\cong}{\,{}^F \!\!\mathcal{A}}_n \subseteq {\mathcal{A}}_n$ of $(\ref{lambdadefn})$ is defined explicitly by $$\begin{aligned} \label{lambdaexplicit} \lambda_{n}\frac{dx_{m}}{x_{m}} & = & \frac{dx_{m}}{x_{m}},\\ \lambda_{n}\frac{\left(\prod_{a\leq i\leq m-1}x_{i}\right)dx_{m}}{\prod_{a\leq i\leq m}x_{i}-1} & = & \frac{d\left(\prod_{a\leq i\leq m}x_{i}\right)}{\prod_{a\leq i\leq m}x_{i}-1}\textrm{ for }1\leq a \leq m. \nonumber \end{aligned}$$ According to these chosen bases, we refer to the vector-spaces $V(C_{n}),\, V(\mathcal{M}_{0,n})$ by $V(\Omega^F_{m}),\, V(\Omega_{m})$ respectively. Iterated integrals are written as linear combinations of words $[\omega_{1}|...|\omega_{k}]$, whose letters are 1-forms in these sets. Note that $\Omega_m $ is the disjoint union of $\Omega_{m-1}$ and $\Omega_{m}^F$. Arnold relations {#sub:Arnold-relations} ---------------- With the above choices, the Arnold relations of $(\ref{Wedgedecomp})$ read explicitly: $$\begin{aligned} \frac{dx_{m}}{x_{m}}\wedge\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1} & = & -\sum_{k=i}^{m-1}\frac{dx_{k}}{x_{k}}\wedge\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1},\\ \frac{d\left(x_{j}...x_{m}\right)}{x_{j}...x_{m}-1}\wedge\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1} & = & \frac{d\left(x_{i}...x_{j-1}\right)}{x_{i}...x_{j-1}-1}\wedge\left(\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1}-\frac{d\left(x_{j}...x_{m}\right)}{x_{j}...x_{m}-1}\right)-\sum_{k=i}^{j-1}\frac{dx_{k}}{x_{k}}\wedge\frac{d\left(x_{i}...x_{m}\right)}{x_{i}...x_{m}-1}\end{aligned}$$ for $1\leq i\leq j\leq m.$ For the implementation on a computer, it is efficient to generate these equations to a desired number of variables once and for all, and to store them as a look-up table since they are used very frequently by the algorithms below. The splitting of theorem \[thmVdecomp\] is realised by a certain application of the Arnold relations. We define an auxiliary map $\rho_{i}$ by the following operations. For a word $\xi=[\omega_{1}|...|\omega_{k}]$ with letters in $\bar{\Omega}^F_{m}$ and some $1\leq i<k$ we consider the neighbouring letters $\omega_i | \omega_{i+1}$ and consider the wedge-product of their images in $\Omega^F_{m}$. By the corresponding Arnold relation, we express this product as a $\mathbb{Q}$-linear combination of wedge-products, with one factor in the base $\Omega_{m-1}$ and one in the fiber $\Omega^F_{m}$. We replace the letters $\omega_i | \omega_{i+1}$ in $\xi$ by the factor in $\Omega^F_{m}$ and pull the base-term in $\Omega_{m-1}$ and rational pre-factors out of the word. In summary, this defines the auxiliary map $$\rho_{i}:\, V\left(\bar{\Omega}^F_{m}\right)\rightarrow\Omega_{m-1}\otimes V\left(\bar{\Omega}^F_{m}\right)$$ by $$\rho_{i}\left[a_{1}|...|a_{k}\right]=\sum_{j}c_{j}\eta_{j}\otimes\left[a_{1}|...|a_{i-1}| \overline{\alpha}_{j}|a_{i+2}|...|a_{k}\right]$$ where $\eta_{j}\in\Omega_{m-1},\,\alpha_{j}\in\Omega_{m}^{F},\, c_{j}\in\mathbb{Q}$ are determined by the Arnold relation $$\lambda_{n}a_{i}\wedge\lambda_{n}a_{i+1}=\sum_{j}c_{j}\eta_{j}\wedge\alpha_{j}.$$ Note that these are the same operations as in our definition of the Gauss-Manin connection $\nabla$ above, which we obtain by summing the $\rho_i$ over $i$. This is because the first sum on the right-hand side of $(\ref{integrability})$ vanishes in our set-up, as all our 1-forms are closed, and the operations on the terms of the second sum correspond to the definition of $\rho_i$. \[exampleArnoldn=5\] For $n=5, m=2$ we have the Arnold relations $$\begin{aligned} \frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}\wedge\frac{dx_{2}}{x_{2}} & = & \frac{dx_{1}}{x_{1}}\wedge\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1},\\ \frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}\wedge\frac{dx_{2}}{x_{2}-1} & = & \left(\frac{dx_{1}}{x_{1}}-\frac{dx_{1}}{x_{1}-1}\right)\wedge\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}+\frac{dx_{1}}{x_{1}-1}\wedge\frac{dx_{2}}{x_{2}-1}.\end{aligned}$$ For the words $\kappa=\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right],$ $\xi=\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]$ in $V(\bar{\Omega}^F_{2})$ we compute $$\begin{aligned} \rho_{1}\kappa & = & \left[\frac{dx_{1}}{x_{1}}\right]\otimes\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}\right] -\left[\frac{dx_{1}}{x_{1}-1}\right]\otimes\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}\right]+\left[\frac{dx_{1}}{x_{1}-1}\right] \otimes\left[\frac{dx_{2}}{x_{2}-1}\right],\\ \rho_{1}\xi & = & \left[\frac{dx_{1}}{x_{1}}\right]\otimes\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right],\\ \rho_{2}\xi & = & 0.\end{aligned}$$ The symbol map {#sub:The-symbol-map} -------------- Both the total connection and the symbol map can be computed conveniently by use of the maps $\rho_{i}$. The total connection (see $(\ref{TotalConnection})$) is computed as $$\nabla_{T}\left[a_{1}|...|a_{k}\right]=d\left[a_{1}|...|a_{k}\right]-\sum_{1\leq i<k}\rho_{i}\left[a_{1}|...|a_{k}\right]$$ where (by $(\ref{BdRdifferential})$) $$d\left[a_{1}|...|a_{k}\right]=a_{1}\otimes\left[a_{2}|...|a_{k}\right].$$ The symbol map $\Psi$ (see $(\ref{SymbolMap})$) is applied to a word in $V(\bar{\Omega}^F_{m})$ by the recursive algorithm $$\begin{aligned} \Psi\left(\left[a_{i}\right]\right) & = & \left[\lambda_{n}\left(a_{i}\right)\right],\nonumber \\ \Psi\left(\left[a_{i_{1}}|a_{i_{2}}|...|a_{i_{k}}\right]\right) & = & \lambda_{n}\left(a_{i_{1}}\right)\sqcup\Psi\left(\left[a_{i_{2}}|...|a_{i_{k}}\right]\right)-\sum_{1\leq i<k}\sqcup\left(\left(\textrm{id}\otimes\Psi\right)\rho_{i}\left[a_{i_{1}}|...|a_{i_{k}}\right]\right),\;1<k,\label{eq:symbol map-1}\end{aligned}$$ where $\xi_1\sqcup \xi_2\equiv\sqcup(\xi_1\otimes \xi_2)$ denotes the concatenation of two words $\xi_1,\, \xi_2.$ Note that on the right hand side of $(\ref{eq:symbol map-1})$ the map $\Psi$ acts on words of length $k-1.$ Making use of the relations derived in example \[exampleArnoldn=5\], we compute $$\begin{aligned} \Psi\left(\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]\right) & = & \left[\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]-\left[\frac{dx_{1}}{x_{1}}\right]\sqcup\Psi\left(\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right]\right)\\ & = & \left[\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]-\left[\frac{dx_{1}}{x_{1}}|\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right]\\ & & -\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}\right]+\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{dx_{2}}{x_{2}-1}\right]\\ & & +\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}}|\frac{x_{1}dx_{2}+x_{2}dx_{1}}{x_{1}x_{2}-1}\right].\end{aligned}$$ The map $\Psi$ is defined such that for any $\xi\in V\left(\bar{\Omega}^F_{m}\right)$ we have $D\Psi\left(\xi\right)=0$ and therefore $\Psi\left(\xi\right)\in V\left(\Omega_{m}\right).$ The vector space $V\left(\Omega_{m}\right)$ is generated, over $V(\Omega_{m-1})$, by the image of $V\left(\bar{\Omega}^F_{m}\right)$ under $\Psi.$ We furthermore note the property $$\Psi\left(\xi_{1}{\, \hbox{\rus x} \,}\xi_{2}\right)=\Psi(\xi_{1}){\, \hbox{\rus x} \,}\Psi(\xi_{2})$$ for any $\xi_{1},\,\xi_{2}\in V\left(\bar{\Omega}^F_{n}\right).$ A slightly different algorithm for $\Psi$ in terms of differentiation under the integral was already given in [@Bog1]. For related constructions, also see references [@Duh1; @Gon1; @Gon2]. In section \[sec:Feynman-type-integrals\] we will make use of $\Psi$ as a part of a procedure to map hyperlogarithms in Schwinger parameters to multiple polylogarithms of cubical variables. We expect the map $\Psi$ also to be useful in different contexts such as [@Druetal]. Primitives {#sub:Primitives} ---------- Let $\omega\in\bar{\Omega}^F_m$ and let $\xi=\sum_{I}c_{I}\left[\omega_{i_{1}}|...|\omega_{i_{k}}\right]$ be an iterated integral in $V(\Omega_{m})$. In subsection \[subMixedPrimitives\], we discussed the strategy of building up the mixed primitive $\omega\star\xi$ by naive left-concatenation of the form $\omega$ to the word $\xi$, yielding $$\label{xi0ofmixedprimitives} \sum_{I}c_{I}\left[\lambda_{n}(\omega)|\omega_{i_{1}}|...|\omega_{i_{k}}\right],$$ and the addition of correction terms until the resulting combination satisfies the integrability condition of $(\ref{intcond})$. For the explicit computation of the correction terms, let us introduce some auxiliary notation. For all $0\leq i < k$ let $C_{i}\left(\Omega_{m}\right)_k= \Omega_{m-1}^{\otimes i} \otimes \Omega_m^F \otimes \Omega_m^{\otimes(k-i-1)}$ be the $\mathbb{Q}$-vector space of words of length $k$ with letters in $\Omega_{m}$, whose first $i$ letters, counted from the left, are in the base $\Omega_{m-1}$, and whose $(i+1)^{\mathrm{th}}$ letter is in the fiber $\Omega_m^F$. The members of these auxiliary sets of words do not necessarily stand for homotopy invariant iterated integrals. We define the auxiliary maps $$\star_{i}:\, C_{i-1}\left(\Omega_{m}\right)_k\rightarrow C_{i}\left(\Omega_{m}\right)_k$$ for $i<k$ by the following recipe $$\begin{aligned} \star_{i}[a_{1}|...|a_{i-1}|a_{i}|a_{i+1}|...|a_{k}] & = & [a_{1}|...|a_{i-1}|a_{i+1}|a_{i}|...|a_{k}]\qquad \textrm{ if } a_{i+1}\in\Omega_{m-1} , \nonumber \\ \star_{i}[a_{1}|...|a_{i-1}|a_{i}|a_{i+1}|...|a_{k}] & = & -\sum_{j}c_{j}[a_{1}|...|a_{i-1}|\eta_{j}|\alpha_{j}|a_{i+2}|...|a_{k}]\qquad \textrm{ if }a_{i+1}\in\Omega_{m}^{F},\, \label{eq:star i}\end{aligned}$$ where the forms $\eta_{j}\in\Omega_{m-1},\,\alpha_{j}\in\Omega_{m}^{F}$ and constants $\, c_{j}\in\mathbb{Q}$ are determined by an Arnold relation $$a_{i}\wedge a_{i+1}=\sum_{j}c_{j}\eta_{j}\wedge\alpha_{j}.$$ Note that indeed, in each word on the right-hand side of $(\ref{eq:star i})$ the 1-forms in the first $i$ positions are in $\Omega_{m-1}$ and the form in the $(i+1)$-th position is in $\Omega_{m}^{F}$. This procedure can be iterated. Since $\ref{xi0ofmixedprimitives}$ lies in $C_{0}\left(\Omega_{m}\right)_ {k+1}$, we repeatedly apply $\star_{\bullet}$ to obtain the following formula for the mixed primitive $$\begin{aligned} \label{eq:Primitive} \omega\star[a_{1}|...|a_{k}]=(1+\star_{1}+\star_{2}\star_{1}+...+\star_{k}...\star_{1})[\lambda_{m}(\omega)|a_{1}|...|a_{k}].\end{aligned}$$ The construction satisfies the relations $(\ref{starproperty1})$ and $(\ref{starproperty2})$. We consider the 1-form $\omega = \frac{dx_2}{x_2}$, the iterated integral $$\begin{aligned} \xi = \Psi \left(\left[ \frac{x_1 d(x_2)}{x_1 x_2 -1} | \frac{dx_2}{x_2} \right] \right) = \left[ \frac{d(x_1 x_2)}{x_1 x_2 -1} | \frac{dx_2}{x_2} \right] - \left[ \frac{dx_1}{x_1} | \frac{d(x_1 x_2)}{x_1 x_2 -1} \right],\end{aligned}$$ and the concatenation $$\begin{aligned} \xi_0 & = & \lambda_2 (\omega) \sqcup \xi = \left[ \frac{dx_2}{x_2} | \frac{d(x_1 x_2)}{x_1 x_2 -1} | \frac{dx_2}{x_2} \right] - \left[ \frac{dx_2}{x_2} | \frac{dx_1}{x_1} | \frac{d(x_1 x_2)}{x_1 x_2 -1} \right].\end{aligned}$$ Following $(\ref{eq:Primitive})$, we compute the primitive $$\begin{aligned} \omega \star \xi = \xi_0 + \xi_1 + \xi_2\end{aligned}$$ where $\xi_1 = \star_1 \xi_0$ and $\xi_2 = \star_2 \star_1 \xi_0$. We obtain $$\begin{aligned} \\ \xi_1 & = & \left[ \frac{dx_1}{x_1} | \frac{d(x_1 x_2)}{x_1 x_2 -1} | \frac{dx_2}{x_2} \right] - \left[ \frac{dx_1}{x_1} | \frac{dx_2}{x_2} | \frac{d(x_1 x_2)}{x_1 x_2 -1} \right], \\ \xi_2 & = & -2\left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1} | \frac{d(x_1 x_2)}{x_1 x_2 -1} \right]\end{aligned}$$ by use of the Arnold relations given in the example of section \[sub:Arnold-relations\]. Limits {#sub:Limits} ------ We consider limits at $x_{l}=u$, $l\in\{1,\,...,\, m\}$ where $u\in\{0,\,1\}.$ By definition, any $\xi\in V\left(\Omega_{m}\right)$ vanishes along $x_{l}=0.$ Limits at $0$ and $1$ are computed as follows. As in the previous sections, let $\mathcal{Z}$ be the $\mathbb{Q}$-vector space of multiple zeta values. It was shown in [@Bro2] that for any $\xi\in V\left(\Omega_{m}\right)$ the limits $\lim_{x_{l}\rightarrow 1}\xi$ are $\mathcal{Z}$-linear combinations of elements of $V\left(\Omega_{m-1}\right) $ (after a possible renumbering of the cubical coordinates: $(x_{l+1},\ldots, x_m) \mapsto (x_l,\ldots, x_{m-1}$).) Our algorithm for the computation of limits proceeds in two steps: - Expand the function $\xi$ at $x_{l}=u$ as a polynomial in $\log(x_{l}-u)$, whose coefficients are power series in $x_{l}-u$, and - Evaluate the constant term (coefficient of $\log(x_l-u)^0$) at $x_{l}=u.$ The series expansion is the non-trivial part in this computation while the evaluation of the series is immediate. Let $\textrm{Exp}_{x_{l}=u}\xi(x_{l})$ denote the expansion of the function $\xi(x_{l})$ at $x_{l}=u.$ We compute the expansion recursively as $$\label{eq:Expansion} \textrm{Exp}_{x_{l}=u}\xi(x_{l})=\textrm{Reg}_{x_{l}=u}\xi(x_{l})+\int dx'_{l}\,\textrm{Exp}_{x'_{l}=u}\frac{\partial}{\partial x'_l}\xi(x'_{l}).$$ where the integral on the right is the regularised integral from the tangential base point ${\partial \over \partial x_l}$ at $x_l=u$ to $x_l$, or equivalently, is an indefinite integral in $x_l$ whose constant of integration is fixed by declaring that its regularised limit at $x_l=u$ vanishes. Note that if $\xi(x_{l})$ is a linear combination of words of length $k$, then in the integrand on the right-hand side of $(\ref{eq:Expansion})$, $\textrm{Exp}_{x'_{l}=u}$ is computed on words of length $k-1$. Rational prefactors are trivially expanded as power series in $x_l=u$ also. The notation $\textrm{Reg}_{x_{l}=u}\xi(x_{l})$ stands for the operation of taking the regularised limit of $\xi$ at $x_{l}=u.$ For $u=0$ we define $\textrm{Reg}_{x_{l}=0}$ to be the identity-map on terms of weight $0$ and $$\textrm{Reg}_{x_{l}=0}\xi(x_{l})=0$$ for $\xi(x_{l})$ with all terms of weight greater than $0$. For $u=1$ regularised limits are defined and computed in the remainder of this subsection. Let us start by computing regularised limits of iterated integrals in only one variable and then extend to the $n$-variable case. We consider $\Omega_{1}=\left\{ \frac{dx_{1}}{x_{1}},\,\frac{dx_{1}}{x_{1}-1}\right\} $ and for $\xi\in V\left(\Omega_{1}\right)$ we use a simplified notation where in each word we symbolically replace $\frac{dx_{1}}{x_{1}}\rightarrow0$ and $\frac{dx_{1}}{x_{1}-1}\rightarrow1$ and multiply the word with $(-1)^{s}$ where $s$ is the number of 1-forms $\frac{dx_{1}}{x_{1}-1}.$ Following [@Bro1] we define the map $$\textrm{Reg}_{x_{1}=1}:\, V\left(\Omega_{1}\right)\rightarrow\mathcal{Z}$$ by the following relations for different cases of words $\xi=\left[a_{1}|...|a_{k}\right],\, a_{i}\in\{0,\,1\},\, i=1,\,...,\, k$: - Case 1: If all letters are equal, $a_{1}=a_{2}=...=a_{k}$, we have $$\textrm{Reg}_{x_{1}=1}\left[a_{1}|...|a_{k}\right]=0.$$ - Case 2: If the word begins with 0 and ends with 1 (from left to right), we have $$\textrm{Reg}_{x_{1}=1}[\underbrace{0|...|0|1|}_{n_{r}}...|1|\underbrace{0|...|0|1}_{n_{1}}]=\zeta(n_1,\,...,\, n_{r})\textrm{ for }n_{r}\geq2,\, n_{i}\geq1,\, n_{1}+...+n_{r}=k.$$ - Case 3: If the word begins in 1 we apply the relation $$\textrm{Reg}_{x_{1}=1}\left[a_{1}|...|a_{k}\right]=\textrm{Reg}_{x_{1}=1}\left[1-a_{k}|...|1-a_{1}\right]$$ which is also true in all other cases. - Case 4: If the word ends with 0 we use the relation $$\textrm{Reg}_{x_{1}=1}[\underbrace{0|...|0|1|}_{n_{1}}...|1|\underbrace{0|...|0|1}_{n_{r}}|\underbrace{0|...|0}_{q}]=$$ $$(-1)^{q}\sum_{i_{1}+...+i_{r}=q}\left(\begin{array}{c} n_{1}+i_{1}-1\\ i_{1} \end{array}\right)...\left(\begin{array}{c} n_{r}+i_{r}-1\\ i_{r} \end{array}\right)\textrm{Reg}_{x_{1}=1}[\underbrace{0|...|0|1|}_{n_{1}+i_{1}}...|1|\underbrace{0|...|0|1}_{n_{r}+i_{r}}],\label{eq:case 4}$$ where $q,\, n_{1},\,...,\, n_{r}\geq1.$ By these relations, implementing the well-known shuffle-regularization, the regularized value of any $\xi\in V\left(\Omega_{1}\right)$ can be expressed as a $\mathbb{Q}$-linear combination of expressions as in case 2, which are multiple zeta values. We consider $\xi=\left[\frac{dx_{1}}{x_{1}-1}|\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}}\right]$ which in short-hand notation reads $\xi=-[1|0|0]$ and falls into the above case 4. By $(\ref{eq:case 4})$ we have $\textrm{Reg}_{x_{1}=1}(-[1|0|0])=\textrm{Reg}_{x_{1}=1}(-[0|0|1])$ and obtain by case 2: $$\textrm{Reg}_{x_{1}=1}\xi=-\zeta(3).$$ Now we extend the definition of regularized limits to $V\left(\Omega_{m}\right).$ Let us first define the auxiliary restriction maps $$\begin{aligned} R_{x_{l}}:\, V\left(\Omega_{m}\right) & \rightarrow & V\left(\Omega_{1}\right)\end{aligned}$$ by $$R_{x_{l}}\xi=\xi|_{dx_{i}=0,\, x_{i}=0\textrm{ for all }i\in\{1,\,...,\, m\},\, i\neq l}\label{eq:restriction R}$$ and $$L_{x_{l}}:\, V\left(\Omega_{m}\right)\rightarrow V\left(\Omega_{m-1}\right)$$ by $$L_{x_{l}}\xi=\xi|_{dx_{l}=0,\, x_{l}=1}.\label{eq:restriction L}$$ and relabelling cubical coordinates as mentioned above. Note that the map $R_{x_{l}}$ is the projection onto words all of whose 1-forms are $\frac{dx_l}{x_l}$ and $\frac{dx_l}{x_l-1}$. These maps play a similar role as the restrictions $E_\gamma$ in section \[sectbardeRham\]. The map $R_{x_{l}}$ restricts the iterated integral to the straight line from the origin to $1_l$ (called $\gamma_2$ in section \[sectbardeRham\]) and $L_{x_{l}}$ restricts to the divisor of $\overline{{\mathcal{M}}}_{0,n}$ defined by $x_l=1$ (in which $\gamma_1$ of section \[sectbardeRham\] lives). According to $(\ref{convolution})$, we take the deconcatenation co-product $\Delta$ of $\xi\in V\left(\Omega_{m}\right)$ and apply $L_{x_{l}}$ and $R_{x_{l}}$ to the left and right part of the tensor product respectively. The right-hand side of the tensor product is then in $V\left(\Omega_{1}\right)$ and we apply the above map of regularized values to this part. In summary, we extend the definition of regularized values to $$\textrm{Reg}_{x_{l}=1}:\, V\left(\Omega_{m}\right)\rightarrow\mathcal{Z}\otimes V\left(\Omega_{m-1}\right)$$ by $$\label{eq:Reg} \textrm{Reg}_{x_{l}=1}\xi=m\left(L_{x_{l}}\otimes \textrm{Reg}_{x_{l}=1} R_{x_{l}}\right)\circ \Delta\xi.$$ This completes our algorithm for computing limits of $\xi\in V\left(\Omega_{m}\right)$ at $x_{l}=0,\,1.$ We consider the iterated integral $$\begin{aligned} \xi & = & \Psi\left(\left[\frac{x_{1}dx_{2}}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]\right)\\ & = & \left[\frac{d(x_{1}x_{2})}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}}|\frac{dx_{2}}{x_{2}-1}\right]-\left[\frac{dx_{1}}{x_{1}}|\frac{d(x_{1}x_{2})}{x_{1}x_{2}-1}|\frac{dx_{2}}{x_{2}-1}\right]-\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{d(x_{1}x_{2})}{x_{1}x_{2}-1}\right]\\ & & +\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{dx_{2}}{x_{2}-1}\right]+\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}}|\frac{d(x_{1}x_{2})}{x_{1}x_{2}-1}\right]\\ & \in & V\left(\Omega_{2}\right).\end{aligned}$$ In this case, the only contibutions to the limit at $x_2=1$ are given by the term $\textrm{Reg}_{x_2 =1} \xi (x_2)$ of $(\ref{eq:Expansion})$, which we compute by use of $(\ref{eq:Reg})$. The coproduct of $\xi$ involves 20 terms, most of which vanish after applying $L_{x_2}$ to the left and $R_{x_2}$ to the right part. From the non-vanishing terms we obtain $$\begin{aligned} \textrm{Reg}_{x_2 =1} \xi (x_2) & = & m \left( \left[ \frac{dx_1}{x_1 -1} \right] \otimes \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2} | \frac{dx_2}{x_2 -1} \right] - \left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1 -1} \right] \otimes \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2 -1} \right] \right. \\ & & + \left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1} | \frac{dx_1}{x_1-1} \right] \otimes 1 - \left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1-1} | \frac{dx_1}{x_1-1} \right] \otimes 1 \\ & & \left.+\left[ \frac{dx_1}{x_1} | \frac{dx_1}{x_1-1} \right] \otimes \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2 -1} \right] \right).\end{aligned}$$ Due to $$\begin{aligned} \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2 -1} \right]=0 \textrm{ \ \ and \ \ } \textrm{Reg}_{x_2 =1} \left[ \frac{dx_2}{x_2} | \frac{dx_2}{x_2 -1} \right] = -\zeta(2)\end{aligned}$$ or by cancellation of the second and fifth terms, we obtain the limit $$\begin{aligned} \lim_{x_{2}\rightarrow1}\xi=\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}\right]-\left[\frac{dx_{1}}{x_{1}}|\frac{dx_{1}}{x_{1}-1}|\frac{dx_{1}}{x_{1}-1}\right]-\zeta(2)\left[\frac{dx_{1}}{x_{1}-1}\right].\end{aligned}$$ Feynman-type integrals {#sec:Feynman-type-integrals} ====================== In this section we consider finite integrals derived from (linearly reducible, unramified) Feynman integrals. We present an algorithm to map such integrals to hyperlogarithms in cubical variables (corresponding to the morphism $X\rightarrow \overline{{\mathcal{M}}}_{0,n+1}$ in the diagram \[Square\]). The integration over one chosen Schwinger parameter maps to the integration over one cubical variable. Then this integration can be computed by the algorithms of section \[sec:Computing-on-the\]. After the integration, as a preparation for the integration over a next Schwinger parameter, we map back to iterated integrals in Schwinger parameters. Schwinger parameters -------------------- In dimensional regularization, scalar Feynman integrals of Feynman graphs $G$ with $N$ edges and loop-number $L,$ can be written in the Feynman parametric form $$I_{G}(D)=\frac{\Gamma(\nu-LD/2)}{\prod_{j=1}^{n}\Gamma(\nu_{j})}\int_{\alpha_{j}\geq0}\delta\left(1-\alpha_{N}\right)\left(\prod_{j=1}^{N}d\alpha_{j}\alpha_{j}^{\nu_{j}-1}\right)\frac{\mathcal{U}_{G}^{\nu-(L+1)D/2}}{\mathcal{F}_{G}{}^{\nu-LD/2}},$$ where $\nu=\sum_{i=1}^{N}\nu_{i}$ is the sum of powers of Feynman propagators and $D\in\mathbb{C}$. We refer to the variables $\alpha_{1},\,...,\,\alpha_{N}$ as Schwinger parameters and the above integration is over the positive range of each of these variables. The functions $\mathcal{U}_{G}$ and $\mathcal{F}_{G}$ are the first and second Symanzik polynomials respectively. They are polynomials in the Schwinger parameters and $\mathcal{F}_{G}$ is furthermore a polynomial of kinematical invariants, which are quadratic functions of particle masses and external momenta of $G.$ For more details we refer to [@Itz; @Nak; @Bog2]. Assume that we want to compute $I_{G}(2n)$ for some $n\in\mathbb{N}.$ There are different scenarios in which our algorithms may be useful. In the simplest case, the integral $I_{G}(2n)$ is finite and we may attempt to compute it without further preparative steps. If $I_{G}(2n)$ is divergent there may be a $n\neq m\in\mathbb{N}$ such that $I_{G}(2m)$ is finite and the method of [@Tar1; @Tar2] may provide useful relations between $I_{G}(2n)$ and $I_{G}(2m).$ These relations however may involve further integrals to be computed. The method of [@Bro3] allows for a subtraction of UV divergent contributions by a renormalization procedure on the level of the integrand. Alternatively, for a possibly UV and IR divergent integral, we may attempt to expand $I_{G}$ as $$I_{G}=\sum_{j=-2L}^{\infty}c_{j}\epsilon^{j}$$ where $\epsilon=(2n-D)/2$ and the $c_{j}$ are finite integrals. In principle such an expansion can be computed by sector decomposition [@Bin], however in this case, a use of our algorithms may be prohibited by the type of polynomials appearing in the integrands of the resulting $c_{j}.$ Recently, an alternative approach, where the latter polynomials are given by Symanzik polynomials of $G$ and its minors was suggested in [@Pan2]. Let us assume that these or alternative methods have led us to an integral over the positive range of Schwinger parameters where the integrand is of the form $$\label{hypIntegrand} f(\alpha_{1},\,...,\,\alpha_{N})=\frac{\prod_{Q_{i}\in\mathcal{Q}}Q_{i}^{\delta_{i}}\textrm{ hyperlogarithms}({P_{i}})}{\prod_{P_{i}\in\mathcal{P}}P_{i}^{\beta_{i}}}$$ where all $\delta_{i},\,\beta_{i}\in\mathbb{N}_{0}$ and where $\mathcal{P}=\{P_{1},\,...,\, P_{r}\}$ and $\mathcal{Q}=\{Q_{1},\,...,\, Q_{q}\}$ are finite sets of irreducible polynomials in Schwinger parameters. We assume furthermore that all $P_i$ are homogeneous and positive or negative definite. This is the case for all Symanzik polynomials in the Euclidean momentum region and in the massless case, and also for the polynomials arising from their linear reduction in a large class of situations. This simplifying assumption allows us to apply the particular change of variables constructed below. However, the general method is not restricted to this case. A more precise description of the numerator is given below. For our algorithms to be applicable we furthermore have to assume that there is an ordering on the Schwinger parameters such that the set $\mathcal{P}$ is linearly reducible and unramified [@Bro5; @Bro4]. In the following let $\alpha_N, \alpha_{N-1},...,\alpha_1$ be such a fixed ordering. From Schwinger parameters to cubical variables\[sub:From-Feynman-parameters\] ----------------------------------------------------------------------------- In the following, we transform a given integrand $f$ of the type given by $(\ref{hypIntegrand})$ to an integrand in cubical variables. According to our fixed ordering on the Schwinger parameters, let $\alpha_N$ be the parameter to be integrated out in the present step. Linear reducibility implies that the polynomials in $\mathcal{P}$ are of degree at most 1 in $\alpha_{N},$ while there are no implications for $\mathcal{Q}.$ We write $\mathcal{P}=\mathcal{P}_{N}\cup\mathcal{P}_{\backslash N}$ where $\mathcal{P}_{N}\subset\mathcal{P}$ is the subset of polynomials linear in $\alpha_{N}$ and $\mathcal{P}_{\backslash N}\subset\mathcal{P}$ is the set of polynomials independent of $\alpha_{N}.$ Let us fix the numbering on the $P_{i}$ such that $\mathcal{P}_{N}=\left\{ P_{1},\,...,\, P_{n}\right\} $ with $n\leq r.$ We also write the set of all polynomials $Q_{i}$ as $\mathcal{Q}=\mathcal{Q}_{N}\cup\mathcal{Q}_{\backslash N}$ where the polynomials in $\mathcal{Q}_{N}$ depend on $\alpha_{N}$ and the ones in $\mathcal{Q}_{\backslash N}$ do not. Now let us be more specific about the functions occurring in the numerator of $(\ref{hypIntegrand})$. We write $L_{w}(\alpha_{N})$ for a hyperlogarithm in $\alpha_{N},$ given by a word $w$ in differential 1-forms in the alphabet $$\Omega_{N}^{\textrm{Feynman}}=\left\{ \frac{d\alpha_{N}}{\alpha_{N}},\,\frac{d\alpha_{N}}{\alpha_{N}-\rho_{i}}\textrm{ where }\rho_{i}= -\frac{ P_{i}|_{\alpha_{N}=0} }{\frac{\partial P_{i}}{\partial \alpha_{N}}}\textrm{ for }i=1,\,...,\, n\right\} .\label{eq: Omega Feyn}$$ Here $\rho_{i}$ is a rational function such that $P_{i}$ vanishes for $\alpha_N=\rho_i$. Throughout this section, we shall assume the Feynman integral we are considering is linearly reducible and unramified. The condition for being unramified was defined in [@Bro5], definition 16, and discussed in [@Bro4], §9.3. It implies in particular that if $\rho_i$ is a constant independent of all $\alpha_i$, then it must be equal to $0$ or $-1$. We assume as an induction hypothesis that the functions in the numerator of the integrand are of a certain type. We will see in section \[sub:Back-to-Feynman\] that this assumption will be satisfied after integration and will be the starting point for the next integration. The numerator of the integrand $f$ is assumed to be a linear combination of hyperlogarithms in $\alpha_{N}:$ $$\textrm{numerator}(f)=\sum_{k}a_{k}b_{k}(\alpha_{N})L_{w_{k}}(\alpha_{N}),\label{eq:Feyn integrand hyperlog}$$ where the $w_{k}$ are words in the alphabet $\Omega_{N}^{\textrm{Feynman}}$ and where we denote the $\alpha_{N}$-dependent and $\alpha_{N}$-independent factor of the $k$-th coefficient by $b_{k}(\alpha_{N})$ and $a_{k}$ respectively. The $\alpha_{N}$-dependent factor $b_{k}(\alpha_{N})$ is a product of $Q_{i}\in\mathcal{Q}_{N}$ while the $\alpha_{N}$-independent factor $a_{k}$ is allowed to be a product of $Q_{i}\in\mathcal{Q}_{\backslash N}$ and of hyperlogarithms which do not depend on $\alpha_{N}.$ As $\alpha_{N}$-independent factors of the numerator remain unchanged in the integration procedure, we restrict our attention to integrals of the type $$\label{eq:LIntegral} \int_{0}^{\infty}d\alpha_{N}f(\alpha_{1},\,...,\,\alpha_{N})=\int_{0}^{\infty}d\alpha_{N}\frac{\prod_{Q_{i}\in\mathcal{Q}_{N}} Q_{i}^{\delta_{i}}L_{w}(\alpha_{N})}{\prod_{P_{i}\in\mathcal{P}}P_{i}^{\beta_{i}}}.$$ Let us now express the integral of $(\ref{eq:LIntegral})$ as an integral over cubical coordinates such that the algorithms of section \[sec:Computing-on-the\] apply. Let $\mathbb{R}_{+}^{N}$ be the subspace of $\mathbb{R}^{N}$ where all Schwinger parameters are greater than or equal to zero and let $\mathbb{R}_{\textrm{cube}}^{n}$ be the unit cube in $n$ cubical variables, i.e. $$\begin{aligned} \mathbb{R}_{+}^{N} & = & \left\{ (\alpha_{1},...,\,\alpha_{N})\in\mathbb{R}^{N}|0\leq\alpha_{i},\, i=1,\,...,\, N\right\} ,\\ \mathbb{R}_{\textrm{cube}}^{n} & = & \left\{ (x_{1},...,\, x_{n})\in\mathbb{R}_{n}|0\leq x_{i}\leq1,\, i=1,\,...,\, n\right\} .\end{aligned}$$ Consider the $\alpha_{N}$-dependent polynomials $\mathcal{P}_{N}=\{P_{1},\,...,\, P_{n}\}$ and the corresponding $\rho_{i}=-\frac{P_{i}|_{\alpha_{N}=0}}{ \frac{\partial P_{i}}{\partial\alpha_{N}} }$ for $i=1,\,...,\, n.$ We introduce an ordering on the set $\mathcal{P}_{N}$ as follows. A sufficiently small open region of the form $0\leq \alpha_{N-1} \ll \alpha_{N-2} \cdots \ll \alpha_1 \ll \epsilon$ (where $x\ll y $ denotes $x<y^M$ for some large $M$) does not intersect the hypersurfaces $\rho_i-\rho_j=0$. Therefore number the polynomials in $\mathcal{P}_{N}=\{P_{1},\,...,\, P_{n}\}$ such that everywhere in this region we have $$0 > \rho_{n}>\rho_{1}>\rho_{2}>...>\rho_{n-2}>\rho_{n-1}.\label{eq:ordered zeroes}$$ For the given, ordered set $(P_{1},\,...,\, P_{n})$, consider the rational map between affine spaces $$\phi:\,\mathbb{A}^{N}\rightarrow\mathbb{A}^{n},$$ (equivalently a homomorphism $\phi^*: \mathbb{Q}(x_1,\ldots, x_n) \rightarrow \mathbb{Q}(\alpha_1,\ldots, \alpha_N)$) given by $$\begin{aligned} \phi^*(x_{n}) & = & \frac{\alpha_{N}}{\alpha_{N}-\rho_{n}}\nonumber, \\ \phi^*(x_{n-1}) & = & 1-\frac{\rho_{n}}{\rho_{n-1}},\nonumber \\ \phi^*(x_{k}) & = & \frac{1-\frac{\rho_{n}}{\rho_{k}}}{1-\frac{\rho_{n}}{\rho_{k+1}}}\textrm{ for }1\leq k\leq n-2.\label{eq: change variables}\end{aligned}$$ These variables $x_i$ will be our cubical coordinates and we construct the set of 1-forms $\bar{\Omega}^F_n$ as above. Note that the restriction of $\phi$ to the first $N-1$ coordinates defines a rational map $\phi: \mathbb{A}^{N-1} \rightarrow \mathbb{A}^{n-1}$, since $\rho_1,\ldots, \rho_{n}$ do not depend on $\alpha_N$. For fixed $\alpha_1,\ldots, \alpha_{N-1}$, the curve ${\mathbb{P}}^1$ with coordinate $\alpha_N$ and punctures at $\{0,\rho_1,\ldots, \rho_{n},\infty \}$ (i.e., the fiber of $\mathbb{A}^N \rightarrow \mathbb{A}^{N-1}$), is isomorphic, via $\ref{eq: change variables}$, to the curve with coordinate $x_n$ and punctures at $\{0,(x_1\ldots x_{n-1})^{-1}, (x_2\ldots x_{n-1})^{-1} , \ldots , x_{n-1}^{-1},\infty, 1\}$, in that order. Via such a (family of) isomorphisms, we can explicitly express all 1-forms in $\Omega_{N}^{\textrm{Feynman}}$ as $\mathbb{Q}$-linear combinations of 1-forms in $\bar{\Omega}^F_n$ in cubical coordinates. We obtain $$\begin{aligned} \label{dalphas} \frac{d\alpha_N}{\alpha_N} & = & \frac{dx_n}{x_n} - \frac{dx_n}{x_n-1} ,\\ \frac{d\alpha_N}{\alpha_N-\rho_n} & = & -\frac{dx_n}{x_n-1} \ , \nonumber \\ \frac{d\alpha_N}{\alpha_N-\rho_i} & = & \frac{x_i ... x_{n-1}dx_n}{x_i... x_n-1} - \frac{dx_n}{x_n-1} , \nonumber \end{aligned}$$ since the $\rho_i$ are constant, for $i=1,...,n-1$. As a consequence, we can express each hyperlogarithm $L_{w}(\alpha_{N})$ as a $\mathbb{Q}$-linear combination of hyperlogarithms in cubical variables $\xi\in V\left(\bar{\Omega}^F_{n}\right).$ For simplicity, we make the following assumption (which is slightly stronger than assuming that the linear reduction of the Feynman integral is unramified): $$\label{eq:xlimits} \lim_{\alpha_{1}\rightarrow0}...\lim_{\alpha_{N}\rightarrow0}x_{k}(\alpha_{1},\,...,\,\alpha_{N})\in\{0,\,1\},\, k=1,\,...,\, n,$$ where these limits are approached from inside the cube $\mathbb{R}_{\textrm{cube}}^{n}.$ The domain of the $\alpha_{N}$-integration is mapped to the domain $0\leq x_{n}\leq1.$ The Jacobian is $J=-\frac{\rho_{n}}{(x_{n}-1)^{2}}.$ Up to rational functions which do not depend on $x_{n},$ we can now express integrals of the type of $(\ref{eq:LIntegral})$ as integrals of the type $$\label{eq:IntCubical} \int_{0}^{1}dx_{n}\frac{\prod q_{i}^{\gamma_{i}}}{\prod f_{i}^{\delta_i}} \xi$$ where $\gamma_{i},\delta_{i}\in\mathbb{N},$ and where each $q_{i}$ is a polynomial in Schwinger parameters without $\alpha_{N}$ or in cubical variables, and the integrand involves functions $f_{i}\in\{x_{n},\, x_{n}-1,\, x_{n-1}x_{n}-1,\,...,\, x_{1}\cdot\cdot\cdot x_{n}-1\}$ and hyperlogarithms $\xi\in V\left(\bar{\Omega}^F_{n}\right).$ Before we can apply our algorithm of subsection \[sub:Primitives\] for the computation of primitives, we use a standard procedure of applying finitely many successive partial fraction decompositions and partial integrations until all powers $\delta_i$ are equal to 1. As a last step of preparation, we apply the symbol map $\Psi$ of subsection \[sub:The-symbol-map\] to $\xi$. We obtain an integral as in $(\ref{eq:IntCubical})$ where now $\xi\in V\left(\Omega_{n}\right).$ Now we compute the definite integral $(\ref{eq:IntCubical})$ by use of the algorithms of subsections \[sub:Primitives\] and \[sub:Limits\]. Up to rational prefactors, we obtain a $\mathcal{Z}$-linear combination of functions in $V\left(\Omega_{n-1}\right).$ Back to Schwinger parameters\[sub:Back-to-Feynman\] ---------------------------------------------------- Note that after the integration, we have a function in terms of both types of variables, the Schwinger parameters and the cubical coordinates. In order to proceed with the integration over a next Schwinger parameter and apply the same steps again, we firstly have to express the integrand only in terms of Schwinger parameters again. Let $I$ be the result of the $\alpha_{N}$-integration, expressed as a linear combination $$I=\sum a_{i}\xi_{i}$$ of multiple polylogarithms $\xi_{i}\in V\left(\Omega_{n-1}\right)$. The coefficients $a_{i}$ are trivially expressed by Schwinger parameters by application of $\phi^*.$ However, expressing the multiple polylogarithms $\xi_{i}$ in terms of Schwinger parameters is more subtle, as we have to respect the limiting conditions of iterated integrals in both sets of variables. For any function $f$ of variables $y_{1},\,...,\, y_{n}$ and numbers $c_{1},\,...,\, c_{n}$ let us introduce the notation $$\lim_{(y_{1},\,...,\, y_{n})\rightarrow(c_{1},\,...,\, c_{n})}f=\mathrm{Reg}_{y_{n}\rightarrow c_{n}}...\mathrm{Reg}_{y_{1}\rightarrow c_{1}}f.$$ where in the right-hand side, $\mathrm{Reg}$ denotes the regularised limits with respect to unit tangent vectors in either cubical coordinates $x_i$ (or $1-x_i$), or Schwinger parameters $\alpha_i$. In the following let us write $0_{n}$ for the vector $(0,\,...,\,0)$ with $n$ components. We consider the vector $x_{p}=(x{}_{p(1)},\,...,\, x_{p(n-1)})$ of the remaining cubical coordinates, where the ordering is given by a permutation $p$ on the set $\{1,\,...,\, n-1\}$. We furthermore consider the vector of remaining Schwinger parameters $\alpha=(\alpha{}_{N-1},\,...,\,\alpha_{1})$ in the ordering in which we integrate over them, as fixed above. Consider a multiple polylogarithm $\xi\in V\left(\Omega_{n-1}\right).$ By definition, it satisfies $$\lim_{x_{\sigma}\rightarrow0_{n-1}}\xi= \epsilon ( \xi) \label{eq:limit condition cubical}$$ for every permutation $\sigma$ on $\{1,\,...,\, n-1\}$, where $\epsilon$ is the augmentation map (projection onto components of length $0$). We want to express each $\xi$ as iterated integrals $\eta$ in Schwinger parameters, for which we impose the condition $$\lim_{\alpha\rightarrow0_{N-1}}\eta=\epsilon ( \eta) . \label{eq:limit condition Feynman}$$ Condition $\ref{eq:limit condition Feynman}$ corresponds to a vanishing condition for the iterated integral $\xi \in V\left(\Omega_{n-1}\right) $ at a tangential base point on ${\mathcal{M}}_{0,n+2}$ (strictly speaking, on a related space ${\mathcal{M}}_{0,n+2}^{\dag}$ ([@Bro4], §8.2) which can be read off from the linear reduction algorithm and involves removing from $\mathbb{A}^{n-1}$ only those hypersurfaces $x_i=0$, $x_ix_{i+1}\ldots x_j =1$ which correspond to singularities actually occurring in the integrand), which is on the boundary of the connected component of ${\mathcal{M}}_{0,n+2}(\mathbb{R})$ defined by the unit hypercube $0\leq x_1,\ldots, x_{n-1} \leq 1$. One can verify that such a point can always be represented by a permutation $p$ on $\{1,\,...,\, n-1\}$ (non-uniquely) and a vector $c=(c_{1},\,...,\, c_{n-1})$ (uniquely) with all $c_{i}\in\{0,\,1\}$ such that for any rational function $g$ in the $x_i$ which is regular on ${\mathcal{M}}^{\dag}_{0,n+2}$, we have $$\lim_{x_{p}\rightarrow c}g=\lim_{\alpha\rightarrow0_{N-1}}\phi^\star g,\label{eq:limits}$$ where on the left-hand side $c$ is approached inside $\mathbb{R}_{\textrm{cube}}^{n-1}$ and on the right-hand side $(0,\,...,\,0)$ is approached inside $\mathbb{R}_{+}^{N-1}.$ Such a point $c$ and permutation $p$ determine the procedure to express $\xi$ in terms of Schwinger parameters. The components of $c$ are computed by $$c_{i}=\lim_{\alpha \rightarrow0_{N-1}}x_{i},$$ where $i\in\{1,\,...,\, n-1\}$, and lies in $\{0,1\}$, by assumption $(\ref{eq:xlimits})$. In the case when ${\mathcal{M}}_{0,n+2}^{\dag}= {\mathcal{M}}_{0,n+2}$ (i.e., all possible singularities which can occur actually do occur), a permutation $p$ satisfying $(\ref{eq:limits})$ can easily be computed with the help of dihedral coordinates $u_{ij}$, which are related to the cubical coordinates as discussed in [@Bro2]. A permutation $p$ satisfies $(\ref{eq:limits})$ for any regular function $g$ on ${\mathcal{M}}_{0,n+2}$ (expressed as a rational function of cubical coordinates) if it satisfies $$\lim_{x_{p}\rightarrow c}u_{ij}=\lim_{\alpha \rightarrow0_{N-1}}\phi^\star u_{ij}$$ for all dihedral coordinates $u_{ij}$. This condition determines $p$ in this case. Suppose ${\mathcal{M}}_{0,5}^{\dag}={\mathcal{M}}_{0,5}$. Let $x_1,x_2$ be cubical coordinates. Suppose that $x_1=1-\alpha_2, x_2=1-{\alpha_2 \over \alpha_1}$. Then the five dihedral coordinates $(x_1,x_2, 1-x_1x_2, {1-x_1 \over 1-x_1x_2}, {1-x_2 \over 1-x_1x_2})$ in the limits $\alpha_2\rightarrow 0$ then $\alpha_1 \rightarrow 0$ tend to $(1,1,0,0,1)$ respectively. This corresponds to taking first the limit as $x_1\rightarrow 1$ and then $x_2 \rightarrow 1$. On the other hand, suppose that $x_1 = 1-\alpha_1, x_2 = 1-\alpha_1$. Then the limit of the five dihedral coordinates above as $\alpha_1 \rightarrow 0$ are $(1,1,0,{1\over 2}, {1\over 2})$, which could potentially produce a $\log 2$ in the iterated integrals (ramification at prime $2$). In such a case, the condition of being unramified will ensure that $1-x_1x_2=\alpha_1(2-\alpha_1)$ does not occur as a singularity of the integrand. Thus ${\mathcal{M}}_{0,5}^{\dag} = \mathbb{A}^2 \backslash \{x_1,x_2=0,1\} = {\mathcal{M}}_{0,4} \times {\mathcal{M}}_{0,4}$ strictly contains ${\mathcal{M}}_{0,5}$. The limit as $\alpha_1\rightarrow 0$ can be obtained as the limit as $x_1\rightarrow 1, x_2\rightarrow 1$ in either order. Now let $x_{p}$ and $c$ be vectors satisfying $(\ref{eq:limits})$. We define $\eta$ by the following equation, where $\xi\in V\left(\Omega_{n-1}\right)$ is the result of the integration of $(\ref{eq:IntCubical})$, $$\eta=m\left(\phi^{\star}\otimes\phi^{\star}\lim_{x_{p}\rightarrow c}\right)\Delta\xi \label{eq:limit procedure}$$ and $m$ is multiplication. Note that this is an application of $(\ref{pathconcat})$. Then $\eta $ is the desired expression in terms of Schwinger parameters. As a last step, we express each iterated integral in terms of hyperlogarithms, such that we arrive at the starting point for the next integration over the variable $\alpha_{N-1}$. As a consequence of linear reducibility, all iterated integrals are now given by differential forms of the type $\omega=dP/P$ where $P$ are polynomials in the Schwinger parameters which are of degree $\leq1$ in the variable $\alpha_{N-1}$. In analogy to the construction of the unshuffle map we define the auxiliary restriction operations $$\pi_{\alpha_{i}}\omega=\omega|_{d\alpha_{i}=0,\,\alpha_{i}=0}$$ and $$r_{\alpha_{i}}\omega=\omega|_{d\alpha_{j}=0\textrm{ for all }j\neq i.}$$ By $$\eta'=m\left(r_{\alpha_{N-1}}\otimes \pi_{\alpha_{N-1}}\right)\Delta \eta \label{eq:write hyperlog}$$ we finally arrive at a linear combination of hyperlogarithms $L_{w}(\alpha_{N-1})$ whose coefficients are products of rational functions in Schwinger parameters, multiple zeta values and iterated integrals independent of $\alpha_{N-1}.$ Iterating the computation of $(\ref{eq:write hyperlog})$ for the remaining Schwinger parameters we can express all iterated integrals as hyperlogarithms. With this expression we can repeat the above steps to integrate out $\alpha_{N-1}$, and so on. Summary of the integration algorithm ------------------------------------ Let us summarize the above steps for integrating over one Schwinger parameter $\alpha_{N}.$ We start from a finite integral $I=\int_{0}^{\infty}d\alpha_{N}f$ whose integrand $f$, as in $(\ref{eq:Feyn integrand hyperlog})$, is a linear combination of hyperlogarithms $L_{w}(\alpha_{N})$ as functions of $\alpha_{N}$, and whose coefficients are products of rational functions $b(\alpha_{N})$ of the Schwinger parameters including $\alpha_{N},$ and further functions (possibly hyperlogarithms) not depending on $\alpha_{N}.$ As above, we write $\mathcal{P}_{N}$ for the set of $n$ polynomials depending linearly on $\alpha_{N},$ which are in the denominators of $b(\alpha_{N})$ and define the differential forms of $L_{w}(\alpha_{N})$ by $(\ref{eq: Omega Feyn})$. The set $\mathcal{P}_{N}$ is linearly reducible with respect to an ordered set $(\alpha_N,...,\alpha_1)$ of all Schwinger parameters, and unramified. The main steps of the algorithm are combined as follows: - Define the $n$ cubical variables $x_{1},\,...,\, x_{n}$, and express the integrand $f$ via the map $\ref{dalphas}$ as a linear combination of hyperlogarithms in $V\left(\bar{\Omega}^F_{n}\right)$. The integration over $\alpha_{N}$ is mapped via \[eq: change variables\] to the integration over $x_{n}$ from 0 to 1. - Apply the symbol map $\Psi$ of subsection \[sub:The-symbol-map\] to lift each function in $V\left(\bar{\Omega}^F_{n}\right)$ to multiple polylogarithms in $V\left(\Omega_{n}\right).$ - Use iterated partial integration and partial fraction decomposition to bring the integrand into the appropriate form. Then use the map $\star$ of subsection \[sub:Primitives\] to compute the primitive of $f.$ - Take the limits of the primitive at $x_{n}=0$ and $x_{n}=1$ to obtain the definite integral from 0 to 1, using the algorithm of subsection \[sub:Limits\]. The result is a linear combination of multiple polylogarithms in $V\left(\Omega_{n-1}\right)$ with coefficients possibly involving multiple zeta values. - Apply the change of variables to obtain an expression only in Schwinger parameters again. For iterated integrals, apply $(\ref{eq:limit procedure})$ such that the regularised limit at $\alpha\rightarrow0_{N-1}$ is preserved. - Write the result as a combination of hyperlogarithms in the next integration variable by $(\ref{eq:write hyperlog})$. Examples for the application of this algorithm by use of our computer program are given below. Applications {#sec:Applications} ============ Cellular integrals ------------------ A particular instance of period integrals on moduli spaces are given by the cellular integrals defined in [@AperyVar] in relation to irrationality proofs. The basic construction is to consider a permutation $\sigma $ of $\{1,\ldots, n\}$ and define a rational function and differential form $$\widetilde{f}_{\sigma} = \prod_{i} {z_i -z_{i+1} \over z_{\sigma(i)} - z_{\sigma(i+1)} } \qquad \hbox{ and } \qquad \widetilde{\omega}_{\sigma} = \prod_{i} {dz_i\over z_{\sigma(i)} - z_{\sigma(i+1)} } \nonumber\ ,$$ on the configuration space $C^n ({\mathbb{P}}^1)$ of $n$ distinct points $z_1,\ldots, z_n$ in ${\mathbb{P}}^1$, where the product is over all indices $i$ modulo $n$. Now $\mathrm{PGL}_2$ acts diagonally on $C^n ({\mathbb{P}}^1)$, and the quotient is $$\mathcal{M}_{0,n} \cong C^n( {\mathbb{P}}^1) / \mathrm{PGL}_2\ .$$ The rational function $\widetilde{f}_\sigma$ and the form $\widetilde{\omega}_{\sigma}$ are $\mathrm{PGL}_2$-invariant, and therefore descend in the standard way to a rational function and form $f_{\sigma}, \omega_{\sigma}$ on $\mathcal{M}_{0,n}$. Because $\mathrm{PGL}_2$ is triply-transitive, we can put $z_1=0, z_{n-1}=1, z_n = \infty$, and replace $z_{i+1}$ by $x_ix_{i+1}\ldots x_{n-3}$ for $i=1,\ldots, n-3$, where $x_1,\ldots, x_{n-3}$ are cubical coordinates on ${\mathcal{M}}_{0,n}$. Therefore we can formally write $$f_{\sigma} = \prod_{i} {z_i -z_{i+1} \over z_{\sigma(i)} - z_{\sigma(i+1)} } \qquad \hbox{ and } \qquad \omega_{\sigma} = { dx_1\ldots dx_{n-3} \over \prod_i z_{\sigma(i)} - z_{\sigma(i+1)} } \nonumber\ ,$$ where the product is over all indices $i$ modulo $n$, and all factors involving $z_n=\infty$ are simply omitted. For all $N\geq 0$, consider the family of basic cellular integrals $$I_N^{\sigma} = \int_{[0,1]^{n-3}} f_{\sigma}^N \omega_{\sigma}$$ where the domain of integration is the unit hypercube in the cubical coordinates $x_i$. Conditions for the convergence of the integral are discussed in [@AperyVar]. When it converges, this integral is a rational linear combination of multiple zeta values of weights $\leq n-3$ and can be computed with our program. In the case $n=5,6$ and $\sigma(1,2,3,4,5) = (1,3,5,2,4)$, and $\sigma(1,2,3,4,5,6)=(1,3,6,4,2,5) $ it gives back exactly the linear forms involved in Apéry’s proofs of the irrationality of $\zeta(2)$ and $\zeta(3)$. A systematic study of examples for higher $n$ (described in [@AperyVar]) was undertaken using the algorithms described in this paper. Expansion of generalized hypergeometric functions ------------------------------------------------- Many Feynman integrals can be expressed in terms of generalized hypergeometric functions $$_{p}F_{q}(a_{1},\,...,\, a_{p};\, b_{1},\,...,\, b_{q};\, z)=\sum_{k=0}^{\infty}\frac{\prod_{j=1}^{p}\left(a_{j}\right)_{k}z^{k}}{\prod_{j=1}^{q}\left(b_{j}\right)_{k}k!},$$ converging everywhere in the $z$-plane if $q\geq p,$ and in the case $q=p-1$ for $|z|<1$ or at $|z|=1$ if the real part of $\sum_{j=1}^{p-1}b_{j}-\sum_{j=1}^{p}a_{j}$ is positive. Here we used the Pochhammer symbol $$(a)_n=\frac{\Gamma (a+n)}{\Gamma (a)}.$$ Multi-variable generalizations, such as Appell and Lauricella functions, play a role in Feynman integral computations as well. If the Feynman integral is considered in $D=4-2\epsilon$ dimensions, the parameters take the form $$\label{eq:HyperParameters} a_{i}=A_{i}+\epsilon\alpha_{i},\, b_{i}=B_{i}+\epsilon\beta_{i}\;\textrm{where }\alpha_{i},\,\beta_{i}\in\mathbb{R}.$$ In the case of massless integrals, the numbers $A_{i},\, B_{i}$ are integers while in the case of non-vanishing masses, some of them are half-integers. In order to arrive at a result for the Feynman integral where pole-terms in $\epsilon$ can be separated, one has to expand these functions near $\epsilon=0.$ Several computer programs are available for this task. The programs of [@MocUwe; @Wei] use algorithms for the expansion of very general types of nested sums [@MocUweWei] while the program [@HubMai2] writes an Ansatz in harmonic polylogarithms and determines the coefficients from differential equations. A method using systems of differential equations was presented in [@Kal; @KalWarYos1; @KalWarYos2]. We also refer to [@Kal2; @Grey; @AblX] for recent progress in the field. Alternatively, we can start from an integral representation of the function, expand the integrand and compute the resulting integrals explicitly. This approach was applied in [@HubMai1] for the expansion of $_{2}F_{1}.$ The algorithms presented above are very well suited for this method and can be used to extend it to more general functions. As a first example we still consider $_{2}F_{1}.$ We have the integral representation $$_{2}F_{1}(a_{1},\, a_{2};\, b;\, z_{1})=\frac{\Gamma(b)}{\Gamma(a_{2})\Gamma(b-a_{2})}\int_{0}^{1}z_{2}^{a_{2}-1}(1-z_{2})^{b-a_{2}-1}(1-z_{1}z_{2})^{-a_{1}}dz_{2}$$ for $\textrm{Re}(b)>\textrm{Re}(a_{2})>0$ and $z_1\notin [1,\infty).$ The parameters $a_i$ and $b$ may depend on $\epsilon$ as in $(\ref{eq:HyperParameters})$. If we exclude the case of half-integers mentioned above, the expansion at $\epsilon=0$ leads to integrands whose denominators are products of $z_2, (1-z_2), (1-z_1 z_2)$ and whose numerators may involve powers of logarithms of these functions. We can view the variables $z_1, z_2$ as cubical coordinates and apply the algorithms of section \[sec:Computing-on-the\] to integrate over $z_{2}$ analytically. Example: $$\begin{aligned} _{2}F_{1}(1,\,1+\epsilon;\,3+\epsilon;\, z_{1}) & = & \frac{\Gamma(3+\epsilon)}{\Gamma(1+\epsilon)}\int_{0}^{1}\frac{z_{2}^{\epsilon}(1-z_{2})}{1-z_{1}z_{2}}dz_{2}\\ & = & \int_{0}^{1}\frac{2(z_{2}-1)}{z_{1}z_{2}-1}dz_{2}+\epsilon\int_{0}^{1}\frac{\left(2\ln(z_{2})+3\right)(z_{2}-1)}{z_{1}z_{2}-1}\\ & & +\epsilon^{2}\int_{0}^{1}\frac{\left(\ln(z_{2})^{2}+3\ln(z_{2})+1\right)(z_{2}-1)}{z_{1}z_{2}-1}dz_{2}+\mathcal{O}\left(\epsilon^{3}\right)\\ & = & \frac{2}{z_{1}^{2}}\left(z_{1}+(1-z_{1})\ln(1-z_{1})\right)+\frac{\epsilon}{z_{1}^{2}}\left(z_{1}+3(1-z_{1})\ln(1-z_{1})\right.\\ & & \left.+2(1-z_{1})\textrm{Li}_{2}(z_{1})\right)+\frac{\epsilon^{2}}{z_{1}^{2}}\left((1-z_{1})\ln(1-z_{1})\right.\\ & & \left.+3(1-z_{1})\textrm{Li}_{2}(z_{1})-2(1-z_{1})\textrm{Li}_{3}(z_{1})\right)+\mathcal{O}\left(\epsilon^{3}\right)\end{aligned}$$ We extend the approach to generalized hypergeometric functions, starting from the integral representation $$_{p}F_{q}(a_{1},\,...,\, a_{p};\, b_{1},\,...,\, b_{q};\, z)$$ $$=\frac{\Gamma(b_{q})}{\Gamma(a_{p})\Gamma(b_{q}-a_{p})}\int_{0}^{1}t^{a_{p}-1}(1-t)^{b_{q}-a_{p}-1}\,_{p-1}F_{q-1}(a_{1},\,...,\, a_{p-1};\, b_{1},\,...,\, b_{q-1};\, zt)dt$$ in the region where it converges. Here again the expansion of the integrand leads to integrals over cubical coordinates which can be computed by the algorithms of section \[sec:Computing-on-the\]. Example: $$\begin{aligned} _{3}F_{2}(2,\,1+\epsilon,\,1+\epsilon;\,3+\epsilon,\,2+\epsilon;\, z_{1}) & = & \frac{\Gamma(3+\epsilon)\Gamma(2+\epsilon)}{\Gamma(1+\epsilon)^{2}}\int_{0}^{1}\int_{0}^{1}\frac{z_{2}z_{3}^{\epsilon}(1-z_{2})^{\epsilon}}{(1-z_{1}z_{2}z_{3})^{1+\epsilon}}dz_{2}dz_{3}\\ & = & \frac{2}{z_{1}^{2}}\left((1-z_{1})\ln(1-z_{1})+z_{1}\right)+\frac{\epsilon}{z_{1}^{2}}\left(7(1-z_{1})\ln(1-z_{1})\right.\\ & & \left.+5z_{1}+(2-4z_{1})\textrm{Li}_{2}(z_{1})\right)+\frac{\epsilon^{2}}{z_{1}^{2}}\left(9(1-z_{1})\ln(1-z_{1})\right.\\ & & \left.+(7-12z_{1})\textrm{Li}_{2}(z_{1})+(6z_{1}-2)\textrm{Li}_{3}(z_{1})+4z_{1}\right)+\mathcal{O}\left(\epsilon^{3}\right)\end{aligned}$$ While for these functions the integral representations are readily given in cubical coordinates, an extension to further cases may require a change of variables. For example the first Appell function $$F_{1}(a;\, b_{1},\, b_{2};\, c;\, x,\, y)=\sum_{m\geq0}\sum_{n\geq0}\frac{\left(a\right)_{m+n}\left(b_{1}\right)_{m}\left(b_{2}\right)_{n}}{m!n!\left(c\right)_{m+n}}x^{m}y^{n}\;\textrm{where }|x|,\,|y|<1$$ with the integral representation [@Pic] $$F_{1}(a;\, b_{1},\, b_{2};\, c;\, x,\, y)=\frac{\Gamma\left(c\right)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-tx)^{-b_{1}}(1-ty)^{-b_{2}}dt$$ for $\textrm{Re}(c)>\textrm{Re}(a)>0$ can be expressed in the appropriate form after introducing cubical coordinates $z_{3}=t,$ $z_{2}=y,$ $z_{1}=x/y.$ Example: $$\begin{aligned} F_{1}(1;\, 1,\, 1;\, 2+\epsilon;\, x,\, y) & = & \frac{\Gamma(2+\epsilon)}{\Gamma(1+\epsilon)}\int_{0}^{1}\frac{(1-z_{3})^{\epsilon}}{(1-z_{1}z_{2}z_{3})(1-z_{2}z_{3})}dz_{3}\\ & = & \frac{1}{x-y}\left(\ln(1-y)-\ln(1-x)\right)\\ & & +\frac{\epsilon}{x-y}\left(\ln(1-y)-\ln(1-x)+\frac{1}{2}\ln(1-y)^{2}-\frac{1}{2}\ln(1-x)^{2}\right.\\ & & -\left.\textrm{Li}_{2}(x)+\textrm{Li}_{2}(y)\right)+\mathcal{O}\left(\epsilon^{2}\right)\end{aligned}$$ We checked the examples with $_{2}F_{1}$ and $_{3}F_{2}$ analytically with the program of [@HubMai2] and the example with $F_{1}$ numerically with the built-in first Appell function in Mathematica. Feynman integrals ----------------- As a third application we turn to the computation of Feynman integrals by direct integration over their Schwinger parameters. As a first example we consider the period integral (in the sense of [@BEK]) of the four-loop vacuum-type graph of figure 5.1 a). The integral is finite in $D=4$ dimensions and is given in terms of Schwinger parameters as $$I_{1}=\int_{\alpha_{i}\geq0}\prod_{i=1}^{8}d\alpha_{i}\delta(1-\alpha_{8})\frac{1}{\mathcal{U}^{2}}.$$ The first Symanzik polynomial $\mathcal{U}$ is linearly reducible in this case. We use our implementation of the algorithms of sections \[sec:Computing-on-the\] and \[sec:Feynman-type-integrals\] to integrate over $\alpha_{1},\,...,\, \alpha_{7}$ in an appropriate ordering and to compute the limit at $\alpha_{8}=1$ in the last step. The computation time per integration grows at first due to the increasing weight and complexity of the functions involved, but decreases in the end as fewer variables remain. Here we compute with multiple polylogarithms of weight 2, 3, 4 and 5 in the fourth, fifth, sixth and seventh integration respectively. We obtain the result $I_{1}=20\zeta(5)$ which is well-known [@CheTka]. Period integrals of this type appear as coefficients of two-point integrals corresponding to graphs obtained from breaking open one edge in the vacuum-graph (see [@Bro5; @CheTka]). As an example for a Feynman integral with non-trivial dependence on masses and external momenta, we consider the hexagon-shaped one-loop graph of figure 5.1 b) with incoming external momenta $p_{1},\,...,\, p_{6}.$ Introducing one particle mass with $m^{2}<0$ we impose the on-shell condition $p_{1}^{2}=m^{2},\, p_{i}^{2}=0,\, i=2,\,...,\,6.$ In $D=6$ dimensions the Feynman integral reads $$I_{2}=\int_{\alpha_{i}\geq0}\prod_{i=1}^{6}d\alpha_{i}\delta(1-\alpha_{6})\frac{2}{\mathcal{F}^{3}}$$ with the second Symanzik polynomial $$\mathcal{F}=\sum_{1\leq i<j\leq 6}\alpha_{i}\alpha_{j}\left(-s_{ij}^{2}\right)$$ and kinematical invariants $$s_{ij}=\left(\sum_{k=i+1}^{j}p_{k}\right)^2.$$ This integral is computed in [@Del3]. In a first step in this computation, the integral is expressed in terms of the cross-ratios $$u_{1}=\frac{s_{26}^{2}s_{35}^{2}}{s_{25}^{2}s_{36}^{2}},\, u_{2}=\frac{s_{13}^{2}s_{46}^{2}}{s_{36}^{2}s_{14}^{2}},\, u_{3}=\frac{s_{15}^{2}s_{24}^{2}}{s_{14}^{2}s_{25}^{2}},\, u_{4}=\frac{s_{12}^{2}s_{36}^{2}}{s_{13}^{2}s_{26}^{2}}$$ as $$I_{2}=\frac{1}{s_{14}^{2}s_{25}^{2}s_{36}^{2}}\int_{\alpha_{i}\geq0}\prod_{i=1}^{3}d\alpha_{i}\frac{1}{(u_{2}+\alpha_{1}+\alpha_{2})(u_{3}\alpha_{1}+u_{1}\alpha_{3}+\alpha_{2})(u_{4}\alpha_{1}\alpha_{2}+\alpha_{2}+\alpha_{1}\alpha_{3}+\alpha_{3})}.$$ We choose the parametrization $$u_{1}=\frac{1}{1+y},\, u_{2}=\frac{1+v}{1+v-u},\, u_{3}=\frac{(1-u)(-y-x)}{(1+y)(-1+u-v)},\, u_{4}=\frac{1+v-x}{1+v}$$ which differs from the one in [@Del3]. This parametrisation is not pulled from thin air: it is constructed recursively out of the polynomials occurring in the linear reduction algorithm, applied to the integrand. With this choice each $u_{i}$ tends to either 0 or 1 at the tangential base-point which we choose by the ordering $(\alpha_{2},\, \alpha_{3},\, \alpha_{1},\, u,\, v,\, x,\, y)$ and furthermore the polynomials in the denominator of the re-written integrand of $I_{2}$ are linearly reducible for the ordering $(\alpha_{2},\, \alpha_{3},\, \alpha_{1})$. Therefore we can apply our implementation to integrate over the $\alpha_{i}$ in this order and we obtain a result for positive $u,\, v,\, x,\, y$. We checked the result analytically with the program of [@Pan3]. \[fig:Graphs\] (220, 120)(2, 1) (5.,10.)(10.,15.)(-0.4,)[Straight]{}[0]{} (10.,15.)(15.,10.)(-0.4,)[Straight]{}[0]{} (15.,10.)(10.,5.)(-0.4,)[Straight]{}[0]{} (10.,5.)(5.,10.)(-0.4,)[Straight]{}[0]{} (5.,10.)(10,10.)(0.,)[Straight]{}[0]{} (10.,15.)(10.,10.)(0.,)[Straight]{}[0]{} (15.,10.)(10.,10.)(0.,)[Straight]{}[0]{} (10.,5.)(10.,10.)(0.,)[Straight]{}[0]{} (7.5,15.)(12.5,15.)(0.,)[Straight]{}[0]{} (12.5,15.)(15.,10.)(0.,)[Straight]{}[0]{} (15.,10.)(12.5,5.)(0.,)[Straight]{}[0]{} (12.5,5.)(7.5,5.)(0.,)[Straight]{}[0]{} (7.5,5.)(5.,10.)(0.,)[Straight]{}[0]{} (5.,10.)(7.5,15.)(0.,)[Straight]{}[0]{} (15.,10.)(18.,10.)(0.,)[Straight]{}[0]{} (12.5,15.)(14.,18.)(0.,)[Straight]{}[0]{} (7.5,15.)(6.,18.)(0.,)[Straight]{}[0]{} (5.,9.95)(2.,9.95)(0.,)[Straight]{}[0]{} (5.,10.05)(2.,10.05)(0.,)[Straight]{}[0]{} (5.,9.9)(2.,9.9)(0.,)[Straight]{}[0]{} (5.,10.1)(2.,10.1)(0.,)[Straight]{}[0]{} (5.,9.85)(2.,9.85)(0.,)[Straight]{}[0]{} (5.,10.15)(2.,10.15)(0.,)[Straight]{}[0]{} (5.,9.8)(2.,9.8)(0.,)[Straight]{}[0]{} (5.,10.2)(2.,10.2)(0.,)[Straight]{}[0]{} (7.5,5.)(6.,2.)(0.,)[Straight]{}[0]{} (12.5,5.)(14.,2.)(0.,)[Straight]{}[0]{} Conclusions =========== In this article we have presented explicit algorithms for symbolic computation of iterated integrals on moduli spaces ${\mathcal{M}}_{0,n+3}$ of curves of genus $0$ with $n+3$ ordered marked points, based on [@Bro2]. These algorithms include the total differential of these functions, computation of primitives and the exact computation of limits at arguments equal to 0 and 1. The algorithms are formulated by use of operations on an explicit model for the reduced bar construction on ${\mathcal{M}}_{0,n+3}$ in terms of cubical coordinates $x_i$. In this formulation, the algorithms are well suited for an implementation on a computer. We have furthermore presented an algorithm for the symbol map, out of which the vector space of homotopy invariant iterated integrals on ${\mathcal{M}}_{0,n+3}$ can be constructed. We expect the algorithms to apply to a variety of problems in theoretical physics and pure mathematics. Here we have concentrated on two main applications. As a first application, we have considered the computation of periods on ${\mathcal{M}}_{0,n+3}$, for which our algorithms are readily applicable. 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ArXiv

--- abstract: 'When a highly charged globular macromolecule, such as a dendritic polyelectrolyte or charged nanogel, is immersed into a physiological electrolyte solution, monovalent and divalent counterions from the solution bind to the macromolecule in a certain ratio and thereby almost completely electroneutralize it. For charged macromolecules in biological media, the number ratio of bound mono- versus divalent ions is decisive for the desired function. A theoretical prediction of such a sorption ratio is challenging because of the competition of electrostatic (valency), ion-specific, and binding saturation effects. Here, we devise and discuss a few approximate models to predict such an equilibrium sorption ratio by extending and combining established electrostatic binding theories such as Donnan, Langmuir, Manning as well as Poisson–Boltzmann approaches, to systematically study the competitive uptake of mono- and divalent counterions by the macromolecule. We compare and fit our models to coarse-grained (implicit-solvent) computer simulation data of the globular polyelectrolyte dendritic polyglycerol sulfate (dPGS) in salt solutions of mixed valencies. The dPGS has high potential to serve in macromolecular carrier applications in biological systems and at the same time constitutes a good model system for a highly charged macromolecule. We finally use the simulation-informed models to extrapolate and predict electrostatic features such as the effective charge as a function of the divalent ion concentration for a wide range of dPGS generations (sizes).' author: - Rohit Nikam - Xiao Xu - Matej Kanduč - Joachim Dzubiella title: 'Competitive sorption of mono- versus divalent ions by highly charged globular macromolecules' --- \[intro\] Introduction ====================== Polyelectrolytes in polar solvents such as water are important and ubiquitous in biological as well as in synthetic matter. [@Muthukumar2017; @Katchalsky1964; @Rubinstein2012; @Boroudjerdi2005; @Forster1995; @Dobrynin2005; @Liu2003] In these systems, electrostatic interactions, regulated by free ions and water, play a dominant role in shaping the structural and electrostatic characteristics of the polyelectrolyte, and the subsequent function of the system. [@Muthukumar2017; @Rubinstein2012; @Boroudjerdi2005] The electrostatic attraction between the isolated polyelectrolyte molecule and the oppositely charged counterions in the solution leads to strong counterion condensation on the molecule. This significantly modifies its interaction with other charged molecules (*e.g.*, proteins, DNA, etc.) and its electric properties such as the electrophoretic mobility in an external electric field. [@Boroudjerdi2005; @Forster1995; @Liu2003] Therefore, understanding counterion condensation is of utmost importance in order to understand the properties of polyelectrolytes and their implications in the biological and synthetic environments. [@Chremos2016; @Boroudjerdi2005] Condensation effectively leads to neutralizing an equivalent amount of the structural charge ${Z_\mathrm{d}}$ of the macromolecule. [@alexander1984charge; @belloni1998ionic] Hence, the charged substrate plus its confined counterions may be considered as a single entity with an effective (or renormalized) charge ${Z_\mathrm{eff}}$, which is significantly lower than the bare structural charge ${Z_\mathrm{d}}$. One can then identify the difference ${Z_\mathrm{d}}-{Z_\mathrm{eff}}$ as the amount of counterions condensed in the surface region. [@Bocquet2002] The phenomenon of counterion condensation and the effect of ionic strength on the configurational properties of different types of polyelectrolyte molecules such as chains, [@Forster1992; @dobrynin1995; @Wenner2002; @Raspaud1998; @Dobrynin2005; @Liu2002; @Liu2003; @Muthukumar2004; @Chremos2016] brushes, [@ruhe2004polyelectrolyte; @pincus1991colloid; @borisov1991collapse; @Zhulina1995; @Zhulina2000] or polyelectrolyte nanogels [@nanogel1; @nanogel2; @nanogel3; @arturo2017; @arturo2018] have been studied extensively in the past. Through the knowledge of the distribution of the salt ions around the polyelectrolyte, *e.g.*, measured in terms of the radial distribution function in simulations and experiments, it is possible to derive important properties such as charge–charge correlation, osmotic compressibility and shear viscosity of the system. [@forster1995polyelectrolytes] Muthukumar, in his extensive and comprehensive review of the experimental, theoretical and simulation based research done on polyelectrolyte chains, described the effect of salt concentration, valency of counterions, chain length and polyelectrolyte concentration on counterion condensation. [@Muthukumar2017; @manning2012poisson] Besides the properties of a single isolated polyelectrolyte molecule, the ionic strength of the solution also influences the interaction of polyelectrolytes with other entities, such as adsorption on substrates, [@VandeSteeg1992; @Dahlgren1993; @Netz1999; @Hariharan1998; @Gittins2001; @caruso2000hollow] formation of ultra-thin polyelectrolyte multilayer membranes, [@Decher1992; @Decher1997; @Ladam2000; @McAloney2001; @Dubas2001] the structure and solubility of polyelectrolyte complexes [@hugerth1997effect; @rusu2003formation; @winkler2002complex; @Kudlay2004a; @Mende2002] or coacervates. [@spruijt2010binodal; @Gucht2011; @biesheuvel2004electrostatic; @Perry2014] As an emerging class of functional polyelectrolytes, polyelectrolyte nanogels [@nanogel1; @nanogel2; @nanogel3; @arturo2017; @arturo2018] and dendritic or hyperbranched polyelectrolytes [@JensDernedde2010; @Khandare2012; @Groeger2013; @Maysinger2015; @Reimann2015] have attracted considerable interest in the scientific community in the last years due to their multifaceted bioapplications, such as biological imaging, drug delivery and tissue engineering. [@Leereview; @Ballauff2004; @Tian2013] In particular, the hyperbranched or dendritic polyglycerol sulfate molecules (hPGS or dPGS, respectively) are found to possess strong anti-inflammatory properties,[@Maysinger2015; @Reimann2015] act as a transport vehicle for drugs towards tumor cells,[@Sousa-Herves2015; @Groeger2013; @Vonnemann2014] and can be used as imaging agents for the diagnosis of rheumatoid arthritis. [@Vonnemann2014] This wide variety of applications, thus, have proven them to be high potential candidates for the use in medical treatments. [@Khandare2012] Hence, the understanding of dPGS interaction with the *in vivo* environment becomes important. The highly symmetric dendritic topology, terminated with monovalent negatively charged sulfate groups, makes dPGS also an excellent representative model in the class of highly charged globular polyelectrolytes. [@xu2017charged; @nikam2018charge] Because of the charged terminal groups, dPGS mainly interacts through electrostatics, rendering counterion condensation and subsequent charge renormalization effects to become substantial for function. There have been past efforts to investigate the counterion condensation and to define the effective charge as a result of the charge renormalization on charged hard-sphere colloids. [@Ohshima1982; @Zimm1983; @alexander1984charge; @Belloni1984; @belloni1998ionic; @Ramanath1988; @Manning2007; @Bocquet2002; @Gillespie2014] However, the characterization of open-structure nanogel particles or dendrites such dPGS, which in part are penetrable to ions and a surface is not well defined, remains challenging. [@Ohshima2008] Recently, Xu *et al.* implemented a simple but accurate scheme to define and determine the effective surface potential and its location for dPGS, by mapping potentials obtained from simulations to the Debye–Hückel potential in the far-field regime. [@xu2017charged] This scheme is widely known as the Alexander prescription. [@alexander1984charge; @Trizac2002; @bocquet2002effective; @Levin2004] Based on this criterion, a systematic electrostatic characterization of dPGS has been performed via coarse-grained [@xu2017charged] and all-atom [@nikam2018charge] simulations by defining the number of condensed (bound) ions. It was then established that the strong binding of dPGS to lysozyme – an abundant protein in the human biological environment – and a sequential formation of a protein corona around dPGS in the presence of NaCl salt solution, is dominantly governed by the entropic gain due to the release of a few Na$^+$ counterions during binding. [@xu:biomacro] Proteins typically bind strongly to the macromolecular surface, thereby forming a protein ‘corona’, a dense shell of proteins that can entirely coat the macromolecule. [@Owens2006; @Cedervall2007; @Lindman2007; @Monopoli2012; @wang2013biomolecular; @LoGiudice2016; @Boselli2017] Considering the medicinal applications of dPGS, it is important to study its interactions with [*divalent metal cations*]{}, *viz.* magnesium(II) and calcium(II) ions, which are key constituents of the human blood serum. Mg$^{2+}$ is essential for the stabilization of proteins, polysaccharides, lipids and DNA/RNA molecules, while Ca$^{2+}$ is critical for bone formation and plays a key role in signal transduction. [@Friesen2019a; @da2001biological] Human serum blood contains approximately $0.75-0.95$mM Mg$^{2+}$ ions, $1-4$mM Ca$^{2+}$ ions and around $150$mM NaCl salt in a dissociated form. [@meyers2004encyclopedia; @kretsinger2013encyclopedia] Thus, upon the administration of dPGS into the human biological environment, it is imperative for the competitive adsorption between the divalent (Mg$^{2+}$/Ca$^{2+}$) and monovalent (Na$^+$) ions to establish on the dPGS molecule, which can change the effective charge, and subsequently the interaction properties of dPGS with other charged entities such as proteins. This microscopic mechanism has a potential to significantly alter the attributes of protein corona around dPGS, [@XiaoCPS] thus, the biological immune response to the dPGS–protein corona complex, its metabolic fate, and the function of such a complex in biomedical and biotechnological applications. The competitive ion binding can be observed also in a wide variety of the biological and industrial ion-exchange processes such as the alkaline-earth/alkali-metal ion-exchange onto polyelectrolytes, [@Pochard1999] desalination of saline water to produce potable water, [@Birnhack2019] demineralization of whey, acid and alkali recovery from waste acid [@kobuchi1986application] and alkali solutions [@sata1993new] by diffusion dialysis, [@Sata2002] etc. Interactions of multivalent ions with polyelectrolyte solutions have been theoretically studied in the past, in terms of their thermodynamic properties, [@Kuhn1999] ionic and potential distributions, [@Gavryushov1997] accurate calculation of the effective charge, [@DosSantos2010] and the effect on the interaction between polyelectrolyte macromolecules. [@Arenzon1999; @Naji2004; @Kanduc2010; @Rudi2016] In this paper, the focus is to theoretically analyze the competitive sorption of [*mono- versus divalent counterions*]{} by highly charged spherical dPGS-like polyelectrolytes with the help of mean-field continuum and discrete binding site models, informed by coarse-grained computer simulations of dPGS of various generations. The theoretical models are generally formulated for globular charged macromolecules and include ion-specific effects in a parametric way and can thus be straightforwardly modified or adapted to other charged globules, where mono-/divalent ion-exchange plays a role. In particular, we begin with the simple Donnan model, modified for ion-specific uptake, assuming that the electrostatic potential and the ionic concentrations are constant within the macromolecule phase and the bulk phase. [@Basser1993; @arturo2016; @Ahualli2014] Despite being simple, still, for the mixed case of monovalent and divalent ions the resultant composition is a non-trivial outcome. We continue with the mean-field Poisson–Boltzmann (PB) model, widely used in colloidal science and electrochemistry, [@Rubinstein2012; @israelachvili2011intermolecular; @adamson1967physical; @verwey1947theory; @Borukhov2000] and with the limitations well known and discussed, in particular the neglect of electrostatic and steric correlations, [@eigen1954; @kralj1996; @Cuvillier1997; @Borukhov1997; @DosSantos2010] or ion-specific sorption effects. [@Kalcher2010; @Kalcher2010a; @Chudoba2018; @arturo2014; @Yaakov2009a; @Lima2008; @Koelsch2007; @okur2017beyond; @LoNostro2012; @Schwierz2013] The PB model has also been implemented to address the problem of competitive counterion binding in a mixed salt for the cases of linear polyelectrolytes such as DNA [@Burak2004; @Chen2002; @Rouzina1997; @Misra1994; @Paulsen1987] and planar geometries. [@Rouzina1994] We also devise a two-state approximation model for an ion condensation around a charged globule. {width="14cm" height="14cm"} The two-state approach was firstly used in the Oosawa–Manning model [@Manning1969; @Oosawa] for the counterion condensation around polyelectrolyte chains, according to which, counterions in a solution can be classified into two categories: ‘free’ counterions, which are able to explore the whole solution volume $V$ and the ‘condensed’ (or ‘bound’) counterions, which are localized within a small volume surrounding the polyelectrolyte macromolecule. An equivalent model for an impenetrable sphere with a surface charge was developed by Manning, where the number of condensed counterions on the macromolecule per bare unit surface charge is obtained by a free energy minimization, pointing to the competition between the electrostatic binding of counterions to the macromolecule and their dissociation entropy. [@Manning2007] We extend this model by introducing a discrete binding site model by considering the finite configurational volume of the ion in the condensed state and that the macromolecule has a finite number of charged binding sites by adopting the mixing entropy from the works of McGhee and von Hippel. [@Mcghee1974] Ion-binding models in the same spirit have been developed in the past to describe the ionization equilibrium of [*linear*]{} polyelectrolytes in monovalent salt [@flory1953molecular; @Raphael1990; @Muthukumar2004], multivalent salt, [@Friedman1984] and in mixtures of mono- and divalent salts. [@Kundagrami2008] All our models are compared to molecular simulations and used to study systematically the key electrostatic features of a highly charged globule, such as the effect of competitive adsorption on the variation of the number of condensed monovalent and divalent counterions, effective charge, and its variation with divalent ion concentration. Coarse-grained computer simulations {#cg_sim} =================================== Simulation methods, force fields, and systems {#sec:cg_ff} --------------------------------------------- The coarse-grained (CG) monomer-resolved models of the dPGS macromolecule have been developed previously [@xu2017charged] and maintain the essential dPGS structural and electrostatic features with affordable computing expense. In brief, the dPGS branching units (–) and inner core (–) (both of which are a part of the glycerol chemical group, respectively), and the terminal sulfate groups (–) are individually represented by the CG segments of specific type. The gross number of the CG segments is equal to the dendrimer polymerization $N_g = 3 \times 2^{n+1} - 2$ of generation index $n$. Only the terminal segments are charged with $-1e$ (where $e$ is the elementary charge), leading to the dPGS bare valency $|Z_n| = 3 \times 2^{n+1}$. The CG segments are connected by bonded and angular potentials both in harmonic form. In the previous work [@xu2017charged] we only studied monovalent ions. Here we extend it to study the competitive uptake of mono- and divalent ions for generations 2 and 4. The bare charge valencies of the G$_2$-dPGS are thus ${Z_\mathrm{d}}= Z_{n=2} = -24$ and ${Z_\mathrm{d}}= Z_{n=4} = -96$. Snapshots are shown in Fig. \[snap\]. The non-bonded interactions between CG beads are described by the Lennard-Jones (LJ) potential together with the Lorentz–Berthelot mixing rules. In particular, the energy parameter $\epsilon_{\rm LJ} = 0.1\,{k_\mathrm{B}T}$ and the diameter $\sigma_{\rm LJ} = 0.4$ nm are set identical for all ions (mono- and divalent) and thus any ion-specific effects are not explicitly included. In our simulations we place the dPGS in the center of a periodically repeated cubic box with a volume of $V$ (side-length of $L = 30$ nm). The solvent is implicitly assumed as a dielectric continuum with a dielectric constant $\epsilon_{\rm w} = 78$. The CG simulations employ the stochastic dynamics (SD) integrator in Gromacs $4.5.5$ as in our previous work. [@xu2017charged] {width="12.5cm" height="12.5cm"} {width="13cm" height="13cm"} All simulations are performed in the canonical ensemble. The divalent cations (DCs), monovalent cations (MCs) and monovalent anions in the system are referred to with subscripts $++,\; +$ and $-$, respectively. The dPGS is accompanied by the corresponding number of monovalent counterions ${N_\mathrm{s}}$ ($24$ for G$_2$-dPGS and $96$ for G$_4$-dPGS) electrically neutralizing the macromolecule and having the same chemical identity as the MCs of the salt. The number of salt ions $i$ ($i = +\!+,+,-$) is denoted as $n_i$, while the corresponding total salt concentrations are denoted as ${c^0_{i}}=n_i/V$. Bulk concentrations are defined as ${c^{\mathrm{b}}_{i}}=(n_i - {N^\mathrm{b}_{i}})/(V - {v_\mathrm{eff}})$ (for $i=+\!+,-$) and ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}=(n_{{\scalebox{.8}{$\scriptscriptstyle +$}}} +{N_\mathrm{s}}- {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})/(V - {v_\mathrm{eff}})$, where ${v_\mathrm{eff}}= 4\pi {r_\mathrm{eff}}^3/3$ is the volume enclosed by the effective radius ${r_\mathrm{eff}}$ of dPGS and ${N^\mathrm{b}_{i}}$ is the number of ions $i$ condensed (bound) on the dPGS. The definitions of both ${r_\mathrm{eff}}$ and ${N^\mathrm{b}_{i}}$ are discussed in Section \[sim\_analysis\]. The simulations are performed at the total DC concentrations ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ of 0.98, 2.95, 3.75, 9.96 and 14.94mM. G$_2$-dPGS simulation snapshots for different ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ values are shown in Fig. \[snap\](a)-(c), while the whole simulation box is displayed in Fig. \[snap\](d). The MC concentration ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ is fixed to $150.37$mM and the monovalent anion concentration is adjusted in a way to ensure electroneutrality in the simulation box. The bulk ionic strength $I = \frac{1}{2}\sum_i z^2_i {c^{\mathrm{b}}_{i}}$ ($i = +,+\!+,-$ with the charge valency $z_i$) ranges from $150.5$mM to $195$mM. The corresponding Debye screening length $\kappa^{-1} = \left(8\pi {l_\mathrm{B}}I \right)^{-1/2}$ (where ${l_\mathrm{B}}$ is the Bjerrum length) ranges from $0.8$nm (${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}=0$ and ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}=150.5$mM) to $0.7$nm (${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}=14.94$ and ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}=150.5$mM). As a reference, we also perform CG simulations in the limit of only monovalent salt, with total concentrations ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ of 10.02, 25.06 and 150.37mM. Simulation results: radial density distributions {#sim_results} ------------------------------------------------ The dPGS structure and its response to the addition of the DCs, is examined by the density distribution of the terminal sulfate beads ${c_\mathrm{s}}(r)$ as a function of the distance $r$ from the center-of-mass (COM) of the dPGS, for different DC concentrations ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$, as shown in Fig. \[rho\_sulfate\]. Interestingly, the presence of DCs does not lead to a notable change in the dPGS structure. Instead, the ${c_\mathrm{s}}(r)$ profiles in the operated range of ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and for both G$_2$-dPGS and G$_4$-dPGS are reasonably coincident. Fig. \[rho\_sulfate\](a) shows that for G$_2$-dPGS, a single-peak distribution is found, indicating that most of the sulfate beads reside on the molecular surface. However, in Fig. \[rho\_sulfate\](b), a bimodal distribution is seen for G$_4$-dPGS with a small peak at $r \simeq 0.6$nm. This backfolding phenomenon, contributing to a dense-core arrangement due to the dense macromolecular shell, [@Ballauff2004] is also found in our previous works [@xu2017charged; @nikam2018charge] and has been detected for other terminally charged CG dendrimer models. [@Huismann2010; @Huismann2010B; @Klos2010; @Klos2011] After the major peak, ${c_\mathrm{s}}(r)$ gradually subsides to zero. The location where the charge density ${c_\mathrm{s}}(r)$ falls to $150$mM, which we set as the physiological NaCl concentration, is defined as the bare (intrinsic) radius of dPGS ${r_\mathrm{d}}$, [^1] shown as vertical dashed blue lines in Fig. \[rho\_sulfate\]. The ${r_\mathrm{d}}$ values for G$_2$-dPGS and G$_4$-dPGS are obtained as $1.40$nm and $2.11$nm, respectively. Fig. \[rho\_sulfate\](b) also shows that a slight shift in the location of the major peak and an enrichment of the lower peak appears as ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases, indicating a slow shrinking of the dPGS molecule due to the condensation of DCs (see Fig. \[density\_ions\]). Figs. \[density\_ions\](a) and (b) show the cation density distributions $c_i(r)$ ($i =+,+\!+$) for G$_2$-dPGS and G$_4$-dPGS, respectively. Let us focus first on G$_2$ in Fig. \[density\_ions\](a). The MC distribution $c_{{\scalebox{.8}{$\scriptscriptstyle +$}}}(r)$ shows a high accumulation of counterions close to the sulfate groups, with a global maximum at distances $r \sim 1.2$nm, slightly larger than the sulfate peak (peaking roughly at $\sim 1$nm). This means that the most strongly bound ‘condensed’ MCs are thus distributed more on the surface layers of the dPGS. At larger distances, $r \sim 2$nm, a Debye–Hückel like decay is observed. Adding more DCs, the MC distribution gradually diminishes, as expected from the exchange of MCs with DCs within the dPGS. However, interestingly, the DC distribution peaks at distances distinctively smaller than the location of the sulfate peak, roughly $0.5-0.6$nm shifted towards the dPGS center away from the peak of the MC distribution. This more interior binding might be attributed to different binding mechanisms between DCs and sulfate, *e.g.*, bridging of two sulfate groups by one DC, which might be sterically favored closer to the dPGS core. These subtle structural effects may have important consequences in the context of the counterion-release mechanism driving the dPGS–protein binding, [@xu:biomacro] which should be interesting for future studies. The ion profiles for G$_4$ shown in Fig. \[density\_ions\](b) show qualitatively the same behavior but are broader and double-peaked because of the significant sulfate backfolding as previously presented in Fig. \[rho\_sulfate\](b). It is worth noting that simulations of DCs in general are more challenging than for MCs only. DC are more heavily hydrated than MCs (*e.g.*, Mg$^{2+}$ and Na$^+$ ions), [@Stokes1948; @Marcus2006] therefore future studies should scrutinize the ionic size used in the implicit solvent. Furthermore, quantum mechanical charge transfer effects as a result of the ion-induced powerful electronic polarization of the surrounding media, [@Yao2015] which are much more prevalent in the case of DCs [@Pavlov1998; @Kohagen2014] than MCs, may also be subsumed in ionic sizes in the implicit water. These model details may subtly change the density profiles shown in Figs. \[density\_ions\](a) and (b). However, the effects on total competitive uptake should be relatively minor as they are dominantly driven by valency and electrostatic correlations, and size effects are typically of second order importance. Using the density distributions of the charged entities shown above, the electrostatic properties of dPGS can be studied in the presence of the mixture of DCs and MCs. The analysis methods described in Section \[sim\_analysis\] are used to define the effective radius ${r_\mathrm{eff}}$, charge valency ${Z_\mathrm{eff}}$ and potential ${\phi_\mathrm{eff}}$ of dPGS. Electrostatic properties of dPGS {#sim_analysis} -------------------------------- **dPGS effective radius** The first step to study the ion condensation behavior is to adopt a characteristic distance ${r_\mathrm{eff}}$ to distinguish a bound ion from an unbound one. A practical method in that respect has been summarized in our previous work. [@xu2017charged] In short, we first consider the dPGS radial electrostatic potential profile $\phi$ (scaled by ${k_\mathrm{B}T}/e$), through the framework of the Poisson’s equation $$\nabla^2 \phi = -4\pi {l_\mathrm{B}}\sum_{i} z_i c_i(r) \qquad i= s, ++, +, - \label{possion}$$ Here, $c_i(r)$ refers to the radial number density profiles with respect to the distance to the dPGS-COM $r$ for all charged species in the CG simulation, namely, sulfates ($s$), DCs, MCs, and monovalent anions. For all ionic species, $c_i(r)$ reaches the bulk number density ${c^{\mathrm{b}}_{i}}$ in the far-field. The simulation results for the profiles are shown in Figs. \[rho\_sulfate\] and \[density\_ions\]. The Poisson’s equation is numerically integrated twice to obtain $\phi(r)$, which is then compared with the dimensionless Debye–Hückel potential $\phi^{}_\mathrm{DH}$, given by [@alexander1984charge; @xu2017charged; @nikam2018charge] $$\phi^{}_\mathrm{DH}(r) = {Z_\mathrm{eff}}{l_\mathrm{B}}\frac{\mathrm{e}^{\kappa {r_\mathrm{eff}}}}{1+\kappa {r_\mathrm{eff}}} \frac{\mathrm{e}^{-\kappa r}}{r}. \label{DH_eq}$$ $\phi^{}_{\rm DH}$ is applicable to a charged sphere with radius ${r_\mathrm{eff}}$ and valency ${Z_\mathrm{eff}}$. It approaches to $\phi$ only after the distance $r^*$ where non-linear effects, including the correlation and condensation of ions, subside. Thus, $r^* = {r_\mathrm{eff}}$ is eligible to serve as the dPGS effective radius to define the bound ions. The effective surface potential of dPGS obtained from simulations is then defined as ${\phi_\mathrm{eff}}= \phi({r_\mathrm{eff}})$, which is shown in Table \[cg\_table\]. Comparing Eq.  to the radial electrostatic potentials from the simulations, [@xu2017charged] the value of ${r_\mathrm{eff}}$ for dPGS in the simulations for G$_2$ and G$_4$ was found to be $1.65$nm and $2.40$nm, respectively, under the operated concentration range in the mixture of DCs with MCs as well as in the monovalent limit, as shown in Table \[cg\_table\]. These values are different than the ones obtained in our previous work, [@xu2017charged] which operates at ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 10$mM, unlike the current work where ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 150.37$mM. The newly obtained ${r_\mathrm{eff}}$ values in this work are then used as an input for the MMvH model, as discussed in Section \[mvh\], to describe the competitive sorption. It is thus implicitly assumed that ${r_\mathrm{eff}}$ does not depend on the sorption of DCs, within the operated range of DC concentrations. The same prescription will be used to define ${r_\mathrm{eff}}$ (denoted as ${r_\mathrm{eff}}^\mathrm{PB}$) from the solutions of the PPB model, as discussed in Section \[sec:ppbl\]. The results for ${r_\mathrm{eff}}^\mathrm{PB}$ are also shown in Table \[cg\_table\]. ---------------------------------------------------- ------------------ -------------------- -------------------- ----------------------- ------------------ -------------------- -------------------- ----------------------- -------------------------------- -------------------------------- ----------------------------------- -------------------------------- -------------------------------- ----------------------------------- ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ ${r_\mathrm{d}}$ ${r_\mathrm{eff}}$ ${Z_\mathrm{eff}}$ ${\phi_\mathrm{eff}}$ ${r_\mathrm{d}}$ ${r_\mathrm{eff}}$ ${Z_\mathrm{eff}}$ ${\phi_\mathrm{eff}}$ ${r_\mathrm{eff}}^\mathrm{PB}$ ${Z_\mathrm{eff}}^\mathrm{PB}$ ${\phi_\mathrm{eff}}^\mathrm{PB}$ ${r_\mathrm{eff}}^\mathrm{PB}$ ${Z_\mathrm{eff}}^\mathrm{PB}$ ${\phi_\mathrm{eff}}^\mathrm{PB}$ ([mM]{}) ([nm]{}) ([nm]{}) ([nm]{}) ([nm]{}) ([nm]{}) ([nm]{}) $0.00$ $-10.09$ $-1.26$ $-20.04$ $-1.27$ $-11.72$ $ -1.32$ $-23.60$ $-1.56$ $0.98$ $-8.85$ $-1.15$ $-17.75$ $-1.14$ $-9.79$ $-1.12$ $-20.03$ $-1.38$ $2.95$ $-7.40$ $-0.98$ $-14.21$ $-0.93$ $-8.89$ $-0.88$ $-15.54$ $-1.05$ $3.75$ $-6.84$ $-0.85$ $-12.25$ $-0.77$ $-8.29$ $-0.83$ $-14.34$ $-0.97$ $9.96$ $-6.33$ $-0.75$ $-10.11$ $-0.62$ $-7.03$ $-0.57$ $-8.86$ $-0.60$ $14.94$ $-5.86$ $-0.68$ $-9.65$ $-0.55$ $-6.36$ $-0.46$ $-6.13$ $-0.44$ ---------------------------------------------------- ------------------ -------------------- -------------------- ----------------------- ------------------ -------------------- -------------------- ----------------------- -------------------------------- -------------------------------- ----------------------------------- -------------------------------- -------------------------------- ----------------------------------- **Number of bound ions and effective charge** The cumulative number of ions of species $i$ as a function of the distance $r$ from the COM of dPGS is calculated as $$N_{\mathrm{acc},i}(r) = \int^{r}_{0} c_i(r') 4\pi {r'}^2 \mathrm{d}r' \quad \quad i = ++,+,-. \label{nbi}$$ Summing up the contribution of all charged species, the cumulative charge valency of the system as a function of the distance $r$ reads $$Z_\mathrm{acc}(r) = {Z_\mathrm{d}}(r) + 2N_{\mathrm{acc},{\scalebox{.8}{$\scriptscriptstyle ++$}}}(r) + N_{\mathrm{acc},{\scalebox{.8}{$\scriptscriptstyle +$}}}(r) - N_{\mathrm{acc},{\scalebox{.8}{$\scriptscriptstyle -$}}}(r), \label{neff}$$ where ${Z_\mathrm{d}}(r)$ denotes the spatial distribution of bare charge valency of the dPGS, obtained from the simulation. With that, the number of the bound ions and the effective charge valency of the dPGS follow as ${N^\mathrm{b}_{i}}= N_{\mathrm{acc},i}({r_\mathrm{eff}})$ and ${Z_\mathrm{eff}}= Z_{\mathrm{acc}}({r_\mathrm{eff}})$, respectively. The values are shown in Table \[cg\_table\]. ${Z_\mathrm{eff}}$ and ${\phi_\mathrm{eff}}$ exhibit a strong decrease in the magnitude with higher ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$, indicating an enhanced dPGS charge renormalization. Theoretical models ================== Basic model {#basic} ----------- In our theoretical models, the macromolecule is represented as a perfect sphere with the bare radius ${r_\mathrm{d}}$, the bare charge valency ${Z_\mathrm{d}}$, the effective radius ${r_\mathrm{eff}}$ and the effective charge valency ${Z_\mathrm{eff}}$, enclosed in a spherical domain of radius $R$ and volume $V$, as shown in Fig. \[fig:theoretical-model\]. The total number of charged monomers in the macromolecule is ${N_\mathrm{s}}$, each of which is negatively charged with a charge valency ${z_\mathrm{s}}$. ![\[fig:theoretical-model\] Schematic of a theoretical model representing the system shown in Fig. \[snap\](d). The computational cell domain (blue) is assumed to be spherical with the same volume as that of the simulation box, $V$, and with a uniform dielectric constant of water $\epsilon_\mathrm{w}=78$. dPGS is assumed to be a perfect sphere (orange) at the center of the domain. The dPGS bare and effective charge valencies are ${Z_\mathrm{d}}$ and ${Z_\mathrm{eff}}$, respectively. ${r_\mathrm{d}}$ is the bare radius of dPGS, while ${r_\mathrm{eff}}$, the effective radius, representing the distance separating the electric double layer regime ($r > {r_\mathrm{eff}}$) from the non-linear counterion ’condensation’ regime ($r < {r_\mathrm{eff}}$).](images/fig4.pdf){width="5.5cm" height="5.5cm"} All ionic species and the macromolecule are assumed to be in an aqueous bath with an implicitly modeled solvent, having a uniform dielectric constant $\epsilon_\mathrm{w}=78$ at a temperature $T=298$K. \[dm\] The Donnan model (DM) ---------------------------- The arguably simplest model for competitive uptake is the Donnan model. The Donnan equilibrium assumes two strictly electroneutral and mutually exclusive regions, *i.e.*, the macromolecule region with the Donnan radius set to be the bare radius ${r_\mathrm{d}}$ taken from simulations (*i.e.*, with a bare macromolecular volume ${v_\mathrm{d}}=4\pi {r_\mathrm{d}}^3 /3$) and total homogeneously distributed bare charge of valency ${Z_\mathrm{d}}= {z_\mathrm{s}}{N_\mathrm{s}}$ with a concentration ${c_\mathrm{s}}= {N_\mathrm{s}}/{v_\mathrm{d}}$ of charged groups of the macromolecule, and the bulk region outside the molecule with a bulk ion concentration ${c^{\mathrm{b}}_{i}}$ ($i=+,+\!+,-$). Charge neutralization of the macromolecule by the counterions leads to the Donnan potential, which is a potential having a constant non-zero value in the macromolecule region. The potential in the bulk region is set to zero. The equilibrium distribution (partitioning) of ions among the regions results in the concentrations of ionic species $i$ as ${c^\mathrm{m}_{i}}$ and ${c^{\mathrm{b}}_{i}}$ in the macromolecule and bulk regions, respectively. These concentrations are related via the partition coefficient ${\mathcal{K}_{i}}$, given by $${\mathcal{K}_{i}}= \frac{{c^\mathrm{m}_{i}}}{{c^{\mathrm{b}}_{i}}} \quad \quad i=+\!+,+,- \label{ki-don1}$$ Neglecting ion–ion correlations, an approximate expression for ${\mathcal{K}_{i}}$ can be obtained using the condition that the equilibrium electrochemical potential of ion $i$ is equal in both the macromolecule and bulk regions, implying that $${\mathrm{ln}}\,{c^{\mathrm{b}}_{i}}= z_i{\phi^{}_\mathrm{D}}+ {\mathrm{ln}}\,{c^\mathrm{m}_{i}}+ \beta {\Delta \mu_{\mathrm{int},\,i}}\label{phase_eqm}$$ where ${\phi^{}_\mathrm{D}}$ is the dimensionless Donnan potential (scaled by ${k_\mathrm{B}T}/e$) in the macromolecule region and $\beta^{-1} ={k_\mathrm{B}T}$ is the thermal energy. With ${\Delta \mu_{\mathrm{int},\,i}}$ we account for additional non-electrostatic effects that can drive adsorption, *e.g.*, dispersion and hydrophobic forces in the net ion–macromolecule interaction, and is termed the ion-specific binding chemical potential of the condensed ion. The inclusion of ${\Delta \mu_{\mathrm{int},\,i}}$ has been considered in previous work, for example, as a term reflecting the steric ion–ion packing effects in a Donnan model for ion binding by polyelectrolytes or charged hydrogels. [@Chudoba2018; @arturo2014; @Ahualli2014] Eq.  with the help of Eq.  then leads to $${\mathcal{K}_{i}}= \frac{{c^\mathrm{m}_{i}}}{{c^{\mathrm{b}}_{i}}} = \mathrm{e}^{-\beta {\Delta \mu_{\mathrm{bind},\,i}}}= \mathrm{e}^{-\beta {\Delta \mu_{\mathrm{int},\,i}}} \mathrm{e}^{-z_i {\phi^{}_\mathrm{D}}} \label{ki_don2}$$ where ${\Delta \mu_{\mathrm{bind},\,i}}$ is the total transfer chemical potential for ion $i$ from the bulk to the macromolecule region. This allows us to define the intrinsic partition ratio for ionic species $i$ as $${\mathcal{K}_{\mathrm{int},\,{i}}}= \mathrm{e}^{-\beta {\Delta \mu_{\mathrm{int},\,i}}} \quad \quad i=+,+\!+ \label{kiz}$$ and the Donnan partition ratio as a contribution from pure electrostatic interaction between the ion and the macromolecule environment as $${\mathcal{K}_{\mathrm{el},\,{i}}}= \mathrm{e}^{-z_i {\phi^{}_\mathrm{D}}} \quad \quad i=+,+\!+ \label{kid}$$ The electrostatic component of total binding chemical potential of a counterion $i$ is then defined as $\beta {\Delta \mu_{\mathrm{el},\,i}}= -{\mathrm{ln}}\,{\mathcal{K}_{\mathrm{el},\,{i}}}= z_i{\phi^{}_\mathrm{D}}$. Eq.  can then be conveniently shortened as $${\mathcal{K}_{i}}= {\mathcal{K}_{\mathrm{el},\,{i}}}\, {\mathcal{K}_{\mathrm{int},\,{i}}}\label{ki-don3}$$ where ${\mathcal{K}_{i}}$ is shown as a composition of intrinsic and electrostatic effects. The signature assumption behind the Donnan model is the electroneutrality in the macromolecule region expressed as $${z_\mathrm{s}}{c_\mathrm{s}}+ \sum_i z_i {c^{\mathrm{b}}_{i}}\,{\mathcal{K}_{i}}= 0 \label{neutral}$$ Solving Eq.  for ${\phi^{}_\mathrm{D}}$ enables us to evaluate the net partition coefficient ${\mathcal{K}_{i}}$. Eq.  has no closed solution for multivalent ions, but it exists for the case of only monovalent ions in the system, ($i=\pm$) and is given as [@cemil2] $${\phi^{}_\mathrm{D}}= -{\mathrm{ln}}\left(-\frac{ \sqrt{1 + \chi_{{\scalebox{.8}{$\scriptscriptstyle +$}}} \chi_{{\scalebox{.8}{$\scriptscriptstyle -$}}}} + 1 }{\chi_{{\scalebox{.8}{$\scriptscriptstyle +$}}}} \right) \label{donnan-mono-expr}$$ where $\chi^{}_i = 2{\mathcal{K}_{\mathrm{int},\,{i}}}\, {c^{\mathrm{b}}_{i}}/{z_\mathrm{s}}{c_\mathrm{s}}$. Note that $\chi^{}_i<0$, since the valency of charged groups ${z_\mathrm{s}}$ is negative. Using Eqs. , and , for the monovalent-only case, the number of ions of species $i(=\pm)$ partitioned into the macromolecule region is then given as $$N^\mathrm{b}_{\pm} = c^0_{\pm} {v_\mathrm{d}}\, \mathcal{K}_{\mathrm{int}, \pm} \left( -\frac{ \sqrt{1 + \chi_{{\scalebox{.8}{$\scriptscriptstyle +$}}} \chi_{{\scalebox{.8}{$\scriptscriptstyle -$}}}} + 1 }{\chi_{{\scalebox{.8}{$\scriptscriptstyle +$}}}} \,\right)^{\pm 1} \label{Nb-don}$$ To evaluate the competition between MCs and DCs in the Donnan model we evaluate Eqs.  and numerically, *cf.* section \[numerical\]. Because of the electroneutrality assumption, the Donnan prediction for the amount of counterion sorption by the macromolecule in the monovalent-only case is given by ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= |Z| + N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle -$}}}$. For highly charged macromolecules, *i.e.*, $\chi_i \rightarrow 0$, Eq.  this trivially yields ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq |Z|$. For the competitive sorption case, however, the result is non-trivial and can give a useful orientation with little effort. The Donnan model should become more quantitative for large dPGS generations, i.e., large size and/or high salt concentrations, so that $\kappa {r_\mathrm{d}}\gg 1$, for which the electroneutrality assumption is then very well justified. \[sec:ppbl\] Ion-specific Penetrable Poisson–Boltzmann (PPB) model ------------------------------------------------------------------ We now put forward a penetrable PB (PPB) model in which the charge profiles can be resolved in $r$, the radial distance from the macromolecular center. Since our charged macromolecules we have in mind (dPGS above and similar) are polymer-based with open structures and typically internally smeared out charge distributions, we opted (as in the Donnan model) for a penetrable model instead of a PB model for surface adsorption as typically used in studies of colloidal charge renormalization. [@Wall1957; @Ohshima1982; @Ohshima2008] Based on the parametrization described in the basic model (Section \[basic\]), we assume the macromolecule as a perfect penetrable sphere with a charge valency ${Z_\mathrm{d}}= {z_\mathrm{s}}{N_\mathrm{s}}$ and radius ${r_\mathrm{d}}$, as shown in Fig. \[fig:theoretical-model\]. ${r_\mathrm{d}}$ is taken from the dPGS internal charge distribution obtained from simulations, *cf.* Section \[sim\_results\] and Fig. \[rho\_sulfate\]. The charged monomers of the macromolecule, thus, have a uniform number distribution ${c_\mathrm{s}}= {N_\mathrm{s}}/{v_\mathrm{d}}$ (where ${v_\mathrm{d}}= 4\pi {r_\mathrm{d}}^3/3$) within the volume ${v_\mathrm{d}}$. ${c_\mathrm{s}}$ is applicable only within the macromolecule domain, *i.e.*, ${c_\mathrm{s}}(r) = {c_\mathrm{s}}\left(1-H(r-{r_\mathrm{d}})\right)$, where $H(r)$ is the Heaviside-step function. As an improvement to the standard PB model, here we also consider a contribution of the intrinsic non-electrostatic ion-specific interaction ${\Delta \mu_{\mathrm{int},\,i}}$ between the ion and the macromolecule, [@Kalcher2010; @Kalcher2010a] analogous to Eq.  in the Donnan model above. Assuming the electrostatic potential far away from the macromolecule, $\phi\left(r \rightarrow R \right)=0$, we first balance the chemical potential for each ion, between the bulk regime far from the macromolecule and the regime at the finite distance $r$ from the center of the macromolecule $${\mathrm{ln}}\,{c^{\mathrm{b}}_{i}}= z_i\phi(r) + {\mathrm{ln}}\,c_{i}(r) + \beta {\Delta \mu_{\mathrm{int},\,i}}(r), \label{pb_chem_pot_bal}$$ which is similar to Eq. , but in a distance-resolved manner. ${\Delta \mu_{\mathrm{int},\,i}}$ is considered on a local level, *i.e.*, ${\Delta \mu_{\mathrm{int},\,i}}(r) = {\Delta \mu_{\mathrm{int},\,i}}\left(1-H(r-{r_\mathrm{d}})\right)$. The Boltzmann ansatz then becomes $$c_i(r) = {c^{\mathrm{b}}_{i}}\,\mathrm{e}^{-z_i\phi(r) - \beta {\Delta \mu_{\mathrm{int},\,i}}(r) } \label{pb_ansatz0}$$ The distance-resolved electrostatic potential can be calculated from Eq.  together with the Poisson’s equation as $$\nabla^2 \phi(r) = -4\pi {l_\mathrm{B}}\left(\sum_{i} z_i c_i(r) + {z_\mathrm{s}}{c_\mathrm{s}}(r) \right) \quad \quad i = +\!+,+,- \label{pb-pot}$$ which establishes the PPB model including ion-specific binding effects. The boundary conditions used are $({{\mathop{}\!\mathrm{d}}}\phi/{{\mathop{}\!\mathrm{d}}}r)\left(r \rightarrow 0 \right) = 0$ and $({{\mathop{}\!\mathrm{d}}}\phi/{{\mathop{}\!\mathrm{d}}}r)\left(r \rightarrow R \right)= 0$. An effective radius for dPGS is calculated independently for this model (labeled ${r_\mathrm{eff}}^\mathrm{PB}$) using the Alexander prescription [@alexander1984charge; @Trizac2002; @bocquet2002effective; @Levin2004] on the obtained potential $\phi$, the same recipe used to calculate ${r_\mathrm{eff}}$ from simulations, *cf.* Section \[sim\_analysis\]. The values of ${r_\mathrm{eff}}^\mathrm{PB}$ for G$_2$-dPGS and G$_4$-dPGS are obtained as $1.42$nm and $2.36$nm, respectively, under the operated range of ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and at ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}=150.37$mM. The ${r_\mathrm{eff}}^\mathrm{PB}$ values are thus found to be close to those obtained from the simulations, as shown in Table \[cg\_table\]. The effective surface potential of the macromolecule is then given by ${\phi_\mathrm{eff}}^\mathrm{PB} = \phi({r_\mathrm{eff}}^\mathrm{PB})$. The number of bound ions of species $i$ within ${r_\mathrm{eff}}$, is then given by $${N^\mathrm{b}_{i}}= \int_0^{{r_\mathrm{eff}}} c_i(r)\, 4 \pi r^2 {\mathop{}\!\mathrm{d}}r \quad \quad i=+,+\!+ \label{Nib}$$ The corresponding effective charge valency ${Z_\mathrm{eff}}^\mathrm{PB}$ is calculated using Eq. . The PPB equations are solved numerically, *cf.* Section \[numerical\]. \[mvh\] Manning–McGhee–von Hippel binding model (MMvH) ------------------------------------------------------ In this section, we introduce a model based on a discrete two-state (condensed or free) perspective for the counterions, built to capture the essential physics of polyelectrolyte–ion binding in an accurate but minimalistic fashion. The model is an extension of ideas by Manning, [@Manning2007] in which ion-condensation on charged spherical surfaces was described on a mean-field free energy level as a competition between the charging (Born) self-energy of the macromolecule in salt solution and the entropy cost of binding for one-component counterions. Here, we extend this model to the case of mixtures of MCs and DCs, including binding saturation for a fixed number of binding sites like in Langmuir isotherms. The extension of the latter to binary binding of one or two binding sites by mono- or divalent solutes, respectively, was put forward buy McGhee and von Hippel. [@Mcghee1974] Therefore, we name the model Manning–McGhee–von Hippel binding model (MMvH). Following Manning, [@Manning2007] we treat the macromolecule as an impenetrable sphere of radius ${r_\mathrm{eff}}$ and charge valency ${Z_\mathrm{d}}= z_s{N_\mathrm{s}}$ taken from simulations, and extend the Manning’s model into a discrete binding site model, where the ${N_\mathrm{s}}$ charged monomers act as a finite collection of discrete binding sites for both the MCs and DCs. For the case of the DCs, two adjacent charged monomers can collectively act as a single binding site for a DC. The resulting combinatorial ways to arrange the bound MCs and DCs lead to mixing entropies worked out by McGhee and von Hippel. [@Mcghee1974] Pertaining to the canonical ensemble, we fix the total number of salt ions $n_i$, the corresponding concentrations ${c^0_{i}}$ ($i= +\!+,+,-$), the number of monovalent counterions ${N_\mathrm{s}}$ to the macromolecule, the total number of binding sites on the macromolecule and the total domain volume $V$. The coions in this model simply serve the function of maintaining electroneutrality in the total domain and their explicit adsorption is neglected. A counterion $i$ ($=+,+\!+$) is assumed to bind to the macromolecule and to occupy $f_i$ consecutive (spatially adjacent) charged terminal groups of the macromolecule. We designate $f_{{\scalebox{.8}{$\scriptscriptstyle +$}}}=1$ and $f_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}=2$ for MCs and DCs, respectively, implying that, in a bound state, one MC occupies only one charged terminal group, while one DC occupies two consecutive charged terminal groups, owing to the fact that each terminal group has a charge valency ${z_\mathrm{s}}= -1$. Consider at a given state, ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ MCs and ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ DCs are bound to the macromolecule. The *binding density*, *i.e.*, the number of bound counterions per charged terminal group is then ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/{N_\mathrm{s}}$ and ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}/{N_\mathrm{s}}$ for MCs and DCs, respectively. By multiplying with $f_i$, we then define the fraction of the binding sites occupied by the counterions, *i.e.*, coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= f_{{\scalebox{.8}{$\scriptscriptstyle +$}}} {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/ {N_\mathrm{s}}= {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/ {N_\mathrm{s}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}= f_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}/{N_\mathrm{s}}= 2{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}/{N_\mathrm{s}}$. Thus, the total number of binding sites on the macromolecule available for MCs, is $N_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = {N_\mathrm{s}}/f_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = {N_\mathrm{s}}$, and those available for DCs, is $N_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} = {N_\mathrm{s}}/f_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} = {N_\mathrm{s}}/2$. The effective charge valency of the macromolecule is then ${Z_\mathrm{eff}}= -{N_\mathrm{s}}+ {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}+ 2{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}= {-{N_\mathrm{s}}(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})}$. The total Helmholtz free energy ${\mathcal{F}_\mathrm{tot}}$ depends on the coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and the ionic concentrations ${c^0_{i}}$. The coverages can then be obtained by minimizing ${\mathcal{F}_\mathrm{tot}}$ simultaneously with respect to ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$. The total Helmholtz free energy ${\mathcal{F}_\mathrm{tot}}$ is given by the expression $${\mathcal{F}_\mathrm{tot}}= {\mathcal{F}_\mathrm{el}}+ {\mathcal{F}_\mathrm{tr}}+ {\mathcal{F}_\mathrm{mix}}+ {\mathcal{F}_\mathrm{int}}\label{eq:mvh_ftot}$$ where the four additive contributions, ${\mathcal{F}_\mathrm{el}}$, ${\mathcal{F}_\mathrm{tr}}$, ${\mathcal{F}_\mathrm{mix}}$ and ${\mathcal{F}_\mathrm{int}}$ are defined respectively as (i) electrostatic (Born) self-energy of charge renormalized macromolecule, (ii) ideal gas entropy of free ions in the bulk regime, (iii) mixing entropy of the condensed counterions in the macromolecule, and (iv) the non-electrostatic ion-specific binding free energy between the condensed counterion and the corresponding binding site on the macromolecule. The Born charging self-energy of the macromolecule immersed in an electrolyte solution associated with the Debye screening length $\kappa^{-1}$, refers to the work required to charge the macromolecule from its electroneutral to a certain charged state. Following Manning, such a charged state is associated with the effective charge ${Z_\mathrm{eff}}e$, corresponding to the sum of the intrinsic bare charge of the macromolecule ${Z_\mathrm{d}}$ and its captive, neutralizing counterions. [@Manning2007] Thus, the expression for the Born charging free energy of the macromolecule (or the self energy of the charge renormalized macromolecule) per monovalent binding site is thus expressed as $$\beta {\mathcal{F}_\mathrm{el}}= \frac{{Z_\mathrm{eff}}^2 {l_\mathrm{B}}}{2 {N_\mathrm{s}}{r_\mathrm{eff}}(1+\kappa {r_\mathrm{eff}})} = \frac{\zeta}{2} (1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})^2 \label{fel1}$$ where $\zeta/2$ is the Born free energy per monovalent binding site in the absence of counterion condensation, and $\zeta$ is given for surface charging by[@mcquarrie2000statistical] $$\zeta = \frac{{N_\mathrm{s}}{l_\mathrm{B}}}{{r_\mathrm{eff}}(1 + \kappa{r_\mathrm{eff}})} \label{zeta}$$ Considering the effective volume of dPGS ${v_\mathrm{eff}}$ to be very small compared to the total volume $V$ (${v_\mathrm{eff}}\ll V$), the bulk concentrations of MCs and DCs are given by $$\begin{aligned} \begin{split} &{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= {c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}+ \frac{{N_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})}{V} \\ &{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}= {c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}- \frac{{N_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{V} \end{split} \label{eq:bulk_conc_mvh}\end{aligned}$$ owing to the depletion of the ions in the bulk due to partitioning. ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ above is calculated considering the monovalent counterions remaining in the solution, in the salt-free limit. We assume that no anions are bound to the macromolecule binding sites, hence their bulk concentration is assumed to be the same as their salt concentration, *i.e.*, ${c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle -$}}}}= {c^0_{{\scalebox{.8}{$\scriptscriptstyle -$}}}}$. The ideal gas free energy of free cations in the bulk, normalized by the number of monovalent binding sites ${N_\mathrm{s}}$, is given as\ $$\begin{aligned} \begin{split} &\beta {\mathcal{F}_\mathrm{tr}}= -\frac{S_\mathrm{id}}{{N_\mathrm{s}}{k_\mathrm{B}}} = \sum_{i=+,++} \left(\frac{n_i - {N^\mathrm{b}_{i}}}{{N_\mathrm{s}}}\right)\left({\mathrm{ln}}\,{c^{\mathrm{b}}_{i}}\Lambda^3_i -1\right) \\ &= \sum_{i=+,++} \left(\frac{n_i - N_i {\Theta_{i}}}{{N_\mathrm{s}}}\right)\left[{\mathrm{ln}}\left({c^0_{i}}\Lambda^3_i - \frac{N_i {\Theta_{i}}\Lambda_i^3}{V}\right) -1\right] \end{split}\end{aligned}$$ where $\Lambda_i$ and $n_i$ are the thermal (de Broglie) wavelength and the number of salt ions $i$. The bound DCs and MCs can occupy the binding sites on the macromolecule in different proportions, and can distribute among the occupied sites in multiple ways at a certain bound coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$. We exert constraints to such possibilities of binding compositions and configurations, such that, ($i$) one bound DC can only bind to two adjacent monovalent binding sites, ($ii$) all non-overlapping configurations between the bound ions are possible, ($iii$) there are no designated binding sites for DCs, and ($iv$) the position of the bound DC can be shifted by a single adjacent monovalent binding site. The number of possible combinatorial binding arrangements under these constraints, adopted from the work by McGhee and von Hippel, [@Mcghee1974] is given by $$W = \frac{ \gamma^{{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}} \gamma^{{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} ({N_\mathrm{s}}- {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})!}{{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}!{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}!({N_\mathrm{s}}- 2{N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}- {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})!} \label{w}$$ where we define $\gamma_i={v^0_{i}}/\Lambda^3_i$ in terms of the effective configurational volume ${v^0_{i}}$ in the bound state. [@XiaoCPS] ${v^0_{i}}$ takes into account the rotational and vibrational degrees of freedom of a bound counterion $i$. We now define the free energy associated with the partition function $W$, normalized by the number of monovalent binding sites ${N_\mathrm{s}}$, as the free energy of mixing of the bound ions per binding site, $$\begin{aligned} \begin{split} &\beta {\mathcal{F}_\mathrm{mix}}= -\frac{S_\mathrm{mix}}{{N_\mathrm{s}}k_{\rm B}}= -\frac{1}{{N_\mathrm{s}}}{\mathrm{ln}}\, W\\ & \simeq {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}{\mathrm{ln}}{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}+ \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2}{\mathrm{ln}}\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2} -\left(1-\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2}\right){\mathrm{ln}}\left(1-\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2}\right) \\ &+ (1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}){\mathrm{ln}}(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}) \\ &- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}{\mathrm{ln}}\frac{{v^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{\Lambda_+^3} - \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2} {\mathrm{ln}}\frac{{v^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{\Lambda_{++}^3} \end{split} \label{fmx}\end{aligned}$$ where the Stirling approximation has been used for the logarithm of the factorials. This description of condensed counterion entropy is different than the ion-binding models proposed in previous works for linear polyelectrolytes [@flory1953molecular; @Raphael1990; @Muthukumar2004] in terms of the localization of counterions within volume ${v^0_{i}}$. We express this intrinsic interaction ${\mathcal{F}_\mathrm{int}}$ by the intrinsic binding chemical potential ${\Delta \mu_{\mathrm{int},\,i}}$ of each bound ion $i$. The sum of such interactions for all bound ions, normalized by the total number of monovalent binding sites gives $$\begin{aligned} \begin{split} \beta {\mathcal{F}_\mathrm{int}}&= \frac{1}{{N_\mathrm{s}}}\left({N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\beta {\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}+ {N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\beta {\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}\right) \\ &= {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\beta {\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}+ \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}{2}\beta {\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}\end{split}\end{aligned}$$ The equilibrium coverages $\Theta_i$ are then obtained by the minimization condition $$\frac{\partial}{\partial {\Theta_{i}}} {\mathcal{F}_\mathrm{tot}}\overset{!}{=} 0 \qquad \quad i=+,+\!+$$ This leads to the relation $${\Delta \mu_{\mathrm{tr},\,i}}+ {\Delta \mu_{\mathrm{el},\,i}}+ {\Delta \mu_{\mathrm{mix},\,i}}+ {\Delta \mu_{\mathrm{int},\,i}}= 0 \quad \quad i=+,+\!+ \label{k_and_g_all}$$ where ${\Delta \mu_{\mathrm{tr},\,i}}$ denotes the translational entropy change associated with one ion $i$ when it transfers from the bulk environment to the bound state in the macromolecule. ${\Delta \mu_{\mathrm{el},\,i}}$ is the *electrostatic binding chemical potential* and ${\Delta \mu_{\mathrm{mix},\,i}}$ is the *mixing chemical potential*. Eq. , similar to the PPB (Eq. ) and DM (Eq. ) models, indicates the counterion chemical potential components contributing to its condensation on the macromolecule. The expressions for the constituent chemical potential contributions in Eq.  are given by $$\begin{aligned} \begin{split} &\beta {\Delta \mu_{\mathrm{tr},\,i}}=-{\mathrm{ln}}\,{c^{\mathrm{b}}_{i}}{v^0_{i}}\qquad \qquad \qquad \quad i=+,+\!+\\ &\beta {\Delta \mu_{\mathrm{el},\,i}}=-z_i\zeta(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}) \quad \quad i=+,+\!+\\ &\beta {\Delta \mu_{\mathrm{mix},\,i}}= \begin{cases} \begin{aligned} {\mathrm{ln}}\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\left(2-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\right)}{4(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})^2} \quad \quad i=++\\[1.1ex] {\mathrm{ln}}\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})} \quad \quad i=+ \end{aligned} \end{cases} \end{split} \label{all_free_energies}\end{aligned}$$ Using Eqs.  and leads to the final form of the MMvH model, given by $$K_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} = v^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} {\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}\mathrm{e}^{2\zeta \left( 1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\right)} = \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}(2 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})}{4 {c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}{(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})}^2} \label{Kb}$$ $$K_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = v^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}} {\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}\mathrm{e}^{\zeta \left(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\right)} = \frac{ {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})} \label{Ka}$$ where $K_i$ are the *equilibrium binding constant* associated with the binding of ion $i$ to its corresponding binding site on the macromolecule. The relationship between $K_i$, the total binding chemical potential ${\Delta \mu_{\mathrm{bind},\,i}}$ and the total partition ratio ${\mathcal{K}_{i}}$ is given as $$\beta {\Delta \mu_{\mathrm{bind},\,i}}= -{\mathrm{ln}}\,\frac{K_i}{{v^0_{i}}} = -{\mathrm{ln}}\,{\mathcal{K}_{i}}\quad \quad i=+,+\!+ \label{kbind}$$ Or in other words, referring back to Eq. , $${\mathcal{K}_{i}}= {\mathcal{K}_{\mathrm{int},\,{i}}}\,{\mathcal{K}_{\mathrm{el},\,{i}}}= {\mathcal{K}_{\mathrm{int},\,{i}}}\, \mathrm{e}^{z_i \zeta \left(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}- {\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\right)}$$ where the electrostatic contribution of the total partition ratio is defined as $${\mathcal{K}_{\mathrm{el},\,{i}}}= \mathrm{e}^{-\beta {\Delta \mu_{\mathrm{el},\,i}}} = \mathrm{e}^{z_i\zeta(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}})} \quad \quad i=+,+\!+ \label{kiel-mvh}$$ From Eq. , for a given magnitude of $K_i$, the absolute magnitude of ${\Delta \mu_{\mathrm{bind},\,i}}$ depends on ${v^0_{i}}$, which we calculate from our simulations and predict respective values of ${\Delta \mu_{\mathrm{bind},\,i}}$. Finally, we consider the limit of the MMvH model for vanishing DCs (MCs only). Without DCs, we have $$\begin{aligned} \begin{split} &\beta {\Delta \mu_\mathrm{tr}}=-{\mathrm{ln}}\,{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}v^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}} \\ &\beta {\Delta \mu_\mathrm{el}}=-\zeta(1-{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}) \\ &\beta {\Delta \mu_\mathrm{mix}}= {\mathrm{ln}}\,\frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})} \label{g_langmuir} \end{split}\end{aligned}$$ Combining Eqs.  and leads to the “Manning–Langmuir" (ML) model $$K_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = v^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}} {\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}\mathrm{e}^{\zeta (1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})} = \frac{{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}{{c^{\mathrm{b}}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}(1 - {\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}})} \label{manning-langmuir}$$ The McGhee–von Hippel combinatorics here reduces to the standard one-component Langmuir picture, *i.e.*, the right-hand-side of Eq.  reflects the Langmuir isotherm. The standard Langmuir model is thus extended to include charging free energies by ion condensation (charge renormalization) and ion-specific binding. From another perspective, it extends the Manning model for the counterion condensation on spheres [@Manning2007; @Gillespie2014] to include ion-specific effects as well as the saturation of binding sites in terms of the translation entropy of the condensed ions. Numerical evaluation {#numerical} -------------------- The PPB model, with the assumption of the uniform intrinsic macromolecular volume charge distribution ${c_\mathrm{s}}(r)e$ and with the knowledge of the bare radius ${r_\mathrm{d}}$ of the macromolecule inherited from simulations, generates the distance-resolved number density profiles of charged species, similar to Fig. \[density\_ions\]. Hence, it performs the same analysis as that for simulations (*cf.* Section \[sim\_analysis\]), to calculate the effective radius ${r_\mathrm{eff}}$ and other electrostatic properties of the macromolecule, such as ${Z_\mathrm{eff}}$, ${\phi_\mathrm{eff}}$, etc. The DM model also assumes uniform ${c_\mathrm{s}}(r)e$ and requires the knowledge of the electroneutrality radius, which is taken as ${r_\mathrm{d}}$ from simulations as an input parameter, similar to the PPB model. The MMvH (ML) model, on the other hand, assumes the macromolecule as a hard sphere with a uniform surface charge distribution. The effective radius of the hard sphere ${r_\mathrm{eff}}$ is taken from simulations as an input parameter. The results from the DM, PPB and MMvH (ML) models and simulations are compared in terms of the coverages ${\Theta_{i}}$ ($i=+\!+, +$), which are defined as ${\Theta_{i}}={N^\mathrm{b}_{i}}/{N_{i}}$, where ${N^\mathrm{b}_{i}}$ is the number of condensed counterions $i$ and ${N_{i}}$ is the corresponding number of binding sites available on dPGS, defined in Section \[mvh\]. Since the PPB model deals with a volume sorption, while the DM model deals with the ion partitioning between two electroneutral phases, “coverage" ${\Theta_{i}}$ in these cases are interpreted as a load or an extent of neutralization of dPGS. For the DM, PPB and MMvH (ML) models, the intrinsic partition coefficients ${\mathcal{K}_{\mathrm{int},\,{i}}}$ for both ions ($i=+\!+,+$) are unknowns and taken as fitting parameters in order to match the coverages from the simulations, which are described in Section \[sec:cg\_ff\]. Regarding the PPB and DM models, we make a further assumption that intrinsic non-electrostatic ion–binding site interaction for the MCs is identical to that for the monovalent anions, *i.e.*, ${\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}= {\mathcal{K}_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle -$}}}}}$. Mathematically, the PPB model represents a boundary-value problem having a second order differential equation (Eq. ) non-linear in the electrostatic potential paired with the boundary conditions, while the MMvH model (Eqs.  and ) represents two non-linear simultaneous equations in coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$. Both PPB and MMvH models are evaluated self consistently for the potential and coverages, respectively. To solve Eq. , we employ the `solve_bvp` function in the SciPy library (version 1.3.1) from Python (version 3.7.4), which solves a boundary-value problem for a system of ordinary differential equations using the fourth order collocation algorithm. [@kierzenka2001bvp] The bulk concentration ${c^{\mathrm{b}}_{i}}$ is obtained using the law of conservation of mass, in an iterative manner. Eqs.  and are solved using `fsolve` function from the SciPy library, which is also used to evaluate the DM model (Eq. ) representing the single non-linear equation in the Donnan potential ${\phi^{}_\mathrm{D}}$. The effective configurational volume ${v^0_{i}}$ of bound counterions, used in the MMvH model is assumed to be equal for both counterions, *i.e.*, $v_{{\scalebox{.8}{$\scriptscriptstyle ++$}}} = v_{{\scalebox{.8}{$\scriptscriptstyle +$}}} = v_0$. It is worth considering that the volume $v_0$ depends on the precise nature of the bound state and it is infeasible to have its knowledge in experiments due to unknown microscopic details, although it can be computed using simulations. [@xu:biomacro; @Yu2015] According to the convention in experiments, the standard volume is defined as $v_0 = 1\,\mathrm{M}^{-1} \simeq 1.6$nm$^{3}$, corresponding to the standard concentration $c^\mathrm{std} = 1$M. [@atkins; @Gilson2007; @General2010a] In this case, the total binding chemical potential ${\Delta \mu_{\mathrm{bind},\,i}}$ can be referred to as the standard binding energy $\Delta G^0$. [@Gilson2007; @General2010a] Results and Discussion ====================== Monovalent limit: theoretical comparison and best fit to simulations -------------------------------------------------------------------- {width="12cm" height="12cm"} Considering the monovalent limit as reference case, we now start with the application of aforementioned theoretical binding models. Fig. \[monocg\](a) shows the predictions of the PPB and ML (monovalent-only limit of MMvH) models for the variation of the binding coverage of MCs, ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, as a function of the MC concentration, ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$. It can be observed that ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ increases sharply for a small increase in ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ from $0$ to $\sim 10$mM, while it increases gradually for larger ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$. This is attributed to the combined contribution of the electrostatics and an entropy of a bound counterion, facilitating condensation. In the low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ regime, the bare charge of G$_2$-dPGS is weakly renormalized, and some of the dPGS binding sites are unoccupied. This leaves a high propensity of condensation for new incoming counterions. This can be conveniently explained via the ML model. Referring to Eq. , the increase in the condensation of MCs at the limit of low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, $\lim_{{c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\to 0} {{\mathop{}\!\mathrm{d}}}{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/{{\mathop{}\!\mathrm{d}}}{c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ is directly proportional to the total binding constant $K_{{\scalebox{.8}{$\scriptscriptstyle +$}}}$, while at high ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, $\lim_{{c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\to \infty} {{\mathop{}\!\mathrm{d}}}{\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}/{{\mathop{}\!\mathrm{d}}}{c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 0$. This implies that at low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, the resultant low coverage ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ leads to a high electrostatic driving force for condensation as well as entropy of a bound counterion, thus a high amount of condensation. On the other hand, at high ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, the macromolecule charge is almost entirely renormalized and most of the binding sites are occupied, resulting in hardly any increase in condensation. Comparing the coverage profiles from PPB and ML models that neglect ion-specific effects, *i.e.*, with ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}=0$ (dotted curves), we find that the PPB coverage values are close to the ML values in the low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ regime, however, attain higher values than the ML counterpart at high ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$. This is attributed to the effects of discrete binding sites incorporated in the ML model, in the form of the configurational volume $v_0$ (here, we used $v_0=1.04$ M$^{-1}$ obtained from our previous simulations [@xu:biomacro]). The PPB model, on the other hand, assumes the condensed ions as point charges, leaving no entropic penalty for new incoming counterions as they condense on the binding sites. Another reason is that the PPB model also incorporates, to some extent, the non-linear effects in the electrostatic interactions, which are not considered in the DH-level Born energy used in the ML model. Both models, however, underestimate the simulations if we do not include corrections via ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$. The reason is likely the approximative treatments of the electrostatic energy in both models, PPB and ML, which are mean-field and do not include the discrete nature of the charged binding sites and the complex spatial charge correlations inside the macromolecule. The DM model, in addition to these assumptions, takes the macroscopic view of macromolecule and bulk phases in a segregated form. The model then predicts the ion partitioning while imposing electroneutralities of phases. In that respect, for highly charged macromolecules like dPGS, the DM model predicts ${N^\mathrm{b}_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq {N_\mathrm{s}}$, implying ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq 1$. This plot is not shown, since it does not provide a useful insight for us in the context of counterion condensation. The case of salt concentration ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 0$ is referred to as the counterion-only case, and gives ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\sim 0.28$ for the PPB model. Note that ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ in this limit is system specific, since the size of the simulation box/computational domain determines the counterion concentration and subsequently the coverage. The coverage ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ in the ML model in this limit is undefined, since the electrostatic binding energy of MCs depends on the screening length $\kappa^{-1}$, which is undefined in this model in the absence of the salt. In the next step, ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ values for PPB and ML models are fitted (bold curves in Fig. \[monocg\](a)) to the simulation results for G$_2$-dPGS in the monovalent limit by allowing ion-specific effects in the counterion–macromolecule binding, *i.e.*, ${\Delta \mu_{\mathrm{int},\,{+}}}$ as a fitting parameter. The values of ${\Delta \mu_{\mathrm{int},\,{+}}}$ are found to be $-0.45\,{k_\mathrm{B}T}$ and $-1.81\,{k_\mathrm{B}T}$ for PPB and ML models, respectively. Recall that the simulations have not really included ion-specific effects in terms of specific hydration phenomena, etc., still, they include excluded-volume, dispersion attraction, and importantly, all electrostatic charge–charge correlations, not captured in the mean-field theories. Hence, the ion-specific fitting parameters can be viewed in general as correction factors, including all ionic contributions that are beyond the mean-field treatment of the PPB and ML models. The larger fitting parameter for ML than PPB (in the absolute value) may indicate the higher level of approximations in the ML model. Having the models now informed using the benchmark data from simulations, they can be utilized to predict the binding at other ion concentrations. Fig. \[monocg\](b) shows the numerical fitting of ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ values (dashed curves) to those obtained from G$_4$-dPGS simulations. The values of ${\Delta \mu_{\mathrm{int},\,{+}}}$ as a fitting parameter are $-0.56\,{k_\mathrm{B}T}$ and $-1.85\,{k_\mathrm{B}T}$ for PPB and ML models, respectively, which are close to those obtained for G$_2$-dPGS, within the error difference of $\sim 0.1\,{k_\mathrm{B}T}$. The ML model fits better to both G$_2$-dPGS and G$_4$-dPGS CG results than the PPB model at large ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$, which may indicate that the dPGS charge in the simulations acts more as finite binding sites, as assumed in the ML model. Divalent case: theoretical comparison and best fit to simulations ----------------------------------------------------------------- {width="14cm" height="14cm"} We now aspire to use the obtained ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ to inform the MMvH and PPB models with the help of the reference data obtained from simulations, in order to capture the competitive ion binding in a mixture of MCs and DCs. The models fitted to the benchmark data can then be used to predict the binding coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ for different dPGS generations and salt concentrations. In practice, we perform the numerical fitting of ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ obtained from the MMvH and PPB models to those from simulations, by fixing ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ for MCs obtained from the monovalent-only case, and then subsequently fitting ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ for DCs. The values of ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ for MCs obtained from the monovalent limit are, for a given binding model (ML or PPB), found to be approximately independent of the dPGS generation (with $\sim 0.1\,{k_\mathrm{B}T}$ as margin of error). Therefore, ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ is averaged over generations (G$_2$ and G$_4$), as shown in Table \[mvh\_ppb\_kint\_table\]. Fig. \[divalent\] depicts the behavior of MMvH, PPB and the DM model in terms of the binding coverages ${\Theta_{i}}$, in a mixture of DCs and MCs. The MMvH model uses the effective configurational volumes $v_0=1.04$ M$^{-1}$ and $0.57$ M$^{-1}$ for G$_2$-dPGS and G$_4$-dPGS, respectively, as obtained from our previous simulations. [@xu:biomacro] At low DC concentration, *i.e.* in the monovalent limit, MCs act as the only counterions to the macromolecule, resulting in the highest MC coverage ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$. In this limit at ${c^0_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}= 150.37$mM, both MMvH and PPB models show ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq 0.57$ for G$_2$-dPGS, and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}\simeq 0.8$ for G$_4$-dPGS. As ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases, more DCs bind to the macromolecule and more of the previously bound MCs get released into the bulk. Table \[cg\_table\] shows the resultant effective charge valency ${Z_\mathrm{eff}}^\mathrm{PB}$ and potential ${\phi_\mathrm{eff}}^\mathrm{PB}$ of G$_2$-dPGS and G$_4$-dPGS evaluated by the PPB model. Quantitatively consistent with the ${Z_\mathrm{eff}}$ and ${\phi_\mathrm{eff}}$ obtained from simulations, ${Z_\mathrm{eff}}^\mathrm{PB}$ and ${\phi_\mathrm{eff}}^\mathrm{PB}$ show a strong decrease in magnitude with a higher ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$, depicting higher dPGS charge renormalization. [cdd]{} &\ & ++ & +\ & 5.13 & 3.37\ & -1.87 & -0.50\ ($v_0$ [CG]{}) & -2.85 & -1.83\ ($v_0$ [Std.]{}) & -2.86 & -1.44\ {width="11.8cm" height="11.8cm"} Corresponding to the fitting of binding coverages ${\Theta_{i}}$ on G$_2$-dPGS and G$_4$-dPGS binding sites, as shown in Fig. \[divalent\], the resulting ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ values are calculated as $-2.73\,{k_\mathrm{B}T}$ (G$_2$) and $-2.98\,{k_\mathrm{B}T}$ (G$_4$) for the MMvH model, whereas $-1.77\,{k_\mathrm{B}T}$ (G$_2$) and $-1.98\,{k_\mathrm{B}T}$ (G$_4$) for the PPB model. Table \[mvh\_ppb\_kint\_table\] shows the values of ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ averaged over G$_2$-dPGS and G$_4$-dPGS cases. It can be observed that both ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ and ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ values from the MMvH model exceed (in magnitude) those from the PPB model across the whole ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}\sim 0-25$mM range. This can again be attributed to higher approximations in the electrostatic partition coefficient designed in the MMvH model, based on the Debye–Hückel charging free energy, as compared to that from the PPB model, incorporating non-linear effects in the electrostatic potential near the macromolecule vicinity. The standard intrinsic chemical potentials ${\Delta \mu_{\mathrm{int},\,i}}^0$ after fitting the MMvH model ${\Theta_{i}}$ with those from simulations are also given in Table \[mvh\_ppb\_kint\_table\]. Unlike the other models, we simultaneously fit both ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ and ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ to perform numerical fitting of ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ obtained from the DM model with the simulation data. As shown in Table \[mvh\_ppb\_kint\_table\], the values of ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle +$}}}}}$ and ${\Delta \mu_{\mathrm{int},\,{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}}$ for the model turn out large and positive compared with those from other models, since the DM model tries to neutralize the entire dPGS charge via the electroneutrality condition in the dPGS phase. The DM fits for ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ differ to an extent with those from simulations, while those for ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ are found to be reasonably good. The DM provides much better fits for ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ in the case of G$_4$-dPGS as compared to G$_2$-dPGS. This is attributed to the bigger size of G$_4$-dPGS, which better satisfies the criterion $\kappa {r_\mathrm{d}}\gg 1$, under which the DM electroneutrality condition holds comparatively well. Having established the model frameworks by informing ${\Delta \mu_{\mathrm{int},\,i}}$ by fitting the coverages ${\Theta_{i}}$ to those from simulations and averaging the values of obtained ${\Delta \mu_{\mathrm{int},\,i}}$ over generations (See Table \[mvh\_ppb\_kint\_table\]), we finally utilize their predictive ability to explore the electrostatic characterization of dPGS for different generations and salt concentrations. As an example, Fig. \[prediction\](d) shows the MMvH model predictions for the binding coverages ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle +$}}}}$ and ${\Theta_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ for the case of a competitive ion binding on G$_6$-dPGS, similar to Fig. \[divalent\] on G$_2$-dPGS and G$_4$-dPGS. We also study the effective charge valency ${Z_\mathrm{eff}}$ of dPGS along with the composition of condensed ions on the molecule. Figs. \[prediction\](a) and \[prediction\](b) show the variation of the effective charge valency ${Z_\mathrm{eff}}$ of G$_2$-dPGS and its normalized form ${Z_\mathrm{eff}}/{Z_\mathrm{d}}$, respectively, as a function of the DC concentration ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$, as predicted by the MMvH model. It can be clearly seen from Fig. \[prediction\](a) that the introduction of DCs leads to a net charge renormalization of dPGS, which further decreases its ${Z_\mathrm{eff}}$. The inset shows that, with reference to the monovalent limit, the dPGS effective charge is $30-35\%$ further renormalized upon introducing DCs in the range of $1-4$mM, which is close to the physiological concentration range for calcium(II) ions. Fig. \[prediction\](b) shows that the fraction of the bare dPGS charge that gets renormalized increases with the dPGS generation. The inset shows the variation for ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ varying from $0$mM to $10$mM. The rate of dPGS charge renormalization with respect to ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ is the highest at the low ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ regime and subsides as ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases, since the charge renormalized dPGS results in lower electrostatic binding chemical potential ${\Delta \mu_{\mathrm{el},\,i}}$. The reduced amount of renormalization is not attributed to the ion packing, which is evident from Fig. \[prediction\](c) showing the total number of condensed ions (including both DCs and MCs) per dPGS sulfate group. As ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases, the total number of condensed ions decreases, indicating that the ion packing effects diminish as ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases. The decrease in the amount of renormalization thus predominantly has electrostatic origin. Fig. \[prediction\](a) shows that $80-90\%$ of the dPGS bare charge is renormalized as ${c^0_{{\scalebox{.8}{$\scriptscriptstyle ++$}}}}$ increases from $0-100$mM, however, the total number of condensed counterions effectively decreases, according to Fig. \[prediction\](c). This in effect would significantly hamper the binding affinity of protein with dPGS. It has been well established through our previous works that the dPGS–protein complexation is dominantly influenced by the release of a few MCs that were highly confined due to strong charge renormalization. [@xu:biomacro] The introduction of DCs, however, decreases the confinement of these condensed counterions, thus less counterions to be released during dPGS-protein binding. In addition, the strongly charge renormalized dPGS leads to lower electrostatic contribution to its overall binding affinity with the protein or any other multivalent ligand. Conclusion ========== In this paper, we have addressed the biologically and industrially relevant problem of the competitive sorption of mono- and divalent counterions into a highly charged globular polyelectrolyte, with direct comparison to CG simulations of the dendritic macromolecule dPGS. Beyond simple Donnan and ion-specific penetrable PB models, we introduced a two-state discrete binding site model (MMvH) applicable for heterogeneous ligand systems (counterions with mixed valencies/stoichiometries). The broad classification of surrounding counterions as “bound" and “free" gives the MMvH model a computationally unique advantage over the PPB model, which involves the calculation of the distance-resolved counterion density profiles. The fitting results with simulations highlight the key differences in the MMvH and PPB models. Although being on a mean-field level, the PPB model incorporates non-linear electrostatic effects, which become more prominent near the surface of dPGS, delivering a relatively accurate picture of the dPGS–counterion electrostatic binding affinity, compared to the MMvH model, which approximates dPGS–counterion electrostatic interaction on a linearized PB (DH) level by absorbing these non-linear electrostatic effects into the effective charge valency ${Z_\mathrm{eff}}$ of dPGS. On the contrary, the MMvH model provides more accurate values of the extent of counterion adsorption $\Theta$ at high concentrations (*i.e.*, in the binding site saturation regime) than the PPB model. The reason is that the MMvH model assumes discrete binding sites, whereas the PPB model treats dPGS charge as continuum and allows an unlimited uptake of counterions, which is not realistic. Future extensions of the MMvH model could include an extra level of competition between adsorbed ions explicitly, namely through a non-linear term in Eq.  (of the type used in the regular solution theory or the Flory–Huggins approximation in polymer theories) that describes the interaction between two adsorbed ions in proximal positions (sites). The effects of this generalization in a different context can be found in a study on ion induced lamellar-lamellar phase transition in charged surfactant systems. [@Harries2006] In general, this type of competition results in non-continuous adsorption equilibria and could be interesting in the present context. The simplest presented model, the Donnan model (DM) extended for ion-specific effects, is also useful for a quick, qualitative prediction of the adsorption ratio. Per construction it should become more accurate for large globules and/or large salt concentrations (for which the globule size becomes larger than the DH screening length), where the electroneutrality condition is better justified. The models presented in this work can be used to accurately extrapolate and predict the competitive ionic sorption in experiments for a wide range of salt concentrations and salt compositions. They can be also easily generalized to more ionic components and valencies. The electroneutrality radius required for the DM model and the intrinsic macromolecular charge distribution required for the PPB model as an input parameter (in the form of the bare radius ${r_\mathrm{d}}$), are taken from simulations. However, they can also be derived by measuring the form factors from, *e.g.*, neutron scattering. [@Boris1996; @Berndt2006] The MMvH (ML) model requires the effective radius ${r_\mathrm{eff}}$ of the macromolecule as an input parameter, which besides simulations, can also be derived from independent experiments such as electrophoresis and fitting structure factors (of non-dilute colloidal suspensions) by DLVO interactions. [@hunter; @israelachvili2011intermolecular] As we showed, ${r_\mathrm{eff}}$ can also be obtained using PB models and related theories provided the intrinsic macromolecular charge distribution is available. The authors are indebted to Matthias Ballauff for insightful discussions. R.N. thanks Jacek Walkowiak for fruitful discussion. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 646659). X. 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ArXiv

--- author: - 'Steven V. Fuerst and Kinwah Wu' date: 'Received: ' title: 'Radiation Transfer of Emission Lines in Curved Space-Time' --- Introduction ============ The strong X-rays observed in active galactic nuclei (AGN) and some X-ray binaries are believed to be powered by accretion of material into black holes. The curved space-time around the black hole influences not only the accretion hydrodynamics but also the transport of radiation from the accretion flow. Emission lines from thin Keplerian disks around non-relativistic stellar objects generally have two symmetric peaks (Smak 1969), corresponding to the approaching and receding line-of-sight velocities due to disk rotation. Because of various relativistic effects, lines from accretion disks around black holes do not always have symmetrical double-peak profiles. The accretion flow near a black hole is often close to the speed of light, and emission is relativistically boosted. The blue peak of the line therefore becomes stronger and sharper. Moreover, the strong gravity near the black-hole event horizon causes time dilation, which shifts the line to lower energies. Emission lines from accretion disks around black holes appear to be broad, with a very extended red wing and a narrow, sharp blue peak (see e.g. the review by Fabian et al. (2000) and references therein). Furthermore, gravitational lensing can produce multiple images and self-occultation, further modifying the emission line profile. Various methods have been used to calculate the profiles of emission lines from accretion disks around black holes. The methods can be roughly divided into three categories. We now discuss each of them briefly. The first method uses a transfer function to map the image of the accretion disk onto a sky plane (Cunningham 1975, 1976). The accretion disk is assumed to reside in the equatorial plane. It is Keplerian and geometrically thin, but optically thick. The space-time metric around the black hole is first specified, and the energy shift of the emission (photons) from each point on the disk surface is then calculated. A parametric emissivity law for the disk emission is usually used — typically, a simple power-law which decreases radially outward. The specific intensity at each point in the sky plane is determined from the energy shift and the corresponding specific intensity at the disk surface, using the Lorentz-invariant property. The transfer-function formulation (Cunningham 1975, 1976) has been applied to line calculations in settings ranging from thin accretion rings (e.g. Gerbal & Pelat 1981) and accretion disks around Schwarzschild (e.g. Laor 1991) and rotating (Kerr) black holes (e.g. Bromley, Chen & Miller 1997). The second method makes use of the impact parameter of photon orbits around Schwarzschild black holes (e.g. Fabian et al. 1989; Stella 1990; Kojima 1991). The transfer function in this method is described in terms of elliptical functions, which are derived semi-analytically. The Jacobian of the transformation from the accretion disk to sky plane is, however, determined numerically via infinitesimal variations of the impact parameter (Bao 1992). The method can be generalized to the case of rotating black holes by using additional constants of motion (Viergutz 1993; Bao, Hadrava & Ostgaard 1994; Fanton et al. 1997; Cadez, Fanton & Calvani 1998). The third method simply considers direct integration of the geodesics to determine the photon trajectories and energy shifts (Dabrowski et al. 1997; Pariev & Bromley 1998; Reynolds et al. 1999). These calculations have shown how the dynamics of the accretion flow around the black hole and the curved space-time shape the line profiles. Various other aspects of the radiation processes, e.g. reverberation and reflection (Reynolds et al. 1999) and disk warping (Cadez et al. 2003) were also investigated using the methods described above. The results obtained from these calculations have provided us with a basic framework for interpreting X-ray spectroscopic observations, in particular, the peculiar broad Fe K$\alpha$ lines in the spectra of AGN, e.g. MCG-6-30-15 (Tanaka et al. 1995). While existing studies have put emphasis on the energy shift of the emission, transport effects such as extinction have been neglected. Resonant absorption (scattering) by ambient material can greatly modify the disk emission line profile. This effect was already demonstrated in a study by Ruszkowski & Fabian (2000), in which a simple rotating disk-corona provides the resonant scattering. Here, we present ray-tracing calculations of spectra from relativistic flows in curved space-time. We include line-of-sight extinction and emission explicitly in the formulation. The radiative-transfer equation is derived from the Lorentz-invariant form of the conservation law. It reduces to the standard classical radiative-transfer equation in the non-relativistic limit. The formulation can incorporate dynamical and geometric models for the line-of-sight absorbing and emitting material. As an illustration, we calculate the from thin accretion disks and thick accretion tori around rotating black holes. The emitted spectra include a power law continuum together with a line. This emission is resonantly scattered by the line-of-sight-material. We include the contribution from higher-order images and allow for self-occultation. We organize the paper as follows. In §2 we show the derivation of the transfer equation. In §3 we construct the equation of motion for free particles in a Kerr space-time and for force-constrained particles for some simple parametric models. In §4 we construct a thin disk and a thick torus model. In §5 we generalize this by adding in absorption due to a distribution of absorbing clouds. In §6 we present the results from the models where either emission geometry (tori), or absorption (clouds) are important. Radiative-Transfer Equation =========================== Throughout this paper, we adopt the usual convention $c=G=h=1$ for the speed of light, gravitational constant and Planck constant. The interval in space-time is specified by $$\label{metric} d\tau^2 = g_{\alpha \beta} dx^{\alpha} dx^{\beta} $$ where $g_{\alpha \beta}$ is the metric. Consider a bundle of particles which fill a phase-space volume element $$\label{bundle} {d\cal{V}} \equiv dx\,dy\,dz\,dp^x\,dp^y\,dp^z\ ,$$ where $dx\,dy\,dz (\equiv dV)$ is the three-volume and $dp^x\ dp^y\ dp^z$ is the momentum range, at a given time $t$. Liouville’s Theorem reads $$\frac{d{\cal V}}{d\lambda} = 0$$ (see Misner, Thorne & Wheeler 1973), with $\lambda$ here being the affine parameter for the central ray in the bundle. The volume element $d{\cal V}$ is thus Lorentz invariant. The distribution function for the particles in the bundle, $F(x^i, p^i)$ is given by $$F(x^i, p^i) = {dN \over d{\cal V}}\ ,$$ where $dN$ is the number of particles in the three-volume. Since $dN/d{\cal V}$ is Lorentz invariant, $F(x^i, p^i)$ is Lorentz invariant. From equation (\[bundle\]), we have $$\label{rawfluxdefn} F={dN\over p^2 dV\,dp\,d\Omega}\ ,$$ where $p^2\,dp\,d\Omega=dp^x\,dp^y\,dp^z$. For massless particles, $v = c = 1$ and $\vert p\vert=E$. The number of photons in the given spatial volume is therefore the number of photons flowing through an area $dA$ in a time $dt$. It follows that $$\label{flux_inv} F={dN\over E^2dA\,dt\,dE\,d\Omega}\ .$$ Recall that the specific intensity of the photons is $$\label{inten_inv} I_\nu={E dN\over dA\,dt\,dE\,d\Omega}\ ,$$ By inspection of equations (\[flux\_inv\]) and (\[inten\_inv\]), we obtain $$F=\frac{I_\nu}{E^3}=\frac{I_\nu}{\nu^3}\ ,$$ where $\nu ~(= E)$ is the frequency of the photon. We will use this Lorentz invariant intensity, ${\cal I}\equiv F$, in the radiative transfer formulation. In a linear medium, extinction is proportional to the intensity, and the emission is independent of the intensity of the incoming radiation. The radiative transfer equation is therefore $$\label{classradtrans} \frac{d{\cal I}}{d s}=-\chi{\cal I} + \eta\left(\frac{\nu_0}{\nu}\right)^3 \ ,$$ where $\chi$ is the absorption coefficient, $\eta$ is the emission coefficient and $ds$ is the length element the ray traverses. The equation in this form is defined in the observer’s frame, and the absorption and emission coefficients are related to their counterparts in the rest frame with respect to the medium via $$\begin{aligned} \label{chframe} \chi&=&\left(\frac{\nu_0}{\nu}\right)\chi_0 \ , \\ \eta&=&\left(\frac{\nu}{\nu_0}\right)^2\eta_0 \ , \end{aligned}$$ where the subscript “0” denotes quantities in the local rest frame. The relative energy / frequency shift in a moving medium with respect to an observer at infinity is given by $$\label{freqshift} \frac{E_0}{E}= \frac{\nu_0}{\nu}= \frac{p^\alpha u_\alpha\vert_\lambda}{p^\alpha u_\alpha\vert_\infty}\ ,$$ where $u^\alpha$ is the four-velocity of the medium as measured by an observer, and $$\begin{aligned} \label{chframe2} \frac{ds}{d\lambda}&=&-p^\alpha u_\alpha\vert_\infty\ . \end{aligned}$$ The radiative transfer equation (equation \[\[classradtrans\]\]) in the co-moving frame is therefore $$\label{radtrans} \frac{d{\cal I}}{d\lambda} =-p^\alpha u_\alpha\vert_\lambda \left[-\chi_0(x^\beta,\nu){\cal I}+\eta_0(x^\beta, \nu)\right]$$ (see Baschek et al. 1997). The results in the co-moving frame can be used to determine the intensity and frequency in the other reference frames. The ray is specified by choosing $x^\alpha(\lambda_0)$ and $p^\alpha(\lambda_0)$. From the geodesic equation, we have $d p^\alpha/d \lambda+\Gamma^\alpha_{\beta\gamma}p^\beta p^\gamma=0$, where we have scaled $\lambda$ by $m$ for massive, and by $1$ for massless particles. The derivative of $\cal I$ is therefore $$\begin{aligned} \label{fullideriv} \frac{d{\cal I}}{d\lambda} &=& \frac{\partial{\cal I}}{\partial x^\alpha}\frac{d x^\alpha}{d \lambda} + \frac{\partial{\cal I}}{\partial p^\alpha} \frac{d p^\alpha}{d \lambda} \nonumber\ , \\ &=&p^\alpha\frac{\partial{\cal I}}{\partial x^\alpha} -\Gamma^\alpha_{\beta\gamma}p^\beta p^\gamma \frac{\partial{\cal I}}{\partial p^\alpha} \ .\end{aligned}$$ This, combined with equation (\[radtrans\]), yields $$\begin{aligned} \label{noray} & & -p^\alpha u_\alpha\vert_\lambda \left[-\chi_0(x^\beta,\nu){\cal I}+\eta_0(x^\beta, \nu)\right] \nonumber\\ & &\hspace{3cm}=p^\alpha\frac{\partial{\cal I}}{\partial x^\alpha} -\Gamma^\alpha_{\beta\gamma}p^\beta p^\gamma \frac{\partial{\cal I}}{\partial p^\alpha}\ , \end{aligned}$$ which is the same as that derived by Lindquist (1966) from the Boltzmann Equation. The metric and the initial conditions define the rays (the photon trajectories in 3D space), and the solution can be obtained by direct integration along the ray. For simplicity, we assume the refractive index $n=1$ throughout the medium. The solution to equation (\[radtrans\]) is then $$\begin{aligned} \label{anaray} {\cal I}(\lambda) &=& {\cal I}(\lambda_0) \exp\left(\int_{\lambda_0}^{\lambda} \chi_0(\lambda',\nu_0) u_\alpha p^\alpha d\lambda'\right)\\ & &\hspace{-1cm} -\int_{\lambda_0}^{\lambda} \exp\biggl(\int_{\lambda'}^{\lambda}\chi_0(\lambda'', \nu_0) u_\alpha p^\alpha d\lambda''\biggr) \eta_0(\lambda', \nu_0) u_\alpha p^\alpha d\lambda'\ . \nonumber \end{aligned}$$ In the non-relativistic limit, $u_\alpha p^\alpha = 1$, and the equation recovers the conventional form (see Rybicki & Lightman 1979). Particle Trajectories ===================== Free particles -------------- To determines the photon trajectories we need to specify the metric of the space-time. We consider the Boyer-Lindquist coordinates: $$\begin{aligned} d\tau^2 &=& \biggl( 1- {{2Mr} \over \Sigma}\biggr)dt^2 + {{4aMr \sin^2\theta} \over \Sigma}dtd\phi - {\Sigma \over \Delta}dr^2 \nonumber \\ & & \hspace{-0.25cm} - \Sigma d\theta^2 - \biggl( r^2+a^2 + {{2a^2Mr \sin^2\theta} \over \Sigma} \biggr) \sin^2\theta d\phi^2 \ , \end{aligned}$$ where $M$ is the black hole mass, $\Sigma = r^2+a^2\cos^2\theta$ and $\Delta = r^2 -2Mr +a^2$. The dimensionless parameter $a/M$ specifies the spin of the black hole, with $a/M = 0$ corresponding to a Schwarzschild (non-rotating) black hole and $a/M = 1$ to a maximally rotating Kerr black hole. The motion of a free particle is described by the Lagrangian: $$\begin{aligned} {\cal L} & = & \frac{1}{2} \biggl[ -\left(1-\frac{2Mr}{\Sigma}\right)\dot{t}^2 - \frac{4aMr\sin^2\theta}{\Sigma}\dot{t} \dot{\phi} + \frac{\Sigma}{\Delta}\dot{r}^2 \nonumber \\ && + \Sigma \dot{\theta}^2 + \left( r^2+a^2 + \frac{2a^2M r\sin^2\theta}{\Sigma} \right) \sin^2\theta \dot{\phi}^2 \biggr] \ \label{lagkerreqn}\end{aligned}$$ (here $\dot x^{\alpha} = d x^{\alpha}/ d\lambda$). The Lagrangian does not depend explicitly on the $t$ and $\phi$ coordinates. The momenta in the four coordinates are therefore $$\begin{aligned} p_{\rm t} = \partial{\cal L}/\partial \dot t &=& -E\ ,\\ \label{pr} p_{\rm r} = \partial{\cal L}/\partial \dot r &=& \frac{\Sigma}{\Delta} \dot r\ , \\ \label{ptheta} p_{\rm \theta} = \partial{\cal L}/\partial \dot \theta &=& \Sigma \dot \theta\ , \\ p_{\rm \phi} = \partial{\cal L}/\partial \dot \phi &=& L\ .\end{aligned}$$ with $E$ being the energy of the particle at infinity and $L$ the angular momentum in the $\phi$ direction. The corresponding equations of motion are $$\begin{aligned} \label{tdot} \dot t & = & E + \frac{2 r(r^2+a^2)E-2a L}{\Sigma\Delta}\ ,\\ \dot r^2 & = & {\Delta \over \Sigma} \big(H + E \dot t - L \dot \phi- \Sigma \dot \theta^2 \big)\ , \\ \dot \theta^2 & = & {1 \over \Sigma^2} \big( Q + (E^2 + H)a^2 \cos^2\theta - L^2 \cot^2 \theta \big)\ , \\ \label{phidot} \dot \phi & = & {{2a rE + (\Sigma - 2 r)L/\sin^2\theta} \over {\Sigma\Delta} } \ , \end{aligned}$$ where $Q$ is Carter’s constant (Carter 1968), and $H$ is the Hamiltonian, which equals 0 for photons and massless particles and equals $-1$ for particles with a non-zero mass. (See Reynolds et al. (1999) for more details.) For simplicity, we have set the black-hole mass equal to unity ($M =1$) in the equations above. This is equivalent to normalizing the length to the gravitational radius of the black hole (i.e., set $R_{\rm g} \equiv GM/c^2 = 1$), and we will adopt this normalization hereafter. There are square terms of $\dot r$ and $\dot \theta$ in two equations of motion. They could cause problems when determining the turning points at which $\dot r$ and $\dot \theta$ change sign in the numerical calculations. To overcome this, we consider the second derivatives of $r$ and $\theta$ instead. From the Euler-Lagrange equation, we obtain $$\begin{aligned} \ddot r & = & \frac{\Delta}{\Sigma} \bigg\{\frac{\Sigma-2r^2}{\Sigma^2}\dot t^2+\frac{(r-1) \Sigma-r\Delta}{\Delta^2}\dot r^2+ r\dot \theta^2 \nonumber \\ & & \hspace{1cm} +\sin^2\theta \left(r+\frac{\Sigma-2r^2}{\Sigma^2} a^2\sin^2\theta\right)\dot \phi^2 \nonumber \\ & & \hspace{1cm} -2a\sin^2\theta\frac{\Sigma-2r^2}{\Sigma^2}\dot t \dot \phi +\frac{2a^2\sin\theta\cos\theta}{\Delta}\dot r \dot \theta\bigg\}\ , \\ \ddot \theta & = & {1 \over \Sigma} \bigg\{ \sin\theta\cos\theta \bigg[ \frac{2a^2r}{\Sigma}\dot t^2 -\frac{4ar(r^2+a^2)}{\Sigma}\dot t \dot \phi-\frac{a^2}{\Delta}\dot r^2\nonumber \\ & & \hspace*{1cm}+a^2\dot \theta^2 +\frac{\Delta +2r(r^2+a^2)^2}{\Sigma^2}\dot \phi^2 \bigg] -2r\dot r\dot \theta \bigg\}\ .\end{aligned}$$ In terms of the momenta and the Hamiltonian, the equations above can be expressed as $$\begin{aligned} \label{rdot} \dot p_{\rm r} &=& \frac{1}{\Sigma \Delta} \left[(r-1)\left((r^2+a^2)H-\kappa\right) +r\Delta H\right.\nonumber\\ & &\hspace{0.5cm}\left.+2r(r^2+a^2)E^2-2aEL\right]-\frac{2{p_r}^2(r-1)}{\Sigma}\ ,\\ \label{thetadot} \dot p_{\rm \theta} &=& \frac{\sin\theta\cos\theta}{\Sigma} \left[\frac{L^2}{\sin^4\theta}-a^2(E^2+H)\right]\ ,\end{aligned}$$ where $\kappa = Q+L^2+a^2(E^2+H)$. Equations (\[pr\]), (\[ptheta\]), (\[tdot\]), (\[phidot\]), (\[rdot\]) and (\[thetadot\]) are the equations of motion. Motion in the presence of external forces ----------------------------------------- The equations of motion obtained in the previous section are applicable to free particles only. In a general situation external (non-gravitational) forces may be present and we need to specify the external force explicitly in deriving the equations of motion. However, in the setting of accretion disks around black holes we can often treat the effect of the external force implicitly which we will discuss in more detail in the following subsections. ### Rotational and Pressure Supported Model Here we consider a simple model such that $$\dot{t} > \dot{\phi} \gg \dot{r} \gg \dot{\theta}\ .$$ As $\dot{r}$ and $\dot{\theta}$ are small in comparison with other quantities, they can be neglected as a first approximation. The equation of motion reads $$\frac{d^2 x^\nu}{d\lambda^2} +\Gamma^{\nu}_{\alpha \beta}u^{\alpha}u^{\beta}=a^{\nu}\ ,$$ where $a^{\nu}$ is the four-acceleration due to an external force per unit mass. For axisymmetry (which is a sensible assumption for accretion onto rotating black holes), $d/d\phi=0$ and $a^{\phi}=0$. The identity $u^\alpha a_\alpha = 0$ together with $\dot r = 0$ and $\dot \theta=0$ imply that $a^{t}=0$. We may also set $a^{r}=0$ for simplicity. Because we have an extra equation from the identity $u^{\alpha}u_{\alpha}=-1$, $a^{\theta}$ can be determined self-consistently under the approximation $\dot{\theta}=0$. This scenario thus corresponds to flows supported by rotation in the $\hat{r}$ direction and by pressure in the $\hat{\theta}$ direction. Inserting the affine connection coefficients for the Kerr metric into the equation of motion yields quantities identical to zero on the left hand side of the equations for the $\hat{t}$ and $\hat{\phi}$ directions. This leaves only the non-trivial momentum equation in the radial direction: $$\begin{aligned} 0&=&-\left(\frac{\Sigma-2r^2}{\Sigma^2}\right)\dot{t}^2 +2\left(\frac{\Sigma-2r^2}{\Sigma^2}\right) a\sin^2\theta\dot{t}\dot{\phi}\nonumber\\ &&\hspace{0.5cm}- \left(r+a^2\sin^2\theta\left(\frac{\Sigma-2r^2}{\Sigma^2}\right)\right) \sin^2\theta\dot{\phi}^2\ ,\end{aligned}$$ which further simplifies to $$\frac{r\Sigma^2\sin^2\theta}{2r^2-\Sigma}\dot{\phi}^2 =\left(\dot{t}-a\sin^2\theta\dot{\phi}\right)^2 \ .$$ Here we choose the positive solution $$\label{flowtphi} \dot{t}=\left(\frac{\sqrt{r}\Sigma\sin\theta} {\sqrt{2r^2-\Sigma}}+a\sin^2\theta\right)\dot{\phi} \ ,$$ which corresponds to the same rotation as the black hole. This solution thus allows the flow to match the rotation of a prograde accretion disk. From the metric we have $$\begin{aligned} \label{mertic1} 1&=&\left(1-\frac{2r}{\Sigma}\right)\dot{t}^2 +\frac{4ar\sin^2\theta}{\Sigma}\dot{t}\dot{\phi}\nonumber\\ &&\hspace{0.5cm}-\left(r^2+a^2+\frac{2a^2r\sin^2\theta}{\Sigma}\right) \sin^2\theta\dot{\phi}^2 \ .\end{aligned}$$ Combining equations (\[flowtphi\]) and (\[mertic1\]) yields $$\Sigma\sin^2\theta\left(\frac{r(\Sigma-2r)}{2r^2-\Sigma} +\frac{2\sqrt{r}a\sin\theta} {\sqrt{2r^2-\Sigma}}-1\right)\dot{\phi}^2=1 \ .$$ It follows that the components of the four-velocity are $$\begin{aligned} \label{mediummotion} \dot{t}&=&\frac{1}{\zeta} \left(\Sigma\sqrt{r}+a\sin\theta \sqrt{2r^2-\Sigma}\right)\ , \nonumber\\ \dot{r}&=&0\ , \nonumber\\ \dot{\theta}&=&0\ ,\nonumber\\ \dot{\phi}&=&\frac{\sqrt{2r^2-\Sigma}}{\zeta\sin\theta} \ ,\end{aligned}$$ where $$\label{zeta} \zeta=\sqrt{\Sigma\left(\Sigma(r+1) -4r^2+2a\sin\theta\sqrt{r(2r^2-\Sigma)}\right)} \ .$$ The marginally stable orbit for particles is defined by the surface where $$\label{margin_stable} \frac{\partial E}{\partial r} = 0 \ .$$ From equations (\[tdot\]), (\[phidot\]) and (\[mediummotion\]), we have $$E = \frac{1}{\zeta}\left((\Sigma-2r) \sqrt{r}+a\sin\theta\sqrt{2r^2-\Sigma}\right) \ .$$ After differentiation, we remove the non-zero factors in the expression and obtain the condition $$\label{marg_stable} \Delta\Sigma^2-4r(2r^2-\Sigma) \left(\sqrt{2r^2-\Sigma}-a\sin\theta\sqrt{r}\right)^2=0\ .$$ Setting $a=0$ gives $r=6$, which is often regarded as the limit for the inner boundary of an accretion disk around a Schwarzschild black hole. This value is the same as that derived by Bardeen, Press & Teukolsky (1972) using $\partial^2 p_{\rm r}^2 / \partial r^2 = 0$. Before we proceed further, we must note that the expressions for the velocity components in equations (\[mediummotion\]) hold only for regions “sufficiently” far from the black-hole event horizon. The approximation that we adopt in the model breaks down when the square root in the denominator approaches zero. This occurs at the light circularisation radius $r_{\rm cir}$, which is given by $\zeta = 0$, or equivalently $$\label{light_circ} \Sigma(r+1)-4r^2+2a\sin\theta\sqrt{r(2r^2-\Sigma)}\ \bigg|_{r = r_{\rm cir}} = 0 \ .$$ Moreover, the assumption of $\dot{\theta}=0$ is also invalid for radii smaller than the radius of the marginally stable orbit — the flow is neither rotational nor pressure supported and it follows a geodesic into the event horizon. ### Isobaric Surfaces In a stationary accretion flow, the acceleration must be balanced by some forces, e.g. the gradient of gas or radiation pressure. As the local acceleration $a^\alpha$ can be calculated from the rotation law $\omega(r,\theta)$ we can derive a set of isobaric surfaces when a rotation law is given. For a barotropic equation of state of the accreting matter the isobaric surfaces coincide with the isopicnic (constant-density) surfaces. The accelerations in the ${\hat r}$ and ${\hat \theta}$ directions are $$\begin{aligned} -\frac{\Sigma}{\Delta}a^r&=&\frac{\Sigma-2r^2}{\Sigma^2} \left(\dot{t}-a\sin\theta\dot{\phi}\right)^2+r \sin^2\theta \dot{\phi}^2\ , \\ -\Sigma a^\theta&=&\sin\theta\cos\theta \left[\frac{2r}{\Sigma^2}\left(a\dot{t}-(r^2+a^2)\dot{\phi}\right)^2 +\Delta\dot{\phi}^2\right]\ .\end{aligned}$$ The surface of constant acceleration is given by $$\label{tsurface} a_\alpha \frac{dx_{\rm surf}^\alpha}{d\lambda} = 0$$ (here and hereafter $dx_{\rm surf}^{\alpha}/{d\lambda} \equiv dx^{\alpha}/{d\lambda}|_{x_{\rm surf}}$). The stationary condition implies $d t/{d\lambda}= 0$, and axisymmetry implies $d \phi/{d\lambda}=0$. Without losing generality, we can choose $t = \phi = 0$ on the surface. Thus, equation (\[tsurface\]) becomes $$\begin{aligned} \label{tsurface2} 0 &=& \frac{\Sigma a^r}{\Delta} \frac{d r_{\rm surf}}{d \lambda} + \Sigma a^\theta \frac{d \theta_{\rm surf}}{d \lambda}\ ,\nonumber\\ &=& \beta_1 \frac{d r_{\rm surf}}{d \lambda} + \beta_2 \frac{d \theta_{\rm surf}}{d \lambda}\ ,\end{aligned}$$ where $$\begin{aligned} \beta_1&=&\frac{\Sigma-2r^2}{\Sigma^2} \left(\frac{1}{\omega}-a\sin\theta\right)^2+r\sin^2\theta \ , \nonumber\\ \beta_2&=&\sin\theta\cos\theta \left[\Delta+\frac{2r}{\Sigma^2} \left(\frac{a}{\omega}-(r^2+a^2)\right)^2\right]\ , \end{aligned}$$ and $d r_{\rm surf}/d \lambda$ and $d \theta_{\rm surf}/d \lambda$ determine the intersection of the isobaric surfaces and the $(r,\theta)$ plane. By rescaling equation (\[tsurface2\]) with a factor of $\sqrt{\Delta/\Sigma}$ and making use of the invariance $$- \left(\frac{d\tau}{d\lambda}\right)^2 = \frac{\Sigma}{\Delta}{\dot r}^2 + \Sigma {\dot \theta}^2 \ ,$$ we obtain $$\begin{aligned} \frac{d r_{\rm surf}}{d \lambda'} &=& \frac{\beta_1}{\sqrt{\beta_2^2+\Delta\beta_1^2}}\ , \nonumber \\ \frac{d \theta_{\rm surf}}{d \lambda'} &=& \frac{-\beta_2}{\sqrt{\beta_2^2+\Delta\beta_1^2}}\ . \label{isobaric} \end{aligned}$$ These two differential equations can be solved numerically and yield the isobaric surface as a parametric function of $\lambda'$. Model Accretion Disks and Tori ============================== We now demonstrate using the equations of motion above to construct the emitter models. The first is a geometrically thin accretion disk, in which the emitting particles are in Keplerian motion. The second is a torus, a 3-dimensional object with non-negligible thickness. Accretion Disk -------------- When space-time curvature is important, the Keplerian angular velocity of a test particle around a gravitating object is no longer $\omega_k= r^{-3/2}$, the expression in flat space-time. Instead, the Keplerian angular velocity in a plane containing the gravitating object can be obtained by setting $\theta=\pi/2$ in equations (\[mediummotion\]) and (\[zeta\]). Hence, the components of the four-velocity of the particles in the disk are $$\begin{aligned} \label{diskvel} \dot{t}&=&\frac{r^2+a\sqrt{r}}{r\sqrt{r^2-3r+2a\sqrt{r}}}\ , \nonumber \\ \dot{r}&=&0 \ ,\nonumber \\ \dot{\theta}&=&0 \ ,\nonumber \\ \dot{\phi}&=&\frac{1}{\sqrt{r}\sqrt{r^2-3r+2a\sqrt{r}}}\ ,\end{aligned}$$ and the rotational velocity of a Keplerian accretion disk around a black hole is $$\omega_k=\frac{1}{r^{3/2}+a},$$ (Bardeen, Press & Teukolsky 1972). The relative energy shift of the emission between the disk particle and an observer at a large distance is determined by equation (\[freqshift\]), with $u^\alpha$ as given in equation (\[diskvel\]). Keplerian disk images can be found in many existing works (e.g.  Bromley, Miller & Pariev 1998), and we do not show disk images here. The general characteristics are that a disk image is asymmetric, with the separatrix for the energy shift of the emission no longer bisecting the disk image into two equal sectors, one for red shift and another for blue shift. The whole disk appears to be reddened, especially at the inner rim. Accretion Torus --------------- To determine the geometry and structure of an accretion torus self-consistently is beyond the scope of this paper. Here, we consider a simple parametric model, with an angular velocity profile given by $$\omega=\frac{1}{(r\sin\theta)^{3/2}+a} \left(\frac{r_{\rm k}}{r\sin\theta}\right)^n \ . \label{rk-equation}$$ The quantity $r_{\rm k}$ is the radius (on the equatorial plane) at which the material moves with a Keplerian velocity. The parameter $n$ adjusts the force term, such as a pressure gradient, to keep the disk particles in their orbits, and it determines the thickness of the torus. In this study we just take $n=0.21$ without losing generality. If the torus is supported by radiation pressure, its inner edge is determined by the intersection of the isobaric surface with either one of two surfaces. These two surfaces provide the constraints, inside which the pressure-supported solution does not hold. The first is a surface defined by the orbits of marginal stability. For $\omega(r,\theta)$, it is given by $$\begin{aligned} 0 &=& 2a\sin^4\theta\left[\frac{r^2}{\Sigma} -\left(r^2+a^2+\frac{a^2r\sin^2\theta}{\Sigma} \right) \frac{\Sigma-2r^2}{\Sigma^2} \right]\omega^3 \nonumber\\ && +\sin^2\theta \bigg[\left(\frac{6r(r^2+a^2)}{\Sigma}+3\Delta-\Sigma\right) \frac{\Sigma-2r^2}{\Sigma^2} \nonumber \\ && \hspace*{0.5cm} +r\left(1-\frac{2r}{\Sigma}\right)\bigg]\omega^2 -\frac{6ar\sin^2\theta}{\Sigma} \left(\frac{\Sigma-2r^2}{\Sigma^2}\right)\omega \nonumber\\ && +\Delta\sin^2\theta ~\omega\frac{\partial\omega}{\partial r} -\left(1-\frac{2r}{\Sigma}\right)\frac{\Sigma-2r^2}{\Sigma^2} \ . \end{aligned}$$ The second is the limiting surface where the linear velocity approaches the speed of light. It is given by $$\begin{aligned} 0 &=& \Sigma-2r+4ar\omega\sin^2\theta \nonumber\\ & &\hspace{1cm} -\left((r^2+a^2)\Sigma+2a^2r\sin^2\theta \right)\omega^2\sin^2\theta \ . \end{aligned}$$ Usually the former is larger than the latter. The outermost of these two surfaces determines the inner boundary and hence the critical surface of the torus. Figure \[various\_surface\] shows the critical density surfaces of two tori. The first torus is around a Schwarzschild black hole and the second torus is around a maximally rotating black hole. The tori are constructed such that their specific angular momentum has a profile similar to those of the simulated accretion disks in Fig. 3. of Hawley & Balbus (2002). In our model the boundary surface of the torus is determined by a single parameter, $n$, which specifies the index of the angular-velocity power law. Its value is selected such that the angular-velocity profile matches the profiles obtained by the numerical simulations — here we consider that of Hawley & Balbus (2002). Model tori can be constructed using various different methods. An example is that in a study of dynamical stability of tori around a Schwarzschild black hole carried out by Kojima (1986), the model parametrizes the angular momentum instead of the angular velocity. We note that the aspect ratios of the torus surfaces obtained by Kojima (1986) and those shown in Fig. \[various\_surface\]. are similar. Extinction ========== The generic setting of the system under our investigation is that emitters with various strengths are distributed in space in a curved space-time, and the radiation is attenuated, and may be re-emitted, when propagating. The emitters and the line-of-sight material are in relativistic motion with respect to the observer and also with respect to each other. An example is that shown in Fig. \[cloud\_model\]., in which the emitters are the surface elements of an accretion disk and the absorbers are some clouds in the vicinity of the disk. The photon trajectories and the motion of the emitters and absorbers are affected by the space-time distorted by the central black hole. To construct the model we need to determine - the rays that connect the emitters, absorbers and observer, - the four-velocities of the emitters and absorbers /scatterers, - the spatial distributions of the emitters and the absorbers/scatterers, and - the effective cross section of the absorbers/scatterers. In the previous section, we have shown how to obtain (i) and (ii); in this section we incorporate (iii) and (iv) into the radiative-transfer calculations. An illustrative model --------------------- We consider a model with the geometry shown in Fig. \[cloud\_model\]. The photons are emitted from the elements on the top and bottom surfaces of a geometrically thin disk in a Keplerian rotation around a Kerr black hole. The radiation is resonantly scattered (absorbed) by plasma clouds and is attenuated in its propagation. The size of the clouds is small in comparison with the length scale of the system. They are not confined to be in the equatorial plane and are in orbital motion, supported by some implicit forces (which may be radiation, kinematic or magnetic pressure gradients). These clouds have a large (thermal) distribution of velocities, in addition to their collective bulk velocity. We assume that the radiation scattered into the energies of the lines is insignificant and ignore the photons that are scattered into the line-of-sight. Under this approximation, scattering simply removes the line photons and causes extinction similar to true absorption. Thus, for simplicity, hereafter we do not distinguish between scattering and absorption, [^1] and the two terms are interchangeable, unless otherwise stated explicitly. The rays originating from the accretion-disk surface that can reach the observer are determined by the 4-momenta of the photons, which are calculated using equations (\[tdot\]), (\[phidot\]) and (\[rdot\]). The 4-velocities of the emitting surface elements on the accretion disk are given by equations (\[diskvel\]). These determine the relative energy shifts of the photons between the emitters and the observer. What we need next is to determine the relative energy shifts between the emitters and the absorbers. Then, we need a model mechanism by which the absorption takes place, and to derive the resonant absorption condition for the absorption coefficient. Now we construct a model for the spatial distribution and the velocities of the absorbers. Consider a parametric model in which the bulk 4-velocities of the clouds are given by equations (\[mediummotion\]) and (\[zeta\]). In this model the bulk velocities of the clouds in the equatorial plane matches the 4-velocities of the accretion disk. The clouds themselves are cold, and the thermal velocity of the gas particles inside are much smaller than their bulk motion and root-mean-square velocity dispersion. However, the clouds have a large velocity dispersion, given by the local virial temperature, which is comparable to the energy of the emission lines of interest. Therefore, the clouds can be considered as relativistic particles in the calculation. Using the bulk-motion velocities obtained by (\[mediummotion\]) and (\[zeta\]) together with the virial theorem, we can derive this temperature and determine the velocity distribution of the absorbing clouds. The clouds fill most of space, with a radially dependent number density. However, close to the black hole, the assumptions above break down, and the axial force cannot support the clouds out of the equatorial plane. When this happens, they will flow along geodesics directly into the hole. In the numerical calculation, we determine the asymptotic boundaries at which the left hand sides of equations (\[light\_circ\]) and (\[marg\_stable\]) vanish. This is done by evaluating these expressions and testing to see if they are negative at each point along the photon rays. Inside that surface, the number density of the clouds will be much less than that outside, which we approximate by setting it to zero in this zone. The absorption coefficient -------------------------- We assume that the absorption is due to “cold” cloudlets with high virial velocities. The absorption coefficient of individual cloudlets is $$\begin{aligned} \chi_{\rm i} & \propto & \sigma ~ \delta\left(\frac{u^\alpha k_\alpha+{E_{\rm line}}}{{E_\gamma}}\right) \end{aligned}$$ (with $\sigma$ as the effective absorption cross section of the cloudlet, and $k_\alpha$ the photon four-momentum). The absorption rest frequency is ${E_{\rm line}}$, and ${E_\gamma}$ is the energy of the photon in the bulk rest frame. The total effective absorption coefficient $\chi_0$ is the sum of the contribution of these cloudlets, i.e., $$\begin{aligned} \chi_0 & = & \sum_{\rm i} ~\chi_{\rm i} \ . \end{aligned}$$ Converting the sum into an integral in momentum space yields the absorption per unit length in the rest frame as $$\begin{aligned} \label{chieqn} \chi_0&=&\frac{-2\pi\lambda\sigma}{{E_{\rm line}}^2}\times \nonumber\\ &&\!\!\!\int\!\!\!\int\!\! p^2dpd\mu \exp(-E/\Theta) u^\alpha k_\alpha \delta\left(\frac{u^\alpha k_\alpha + {E_{\rm line}}}{{E_\gamma}}\right),\end{aligned}$$ where we have defined $\mu = \cos\theta$, $\lambda$ is a normalization constant, $\Theta$ is the temperature in relativistic units (with $k_B=1$), and $E$ and $p$ are the energy and momentum of a gas particle in the bulk rest frame. This form assumes isotropic thermal motion in the rest frame. There are three terms that we need to determine before we can evaluate the integral, and they are the normalisation parameter $\lambda$, the temperature $\Theta$ and the photon energy in the rest frame of the absorbing particle $u^\alpha k_\alpha$. When these variables are determined, we can then parametrise $\sigma$ after integration is carried out. ### The normalisation parameter $\lambda$ We now derive $\lambda$ starting from $$N=4\pi\lambda\int^\infty_0 p^2dp \exp\left(-E/\Theta\right)\ ,$$ where $N$ is the number density of absorbing clouds. Integrating yields $$\lambda=\frac{N\frac{m}{\Theta}}{4\pi m^3K_2(\frac{m}{\Theta})}\ ,$$ where $K_\nu(x)$ is a modified Bessel function, and $m$ is the average cloud mass. This is called the Jüttner distribution and corresponds to Maxwell’s distribution except in the case of a relativistically high temperature. ### The temperature $\Theta$ The total energy in the distribution of clouds is given by $$E_{\rm tot}=4\pi\lambda\int^\infty_0 p^2dp E \exp\left(-E/\Theta\right)\ .$$ Integrating this yields the energy per unit mass as $$\label{Etot2} \frac{E_{\rm tot}}{Nm}=\frac{K_3(\frac{m}{\Theta})} {K_2(\frac{m}{\Theta})}-\frac{\Theta}{m}\ .$$ Using conservation of energy and angular momentum, we calculate the thermal energy of the virialised relativistic gas of absorbing clouds. At infinity the medium has $$\begin{aligned} E_{\rm init}&=&Nm \ , \\ L_{\rm init}&=&L_{\rm fin} \ .\end{aligned}$$ Close to the black hole it has $$\begin{aligned} E_{\rm fin} &=&\frac{Nm}{\zeta}\left[(\Sigma-2r)\sqrt{r} +a\sin\theta\sqrt{2r^2-\Sigma}\right]\ , \\ L_{\rm fin}&=&\frac{Nm}{\zeta}\left[2ar\sqrt{r}\sin^2\theta -(r^2+a^2)\sin\theta\sqrt{2r^2-\Sigma}\right]\end{aligned}$$ The energy released by the gas falling from infinity and slamming into a wall moving with a velocity given by equation (\[mediummotion\]) is $$-E_{\rm tot}=u^\alpha p_\alpha =-E_{\rm fin}\dot{t}_{\rm init}-L_{\rm fin}\dot{\phi}_{\rm init}\ .$$ After simplification, this becomes $$\begin{aligned} \label{Etot1} -\frac{E_{\rm tot}}{Nm}\!&=&\!\frac{1}{\zeta^2} \left[(2r^2-\Sigma)(r^2+a^2) -2ar\sin\theta\sqrt{r}\sqrt{2r^2-\Sigma}\right. \nonumber\\ &&\hspace{0.5cm}\left.-\left(a\sin\theta\sqrt{2r^2-\Sigma} +\Sigma\sqrt{r}\right)\zeta\right]\ .\end{aligned}$$ Thus the temperature of the media can be derived using equations (\[Etot2\]) and (\[Etot1\]). Unfortunately, this yields an implicit relation of $m/\Theta$ that contains transcendental functions. The modified Bessel functions can be expanded in the limit where $\Theta \ll m$ which corresponds to an “almost relativistic” gas. Since the potential energy released in accretion is of the order of a few percent of the rest mass of the infalling material, this approximation should hold in AGN. Expanding to second order in $\Theta / m$, cancelling the exponential factors, and then solving the resulting quadratic yields $$\frac{\Theta}{m} = \frac{2}{5}\left(-1+\sqrt{1+\frac{10}{3} \left(\frac{E_{\rm tot}}{Nm}-1\right)}\right)\ .$$ Thus we have an explicit description of how the kinematic temperature varies with position. ### The photon energy $u^\alpha k_\alpha$ In the rest frame, the motion of the thermalised medium is isotropic. Thus we can simplify the problem by aligning an axis along the photon propagation vector and working in a local Lorentz frame so that $$k_\alpha = {E_\gamma}(-1, 1, 0, 0)\ ,$$ and $$p^\alpha = m u^\alpha =(E, p \mu, p_{\rm y}, p_{\rm z})\ .$$ Thus, $$u^\alpha k_\alpha = \frac{{E_\gamma}}{m}(p\mu - E)\ .$$ ### Evaluation of the $\delta-$function Using the relation that $$\frac{d}{d\mu}\ \left[\frac{u^\alpha k_\alpha + {E_{\rm line}}}{{E_\gamma}}\right] = -\frac{p}{m} \ ,$$ we obtain $$\begin{aligned} \int d\mu \ u^\alpha k_\alpha \delta\left(\frac{u^\alpha k_\alpha + {E_{\rm line}}}{{E_\gamma}}\right) & & \nonumber \\ & & \hspace{-2cm} = {\frac{u^\alpha k_\alpha}{\big\vert \frac{d}{d\mu}\left[\frac{u^\alpha k_\alpha + {E_{\rm line}}}{{E_\gamma}}\right]\big\vert}}\bigg\vert_{u^\alpha k_\alpha =- {E_{\rm line}}} \nonumber \\ & & \hspace{-2cm} = -{E_{\rm line}}\frac{m}{p}\bigg\vert_{u^\alpha k_\alpha =- {E_{\rm line}}} \ . \end{aligned}$$ We change the variable $p dp$ to $EdE$. The integral in equation (\[chieqn\]) can now be simplified to $$\chi_0=\frac{2\pi\lambda \sigma m}{{E_{\rm line}}}\int^\infty_{\frac{m}{2}\left(\frac{{E_\gamma}}{{E_{\rm line}}}+\frac{{E_{\rm line}}}{{E_\gamma}}\right)} EdE \exp(-E/\Theta)$$ Integrating this gives the absorption coefficient in the rest frame of the gas $$\begin{aligned} \label{abschi0} \chi_0&=&\frac{N\sigma}{2 K_2(\frac{m}{\Theta})}\left[\frac{1}{2}\left(\frac{{E_\gamma}}{{E_{\rm line}}}+\frac{{E_{\rm line}}}{{E_\gamma}}\right)+\frac{\Theta}{m}\right]\times\nonumber\\ &&\hspace{1.5cm}\exp\left[-\left(\frac{m}{2\Theta}\right)\left(\frac{{E_\gamma}}{{E_{\rm line}}}+\frac{{E_{\rm line}}}{{E_\gamma}}\right)\right]\end{aligned}$$ This equation can be recast in terms of $\chi$ by using equations (\[chframe\]) and (\[freqshift\]). ### The effective absorption $N\sigma$ The absorption coefficient depends upon $N$, the number density of the clouds, and on $\sigma$, the absorption cross-section per cloud. These are in general a function of position. Since the pressure gradient and the inflow velocity in the $\hat{r}$ and $\hat{\theta}$ directions are ignored in our approximation of the flow, the density profile cannot be determined in a fully self-consistent manner via the mass continuity equation. To overcome this, we assume a simple two-parameter profile $$N\sigma=\sigma_0 r^{-\beta}\ .$$ Where $\sigma_0$ can be considered as a proportionality constant fixing the density and opacity scales. Without losing generality, we adopt a value $\beta = 3/2$. The results can then easily be generalized to other values of $\beta$. For the case of a swarm of absorbing clouds around a black hole, the optical depth is approximately one. This corresponds to the case where $\sigma_0$ is of order $0.5$ or greater, with the integration proceeding radially to the event horizon. The effective optical depth depends greatly on the paths the photons take. Spectral Calculations ===================== No absorption ------------- ### Accretion disks We calculate the observed energy of the flux from a point on the planar disk using equation (\[freqshift\]). We ignore absorption. The line emissivity has a power law profile, which decreases radially from disk centre. The intensity is proportional to the third power of the relative shift; the flux has an extra factor of $\nu_0/\nu$ due to time dilation, i.e. it is proportional to the fourth power of the relative frequency shift. (Note that $F$ in §2 is the distribution function, not the flux of the emission.) Our calculation reproduces the line profiles of direct images of accretion disks as those shown in Fabian et al (1989), Kojima (1991), Fanton et al (1997), Bromley et al (1997), and Reynolds et al (1999). Figure \[fantonfig\] shows two example line spectra calculated using the method described above. The spectra contain only emission from the direct image. We also show line obtained using the method by Fanton et al (1997) for comparison. The results in the two ca lculations are in excellent argeement. We also carry out spectral line calculations which include contribution from higher-order disk images. (Here and hereafter we assume that the emissivity powerlaw has an index of $-$2, except where otherwise stated explicitly. ) Our calculations show that the contribution of the higher-order images are significant only at high inclination angles (see Fig. \[multi\_orders\]). The emission from high-order images is mostly at frequencies close to the rest frequency of the line, because the region where highly red and blue-shifted emission originates is obscured. ### Rotational Torus We now investigate the emission from an accretion torus. We consider a model in which the inner radius of the torus is determined by the marginally stable orbits of the particles. These marginally stable orbits, which depend on $\omega$ and $d \omega/dr$, form a surface in a three-dimensional space. The marginally stable orbit for particles in Keplerian motion in the equatorial plane is 6 $R_{\rm g}$ around a Schwarzschild black hole and is 1.23 $R_{\rm g}$ around a Kerr black hole with $a = 0.998$. We use a surface-finding algorithm to determine the boundary of the torus (see Appendix \[surface\_finding\]). In Fig. \[torus\_image2\]. we show three-dimensional images of the model torus around a Kerr black hole. We include the first four image orders. The inclusion of high order images is mandatory due to the ‘mixing’ caused by the extension out of the equatorial plane (Viergutz 1993). The torus is viewing inclination angles of $45^\circ$ and $85^\circ$ (top and bottom panels respectively). The left-right asymmetry is caused by inertial-frame dragging. The multiple images are consequences of gravitational lensing. At small inclination angles, only the surface above the equatorial plane of the torus is seen in the direct image. At very large inclination angles, the surface below the equatorial plane is severely lensed and also becomes visible. The false-colour map laid on the torus surface show the energy shift of the emitted photons (determined by equation \[\[freqshift\]\]), as viewed by a distant observer. The separatrix, which corresponds to zero energy shift, divides the torus surface into regions of blue energy shift and regions of red energy shift. At large inclination angles the inner surface of the near side of the torus is not visible, and the inner surface of the far side is obscured by the near side of the torus. Thus, emission with the largest energy shifts is hidden. This is very different to the situation for a planar accretion disk – regardless of the viewing inclination and the visual distortion, the emission from the innermost part of the disk is always visible. Figure \[torus\_line1\]. shows the resulting line profiles obtained by integrating the emission over the images shown in Fig. \[torus\_image2\]. The line profile of a torus viewed at $45^\circ$ is similar to that of the planar accretion disk. It has a sharp blue peak and smaller red peak. It also has an extended red wing. This is due to the fact that the projections of a torus and a disk on the sky plane are very similar at low inclination angles. However, our calculations show that geometric effects are very important for large viewing inclination angles. When some part of the emission region is self-obscured. the resulting profile, as observed from infinity, to be completely different from that of the flat disk (see Fig. \[torus\_line2\].). For a torus with large viewing inclination angles, the inner surface of the torus tends to be obscured. This corresponds to the region where the most redshifted flux of the line is emitted (due to large transverse red shift and gravitational red shift). This makes the red wing less prominent. The outer surface of the torus is visible from all inclinations. As a result the line profile tends to be singly peaked, with the maximum at the unshifted line frequency due to the emission from the outer surface dominating. By altering the geometry of the emitter a wide variety of emission profiles can be obtained. Resonant Scattering ------------------- We use a disk model to illustrate the resonant scattering effects. We assume the inner edge of the accretion disk is given by the marginally stable orbit. We use an outer disk radius of $20 R_{\rm g}$ in all the disk simulations. This was chosen to accentuate the relativistic effects. The emission line profile is assumed to be a delta function. We collate the light from the first four image orders of the accretion disk. The line profiles are obtained from $750\times750$ pixel images. The intensity scale on the graphs is in arbitrary units. (It is just the log of the sum of pixel intensity over the image, as a function of frequency.) The x-axis of the graph is in units of the line rest energy, with $E/E_0 = 1$ corresponding to the unshifted line. We bin the injection spectrum and the absorption coefficient linearly with energy. There are 1000 bins from $E=0$ to $E=2E_0$. We investigated two space-time models. One with $a=0$ corresponding to a Schwarzschild black hole, and one with $a=0.998$, corresponding to a maximally spinning Kerr black hole. We have plotted spectra containing the continuum and the continuum plus line as we have varied the opacity of the absorbing clouds. We have also shown how the spectra change with inclination due to geometric effects. We have included a nearly edge-on model ($i=85^\circ$), and a model with a moderate inclination of $45^\circ$. See Fig. \[contnoline\]. for the results of absorbing the power law continuum, and Fig. \[contline\]. for the results of absorbing both the line and continuum. Discussion ========== We have modelled line profiles from accretion disks to demonstrate the use of a general formulation for transfer of radiation through relativistic media in arbitrary space-times. In this paper, we used the transfer of emission from AGN as an illustration. In this model, we parametrized the disk/torus to describe the emitters and the space-density distribution of the absorbers. We took into account relativistic effects on the bulk dynamics and the microscopic kinematic properties of the absorbing medium. Resonant absorption/scattering of line emission from accreting black holes in the general relativistic framework had been investigated previously by Ruszkowski & Fabian (2000). In their study, a thin Keplerian disk was assumed and the absorbing medium is a spherical corona of constant density centred on the black hole. The corona is rotating, with local rates obtained by linear interpolation from the rotation rate of a planar Keplerian accretion disk and the rotation rate at the polar region caused by frame dragging due to the Kerr black hole. The Sobolov approximation was used in the resonant absorption calculations, and a Monte Carlo method determined the re-emission/scattering. Our calculation is different to that of Ruszkowski & Fabian (2000) in the following ways. Firstly, the emitters are not confined to the equatorial plane, i.e. they can be thin accretion disks or thick tori. Secondly, the absorbing medium is a collection of (cold) clouds with relativistic motions. The number density distribution of the clouds is parametrized by a powerlaw decreasing radially. The local bulk (rotational) velocity of the clouds is determined by general relativistic dynamics, and the velocity dispersion is calculated from the Virial theorem. Thirdly, we do not assume the Sobolov approximation. The resonant condition for the absorption coefficient is derived directly from the kinematics of the absorbing cloud particles. Fourthly, we ignore the contribution from re-emission to the line flux. However, we include emission from higher order disk images, in addition to the direct image. One of the main differences between the two studies is the treatment of resonant absorption. In the Sobolov approximation, the absorption takes place locally (see Rybicki and Lightman 1979). The line profile is practically a delta function; otherwise, the assumption of quasi-local absorption breaks down. Moreover, it requires that the absorbing medium is a radial flow. The emission lines are because of relativistic effects, and the motion of the medium is rotationally dominated. The locality of the absorption (required by the Sobolov approximation) therefore breaks down. To overcome these difficulties, we abandon the Sobolov approximation but, instead, employ full ray tracing. In modelling the bulk flow, we derived the equation of motion of the absorbers using the rotational-support approximation for the accretion disk. In addition, we assume that the flow is supported out of the equatorial plane. There is negligible radial force in the equations of motion, and the radial velocity can be neglected. To include outflowing as a wind, or inflowing, requires additional components in the equation of motion. This complicates the formulation, in particular, when matching the flow boundary condition at the surfaces of the accretion disk/torus. A self-consistent boundary condition requires a dissipation mechanism in the boundary regions of the disk/torus and absorbing clouds. The inclusion of such dissipation is beyond the scope of this paper and this issue will be addressed in future works. As the rays propagate from the accretion disk to the observer, they experience position-dependent absorption. The absorption depends upon the velocity profile of the material as well as its density and the line profile function of the absorption coefficient. Since the absorbers are moving in relativistic speeds, the bulk velocity is an important factor. Lorentz contraction increases the absorption coefficient accordingly, and Doppler shift alters the frequency of the emission as seen by the absorbers. The potential energy liberated by material infalling into a black hole is of order a few percent of the material’s rest mass, and the energy corresponding to ‘thermal’ kinematic velocity dispersion approaches this rest mass energy. Thus, we replace the conventional Maxwellian distribution by the Jüttner distribution for relativistic particles in deriving the resonant line absorption coefficient. This distribution does not give a Gaussian absorption profile even in the bulk rest frame of the absorbers. The line profiles of the direct images show a dip around the line rest frequency (energy) $(E/E_0 =1)$. Absorption can change the line profile significantly. The flux around the rest frequencies is, however, augmented by the flux from shifted lines from the higher order images, especially at high disk inclination. These two effects compete, and when the line-of-sight optical depth is high, the contribution of the high-order images is masked. Conclusion ========== We present a numerical ray-tracing method for radiative-transfer calculations in curved space time and apply the method to calculate line emission from accretion disks and tori around black holes. Our calculations have shown that lines from relativistic accretion tori have profiles very different to lines from relativistic thin planar accretion disks for the same system parameters, such as the spin of the black hole and the viewing inclination. The self-obscuration of the inner region of the accretion torus leads to weaker red wing in the emission lines when compared with the lines emitted from a thin planar accretion disk. At high inclination angles the strong blue peak is also absent in the line emission from the tori. We also investigate the effects of resonant absorption/scattering by the line-of-sight material in relativistic motion with respect to the emitters in the disk/torus, and the observer. Our method does not invoke the Sobolov approximation, and the resonant absorption/scattering condition is derived directly. We have shown that absorption effects are important in shaping the profiles of emission lines. The interpretation of observations of relativistic lines from AGN is non-trivial when absorption is present. We thank Mat Page for discussion, and Alex Blustin for comments on the manuscript. SVF acknowledges the support by a UK government Overseas Research Students Award and a UCL Graduate School Scholarship. Bao, G. 1992, A&A, 257, 594 Bao, G., Hadrava, P., & Ostgaard, E. 1994, ApJ, 435, 55 Bardeen, J. M., Press, W. H., & Teukolsky, S. 1972, ApJ, 178, 347 Baschek, B., Efimov, G. V., Waldenfels, W. von, & , R. 1997, A&A, 317, 630 Bromley, B. C., Chen, K., & Miller, W. A. 1997, ApJ, 475, 57 Cadez, A., Fanton, C., & Calvani, M. 1998, New Astronomy, 3, 647 Cadez, A., Brajnik, M., Comboc, A., Calvani, M., & Fanton, C. 2003, A&A, 403, 29 Carter, B. 1968, Phys. Rev., 174, 1559 Cunningham, C. T. 1975, ApJ, 202, 788 Cunningham, C. T. 1976, ApJ, 208, 534 Dabrowski, Y., Fabian, A. C., Iwasawa, K., Lasenby, A. N., & Reynolds, C. S. 1997, MNRAS, 288, L11 Fabian, A. C., Rees, M. J., Stella, L., & White, N. E. 1989, MNRAS, 238, 729 Fabian, A. C., Iwasawa, K., Reynolds, C. S., & Young, A. J. 2000, PASP, 112, 1145 Fanton, C., Calvani, M., De Felice, F., & Cadez, A. 1997, PASJ, 49, 159 Gerbal, D., & Pelat, D. 1981, A&A, 95, 18 Hawley, J. F., & Balbus, S. A. 2002, ApJ, 573, 738 Kojima, Y. 1986, Prog. Theor. Phys, 75, 1464 Kojima, Y. 1991, MNRAS, 250, 629 Laor, A. 1991, ApJ, 376, 90 Lindquist, R. 1966, Annals of Physics, 37, 487 Matt, G., Perola, G. C., & Stella, L. 1993, A&A, 267, 643 Mihalas, D. 1980, ApJ, 237, 574 Misner, C. W., Thorne, K. S., & Wheeler, J. A. 1970, Gravitation, W. H. Freeman, San Francisco Nobili, L., Turolla, R., & Zampieri, L. 1991, ApJ, 383, 250 Pariev, V. I., & Bromley, B. C., 1998, ApJ, 508, 590 Park, M., 1990, ApJ, 354, 64 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical Recipes in Fortran, 2nd Ed., Cambridge University Press, Cambridge Reynolds, C. S., Young, A. J., Begelman, M. C., & Fabian, A. C., 1999, ApJ, 514, 164 Ruszkowski, M., & Fabian, A. C. 2000, MNRAS, 315, 223 Rybicki, G. B., & Lightman, A. P. 1979, Radiative Processes in Astrophysics, Wiley, New York Smak, J. 1969, Acta Astronomica, 19, 155 Stella, L. 1990, Nature, 344, 747 Tanaka, Y., et al. 1995, Nature, 375, 659 Viergutz, S. U. 1993, A&A, 272, 355 Zane, S., Turolla, R., Nobili, L., & Erna, M. 1996, ApJ, 466, 871 Frequency Bins ============== Using an astrophysical model together with the metric, we can obtain an equation describing the flow of the medium through which the ray is propagating; $u^\alpha(x^\alpha(\lambda))$. The properties and distribution of the medium can be used to derive the absorption and emission terms $\chi_0(x^\alpha(\lambda))$ and $\eta_0(x^\alpha(\lambda))$. These can then be used to calculate what is observed from infinity via equation (\[anaray\]). In practice this simple method is altered with a ray-tracing numerical algorithm. The radiative transfer equation is not time-symmetric, so if one follows rays back from the observer to the emitter the above formulation cannot be used. The emission frequency and intensity is not known until the emitter is reached along the path. This problem can be solved using one of two approaches. The first is to collate the path from emitter to observer, and then to integrate along that path forward in time using equation (\[anaray\]). The second approach is to collate the optical depth for $\cal I$ for a table of frequencies as one travels back in time. This optical depth can then be used to calculate the observed intensity, $I$, for each binned frequency. The results for each emission element along the path are simply summed. Since, in general, the resulting frequencies are binned anyway to get a spectrum, we choose the second method. It requires less storage, and is a simple extension of the grey-absorber case where $\chi_0$ and $\eta_0$ are no longer functions of $\nu$. In our treatment, we sum over the observed frequencies rather than those in the rest frame, avoiding the need to transform the distribution of the frequencies along the path due to the gravitational redshift factor. The act of going from the emission frame to the observer’s frame also affects the intensity $I$, so we use the invariant $\cal I$ instead, only converting to specific intensity just before outputting the results. Binning the optical depth with frequency along a path also allows us to model the radiative transport of the continuum as well as the lines, provided that one is in the limit where scattering is unimportant. (Stimulated emission can be modeled by using a negative absorption coefficient.) Depending on the relative intensities of the continuum and the line, absorption may cause the emission feature to be converted into an absorption feature. We assume that the continuum is a power law, which we parametrize by a slope and intensity at the line rest frequency. We have assumed that the continuum is emitted with an intensity that scales with the line emissivity. This means that the equivalent width of the emission line is constant across the disk. We set this to be $0.05$ of the rest line frequency. (This value is roughly what is seen in AGN.) To simplify things further, we will fix the power law index of the continuum to be $\gamma=0.5$, where $\gamma$ is defined by $$I=C E^{-\gamma},$$ in which $I$ is the continuum intensity as a function of energy, $E$, and $C$ is derived from the given equivalent width. This rather hard spectrum was chosen to emphasise the line. The treatment assumes that the continuum is created in a relatively thin planar structure above the accretion disk. In effect, we treat the emission from the disk corona as part of the injection spectrum, together with the emission line, which we propagate through the absorbing material suspended much higher above. Ray Tracing Algorithm {#rtalgorithm} ===================== A direct ray-tracing method is used instead of the conventional transfer-function method, as it is easier to incorporate the numerical radiative-transport calculations. The ray-tracing algorithm is as follows: 1. Integrate the equations governing the geodesics, and those describing the optical depth for each frequency, from the observer to the emitting surface; 2. At each crossing of the equatorial plane / torus surface, collate the position and the direction of the photon; 3. Construct the image, and determine the observed frequency/energy shift; 4. Integrate the emission over the images of each order to produce the line profiles. The foot points of the null geodesics (photon trajectories) on the disk surface are calculated by a root-finding algorithm (see e.g. Press et al. 1992, p.343). The first four intersections of the null geodesics and the disk plane (corresponding to the direct and first three higher-order images), and the four-vectors of the photons emitted from there are recorded. The incorporation of an emissivity law is therefore straightforward, as it is defined in terms of the spatial coordinates on the disk plane. Since the trajectory of the photon is also saved at each crossing point, it is also possible to include the effects of limb darkening, and to model a semi-transparent disk. Since the disk is imaged upon a sky plane, all gravitational lensing effects on the intensity of the light are implicitly included in the calculation of the image itself. If a region of the disk is magnified then it will cover more area in the image, and thus will appear brighter than a non-magnified region. Inclination effects are also included implicitly. An inclined disk will cover less pixels than a face-on disk, where the number of pixels is roughly proportional to $\cos i$, where $i$ is the inclination angle. (Light bending causes this Euclidean formula to be only an approximation.) Gravitational lensing does not alter the observed surface brightness of a point. This implicit inclusion of changing areas of photon flux-tubes linking the observer to the emitter vastly simplifies the calculation of the observed flux. All that is required is to integrate over each pixel on the image, taking into account the redshift of the emission regions corresponding to the pixels. If one were integrating over the surface of the disk, instead of over the image, then the Jacobian of the transformation from the disk to the image plane coordinates would be required. This is numerically difficult to obtain, and would require a separate transformation for each image order. Surface Finding Algorithm {#surface_finding} ========================= We consider the following algorithm to determine the torus surface. We integrate equation (\[isobaric\]) and tabulate the resulting points ($r,\theta$) along the path of the integration. Then we interpolate ($r,\theta$) and construct the torus surface, where the emission originates. We use spline interpolation between the surface points. When ray tracing the photon paths, we determine the intersection of the trajectory and the torus surface. As the photon trajectory calculations may take large spatial steps, there is a possibity that the torus is not ’detected’. To prevent this from happening, we consider the following procedure. We take note of the region where the photon is located during the trajectory calculation: either inside or outside the torus and either above or below the equatoral plane. Whenever the photon leaves one of the four regions, and enters another region, we use a boundary-searching algorithm to find the exact location where the transit occurs. If the trajectory hits the equatorial plane outside the torus, the integration will continue. If the trajectory hits the torus, then integration is terminated. By taking smaller steps required by the boundary-finding algorithm, we can prevent the integrator from missing the torus entirely. This algorithm can be used for more complicated surfaces. It works well, because the integrator will only miss intersections when the trajectory is close to tagential to a surface. This happens close to the equatorial plane in the torus models. Adding in a fake boundary there, and thus decreasing the step size, helps in preventing missed intersections. [^1]:

ArXiv

--- abstract: 'Quantum pigeonhole principle states that if there are three pigeons and two boxes then there are instances where no two pigeons are in the same box which seems to defy classical pigeonhole counting principle. Here, we investigate the quantum pigeonhole effect on the ibmqx2 superconducting chip with five physical qubits. We also observe the same effect in a proposed non-local circuit which avoid any direct physical interactions between the qubits which may lead to some unknown local effects. We use the standard quantum gate operations and measurement to construct the required quantum circuits on IBM quantum experience platform. We perform the experiment and simulation which illustrates the fact that no two qubits (pigeons) are in the same quantum state (boxes). The experimental results obtained using IBM quantum computer are in good agreement with theoretical predictions.' author: - 'Narendra N. Hegade' - Antariksha Das - Swarnadeep Seth - 'Prasanta K. Panigrahi' date: 'Received: date / Accepted: date' title: Investigation of quantum pigeonhole effect in IBM quantum computer --- [example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore [Quantum Pigeonhole Effect, IBM Quantum Experience]{} Introduction \[I\] ================== Quantum mechanics is well known for its counter-intuitive results which pose a conceptual conflict of our regular understanding. There are many quantum mechanical phenomena such as the EPR paradox, the no-cloning theorem, quantum Zeno effect, quantum teleportation, quantum tunneling etc. that can not be answered by classical physics. Quantum pigeonhole effect is one of these. In number theory, classical pigeonhole principle [@Allenby2011Howtocount] states that if $n$ objects are distributed among $m$ boxes, with the condition $m < n$, then there is at least one box where we can find more than one object. In other words, if there are more objects than the number of boxes then there is at least one box which must contain more than one object. Aharonov et al. [@Aharonov2016QPP] first proposed the idea of quantum pigeonhole effect(QPHE) where they have shown that in some scenarios the classical pigeonhole counting principle is violated. It is shown that for a particular choice of pre- and post-selected state, three quantum particles which can take two quantum states could end up in a situation where no two particles can be found in the same quantum states. To observe the quantum pigeonhole effect, the former designed and performed an interferometric experiment shown in Figure \[fig1\]. In their set-up three quantum particles (pigeons) pass simultaneously through the two arms (Pigeonholes/boxes) of the Mach-Zehnder interferometer (MZI) which characterizes two distinct quantum state ($\Ket{0}$ and $\Ket{1}$) of the quantum particles. Now because of the Coulomb repulsion between the quantum particles if at least two or three particles are in the same arm of the interferometer then the particles will repel each other and expected to get deflected and the pattern of the detector will give information whether any of the quantum particles are in the same path of the interferometer or not. The particles have equal probability of arriving at either of the two detectors. It is shown that if one post-select those cases where all three quantum particles are detected at the same detector then the pattern of the detector indicates that none of the quantum particles get deflected thus no such interaction has taken place. It suggests that no two quantum particles can take the same path which contradict the classical pigeonhole principle. This phenomenon has already drawn a fair amount of attention. Recently, Chen et al. [@chen2019ExpParadox] experimentally demonstrated the quantum pigeonhole paradox using three single photons, transmitting through two distinct polarization channels under appropriate pre and post selections of the polarization states. They used the weak measurement technique [@weakmeasure1; @weakmeasure2; @weakmeasure3] to probe the underlying mechanism. Mahesh et al. [@mahesh2016NMR] experimentally simulated the quantum pigeonhole principle using four-qubit NMR quantum simulator where the quantum pigeons are mimicked by three spin-1/2 nuclei whose states are probed by another ancillary spin. It was also argued that the effect arises from the quantum contextuality in quantum physics. Later, Rae and Forgan describe, the effect is observed due to the interference between the wavefunctions of weakly interacting quantum particles [@RaeandForgan]. In order to close any conceptual loophole that may arise from the unknown local interactions of the physical qubits, Paraoanu illustrated the violation based on non-local schemes by designing two different quantum circuit using standard gates and measurement [@paraoanu2018nonlocal]. IBM’s cloud-based quantum computing platform has opened a new window of opportunity to perform experiments with quantum states. It allows testing various quantum mechanical phenomena [@GarciaJAMP2018; @SisodiaQIP2017; @HuffmanPRA2017; @VishnuQIP2018; @AlsinaPRA2016; @YalcinkayaPRA2017; @KandalaNAT2017; @SisodiaPLA2017]. Here, we present an equivalent quantum circuit design using IBM’s real quantum processor ‘ibmqx2’ to investigate the quantum pigeonhole effect. In order to get rid of any kind of local interactions we implement two similar non local circuits proposed by Paraoanu [@paraoanu2018nonlocal]. We show that by standard quantum gate operations and measurements, it is indeed possible to observe quantum pigeonhole effect. We perform simulation to verify the theoretical predictions. ![**Schematic diagram of the Mach-Zehnder interferometer**. Three quantum particles are injected simultaneously. They are split into the two arms of the interferometer ($\Ket{0}$ and $\Ket{1}$) after the first beam splitter $BS1$. There is a phase shifter in one path of the interferometer. The particles are detected at detector $D_0$ and $D_1$ after another beam splitter $BS2$.[]{data-label="fig1"}](fig1.eps){width="\linewidth"} This paper is organized in the following way. In Section \[II\], the quantum pigeonhole effect is discussed briefly. In Section \[III\], we present the implementation of the quantum circuits on ‘ibmqx2’ superconducting chip to investigate the quantum pigeonhole effect and discuss the experimental outcome and its significance. In Section \[IV\], we give the conclusion about the work with some remarks. Theory \[II\] ============= In our experiment, we model the quantum pigeonhole effect using superconducting qubits in IBM quantum experience platform as shown in Figure \[fig2\]). We consider a three qubit system which corresponds to three pigeons and two orthogonal states $\ket{0}$ and $\Ket{1}$, represents two boxes. We prepare the initial state by applying Hadamard gate on the three qubits $$\Ket{\psi_i} =\Ket{+}_1 \Ket{+}_2 \Ket{+}_3 .$$ where, $\Ket{+}=\frac{\ket{0}+\Ket{1}}{\sqrt{2}}$ and the indices 1,2,3 refer to the qubits one, two and three respectively. A phase-shifter is then operated on the initial state $\Ket{\psi_i}$ and the state transforms into $\Ket{+i}_1 \Ket{+i}_2 \Ket{+i}_3$ where, $\Ket{+i}=\frac{\ket{0}+i \Ket{1}}{\sqrt{2}}$. Then, after applying Hadamard gate, the three qubit state becomes $$\begin{aligned} \Ket{\psi_f} &= \left( \frac{1+i}{2}\Ket{0} + \frac{1-i}{2}\Ket{1} \right) \otimes \left( \frac{1+i}{2}\Ket{0} + \frac{1-i}{2}\Ket{1} \right) \nonumber \\ & \hspace{2.9cm} \otimes \left( \frac{1+i}{2}\Ket{0} + \frac{1-i}{2}\Ket{1} \right).\end{aligned}$$ So, each qubit has equal probability to be found in either $\Ket{0}$ or $\Ket{1}$ state. The $\Ket{+}$ state can also be written as $$\Ket{+}=\frac{1-i}{2} \Ket{+i} + \frac{1+i}{2} \Ket{-i}.$$ After the phase shift operator, $\Ket{+i}$ will transform to $\Ket{-}= \frac{\Ket{0} - \Ket{1}}{\sqrt{2}} $ and finally to $\Ket{1}$, after the Hadamard operation. Similarly, $\Ket{-i}$ will transform to $\Ket{+}$ and then to $\Ket{0}$, after the Hadamard operation. From this we can infer that after the measurement if we get $\Ket{0}$, then it corresponds to a post-selected state $\Ket{-i} = \frac{\ket{0}-i \Ket{1}}{\sqrt{2}}$ just before the phase-shift operator. In the same way $\Ket{1}$ will corresponds to the post-selected state $\Ket{+i}$. \[H\] ![**The circuit schematic for investigation of the Quantum Pigeonhole Effect.** Hadamard gates are used to prepare the initial state of the qubits. The phase shifter perform the phase shift operation on the qubits. The 2-qubit parity measurement $W_{lm}$ is performed in order to retrieve the intermediate state information of the qubits. Another Hadamard operation is performed before the qubits are measured at the end of the circuit.[]{data-label="fig2"}](fig2.eps "fig:"){width="\linewidth"} In order to probe whether any two qubits are in the same quantum state or not, we need to perform a 2-qubit parity measurement $W_{lm} (l,m=1,2,3;l\neq m)$ of any two of the three qubits. The intermediate state of the qubits are measured through an ancilla qubit. For the intermediate situation thus it is convenient to define projection operators for various pairing combinations of three qubits. $$\begin{aligned} \Pi_{12}= \ket{0}\bra{0}\otimes \Ket{0}\bra{0}\otimes \mathbbm{1} + \Ket{1}\bra{1}\otimes \Ket{1}\bra{1}\otimes \mathbbm{1} \nonumber \\ \Pi_{23}= \mathbbm{1} \otimes \Ket{0}\bra{0}\otimes \Ket{0}\bra{0}+ \mathbbm{1} \otimes \Ket{1}\bra{1}\otimes \Ket{1}\bra{1} \nonumber \\ \Pi_{13}=\Ket{0}\bra{0}\otimes \mathbbm{1} \otimes \Ket{0}\bra{0} + \Ket{1}\bra{1}\otimes \mathbbm{1} \otimes \Ket{1}\bra{1}. \label{eqn4}\end{aligned}$$ The projection operators in equation \[eqn4\] tell us whether any two of the three qubits are in the same state or not. $$\Pi_{lm}\Ket{\psi_i}=\Ket{\psi^{same} _{l,m}} \hspace{0.3cm} (l,m= 1,2,3 \hspace{0.1cm}; l \neq m).$$ There are eight possible measurement outcome with equal probability {$\Ket{000}, \Ket{001},\Ket{010},\Ket{011},\Ket{100},\Ket{101},\Ket{110},\Ket{111}$} Now we would have expected from pigeonhole principle that at least two of the three qubits take the same quantum state but we can see that for a particular instance where the measurement outcome is $\Ket{000}$ (or $\Ket{111}$), which corresponds to the post-selected state $\Ket{-i-i-i}$ (or $\Ket{+i+i+i}$) $\ket{\psi^{same} _{l,m}}$ is orthogonal to the post selected state $\Ket{-i-i-i}$ (and $\Ket{+i+i+i}$) prior to the phase-shift operator. $$\begin{aligned} \braket{{-i-i-i}|{\psi^{same} _{l,m}}} = 0 \nonumber \\ \braket{{+i+i+i}|{\psi^{same} _{l,m}}} = 0,\end{aligned}$$ So we can infer that no two of the three qubits are found in identical states for the given particular post selected states $\Ket{-i-i-i}$ and $\Ket{+i+i+i}$. This result can be interpreted as Quantum Pigeonhole Effect (QPHE).\ Circuit implementation and Results \[III\] ========================================== For the experimental realization of Quantum Pigeonhole Effect we have used IBM’s 5 qubit quantum computing interface ibmqx2. The circuit implementation for QPHE on ibmqx2 is shown in Figure \[fig3\]. Here q\[0\], q\[1\] and q\[3\] are the three superconducting qubits. The information about the state of any qubit can be measured by using an ancilla q\[2\], another superconducting qubit. ![**The circuit implementation in IBM qauntum computer to investigate QPHE.** Hadamard gate is used to prepare the initial state. The parity measurement of the second (q\[1\]) and third(q\[3\]) qubit is performed using two consequtive CNOT gates with second (q\[1\]) and third (q\[3\]) qubit as control qubits and the ancilla qubit (q\[2\]) as the common target qubit. The phase gate $S$ introduce a phase shift. Hadamard operations is performed before all the qubits are measured at the end. []{data-label="fig3"}](fig3.eps){width="\linewidth"} -------------------- --- --- --- $\Ket{{+i +i +i}}$ 1 1 1 $\Ket{+i +i -i}$ 1 0 0 $\Ket{{+i -i +i}}$ 0 0 1 $\Ket{{+i -i -i}}$ 0 1 0 $\Ket{{-i +i +i}}$ 0 1 0 $\Ket{{-i +i -i}}$ 0 0 1 $\Ket{{-i -i +i}}$ 1 0 0 $\Ket{{-i -i -i}}$ 1 1 1 -------------------- --- --- --- : 2-qubit parity measurement for all possible post-selected state. $W_{lm}$ represents the 2-qubit parity measurement on $l th$ and $m th$ qubits. For the post selected state $\Ket{+i+i+i}$ (or $\Ket{-i-i-i}$) no two qubits are found in same state.[]{data-label="tab1"} To probe the intermediate state information of any two ($(l,m= q[0],q[1],q[3]; l\neq m)$) of the three qubit prior to the phase-shifter, we perform a 2-qubit parity measurement $W_{lm}$ using a pair of CNOT gates ($C_l NOT_{q[2]},C_m NOT_{q[2]}$) where the ancilla q\[2\] acts as a common target bit. The parity measurement operator $W_{lm}$ preserve the state of the ancilla if both $l th$ and $m th$ qubits are in the same quantum state and inverts otherwise. The intermediate information about the states of any pair ($l,m= 1,2,3 \hspace{0.1cm}; l \neq m$) of qubits can be obtained by measuring the state of ancilla. The state of the ancilla $0$ indicates that the pair of qubits are in the same state and the state of the ancilla $1$ corresponds to the situation where the two qubits are in different state. Table \[tab1\] shows the results of the outcome of the parity measurement for all possible post-selected states. For the post-selected states $\Ket{+i+i+i}$ and $\Ket{-i-i-i}$ we can see that no pair of qubits are in the same state, thus shows QPHE. In the remaining cases we can also observe some interesting effects, e.g for the post-selected state $\Ket{-i+i-i}$ we find that the qubit 1 and 2 are in same state, qubit 2 and 3 are in same state but qubit 1 and 3 are in different state. Similar effects are observed for all the other cases. The qubits might get disturbed due to some local interactions[@locality; @realism] or direct physical interactions between qubits while performing the parity measurement using CNOT gates which can change the pre-existing values of the qubits. To eliminate such local interaction between the qubits, we consider two non-local set-ups as shown in Figure \[fig4\] and Figure \[fig6\]. \[H\] ![**The circuit illustrating QPHE based on entanglement distillation.** Two ancilla qubits are used for measuring the parity of the qubits, which are initially prepared in an entangled state $\frac{\Ket{00}+\Ket{11}}{\sqrt{2}}$. Here, the double dotted line represents the entanglement between the qubits and the single dotted line represents the classical channel.[]{data-label="fig4"}](fig4.eps "fig:"){width="\linewidth"} \[H\] ![**The circuit implementation for QPHE based on entanglement distillation in IBM quantum computer.** Here, q\[0\], q\[1\], and q\[3\] are considered as system qubits, q\[2\] and q\[4\] are used as ancilla qubits for parity measurement.[]{data-label="fig5"}](fig5.eps "fig:"){width="\linewidth"} -------------------- --- --- --- $\Ket{{+i +i +i}}$ 1 1 1 $\Ket{+i+i-i}$ 1 0 0 $\Ket{{+i -i +i}}$ 0 0 1 $\Ket{{+i -i -i}}$ 0 1 0 $\Ket{{-i +i +i}}$ 0 1 0 $\Ket{{-i +i -i}}$ 0 0 1 $\Ket{{-i -i +i}}$ 1 0 0 $\Ket{{-i -i -i}}$ 1 1 1 -------------------- --- --- --- : 2-qubit non-local parity measurement based on entanglement distillation for all possible post-selected state. $W_{lm}$ represents the 2-qubit non-local parity measurement on $l th$ and $m th$ qubits. For the post selected state $\Ket{+i+i+i}$ (or $\Ket{-i-i-i}$) no two qubits are in same state.[]{data-label="tab2"} In the scheme shown in Figure \[fig4\] the parity measurement is realized by two local CNOT gate operations which are converted into a classical parity assessment using classical XOR gate [@EntgDistil]. Here two ancilla qubits are entangled in the $\Ket{\Phi^+}=\frac{\ket{00}+\Ket{11}}{\sqrt{2}}$ Bell state which act as the target qubits while performing the parity measurement using CNOT gates. The ancilla qubits are then measured and the measurement outcome is transmitted to a XOR gate as classical bits. If the output of the XOR gate is 0 (1) then it indicates that the two qubits are in the same (different) quantum state. The IBM circuit implementation is shown in Figure \[fig5\] In another possible non-local scheme, shown in Figure \[fig6\] which is based on the idea of teleportation of CNOT gates [@TeleportCnot]. In the previous setup two entangled ancilla are used for measuring the parity. Now it is possible that some unknown effects from the first qubit which can propagate to the ancilla, and then to the second qubit. To avoid this, two ancilla are used for measuring the parity and another ancilla as their common target. If the outcome of third ancilla is $0 (1)$, it corresponds that the qubits are in the same (different) state. The IBM circuit implementation is demonstrated in Figure \[fig7\] \[H\] ![**The circuit illustrating QPHE based on teleportation of CNOT gates.** The double dotted line represents the entanglement between the qubits, and the single dashed line indicates classical communication. The measurement outcome of ancilla, 0 (or 1) corresponds to qubits being in the same state (or different state).[]{data-label="fig6"}](fig6.eps "fig:"){width="\linewidth"} \[H\] ![**The circuit implementation for QPHE based on teleportation of CNOT gates in IBM quantum computer.** q\[0\], q\[1\] and q\[2\] are the system qubits and q\[3\], q\[4\], q\[5\] and q\[6\] are the ancilla qubits.[]{data-label="fig7"}](fig7.eps "fig:"){width="\linewidth"} -------------------- --- --- --- $\Ket{{+i +i +i}}$ 1 1 1 $\Ket{+i+i-i}$ 1 0 0 $\Ket{{+i -i +i}}$ 0 0 1 $\Ket{{+i -i -i}}$ 0 1 0 $\Ket{{-i +i +i}}$ 0 1 0 $\Ket{{-i +i -i}}$ 0 0 1 $\Ket{{-i -i +i}}$ 1 0 0 $\Ket{{-i -i -i}}$ 1 1 1 -------------------- --- --- --- : 2-qubit non-local parity measurement based on teleportation of CNOT gates for all possible post-selected state. $W_{lm}$ represents the 2-qubit non-local parity measurement on $l th$ and $m th$ qubits. For the post selected state $\Ket{+i+i+i}$ (or $\Ket{-i-i-i}$) no two qubits are in same state[]{data-label="tab3"} Table \[tab2\] and \[tab3\] shows the results obtained using aforementioned two non-local schemes based on entanglement distillation and teleportation of CNOT gates respectively. From both of the table \[tab2\] and \[tab3\] it is pretty evident that for the post-selected state $\Ket{+i+i+i}$ and $\Ket{-i-i-i}$ the outcome of the non-local parity measurement indicate that none of the two qubits are in the same quantum state and it also explins the fact that the effect is non-local. For other post-selected states the results resembles similar patterns as we have observed in \[tab1\]. Overall, we can see a consensus among the results from all the three schemes which match well with the theoretical predictions of the quantum pigeonhole effect.\ Conclusion \[IV\] ================= To summarize, we have designed and successfully implemented a suitable quantum circuit to observe the quantum pigeonhole effect. We have performed experimental simulation using IBM’s real quantum processor ‘ibmqx2’. Each qubit is prepared and post-selected individually. If we measure the state of each qubit separately, they appear to be completely uncorrelated, but when we make a joint measurement on the pairs of qubit we find them to be correlated. This correlation is a manifestation of the quantum pigeonhole effect. We have also shown that the correlation exist in a non-local set-up where any possible unknown local interactions is eliminated using non-local parity measurement technique. Acknowledgements {#acknowledgements .unnumbered} ================ The authors acknowledge the use of IBM Quantum Experience platform for this work and grateful to IBM team. The discussions and opinions developed in this paper are only those of the authors and do not reflect the opinions of IBM or IBM Q experience team. Author contributions {#author-contributions .unnumbered} ==================== The first three authors NNH, AD, SS contributed equally to the work. NNH designed the quantum circuit and implemented on the IBM’s quantum processor along with AD and SS. NNH, AD and SS performed experimental simulations and completed the work under the guidance of PKP. All authors discussed the results and contributed to the final manuscript. [99]{} R. B. J. T. Allenby, *How to count : an introduction to combinatorics* (CRC Press, Boca Raton, 2011), 2nd Ed. . . . . . . . . . . . . . . . . . . .

ArXiv

--- abstract: 'We develop a framework for approximating collapsed Gibbs sampling in generative latent variable cluster models. Collapsed Gibbs is a popular MCMC method, which integrates out variables in the posterior to improve mixing. Unfortunately for many complex models, integrating out these variables is either analytically or computationally intractable. We efficiently approximate the necessary collapsed Gibbs integrals by borrowing ideas from expectation propagation. We present two case studies where exact collapsed Gibbs sampling is intractable: mixtures of Student-$t$’s and time series clustering. Our experiments on real and synthetic data show that our approximate sampler enables a runtime-accuracy tradeoff in sampling these types of models, providing results with competitive accuracy much more rapidly than the naive Gibbs samplers one would otherwise rely on in these scenarios.' author: - 'Christopher Aicher[^1] and Emily B. Fox[^2]' bibliography: - 'bib.bib' title: Approximate Collapsed Gibbs Clustering with Expectation Propagation --- Introduction ============ Background {#sec:background} ========== Approximate Collapsed Gibbs Sampling ==================================== \[sec:inference\] Case Studies ============ \[sec:case\_studies\] We consider two motivating examples for the use of our EP-based approximate collapsed Gibbs algorithm. The first is a mixture of Student-$t$ distributions, which can capture heavy-tailed emissions crucial in robust modeling (i.e., reducing sensitivity to outliers). The second example is a time series clustering model. Mixture of Multivariate Student-$t$ ----------------------------------- \[sec:student\] Time Series Clustering ---------------------- \[sec:tscluster\] Experiments =========== \[sec:experiments\] Conclusion ========== We presented a framework for constructing approximate collapsed Gibbs samplers for efficient inference in complex clustering models. The key idea is to approximately marginalize the nuisance variables by using EP to approximate the conditional distributions of the variables with an individual observation removed; by approximating this conditional, the required integral becomes tractable in a much wider range of scenarios than that of conjugate models. Our use of this EP approximation takes two steps from its traditional use: (1) we approximate a (nearly) full conditional rather than directly targeting the posterior, and (2) our targeted conditional changes as we sample the cluster assignment variables. For the latter, we provided a brief analysis and demonstrated the impact of the changing target, drawing parallels to previously proposed samplers that use stale sufficient statistics. We demonstrated how to apply our EP-based approximate sampling approach in two applications: mixtures of Student-$t$ distributions and time series clustering. Our experiments demonstrate that our EP approximate collapsed samplers mix more rapidly than naive Gibbs, while being computationally scalable and analytically tractable. We expect this method to provide the greatest benefit when approximately collapsing large parameter spaces. There are many interesting directions for future work, including deriving bounds on the asymptotic convergence of our approximate sampler [@pillai2014ergodicity; @dinh2017convergence], considering different likelihood approximation update rules such as *power EP* [@minka2004power], and extending our idea of approximately integrating out variables to other samplers. For the analysis, [@dehaene2015expectation] showed that EP with Gaussian approximations is exact in the large data limit; one could extend these results to consider the case of data being allocated amongst *multiple* clusters. Another interesting direction is to explore our EP-based approximate collapsing within the context of variational inference, possibly extending the set of models for which collapsed variational Bayes [@teh2007collapsed] is possible. Finally, there are many ways in which our algorithm could be made even more scalable through distributed, asynchronous implementations, such as in [@ahmed2012scalable]. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Nick Foti, You “Shirley” Ren and Alex Tank for helpful discussions. This paper is based upon work supported by the NSF CAREER Award IIS-1350133 This paper is an extension of our previous workshop paper [@aicher2016scalable]. **Appendix** Mixture of Multivariate Student-$t$ {#app:student} =================================== Time Series Clustering {#app:tscluster} ====================== EP Convergence {#app:ep_converge} ============== Synthetic Time Series Trace Plots {#app:traceplots} ================================= Seattle Housing Data {#app:housing_data} ==================== [^1]: Department of Statistics, University of Washington, `[email protected]` [^2]: Department of Computer Science and Statistics, University of Washington, `[email protected]`

ArXiv

--- abstract: 'We re-examine the physics of supercritical nuclei, specially focusing on the scattering phase $\delta_{\varkappa}$ and its dependence on the energy $\varepsilon$ of the diving electronic level, for which we give both exact and approximate formulas. The Coulomb potential $Z\alpha/r$ is rounded to the constant $Z\alpha/R$ for $r < R$. We confirm the resonant behavior of $\delta_{\varkappa}$ that we investigate in details. In addition to solving the Dirac equation for an electron, we solve it for a positron, in the field of the same nucleus. This clarifies the interpretation of the resonances. Our results are compared with claims made in previous works.' author: - 'S.I. Godunov' - 'B. Machet' - 'M.I. Vysotsky' bibliography: - 'references.bib' title: 'Resonances in positron scattering on a supercritical nucleus and spontaneous production of $e^{+}e^{-}$ pairs' --- Introduction ============ The Coulomb problem for a nucleus with charge $Z>Z_{\rm cr}$ was recently analysed [@Kuleshov] by solving the Dirac equation for an electron in the external field of this nucleus. Because of the specificity of the Dirac equation that accounts simultaneously for electrons and positrons this problem gets connected to the scattering of positrons (holes in the Dirac sea) on the nucleus (see below). The behavior of the scattering amplitude was found to be very peculiar: it contains resonances and their energies, obtained from an analytical formula found in [@Kuleshov], $$\varepsilon = -\xi + \frac{i}{2} \gamma, \;\; \xi > m, \;\; \gamma > 0, \label{eq:1}$$ correspond to poles of the $S$ matrix located above the left cut, on the second (unphysical) sheet of the energy plane. The resonances in positron scattering were discussed in Refs. [@MRG1:1972; @MRG2:1972]. At $Z<Z_{\rm cr}$, the width $\gamma$ vanishes, and this equation describes the usual bound states of electrons in the Coulomb field of the nucleus. When $Z>Z_{\rm cr}$, $\gamma\neq0$ makes these states quasistationary [@Mur:1976wh; @Popov:1976dh]. For electrons, as $Z$ increases, the transition from bound states to resonant states corresponds to the diving of the bound states, which start at $\varepsilon=+m$, downwards into the lower continuum. In the present paper, in order to clarify the situation, we will also study “the Dirac equation for positron”. By this we mean here the standard Dirac equation with the substitution of electron charge $e$ by $-e$. Now, as $Z$ increases, bound states raise up from $\varepsilon=-m$ and become resonant in the upper continuum. For $Z<Z_{\rm cr}$, the interpretation of these bound states (also noted in [@GMR] chapter 4.3) is the following. For obvious reasons they cannot be $\left(e^{+}N^{+}\right)$ bound states, but are just our previous $\left(e^{-}N^{+}\right)$ bound states. There is no more information in there[^1]. For $Z>Z_{\rm cr}$, we find that $\left(e^{+}N^{+}\right)$ resonances occur at the energies $$\varepsilon_{\rm p} = \xi - \frac{i}{2} \gamma, \;\; \xi > m, \;\; \gamma > 0, \label{eq:2}$$ which now correspond to poles of the $S$ matrix below the right cut of the energy plane, also, as it should be, on the second, unphysical, sheet. This result confirms the proposal made in [@Kuleshov] that the sign of the energy in (\[eq:1\]) should be reversed. This change of sign we are accustomed to when dealing with holes in the lower continuum: the absence of an electron with energy $-\varepsilon$ is then interpreted as the presence of a positron with energy $\varepsilon$. It is now to be operated on the empty states of the energy levels that dive into the lower continuum. Our consideration of the Dirac equation for positrons therefore helps to clarify the nature and position of the resonances. No physical interpretation for them was suggested in [@Kuleshov]. It was only claimed that spontaneous $e^{+}e^{-}$ pair production by naked nuclei at $Z>Z_{\rm cr}$, as discussed in [@MRG1:1972; @MRG2:1972; @Voronkov:1961; @Gershtein:1969; @Greiner:1969; @Popov:1970-1; @Popov:1970-2; @Gerstein:1969-lett; @Popov:1970nz; @Popov:1970-ZhETF-2; @Zeldovich:1972; @Zeldovich:1971; @KP:2014; @Gershtein1973; @Okun:1974rza; @GMR; @GMM], does not occur. We, however, do not see any sensible objection to the occurrence of this process: an empty state diving into the lower continuum gets filled by one electron of the Dirac sea; the resulting hole in the sea is the positron that gets ejected by the nucleus the charge of which has become $Z-1$. The characteristic time of this emission process is $1/\gamma$, in agreement with the results obtained in [@MRG1:1972; @MRG2:1972; @Voronkov:1961; @Gershtein:1969; @Greiner:1969; @Popov:1970-1; @Popov:1970-2; @Gerstein:1969-lett; @Popov:1970nz; @Popov:1970-ZhETF-2; @Zeldovich:1972; @Zeldovich:1971; @KP:2014; @Gershtein1973; @Okun:1974rza; @GMR; @GMM]. Furthermore, spontaneous production of $e^+e^-$ pairs was recently observed in the numerical solution of the Dirac equation in the case of heavy ion collisions [@Maltsev:2014qna; @Maltsev2017]. The plan of the paper is as follows. In Section \[sec:lower\], following [@Kuleshov] and using the Dirac equation, we study the scattering of states of the lower continuum on a supercritical nucleus. In addition to reproducing the approximate results obtained in [@Kuleshov] we get explicit results without using an expansion over the parameter $m\times R$, where $R$ is the nucleus radius. Such an expansion being good for electrons does not work for heavy particles, for example, muons [@Mur:1976wh; @Popov:1976dh]. In Section \[sec:upper\], we use instead the Dirac equation for positrons (see above) and study the scattering of states of its upper continuum on a supercritical nucleus. We conclude in Section \[sec:conclusions\]. Lower continuum wave functions and scattering phases in the Coulomb field of a supercritical nucleus {#sec:lower} ==================================================================================================== The radial functions of the Dirac equation $F(r) \equiv rf(r)$ and $G(r) \equiv rg (r)$ are determined by the following differential equations [@Bethe; @Bethe2; @BLP]: $$\left\{ \begin{aligned} &\frac{dF}{dr} + \frac{\varkappa}{r}F - \left(\varepsilon + m - V(r)\right)G = 0,\\ &\frac{dG}{dr} - \frac{\varkappa}{r}G + \left(\varepsilon - m - V(r)\right)F = 0, \end{aligned} \right. \label{eq:3}$$ where $\varkappa = -(j+1/2) = -1, -2,\dots$ for $j = l + 1/2$ and $\varkappa = (j +1/2)= 1,2,3\dots$ for $j = l-1/2$ and the ground state corresponds to $\varkappa = -1$ (let us note that in [@Kuleshov] the Dirac equation with the substitution $F\Rightarrow -F$ is used). In order to deal with the case $Z\alpha >1$ the Coulomb potential should be regularised at $r=0$ [@PomSmo:1945]. To do this we shall approximate the nucleus as a homogeneous charged sphere with radius $R$ (the so-called rectangular cutoff). Thus, the potential in which the Dirac equation should be solved looks like: [V(r)=]{} -, & $r < R$, \[eq:potential\_r&lt;R\]\ -, & $r > R$. \[eq:potential\_r&gt;R\] \[eq:potential\] At small distances $r< R$, substituting expression (\[eq:potential\_r&lt;R\]) into (\[eq:3\]), we obtain the Dirac equation with a constant potential, the solution of which is expressed through Bessel functions. In order to obtain finite $f$ and $g$ at $r=0$ among the two sets of solutions the one with a positive index of the Bessel function should be selected[^2]: $$\left( \begin{array}{l} F \\ G \end{array}\right) = {\rm const}\cdot\sqrt{\beta r}\cdot \left( \begin{array}{l} \mp J_{\mp(1/2 + \varkappa)} (\beta r)\\ J_{\pm(1/2 - \varkappa)} (\beta r) \frac{\beta}{\varepsilon+m+\frac{Z\alpha}{R}} \end{array} \right),\; r < R, \label{5}$$ where $\beta = \sqrt{(\varepsilon + Z\alpha /R)^2 - m^2}$. Upper (lower) signs should be taken for $\varkappa < 0$ ($\varkappa > 0$). For $r>R$, we need the solution of the Dirac equation for the Coulomb potential. We introduce the standard quantity $\lambda$ which, for $-m < \varepsilon < m$, equals $\lambda = \sqrt{(m-\varepsilon)(m+\varepsilon)} \equiv -ik$, where $k$ is the electron momentum. Here we have to make an important remark. Since later we are going to look for resonances in the complex $\varepsilon$ plane, we must carefully define the square roots used here. Each of them, $\sqrt{m-\varepsilon}$ and $\sqrt{m+\varepsilon}$, are defined on two Riemann sheets of the complex $\varepsilon$ plane. To avoid ambiguous expressions let us introduce a uniquely defined function ${\rm sqrt(z)}$ as follows: $$\label{eq:sqrt_general_0} {\rm sqrt}\left(|z|e^{i{\rm Arg}(z)}\right)=\sqrt{|z|}e^{i{\rm Arg}(z)/2},\text{ for } {\rm Arg}(z)\in(-\pi;\pi].$$ For example $$\label{eq:sqrt_0} {\rm sqrt}\left(z\right)= \begin{cases} i& \text{for }z=-1+i\cdot0,\\ i& \text{for }z=-1,\\ -i& \text{for }z=-1-i\cdot0. \end{cases}$$ It is therefore the first branch of the function $\sqrt{z}$ with the cut $(-\infty;0)$. The second branch is given by $-{\rm sqrt}(z)$. This definition is also very convenient because the square root is defined in this way in many numerical tools for computers. Switching branches of both square roots, $\sqrt{m-\varepsilon}$ and $\sqrt{m+\varepsilon}$, leads to the same value of $\lambda$. Therefore, $\lambda$ is defined on the two Riemann sheets according to: $$\label{eq:lambda_sheets} \lambda= \begin{cases} {\rm sqrt}\left(m-\varepsilon\right)\cdot {\rm sqrt}\left(m+\varepsilon\right)& \text{on the physical sheet,}\\ -{\rm sqrt}\left(m-\varepsilon\right)\cdot {\rm sqrt}\left(m+\varepsilon\right)& \text{on the unphysical sheet,} \end{cases}$$ with two cuts originating, respectively, from each of the square roots (see Fig. \[fig:cuts\])[^3]. From general arguments of scattering theory, we know that electron bound states are located at real $\varepsilon$ in the interval $-m < \varepsilon < m$. Unbound electron states are located above the right cut and unbound positron states below the left cut. ![The plane of complex energy $\varepsilon$.[]{data-label="fig:cuts"}](pics/cuts.pdf){width="4.5in"} In what follows we shall use the following conventions for the “$\sqrt{~}$” symbol: $$\begin{aligned} \label{eq:roots} \sqrt{m+\varepsilon}&= \begin{cases} {\rm sqrt}\left(m+\varepsilon\right)&\text{on the physical sheet,}\\ -{\rm sqrt}\left(m+\varepsilon\right)&\text{on the unphysical sheet,} \end{cases}\\ \sqrt{m-\varepsilon}&={\rm sqrt}\left(m-\varepsilon\right)\text{ on both sheets.}\end{aligned}$$ It does not matter which root changes sign when we go to the second sheet since we can always also change the signs of both. We are looking for solution written in the standard form [@BLP][^4]: $$\left( \begin{array}{c} F \\ G \end{array} \right) = \left( \begin{array}{c} \sqrt{m+\varepsilon} \\ -\sqrt{m-\varepsilon} \end{array} \right) {\rm exp} (-\rho/2)\rho^{i\tau} \left( \begin{array}{c} Q_1 + Q_2 \\ Q_1 - Q_2 \end{array} \right), \label{8}$$ where $\tau=\sqrt{\left(Z\alpha\right)^{2}-\varkappa^{2}}$, $\rho = 2\lambda r = -2ikr$, $Q_1$ and $Q_2$ are determined by differential equations, the solutions of which are Kummer confluent hypergeometric functions $_1F_1(\alpha,\beta,z)$ (also sometimes noted $F(\alpha,\gamma,z)$ like in [@BLP]). In textbooks dealing with the case $Z \alpha <1$, $R=0$, only solutions regular at $r=0$ are considered. We must instead here take into account both type of solutions of the equations for $Q_1$ and $Q_2$. The formulas for the $Q_i$ are derived in Appendix \[sec:Q1Q2\]. From (\[A6\]) to (\[A8\]) we get: $$\hspace{-7mm}\left\{ \begin{aligned} Q_1 &= C\cdot\frac{-\frac{iZ\alpha m}{k}+\varkappa} {-i\tau+\frac{iZ\alpha\varepsilon}{k}} \cdot {_1F_{1}}\left(i\tau-\frac{iZ\alpha\varepsilon}{k},2i\tau +1,\rho\right)+\\ &\hspace{40mm}+D\cdot\frac{-\frac{iZ\alpha m}{k} +\varkappa} {i\tau+\frac{iZ\alpha\varepsilon}{k}} \rho^{-2i\tau}{_{1}F_{1}}\left(-i\tau - \frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho\right),\\ Q_2 &= C\cdot{_{1}F_{1}}\left(1+i\tau - \frac{iZ\alpha\varepsilon}{k}, 2i\tau+1, \rho\right) + D\rho^{-2i\tau}{_{1}F_{1}}\left(1-i\tau - \frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho\right), \end{aligned}\right.\label{eq:9}$$ where $C$ and $D$ are arbitrary coefficients[^5]. The scattering phase $\delta_{\varkappa}(\varepsilon,Z)$ is determined by investigating the behavior of the wave function at large $r$. To this purpose, the asymptotic expansion of $_1F_1$ at large $|z|$ $$_{1}F_{1}(\alpha, \gamma, z)\Big\rvert_{|z| \to \infty} = \frac{\Gamma(\gamma)}{\Gamma(\gamma-\alpha)}(-z)^{-\alpha} [1+ O(1/z)] + \frac{\Gamma(\gamma)}{\Gamma(\alpha)} e^z z^{\alpha -\gamma} [1+O(1/z)] \label{11}$$ is very useful. Using the asymptotic expansion (\[11\]) for the Kummer functions occurring in (\[eq:9\]) gives: $$\begin{aligned} \left.\left( \begin{array}{c} F \\ G \end{array} \right)\right|_{r\to\infty} &= A\cdot\left( \begin{array}{c} \sqrt{m+\varepsilon} \\ -\sqrt{m-\varepsilon} \end{array} \right)\times\\ \times\Biggl(C&\left[ e^{-\frac{\rho}{2}}\frac{\Gamma\left(2i\tau+1\right)} {\Gamma\left(1+i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} \frac{\frac{iZ\alpha m}{k}-\varkappa}{i\tau-\frac{iZ\alpha\varepsilon}{k}} \rho^{i\tau}\left(-\rho\right)^{-i\tau} \left(-\rho\right)^{\frac{iZ\alpha\varepsilon}{k}} \pm e^{\frac{\rho}{2}}\frac{\Gamma\left(2i\tau+1\right)} {\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} \rho^{-\frac{iZ\alpha\varepsilon}{k}} \right] +\\ +D&\left[ e^{-\frac{\rho}{2}}\frac{\Gamma\left(-2i\tau+1\right)} {\Gamma\left(1-i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} \frac{\frac{iZ\alpha m}{k}-\varkappa}{-i\tau-\frac{iZ\alpha\varepsilon}{k}} \rho^{-i\tau}\left(-\rho\right)^{i\tau} \left(-\rho\right)^{\frac{iZ\alpha\varepsilon}{k}} \pm e^{\frac{\rho}{2}}\frac{\Gamma\left(-2i\tau+1\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} \rho^{-\frac{iZ\alpha\varepsilon}{k}} \right]\Biggr), \label{12}\end{aligned}$$ where the upper sign corresponds to $F$ and the lower sign corresponds to $G$. The ratio $$\begin{aligned} \frac{\left(-\rho\right)^{\frac{iZ\alpha\varepsilon}{k}}} {\rho^{-\frac{iZ\alpha\varepsilon}{k}}}\end{aligned}$$ yields the Coulomb logarithm (for real $\varepsilon$ below the left cut it gives $\exp\left[\frac{2iZ\alpha\varepsilon}{k}\ln\left(2kr\right)\right]$). Since the latter does not contribute to the differential scattering cross section at nonzero angle $\theta$, we will omit this term in our further calculations. From the general formula $$\begin{aligned} \left.\left( \begin{array}{c} F \\ G \end{array} \right)\right|_{r\to\infty} \propto\left( \begin{array}{c} \sqrt{m+\varepsilon} \\ -\sqrt{m-\varepsilon} \end{array} \right) \left\{e^{i(kr+\frac{Z\alpha\varepsilon}{k}\ln(2kr))} e^{2i\delta} \pm e^{-i(kr+\frac{Z\alpha\varepsilon}{k}\ln(2kr))}\right\}\end{aligned}$$ it follows that the ratio of the remaining coefficients define the scattering phase $\delta_{\varkappa}$ (on the real axis below the left cut $e^{-\rho/2}\equiv e^{ikr}$ corresponds to the outgoing wave and $e^{\rho/2}\equiv e^{-ikr}$ corresponds to the incoming wave): $$\begin{aligned} \label{eq:phase} e^{2i\delta_{\varkappa}} &=-\frac{1}{\varkappa+\frac{iZ\alpha m}{k}}\cdot \frac {\frac{C}{D}\cdot\frac{\Gamma\left(2i\tau\right)} {\Gamma\left(i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} \rho^{i\tau}\left(-\rho\right)^{-i\tau} - \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(-i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} \rho^{-i\tau}\left(-\rho\right)^{i\tau}} {\frac{C}{D}\cdot\frac{\Gamma\left(2i\tau\right)} {\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} - \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)}},\end{aligned}$$ where $$\begin{aligned} \rho^{i\tau}\left(-\rho\right)^{-i\tau} &= \exp\left[i\tau\ln\left(\rho\right)-i\tau\ln\left(-\rho\right)\right] =\exp\left[-\tau\left( {\rm Arg}\left[\rho\right]-{\rm Arg}\left[-\rho\right] \right)\right]=\\ &=e^{-\pi\tau\cdot{\rm sign}\left[{\rm Arg}\left[\rho\right]\right]}.\nonumber\end{aligned}$$ The resonance of the scattering amplitude corresponds to the pole of the $S$-matrix element $S\equiv e^{2i\delta}$ and from (\[eq:phase\]) we immediately get an equation for the position of this pole in the $\varepsilon$-plane: $$\begin{aligned} \label{eq:pole} \frac{C}{D}\cdot\frac{\Gamma\left(2i\tau\right)} {\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} - \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)}=0.\end{aligned}$$ In what follows we will match the solutions at $r<R$ and $r>R$ to obtain the ratio $C/D$, such that we can calculate the phase $\delta_{\varkappa}$ and find the poles of the $S$ matrix which correspond to the energy levels. This procedure can be performed both exactly and approximately. Exact results ------------- With the help of the exact formulas (\[5\]), (\[8\]), and (\[eq:9\]) we get the ratio $C/D$ from matching $F/G$ at $r=R+0$ and $r=R-0$: $$\label{eq:CD_exact} \frac{C}{D}= -\rho_{0}^{-2i\tau}\cdot\frac{F_{g}^{-}-MF_{f}^{-}}{F_{g}^{+}-MF_{f}^{+}},$$ where $$\begin{aligned} M&=\pm\frac{\sqrt{m+\varepsilon}}{\sqrt{m-\varepsilon}} \cdot \frac{J_{\pm\left(1/2-\varkappa\right)}\left(\beta R\right)} {J_{\mp\left(1/2+\varkappa\right)}\left(\beta R\right)} \cdot \frac{\beta}{\varepsilon+m+\frac{Z\alpha}{R}},\label{eq:M}\\ \label{eq:Ff} F_{f}^{\pm}&= {_{1}F_{1}}\left(\alpha_{1}^{\pm},\gamma^{\pm},\rho_{0}\right) \frac{\frac{iZ\alpha m}{k}-\varkappa}{\alpha_{1}^{\pm}} +{_{1}F_{1}}\left(\alpha_{2}^{\pm},\gamma^{\pm},\rho_{0}\right),\\ \label{eq:Fg} F_{g}^{\pm}&= {_{1}F_{1}}\left(\alpha_{1}^{\pm},\gamma^{\pm},\rho_{0}\right) \frac{\frac{iZ\alpha m}{k}-\varkappa}{\alpha_{1}^{\pm}} -{_{1}F_{1}}\left(\alpha_{2}^{\pm},\gamma^{\pm},\rho_{0}\right),\end{aligned}$$ and $$\label{eq:args} \alpha_{1}^{\pm}=\pm i\tau-\frac{iZ\alpha\varepsilon}{k}, \alpha_{2}^{\pm}=1\pm i\tau-\frac{iZ\alpha\varepsilon}{k}, \gamma^{\pm}=\pm 2i\tau+1, \rho_{0}=-2ikR.$$ The numerical evaluation of the square roots in (\[eq:M\]) and of $k$ in (\[eq:args\]) for *real* $\varepsilon$ is somewhat tricky since one should carefully choose the side of the cut to use. Due to the definition (\[eq:sqrt\_general\_0\])–(\[eq:sqrt\_0\]) of the ${\rm sqrt}()$ function the expression ${\rm sqrt}\left(m+\varepsilon\right)$ gives, for *real* $\varepsilon$, the values above the cut such that $-{\rm sqrt}\left(m+\varepsilon\right)$ should be used. It corresponds *formally* to calculating the scattering phase on the second (unphysical) sheet. The same holds for $k$. For any *real* $\varepsilon$ it is also possible to use $k={\rm sqrt}\left(\varepsilon^{2}-m^{2}\right)$ which chooses the correct side of the cut; then $\sqrt{m+\varepsilon}/\sqrt{m-\varepsilon}=-ik/\left(m-\varepsilon\right)$. With the help of (\[eq:CD\_exact\]) we can calculate the scattering phase $\delta_{\varkappa}$ defined by (\[eq:phase\]). In the domain $\varepsilon<-m$, $\delta_{\varkappa}(\varepsilon,Z)$ gives the scattering phase of a positron with energy $\varepsilon_{\rm p}=-\varepsilon>m$ on the nucleus (for real $\varepsilon<-m$ we get ${\rm Arg}\left[\rho\right]<0$). Its dependence on $\varepsilon_{\rm p}$ for $\varkappa = -1$ and $Z = 232$ is shown in Fig. \[fig:phase\] (compare with Fig. 3 of [@Kuleshov]). The scattering phase $\delta_{\varkappa}$ exhibits a resonance behavior; it goes through $\pi/2$ at $\varepsilon_{\rm p}/m\approx 5.06$. We obtain the equation for the position of the poles by substituting (\[eq:CD\_exact\]) into (\[eq:pole\]) $$\begin{aligned} \label{eq:pole_detailed} \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(2i\tau\right)}\cdot \frac{\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)}= -\rho_{0}^{-2i\tau}\cdot\frac{F_{g}^{-}-MF_{f}^{-}}{F_{g}^{+}-MF_{f}^{+}}.\end{aligned}$$ The solutions of (\[eq:pole\_detailed\]) can be found by scanning the complex $\varepsilon$ plane. This is the method that we used to find the exact positions[^6] of the $S$-matrix poles (see Fig. \[fig:e\_of\_Z\] and Table \[tab:e\_of\_Z\]). The energies $\varepsilon$ of the quasistationary states are located above the left cut on the second sheet of the complex $\varepsilon$ plane[^7]: $$\varepsilon = -\xi + \frac{i}{2}\gamma,~\xi>m,~\gamma>0. \label{eq:resonances}$$ Approximate results ------------------- In [@Kuleshov] the approximation $1/R\gg\varepsilon,m$ was used. In this section we are going to reproduce their results and compare them to the exact ones. Being interested in the case $Z\alpha \gtrsim 1$ and taking into account the smallness of the nucleus radius in comparison with the electron Compton wavelength $1/m$ we obtain that $\beta \approx Z\alpha/R$ in (\[5\]). The solution of the system (\[eq:3\]) at $r > R$ should match (\[5\]) at $r=R$, in particular the ratio $F/G$ of both solutions at $r=R$ should coincide. Substituting (\[eq:potential\_r&gt;R\]) in (\[eq:3\]) at $r\to 0$ we easily get $$\left.\left( \begin{array}{l} F \\ G \end{array}\right)\right|_{r \to 0} = \eta_\sigma r^\sigma \left( \begin{array}{c} -1 \\ \frac{Z\alpha}{\sigma - \varkappa} \end{array} \right) + \eta_{-\sigma} r^{-\sigma} \left( \begin{array}{c} -1 \\ \frac{Z\alpha}{-\sigma - \varkappa} \end{array} \right), \label{6}$$ where $\sigma = \sqrt{\varkappa^2 - Z^2\alpha^2}$ and $\eta_\sigma$ and $\eta_{-\sigma}$ are arbitrary constants. Matching the ratios $F/G$ from (\[6\]) and (\[5\]) at $r=R$ we obtain $$\frac{\eta_\sigma}{\eta_{-\sigma}} = \frac{\sigma -\varkappa}{\sigma +\varkappa}\cdot \frac{Z\alpha J_{\mp(1/2 +\varkappa)}(Z\alpha) \pm (\sigma + \varkappa)J_{\pm(1/2 -\varkappa)}(Z\alpha)} {Z\alpha J_{\mp(1/2 +\varkappa)}(Z\alpha) \mp (\sigma - \varkappa) J_{\pm(1/2 -\varkappa)}(Z\alpha)} \cdot\frac{R^{-\sigma}}{R^\sigma}=\tan\theta, \label{7}$$ which coincides with Eq. (13) from [@Kuleshov]. In the case $Z\alpha > |\varkappa|$ one should substitute $\sigma$ by $i\tau$ (where, as before, $\tau = \sqrt{Z^2\alpha^2 - \varkappa^2}$): $$\frac{\eta_\tau}{\eta_{-\tau}} = \frac{i\tau -\varkappa}{i\tau +\varkappa}\cdot \frac{Z\alpha J_{\mp(1/2 +\varkappa)}(Z\alpha) \pm (i\tau + \varkappa) J_{\pm(1/2 -\varkappa)}(Z\alpha)} {Z\alpha J_{\mp(1/2 +\varkappa)}(Z\alpha) \mp (i\tau - \varkappa) J_{\pm(1/2 -\varkappa)}(Z\alpha)} \cdot\frac{R^{-i\tau}}{R^{i\tau}}= e^{2i\theta}. \label{77}$$ The modulus of the r.h.s of (\[77\]) can be easily checked to be unity, this is why we can rewrite it as an $\exp{\left(2i\theta\right)}$ with real $\theta$. The expansion of (\[8\]) at small $\rho$ contains terms $\sim\rho^{i\tau}$ and $\rho^{-i\tau}$. Comparing this expansion with (\[6\]) and substituting $\sigma \to i\tau$, $\eta_\sigma \to \eta_\tau$, $\eta_{-\sigma}\to \eta_{-\tau}$ yields: $$\eta_\tau = C\cdot(-2ik)^{i\tau} \frac{i\tau - \varkappa + Z\alpha\sqrt{\frac{m-\varepsilon} {m+\varepsilon}}}{i\tau - \frac{iZ\alpha\varepsilon}{k}}, \; \eta_{-\tau} = D\cdot(-2ik)^{-i\tau} \frac{-i\tau - \varkappa + Z\alpha\sqrt{\frac{m-\varepsilon} {m+\varepsilon}}}{-i\tau - \frac{iZ\alpha \varepsilon}{k}}. \label{10}$$ Getting an equation for $C/D$ needs matching (\[77\]) with $\eta_{\tau}/\eta_{-\tau}$ obtained from (\[10\]): $$\begin{aligned} \label{eq:CD} \frac{C}{D}=e^{2i\theta}\cdot \frac{\left(-2ik\right)^{-i\tau}}{\left(-2ik\right)^{i\tau}}\cdot \frac{Z\alpha\sqrt{m-\varepsilon}+\left(-i\tau-\varkappa\right)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon}+\left(i\tau-\varkappa\right)\sqrt{m+\varepsilon}}\cdot \frac{i\tau-\frac{iZ\alpha\varepsilon}{k}}{-i\tau-\frac{iZ\alpha\varepsilon}{k}}.\end{aligned}$$ Two sets of approximations were made in deriving (\[eq:CD\]): i. to get $\eta_\tau/\eta_{-\tau}$ at $r=R-0$ we replaced $\beta R$ with $Z \alpha$ and used (\[6\]) which was itself derived for $Z \alpha /r \gg \varepsilon,m$; ii. to get $\eta_\tau/\eta_{-\tau}$ at $r=R+0$ we expanded (\[8\]) and (\[eq:9\]) at $\rho\ll 1$. For $m\cdot R = 0.031$ one cannot expect an accuracy better than 3% and, with growing $|\varepsilon|$ it can even get worse. The accuracy of the final result is not easy to guess from the start, and the best way is to compare it with the exact solution which was found in the previous subsection. Note that all results in [@Kuleshov] are based on the asymptotic behavior (\[6\]) and are therefore approximate by default. Substituting (\[eq:CD\]) into (\[eq:phase\]) we obtain the approximate expression for the scattering phase $\delta_{\varkappa}$. Its dependence on $\varepsilon_{\rm p}\equiv-\varepsilon$ for $\varkappa = -1$ and $Z = 232$ is shown in Fig. \[fig:phase\] (compare with Fig. 3 of [@Kuleshov]). The scattering phase $\delta_{\varkappa}$ exhibits a resonance behavior; it goes through $\pi/2$ at $\varepsilon_{\rm p}/m\approx 4.88$. Let us note that on the real axis of $\varepsilon$ the expression for the scattering phase $\delta_{\varkappa}$ can be written in the same form as in [@Kuleshov] (see Appendix \[sec:real\_phase\]). ![Dependence on $\varepsilon_{\rm p}$ of the scattering phase $\delta_{-1}(\varepsilon_{\rm p}, 232)$ ($Z=232$ and $\varkappa=-1$) for a nucleus with radius $R=0.031/m$. The blue solid line corresponds to the exact phase, the green dashed line corresponds to the approximate one.[]{data-label="fig:phase"}](pics/phase.pdf){width="6.5in"} The positions of the $S$ matrix poles are defined by the same equality (\[eq:pole\]) with $C/D$ given by (\[eq:CD\]): $$\begin{aligned} e^{2i\theta}= \frac{\left(-2ik\right)^{i\tau}}{\left(-2ik\right)^{-i\tau}}\cdot \frac{\Gamma\left(-2i\tau\right)}{\Gamma\left(2i\tau\right)}\cdot \frac{\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)}\cdot \frac{Z\alpha\sqrt{m-\varepsilon}+\left(-i\tau+\varkappa\right)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon}+\left(i\tau+\varkappa\right)\sqrt{m+\varepsilon}}, \label{eq:eq_poles}\end{aligned}$$ where the l.h.s. is defined by (\[77\]). The r.h.s. coincides with Eq. (26) from [@Kuleshov]. The exact expression (\[eq:pole\_detailed\]) is, of course, more complicated, but, anyhow, special functions have to be evaluated numerically in both cases. The accuracy of ${\rm Re}[\varepsilon]$ obtained by the approximate procedure is quite reasonable (see Fig. \[fig:e\_of\_Z\] and Table \[tab:e\_of\_Z\]); however it is much worse for ${\rm Im}[\varepsilon]$, for example $\approx15\%$ at $Z=186$. This is why it is worth getting the exact values of the energies $\varepsilon$. The question that we want to address now is the origin of the resonance and how it transforms for $Z < Z_{\rm cr}$. At $Z < Z_{\rm cr}$ the resonances become bound states, the energies of which are determined by the same type of matching at $r=R$ as before (it is convenient to replace now, in (\[eq:eq\_poles\]), $k$ by $i\lambda$, since on the real axis, for $-m<\varepsilon<+m$, $\lambda$ is real positive): $${\rm exp}(2i\theta) = \frac{(2\lambda)^{i\tau}}{(2\lambda)^{-i\tau}}\cdot \frac{\Gamma(-2i\tau)}{\Gamma(2i\tau)}\cdot \frac{\Gamma\left(1+i\tau-\frac{Z\alpha\varepsilon}{\lambda}\right)} {\Gamma\left(1-i\tau-\frac{Z\alpha\varepsilon}{\lambda}\right)}\cdot \frac{Z\alpha\sqrt{m-\varepsilon}+(\varkappa-i\tau)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon}+(\varkappa+i\tau)\sqrt{m+\varepsilon}}. \label{171}$$ Last, for $Z\alpha < |\varkappa|$ we must change $i\tau$ into $\sigma = \sqrt{\varkappa^2 - Z^2\alpha^2}$: $$\tan\theta = \frac{(2\lambda)^\sigma}{(2\lambda)^{-\sigma}}\cdot \frac{\Gamma(-2\sigma)}{\Gamma(2\sigma)}\cdot \frac{\Gamma\left(1+\sigma-\frac{Z\alpha\varepsilon}{\lambda}\right)} {\Gamma\left(1-\sigma-\frac{Z\alpha\varepsilon}{\lambda}\right)}\cdot \frac{Z\alpha\sqrt{m-\varepsilon}+(\varkappa-\sigma)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon}+(\varkappa +\sigma)\sqrt{m+\varepsilon}}. \label{181}$$ ![The dependence of the ground state energy on $Z$. The square markers are for the exact values of the energy (see (\[eq:pole\_detailed\])) and the round markers are for the approximate ones calculated with the help of (\[eq:eq\_poles\]). The correspondence between color and $Z$ is shown in the legend (the real part of the energy is monotonically decreasing). At $Z=Z_{\rm cr}$ the bound states become resonances with positive ${\rm Im}[\varepsilon]$.[]{data-label="fig:e_of_Z"}](pics/ground_level_210.pdf){width="6.5in"} [|c|c|c|c|c|]{} $Z$ & ${\rm Re}\left(\varepsilon_{\rm appr}\right)$ & ${\rm Im}\left(\varepsilon_{\rm appr}\right)$ & ${\rm Re}\left(\varepsilon\right)$ & ${\rm Im}\left(\varepsilon\right)$\ Let us consider for example $Z\alpha<1$, for which taking a point-like nucleus is reliable. At the limit $R\to0$, the r.h.s. of (\[7\]) becomes infinite. Therefore, the spectrum of the Dirac equation is given by the poles of (\[181\]). They are given by the poles of $\Gamma\left(1+\sigma - \frac{Z\alpha\varepsilon}{\lambda}\right)$: $$\sqrt{\varkappa^2-Z^2\alpha^2} - \frac{Z\alpha\varepsilon}{\sqrt{m^2-\varepsilon^2}} = -1, -2, \dots\equiv -n_r, \label{19}$$ to which must be added, for $\varkappa<0$, the zero of the last term in the denominator of (\[181\])[^8]: $$Z\alpha\sqrt{m-\varepsilon}+(\varkappa +\sigma)\sqrt{m+\varepsilon}=0 \;\Rightarrow\;\sqrt{\varkappa^2-Z^2\alpha^2} - \frac{Z\alpha\varepsilon}{\sqrt{m^2 -\varepsilon^2}} = 0 \equiv n_{r}. \label{20}$$ The electron bound states at $Z < Z_{\rm cr}$ become therefore resonances at $Z > Z_{\rm cr}$; the poles of the $S$ matrix corresponding to the latter describe positron-nucleus scattering. The trajectory of the ground state energy with growing $Z$ is shown in Fig. \[fig:e\_of\_Z\] (see also Table \[tab:e\_of\_Z\]). Let us notice the unusual signs of both real and imaginary parts of the resonance energy. It was suggested in [@Kuleshov] that the sign of the energy should be reversed, under the claim that the corresponding state is a resonance in the positron-nucleus system. Such a sign reversal is usual for holes in the lower continuum of the Dirac equation: the absence of an electron of energy $-\varepsilon$ is equivalent to the presence of a positron with energy $\varepsilon$. Advocating for the same procedure in the case at hands looks a priori suspicious since the resonances that we found originate from electron bound energy levels (however, also empty) that dive from $\varepsilon=+m$ downwards into the lower continuum (and will return upwards to $+m$ if $Z$ decreases). An interpretation of the phenomenon in terms of electrons looks therefore more intuitive. In order to resolve this (apparent) puzzle, we shall solve in the next section the Dirac equation for positrons, which describes the scattering of a positron in the upper continuum on a nucleus. The Dirac equation for positrons: upper continuum wave functions and scattering phases in the Coulomb field of a supercritical nucleus {#sec:upper} ====================================================================================================================================== Changing the sign of $Z\alpha$ in (\[eq:potential\]), we get instead of (\[eq:3\]) $$\left\{ \begin{aligned} &\frac{d\tilde F}{dr} + \frac{\varkappa}{r} \tilde F - (\varepsilon +m -\tilde V(r)) \tilde G = 0,\\ &\frac{d\tilde G}{dr} - \frac{\varkappa}{r} \tilde G + (\varepsilon -m -\tilde V(r)) \tilde F = 0, \end{aligned} \right. \label{21}$$ where [V(r) =]{} , & $r > R$, \[eq:pos\_potential\_r&gt;R\]\ , & $r < R$. Notice that (\[eq:3\]) gives (\[21\]) by the set of transformations $\varkappa\to -\varkappa$, $\varepsilon\to -\varepsilon$, $F\to\tilde G$ and $G\to\tilde F$. The states in the upper continuum ($\varepsilon > m$) describe positron scattering on a nucleus. Since a positron cannot form a bound state with a positively charged nucleus, one could think that no resonance at $Z > Z_{\rm cr}$ will occur, nor the resonant behavior of the scattering phase found in [@Kuleshov] and reproduced in Section \[sec:lower\]. The central issue is therefore to investigate whether bound states and resonances arise or not in the Dirac equation for positrons (\[21\]). Solving (\[21\]) at $r < R$ we obtain $$\left( \begin{array}{l} \tilde F \\ \tilde G \end{array}\right) = {\rm const}\cdot\sqrt{\tilde\beta r}\cdot \left( \begin{array}{l} \pm J_{\mp(1/2 + \varkappa)} (\tilde\beta r) \\ J_{\pm(1/2 - \varkappa)} (\tilde\beta r) \frac{\tilde\beta}{\varepsilon+m-\frac{Z\alpha}{R}} \end{array} \right),\; r < R, \label{23}$$ where $\tilde\beta = \sqrt{(\varepsilon - Z\alpha/R)^2 - m^2}$ and, at small distances, where the solution (\[23\]) will be used, $\tilde\beta \approx \beta \approx Z\alpha/R$. The upper (lower) signs in (\[23\]) should be taken for $\varkappa < 0$ ($\varkappa > 0$). Note that the sign of $\tilde F$ is opposite to that of $F$ in (\[5\]), while the signs of $\tilde G$ and $G$ coincide. Substituting in (\[21\]) the Coulomb potential (\[eq:pos\_potential\_r&gt;R\]) and going to the limit $r \to 0$ we get: $$\left.\left( \begin{array}{l} \tilde F \\ \tilde G \end{array} \right)\right|_{r \to 0} = \tilde\eta_\sigma r^\sigma \left( \begin{array}{c} -1 \\ \frac{-Z\alpha}{\sigma -\varkappa} \end{array} \right) + \tilde\eta_{-\sigma} r^{-\sigma} \left( \begin{array}{c} -1 \\ \frac{-Z\alpha}{-\sigma -\varkappa} \end{array} \right). \label{24}$$ Note that the sign of $\tilde G$ is opposite to that of $G$ in (\[6\]), while the signs of $\tilde F$ and $F$ coincide. Thus, when matching the ratios of $\tilde F/\tilde G$ from (\[23\]) and (\[24\]) at $r=R$ we obtain equations identical to (\[7\]), (\[77\]) with the change $\eta\to\tilde\eta$. Like in (\[8\]), we look for solutions of the form $$\left( \begin{array}{l} \tilde F \\ \tilde G \end{array} \right) = \left( \begin{array}{c} \sqrt{m+\varepsilon} \\ -\sqrt{m-\varepsilon} \end{array} \right) {\rm exp}(-\rho/2) \rho^{i\tau} \left( \begin{array}{c} \tilde Q_1 + \tilde Q_2 \\ \tilde Q_1 - \tilde Q_2 \end{array} \right), \label{25}$$ where as before $\rho = 2\lambda r = -2ikr$. The expressions for $\tilde Q_1$ and $\tilde Q_2$ are given by (\[eq:9\]), where $Z\alpha$ should be substituted by $-Z\alpha$: $$\hspace{-7mm}\left\{ \begin{aligned} \tilde Q_1 &= C\cdot\frac{\frac{iZ\alpha m}{k} + \varkappa} {-i\tau-\frac{iZ\alpha\varepsilon}{k}}\cdot{_{1}F_{1}}\left( i\tau + \frac{iZ \alpha \varepsilon}{k}, 2i\tau + 1, \rho \right) + \\ &\hspace{40mm} + D\cdot\frac{\frac{iZ\alpha m}{k} + \varkappa}{i\tau-\frac{iZ\alpha\varepsilon}{k}}\cdot \rho^{-2i\tau}\cdot {_{1}F_{1}}\left(-i\tau + \frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho\right), \\ \tilde Q_2 &= C\cdot{_{1}F_{1}}\left(1+i\tau + \frac{iZ\alpha\varepsilon}{k}, 2i\tau + 1, \rho \right) + D\rho^{-2i\tau}\cdot{_{1}F_{1}}\left( 1-i\tau + \frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho \right). \end{aligned} \right. \label{26}$$ Since we are interested in resonant states, we demand that only terms $\propto\exp[ikr]$ (outgoing waves) survive at $r\to\infty$. For $\varepsilon<m$, $\exp[ikr]$ becomes $\exp[-\lambda r]$, which describes bound states. In $\tilde Q_1$, the coefficient of the $\exp[-ikr]$ term, being damped by an extra $1/r$, does not contribute, and the condition for the terms proportional to $\exp[-ikr]$ to be absent in $\tilde Q_2$ is $$C\cdot\frac{\Gamma(2i\tau)} {\Gamma\left(1+i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} - D\cdot\frac{\Gamma(-2i\tau)} {\Gamma\left(1-i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} = 0. \label{27}$$ Substituting (\[26\]) into (\[25\]) at the limit $r\to 0$, we reproduce (\[24\]) for $$\begin{aligned} \frac{\tilde\eta_\tau}{\tilde\eta_{-\tau}} & = \frac{(-2ik)^{i\tau}}{(-2ik)^{-i\tau}}\cdot \frac{\Gamma(-2i\tau)}{\Gamma(2i\tau)}\cdot \frac{\Gamma\left(1+i\tau+\frac{iZ\alpha\varepsilon} {k}\right)} {\Gamma\left(1-i\tau+\frac{iZ\alpha \varepsilon} {k}\right)}\cdot \frac{Z\alpha\sqrt{m-\varepsilon} -(-i\tau+\varkappa)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon} -(i\tau+\varkappa)\sqrt{m+\varepsilon}}. \end{aligned} \label{28}$$ Matching Eqs. (\[28\]) and (\[77\]) yields an equation for the energies of the resonant states. After the substitution of ($\varkappa, \varepsilon$) by ($-\varkappa, -\varepsilon$), it coincides with the similar equation that we obtained in Section \[sec:lower\]. Thus, resonances also arise as solutions of the Dirac equation for positrons, at energies $\varepsilon=\xi-\frac{i}{2}\gamma,~\xi>m,~\gamma>0$[^9]. After making the same substitutions as in Section \[sec:lower\], we get equations that coincide with (\[171\]), (\[181\])). This clears the mystery concerning the resonances that we have found there. Positrons states of negative energies should be interpreted in terms of electrons. At $Z < Z_{\rm cr}$ we just found electron–nucleus bound states — with growing $Z$, the energy of the bound particle moves from $-m$ (at $Z=0$) to $+m$ (see also footnote \[footnote:pos\_bound\_states\]) and, at $Z > Z_{\rm cr}$ it becomes complex and located on the second sheet below the right cut. Equations for the scattering phase $\delta_{\varkappa}$ analogous to (\[eq:phase\]), (\[77\]), (\[eq:CD\]) in Section \[sec:lower\] can be written. They coincide with these equations after changing $\varkappa\to-\varkappa$ and $\varepsilon\to-\varepsilon$. It is therefore not necessary to solve the Dirac equation for positrons as we did in this section. It is enough to note that, after substitution of $\varepsilon$ by $-\varepsilon$, $\varkappa$ by $-\varkappa$, $F$ by $\tilde G$ and $G$ by $\tilde F$, Eq. (\[eq:3\]) becomes (\[21\]) with $V(r)$ converted to $\tilde V(r)$. In this way, the formulas of Section \[sec:upper\] can be directly deduced from the ones of Section \[sec:lower\]. Conclusions {#sec:conclusions} =========== In Sections \[sec:lower\] and \[sec:upper\], the scattering of positrons on a supercritical nucleus was studied. It has the spectacular resonance behavior discovered in [@MRG1:1972; @MRG2:1972; @Kuleshov]. In the present paper, results with an exact dependence on the parameter $m\times R$ have been obtained on both sheets of the complex energy plane in the form convenient for numerical evaluation. However, one can hardly hope to study this phenomenon experimentally: even if a supercritical nucleus can be produced in heavy ions collisions, its life time will be so short that one cannot scatter a positron on it, not to mention the still bigger challenge of making a target with supercritical nuclei. Let us note that since the elastic scattering matrix was found to be unitary (the scattering phase is real) there are no inelastic processes in the positron scattering on supercritical nucleus. More realistic is the hope to detect the emission of positrons from a short-lived supercritical nucleus eventually produced in heavy ions collisions. Indeed we do not agree with the claim made in the abstract of [@Kuleshov] (and in contradiction with [@Zeldovich:1971] in particular) that the spontaneous production of $e^+e^-$ pairs from a supercritical nucleus does not occur. On the contrary, we believe that the resonance found in [@Kuleshov] in the system positron—supercritical nucleus is precisely the signal for pair production. It occurs when, as $Z$ grows, an empty electron level dives into the lower continuum of the Dirac equation. In the absence of the nucleus, this empty state in the lower continuum would just mean the presence of a positron. The presence of the nucleus makes the energy of this state complex, and its lifetime is precisely $1/\gamma$. In this lapse of time, an electron from the sea with the same energy $-\xi$ located far from the nucleus can penetrate in its vicinity. It partially screens the charge of the nucleus and, at the same time, an empty electron state arises in the Dirac sea. This is the positron which gets repulsed to infinity by the nucleus. Let us suppose that solutions of the Dirac equation we get are approximately valid also when an electron screens nuclear potential, being embedded in the lower continuum. It means our solutions for the resonance energy and width are almost valid. It well can be so, since electric charge of one electron is small and it is situated far from nucleus, $r \approx 1/m$. So, the obtained width (imaginary part of energy) is the lifetime of positron in the vicinity of nucleus, which is already surrounded by diving electron. Therefore this is the lifetime of the system of nucleus, electron and positron with respect to positron emission to infinity, so it is an average time of $e^{+}e^{-}$ pair production (in reality two independent pairs are produced because of electron spin degeneracy). The potential barrier which holds the positron in the vicinity of the nucleus is shown in Fig. 2 of [@Zeldovich:1971]; its penetration time is given by the analytical formulas (4.14, 4.15), and the results of numerical calculations are shown in Fig. 13 of the same review paper. We reproduced the curve shown in Fig. 13 from the dependence $\gamma(Z)$ that we obtained in Section \[sec:lower\] for the energy of the Gamov (quasistationary) state. Let us finally mention that we agree with the description of the stable states of a supercritical nucleus made in Section 6 of [@Kuleshov]: empty states in the upper continuum, empty discrete levels, and occupied states in the lower continuum. The levels of the lower continuum that get occupied by electrons after the diving process form the so-called “charged vacuum”; it has charge $-n$, where $n$ is the number of these levels. The $n$ positrons that get emitted compensate for this negative charge. A supercritical nucleus is no longer naked and its electric charge is partially screened by these electrons. Indirect evidence of such a phenomenon is found in graphene physics [@Wang734; @NaturePhysics]. We thank O.V. Kancheli, V.D. Mur, V.A. Novikov, and M.I. Eides for useful discussions. We are grateful to V.M. Shabaev who provided us with references [@Maltsev:2014qna; @Maltsev2017]. S.G. is supported by RFBR under grants 16-32-60115 and 16-32-00241, by the Grant of President of Russian Federation for the leading scientific Schools of Russian Federation, NSh-9022-2016, and by the “Dynasty Foundation”. M.V. is supported by RFBR under grant 16-02-00342. M.V. is grateful to LPTHE, CNRS and Sorbonne Univesité for hospitality and funding during the first steps (projet IDEX PACHA OTP-53897) and the last steps of this work. Functions $Q_1$ and $Q_2$ {#sec:Q1Q2} ========================= Substituting (\[8\]) into the Dirac equations (\[eq:3\]) we get: $$\begin{aligned} &\rho(Q_1^\prime + Q_2^\prime) + (i\tau +\varkappa)(Q_1 +Q_2)-\rho Q_2 + Z\alpha \sqrt{\frac{m-\varepsilon}{m+\varepsilon}} (Q_1 - Q_2) = 0,\\ &\rho(Q_1^\prime - Q_2^\prime) + (i\tau -\varkappa)(Q_1 - Q_2)+\rho Q_2 -Z\alpha \sqrt{\frac{m+\varepsilon}{m-\varepsilon}} (Q_1 + Q_2) = 0, \end{aligned} \label{A1}$$ where a prime means the derivative with respect to $\rho$. The sum and difference of the two equations (\[A1\]) give (compare with Eq. (36.5) from [@BLP]): $$\begin{aligned} &\rho Q_1^\prime + \left(i\tau - \frac{iZ\alpha\varepsilon}{k}\right)Q_1 + \left(\varkappa - \frac{iZ\alpha m}{k}\right) Q_2 = 0,\\ &\rho Q_2^\prime + \left(i\tau - \rho +\frac{iZ\alpha\varepsilon}{k}\right)Q_2 + \left(\varkappa + \frac{iZ\alpha m}{k}\right) Q_1 = 0. \end{aligned} \label{A2}$$ Eliminating $Q_1$ or $Q_2$ gives $$\begin{aligned} &\rho Q_1^{\prime\prime} + (2 i\tau +1 -\rho)Q_1^\prime + \left(\frac{iZ\alpha \varepsilon}{k} - i\tau\right) Q_1 = 0,\\ &\rho Q_2^{\prime\prime} + (2 i\tau +1 - \rho)Q_2^\prime + \left(\frac{iZ\alpha \varepsilon}{k} - 1 - i\tau\right) Q_2=0. \end{aligned} \label{A3}$$ Unlike in the case of a point-like nucleus, we do not demand here that the solutions of (\[A3\]) be regular at $\rho=0$. We accordingly consider linear superpositions of the two independent solutions of the second order differential equations (\[A3\]) with arbitrary coefficients. First let us recall that the general solution of the equation $$zu^{\prime\prime} + (\gamma - z)u^\prime -\alpha u = 0 \label{A4}$$ is: $$u = C_1\cdot{_{1}F_{1}}(\alpha, \gamma, z) + C_2\cdot z^{1-\gamma}{_{1}F_{1}}(\alpha -\gamma +1, 2-\gamma, z) , \label{A5}$$ where $C_1$ and $C_2$ are arbitrary coefficients while the $_1F_1$ are the Kummer confluent hypergeometric functions. Thus for the solutions of (\[A3\]) we obtain: $$\hspace{-3mm} \begin{aligned} Q_1 & = A\cdot{_{1}F_{1}}\left(i\tau - \frac{iZ\alpha\varepsilon}{k}, 2i\tau + 1, \rho \right)+ B\cdot\rho^{-2i\tau} {_{1}F_{1}}\left( -i\tau -\frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho \right),\\ Q_2 & = C\cdot{_{1}F_{1}}\left(1 +i\tau - \frac{iZ\alpha \varepsilon}{k}, 2i\tau + 1, \rho \right) + D\cdot\rho^{-2i\tau}{_{1}F_{1}}\left(1 -i\tau -\frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho \right), \end{aligned} \label{A6}$$ where $A$, $B$, $C$, and $D$ are arbitrary coefficients. At small $z$, $_1F_1=1+\mathcal{O}(z)$. Substituting the expansions of (\[A6\]) at small $\rho$ into the first equation in (\[A2\]) determines $A$ and $B$, respectively, in terms of $C$ and $D$: $$\begin{aligned} &\left( i\tau - \frac{iZ\alpha\varepsilon}{k}\right) A + \left(\varkappa - \frac{iZ\alpha m}{k}\right) C = 0, \label{A7}\\ &\left( -i\tau - \frac{iZ\alpha\varepsilon}{k}\right) B + \left(\varkappa - \frac{iZ\alpha m}{k}\right) D = 0. \label{A8}\end{aligned}$$ Plugging then $A$ and $B$ obtained from (\[A7\]) and (\[A8\]) into (\[A6\]) yields (\[eq:9\]). The scattering phase according to [@Kuleshov] {#sec:real_phase} ============================================= Considering real $\varepsilon<-m$ below the left cut we can rewrite the expression for the scattering phase in a more compact form. Let us introduce the following notations equivalent to those used in [@Kuleshov]: $$\begin{aligned} \label{eq:kuleshov_notations} \exp(i\varphi) &= e^{2i\theta}\frac{(2k)^{-i\tau}\Gamma(2i\tau)} {(2k)^{i\tau}\Gamma(-2i\tau)},\\ a &= \frac{Z\alpha\sqrt{m-\varepsilon} +(-i\tau+\varkappa)\sqrt{m+\varepsilon}} {\Gamma(1-i\tau - \frac{iZ\alpha\varepsilon}{k})},\\ b &= \frac{Z\alpha\sqrt{m-\varepsilon} -(-i\tau+\varkappa)\sqrt{m+\varepsilon}} {\Gamma(1-i\tau + \frac{iZ\alpha\varepsilon}{k})}.\end{aligned}$$ With these notations the approximate ratio $C/D$ defined by (\[eq:CD\]) can be written in the following way: $$\begin{aligned} \label{eq:CD_Kuleshov_numerator} C/D &=e^{i\varphi-\pi\tau}\cdot\frac{a^{*}}{b}\cdot \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(2i\tau\right)}\cdot \frac{\Gamma\left(i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} {\Gamma\left(-i\tau+\frac{iZ\alpha\varepsilon}{k}\right)}=\\ \label{eq:CD_Kuleshov_denominator} &=e^{i\varphi-\pi\tau}\cdot\frac{b^{*}}{a}\cdot \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(2i\tau\right)}\cdot \frac{\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)},\end{aligned}$$ where we used $$\frac{\left(-i\right)^{-i\tau}}{\left(-i\right)^{i\tau}}=e^{-\pi\tau},$$ and $$\begin{aligned} \left(-i\tau-\frac{iZ\alpha\varepsilon}{k}\right) & \left(Z\alpha\sqrt{m-\varepsilon}+ \left(i\tau-\varkappa\right)\sqrt{m+\varepsilon}\right)=\nonumber\\ \nonumber &=-i\tau\left(Z\alpha\sqrt{m-\varepsilon} +\left(i\tau-\varkappa\right)\sqrt{m+\varepsilon}\right) -iZ\alpha\varepsilon\left(\frac{Z\alpha}{i\sqrt{m+\varepsilon}} +\frac{i\tau-\varkappa}{i\sqrt{m-\varepsilon}}\right)=\\ \nonumber &=\frac{-i\tau\left(i\tau-\varkappa\right)\left(m+\varepsilon\right) -\left(Z\alpha\right)^{2}\varepsilon}{\sqrt{m+\varepsilon}} +\frac{-i\tau\left(Z\alpha\right)\left(m-\varepsilon\right) -Z\alpha\varepsilon\left(i\tau-\varkappa\right)}{\sqrt{m-\varepsilon}}=\\ \nonumber &=\frac{\left(Z\alpha\right)^{2}m +\varkappa\left(i\tau-\varkappa\right)\left(m+\varepsilon\right)} {\sqrt{m+\varepsilon}} +\frac{-\left(i\tau-\varkappa\right)Z\alpha m-\varkappa Z\alpha\left(m-\varepsilon\right)}{\sqrt{m-\varepsilon}}= \nonumber \\ &=Z\alpha m\left(\frac{Z\alpha}{\sqrt{m+\varepsilon}}+ \frac{-i\tau+\varkappa}{\sqrt{m-\varepsilon}}\right) -\varkappa\left(Z\alpha\sqrt{m-\varepsilon} +\left(-i\tau+\varkappa\right)\sqrt{m+\varepsilon}\right)=\nonumber\\ &=\left(Z\alpha\sqrt{m-\varepsilon}+ \left(-i\tau+\varkappa\right)\sqrt{m+\varepsilon}\right)\left(\frac{iZ\alpha m}{k}-\varkappa\right),\end{aligned}$$ where we used the relation $$\begin{aligned} \left(-i\tau-\varkappa\right)\left(i\tau-\varkappa\right)= \tau^{2}+\varkappa^{2}=\left(Z\alpha\right)^{2}.\end{aligned}$$ Then, the scattering phase $\delta_{\varkappa}$ can be written as follows (by substituting (\[eq:CD\_Kuleshov\_numerator\]) and (\[eq:CD\_Kuleshov\_denominator\]) into the numerator and the denominator of (\[eq:phase\]) respectively): $$e^{2i\delta_{\varkappa}} = -\frac{{\rm exp}\left(\frac{\pi\tau}{2}+\frac{i\varphi}{2}\right)a^* -{\rm exp}\left(-\frac{\pi\tau}{2}-\frac{i\varphi}{2}\right)b} {{\rm exp}\left(\frac{\pi\tau}{2}-\frac{i\varphi}{2}\right)a -{\rm exp}\left(-\frac{\pi\tau}{2} +\frac{i\varphi}{2}\right)b^*}. \label{13}$$ In eq.(22) of [@Kuleshov] the phase $\delta_{\varkappa}$ is expressed through the ratio $f^*/f$, where $f$ is the Jost function. Our result differs from eq.(23) of [@Kuleshov] by the substitution $\varphi\to\varphi/2$ (it seems that there is a misprint in [@Kuleshov]). Qualitative explanation of the resonance phenomena in the $e^+ N^+$ system {#sec:eff_potential} ========================================================================== The effective potential for an electron in the field of a supercritical nucleus is derived in [@Zeldovich:1971] from the Dirac equation, for $\varepsilon\approx-m$. It is attractive at short distances, repulsive at large distances, with a Coulomb barrier in between. We derive below, in a similar way, the effective potential for a positron in the field of a similar nucleus, in the vicinity of $\varepsilon=+m$. As already noticed at the beginning of Section \[sec:upper\], the Dirac equation (\[eq:3\]) for electrons becomes (\[21\]) for positrons after the following substitutions: $$\varkappa\to-\varkappa, \;\; \varepsilon\to-\varepsilon, \;\; F \to \tilde G, \;\; G \to \tilde F, \;\; V(r) \to \tilde V(r) = -V(r). \label{C1}$$ To proceed like in [@Zeldovich:1971], we deduce the second order differential equation satisfied by $\tilde G$ from (\[21\]), which is $$\label{eq:G_shred} \tilde G'' +\frac{\tilde V'}{\varepsilon-m-\tilde V}\left(\tilde G'-\frac{\varkappa}{r}\tilde G\right) +\left(\left(\varepsilon-\tilde V\right)^{2}-m^{2} +\frac{\varkappa\left(1-\varkappa\right)}{r^{2}}\right)\tilde G=0,$$ in which “$~'~$” means here derivation with respect to $r$. In order to transform this equation into a Schrödinger-like equation, the following change of variables must be operated $$\label{eq:G_to_chi} \tilde G=\chi\sqrt{m-\varepsilon+\tilde V}.$$ Thus, we get: $$\label{eq:chi_shred} \chi''+ k^{2}\chi=0,$$ where $k^{2}=2m\left(E-U\right)$, $E=\frac{\varepsilon^{2}-m^{2}}{2m}$. The effective potential is seen to be made of two terms: $U=U_{1}+U_{2}$, where: $$\label{eq:U1} U_{1}=\frac{\varepsilon}{m}\tilde V-\frac{1}{2m}\tilde V^{2} -\frac{\varkappa\left(1-\varkappa\right)}{2mr^{2}},$$ and $$\label{eq:U2} U_{2}=\frac{\tilde V''}{4m\left(\varepsilon-m-\tilde V\right)} +\frac{3}{8m}\frac{\left(\tilde V'\right)^{2}}{\left(\varepsilon-m-\tilde V\right)^{2}} +\frac{\varkappa\tilde V'}{2mr\left(\varepsilon-m-\tilde V\right)}.$$ It coincides with the equation obtained in [@Zeldovich:1971] after the substitution $\varepsilon\to-\varepsilon$, $\varkappa\to-\varkappa$ and $\tilde V\to-V$. We are interested in positrons with $\varepsilon\approx m$. At large distances the first term in $U_1$ dominates, and describes the repulsion of the positron by the nucleus. For the ground state $\varkappa=1$ the centrifugal term in $U_1$ vanishes. Finally, for $\varepsilon=m$ and $\varkappa=1$, we get from (\[eq:U1\]) and (\[eq:U2\]) $$\label{eq:eff_potential} U = \frac{Z\alpha}{r} + \frac{3-4(Z\alpha)^2}{8mr^{2}}.$$ At short distances the terms $\propto 1/r^2$ dominates and, for a supercritical nucleus, they lead to attraction, while the Coulomb term dominates at $r\geq1/m$. This attractive force explains the existence of resonances in the $e^{+}N^{+}$ system while bound state cannot exist due to the narrowness of the well. Let us note that the fall to the center occurs only for $Z\alpha>1$ when the coefficient in front of the term $\propto -1/r^2$ becomes larger than $1/8m$ (see [@LL3], eq. (35.10)). We are grateful to V.A. Novikov who brought our attention to this feature. In the problem under consideration the finite nucleus size prevents the fall to the center. [^1]: The reader can be convinced as regards this interpretation as follows. In the present simple formalism, which only uses the Dirac equation, the energy of a bare positron, which is obtained by simply taking the limit $Z\to0$, is found to be $-m$. Since the “production” of such a particle costs at least the energy $+m$, the result that is obtained can only be interpreted in term of an electron with energy $-(-m)=+m$. This is what we mean by the statement that “there is no more information”. A more satisfying description of positrons can only be achieved in the framework of Quantum Field Theory, where creating (annihilating) an electron and annihilating (creating) a positron both occur in the expansion of the field operator $\psi$ in terms of creation and annihilation operators.\[footnote:pos\_bound\_states\] [^2]: Any solution with a negative index of the Bessel function is not normalizable, so it should be discarded. [^3]: The procedure used in [@Kuleshov] amounts to stating that, below the left cut, $\lambda = -i\sqrt{(m-\varepsilon)(-m-\varepsilon)}$. So doing, $\sqrt{-m-\varepsilon}$ is defined with the same cut $(-\infty;-m)$ as $\sqrt{m+\varepsilon}$, with positive values below the cut. With such a definition, $-i\sqrt{-m-\varepsilon}={\rm sqrt}\left(m+\varepsilon\right)$ everywhere on the physical sheet, not only below the left cut. There is no need to rewrite formulas in this way since, when numerical outputs are needed, we should return to the original definition (\[eq:lambda\_sheets\]). Let us note that on the first sheet formulas (17) and (26) from [@Kuleshov] are exactly the same. [^4]: Let us note that changing the signs of both square roots is still permitted since it leads to changing the sign of the full wave function. [^5]: Unlike in [@Kuleshov] we did not feel necessary to use Tricomi functions. [^6]: Due to the unitarity of the $S$ matrix, there is a zero of $e^{2i\delta_{\varkappa}}$ at $\varepsilon=-\xi-\frac{i}{2}\gamma$ that is symmetric to the pole $\varepsilon=-\xi+\frac{i}{2}\gamma$ with respect to the real axis. It corresponds to incoming waves instead of the outgoing waves that we selected. [^7]: It corresponds to ${\rm Re}[k]>0$, ${\rm Im}[k]=\frac{\displaystyle{\rm Re}[\varepsilon]{\rm Im}[\varepsilon]} {\displaystyle{\rm Re}[k]}<0$. In [@MRG1:1972; @MRG2:1972; @Zeldovich:1971] the resonance in positron scattering on a supercritical nucleus was discussed. [^8]: This term is proportional to the sum $\left(\sqrt{\varkappa^{2}-\left(Z\alpha\right)^{2}}-\frac{\displaystyle Z\alpha\varepsilon} {\displaystyle\sqrt{m^{2}-\varepsilon^{2}}}\right)+\left(\varkappa+ \frac{\displaystyle Z\alpha m}{\displaystyle\sqrt{m^{2}-\varepsilon^{2}}}\right)$. It is easy to check that, when the first term vanishes, so does the second. Their sum increasing monotonically when $\varepsilon$ increases can vanish only once. [^9]: One may wonder how a positron, being repelled from the positively charged nucleus, can form a quasistationary resonance state with it. This unusual phenomena is explained in Appendix \[sec:eff\_potential\].

ArXiv

--- abstract: 'A capacity bounded grammar is a grammar whose derivations are restricted by assigning a bound to the number of every nonterminal symbol in the sentential forms. In the paper the generative power and closure properties of capacity bounded grammars and their Petri net controlled counterparts are investigated.' author: - Ralf Stiebe - Sherzod Turaev bibliography: - 'stiebe.bib' title: Capacity Bounded Grammars and Petri Nets --- Introduction {#sec:introduction} ============ The close relationship between Petri nets and language theory has been extensively studied for a long time [@cre:man; @das:pau]. Results from the theory of Petri nets have been applied successfully to provide elegant solutions to complicated problems from language theory [@esp; @hau:jan]. A context-free grammar can be associated with a context-free (communica-tion-free) Petri net, whose places and transitions, correspond to the nonterminals and the rules of the grammar, respectively, and whose arcs and weights reflect the change in the number of nonterminals when applying a rule. In some recent papers, context-free Petri nets enriched by additional components have been used to define regulation mechanisms for the defining grammar [@das:tur; @tur]. Our paper continues the research in this direction by restricting the (context-free or extended) Petri nets with place capacity. Quite obviously, a context-free Petri net with place capacity regulates the defining grammar by permitting only those derivations where the number of each nonterminal in each sentential form is bounded by its capacity. A similar mechanism was discussed in [@gin:spa1] where the total number of nonterminals in each sentential form is bounded by a fixed integer. There it was shown that grammars regulated in this way generate the family of context-free languages of finite index, even if arbitrary nonterminal strings are allowed as left-hand sides. The main result of this paper is that, somewhat surprisingly, grammars with capacity bounds have a greater generative power. This paper is organized as follows. Section \[sec:def\] contains some necessary definitions and notations from language and Petri net theory. The concepts of grammars with capacities and grammars controlled by Petri nets with place capacities are introduced in section \[sec:capacities\]. The generative power and closure properties of capacity-bounded grammars are investigated in sections \[sec:power-gs\] and \[sec:nb-cfg\]. Results on grammars controlled by Petri nets with place capacities are given in section \[sec:PNC\]. Preliminaries {#sec:def} ============= Throughout the paper, we assume that the reader is familiar with basic concepts of formal language theory and Petri net theory; for details we refer to [@das:pau; @han; @rei:roz]. The set of natural numbers is denoted by ${\mathbb{N}}$, the power set of a set S by ${\mathcal{P}({S})}$. We use the symbol $\subseteq$ for inclusion and $\subset$ for proper inclusion. The *length* of a string $w \in X^*$ is denoted by $|w|$, the number of occurrences of a symbol $a$ in $w$ by $|w|_a$ and the number of occurrences of symbols from $Y\subseteq X$ in $w$ by $|w|_Y$. The *empty* string is denoted by ${\lambda}$. A *phrase structure grammar* (due to Ginsburg and Spanier [@gin:spa1]) is a quadruple $G=(V, \Sigma, S, R)$ where $V$ and $\Sigma$ are two finite disjoint alphabets of *nonterminal* and *terminal* symbols, respectively, $S\in V$ is the *start symbol* and is a finite set of *rules*. A string $x\in (V\cup \Sigma)^*$ *directly derives* a string $y\in (V\cup \Sigma)^*$ in $G$, written as $x{\Rightarrow}y$, if and only if there is a rule $u\to v\in R$ such that $x=x_1ux_2$ and $y=x_1vx_2$ for some $x_1, x_2\in (V\cup \Sigma)^*$. The reflexive and transitive closure of the relation ${\Rightarrow}$ is denoted by ${\Rightarrow}^*$. A derivation using the sequence of rules $\pi=r_1r_2\cdots r_k$, $r_i\in R$, $1\leq i\leq k$, is denoted by $\xRightarrow{\pi}$ or $\xRightarrow{r_1r_2\cdots r_k}$. The *language* generated by $G$, denoted by $L(G)$, is defined by $L(G)=\{w\in \Sigma^*{:}S{\Rightarrow}^* w\}.$ A phrase structure grammar $G=(V, \Sigma, S, R)$ is called *context-free* if each rule $u\to v\in R$ has $u\in V$. The family of context-free languages is denoted by $\mathbf{CF}$. A *matrix grammar* is a quadruple $G=(V, \Sigma, S, M)$ where $V, \Sigma, S$ are defined as for a context-free grammar, $M$ is a finite set of *matrices* which are finite strings (or finite sequences) over a set of context-free rules. The language generated by the grammar $G$ consists of all strings $w\in \Sigma^*$ such that there is a derivation $S\xRightarrow{r_1r_2\cdots r_n}w$ where $r_1r_2\cdots r_n$ is a concatenation of some matrices $m_{i_1}, m_{i_2}, \ldots, m_{i_k}\in M$, $k\geq 1$. The family of languages generated by matrix grammars without erasing rules (with erasing rules, respectively) is denoted by $\mathbf{MAT}$ (by $\mathbf{MAT}^{{\lambda}}$, respectively). A *vector grammar* is defined like a matrix grammar, but the derivation sequence $r_1r_2\cdots r_n$ has to be a shuffle of some matrices $m_{i_1}, m_{i_2}, \ldots, m_{i_k}\in M$, $k\geq 1$. A *semi-matrix grammar* is defined like a matrix grammar, but the derivation sequence $r_1r_2\cdots r_n$ has to be the semi-shuffle of some matrices $m_{i_1}, m_{i_2}, \ldots, m_{i_k}\in M$, $k\geq 1$, i.e., from the shuffle of sequences from $\bigcup_{i=1}^t m_i^*$ where $$M=\{m_1,\ldots,m_t\}.$$ The language families generated by vector and semi-matrix grammars are denoted by ${{\bf V}}^{[{\lambda}]}$ and ${{\bf sMAT}}^{[{\lambda}]}$. A *Petri net* (PN) is a construct $N = (P, T, F, \phi)$ where $P$ and $T$ are disjoint finite sets of *places* and *transitions*, respectively, $F \subseteq (P\times T) \cup (T\times P)$ is the set of *directed arcs*, $$\varphi: (P\times T) \cup (T\times P) \rightarrow \{0, 1, 2, \dots\}$$ is a *weight function*, where $\varphi(x,y)=0$ for all $(x,y)\in ((P\times T) \cup (T\times P))-F$. A mapping $$\mu: P \rightarrow \{0,1,2, \ldots\}$$ is called a *marking*. For each place $p\in P$, $\mu(p)$ gives the number of *tokens* in $p$. $^{\bullet}x=\{y{:}\, (y,x)\in F\}$ and $x^{\bullet}=\{y{:}\, (x,y)\in F\}$ are called the sets of *input* and *output* elements of $x\in P\cup T$, respectively. A sequence of places and transitions $\rho=x_1x_2\cdots x_n$ is called a *path* if and only if no place or transition except $x_1$ and $x_n$ appears more than once, and $x_{i+1}\in x^\bullet_{i}$ for all $1\leq i\leq n-1$. We denote by $P_\rho, T_\rho, F_\rho$ the sets of places, transitions and arcs of $\rho$. Two paths $\rho_1$, $\rho_2$ are called *disjoint* if $P_{\rho_1}\cap P_{\rho_2}=\emptyset$ and $T_{\rho_1}\cap T_{\rho_2}=\emptyset$. A path $\rho=t_{1}p_{1}t_{2}p_{2}\cdots p_{k-1}t_{k}$ ($\rho=p_{1}t_{1}p_{1}t_{2}\cdots t_{k}p_{1}$) is called a *chain* (*cycle*). A transition $t \in T$ is *enabled* by marking $\mu$ iff $\mu(p)\geq \phi(p,t)$ for all $p\in P$. In this case $t$ can *occur*. Its occurrence transforms the marking $\mu$ into the marking $\mu'$ defined for each place $p \in P$ by $\mu'(p)=\mu(p)-\phi(p,t)+\phi(t,p)$. This transformation is denoted by $\mu\xrightarrow{t}\mu'$. A finite sequence $t_1t_2\cdots t_k$ of transitions is called *an occurrence sequence* enabled at a marking $\mu$ if there are markings $\mu_1, \mu_2, \ldots, \mu_k$ such that $\mu \xrightarrow{t_1} \mu_1 \xrightarrow{t_2} \ldots \xrightarrow{t_k} \mu_k$. For each $1\leq i\leq k$, marking $\mu_i$ is called *reachable* from marking $\mu$. $\mathcal{R}(N, \mu)$ denotes the set of all reachable markings from a marking $\mu$. A *marked* Petri net is a system $N=(P, T, F, \phi, \iota)$ where $(P, T, F, \phi)$ is a Petri net, $\iota$ is the *initial marking*. Let $M$ be a set of markings, which will be called *final* markings. An occurrence sequence $\nu$ of transitions is called *successful* for $M$ if it is enabled at the initial marking $\iota$ and finished at a final marking $\tau$ of $M$. A Petri net $N$ is said to be $k$-*bounded* if the number of tokens in each place does not exceed a finite number $k$ for any marking reachable from the initial marking $\iota$, i.e., $\mu(p)\leq k$ for all $p\in P$ and for all $\mu\in \mathcal{R}(N, \iota)$. A Petri net is called *bounded* if it is $k$-bounded for some $k\geq 1$. A Petri net with *place capacity* is a system $N=(P, T, F, \phi, \iota,\kappa)$ where $(P, T, F, \phi,\iota)$ is a marked Petri net and $\kappa:P \to {\mathbb{N}}$ is a function assigning to each place a number of maximal admissible tokens. A marking $\mu$ of $N$ is valid if $\mu(p)\leq \kappa(p)$, for each place $p\in P$. A transition $t \in T$ is *enabled* by a marking $\mu$ if additionally the successor marking is valid. A *cf Petri net* with respect to a context-free grammar $G=(V,\Sigma, S, R)$ is a system $$N=(P, T, F, \phi, \beta, \gamma, \iota)$$ where - labeling functions $\beta:P\rightarrow V$ and $\gamma:T\rightarrow R$ are bijections; - $(p,t)\in F$ iff $\gamma(t)=A\rightarrow \alpha$ and $\beta(p)=A$ and the weight of the arc $(p,t)$ is 1; - $(t,p)\in F$ iff $\gamma(t)=A\rightarrow \alpha$, $\beta(p)=x$ where $|\alpha|_x>0$ and the weight of the arc $(t,p)$ is $|\alpha|_x$; - the initial marking $\iota$ is defined by $\iota(\beta^{-1}(S))= 1$ and $\iota(p) = 0$ for all $p\in P-\beta^{-1}(S)$. Further we recall the definitions of extended cf Petri nets, and grammars controlled by these Petri nets (for details, see [@das:tur; @tur]). Let $G=(V, \Sigma, S, R)$ be a context-free grammar with its corresponding cf Petri net $$N=(P, T, F, \phi, \beta, \gamma, \iota).$$ Let $T_1, T_2, \ldots, T_n$ be a partition of $T$. 1\. Let $\Pi=\{\rho_1, \rho_2, \ldots, \rho_n\}$ be the set of disjoint chains such that $T_{\rho_i}=T_i$, $1\leq i\leq n$, and $$\bigcup_{\rho\in\Pi}P_\rho\cap P=\emptyset.$$ An *$h$-Petri net* is a system $N_h=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ where and $E=\bigcup_{\rho\in\Pi}F_\rho$; the weight function $\varphi$ is defined by $\varphi(x,y)=\phi(x,y)$ if $(x,y)\in F$ and $\varphi(x,y)=1$ if $(x,y)\in E$; the labeling function $\zeta:P\cup Q\rightarrow V\cup\{\lambda\}$ is defined by $\zeta(p)=\beta(p)$ if $p\in P$ and $\zeta(p)=\lambda$ if $p\in Q$; the initial marking $\mu_0$ is defined by $\mu_0(p)=\iota(p)$ if $p\in P$ and $\mu_0(p)=0$ if $p\in Q$; $\tau$ is the final marking where $\tau(p)=0$ for all $p\in P\cup Q$. 2\. Let $\Pi=\{\rho_1, \rho_2, \ldots, \rho_n\}$ be the set of disjoint cycles such that $T_{\rho_i}=T_i$, $1\leq i\leq n$, and $$\bigcup_{\rho\in\Pi}P_\rho\cap P=\emptyset.$$ A *$c$-Petri net* is a system $N_c=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ where $Q=\bigcup_{\rho\in\Pi}P_\rho$ and $E=\bigcup_{\rho\in\Pi}F_\rho$; the weight function $\varphi$ is defined by $\varphi(x,y)=\phi(x,y)$ if and $\varphi(x,y)=1$ if $(x,y)\in E$; the labeling function $\zeta:P\cup Q\rightarrow V\cup\{\lambda\}$ is defined by $\zeta(p)=\beta(p)$ if $p\in P$ and $\zeta(p)=\lambda$ if $p\in Q$; the initial marking $\mu_0$ is defined by $\mu_0(p)=\iota(p)$ if $p\in P$, and $\mu_0(p_{i,1})=1$, $\mu_0(p_{i,j})=0$ where $p_{i,j}\in P_i$, $1\leq i\leq n$, $2\leq j\leq k_i$; $\tau$ is the final marking where $\tau(p)=0$ if $p\in P$, and $\tau(p_{i,1})=1$, $\tau(p_{i,j})=0$ where $p_{i,j}\in P_i$, $1\leq i\leq n$, $2\leq j\leq k_i$. 3\. Let $\Pi=\{\rho_1, \rho_2, \ldots, \rho_n\}$ be the set of cycles such that $T_{\rho_i}\!=T_i$, $1\leq i\leq n$, and $$\bigcup_{\rho\in\Pi}P_\rho\cap P=\emptyset.$$ An *$s$-Petri net* is a system $N_s=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ where $Q=\bigcup_{\rho\in\Pi}P_\rho, E=\bigcup_{\rho\in\Pi}F_\rho$; the weight function $\varphi$ is defined by $\varphi(x,y)=\phi(x,y)$ if $(x,y)\in F$ and $\varphi(x,y)=1$ if $(x,y)\in E$; the labeling function $\zeta:P\cup Q\rightarrow V\cup\{\lambda\}$ is defined by $\zeta(p)=\beta(p)$ if $p\in P$ and $\zeta(p)=\lambda$ if $p\in Q$; $\mu_0$ is the initial marking where $\mu_0(p_0)=1$ and $\mu_0(p)=\iota(p)$ if $p\in (P\cup Q)-\{p_0\}$; $\tau$ is the final marking where $\tau(p_0)=1$ and $\tau(p)=0$ if $p\in (P\cup Q)-\{p_0\}$. Figure \[fig:xPNs\] depicts extended cf Petri nets which are constructed with respect to the context-free grammar $G'=(\{S, A, B\}, \Sigma, S, R)$ where $R$ consists of $r_0: S\to AB$, $r_1: A\to \lambda$, $ r_3: A\rightarrow aA$, $r_5: A\to bA$, $r_2: B\to \lambda$, $r_4: B\to aB$, $r_6: B\to bB$.$\diamond$ A *$z$-PN controlled grammar* is a system $G=(V, \Sigma, S, R, N_z)$ where is a context-free grammar and $N_z$ is $z$-Petri net with respect to the context-free grammar $G'$ where $z\in\{h, c, s\}$. The *language* generated by a $z$-Petri net controlled grammar $G$ consists of all strings $w\in \Sigma^*$ such that there is a derivation $S\xRightarrow{r_1r_2\cdots r_k}w\in \Sigma^*$ and a successful occurrence sequence of transitions $\nu=t_1t_2\cdots t_k$ of $N_z$ such that $r_1r_2\cdots r_k=\gamma(t_1t_2\cdots t_k)$. Grammars and Petri nets with capacities {#sec:capacities} ======================================= We will now introduce grammars with capacities and show some relations to similar concepts known from the literature. A *capacity-bounded* grammar is a quintuple $G=(V,\Sigma,S,R,\kappa)$ where $G'=(V,\Sigma,S,R)$ is a grammar and $\kappa: V \to {\mathbb{N}}$ is a capacity function. The language of $G$ contains all words $w \in L(G')$ that have a derivation $S{\Rightarrow}^* w$ such that $|\beta|_A\leq \kappa(A)$ for all $A\in V$ and each sentential form $\beta$ of the derivation. The families of languages generated by arbitrary capacity-bounded grammars (due to Ginsburg and Spanier) and by context-free capacity-bounded grammars are denoted by $\mathbf{GS}_{{\mathit{cb}}}$ and $\mathbf{CF}_{{\mathit{cb}}}$, respectively. The capacity function mapping each nonterminal to $1$ is denoted by $\mathbf{1}$. Capacity bounded grammars are closely related to nonterminal-bounded, deri-vation-bounded and finite index grammars. A grammar $G=(V,\Sigma,S,R)$ is *nonterminal bounded* if $|\beta|_V\leq k$ for some fixed $k \in {\mathbb{N}}$ and all sentential forms $\beta$ derivable in $G$. The *index* of a derivation in $G$ is the maximal number of nonterminal symbols in its sentential forms. $G$ is of *finite index* if every word in $L(G)$ has a derivation of index at most $k$ for some fixed $k\in {\mathbb{N}}$. The family of context-free languages of finite index is denoted by $\mathbf{CF}_{{\mathit{fin}}}$. A *derivation-bounded* grammar is a quintuple $G=(V,\Sigma,S,R,k)$ where $G'=(V,\Sigma,S,R)$ is a grammar and $k \in {\mathbb{N}}$ is a bound on the number of allowed nonterminals. The language of $G$ contains all words $w \in L(G')$ that have a derivation $S{\Rightarrow}^* w$ such that $|\beta|_V\leq k$, for each sentential form $\beta$ of the derivation. It is well-known that the family of derivation bounded languages is equal to $\mathbf{CF}_{{\mathit{fin}}}$, even if arbitrary grammars due to Ginsburg and Spanier are permitted [@gin:spa2]. \[exa:NBLnotCF1\] Let $G=(\{S, A, B, C, D, E, F\}, \{a, b, c\}, S, R,\mathbf{1})$ be the capacity-bounded grammar where $R$ consists of the rules: $$\begin{array}{llll} r_1: S\to ABCD, & r_2: AB\to aEFb, & r_3: CD\to cAD, &r_4: EF\to EC,\\ r_5: EF\to FC, & r_6: AD\to FD, & r_7: AD\to ED, & r_8: EC\to AB,\\ r_9: FD\to CD, & r_{10}: FC\to AF,& r_{11}: AF\to {\lambda}, & r_{12}: ED\to {\lambda}. \end{array}$$ The possible derivations are exactly those of the form $$\begin{array}{ll} S &\xRightarrow{r_1}ABCD\xRightarrow{(r_2r_3r_4r_6r_8r_9)^n}a^nABb^nc^nCD \xRightarrow{r_2r_3}a^{n+1}EFb^{n+1}c^{n+1}AD \\ & \xRightarrow{r_5r_7}a^{n+1}FCb^{n+1}c^{n+1}ED\xRightarrow{r_{10}r_{11}r_{12}}a^nb^nc^n \end{array}$$ (in the last phase, the sequences $r_{10}r_{12}r_{11}$ and $r_{12}r_{10}r_{11}$ could also be applied with the same result). Therefore, $L(G)=\{a^nb^nc^n{:}n\geq 1\}$.$\diamond$ \[exa:NBLnotCF2\] Let $G=(\{S,A,B,C\},\{a,b,c\},S,R,\mathbf{1})$ be the context-free capa-city-bounded grammar where $R$ consists of the rules $r_1: S\to aBbaAb$, $r_2: A\to aBb$, $r_3: B\to C$, $r_4: C\to A$, $r_5: A\to BC$, $r_6: A\to c$, and let $M$ be the regular set $M=\{a^*ccb^*a^*cb^*\}$. The derivations in $G$ generating words from $M$ are exactly those of the form $$\begin{array}{ll} S &\xRightarrow{r_1}aBbaAb\xRightarrow{(r_3r_2r_4r_3r_2r_4)^n}a^nBb^na^nAb^n \xRightarrow{r_6r_3r_4}a^nAb^na^ncb^n\\ &\xRightarrow{(r_2r_3r_4)^m}a^{n+m}Ab^{n+m}a^ncb^n \xRightarrow{r_5r_4r_3r_6r_4r_6}a^{n+m}ccb^{n+m}a^ncb^n \end{array}$$ (one can also apply $r_3r_6r_4$ in the third phase and $r_5r_4r_6r_3r_4r_6$ in the last phase with the same result). Hence, $ L(G)\cap M=\{a^nccb^na^mcb^m{:}n\geq m\geq 1\}\not\in \mathbf{CF}, $ implying that $L(G)$ is not context-free.$\diamond$ The above examples show that capacity-bounded grammars – in contrast to derivation bounded grammars – can generate non-context-free languages. The generative power of capacity-bounded grammars will be studied in detail in the following two sections. The notions of finite index and bounded capacities can be extended to matrix, vector and semi-matrix grammars. The corresponding language families are denoted by ${{\bf MAT}}^{[{\lambda}]}_{{\mathit{fin}}}$, ${{\bf V}}^{[{\lambda}]}_{{\mathit{fin}}}$, ${{\bf sMAT}}^{[{\lambda}]}_{{\mathit{fin}}}$, ${{\bf MAT}}^{[{\lambda}]}_{cb}$, ${{\bf V}}^{[{\lambda}]}_{cb}$, ${{\bf sMAT}}^{[{\lambda}]}_{cb}$. Also control by Petri nets can in a natural way be extended to Petri nets with place capacities. Since an extended cf Petri net $N_z$, $z\in\{h, c, s\}$, has two kinds of places, i.e., places labeled by nonterminal symbols and *control* places, it is interesting to consider two types of place capacities in the Petri net: first, we demand that only the places labeled by nonterminal symbols are with capacities (*weak capacity*), and second, all places of the net are with capacities (*strong capacity*). A $z$-Petri net $N_z=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$ is with *weak capacity* if the corresponding cf Petri net $(P, T, F, \phi, \iota)$ is with place capacity, and *strong capacity* if the Petri net $(P\cup Q, T, F\cup E, \varphi, \mu_0)$ is with place capacity. A grammar controlled by a $z$-Petri net with *weak* (*strong*) *capacity* is a $z$-Petri net controlled grammar $G = (V, \Sigma, S, R, N_z)$ where $N_z$ is with weak (strong) place capacity. We denote the families of languages generated by grammars (with erasing rules) controlled by $z$-Petri nets with weak and strong place capacities by $\mathbf{wPN}_{cz}$, $\mathbf{sPN}_{cz}$ ($\mathbf{wPN}^{\lambda}_{cz}$, $\mathbf{sPN}^{\lambda}_{cz}$), respectively, where $z\in\{h, c, s\}$. The power of arbitrary grammars with capacities {#sec:power-gs} =============================================== It will be shown in this section that arbitrary grammars (due to Ginsburg and Spanier) with capacity generate exactly the family of matrix languages of finite index. This is in contrast to derivation bounded grammars which generate only context-free languages of finite index. First we show that we can restrict to grammars with capacities bounded by $1$. Let $\mathbf{CF}_{{\mathit{cb}}}^{1}$ and $\mathbf{GS}_{{\mathit{cb}}}^{1}$ be the language families generated by context-free and arbitrary grammars with capacity function $\mathbf{1}$. $\mathbf{CF}_{{\mathit{cb}}}=\mathbf{CF}_{{\mathit{cb}}}^{1}$ and $\mathbf{GS}_{{\mathit{cb}}}=\mathbf{GS}_{{\mathit{cb}}}^{1}$. Let $G=(V,\Sigma,S,R, \kappa)$ be a capacity-bounded phrase structure grammar. We construct the grammar $G'=(V',\Sigma,(S,1),R')$ with capacity function $\mathbf{1}$ and $$\begin{aligned} V'&=& \{(A,i) {:}A \in V, 1\leq i\leq \kappa(A)\},\\ R'&=& \{\alpha' \to \beta' {:}\alpha' \in h(\alpha), \beta' \in h(\beta), \mbox{ for some } \alpha \to \beta \in R\}, \end{aligned}$$ where $h:(V\cup \Sigma)^* \to (V' \cup \Sigma)^*$ is the finite substitution defined by $h(a)=\{a\}$, for $a \in \Sigma$, and , for $A \in V$. It can be shown by induction on the number of derivation steps that $S \!{\Rightarrow}^*_{G,\kappa}\! \alpha$ holds iff , for some $\alpha' \in h(\alpha)$. \[lem:GScbSubsetMATfin\] $\mathbf{GS}_{{\mathit{cb}}}\subseteq \mathbf{MAT}_{{\mathit{fin}}}$. Consider some language $L\in \mathbf{GS}_{{\mathit{cb}}}$ and let $G=(V,\Sigma,S,R,\mathbf{1})$ be a capacity-bounded phrase structure grammar (due to Ginsburg and Spanier) such that $L=L(G)$. A word $\alpha\in (V\cup \Sigma)^*$ can be uniquely decomposed as $$\alpha=x_1 \beta_1 x_2 \beta_2 \cdots x_n \beta_n x_{n+1}, x_1,x_{n+1} \in \Sigma^*, x_2,\ldots,x_n \in \Sigma^+, \beta_1,\ldots, \beta_n\in V^+.$$ The subwords $\beta_i$ are referred to as the *maximal nonterminal blocks* of $\alpha$. Note that the length of a maximal block in any sentential form of a derivation in $G$ is bounded by $|V|$. We will first construct a capacity-bounded grammar $G'$ with $L(G')=L$ such that all words of $L$ can be derived in $G'$ by rewriting a maximal nonterminal block in every step. Let $G'=(V,\Sigma,S,R',\mathbf{1})$ where $$\begin{aligned} R'&=& \{\alpha_1 \alpha \alpha_2 \to \alpha_1 \beta \alpha_2 {:}\alpha \to \beta \in R, \alpha_1,\alpha_2 \in V^*, |\alpha_1 \alpha \alpha_2|_A \leq 1, \mbox{ for all } A\in V\}. \end{aligned}$$ The inclusion $L(G) \subseteq L(G')$ is obvious since $R\subseteq R'$. On the other hand, any derivation step in $G'$ can be written as $\gamma_1 \underline{\alpha_1 \alpha \alpha_2} \gamma_2 {\Rightarrow}_{G'} \gamma_1 \underline{\alpha_1 \beta \alpha_2} \gamma_2$, where $\alpha \to \beta \in R$, implying that the same step can be performed in $G$ as $\gamma_1 \alpha_1 \underline{\alpha} \alpha_2 \gamma_2 {\Rightarrow}_{G,1} \gamma_1 \alpha_1 \underline{\beta} \alpha_2 \gamma_2.$ Thus $L(G')\subseteq L(G)$ holds as well. Moreover, any derivation step in $G$, $\gamma_1 \alpha_1 \underline{\alpha} \alpha_2 \gamma_2 {\Rightarrow}_{G,1} \gamma_1 \alpha_1 \underline{\beta} \alpha_2 \gamma_2$, $\alpha_1\alpha\alpha_2$ being a maximal nonterminal block, can be performed in $G'$ replacing the maximal nonterminal block $\alpha_1\alpha\alpha_2$ by $\alpha_1\beta\alpha_2$. In the second step we construct a context-free matrix grammar $H$ which simulates exactly those derivations in $G'$ that replace a maximal nonterminal block in each step. We introduce two alphabets $$\begin{aligned} [V]&=&\{[\alpha] {:}\alpha \in V^+, |\alpha|_A\leq 1, \mbox{ for all } A \in V\}\mbox{ and } \overline{V}=\{\overline{A} {:}A \in V\}. \end{aligned}$$ The symbols of $[V]$ are used to encode each maximal nonterminal block as single symbols, while $\overline{V}$ is a disjoint copy of $V$. Any word $$\alpha=x_1 \beta_1 x_2 \beta_2 \cdots x_n \beta_n x_{n+1}, x_1,x_{n+1} \in \Sigma^*, x_2,\ldots,x_n \in \Sigma^+, \beta_1,\ldots \beta_n\in V^+$$ such that $|\alpha|_A\leq 1$, for all $A \in V$, can be represented by the word $[\alpha]=x_1 [\beta_1] x_2 [\beta_2] \cdots x_n [\beta_n] x_{n+1}$, where the maximal nonterminal blocks in $\alpha$ are replaced by the corresponding symbols from $[V]$. The desired matrix grammar is obtained as $H=(V_H,\Sigma,S',M)$, with $V_H=[V]\cup V \cup \overline{V} \cup \{S'\}$ and the set of matrices defined as follows. For any rule $r=\alpha\to \beta$ in $R'$, $M$ contains the matrix $m_r$ consisting of the rules - $[\alpha] \to [\beta]$ (note that $\alpha \in [V]$, but $\beta\in ([V]\cup\Sigma)^*$), - $A \to \overline{A}$, for all $A \in V$ such that $|\alpha|_A=1$ and $|\beta|_A=0$, - $\overline{A}\to A$, for all $A \in V$ such that $|\alpha|_A=0$ and $|\beta|_A=1$. (The order of the rules in $m_r$ is arbitrary). Additionally, $M$ contains the starting and the terminating matrices $$(S'\to [S] S \overline{A_1} \cdots \overline{A_m}) \mbox{ and } (\overline{S} \to {\lambda},\overline{A_1} \to {\lambda}, \ldots,\overline{A_m} \to {\lambda}),$$ where $V=\{S,A_1,\ldots,A_m\}$. Intuitively, $H$ generates sentential forms of the shape $[\beta] \gamma$ where$[\beta] \in ([V] \cup \Sigma)^*$ encodes a sentential form $\beta$ derivable in $G'$ and $\gamma \in (V\cup \overline{V})$ gives a count of the nonterminal symbols in $\beta$ as follows: $|\gamma|_A+|\gamma|_{\overline{A}}=1$ and $|\gamma|_A=|\beta|_A$. Formally, it can be shown by induction that a sentential form over $V_H \cup \Sigma$ can be generated after applying $k\geq 1$ matrices (except for the terminating) iff it has the form $[\beta] \gamma$ where - $\beta \in (V\cup\Sigma)^*$ can be derived in $G'$ in $k-1$ steps, - $\gamma \in \{S,\overline{S}\} \{A_1,\overline{A}_1\} \cdots\{A_m,\overline{A}_m\}$ and $|\gamma|_A=1$ iff $|\beta|_A=1$. We can also show that the inverse inclusion also holds. \[MATfinInCScb\] $\mathbf{MAT}_{{\mathit{fin}}}\subseteq \mathbf{GS}_{{\mathit{cb}}}$. Capacity-bounded context-free grammars {#sec:nb-cfg} ====================================== In this section, we investigate capacity-bounded context-free grammars. It turns out that they are strictly between context-free languages of finite index and matrix languages of finite index. Closure properties of capacity bounded languages with respect to AFL operations are shortly discussed at the end of the section. As a first result we show that the family of context-free languages with finite index is properly included in ${{\bf CF}}_{{\mathit{cb}}}$. \[thm:hierarchyCapacityBounded1\] ${{\bf CF}}_{{\mathit{fin}}} \subset {{\bf CF}}_{{\mathit{cb}}}$. Any context-free language generated by a grammar $G$ of index $k$ is also generated by the capacity-bounded grammar $(G,\kappa)$ where $\kappa$ is the capacity function constantly $k$. The properness of the inclusion follows from Example \[exa:NBLnotCF2\]. An upper bound for ${{\bf CF}}_{{\mathit{cb}}}$ is given by the inclusion ${{\bf CF}}_{{\mathit{cb}}}\subseteq {{\bf GS}}_{{\mathit{cb}}}={{\bf MAT}}_{{\mathit{fin}}}$. We can prove the properness of the inclusion by presenting a language from ${{\bf MAT}}_{{\mathit{fin}}} \setminus {{\bf CF}}_{{\mathit{cb}}}$. $L=\{a^n b^n c^n {:}n\geq 1\} \notin \mathbf{CF}_{{\mathit{cb}}}$. Consider a capacity-bounded context-free grammar $G=(V,\Sigma,S,R,\mathbf{1})$ such that $L \subseteq L(G)$. For $A \in V$, let $G_A=(V,\Sigma,R,A,\mathbf{1})$. The following holds obviously for any derivation in $G$ involving $A$: If $\alpha A \beta {\Rightarrow}^*_{G} xyz$, where $\alpha,\beta \in (V\cup \Sigma)^*$, $x,y,z \in \Sigma^*$ and $y$ is the yield of $A$, then $y \in L(G_A)$. On the other hand, for all $x,y,z \in \Sigma^*$ such that $y\in L(G_A)$, the relation $xAz {\Rightarrow}^*_{G} xyz$ holds. The nonterminal set $V$ can be decomposed as $V=V_{{\mathit{inf}}} \cup V_{{\mathit{fin}}}$, where $$\begin{aligned} V_{{\mathit{inf}}} &=& \{A \in V {:}L(G_A) \mbox{ is infinite}\}\mbox{ and } V_{{\mathit{fin}}} = \{A \in V {:}L(G_A) \mbox{ is finite}\}. \end{aligned}$$ Let $K$ be a number such that $|w|<K$, for all $w \in \bigcup_{A \in V_{{\mathit{fin}}}} L(G_A)$. Consider the word $w=a^{rK} b^{rK} c^{rK}$, where $r$ is the longest length of a right side in a rule of $R$. There is a derivation $S {\Rightarrow}^*_{G} w$. Consider the last sentential form $\alpha$ in this derivation that contains a symbol from $V_{{\mathit{inf}}}$. Let this symbol be $A$. All other nonterminals in $\alpha$ are from $V_{{\mathit{fin}}}$, and none of them generates a subword containing $A$ in the further derivation process. We get thus another derivation of $w$ in $G$ by postponing the rewriting of $A$ until all other nonterminals have vanished by applying on them the derivation sequence of the original derivation. This new derivation has the form $S{\Rightarrow}^*_{G} \alpha {\Rightarrow}^*_{G} xAz {\Rightarrow}^*_{G} xyz=w.$ The length of $y$ can be estimated by $|y|\leq rK$, as $A$ is in the first step replaced by a word over $(\Sigma \cup V_{{\mathit{fin}}})$ of length at most $r$. By the remarks in the beginning of the proof, any word $xy'z$ with $y' \in L(G_A)$ can be derived in $G$. A case analysis shows that $xy'z$ is not in $L$, for any $y'\neq y$. Hence $L(G) \neq L$. The results can be summarized as follows: \[thm:hierarchyCapacityBounded\] $\mathbf{CF}_{{\mathit{fin}}}\subset \mathbf{CF}_{{\mathit{cb}}} \subset \mathbf{GS}_{{\mathit{cb}}}=\mathbf{MAT}_{{\mathit{fin}}}.$ As regards closure properties, we remark that the constructions showing the closure of $\mathbf{CF}$ under homomorphisms, union, concatenation and Kleene closure can be easily extended to the case of capacity bounded languages. \[thm:closureCapacityBounded\] $\mathbf{CF}_{{\mathit{cb}}}$ is closed under homomorphisms, union, concatenation and Kleene closure. We give here a proof only for the Kleene closure and leave the other cases to the reader. Let $L\in \mathbf{CF}_{{\mathit{cb}}}$ and let $G=(V,\Sigma,S,R,\mathbf{1})$ be a context-free grammar such that $L=L(G)$. We construct $G'=(V\cup\{S'\},\Sigma,S',R\cup \{S'\to SS',S'\to {\lambda}\},\mathbf{1}).$ Any terminating derivation in $G'$ that applies the rule $S'\to SS'$ $k$ times generates a word, where $w_i$ is the yield of the $i$-th symbol $S$ introduced by $S'\to SS'$. The subderivation from $S$ to $w_i$ only uses rules from $R$. Moreover, any sentential form $\beta_i$ in this subderivation is the subword of some sentential form $\beta$ in the derivation of $w$ in $G'$. Hence, $|\beta_i|_A \leq |\beta|_A\leq 1$, for all $1\leq i \leq k$ and all $A \in V$. Consequently, $w_i\in L(G)=L$ and $w \in L^*$. Conversely, any word $w=w_1 w_2 \cdots w_k$ with $w_i\in L$, for $1\leq i\leq k$, can be obtained in $G'$ by the derivation $$S'{\Rightarrow}SS' {\Rightarrow}^* w_1 S' {\Rightarrow}w_1SS' {\Rightarrow}^* w_1w_2S' {\Rightarrow}^* w_1w_2 \cdots w_kS' {\Rightarrow}w_1w_2\cdots w_k$$ where the subwords $w_i$ are derived from $S$ as in $G$. As regards closure under intersection with regular sets and under inverse homomorphisms, the constructions to show closure of $\mathbf{CF}$ cannot be extended, since they do not keep the capacity bound. We suspect that $\mathbf{CF}_{{\mathit{cb}}}$ is not closed under any of these operations. Control by Petri nets with place capacities {#sec:PNC} =========================================== We will first establish the connection between context-free Petri nets with place capacities and capacity-bounded grammars. Later we will investigate the generative power of various extended context-free Petri nets with place capacities. The proof for the equivalence between context-free grammars and grammars controlled by cf Petri nets can be immediately transferred to context-free grammars and Petri nets with capacities: \[thm:CapacityPetriNetGrammar\] Grammars controlled by context-free Petri nets with place capacity functions generate the family of capacity-bounded context-free languages. Let us now turn to grammars controlled by extended cf Petri nets with capacities. We will first study the generative power of capacity-bounded matrix and vector grammars, which are closely related to these Petri net grammars. \[thm:matrixGrammarBounds\] ${{\bf MAT}}_{{\mathit{fin}}}={{\bf V}}^{[{\lambda}]}_{{\mathit{cb}}}={{\bf MAT}}^{[{\lambda}]}_{{\mathit{cb}}}={{\bf sMAT}}^{[{\lambda}]}_{{\mathit{cb}}}$. We give the proof of ${{\bf MAT}}_{{\mathit{fin}}}={{\bf V}}^{{\lambda}}_{{\mathit{cb}}}$. The other equalities can be shown in an analogous way. Since ${{\bf MAT}}_{{\mathit{fin}}}={{\bf V}}_{{\mathit{fin}}}={{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$, it suffices to prove ${{\bf V}}_{{\mathit{fin}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{cb}}}$ and ${{\bf V}}^{{\lambda}}_{{\mathit{cb}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$. The first inclusion is obvious because any vector grammar of finite index $k$ is equivalent to the same vector grammar with capacity function constantly $k$. To show ${{\bf V}}^{{\lambda}}_{{\mathit{cb}}}\subseteq {{\bf V}}^{{\lambda}}_{{\mathit{fin}}}$, consider a capacity-bounded vector grammar $$G=(\{A_0,A_1,\ldots,A_m\},\Sigma,A_0,M,\mathbf{1}).$$ (The proof that it suffices to consider the capacity function $\mathbf{1}$ is like for usual grammars.) To construct an equivalent vector grammar of finite index, we introduce the new nonterminal symbols $B_i,B'_i$, $0\leq i\leq m$, $C$, $C'$. For any rule $r: A\to \alpha$, we define the matrix $\mu(r)=(C\to C',s_0,s_1,\ldots,s_m,r,C'\to C)$ such that $s_i=B_i \to B'_i$ if $A=A_i$ and $|\alpha|_A=0$, $s_i=B'_i \to B_i$ if $A\neq A_i$ and $|\alpha|_{A_i}=1$, and $s_i$ is empty, otherwise. Now we can construct $G'=(V',\Sigma,S',M')$ where $M'$ contains - for any matrix $m=(r_1,r_2, \ldots, r_k)$, the matrix $m'=(\mu(r_1), \ldots, \mu(r_k))$, - the start matrix $(S'\to A_0 B_0 B'_1 \cdots B'_m C)$, - the terminating matrix $(C\to {\lambda}, B'_0\to {\lambda},B'_1\to{\lambda}, \ldots, B'_m\to {\lambda})$, and $V'=V\cup \{B_i,B'_i{:}0\leq i\leq m\} \cup \{S',C,C'\}$. The construction of $G'$ allows only derivation sequences where complete submatrices $\mu(r)$ are applied: when the sequence $\mu(r)$ has been started, there is no symbol $C$ before $\mu(r)$ is finished, and no other submatrix can be started. It is easy to see that $G'$ can generate after applying complete submatrices exactly those words $\beta \gamma C$ such that $\beta \in (V\cup \Sigma)^*$, such that $\beta$ can be derived in $G$ and $|\gamma|_{B_i}=1$ iff $|\beta|_{A_i}=1$. Moreover, $G'$ is of index $2 |V|+1$. By constructions similar to those in [@tur] and Theorem \[thm:matrixGrammarBounds\] we can show with respect to weak capacities: \[lem:VfinInwPNch\] For $z\in \{h,c,s\}$, ${{\bf MAT}}_{{\mathit{fin}}}=\mathbf{wPN}^{[{\lambda}]}_{cz}$. We give only the proof for $z=h$. The other equations can be shown using analogous arguments. By Theorem \[thm:matrixGrammarBounds\] it is sufficient to show the inclusions ${{\bf V}}_{fin}\subseteq\mathbf{wPN}_{ch}$ and $\mathbf{wPN}^{{\lambda}}_{ch}\subseteq {{\bf V}}^{{\lambda}}_{cb}$. As regards the first inclusion, let $L$ be a vector language of finite index (with or without erasing rules), and let $ind(L)=k$, $k\geq 1$. Then, there is a vector grammar $G=(V, \Sigma, S, M)$ such that $L=L(G)$ and $ind(G)\leq k$. Without loss of generality we assume that $G$ is without repetitions. Let $R$ be the set of the rules of $M$. By Theorem 16 in [@tur], we can construct an $h$-Petri net controlled grammar $G'=(V, \Sigma, S, R, N_h)$, $N_h=(P\cup Q, T, F\cup E, \varphi, \zeta, \gamma, \mu_0, \tau)$, which is equivalent to the grammar $G$. By definition, for every sentential form $w\in (V\cup\Sigma)^*$ in the grammar $G$, $|w|_V\leq k$. It follows that $|w|_A\leq k$ for all $A\in V$. By bijection $\zeta:P\cup Q\to V\cup\{{\lambda}\}$ we have $\mu(p)=\mu(\zeta^{-1}(A))\leq k$ for all $p\in P$ and $\mu \in \mathcal{R}(N_h, \mu_0)$, i.e., the corresponding cf Petri net $(P, T, F, \phi, \beta, \gamma, \iota)$ is with $k$-place capacity. Therefore $G'$ is with weak place capacity. On the other hand, the construction of an equivalent vector grammar for an $h$-Petri net controlled grammar, can be extended to the case of weak capacities just by assigning the capacities of the corresponding places to the nonterminal symbols of the grammar. As regards strong capacities, there is no difference between weak and strong capacities for grammars controlled by $c$- and $s$-Petri nets because the number of tokens in every circle is limited by $1$. This yields: \[lem:wPNx=sPNx\] For $z\in \{c,s\}$, ${{\bf MAT}}_{{\mathit{fin}}}=\mathbf{sPN}^{[{\lambda}]}_{cz}$. The only families not characterized yet are $\mathbf{sPN}^{[{\lambda}]}_{ch}$. We conjecture that they are also equal to ${{\bf MAT}}_{{\mathit{fin}}}$. Conclusions {#sec:conclusions} =========== We have introduced grammars with capacity bounds and their Petri net controlled counterparts. In particular, we have shown that their generative power lies strictly between the context-free languages of finite index and the matrix languages of finite index. Moreover, we studied extended context-free Petri nets with place capacities. A possible extension of the concept is to use capacity functions that allow an unbounded number of some nonterminals. The investigation shows that for every grammar controlled by a cf Petri net with $k$-place capacity, $k\geq 1$, there exists an equivalent grammar controlled by a cf Petri net with 1-place capacity, i.e., the families of languages generated by cf Petri nets with place capacities do not form a hierarchy with respect to the place capacities.

ArXiv

--- abstract: 'This paper concerns the numerical approximation of low-energy eigenstates of the linear random Schrödinger operator. Under oscillatory high-amplitude potentials with a sufficient degree of disorder it is known that these eigenstates localize in the sense of an exponential decay of their moduli. We propose a reliable numerical scheme which provides localized approximations of such localized states. The method is based on a preconditioned inverse iteration including an optimal multigrid solver which spreads information only locally. The practical performance of the approach is illustrated in various numerical experiments in two and three space dimensions and also for a non-linear random Schrödinger operator.' address: '${}^{*}$ Department of Mathematics, University of Augsburg, Universitätsstr. 14, 86159 Augsburg, Germany' author: - 'R. Altmann$^*$, D. Peterseim$^*$' date: '. Accepted for publication in SIAM J. Sci. Comput.' title: Localized computation of eigenstates of random Schrödinger operators --- \ Introduction ============ This paper concerns the numerical approximation of essentially localized eigenstates of the linear Schrödinger eigenvalue problem $$\begin{aligned} \label{eq:EVP:strong} -\Delta u + Vu = \lambda u\end{aligned}$$ on a bounded domain $D\subseteq \R^d$ with homogeneous Dirichlet boundary conditions. The non-negative variable coefficient $V$ represents an external potential reflecting a high degree of disorder. This apparently simple problem is relevant in the context of quantum-physical processes related to ultracold bosonic or photonic gases, known as Bose-Einstein condensates [@Bos24; @Ein24; @DalGPS99; @PitS03]. A Bose-Einstein condensate (BEC) is an extreme state of matter formed by a dilute gas of bosons at ultra-cold temperatures, very close to absolute zero. In a BEC, individual particles (i.e., their wave packages) overlap, lose their identity, and form one single super atom. BECs allow to study macroscopic quantum phenomena such as superfluity (i.e., the frictionless flow of a fluid) on an observable scale. When BECs are trapped in a highly oscillatory high amplitude potential $V$ that exhibits a sufficiently large degree of disorder, the low-energy stationary states essentially localize in the sense of an exponential decay of their moduli. This localization is a universal wave phenomenon referred to as Anderson localization [@And58]. For BECs, it has been observed experimentally in [@FalFI08]. On a mathematical level, the formation of stationary quantum states can be modeled by the slightly more involved Gross-Pitaevskii eigenvalue problem (GPEVP). In non-dimensional form, the GPEVP seeks $L^2$-normalised eigenfunctions $u \in H_0^1(D)$ and corresponding eigenvalues (so-called chemical potentials) $\lambda \in \mathbb{R}$ such that $$\begin{aligned} - \triangle u + V u + \delta |u|^2 u = \lambda u.\end{aligned}$$ The parameter $\delta\ge0$ resembles the strength of repulsive particle interactions depending on physical properties of the particles that form the BEC. The linear case   corresponds to the regime of vanishing particle interaction ($\delta=0$). Since the interactions are weak for BECs, the linear case provides very good approximations of actual BEC states [@Rog13]. This is why we will mostly consider the numerical solution of the linear eigenvalue problem . However, the numerical techniques can be generalized to the nonlinear setting which will be demonstrated through numerical experiments at the end of this paper. The numerical approximation of localized Schrödinger eigenstates has recently caused a large interest in the fields of computational physics and scientific computing. Fundamentally novel methodologies have been developed to cope with the intrinsic difficulty coming from the multiscale nature of the potential [@ArnDJMF16; @Ste17; @ArnDFJM19; @XieZO18ppt]. They are based on the link of the groundstate $u_1$, normalized by $\| u_1 \|_{L^{\infty}(D)}=1$, and the so-called landscape function $\psi \in H^1_0(D)$ defined through of the homogeneous elliptic equation $-\Delta \psi + V\psi=1$, cf. [@FilM12]. The landscape function $\psi$ gives rise to surprisingly sharp eigenvalue bounds and its local maxima indicate regions where localization may occur. While the new techniques based on landscape functions are empirically successful in predicting the eigenvalues they lack any control on the accuracy of the approximation. In particular the approximation of the states is rather sketchy. The present paper aims at a more sophisticated method leading not only to the regions of localization but also to actual approximations of the lowermost eigenstates. The starting point is the recent rigorous a priori prediction of exponentially localized low-energy states caused by the interplay of disorder (randomness) and high amplitude (contrast) of the potential trap $V$ [@AltHP18ppt]. The results of [@AltHP18ppt] employ the convergence analysis of a preconditioned inverse iteration in the spirit of iterative numerical homogenization [@KorY16] and thereby provides a role model for efficient simulation. The main idea is to exploit the localization property, meaning that these eigenfunctions can be approximated in a sophisticated manner by functions supported in the union of only a few small sub-domains. The main contribution of this paper is the finding of an appropriate starting subspace and the construction of a preconditioned eigensolver which only relies on local operations. In this way, we are able to approximate the eigenstates of lowest energy roughly at the same costs as the computation of the landscape function $\psi$. In addition, the algorithm is applicable for the nonlinear case. The first step is a finite element discretization of the eigenvalue problem  which we discuss in Section \[sec:evp\]. For this, we consider a uniform refinement of the $\eps$-mesh on which the potential $V$ is defined. This automatically provides a mesh hierarchy for which we define a multigrid based preconditioner. Using only a small number of smoothing steps and no direct solves, one multigrid step preserves locality in the sense that the support of the resulting function is only slightly larger than the initial function. This then leads to a [*localization preserving*]{} eigensolver introduced in Section \[sec:precond\]. In other words, the simple yet efficient trick is to consider a multigrid preconditioner on a hierarchy of meshes starting from the $\eps$-level. This sufficed to be optimal in the sense of an $\mathcal{O}(1)$ condition number of the preconditioned operator. If no direct solver is used on the coarsest level, this preconditioner prevents the otherwise global communication of a standard multigrid involving coarser levels. The overall algorithm consists of two parts. First, we apply in parallel the localization preserving iteration scheme to finite element basis functions on a coarse grid. Based on the energies we select the most promising candidates indicating the regions of localization. Second, we consider a Ritz-Rayleigh iteration on the remaining local functions, i.e., we project the eigenvalue problem on a very small subspace. This procedure is applicable in any dimension and thus, allows to compute eigenfunctions for $d=3$ where standard eigensolvers break. This and further numerical experiments are subject of Section \[sec:numerics\]. Schrödinger Eigenvalue Problem {#sec:evp} ============================== This section is devoted to the linear Schrödinger eigenvalue problem  and the introduction of needed finite element spaces. Further, we discuss localization effects of the lowermost eigenfunctions if the potential contains disorder. Model problem {#sec:evp:model} ------------- In this paper, we consider the $d$-dimensional linear eigenvalue problem of Schrödinger type with a potential being highly oscillatory and of large amplitude. In particular, we assume that the potential includes some kind of disorder such that the first few eigenfunctions of lowest energy localize. The variational formulation corresponding to the eigenvalue problem  reads as follows: Given a non-negative potential $0\leq V\in L^\infty(D)$, find non-trivial eigenpairs $(u, \lambda) \in \V\times\R$ with search and test space $\V := H^1_0(D)$ such that $$\begin{aligned} \label{eq:EVP:weak} a(u, v) := \int_{D} \nabla u(x) \cdot \nabla v(x) + V(x)\, u(x) v(x) \dx = \lambda\, (u, v) \end{aligned}$$ for all test functions $v \in \V$. Here, $(\cdot,\cdot)$ denotes the $L^2$-inner product on $D$ and eigenfunctions are assumed to be normalized in the $L^2$-norm. Further, we assume that the eigenvalue are ordered, i.e., $0 <\lambda_1 < \lambda_2 \le \dots$. As far as the theory is concerned, we focus on a representative class of potentials which are piecewise constant with respect to a mesh $\calT^\eps$ consisting of cubes with side length $\eps\ll 1$. For the sake of simplicity we assume $\eps=2^{-\ell_\eps}$ where $\ell_\eps$ denotes the level of the $\eps$-scale. On each cube in $\calT^\eps$, the potential takes (randomly) a value in the interval $[\alpha, \beta]$ with moderate $0\le \alpha \approx 1$ and large $\beta$ in the sense of $\beta \gtrsim \eps^{-2}$. We emphasize that the resulting potentials are highly oscillatory due to the underlying mesh on $\eps$-scale. Examples of such random potentials are shown in Figure \[fig:pot\]. In the field of matter waves, one often considers disorder potentials created optically by using speckle patterns [@BoiMFGSG99; @LyeFMWFI05; @FalFI08]. This means that a laser beam is transmitted trough a diffusive plate forming randomized and high-contrast patterns. Within this paper, we imitate such speckles by piecewise constant potentials with respect to a quadrilateral mesh consisting of cubes with side length $\eps$. On each of these cubes the potential takes random values following certain statistical assumptions, cf. [@Goo75; @DunKW08]. ![Three examples of prototype disorder potentials: a 1D-tensorized potential with only two values (left), a fully random potential (middle), and a speckle potential (right). In all cases, dark color implies a large value of the potential up to the maximum $\beta$.[]{data-label="fig:pot"}](pics/potentials/tensor "fig:"){width="4.8cm" height="4.8cm"} ![Three examples of prototype disorder potentials: a 1D-tensorized potential with only two values (left), a fully random potential (middle), and a speckle potential (right). In all cases, dark color implies a large value of the potential up to the maximum $\beta$.[]{data-label="fig:pot"}](pics/potentials/random "fig:"){width="4.8cm" height="4.8cm"} ![Three examples of prototype disorder potentials: a 1D-tensorized potential with only two values (left), a fully random potential (middle), and a speckle potential (right). In all cases, dark color implies a large value of the potential up to the maximum $\beta$.[]{data-label="fig:pot"}](pics/potentials/speckle2 "fig:"){width="4.8cm" height="4.8cm"} Within this paper, $\Vert\cdot\Vert:=\sqrt{(\cdot,\cdot)}$ denotes the standard $L^2$-norm on $D$, whereas the $V$-weighted $L^2$-norm is given by $$\Vert v \Vert_V^2 := (Vv,v) = \int_D V(x)\, |v(x)|^2\dx.$$ Corresponding to the Schrödinger operator, we define the energy norm as $$\Vvert v \Vvert^2 := a(v, v) = \Vert \nabla v\Vert^2 + \Vert v\Vert_V^2.$$ The energy of a function is characterized by the Rayleigh quotient, namely $$\begin{aligned} \lambda(v) := \frac{a(v,v)}{\Vert v\Vert^2} = \frac{\Vvert v\Vvert^2}{\Vert v\Vert^2}.\end{aligned}$$ In case $v$ is an eigenfunction of the Schrödinger eigenvalue problem , the energy $\lambda(v)$ is equal to the corresponding eigenvalue. The eigenfunction of minimal energy $\lambda_1 := \min_{v\in \V \setminus \{0\}} \lambda(v)$ is called the groundstate. Exponential decay of the Green’s functions {#sec:evp:green} ------------------------------------------ For certain classes of potentials it is known that the corresponding Green’s function of the Schrödinger operator decays exponentially. For constant potentials $V\equiv \beta$ with sufficiently large $\beta$ this was shown in [@Glo11 Lem. 3.2]. Piecewise constant potentials with two values $\alpha$ and $\beta$ were considered in [@AltHP18ppt]. For this, an operator preconditioner was constructed depending on the geometric structure of the potential, i.e., on the interaction of $\alpha$-valleys and $\beta$-peaks. It was designed in such a way that the application of this operator preconditioner only enlarges the support by a small amount of $\eps$-layers within $\calT^\eps$. With this, one can show that the weak solution $u\in \calV$ of the variational problem $a(u,\cdot) = (f,\cdot)$ decays exponentially fast around the support of $f$ for potentials under certain statistical assumptions (including, e.g., 1D-tensorized potentials as in Figure \[fig:pot\]). More precisely, this means that $$\Vvert u\Vvert_{D\setminus B^\infty_{p\eps L}(\supp f)} \le c\, \gamma^{p}\, \Vvert u\Vvert$$ for constants $c>0$ and $0<\gamma<1$ independent of $\eps$, $L\approx \log(1/\eps)$, and $B^\infty_r(z)$ denoting the ball of radius $r$ around $z$. We emphasize that this result only relies on $\eps$ being small (oscillatory) and $\beta$ being large (high amplitude). Thus, disorder does not play a role for the exponential decay of the Green’s function. Localization of eigenfunctions {#sec:evp:localization} ------------------------------ For the localization of eigenfunctions, the potential needs to include a certain degree of disorder. For a periodic potential the lowermost part of the spectrum of the Schrödinger operator is clustered. In the one-dimensional case, which is illustrated in Figure \[fig:decay\], one can prove that there exist about $\eps^{-d}/2$ (the number of potential valleys) eigenvalues in the energy range of $\eps^{-2}$. Disorder changes the picture dramatically and leads to significant spectral gaps already within the first few eigenvalues. In one space dimension there is a one-to-one correspondence between the largest $\alpha$-valleys of the potential and the smallest eigenvalues. This is illustrated by dotted vertical lines which correspond to the largest $\alpha$-valleys and coincide with the spectral gaps for the random potential. This then leads to the exponential decay of the lowermost eigenfunctions, cf. Figure \[fig:decay\]. ![The first three eigenfunctions for a periodic (left) and random (middle) potential in the one-dimensional case with $\ell_\eps = 10$ and $\beta=2\cdot \eps^{-2}$. The lower part of the spectrum for both cases (periodic [$\times$]{} and random ${\color{red}\cdot}$) is shown on the right, clearly depicting the different nature of the two cases. The dotted vertical lines indicate for the random case that there is a single valley of largest, second largest, and third largest diameter, two valleys of fourth largest diameter, and so forth.[]{data-label="fig:decay"}](pics/spectrum/spectrum_per "fig:"){width="4.8cm" height="4.8cm"} ![The first three eigenfunctions for a periodic (left) and random (middle) potential in the one-dimensional case with $\ell_\eps = 10$ and $\beta=2\cdot \eps^{-2}$. The lower part of the spectrum for both cases (periodic [$\times$]{} and random ${\color{red}\cdot}$) is shown on the right, clearly depicting the different nature of the two cases. The dotted vertical lines indicate for the random case that there is a single valley of largest, second largest, and third largest diameter, two valleys of fourth largest diameter, and so forth.[]{data-label="fig:decay"}](pics/spectrum/spectrum_random "fig:"){width="4.8cm" height="4.8cm"} ![The first three eigenfunctions for a periodic (left) and random (middle) potential in the one-dimensional case with $\ell_\eps = 10$ and $\beta=2\cdot \eps^{-2}$. The lower part of the spectrum for both cases (periodic [$\times$]{} and random ${\color{red}\cdot}$) is shown on the right, clearly depicting the different nature of the two cases. The dotted vertical lines indicate for the random case that there is a single valley of largest, second largest, and third largest diameter, two valleys of fourth largest diameter, and so forth.[]{data-label="fig:decay"}](pics/spectrum/spectrum_zoom "fig:"){width="4.8cm" height="4.8cm"} In [@AltHP18ppt] the localization of eigenfunctions was rigorously proven for the regime $\beta\gtrsim\eps^{-2}$ and certain statistical assumptions on the potential. The key ingredient was to prove the existence of spectral gaps in the presence of disorder in combination with a preconditioned block inverse iteration. For this, particular finite element spaces and quasi-interpolation operators were considered. Further, the connection of valley sizes and eigenvalues was used. More precisely, we considered the first eigenfunctions of the Laplacian in the largest $\alpha$-valleys with homogeneous Dirichlet boundary conditions to obtain eigenvalue estimates of the Schrödinger operator. These Laplace eigenfunctions also serve as stating functions within the preconditioned block inverse iteration. The claimed decay then follows from the exponential convergence with respect to the number of iteration steps. The exponential decay may also be characterized a posteriori in terms of the landscape function $\psi \in \calV$ defined through $$\begin{aligned} \label{eqn:defLandscape} a(\psi,v) = (1, v)\end{aligned}$$ for all test functions $v\in\calV$, cf. [@FilM12]. The connection to the eigenvalue problem  gets visible by the possible reformulation as an eigenvalue problem with the effective confining potential $1/\psi$, which encodes the decay of the eigenfunction in some Agmon measure [@ArnDJMF16]. This means that the eigenfunction reduces by a certain factor when crossing a valley of $1/\psi$. This interplay has also been analyzed in [@Ste17] using averages over local Brownian motion paths. Finite element discretization {#sec:evp:fem} ----------------------------- In order to approximate the (localized) eigenstates of the Schrödinger operator, we consider a finite element discretization of . For this, we consider a uniform refinement of $\calT^\eps$, namely $\calT^h$, consisting of cubes with side length $h=2^{-\ell_h}\le \eps$. The corresponding set of nodes is denoted by $\calN^h$ and the set of interior nodes by $\calN^h_0$. Based on $\calT^h$, we define $V_h := Q_1(\calT^h) \cap \calV \subseteq \calV$ as the conforming $Q_1$-finite element space, cf. [@BreS08 Sect. 3.5]. This space has the dimension $n:=|\calN^h_0|$ and is spanned by the standard $Q_1$ hat functions, which we denote by $\varphi^h_z$ for $z\in\calN^h_0$. Recall that $\varphi^h_z$ is a piecewise polynomial of partial degree one with $\varphi^h_z(z)=1$ and $\varphi^h_z(w)=0$ for any other node $w\in\calN^h\setminus\{z\}$. Clearly, hat functions can also be defined on the original mesh $\calT^\eps$ w.r.t. its set of interior nodes $\calN^\eps_0$. For these hat functions we write $\varphi^\eps_z$ for $z\in\calN^\eps_0$. We now apply a Galerkin ansatz to the eigenvalue problem . Thus, we restrict the trial and test space to $V_h$, which then leads to a discrete eigenvalue problem of the form $$\begin{aligned} \label{eq:EVP:FEM} A_h u_h = \lambda_h M_h u_h.\end{aligned}$$ Here, $A_h \in \R^{n, n}$ and $M_h \in \R^{n, n}$ denote the symmetric stiffness and mass matrices defined through $$(A_h)_{ij} := a(\varphi^h_{z_i}, \varphi^h_{z_j}),\qquad (M_h)_{ij} := (\varphi^h_{z_i}, \varphi^h_{z_j}).$$ Note that the stiffness matrix already includes the potential. Eigenpairs of the matrix eigenvalue problem  approximate eigenpairs of the PDE eigenvalue problem . Due to the min-max principle of eigenvalues based on the Rayleigh quotient, we know that $\lambda_1 \le \lambda_{h,1}$. As mentioned above, we consider highly oscillatory potentials meaning that $\eps$ is small. Further, the mesh size $h$ needs to sufficiently small compared to $\eps$ in order to resolve the oscillations of the potential and thus, guarantees a reasonable approximation of the eigenpairs. Such a condition on the minimal resolution are justified by standard a priori error analysis. If $D$ is convex, any eigenfunction $u\in H^1_0(D)$ is $H^2$ regular. When normalized in $L^2(D)$, $u$ satisfies the bound $$\|D^2u\| \leq \|\Delta u\| = \|\lambda u - Vu\| \leq \lambda+\beta,$$ where $\lambda$ is the eigenvalue it corresponds to. For the lowermost eigenvalue in the regime of [@AltHP18ppt] where $\lambda_1\approx\beta\approx\eps^{-2}$ this leads to an error bound $$\frac{\lambda_{h,1}-\lambda_1}{\lambda_1} \lesssim \frac{h^2}{\eps^2} + \frac{h^4}{\eps^4},$$ see, e.g., [@StrF73 Lem. 6.1]. While the hidden constant in this bound depends only on the quasi-uniformity of the mesh, multiplicative constants in bounds for larger eigenvalues or any eigenfunction may deteriorate with the distance to neighboring eigenvalues in the case of clustered eigenvalues [@MR3107358]. In this sense the resolution condition $h\lesssim\eps$ is minimal for the approximation of the lowermost eigenvalue and may be much more restrictive in other cases. As a result, the eigenvalue problem  is of large dimension, which makes the solution computationally costly. As we are interested in the smallest eigenvalues and the corresponding localized eigenstates, we aim to construct a preconditioner based on local operations. Such a [*localization preserving*]{} preconditioner then allows parallel computations and is subject of the following section. Localization Preserving Preconditioner {#sec:precond} ====================================== In [@AltHP18ppt] we have presented a locally operating preconditioner based on a domain decomposition of $D$, which was a key ingredient for the proof of the exponential decay of the first eigenstates. The construction, however, was tailored to theoretical needs and checkerboard potentials. It included exact projections of local sub-domains. For actual computations, it turns out that already a simple multigrid preconditioned iteration may act as a localized version of a block inverse power method leading to sophisticated approximation results. In order to preserve locality, the coarsest level used within the multigrid cycle is the mesh $\calT^\eps$ on which the potential is defined. This still yields an optimal preconditioner due to the properties of the Green’s function of the Schrödinger operator. Further, we discuss how to find a low-dimensional starting subspace of local functions with which we can initialize the eigenvalue iteration. This is a crucial step in order to obtain local approximations of the lowermost eigenfunctions. Multigrid preconditioner {#sec:precond:multigrid} ------------------------ Recall that we have assumed that the finite element mesh $\calT^h$ is defined by a uniform refinement of the mesh on $\eps$-level on which the potential is defined. Thus, we have a mesh hierarchy given by $\calT^h$, possible intermediate meshes, and $\calT^\eps$. For this hierarchy, we define a standard geometric multigrid method without direct solve on the coarsest level, which will serve as a preconditioner for the eigenvalue iteration. A major aspect is that the multigrid method maintains locality, i.e., for a local right-hand side $b$ the resulting approximate solution of $A_h^{-1}b$ is supported on a domain which is only slightly larger than the support of $b$. Given the mesh hierarchy, we consider a so-called V-cycle with only one smoothing step on each level, cf. [@Hac85; @Mcc87; @Yse93]. Starting with a vanishing starting vector, we consider the residuals of $A_h x = b$ on different levels of the hierarchy. On the coarsest level, i.e., on $\calT^\eps$, we run one single relaxation step. This means that we do not use a direct solver on $\eps$-level with the stiffness matrix $A_\eps$ but use instead a single step of the Jacobi iteration. This yields a reasonable approximation, since the condition number of $A_\eps$ is of order $1$. The reason for this is the shift by the potential in the bilinear form $a$ in  and the fact that $\beta \gtrsim \eps^{-2}$. A computational study of the condition numbers of the stiffness matrices $A_h$ and $A_\eps$ is shown in Figure \[fig:cond\]. We emphasize that a direct solver would ruin the locality immediately. On each level of the multigrid cycle the support of the iterate grows slightly due to the application of the stiffness matrix on the respective level. In total, the application of this preconditioner enlarges the support by strictly less than three layers of $\eps$-cubes. It is well-known that multigrid solvers with a reduced tolerance (we only perform one V-cycle with a single relaxation step per level) can be used efficiently as a preconditioner within an external iterative solver, leading to methods such as [*pcg*]{} or [*lopcg*]{}, cf. [@KnyN03]. To keep things simple, we concentrate on pcg. We denote the application of $j$ steps of pcg with the multigrid preconditioner by $\calP_j$. Thus, $\calP_j(b)$ yields an approximation of $A_h^{-1} b$, where the accuracy can be controlled by the number of iteration steps. We emphasize that $\calP_j$ preserves locality in the following sense. \[thm\_pcg\] Assume that $x\in \R^n$ is the coefficient vector of a local function $u_h \in V_h \subset \calV$. Then, $\calP_j(x)$ is the representative of a function in $V_h$ with support being at most $3j$ layers of $\eps$-cubes larger than $\supp(u_h)$. As already mentioned, the application of a V-cycle as described above affects less than three layers of $\eps$-cubes. More precisely, the support enlarges at most within a ball of radius ${3\eps-h}$ around $\supp(u_h)$. The cg step involves another application of the stiffness matrix on the finest level and thus, adds only one layer of $h$-cubes to the support. In total this leads to a growth of $3$ layers with side length $\eps$ per pcg step. Note that the choice of the preconditioner is by far not unique and may be replaced, e.g., by local variants of hierarchical basis [@Yse86] or BPX [@BraPX90] preconditioners or Gamblets [@XieZO18ppt]. Here, local means that $\calT^\eps$ is the coarsest level in the underlying hierarchy of meshes. Starting subspace {#sec:precond:start} ----------------- For an efficient eigenvalue iteration we also need a suitable starting subspace. Starting for example with a vector which has only positive entries, we will approach the groudstate but every iteration step is ’global’. Instead, we search for a number of local functions at positions where we expect localization of the first eigenfunctions. For the detection of a suitable starting subspace we discuss several approaches. ### Landscape function {#sec:precond:start:Filoche} One possibility to find spots of localization is to compute the already mentioned [*landscape function*]{} $\psi \in \calV$, cf. [@FilM12]. This function is defined as the solution of the Schrödinger source problem with right-hand side $1$ and homogeneous Dirichlet boundary conditions, cf. equation . In other words, $\psi$ is the outcome of one step of the inverse power method in the PDE setting. The corresponding discrete approximation $\psi_h$ satisfies $$\psi_h = A_h^{-1} M_h\, \mathbf{1},$$ where $\mathbf{1} = [1,\ 1,\ \dots, 1]^T \in \R^{n}$. The landscape function shall indicate where the first eigenfunctions may localize. To see this, let $u_1$ denote the normalized ground state of the Schrödinger eigenvalue problem, i.e.,  $\| u_1 \|_{L^{\infty}(D)}=1$. Then, it is shown in [@FilM12] that the landscape function satisfies pointwise $$\begin{aligned} \label{eq:Filoche} |u_1(x)| \le \lambda_1 |\psi(x)|. \end{aligned}$$ This bound implies that in regions where $\psi$ is small compared to the smallest eigenvalue $\lambda_1$, the eigenfunction $u_1$ needs to be small as well. Considering the peaks of the landscape function then provides empirically accurate predictions where to expect localization. However, since we know that $\lambda_1 \gtrsim \eps^{-2}$, the estimate may degenerate quickly to $|u_1(x)|\le 1 =\| u_1 \|_{L^{\infty}(D)}$. Thus, this approach does not allow rigorous predictions a priori but may serve as an indicator. Although the estimate  only includes the groundstate, the landscape function has been used to locate a larger number of eigenfunctions by analyzing the local minima of $\psi$, cf. [@ArnDFJM19]. Further, it may be used to partition the computational domain $D$ into a network of valleys, cf. Figure \[fig:landscape\]. Yet another possibility is to consider multiple applications of the inverse Schrödinger operator, i.e., $(A_h^{-1}M_h)^k\mathbf{1}$ in the discrete setting, cf. [@Ste17]. ![Sum of the first five eigenfunctions (left) for a two-dimensional random potential with $\ell_\eps=7$, $\beta = 5\cdot\eps^{-2}$ and the corresponding landscape function $\psi$ (middle). Four additional steps of the inverse power method leads to a more selective landscape (right).[]{data-label="fig:landscape"}](pics/landscape/A_groundstates "fig:"){width="4.8cm" height="4.8cm"} ![Sum of the first five eigenfunctions (left) for a two-dimensional random potential with $\ell_\eps=7$, $\beta = 5\cdot\eps^{-2}$ and the corresponding landscape function $\psi$ (middle). Four additional steps of the inverse power method leads to a more selective landscape (right).[]{data-label="fig:landscape"}](pics/landscape/A_filoche_1 "fig:"){width="4.8cm" height="4.8cm"} ![Sum of the first five eigenfunctions (left) for a two-dimensional random potential with $\ell_\eps=7$, $\beta = 5\cdot\eps^{-2}$ and the corresponding landscape function $\psi$ (middle). Four additional steps of the inverse power method leads to a more selective landscape (right).[]{data-label="fig:landscape"}](pics/landscape/A_filoche_5 "fig:"){width="4.8cm" height="4.8cm"} ### Randomized landscape function {#sec:precond:start:Steinerberger} A similar approach to detect positions of localization was introduced in [@LuS18]. Therein, the inverse power method is applied to the unit vectors of dimension $n$. More precisely, for a fixed parameter $p\in\N$ and $e_k\in\R^n$ denoting the $k$-th unit vector, one computes for $k=1,\dots,n$ $$f_p(k) = \log\big( \Vert (A_h^{-1}M_h)^p e_k \Vert_2 \big).$$ In [@LuS18] it was shown that highly localized eigenfunctions correspond to metastable states of the power iteration and thus, are directly connected to local maxima of the function $f_p$. Note that in the given setting $f_p$ computes the inverse iteration to all hat functions on the $h$-scale, which is expensive due to the size of the eigenvalue problem. A variant is the following randomized version: Given a random matrix $R\in\R^{n,m}$ with $m\ll n$ we compute $$f_{R,p}(k) = \log\big( \Vert e_k^T (A_h^{-1}M_h)^p R \Vert_2 \big).$$ Thus, we apply the inverse power method to random vectors and consider the means in each component. In other words, we replace the application of $A_h^{-1}M_h$ to $n$ local functions by the application to $m$ global functions [@LuS18]. The outcome of this approach is displayed in Figure \[fig:steinerberger\]. It shows a landscape function which seems more smoothed than the previous approach of Section \[sec:precond:start:Filoche\]. Further, an increase of the number of iterations does not focus purely on the groundstate as this would happen by applying the inverse power method to the vector $M_h\, \mathbf{1}$. ![Examples of randomized landscape functions $f_{R,p}$ using $m$ random vectors and $p$ applications of the inverse power method (left: $m=1$, $p=50$; middle: $m=3$, $p=100$) with the same potential as used in Figure \[fig:landscape\]. The plot on the right shows the result for $m=3$ and $p=1000$ if we replace $A_h^{-1}$ by $I_n - A_h/\Vert A_h\Vert$.[]{data-label="fig:steinerberger"}](pics/landscape/A_steinerberger_m1_p50 "fig:"){width="4.8cm" height="4.8cm"} ![Examples of randomized landscape functions $f_{R,p}$ using $m$ random vectors and $p$ applications of the inverse power method (left: $m=1$, $p=50$; middle: $m=3$, $p=100$) with the same potential as used in Figure \[fig:landscape\]. The plot on the right shows the result for $m=3$ and $p=1000$ if we replace $A_h^{-1}$ by $I_n - A_h/\Vert A_h\Vert$.[]{data-label="fig:steinerberger"}](pics/landscape/A_steinerberger_m3_p100 "fig:"){width="4.8cm" height="4.8cm"} ![Examples of randomized landscape functions $f_{R,p}$ using $m$ random vectors and $p$ applications of the inverse power method (left: $m=1$, $p=50$; middle: $m=3$, $p=100$) with the same potential as used in Figure \[fig:landscape\]. The plot on the right shows the result for $m=3$ and $p=1000$ if we replace $A_h^{-1}$ by $I_n - A_h/\Vert A_h\Vert$.[]{data-label="fig:steinerberger"}](pics/landscape/A_steinerberger_series_m3_p1000.png "fig:"){width="4.8cm" height="4.8cm"} ### Largest valleys Assume that the potential $V$ is given in form of a random checkerboard, i.e., the function values of $V$ are either $\alpha$ or $\beta$, each with probability $0.5$. Then, motivated by the theoretical findings in [@AltHP18ppt], we may start with a number of hat functions distributed in the largest valleys. Here, a valley denotes a cube in which the potential is constant $\alpha$. For special classes of such checkerboard potentials, this approach guarantees to yield an appropriate starting subspace, meaning that the number of initial functions is $\calO(1)$, that these functions are local, and that the inverse power method converges quickly, cf. [@AltHP18ppt]. However, the search for valleys comes with the drawback that one first needs to analyze the structure of the given potential. Further, a generalization of the term valley is needed to consider general potentials. Nevertheless, the idea motivates the subsequent approach using uniformly distributed hat functions. ### Uniformly distributed hat functions {#sec:precond:start:hats} The current approach combines the benefits of the previous subsections. In this manner, we obtain a suitable subspace of local functions which is highly eligible to serve as a starting point within a (preconditioned) eigenvalue iteration. At the same time, we restrict the computational costs such that it is comparable to a single step of the inverse power method. Let $\calT^H$ be a mesh of cubes with side length $H=2^{-\ell_H}\ge\eps$ such that $\calT^\eps$ is a refinement of $\calT^H$ and thus, $\ell_H\le \ell_\eps$. For each interior node $z\in\calN^H_0$ there exists a $Q_1$ hat function $\varphi_z^H$. Similar to the approach in Section \[sec:precond:start:Steinerberger\] we may now apply several steps of the inverse power method. Instead, we apply a single pcg iteration step as introduced in Section \[sec:precond:multigrid\] to each hat function. Note that all these computations may be performed in parallel and that the application of $\calP_1$ maintains locality in the sense that the support of $\calP_1(\varphi_z^H)$ is only slightly larger than the support of $\varphi_z^H$, cf. Theorem \[thm\_pcg\]. Although we perform only one pcg step, we gain sufficient information in order to rank the importance of the basis function. For this we compute the energies $\lambda(\calP_1(\varphi_z^H))$ and sort them in an increasing order. Afterwards, we only keep the functions of lowest energy defined through a fixed ratio $0<\eta<1$. This procedure is then repeated until the number of functions is small enough. We emphasize that the support of each function only grows gently in each step whereas the number of functions decreases by the prescribed factor $\eta$. In other words, the proposed algorithm is approximately as costly as computing the landscape function $\psi$, cf. Section \[sec:precond:start:Filoche\]. Here, however, we directly obtain a number of candidates where the first eigenfunctions will localize without the need of constructing a network structure. An illustration of the algorithm is given in Figure \[fig:hats\_pre\]. ![Illustration of finding an appropriate local and low-dimensional starting subspace. Starting point are the hat functions on a coarser grid $\calT^H$. After each pcg step a fixed amount of functions is selected. Pictures show the results for $\eta=0.2$ after one (left), two (middle), and three (right) pcg steps.[]{data-label="fig:hats_pre"}](pics/hats/A_iter_1 "fig:"){width="4.8cm" height="4.8cm"} ![Illustration of finding an appropriate local and low-dimensional starting subspace. Starting point are the hat functions on a coarser grid $\calT^H$. After each pcg step a fixed amount of functions is selected. Pictures show the results for $\eta=0.2$ after one (left), two (middle), and three (right) pcg steps.[]{data-label="fig:hats_pre"}](pics/hats/A_iter_2 "fig:"){width="4.8cm" height="4.8cm"} ![Illustration of finding an appropriate local and low-dimensional starting subspace. Starting point are the hat functions on a coarser grid $\calT^H$. After each pcg step a fixed amount of functions is selected. Pictures show the results for $\eta=0.2$ after one (left), two (middle), and three (right) pcg steps.[]{data-label="fig:hats_pre"}](pics/hats/A_iter_3 "fig:"){width="4.8cm" height="4.8cm"} Approximation of eigenfunctions {#sec:precond:alg} ------------------------------- In this subsection we finally introduce an algorithm to approximate the eigenfunctions of lowest energy of the linear Schrödinger eigenvalue problem  under disorder potentials. Thus, we do not only intend to find possible regions of localization but actually compute approximations of the smallest eigenvalues and their corresponding eigenstates. Recall the definition of $\calP_j$ in Section \[sec:precond:multigrid\] containing $j$ pcg steps preconditioned my a multigrid cycle. Further, we fix the following parameters. ------------------- ---------------------------------------------- ---------------------------------------------- parameter default value meaning \[0.1\] $\Neigs$ $5$ number of eigenfunctions to compute \[0.15\] $\Kpre$ $3$ number of pre-iteration steps \[0.15\] $\Kpost$ $5$ number of post-iteration steps \[0.15\] $\ell_H$ $\ell_\eps- \lfloor \log(\ell_\eps) \rfloor$ level of $\calT^H$ for initial hat functions \[0.15\] $\eta$ $0.2$ ratio for selection process ------------------- ---------------------------------------------- ---------------------------------------------- The proposed algorithm consists of two parts: In the [*pre-iteration*]{}, we apply the selection process described in Section \[sec:precond:start:hats\]. For this, we consider hat functions on a (coarser) mesh of level $\ell_H$. Using the hat functions corresponding to $\calT^H$ rather than $\calT^\eps$ reduces the number of initial functions. Further, we truncate these hat functions in order improve the locality. After $\Kpre$ iteration steps with ratio $\eta$ we then have a much smaller number of functions, which already give a rough approximation of certain eigenfunctions. Most notably, however, they indicate with high precision where localization will take place without the need of constructing an additional mesh or finding local minima as with the landscape approach. If we are only interested in the groundstate, then one may reduce the number of functions until only a single function is left. Otherwise, we need to keep a sufficiently large amount of functions within the iteration process in order to prevent that only the groundstate is approximated. Further, since we only perform rough approximations of the inverse power method, we cannot guarantee that the functions of lowest energy correspond to the lowermost eigenstates. Therefore, we always keep at least $3\,\Neigs$ functions within the pre-iteration. In the [*post-iteration*]{}, we combine all functions which were selected by the pre-iteration in form of a Ritz-Rayleigh approximation. This means that we first perform three preconditioned cg steps to each function and then project the mass and stiffness matrices to the span of these functions and solve a (small) eigenvalue problem of the form $\mathbf{M} \alpha = \mu \mathbf{A} \alpha$. Note that the dimension of the eigenvalue problem depends on the number of pre-iteration steps. The first eigenvector $\alpha_1$ contains the coefficients of the optimal linear combination of the ansatz functions and provides the approximate groundstate. The second eigenvector $\alpha_2$ then defines the function which serves as approximation of the second eigenstate and so on. Further, we decrease the number of ansatz functions by cutting off the candidates of highest energy. Similar to the pre-iteration, we always keep a certain number of functions, namely twice the amount of eigenfunctions we are interested in, i.e., $2\,\Neigs$. Note that, in contrast to the parallel steps within the pre-iteration, we consider here a combined iteration of all remaining candidates. A summary of the procedure is given in Algorithm \[alg:hats\] and obtained numerical results for the two- and three-dimensional case are discussed in the subsequent section. : $\Neigs$, $\Kpre$, $\Kpost$, $\ell_H$, $\eta$ define mesh $\calT^H$ with $H = 2^{-\ell_H}$ $m = |\calN_0^H|$ coefficient matrix of hat functions $\Phi = [\varphi_1^H, \dots, \varphi_m^H] \in \R^{n,m}$ start pre-iteration $\Phi = \calP_1(M_h \Phi)$ \[line:pinvit\] $\Phi_j = \Phi_j / \Vert \Phi_j\Vert_{M_h}$, $j=1,\dots, m$ $\text{energies} = \text{diag} \big( \Phi^T A_h \Phi \big)$ $[\text{energies}, \text{idx}] = \text{sort} (\text{energies})$ $m = \max(\lfloor\eta m\rfloor, 3\,\Neigs )$ $\Phi = [\Phi_{\text{idx(1)}}, \dots, \Phi_{\text{idx(m)}}]$ start post-iteration $\Phi = \calP_3(M_h \Phi)$ $\Phi_j = \Phi_j / \Vert \Phi_j\Vert_{M_h}$, $j=1,\dots, m$ $\mathbf{M} = \Phi^T M_h \Phi \in \R^{m,m}$ $\mathbf{A} = \Phi^T A_h \Phi \in \R^{m,m}$ solve eigenvalue problem $\mathbf{M} \alpha = \mu \mathbf{A} \alpha$ \[line:evp\] resulting eigenpairs $(\alpha_1, \mu_1), \dots, (\alpha_m, \mu_m)$ with $\mu_1 \le \dots \le \mu_m$ \[line:cutoff\] $m = \max(\lfloor\eta m\rfloor, 2\,\Neigs )$ $\Phi = [\Phi \alpha_1, \dots \Phi\alpha_m]$ The simple pcg steps in the pre-iteration of Algorithm \[alg:hats\] (line \[line:pinvit\]) may also be replaced by $\Kpre$ steps of lopcg, cf. [@Kny00; @Kny01]. Since the main purpose of the pre-iteration is the detection of regions of localization, we consider here only this simple variant. In order to accelerate the iteration and improve the stability of the method, one may consider additional cut-offs. In particular, one may set components of $\alpha_j$ in line \[line:cutoff\] of Algorithm \[alg:hats\] to zero if they are beneath a certain threshold. This also increases the level of locality. The procedure of Algorithm \[alg:hats\] assumes that the first eigenfunctions are indeed localized. Thus, global eigenstates as they would appear for periodic potentials are not well-approximated by this method. However, one may adapt the selection process in the algorithm to such an extent that periodic structures are at least detected. For this, one may substitute the fixed-ratio criterion by a selection step which is based on the energies of the considered functions. In a periodic structure, where all hat functions would have a comparable energy, we would then not cut off any functions, meaning that the parameter $m$ does not decrease and that the eigenvalue problem in line \[line:evp\] of the algorithm is still of large dimension. Numerical Examples {#sec:numerics} ================== We perform several numerical tests in order to explore the performance and feasibility of the proposed method. In the first experiment we consider the two-dimensional random potential, which was already used in Section \[sec:precond\] for the illustration of the approaches to find regions of localization. Second, we consider a three-dimensional speckle potential leading to huge eigenvalue problems which quickly overcharge standard eigenvalue solvers. Finally, we apply the method to the nonlinear Gross-Pitaevskii eigenvalue problem. All computations have been performed with Matlab on an [*HPC Infiniband cluster*]{} (1.7 TB RAM, 2 Tesla V100 32GB GPUs). The reference solutions, which are used for the error plots, are obtained by the Matlab solver [*eigs*]{} with tolerance $10^{-12}$ for the 2D examples and $10^{-10}$ in 3D, both with a maximum of $100$ iterations. The code is available as supplementary material. Random potential in 2D ---------------------- In this first numerical experiment we consider a random potential in two space dimensions with parameters $$\ell_\eps = 7, \quad \ell_h = 9, \quad \ell_H = 6, \quad \alpha = 1, \quad \beta = 5\cdot\eps^{-2}.$$ We are interested in the first five eigenstates of lowest energy. The outcome of the pre-iteration with $\eta=0.2$ was already illustrated in Figure \[fig:hats\_pre\], leading to a small number of regions in which we expect the smallest eigenstates to localize. Applying the Ritz-Rayleigh method and the additional selection process, we end up in a ten-dimensional subspace of $V_h$, where all basis functions have local support. The five functions with lowest energy then serve as approximations of the eigenfunctions $u_1, \dots, u_5$. The results are very convincing and depicted in Figure \[fig:hats\_post\]. For a comparison with the actual eigenstates we refer to Figure \[fig:landscape\] (left). ![Outcome of Algorithm \[alg:hats\] with the default parameters after one (left), three (middle), and five (right) Ritz-Rayleigh steps. The two pictures on the left show the sum of all remaining functions whereas the picture on the right only includes the five functions of lowest energy. Cf. Figure \[fig:landscape\] for a reference solution.[]{data-label="fig:hats_post"}](pics/hats/A_iter_21 "fig:"){width="4.8cm" height="4.8cm"} ![Outcome of Algorithm \[alg:hats\] with the default parameters after one (left), three (middle), and five (right) Ritz-Rayleigh steps. The two pictures on the left show the sum of all remaining functions whereas the picture on the right only includes the five functions of lowest energy. Cf. Figure \[fig:landscape\] for a reference solution.[]{data-label="fig:hats_post"}](pics/hats/A_iter_23 "fig:"){width="4.8cm" height="4.8cm"} ![Outcome of Algorithm \[alg:hats\] with the default parameters after one (left), three (middle), and five (right) Ritz-Rayleigh steps. The two pictures on the left show the sum of all remaining functions whereas the picture on the right only includes the five functions of lowest energy. Cf. Figure \[fig:landscape\] for a reference solution.[]{data-label="fig:hats_post"}](pics/hats/A_iter_25TOP "fig:"){width="4.8cm" height="4.8cm"} As mentioned above, the pre-iteration only provides very rough approximations of the eigenstates and serves more the finding of an appropriate starting subspace for the Rayleigh-Ritz method. As a result, the assignments of the localization regions to the actual eigenstates may be inaccurate in the beginning. In the present example, four Rayleigh-Ritz steps were needed until the fifth eigenstate was correctly assigned. This effect can also be observed in the convergence plot in Figure \[fig:conv\]. Therein, the relative error in $\lambda_5$ drops significantly from step $6$ to step $7$. Such effects are directly influenced by gaps within the spectrum of the Schrödinger operator. Here, the fifth and sixth eigenvalues are given by $2.6076\cdot 10^4$ and $2.6093 \cdot 10^4$ and thus, at close quarters. Figure \[fig:spectrum\] shows that the proposed methods also works if we are interested in a larger number of eigenvalues and eigenfunctions. If we only consider the pre-iteration, the relative error of the eigenvalues seems to be almost constant which is in agreement with the observations in [@ArnDFJM19]. In contrast to the landscape function approach we are able to control the accuracy of the method by the number of post-iterations. Further, the method is flexible enough to recognize when there are several eigenfunctions located in a single valley. Speckle potential in 3D ----------------------- The second example considers the linear Schrödinger eigenvalue problem  in three space dimensions with a speckle-like potential as it is used in actual experiments [@FalFI08]. As parameters we choose $$\ell_\eps = 5, \quad \ell_h = 6, \quad \ell_H = 4, \quad \alpha = 1, \quad \beta = 5\cdot\eps^{-2}.$$ An illustration of the potential is given in Figure \[fig:eigComp3D\]. As for the previous example we compare the outcome of Algorithm \[alg:hats\] with the results obtained by the [*eigs*]{} solver in Matlab. As a consequence, we are not able to exceed level $\ell_h=6$, as this marks the limit of the integrated eigenvalue solver. The approximation of the first eigenfunction and the corresponding result obtained by [*eigs*]{} are compared in Figure \[fig:eigComp3D\]. The displayed approximations are computed with $\Kpre= 3$ pre- and $\Kpost=5$ post-iteration steps. A comparison of computation times for various mesh levels of $\calT^\eps$ and $\calT^h$ are summarized in the following table: --------------------------------- ----------------- ----------------- ---------------------- ----------------- ----------------- ---------------------- $\ell_\eps = 4$ $\ell_\eps = 5$ $\ell_\eps = 6$ $\ell_\eps = 3$ $\ell_\eps = 4$ $\ell_\eps = 5$ \[0.1\] Matlab [*eigs*]{} $7.15\,s$ $279.10\,s$ $\infty\phantom{nn}$ $7.21\,s$ $228.60\,s$ $\infty\phantom{nn}$ \[0.15\] Algorithm \[alg:hats\] $4.86\,s$ $57.65\,s$ $1788.02\,s$ $6.43\,s$ $74.99\,s$ $1242.41\,s$ --------------------------------- ----------------- ----------------- ---------------------- ----------------- ----------------- ---------------------- ![Speckle potential (left) and comparison of the first eigenfunction (middle) and its numerical approximation obtained by Algorithm \[alg:hats\] (right).[]{data-label="fig:eigComp3D"}](pics/speckle3D/potential "fig:"){width="4.8cm" height="4.8cm"} ![Speckle potential (left) and comparison of the first eigenfunction (middle) and its numerical approximation obtained by Algorithm \[alg:hats\] (right).[]{data-label="fig:eigComp3D"}](pics/speckle3D/compare_1_eigs "fig:"){width="4.8cm" height="4.8cm"} ![Speckle potential (left) and comparison of the first eigenfunction (middle) and its numerical approximation obtained by Algorithm \[alg:hats\] (right).[]{data-label="fig:eigComp3D"}](pics/speckle3D/compare_1_hats "fig:"){width="4.8cm" height="4.8cm"} Nonlinear Gross-Pitaevskii eigenvalue problem --------------------------------------------- In this final example we consider the nonlinear version of the Schrödinger equation, describing quantum-physical processes with interaction. This GPEVP has the form $$\begin{aligned} -\Delta u + Vu + \delta|u|^2 u= \lambda u\end{aligned}$$ with $\delta\ge 0$ regulating the nonlinearity. For moderate values of $\delta$ similar localization results for the groundstates can be observed [@AltPV18]. As a result, the here presented localization preserving preconditioner promises good approximation results. We emphasize that this nonlinear case is not easily treated by the landscape function approach. Considering again a finite element discretization as in Section \[sec:evp:fem\], we obtain the nonlinear matrix eigenvalue problem $$A_h u_h + \delta N_h(u_h)u_h = \lambda M_h u_h,$$ where $N_h(u_h)$ equals the finite-dimensional approximation of the nonlinearity. Inspired by the inverse iteration for the linear case, i.e., $u^{n+1}_h = A_h^{-1} M_h u^n_h$, one may consider the iteration $$u^{n+1}_h = (A_h + \delta N_h(u^n_h))^{-1} M_h u^n_h$$ with an additional normalization step. This then leads to the results shown in Figure \[fig:nonlinear\], illustrating that the groundstate can be well approximated by local functions also in the nonlinear case. It goes without saying that the iteration scheme may be replaced by more advanced methods, e.g., based on gradient flows [@BaoD04; @HenP18ppt; @AltHP19ppt]. ![Approximation of the groundstate for the Gross-Pitaevskii eigenvalue problem for $\delta=1$ (left), $\delta=10$ (middle), and $\delta=100$ (right) for a two-dimensional random potential with $\ell_\eps=7$, $\beta = 5\cdot\eps^{-2}$. Results after 100 iteration steps with initial function being constant 1.[]{data-label="fig:nonlinear"}](pics/nonlinear/INVIT_gam1_steps100 "fig:"){width="4.8cm" height="4.8cm"} ![Approximation of the groundstate for the Gross-Pitaevskii eigenvalue problem for $\delta=1$ (left), $\delta=10$ (middle), and $\delta=100$ (right) for a two-dimensional random potential with $\ell_\eps=7$, $\beta = 5\cdot\eps^{-2}$. Results after 100 iteration steps with initial function being constant 1.[]{data-label="fig:nonlinear"}](pics/nonlinear/INVIT_gam10_steps100 "fig:"){width="4.8cm" height="4.8cm"} ![Approximation of the groundstate for the Gross-Pitaevskii eigenvalue problem for $\delta=1$ (left), $\delta=10$ (middle), and $\delta=100$ (right) for a two-dimensional random potential with $\ell_\eps=7$, $\beta = 5\cdot\eps^{-2}$. Results after 100 iteration steps with initial function being constant 1.[]{data-label="fig:nonlinear"}](pics/nonlinear/INVIT_gam100_steps100 "fig:"){width="4.8cm" height="4.8cm"} Conclusion {#sec:conclusion} ========== In this paper, we have constructed a novel iterative scheme using analytical inside and adapting numerical analysis techniques used in localization proofs of the eigenstates [@AltHP18ppt]. 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ArXiv

--- abstract: 'Automated detection of cervical cancer cells or cell clumps has the potential to significantly reduce error and increase productivity in cervical cancer screening. However, most traditional methods rely on the success of accurate cell segmentation and discriminative hand-crafted features extraction. Recently there are emerging deep learning-based methods which train convolutional neural networks (CNN) to classify image patches, but they are computationally expensive. In this paper we propose to exploit contemporary object detection methods for cervical cancer detection. To deal with the limited size of training samples, we develop the comparison classifier into the state-of-the-art two-stage object detection method based on the comparison with the reference images of each category. In addition, we propose to learn the reference images of the background from the data instead of manually choosing them by some heuristic rules. This architecture, called the Comparison detector, shows significant improvement for small size dataset, achieving a mean Average Precision (mAP) 26.3% and an Average Recall (AR) 35.7%, both improving about **20** points compared to baseline model. Moreover, Comparison detector achieves same mAP performance as the current state-of-the-art model when training on the medium size dataset, and improves AR by **4** points. Our method is promising for the development of automation-assisted cervical cancer screening systems.' author: - 'Yixiong Liang, Zhihong Tang, Meng Yan, Jialin Chen, and Yao Xiang [^1] [^2] [^3]' bibliography: - 'IEEEabrv.bib' - 'paper.bib' title: 'Comparison Detector: Convolutional Neural Networks for Cervical Cell Detection' --- Cervical cancer screening, object detection, convolutional neural networks, prototype-based classification Introduction ============ Cervical cytology is the most common and effective screening method for cervical cancer and premalignant cervical lesions [@davey2006effect], which is performed by a visual examination of cytopathological analysis under the microscope of the collected cells that have been smeared on a glass slide and stained and finally giving a diagnosis report according to the descriptive diagnosis method of the Bethesda system (TBS)[@nayar2015bethesda]. Currently in developed countries, it has been widely used and has significantly reduced the number of deaths caused by related diseases, but it is still unavailable for population-wide screening in the developing countries [@saslow2012american], partly due to the fact that it is labor-intensive, time-consuming and expensive [@bengtsson2014screening]. In addition, it is subjective and therefore has motivated lots of automated methods for the automation of cervical screening based on the image analysis techniques. Over the past 30 years extensive research has attempted to develop automation-assisted screening methods [@birdsong1996automated; @chankong2014automatic; @zhang2014automation; @phoulady2016automatic]. Most of them try to classify a single cell into various stages of carcinoma, which often consists three steps: cell (cytoplasm and nuclei) segmentation, feature extraction and classification. The performance of these methods, however, heavily depends on the accuracy of the segmentation and the effectiveness of the hand-crafted features. With the overwhelming success in a broad range of applications such as image classification [@krizhevsky2012imagenet; @he2016deep], semantic segmentation [@long2015fully], object detection [@ren2015faster; @lin2017feature] and medical imaging analysis [@litjens2017survey; @gulshan2016development; @esteva2017dermatologist], CNN has also been applied to the segmentation and classification of cervical cell [@tareef2017optimizing; @song2017accurate; @song2015accurate; @zhang2017combining; @lu2017evaluation; @zhang2017deeppap; @jith2018deepcerv; @gautam2018considerations]. The majority of them [@tareef2017optimizing; @song2015accurate; @zhang2017combining] are trying to take advantage of CNN to improve the segmentation accuracy of cytoplasm and nuclei, but they do not provide the needed segmentation accuracy [@lu2017evaluation; @jith2018deepcerv], whereas once the segmentation error are taken into account, the classification accuracy would drop [@zhang2017deeppap]. To avoid the dependence on accurate segmentation, the patch-based methods try to use CNN to classify the image patches [@jith2018deepcerv; @gautam2018considerations]. However, the extraction of such patches still requires the segmentation of nuclei. The recent work [@zhang2017deeppap] also adopt the patch-based strategy but during the inference the random-view aggregation and multiple crop testing are needed to produce the final prediction results and thereby is time-consuming. In this paper, we propose an efficient and effective strategy to apply CNN for cervical cancer screening, *without* any pre-segmentation step. Specifically, we exploit the contemporary CNN-based object detection methods [@ren2015faster; @lin2017feature] to detect the cervical cytological abnormalities directly. It is straightforward and has been successfully applied for other medical image analysis [@liang2018end; @liang2018object], but we are not aware of any works try to apply CNN-based object detection for automated cervical cytology. We attribute this to the lack of the right cervical cancer microscopic image dataset for the detection task. CNN-based object detection methods often need sufficient annotated data to obtain good generalization, but for cervical cytological abnormalities detection, collecting the large amounts of data with careful and accurate annotation is difficult partially due to the limitation by laws, the scarcity of positive samples and especially the unanimous agreement between cytopathologists [@stoler2001interobserver]. To alleviate the small training dataset size problem, we propose the named *Comparison Detector*, which migrate the idea of *comparison* in one/few-shot learning for image classification [@koch2015siamese; @vinyals2016matching; @snell2017prototypical; @yang2018learning] into CNN-based object detection, for cervical cancer detection. Specifically, we choose the state-of-the-art object detection method, Faster R-CNN [@ren2015faster] with FPN [@lin2017feature], as our baseline model and replace the original parameter classifier with a non-parametric one based on the idea of comparison with the reference images of each category. Furthermore, instead of manually choosing the reference images of the background category by some heuristic rules, we propose to learn them from the data. We also investigate several important factors including generating prototype representations of categories and the design of head model for cervical cell detection. Our algorithm directly operate on the whole image rather than the extracted patches based on the nuclei and hereby only need one forward propagation for each image, making the inference very efficient. In addition, the proposed method is *flexible* to be intergraded into other proposal-based methods. We collect a small size dataset $D_s$ and a medium size dataset $D_f$ which are directly dedicated to cervical cell/clumps detection, on which we evaluate the performance of the proposed Comparison Detector. When the model is learned from the small size dataset, the performance of our method is significantly better than the one of baseline model, i.e. Comparison Detector has an mAP 26.3% and an AR 35.7% but the baseline model only gains an mAP 6.6% and an AR 12.9%. When the model is learn from the medium size dataset, our Comparison Detector achieves almost the same performance with a mAP of 45.3%, but improves nearly 4 points comparing to baseline model with AR. We summarize our contributions as follows: 1) To the best of our knowledge, this is the first application of the CNN-based object detection methods to cervical cancer detection; 2) We propose Comparison Detector method to deal with the small training sample size problem in cervical cell detection; 3) We propose a strategy to directly learn the background reference images and 4) Our method performs much better than the baseline on both small size and medium size dataset and has the potential applications to the real automation-assisted cervical cancer screening systems. Related work ------------ **Cervical cell segmentation and classification.** Traditional cytological criteria for classifying cervical cell abnormalities are based on the changes in nucleus to cytoplasm ratio, nuclear size, irregularity of nuclear shape and membrane, therefore there are numerous works focusing on the segmentation of cell or cell components (nuclei, cytoplasm) [@chankong2014automatic; @zhang2014segmentation; @gencctav2012unsupervised; @chen2014semi; @song2015accurate; @song2017accurate; @zhang2017graph; @lu2015improved; @lee2016segmentation; @li2012cytoplasm; @tareef2018multi]. Although significant progress has been achieved recently, the segmentation of cell or cell components remains an open problem due to the large shape and appearance variation between cells, the poor contrast of cytoplasm boundaries and the overlap between cells [@zhang2017deeppap; @lee2016segmentation; @tareef2018multi; @lu2017evaluation]. On the other hand, cervical cell classification methods try to differentiate the dysplastic cells from the norm cells and classify them into various stages of carcinoma. According to TBS rules [@nayar2015bethesda], a large number of hand-crafted features are designed to describe the shape, texture and appearance characteristics of the nucleus and cytoplasm [@gencctav2012unsupervised; @chen2014semi; @marinakis2009pap; @chankong2014automatic; @phoulady2016automatic; @bora2017automated]. The resulting features are often further organized by feature selection or dimensionality reduction and then are fed into various classifiers (e.g. random forests, SVM, softmax regression, neural network, etc.) to perform the final classification. However, as mentioned above, the extraction of those engineered features depends on the accurate segmentation of cell or cell components. Furthermore, it is also limited by the current understanding of cervical cytology [@zhang2017deeppap]. To reduce the dependency on the accurate segmentation, the CNN are used to learn the features from data recently [@jith2018deepcerv; @gautam2018considerations], but an approximate segmentation or (region of interest) ROI detection is still necessary. Although the DeepPap [@zhang2017deeppap] is claimed totally segmentation-free, it still needs the nucleus centroid information for training and the random-view aggregation and multiple crop testing during the inference stage, which are very time-consuming. There are a handful public available microscopic image datasets dedicated to cervical cell segmentation such as ISBI-14[^4], ISBI-15 [^5], but to our best knowledge for cervical cell classification the only public available microscopic image dataset is the Herlev benchmark dataset [@marinakis2009pap], which consists of 917 single cell images corresponding to four categories of abnormal cell with different severity (namely light dysplastic, moderate dysplastic, severe dysplastic and carcinoma in situ) and three categories of normal cells (normal columnar, normal intermediate and normal superficial). The limited annotated data prevents the applications of traditional object detection methods such as Viola-Jones detector [@viola2004robust] or contemporary CNN-based detectors [@liu2018deep] to cervical cancer screening. **CNN-based object detection.** The Overfeat [@sermanet2013overfeat] made the earliest efforts to apply CNN for object detection and has achieved a significant improvement of more than 50% mAP when compared to the best methods at that time which were based on the hand-crafted features. Since then, a lot of CNN-based methods [@ren2015faster; @liu2016ssd; @girshick2014rich; @girshick2015fast; @he2017mask; @li2017light; @lin2018focal; @singh2018r; @redmon2016you; @redmon2017yolo9000; @zhang2018single; @redmon2018yolov3] have been proposed for high-quality object detection, which can be roughly classified into two categories: object proposal-based and proposal-free. The road-map of proposal-based methods starts from the notable R-CNN [@girshick2014rich] and is improved by Fast-RCNN [@girshick2015fast] in an end-to-end manner and by Faster R-CNN [@ren2015faster] to quickly generate object regions, which has motivated a lot of follow-up improvements [@lin2017feature; @he2017mask; @li2017light; @lin2018focal; @singh2018r] in terms of accuracy and speed. The proposal-free methods [@liu2016ssd; @redmon2016you; @redmon2017yolo9000; @redmon2018yolov3] directly predict the bounding boxes without the proposal generation step. Generally, the proposal-free methods are conceptually simpler and much faster than the proposal-based methods, but the detection accuracy is usually behind that of the proposal-based methods [@zhang2018single]. Here we choose the Faster R-CNN [@ren2015faster] with FPN [@lin2017feature] as our baseline model but our method is compatible with other proposal-based methods.  **One/few-shot learning.** One/few-shot learning is a task of learning from just one or a few training samples per class and has been extensively discussed in the context of image recognition and classification [@koch2015siamese; @vinyals2016matching; @snell2017prototypical]. Recently significant progress has been made for one/few-shot learning tackled by meta-learning or learning-to-learn strategy, which can be roughly divided into three categories: metric-based, memory-based and optimization-based. The metric-based methods [@koch2015siamese; @vinyals2016matching; @snell2017prototypical; @yang2018learning] learn to compare the query image with support set images. The memory-based methods [@santoro2016one] exploit the memory-augmented neural network to quickly store and retrieve sufficient information for each classification task, while the optimization-based methods [@ravi2016optimization; @finn2017model] aim to learning a base-model which can be fine-tuned quickly for a new classification task. All these works only tackle image classification tasks. **Object detection with limited-data.** Most prior works on object detection with limited labels use semi-/weakly-supervised methods or few-example learning [@dong2018few] to make use of abundant unlabeled data, whereas in limited-data regime there are few work focus on using few-shot learning to address this problem [@schwartz2018repmet; @kang2018few]. Kang et al. [@kang2018few] decompose the training into base-model learning and meta-model learning and train a meta-model to reweight the features extracted by the base-model to assist novel object detection. However, the training of base model still needs abundant annotated data for base classes. RepMet [@schwartz2018repmet] introduces a metric learning-based sub-network architecture to learn the embedding space and distribution of the training categories without using external data. However, RepMat involves an alternating optimization between the external class distribution module learning and net parameters updating, whereas our solution is a clean, single-step training framework. Comparison Detector {#section3} =================== Basic Architecture ------------------ Our proposed comparison detector is based on proposal-based detection framework consisting of a region proposal network (RPN) for proposal generating, a backbone network for feature extraction and a head for the proposal classification and bounding box regression. Here we choose the Faster R-CNN with FPN [@lin2017feature] as our baseline. Then we decouple the regression and classification in the head and replace the original parameter classifier with our comparison classifier. Our no-parameters classifier introduces a inductive bias, namely the within-class distance is less than the between-class in the embedding space, into the model and henceforth mitigates the small sample size issue to some extent [@battaglia2018relational]. The framework of the proposed Comparison detector is depicted in Fig.\[fig:1\], which is divided into three stages to describe. At the first stage, as shown in Fig. \[fig:1\](a), the features of both the reference and the object images are computed by backbone network with FPN [@lin2017feature], without using any extra models to encode the reference images. The only difference is there are no RPN operation on the reference images. Assuming that there are $n$ samples per category with $t$ levels pyramid feature in the reference images. Let $F_i^l$ be the $i$-th categories’ prototype representation of the $l$-level pyramid features, which can be computed by average operation as follows $$F_i^l = \frac{1}{n} \sum_jF^l (R_{ij}), \label{equ:1}$$ where $F^l(\cdot)$ and $R_{ij}$ denote the $l$-th level feature extraction function and the $j$-th reference image of class $i$, respectively. The second stage is to generate the prototype representations of each category from the reference images’ pyramid features. We need to find a map function $S(\cdot)$ which use all level pyramid features each category as input to compute the final prototype representation $F_i$ for class $i$ $$F_i = S (\{F_i^l\}). \label{equ:3}$$ The third stage is the design of the head model for classification and bounding box regression (Fig. \[fig:1\](b)), consisting of a few convolutional ($Conv$) and fully connected ($FC$) layers. Let $d(P_m, F_i)$ be a metric function to compute the distance between the feature of $m$-th proposal $P_m$ and prototype representation of the $i$-th category $F_i$. It is important to note that $P_m$ and $F_i$ have the same size. Each proposal’s classification $p_i$ and bounding box regression $b_i$ can be obtained by $$p_i = \frac{e^{-d(P_m, F_i)}}{\sum_ke^{-d(P_m, F_k)}}, \label{equ:4}$$ $$b_i = B(P_m, F_i), \label{equ:5}$$ where $B(\cdot,\cdot)$ denotes the box regression function. The rest of the model is the same as Faster R-CNN with FPN model [@lin2017feature]. ![ The block for learning prototype representations of background class[]{data-label="fig:2"}](figure2.pdf) Learning the reference background --------------------------------- There are many negative proposals generated by RPN, so the R-CNN [@girshick2014rich] adds a background category to represent them. In our Comparison detector, we need to select a number of reference images for each category and therefore we also need to choose reference images for the background category. Due to the overwhelming diversity, selecting background reference is very difficult. Notice that a region is considered to a the proposal indicating that it has certain similarity with categories. Therefore, it can be inferred that its features are a combination of different categories in the most case. So we propose to learn it by combining the prototype representations of all the categories in the reference samples, which can be implemented by a simple $1\times1$ convolution operation, as shown in Fig. \[fig:2\]. Generating prototype representations of categories -------------------------------------------------- As shown in Eq. \[equ:4\], the Comparison detector uses metric function to measure distance or the dissimilarity between the prototype representation of categories and the features of the proposal, then obtains the label of the proposals based on the dissimilarity. Features of proposal may come from any of the four level pyramids, and the prototype representation of the categories is obtained according to Eq. \[equ:3\]. For simplicity, we directly resize the each feature pyramid which is generated by reference images to a fixed size, and then calculate prototype representation by averaging operation, i.e. $$\label{pr_avg} S(\{F_i^l\}) = \frac{1}{t}\sum_{l}r(F_i^l, s),$$ where $t$ is the total number of level feature pyramids, $r(\cdot, \cdot)$ is resize function and $s$ is the size of final features. Different levels pyramid features of the category are resized into fixed size, and then getting the prototype representation by simply averaging them. The head for classification and regression {#section:3.3} ------------------------------------------ As shown in Fig. \[fig:3\](a), the structure of the baseline model’s head is to transform the proposal feature firstly and then one branch is used for classification, and another is used to predict the offset of the bounding box. For our Comparison detector, due to the introduction of the reference images, we need to re-organise the head. The are two choices according to whether the reference images are involved in the box regression branch. One is that the reference prototypes are only used for classification, as shown in Fig. \[fig:1\](b). Unlike the baseline model, the comparison classifier and bounding box regressor in the head of Comparison detector are independent. And the bounding box regressor only uses the features of ROI to predict the offset of the bounding box. It is equivalent to $$\begin{aligned} d(P_m, F_i) & = FC(F(P_m, F_i)), \\ B(P_m, F_i) & = FC(FC(FC(P_m))),\end{aligned}$$ where $F(P_m, F_i) = Conv_3(Conv_1(|F_i - P_m|^2))$. Another choice is to use the reference prototypes for both classification and regression, as shown in Fig. \[fig:3\](b), which means $$B(P_m, F_i) = FC(F(P_m, F_i)).$$ We call this method as shared module. They all achieve good performance in our experiments but the independent module performs slightly better (see Table \[tab:1\]). Reference images sampling ------------------------- In our Comparison detector, we also need to choose the reference images for each category. A intuitive way is to select them according to the Bethesda atlas [@nayar2015bethesda]. However, there are very significant difference between the given atlas and our data due to the variations of the preparation and digitization of slide. Hence we resort to other feasible data-driven alternatives. We randomly select about 150 instances of each category from the training sets. The shortest side of these instances is greater than 16 pixels. Therefore we get a total of 1560 instances and from them, we can select suitable instances in these objects as our reference images. There are two possible way. The first is to randomly choose several instances of each category as the reference images. The second is to first map all 1560 objects into the feature space through the pre-trained model and get the features of each object and then use t-SNE [@maaten2008visualizing] for feature dimension reduction (Fig. \[fig:4\]). Based on the results of t-SNE, we select the most representative samples in 3D space as our reference images. Experiment and Result ===================== Materials and experiments {#sect:4.1} ------------------------- Since there are no established benchmarks for cervical cell object detection in the community, we first establish a database consisting of 7,086 cervical microscopical images and based on which 48,587 object instance bounding boxes were labeled by experienced pathologists. Conforming to TBS categories [@nayar2015bethesda], we divide these objects into 11 categories, namely ASC-US (ascus), ASC-H (asch), low-grade squamous intraepithelial lesion (lsil), high-grade squamous intraepithelial lesion (hsil), squamous-cell carcinoma (scc), atypical glandular cells (agc), trichomonas (trich), candida (cand), flora, herps, actinomyces (actin). Figure \[fig:5\] shows some examples of each category in our database. Then we divide the dataset into training set $D_f$ which contains 6667 images, test set which contains 419 images for experiment. We randomly choose 762 images from the training dataset to form a small dataset of $D_s$. The number of categories in each dataset is shown in the Fig. \[fig:6\] {width=".7\textwidth"}  In all experiments, we used ResNet50 as backbone network with ImageNet pre-trained model. For reference images, we re-scale them such that their side is $w=h=224$ which is coincident with pre-trained model. The initial learning rate is 0.001, and then decreased by a factor of 10 at 35-th and 50-th epoch. Training is stopped after 60 epochs and the other parameters are the same as FPN [@lin2017feature]. The experiment is firstly trained on the $D_f$ to evaluate the performance on sufficient data. In our setting, the reference images are fixed in each training iteration for the stability of the training model. And test stage is the same. As for the cervical cell images, annotators are prone to take a higher threshold when label the objects due to the low discrimination of them. At the same time, multiple nearby objects with the same category will be marked as one, so the performance of the model can not be well reflected by mAP. Therefore, the performance of the model is evaluated by using mAP and AR as a complement on test set. If the mAP does not decrease and the AR improves, it surely signifies the performance is improved. Herein, the results are reported in both mAP and AR. A summary of results can be found in Table \[tab:1\] and some detection results on the test set are shown in Fig.\[fig:7\]. . [|c|c|c|c|c|c|c|c|]{} ------------------------------------------------------------------------ **model & ****** ------------ learning background ------------ & **independent mode & ****** ------------------ using all pyramid features ------------------ & **refine box & **balance loss & **mAP & **AR\ ******** ------------------------------------------------------------------------ `A` &$\surd$ &$\surd$ &$\surd$ &$\surd$ & &34.1 &53.3\ ------------------------------------------------------------------------ `B` & &$\surd$ &$\surd$ &$\surd$ & &31.4 &49.3\ ------------------------------------------------------------------------ `C` &$\surd$ &$\surd$ & &$\surd$ & &32.7 &50.8\ ------------------------------------------------------------------------ `D` &$\surd$ & &$\surd$ &$\surd$ & &41.0 &51.3\ ------------------------------------------------------------------------ `E` &$\surd$ & & &$\surd$ & &38.9 &49.8\ ------------------------------------------------------------------------ `F` &$\surd$ &$\surd$ &$\surd$ & & &37.7 &51.1\ ------------------------------------------------------------------------ `G` &$\surd$ &$\surd$ &$\surd$ & &$\surd$ &38.8 &52.3\ ------------------------------------------------------------------------ `H` &$\surd$ & &$\surd$ &$\surd$ &$\surd$ &43.5 &58.9\ ------------------------------------------------------------------------ `I` &$\surd$ &$\surd$ &$\surd$ &$\surd$ &$\surd$ &**43.7** &**60.7**\ ----------------------------------------------------------------------------------- -- -- -- **comparator & **$\ell_2$-distance &**parameterized $\ell_2$-distance & **concat\ mAP &34.1(43.7)& 38.2(**44.5**) & 40.7(42.5)\ AR &53.3(60.7)& 56.8(**61.6**) & 49.1(58.1)\ ******** ----------------------------------------------------------------------------------- -- -- -- Reference background -------------------- We first evaluate our scheme to learn the background reference. Experiment shows our method is feasible (See model `A` in Table \[tab:1\]). It should be noted that because the prototype representation of the background category is learned from the prototype representation of other categories, the gradient propagation will also have some effect on the optimization of other prototype representation. In order to make sure whether this effect is beneficial, we stop gradient propagation at the fork position in Fig. \[fig:2\]. The performance of the model has declined with an mAP of 33.0% and a AR of 52.6%. In order to compare the effect of learning background, we randomly select some background samples from the proposals to obtain the features of background. The result is model `B` in Table \[tab:1\]. Besides generating the prototype of background, we try to remove it but the training fails to converge. Prototype representations of categories --------------------------------------- In our approach, as shown in Eq. \[pr\_avg\], we use all pyramid features to generate prototype representation of categories. Another choice is to only use the last level pyramid feature as the category of prototype, i.e. $S (\{F_i^l\}) = F^5_i$. As shown the results of model `A` comparing to model `C` and model `D` comparing to model `E` in Table \[tab:1\], using all pyramid features performs much better. Because it can combine features from multiple level pyramids, which not only have rich semantics but also take into account objects of different size. We also fuse different levels of pyramid features by using LSTM [@hochreiter1997long], but the speed is greatly reduced. Head model {#section:4.3} ---------- As mentioned before, in independent module, the box regression function $B(\cdot,\cdot)$ is the same as baseline model because experiment found that removing one layer will make the result worse. The results show shared module (model `D`) performs much better than independent module (model `A`). Furthermore, we drop the operation of refining bounding box from the head. As shown in Table \[tab:1\], it’s weird that model `F` is better than model `A` which goes against common sense that fine-tuning bounding box twice is often better than just once. We conjecture that the importance of classification should be more important than bounding box regression in our model [@liang2018object]. So we add a weight coefficient $\lambda$ to balance the classification loss and bounding box regression loss. Here we select $\lambda=5$. The results of model `G`, `H` and `I` in Table \[tab:1\] confirm our analysis. By analyzing model `H` and model `I`, we find that the difference between them is not only classification and bbox regression is independent, but also the comparison classifier of model `H` is semi-parameter. After changing the comparison classifier of model `I` into semi-parameter(Fig. \[fig:1\] (b)), the results show that it is better than model `H`. ---------------------------------------------------- -- -- -- **method & **fixed mode & **random mode & **t-SNE\ mAP &44.5 &42.8&**45.3**\ AR & 61.6 &61.0& **62.8**\ ******** ---------------------------------------------------- -- -- -- : Different way of selecting the reference images.[]{data-label="tab:3"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} {height="3.2cm" width="4.3cm"} [|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|]{} **method & **dataset & **AR & **mAP & **ascu &**asch &**lsil &**hsil &**scc & **agc & **trich & **cand & **flora & **herps&**actin\ ****************************** ---------- baseline model ---------- & $D_s$ &12.9&6.6&11.0&2.0&23.7&21.6&0.0&3.5&0.0&11.5&0.0&0.0&0.0\ ------------ Comparison detector ------------ &$D_s$ &**35.7**&**26.3**&10.5&1.7&42.8&32.3&0.8&40.5&37.5&24.1&6.9&45.0&46.6\ ---------- baseline model ---------- &$D_f$ &58.9&45.2&27.2&6.7&41.7&35.3&18.6&57.3&46.7&72.2&57.3&83.0&51.4\ ------------ Comparison detector ------------ &$D_f$ &**62.8**&**45.3**&29.1&7.8&43.0&37.8&19.3&56.2&50.4&62.2&59.4&64.4&68.3\ Optimizing comparison classifier -------------------------------- We evaluate three distance metrics in the comparison classifier. The first is $\ell_2$-distance which means $d(P_m, F_i) = M(|F_i-P_m |^2) $. $M(\cdot)$ represents averaging function for tensor. The second is the parameterized $\ell_2$-distance, such as $d(P_m, F_i) = Conv_7(|F_i - P_m|^2)$. Similar to [@yang2018learning], we also try to make the model to learn the metric function instead of the predefined ones. According to the result of Table \[tab:2\], our model ultimately adopts parameterized $\ell_2$-distance. When $\lambda =5$, the result is shown in brackets. Combining with the results shown in Table \[tab:1\], it is universal that the balance trick can improve performance in our model. So we adopt this trick in all the next experiments. Reference images sampling ------------------------- We first evaluate the scheme of randomly choosing reference image which includes two tests. The first is to randomly choose 3 instances of each category (this number is limited by GPU’s memory) as the reference images (`fixed mode`). The second one is to randomly select 5 candidates of each category in those objects. Then the model randomly selected three of the five candidates as templates during training, but five in testing (`random mode`). The results are listed in Table \[tab:3\], which shows that compared with `random mode`, `fixed mode` performs better. Therefore when evaluate the scheme of choosing reference image by applying t-SNE, we also adopt the `fixed mode`. During the training of t-SNE, we adopt the following parameters setting, i.e. the hyper-parameters are 30 for perplexity, 1 for learning rate, and 10 for label supervision. Table \[tab:3\] shows that the selection reference image via t-SNE performs the best. Performance on training dataset $D_f$ and $D_s$ ----------------------------------------------- As shown in Table \[tab:4\], Comparison detector has the almost same mAP as the baseline model when training on the $D_f$ dataset, but improves the AR by near 4 points. Due to the special annotating situation as described in Section \[sect:4.1\], some correct predictions may be identified as false positives. Therefore, there is an significant increase in AR, but little improvement in mAP. When training on the $D_s$, Comparison detector is completely superior to baseline model. It achieves a top result on the test set with a mAP of 26.3% compared to 6.6%, which indicates our method alleviates the over fitting problem to some extent. Prototype representation in this model is generated by reference images, however, it can be generated by other way, such as external memory. In the future work, We expect a better solution for the generation of prototype representations. Conclusion ========== In this work, we propose to apply contemporary CNN-based object detection methods for automated cervical cancer detection. To deal with the limited size of training samples, we develop the comparison classifier into the state-of-the-art two-stage object detection method based on the comparison with the reference images of each category. Instead of manually choosing the reference images of the background by some heuristic rules, we present a scheme to learn them form the data directly. We also investigate several important ingredients including the generation of prototype representations of each class and the design of head model for cervical cell detection. Experimental results show that compared with the baseline, our method improves the mAP by **19.7** points and the AR by **22.8** when trained on the small size training data, and achieves almost the same mAP but improves the AR by **3.9** when trained on the medium size training data. It should be noticed that our algorithm directly operate on the whole image rather than the extracted patches based on the nuclei and hereby only need one forward propagation for each image, making the inference very efficient. In addition, the proposed method is *flexible* to be intergraded into other proposal-based methods. [^1]: This research was partially supported by the National Natural Science Foundation of China under Grant No. 61602522, the Natural Science Foundation of Hunan Province, China under Grant No.14JJ2008 and the Fundamental Research Funds of the Central Universities of Central South University under Grant No. 2018zzts577. (*Corresponding author: Yao Xiang*.) [^2]: Y. Liang, Z. Tang, M. Yan, J. Chen and Y. Xiang are with the School of Computer Science and Engineering, Central South University, Hunan 410083, China. E-mail: {yxliang,yao.xiang}@csu.edu.cn. [^3]: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. [^4]: <https://cs.adelaide.edu.au/~carneiro/isbi14_challenge/index.html> [^5]: <https://cs.adelaide.edu.au/~zhi/isbi15_challenge/index.html>

ArXiv

--- abstract: 'We investigate Monte Carlo Markov Chain (MCMC) procedures for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. We will see that an approach inspired by *optimal transport* allows us to efficiently bound the mixing time of the associated Markov chain. The algorithm is robust and easy to implement, and samples an “almost” uniform path of length $n$ in $n^{3+{\varepsilon}}$ steps. This bound makes use of a certain *contraction property* of the Markov chain, and is also used to derive a bound for the running time of Propp-Wilson’s *Coupling From The Past* algorithm.' author: - Lucas Gerin title: 'Random sampling of lattice paths with constraints, via transportation' --- Lattice Paths with Constraints ============================== Lattice paths arise in several areas in probability and combinatorics, either in their own interest (as realizations of random walks, or because of their interesting combinatorial properties: see [@Ban] for the latter) or because of fruitful bijections with various families of trees, tilings, words. The problem we discuss here is to efficiently sample uniform (or *almost* uniform) paths in a family of paths with constraints. There are several reasons for which one may want to generate uniform samples of lattice paths: to make and try conjectures on the behaviour of a large “typical” path, test algorithms running on paths (or words, trees,...). In view of random sampling, it is often very efficient to make use of the combinatorial structure of the family of paths under study. In some cases, this yields linear-time (in the length of the path) *ad-hoc* algorithms [@MBM; @Duc]. However, the nature of the constraints makes sometimes impossible such an approach, and there is a need for robust algorithms that work in lack of combinatorial knowledge. Luby,Randall and Sinclair [@LRS] design a Markov chain that generate sets of non-intersecting lattice paths. This was motivated by a classical (and simple, see illustrations in [@Des; @Wilson]) correspondence between dimer configurations on an hexagon, rhombae tilings of this hexagon and families of non-intersecting lattice paths. As the first step for the analysis of this chain, Wilson [@Wilson] introduces a peak/valley Markov chain (see details below) over some simple lattice paths and obtain sharp bounds for its mixing time. We present in this paper a variant of this Markov chain, which is valid for various constraints and whose analysis is simple. It generates an “almost” uniform path of length $n$ in $n^{3+{\varepsilon}}$ steps, this bound makes use of a certain *contraction property* of the chain. Appart from the algorithmic aspect, the peak/valley process seems to have a physical relevancy as a simplified model for the evolution of *quasicrystals* (see a discussion on a related process in the introduction of [@Des]). In particular, the mixing time of this Markov seems to have some importance. Notations {#notations .unnumbered} --------- {width="40mm"} We fix three integers $n,a,b>0$, and consider the paths of length $n$, with steps $+a/-b$, that is, the words of $n$ letters taken in the alphabet ${\left\{a,-b\right\}}$. Such a word $s=(s_1,s_2,\dots,s_n)$ is identified to the path $S=(S_1,\dots,S_n):=(s_1,s_1+s_2,\dots,s_1+s_2+\dots +s_n)$. To illustrate the methods and the results, we focus on some particular sub-families ${\mathcal{A}_{n}}\subset {\left\{a,-b\right\}}^n$: 1. Discrete *meanders*, denoted by ${\mathcal{M}_{n}}$, which are simply the non-negative paths: $S\in{\mathcal{M}_{n}}$ if for any $i\leq n$ we have $S_i\geq 0$. This example is mainly illustrative because the combinatorial properties of meanders make it possible to perform exact sampling very efficiently (an algorithm running in $\mathcal{O}(n^{1+{\varepsilon}})$ steps is given in [@MBM], an order that we cannot get in the present paper). 2. Paths with *walls*. A path with a wall of height $h$ between $r$ and $s$ is a path such that $S_i\geq h$ for any $r\leq i\leq s$ (see Fig. \[Fig:CheminMur\] for an example). These are denoted by ${\mathcal{W}_{n}}={\mathcal{W}_{n}}(h,r,s)$, they appear in statistical mechanics as toy models for the analysis of random interfaces and polymers (see examples in [@Walls]). 3. *Excursions*, denoted by ${\mathcal{E}_{n}}$, which are non-negative paths such that $S_n=0$. In the case $a=b=1$, these correspond to well-parenthesed words and are usually called Dyck words. In the general case, Duchon [@Duc] proposes a rejection algorithm which generates excursions in linear time. 4. *Culminating paths* of size $n$, denoted further by ${\mathcal{C}_{n}}$, which are non-negative paths whose maximum is attained at the last step: for any $i$ we have $0\leq S_i\leq S_n$. They have been introduced in [@MBM], motivated in particular by the analysis of some algorithms in bioinformatics. ![A path of steps $+1/-2$, with a wall of height $h=6$ between $i=10$ and $j=15$.[]{data-label="Fig:CheminMur"}](CheminMur.eps){width="65mm"} Sampling with Markov chains {#Sec:Sampling} =========================== We will consider Markov chains in a family ${\mathcal{A}_{n}}$, where all the probability transitions are symmetric. For a modern introduction to Markov chains, we refer to [@Hagg]. Hence we are given a transition matrix $(p_{i,j})$ of size $|{\mathcal{A}_{n}}|\times|{\mathcal{A}_{n}}|$ with $$\begin{aligned} p_{i,j} &=p_{j,i} \mbox{ whenever }i\neq j,\\ p_{i,i} &= 1-\sum_{j\neq i}p_{i,j}.\end{aligned}$$ \[Lem:Unif\] If such a Markov chain is irreducible, then it admits as unique stationary distribution the uniform distribution $\pi=\pi({\mathcal{A}_{n}})$ on ${\mathcal{A}_{n}}$. The equality $\pi(i) p_{i,j}= \pi(j) p_{j,i}$ holds for any two vertices $i,j$. This shows that the probability distribution $\pi$ is reversible for $(p_{i,j})$, and hence stationary. It is unique if the chain is irreducible. This lemma already provides us with a scheme for sampling an almost uniform path in ${\mathcal{A}_{n}}$, without knowing much about ${\mathcal{A}_{n}}$. To do so, we define a “flip” operator on paths, this is an operator $$\begin{array}{r c c c} \phi: & {\mathcal{A}_{n}}\times {\left\{1,\dots,n\right\}}\times {\left\{\downarrow,\uparrow\right\}}\times {\left\{+,-\right\}} &\to &{\mathcal{A}_{n}}\\ & (\mathbf{S},i,{\varepsilon},\delta) &\mapsto & \phi(\mathbf{S},i,{\varepsilon},\delta). \end{array}$$ When $i\in{\left\{1,2,\dots,n-1\right\}}$ the path $\phi(\mathbf{S},i,\uparrow,\delta)$ is defined as follows : if $(s_i,s_{i+1})=(-b,a)=$ {width="7mm"} then these two steps are changed into $(a,-b)=$ {width="7mm"}. The $n-2$ other steps remain unchanged. If $(s_i,s_{i+1})\neq (-b,a)$ then ${\phi(\mathbf{S},i,\uparrow)}{\delta}=\mathbf{S}$. Note that in the case $i\in{\left\{1,2,\dots,n-1\right\}}$ the value of $\phi$ does not depend on $\delta$. For the case $i=n$, if $\delta=+$, we define ${\phi(\mathbf{S},n,{\varepsilon})}{\delta}$ as before as if there would be a $+a$ as the end if the path. For instance, in the case where $S_n=-b$, the path ${\phi(\mathbf{S},n,\uparrow)}{+}$, the $n$-th step is turned into $a$. The path ${\phi(\mathbf{S},i,\downarrow)}{\delta}$ is defined equally: if $i<n$ and $(s_i,s_{i+1})=$ {width="7mm"}, it turns into {width="7mm"}. When $\delta=-$, one flips as if there would be a $-b$ at the end of the path. For culminating paths, we have to take another definition of ${\phi(\mathbf{S},n,\uparrow)}{\delta},{\phi(\mathbf{S},n,\downarrow)}{\delta}$, see Section \[Sec:Analysis\]. We are also given a probability distribution $\mathbf{p}=(p_i)_{1\leq i\leq n}$, and we assume that $p_i>0$ for each $i$. We will consider a particular sequence $\mathbf{p}$ later on, but at this point we can take the uniform distribution in ${\left\{1,\dots,n\right\}}$. We describe the algorithm below in Algorithm \[Algo:CM\]. initialize $\mathbf{S}=(+a,+a,+a,\dots,+a)$ $I_{1},I_{2},\dots\leftarrow$ i.i.d. r.v. with law $\mathbf{p}$ ${\varepsilon}_{1},{\varepsilon}_{2},\dots\leftarrow$ i.i.d. uniform r.v. in ${\left\{\uparrow,\downarrow\right\}}$ $\delta_{1},\delta_{2},\dots\leftarrow$ i.i.d. uniform r.v. in ${\left\{+,-\right\}}$ $\mathbf{S}\leftarrow {\phi(\mathbf{S},I_t,{\varepsilon}_t)}{\delta_t}$ In words, this algorithm performs the Markov chain in ${\mathcal{A}_{n}}$ with transition matrix $P=\left(P_{\mathbf{R},\mathbf{S}}\right)_{\mathbf{R},\mathbf{S}\in{\mathcal{A}_{n}}}$ defined as follows: $$\begin{cases} P_{\mathbf{R},\mathbf{S}}&=p_i/2, \mbox{ if } \mathbf{S}\neq\mathbf{R}\text{ and } \mathbf{S}={\phi(\mathbf{R},i,{\varepsilon})}{\delta} \mbox{ for some }i,{\varepsilon}, \delta\\ P_{\mathbf{R},\mathbf{S}}&=0\text{ otherwise,}\\ P_{\mathbf{R},\mathbf{R}}&=1-\sum_{\mathbf{S}\neq \mathbf{R}} P_{\mathbf{R},\mathbf{S}}.\\ \end{cases}$$ Denote by $S(t)$ the random path obtained after the $t$-th run of the loop in Algorithm \[Algo:CM\]. When $t\to\infty$, the sequence $S(t)$ converges in law to the uniform distribution in ${\mathcal{A}_{n}}$. Moreover, the execution of Algorithm \[Algo:CM\] until time $T$ is linear in $T$. For the first claim, we have to check that the chain is aperiodic and irreducible. Aperiodicity comes from the (many) loops. Irreducibility will follow from Lemma \[Lem:Geodesique\]. For the second claim, notice that the time needed for the test “${\phi(\mathbf{S},I_t,{\varepsilon}_t)}$ is in ${\mathcal{A}_{n}}$” can be considered as constant, since for the families ${\mathcal{M}_{n}}$ and ${\mathcal{E}_{n}}$ we only have to compare $0,S_i$ while for the family ${\mathcal{W}_{n}}$ we only have to compare $S_i$ with the height of the wall at $i$. For the case of the culminating paths, see below in Section \[Sec:Analysis\]. We now choose the distribution $(p_i)$. Instead of $p_i=1/n$, we will use the probability distribution defined by$$\label{Eq:Poids} p_i:=i(2n-i)\kappa_0 +a\quad (\mbox{ for }i=1,\dots,n),$$ where $$\begin{aligned} \kappa_0&=\frac{3}{2n^2(n+1)}\\ a &=1/4n^3.\end{aligned}$$ We let the reader check that $(p_i)_{i\leq n}$ is indeed a probability distribution. The reason for which we use this particular distribution will appear in the proof of Proposition \[Lem:Courbure\]. We will then need the following relation: for each $1\leq i\leq n-1$, $$\label{Eq:kappa} p_i-p_{i-1}/2-p_{i+1}/2 = \kappa_0.$$ For Algorithm \[Algo:CM\] to be efficient, we need to know how $S(T)$ is close in law to $\pi$. This question is related to the spectral properties of the matrix $P$. In particular, the speed of convergence is governed by the spectral gap ([[*i.e.*]{} ]{}$1-\lambda$, where $\lambda$ is the largest of the modulus of the eigenvalues different from one, see [@Mix] for example), but this quantity is not known in general. Some geometrical methods [@Dia] allow to bound from below $1-\lambda$, but they assume a precise knowledge of the structure of the graph defined by the chain $P$. It seems that such results do not apply here. Instead, we will study the metric properties of the chain $P$ with respect to a natural distance on ${\mathcal{A}_{n}}$, and show that it satisfies a certain *contraction property*. The variant of Algorithm \[Algo:CM\] for culminating paths {#Sec:Analysis} ---------------------------------------------------------- Unchanged, our Markov chain $P$ cannot generate culminating paths since the path $\mathbf{S}=(a,a,\dots,a)$ would then be isolated: it has no peak/valley and ${\phi(\mathbf{S},n,\downarrow)}{-}=(a,a,\dots,-b)$ which is not culminating. Thus we propose a slight modification for the family ${\mathcal{C}_{n}}$. We only change the definition of ${\phi(\mathbf{S},i,{\varepsilon})}{\delta}$ when $i=n$ (it won’t depend on $\delta$). Since the maximum is reached at $n$, the $\lceil b/a\rceil +1$ last steps are necessarily $$(a,a,\dots,a) \mbox{ or } (-b,a,\dots,a).$$ We thus define ${\phi(\mathbf{S},n,\uparrow)}{\delta}$ as the path obtained by changing the $\lceil b/a\rceil +1$ last steps into $(a,a,\dots,a)$ (regardless of their initial values in $\mathbf{S}$) and ${\phi(\mathbf{S},n,\downarrow)}{\delta}$ as the path obtained by changing the $\lceil b/a\rceil +1$ last steps into $(-b,a,\dots,a)$. Notice that despite this change the execution time of each loop of Algorithm \[Algo:CM\] is still a $\mathcal{O}(1)$: - If $I_t< n$, the time needed for the test “${\phi(\mathbf{S},I_t,{\varepsilon}_t)}{\delta_t}$ is in ${\mathcal{A}_{n}}$” can be considered as constant, since we only have to compare $0,S_i,S_n$. - If $I_t=n$, the new value $S_n$ is compared with the maximum of $S$, which can be done in $\mathcal{O}(n)$. Fortunately, this occurs with probability $p_n=\mathcal{O}(1/n)$, so that the time-complexity of each loop is, on average, a $\mathcal{O}(1)$. Error estimates with contraction {#Sec:Ricci} ================================ Going back to a more general setting, we consider a Markov chain in a finite set $V$, endowed with a metric $d$. For a vertice $x\in V$ and a transition matrix $P$, we denote by $P\delta_x$ (resp. $P^t\delta_x$) the law of the Markov chain associated with $P$ at time $1$ (resp. $t$), when starting from $x$. For $x,y\in V$, the main assumption made on $P$ is that there is a coupling between $P\delta_x,P\delta_y$ (that is, a random variable $(X_1,Y_1)$ with $X_1\stackrel{\mbox{law}}=P\delta_x,Y_1\stackrel{\mbox{law}}=P\delta_y$) such that $$\label{Eq:Courbure} \mathbb{E}\left[d(X_1,Y_1)\right]\leq (1-\kappa)d(x,y),$$ for some $\kappa >0$, which is called the *Ricci curvature* of the chain, by analogy with the Ricci curvature in differential geometry[^1]. If the inequality holds, then it implies ([@Mix],p.189) that $P$ admits a unique stationary measure $\pi$ and that, for any $x$, $$\label{Eq:Mixing} \parallel P^t\delta_x -\pi\parallel_{\mathrm{TV}} \leq (1-\kappa)^t\mathrm{diam}(V),$$ where $\mathrm{diam}(V)$ is the diameter of the graph with vertices $V$ induced by the Markov chain. The notation $\parallel .\parallel_{\mathrm{TV}}$ stands, as usual, for the *Total Variation* distance over the probability distributions on $V$ defined by $$\parallel \mu_1 -\mu_2\parallel_{\mathrm{TV}}:= \sup_{A\subset V} \left|\mu_1(A)-\mu_2(A)\right|.$$ Hence, a positive Ricci curvature gives the exponential convergence to the stationary measure, with an exact ([[*i.e.*]{} ]{} is non-asymptotic in $t$) bound. In many situations, a smart choice for the coupling between $X_1,X_2$ gives a sharp rate of convergence in eq. (see some striking examples in [@Olli]). Metric properties of $P$ ------------------------ To apply the Ricci curvature machinery, we endow each ${\mathcal{A}_{n}}$ with the $L^1$-distance $$d_1(S,S')=\frac{1}{a+b}\sum_{i=0}^n |S_i-S_i'|.$$ (Notice that $|S_i-S_i'|$ is always a multiple of $a+b$.) For our purpose, it is fundamental that this metric space is *geodesic*. A Markov chain $P$ in a finite set $V$ is said to be *geodesic* with respect to the distance $d$ on $V$ if for any two $x,y\in V$ with $d(x,y)=k$, there exist $k+1$ vertices $x_0=x,x_1,\dots,x_k=y$ in $V$ such that for each $i$ - $d(x_i,x_{i+1})=1$ ; - $x_i$ and $x_{i+1}$ are neighbours in the Markov chain $P$ ([[*i.e.*]{} ]{}$P(x_i,x_{i+1})>0$). This implies in particular that $P$ is irreducible and that the diameter of $P$ is smaller than $\max_{x,y}d(x,y)$. \[Lem:Geodesique\] For each family ${\mathcal{C}_{n}}$,${\mathcal{W}_{n}}$,${\mathcal{E}_{n}}$,${\mathcal{M}_{n}}$ the Markov chain of Algorithm \[Algo:CM\] is geodesic with respect to $d_1$. The proof goes by induction on $k$. We fix $S\neq T$ (and denote by $s,t$ the corresponding words) ; we want to decrease $d_1(S,T)$ by one, by applying the operator $\phi$ with proper $i,{\varepsilon}$. We denote by $i_0\in{\left\{1,\dots,n\right\}}$ the first index for which $S\neq T$. For instance we have $T_{i_0}=S_{i_0}+a+b$. Let $j$ be the position of the left-most peak in $T$ in ${\left\{i_0+1,i_0+2,\dots,n\right\}}$, if such a peak exists. Then $S':={\phi(\mathbf{T},j,\downarrow)}{\delta}$ is also in ${\mathcal{A}_{n}}$: it is immediate for the families ${\mathcal{M}_{n}},{\mathcal{W}_{n}},{\mathcal{C}_{n}},{\mathcal{E}_{n}}$. We have $d_1(S,S')=k-1$. If there is no peak in $T$ after $i_0$, then $(t_{i_0+1},t_{i_0+2},\dots,t_n)=(a,a,\dots,a)$. Hence we try to increase the final steps of $S$ by one. To do so, we choose $S':={\phi(\mathbf{S},n,\uparrow)}{\delta}$ if $S\neq {\phi(\mathbf{S},n,\uparrow)}{\delta}$, or $S'={\phi(\mathbf{S},j,\uparrow)}{\delta}$ where $j$ is the position of the right-most $-b$ otherwise (we choose the right-most one to ensure that ${\phi(\mathbf{S},j,\uparrow)}{\delta}$ remains culminating in the case where ${\mathcal{A}_{n}}={\mathcal{C}_{n}}$.). For meanders, excursions and walls, we will show that the Ricci curvature of $P$ with respect to the distance $d_1$ is (at least) of order $1/n^3$. \[Lem:Courbure\] For the three families ${\mathcal{M}_{n}},{\mathcal{E}_{n}},{\mathcal{W}_{n}}$, the Ricci curvature of the associated Markov chain, with weights $(p_i)$ defined as in , is larger than $\kappa_0$. Fix $\mathbf{S},\mathbf{T}$ in ${\mathcal{A}_{n}}\in{\left\{{\mathcal{M}_{n}},{\mathcal{E}_{n}},{\mathcal{W}_{n}}\right\}}$, we first assume that $\mathbf{S},\mathbf{T}$ are neighbours, for instance $\mathbf{T}={\phi(\mathbf{S},i,\uparrow)}$ for some $i$. {width="35mm"} Let $(\mathbf{S}^1,\mathbf{S}^2)$ be the random variable in ${\mathcal{A}_{n}}\times{\mathcal{A}_{n}}$ whose law is defined by $$(\mathbf{S}^1,\mathbf{S}^2)\stackrel{\mbox{(law)}}{=} \left(\phi(\mathbf{S},\mathcal{I},{E}),\phi(\mathbf{T},\mathcal{I},E)\right),$$ where $\mathcal{I}$ is a r.v. taking values in ${\left\{1,\dots,n\right\}}$ with distribution $\mathbf{p}$ and $E$ is uniform in ${\left\{\uparrow,\downarrow\right\}}$. In other words, we run one loop of Algorithm \[Algo:CM\] simultaneously on both paths. We want to show that $\mathbf{S}^1,\mathbf{S}^2$ are, on average, closer than $\mathbf{S},\mathbf{T}$. Different cases may occur, depending on $\mathcal{I}$ and on the index $i$ where $\mathbf{S},\mathbf{T}$ differ. [**  **]{} [**  **]{} This occurs with probability $p_i$ and, no matter the value of $E$, we have $\mathbf{S}^1=\mathbf{S}^2$. [**  **]{} We consider the case $i-1$. Since $\mathbf{S}$ and $\mathbf{T}$ coincide everywhere but in $i$, we necessarily have one of these two cases: - there is a peak in $\mathbf{S}$ at $i-1$ and neither a peak nor a valley in $\mathbf{T}$ at $i-1$ (as in the figure on the right) ; - there is a valley in $\mathbf{T}$ at $i-1$ and neither a peak nor a valley in $\mathbf{S}$ at $i-1$. In the first case for instance, then we may have $d_1(\mathbf{S}^1,\mathbf{S}^2)=2$ if $E=\downarrow$, while the distance remains unchanged if $E=\uparrow$. The case $\mathcal{I}=i+1$ is identical. This shows that with a probability smaller than $p_{i-1}/2+p_{i+1}/2$ we have $d_1(\mathbf{S}^1,\mathbf{S}^2)=2$. [**  **]{} In this case, $\mathbf{S}$ and $\mathbf{T}$ are possibly modified in $\mathcal{I}$, but if there is a modification it occurs in both paths. It is immediate since for the families ${\mathcal{M}_{n}}$,${\mathcal{W}_{n}}$ and ${\mathcal{E}_{n}}$ since the constraints are local. [**  **]{} In this case, it is easy to check that, because of our definition of ${\phi(\mathbf{S},n,{\varepsilon})}{\delta}$, we have $$\mathbb{E}\left[d_1(\mathbf{S}^1,\mathbf{S}^2)\right] \leq 1-p_{n-1}+p_{n-2}/2+p_{n}/2 =1-\kappa_0.$$ [**  **]{} We have $$\mathbb{E}\left[d_1(\mathbf{S}^1,\mathbf{S}^2)\right] \leq 1+p_{n-1}/2-p_{n}/2=1-\kappa_0.$$ Thus, we have proven that when $\mathbf{S},\mathbf{T}$ only differ at $i$ $$\begin{aligned} \mathbb{E}\left[d_1(\mathbf{S}^1,\mathbf{S}^2)\right] &\leq 2\times(p_{i-1}/2+p_{i+1}/2) +0\times p_i+1\times(1-p_i-p_{i-1}/2-p_{i+1}/2)\label{Eq:Accroissement}\\ &\leq (1-\kappa_0)\times 1=(1-\kappa_0)d_1(S,T)\notag.\end{aligned}$$ What makes Ricci curvature very useful is that if this inequality holds for pairs of neighbours then it holds for any pair, as noticed in [@Bub]. Indeed, take $k+1$ paths $S_0=S,S_1,\dots,S_k=T$ as in Lemma \[Lem:Geodesique\] and apply the triangular inequality for $d_1$: $$\begin{aligned} \mathbb{E}\left[d_1(\phi(S,\mathcal{I},{E}),\phi(T,\mathcal{I},E))\right] &\leq \sum_{i=0}^{k-1} \mathbb{E}\left[d_1(\phi(S_i,\mathcal{I},{E}),\phi(S_{i+1},\mathcal{I},E))\right]\\ &\leq (1-\kappa_0)k=(1-\kappa_0)d_1(S,T).\end{aligned}$$ It is easy to exhibit some $S,T$ such that ineq. is in fact an equality. In the case where $p_i=1/n$, this equality reads $\mathbb{E}\left[d_1(\mathbf{S}^1,\mathbf{S}^2)\right]=d_1(S,T)$, and we cannot obtain a positive Ricci curvature (though this does not prove that there is not another coupling or another distance for which we could get a $\kappa >0$ in the case $p_i=1/n$.). We recall that for each family ${\mathcal{A}_{n}}$, $\mathrm{diam}({\mathcal{A}_{n}})= \max d_1(\mathbf{S},\mathbf{T}) \leq n(n+1)/2$. Hence, combining Proposition \[Lem:Courbure\]with Eq. gives our main result: \[Th:Mix\] For meanders, excursions and path with walls, Algorithm \[Algo:CM\] returns an almost uniform sample of $\pi$, as soon as $T \gg n^3$. Precisely, for any itinialization of Algorithm \[Algo:CM\], $$\parallel \mathbf{S}(T) -\pi\parallel_{\mathrm{TV}} \leq \mathrm{diam}({\mathcal{A}_{n}})(1-\kappa)^T \leq \frac{n(n+1)}{2}\exp\left(-\frac{3}{2n^2(n+1)}T\right).$$ Another formulation of this result is that the mixing time of the associated Markov chain, defined as usual by $$\label{Eq:tmix} t_{\mbox{mix}}:={\left\{\inf\ t\geq 0\ ;\ \sup_{v\in V} \parallel P^t\delta_v -\pi\parallel_{\mathrm{TV}}\leq e^{-1}\right\}}$$ ($e^{-1}$ is here by convention), is smaller than $n^2(n+1)\log n$. For culminating paths, the argument of Case 1c fails and does not hold, we are not able to prove such a result as Theorem \[Th:Mix\]. However, it seems empirically that the mixing time is also of order $n^3\log n$ (with a constant strongly dependent on $a,b$). A way to prove this could be the following observation: take $(\mathbf{S}^0,\mathbf{T}^0)=(\mathbf{S},\mathbf{T})$ two any culminating paths, and define $$(\mathbf{S}^{t+1},\mathbf{T}^{t+1})=(\phi(\mathbf{S}^t,I_t,{\varepsilon}_t,\delta_t),\phi(\mathbf{T}^t,I_t,{\varepsilon}_t,\delta_t)),$$ where $I_t,{\varepsilon}_t,\delta_t$ are those in Algorithm \[Algo:CM\]. The sequence $ \left(\parallel \mathbf{S}^t-\mathbf{T}^t\parallel_\infty \right)_t $ is decreasing throughout the process. Unfortunately we cannot get a satisfactory bound for the time needed for this quantity to decrease by one. Related works {#Sec:Related} ------------- Bounding mixing times via a contraction property over the transportation metric is quite a standard technique, the main ideas dating back to Dobrushin (1950’s). A modern introduction is made in [@Mix]. For geodesic spaces, this technique has been developped in [@Bub] under the name *path coupling*. As mentioned in the introduction, the Markov chain $P$ on lattice paths with uniform weights $p_i=1/n$ has in fact already been introduced for paths starting and ending at zero (sometimes called *bridges*) in [@LRS], and its mixing time has been estimated in [@Wilson]. Wilson also proves a mixing time of order $n^3\log n$, by showing that holds with a different distance (namely, a kind of Fourier transform of the heights of the paths)[^2]. This is the concavity of this Fourier transform which gives a good mixing time, exactly as the concavity of our $p_i$’s speeds up the convergence of our chain. Wilson’s method is developped only for bridges in [@Wilson] and it is not completely straightforward to use it when the endpoints are not fixed. For instance, take $n=7$ and $a=b=1$, and consider the paths $+++--++$ and $---++--$. There are more “bad moves” (moves that take away these paths) than “good moves”. *Coupling From The Past* with $P$ {#Sec:ProppWilson} ================================= Propp-Wilson’s Coupling From The Past (CFTP) [@PW] is a very general procedure for the exact sampling of the stationary distribution of a Markov chain. It is efficient if the chain is monotonous with respect to a certain order relation $\preceq$ on the set $V$ of vertices, with two extremal points denoted $\hat{0},\hat{1}$ ([[*i.e.*]{} ]{}such that $\hat{0}\preceq x\preceq\hat{1}$ for any vertex $x$). This is the case here for each family ${\mathcal{C}_{n}}$,${\mathcal{W}_{n}}$,${\mathcal{E}_{n}}$,${\mathcal{M}_{n}}$ , with the partial order $$\mathbf{S}\preceq \mathbf{T} \mbox{ iff } S_i\leq T_i \mbox{ for any }i.$$ For the family ${\mathcal{M}_{10}}$ with $a=1,b=-2$ for instance, we have $$\begin{aligned} \hat{0}=\hat{0}_{\mbox{meanders}}&=(1,1,-2,1,1,-2,1,1,-2,1),\\ \hat{1}=\hat{1}_{\mbox{meanders}}&=(1,1,1,1,1,1,1,1,1,1).\end{aligned}$$ We describe CFTP, with our notations, in Algorithm \[Algo:CFTP\]. $\mathbf{S}\leftarrow\hat{0}$, $\mathbf{T}\leftarrow\hat{1}$ $\dots,I_{-2},I_{-1}\leftarrow$ i.i.d. r.v. with law $\mathbf{p}$ $\dots,{\varepsilon}_{-2},{\varepsilon}_{-1}\leftarrow$ i.i.d. uniform r.v. in ${\left\{\uparrow,\downarrow\right\}}$ $\dots,\delta_{-2},\delta_{-1}\leftarrow$ i.i.d. uniform r.v. in ${\left\{+,-\right\}}$ $\tau=1$ $\mathbf{S}\leftarrow\hat{0}$, $\mathbf{T}\leftarrow\hat{1}$ ${\phi(\mathbf{S},I_t,{\varepsilon}_t)}$ is in ${\mathcal{A}_{n}}$ [**then** ]{} $\mathbf{S}\leftarrow {\phi(\mathbf{S},I_t,{\varepsilon}_t)}{\delta_t}$ ${\phi(\mathbf{T},I_t,{\varepsilon}_t)}$ is in ${\mathcal{A}_{n}}$ [**then** ]{} $\mathbf{T}\leftarrow {\phi(\mathbf{T},I_t,{\varepsilon}_t)}{\delta_t}$ $\tau\leftarrow 2\tau$ We refer to ([@Hagg],Chap.10) for a very clear introduction to CFTP, and we only outline here the reasons why this indeed gives an exact sampling of the stationary distribution. - The output of the algorithm (if it ever ends!) is the state of the chain $P$ that has been running “since time $-\infty$”, and thus has reached stationnarity. - The exit condition $\mathbf{S}=\mathbf{T}$ ensures that it is not worth running the chain from $T$ steps earlier, since the trajectory of any lattice path $\hat{0}\preceq \mathbf{R}\preceq\hat{1}$ is “sandwiched” between those of $\hat{0},\hat{1}$, and therefore ends at the same value. ![A sketchy representation of CFTP : trajectories starting from $\hat{0},\hat{1}$ at time $-T/2$ don’t meet before time zero, while those starting at time $-T$ do.[]{data-label="Fig:ProppWilson"}](ProppWilson.eps){width="65mm"} \[Prop:CFTP\] Algorithm \[Algo:CFTP\] ends with probability $1$ and returns an exact sample of the uniform distribution over ${\mathcal{A}_{n}}$. For the families ${\mathcal{W}_{n}}$,${\mathcal{E}_{n}}$,${\mathcal{M}_{n}}$, this takes on average $\mathcal{O}(n^3(\log n)^2)$ time units. Let us mention that in the case where the mixing time is not rigorously known, Algorithm \[Algo:CFTP\] (when it ends) outputs an exact uniform sample and therefore is of main practical interest compared to MCMC. It is shown in [@PW] that Algorithm \[Algo:CFTP\] returns an exact sampling in $\mathcal{O}(t_{\mbox{mix}}\log H)$ runs of the chain, where $t_{\mbox{mix}}$ is defined in and $H$ is the length of the longest chain of states between $\hat{0}$ and $\hat{1}$. It is a consequence of the proof of Lemma \[Lem:Geodesique\] that $H=\mathcal{O}(n^2)$. We have seen that $t_{\mbox{mix}}=\mathcal{O}(n^3\log n)$. (Recall that each test in Algorithm \[Algo:CFTP\] takes, on average, $\mathcal{O}(1)$ time units.) We recall that CFTP has a major drawback compared to MCMC. For the algorithm to be correct, we have to reuse the same random variables $I_t,{\varepsilon}_t,\delta_t$, so that space-complexity is in fact linear in $n^3(\log n)^2$. This may become an issue when $n$ is large. Concluding remarks and simulations ================================== [**1.**]{} In Fig.\[Fig:Simus\], we show simulations of the three kinds of paths, for $a=1,b=2,n=600$. We observe that the final height of the culminating path is very low (about $30$), it would be interesting to use our algorithm to investigate the behaviour of this height when $n\to\infty$ ; this question was left open in [@MBM]. ![(Almost) uniform paths of length $600$, with $a=1,b=2$. From top to bottom: a culminating path, a meander, a path with wall (shown by an arch).[]{data-label="Fig:Simus"}](SimuCulmi600.eps "fig:"){width="140mm"}\ ![(Almost) uniform paths of length $600$, with $a=1,b=2$. From top to bottom: a culminating path, a meander, a path with wall (shown by an arch).[]{data-label="Fig:Simus"}](SimuPositif600.eps "fig:"){width="140mm"}\ ![(Almost) uniform paths of length $600$, with $a=1,b=2$. From top to bottom: a culminating path, a meander, a path with wall (shown by an arch).[]{data-label="Fig:Simus"}](SimuWall600.eps "fig:"){width="140mm"}\ [**2.**]{} One may wonder to what extent this work applies to other families ${\mathcal{A}_{n}}$ of paths. The main assumption is that the family of paths should be a geodesic space w.r.t. distance $d_1$. This is true for example if the following condition on ${\mathcal{A}_{n}}$ is fulfilled: $$\left(R,T\in{\mathcal{A}_{n}} \mbox{ and }R\preceq S\preceq T\right) \Rightarrow S\in{\mathcal{A}_{n}}.$$ Notice however that this is quite a strong requirement, and it is not verified for culminating paths for instance. [**3.**]{} A motivation to sample random paths is to make and test guesses for some functionals of these paths, taken on average over ${\mathcal{A}_{n}}$. Consider a function $f:{\mathcal{A}_{n}}\to\mathbb{R}$, we want an approximate value of $\pi(f):={\mathrm{card}}({\mathcal{A}_{n}})^{-1}\sum_{s\in{\mathcal{A}_{n}}}f(s)$, if the exact value is out of reach by calculation. We estimate this quantity by $$\label{Eq:Chapeau} \hat{\pi}(f):=\frac{1}{T} \sum_{t=1}^T f \left(\mathbf{S}(t)\right),$$ (recall that $S(t)$ is the value of the chain at time $t$). For Algorithm \[Algo:CM\] to be efficient in practice, we have to bound $$\label{Eq:AMajorer} \mathbb{P}\left(\left|\pi(f)-\hat{\pi}(f)\right|>r\right),$$ for any fixed $r>0$, by a non-asymptotic (in $T$) quantity. This can be done with ([@JouOlli], Th.4-5), in which one can find concentration inequalities for . The sharpness of these inequalities depends on $\kappa$ and on the geometrical structure of ${\mathcal{A}_{n}}$. [**Aknowledgements.**]{} Many thanks to Frédérique Bassino and the other members of <span style="font-variant:small-caps;">Anr Gamma</span> for the support ; I also would like to thank Élie Ruderman for the English corrections. A referee raised a serious error in the first version of this paper, I am grateful to them. [99]{} C. Banderier and Ph. Flajolet. Basic Analytic Combinatorics of Directed Lattice Paths, *Theoretical Computer Science* [**281**]{} (1):37-80 (2002). M.Bousquet-Mélou and Y. Ponty. Culminating paths, *Discrete Math and Theoretical Computer Science* [**10**]{} (2):125-152 (2008). R.Bubley and M. Dyer. Path coupling: A technique for proving rapid mixing in Markov chains. Proceedings of the *38th Annual Symposium on Foundations of Computer Science*, p.223-231 (1997). N.Destainville. Flip dynamics in octagonal rhombus tiling sets, *Physical Review Letters* [**88**]{} (2002). P.Diaconis and D.W.Stroock. Geometric Bounds for Eigenvalues of Markov Chains, *Annals of Applied Probability* [**1**]{} (1):36-61 (1991). Ph.Duchon. On the enumeration and generation of generalized Dyck words. *Discrete Mathematics* [**225**]{} (1-3)121–135 (2000). M.E.Fisher. Walks, Walls, Wetting, and Melting. *Journal of Statistical Physics* [**34**]{} (5):667-729 (1984). O.Häggström. *Finite Markov Chains and Algorithmic Applications*. London Mathematical Society (2002). A.Joulin and Y.Ollivier. Curvature, concentration, and error estimates for Markov chain Monte Carlo. To appear in *Annals of Probability*, `arXiv:0904.1312` (2009). D.A.Levin, Y.Peres, and E.L.Wilmer. *Markov Chains and Mixing Times*. American Mathematical Society (2009). M.Luby, D.Randall, A.Sinclair. Markov chain algorithms for planar lattice structures. *SIAM Journal on Computing*, [**31**]{} (1):167–192 (2001). Y.Ollivier. Ricci curvature of Markov chains on metric spaces. *Journal of Functional Analysis* [**256**]{} (3):810-864 (2009). J.G. Propp and D.B. Wilson. Exact sampling with coupled Markov chains and applications to statistical mechanics. *Random Structures and Algorithms*, [**9**]{} (1):223-252, (1996). D.B.Wilson. Mixing times of Lozenge tiling and card shuffling Markov chains. *Annals of Applied Probability* [**14**]{} (1):274–325 (2004). [^1]: The Ricci curvature is actually the largest positive number such that holds, for all the couplings of $P\delta_x,P\delta_y$ ; here we should rather say that Ricci curvature is larger than $\kappa$. [^2]: Notice that $a,b$ do not have the same meaning in Wilson’s paper: $a$ (resp. $b$) stands for the number of positive (resp. negative) steps.

ArXiv

--- author: - | Gonz[á]{}lez-Nuevo J., Cueli M. M., Bonavera L., Lapi A., Migliaccio M.,\ Arg[ü]{}eso F. , Toffolatti L. bibliography: - './XCORR\_ZPH.bib' date: 'Received xxx, xxxx; accepted xxx, xxxx' title: 'Cosmological constraints with the sub-millimetre galaxies Magnification Bias after large scale bias corrections.' --- [The study of the magnification bias produced on high redshift sub-millimetre galaxies by foreground galaxies through the analysis of the cross-correlation function was recently demonstrated as an interesting independent alternative to the weak lensing shear as a cosmological probe.]{} [In the case of the proposed observable, most of the cosmological constraints depends mainly on the largest angular separation measurements. Therefore, we aim at studying and correcting the main large scale biases that affect foreground and background galaxy samples in order to produce a robust estimation of the cross-correlation function. Then we analyse the corrected signal in order to derive updated cosmological constraints.]{} [The large scale bias corrected cross-correlation functions are measured using a background sample of H-ATLAS galaxies with photometric redshifts &gt; 1.2 and two different foreground samples (GAMA galaxies with spectroscopic redshifts or SDSS galaxies with photometric ones, both in the range 0.2 &lt; z &lt; 0.8). They are modelled using the traditional halo model description that depends on both halo occupation distribution and cosmological parameters. These parameters are then estimated by performing a Markov chain Monte Carlo under different scenarios to study the performance of this observable and the way to further improve its results.]{} [After the large scale bias corrections, we get only minor improvements with respect to the Bonavera et al. 2020 results, mainly confirming their conclusions: a lower bound on $\Omega_m > 0.22$ at $95\%$ C.L. and an upper bound $\sigma_8 < 0.97$ at $95\%$ C.L. (results from the $z_{spec}$ sample). Neither the much higher surface density of the foreground photometric sample nor the assumption of gaussian priors for the remaining unconstrained parameters improves significantly the derived constraints. However, by combining both foreground samples into a simplified tomographic analysis, we were able to obtain interesting constraints on the $\Omega_m$-$\sigma_8$ plane: $\Omega_m= 0.42_{- 0.14}^{+ 0.08}$ and $\sigma_8= 0.81_{- 0.09}^{+ 0.09}$ at 68% CL.]{} Introduction ============ The apparent excess number of high redshift sources observed near low redshift mass structures is known as Magnification Bias [see e.g. @SCH92]: the deflections produced by the intervening gravitational field (area stretching and amplification) affecting the light rays coming from distant sources increase, in general, their chances of being included in a flux-limited sample [see for example @ARE11]. An unambiguous manifestation of this bias is the existence of a non negligible cross-correlation function between two source samples with non-overlapping redshift distributions. It has been observed in several contexts: galaxy-quasar cross-correlation function [@SCR05; @MEN10], cross-correlation signal between Herschel sources and Lyman-break galaxies [@HIL13] or the CMB [@BIA15; @BIA16] among others. The cross-correlation signal can be enhanced by optimizing the choice of foreground and background samples. In this paper we use the sub-millimetre galaxies (SMGs) as the background sample because some of their features (steep luminosity function, very faint emission in the optical band and typical redshifts above $ z > 1-1.5 $) make them close to the optimal background sample for lensing studies as confirmed by a long series of publications [see for example @BLA96; @NEG07; @NEG10; @GON12; @BUS12; @BUS13; @FU12; @WAR13; @CAL14; @NAY16; @NEG17; @GON19; @BAK20 among the most important ones]. In early works, the magnification bias produced on SMGs was already observed [@WAN11] and measured with high significance, $> 10\sigma$ [@GON14]. In @GON17 the measurements were further improved, allowing a more detailed study with a Halo model. It was concluded that the lenses are massive galaxies or even galaxy groups/clusters, with minimum mass of $M_{lens}\sim10^{13}M_{\odot}$. Moreover, it was demonstrated that it is possible to split the foreground sample in different redshift bins and to perform a tomographic analysis thanks to the better statistics. Finally, @BON19 use the magnification bias to study the mass properties of a different type of lenses, a sample QSOs at $0.2<z<1.0$. It was possible to estimate the halo mass where the QSOs acting as lenses are located in the sky, $M_{min} = 10^{13.6_{-0.4}^{+0.9}} M_{\odot}$. These mass values indicate that we are observing the lensing effect of a cluster size halo signposted by the QSOs. The interest in magnification bias is driven by the fact that it can be used as an additional cosmological probe to address the estimation of the parameters in the standard cosmological model. In fact, the importance of the magnification bias effect depends on the gravitational deflection caused by low redshift galaxies on light travelling close to such lens, which in turn depends on cosmological distances and galaxy halo properties. Features like the anisotropies in the CMB [e.g., @HIN13; @PLA16_XIII; @PLA18_VI], the big bang nucleosynthesis [e.g. @FIE06] and the SNIa observations of the Universe accelerating expansion [e.g. @BET14] are well handled by the current ‘standard cosmological model’. It is also inclusive of some Large Scale Structure (LSS) significant predictions about galaxies distributions (e.g. [@PEA01]), such as baryon acoustic oscillations (BAOs) (e.g. [@ROS15]). Therefore, measurements based on such observables provides independent and complementary constraints on the cosmological parameters [e.g., @PEA94]. The success of the current model is in the fact that results from different probes are in great accordance. However, with the increase in the quality and quantity of the measurements, some ‘tensions’ and small-scale issues have arisen that might indicate the necessity of modifications of the $\Lambda$CDM model. The main tensions are the value of the Hubble’s constant, $H_0$ [$74.03 \pm 1.42$ km/s/Mpc by @RIE19; @PLA18_VI with $67.4 \pm 0.5$ km/s/Mpc], and the usually degenerate relationship between the $\Omega_m$ and $\sigma_8$ parameters [e.g., @HEY13; @PLA16_XXIV; @HIL17; @PLA18_VI]. In this context, @BON20 (hereafter BON20) test the capability of the Magnification Bias produced on high-z SMGs as an additional independent cosmological probe in the effort to resolve the tensions. With this proof of concept analysis $\Omega_m$ and $H_0$ were not well constrained. However, interesting limits were found: a lower limit of $\Omega_m>0.24$ at 95% CL and an upper limit of $\sigma_8<1.0$ at 95% CL (with a tentative peak around 0.75). Although the derived cosmological constraints from the Magnification Bias are relatively weak, it was confirmed as a new, independent observable making it a valuable new technique. Therefore it is worth making some efforts to improve further such results. In this respect, most of the cosmological analysis that can be performed using the measured cross-correlation function (cosmological parameters, mass function, neutrinos, ...) depends mainly on the observed data at the largest angular scales ($\gtrsim20$ arcmin). On the one hand, this data are the most uncertain ones with large error-bars. Large areas and high source densities are needed in order to derive precise measurements. On the other hand, large scale bias, that can be considered negligible at smaller scales, can affect the data, and, as a consequence, the derived cosmological results. For these reasons the main goal of this work is to deeply study and find the optimal strategy to measure and analyse a precise and unbiased cross-correlation function at cosmological scales. The work is organised as follows. In section \[sec:data\] the background and foreground samples are described and in section \[sec:methodology\] the methodology is presented. The large scale biases and how to correct them are described in \[sec:LS\_bias\]. The derived cosmological constraints and conclusions are discussed in sections \[sec:results\] and \[sec:conclusion\] respectively. In Appendix \[sec:corner\_plots\] we show the posteriors distributions of all the cases analysed and discussed in this work. Data {#sec:data} ==== The different galaxy samples used in this work are described in this section: the background sample, consisting of SMGs sources, and the foreground samples, consisting of two independent samples with spectroscopic and photometric redshifts, respectively. ![Normalised redshift distributions of the three catalogues used in this work: the background sample i.e. H-ATLAS high-z SMGs (red solid line), the GAMA spectroscopic foreground sample (blue solid line) and the SDSS photometric foreground sample (magenta dashed line).[]{data-label="Fig:dNdz_hist"}](PLOTS/zhist){width="\columnwidth"} Background sample ----------------- The Herschel Astrophysical Terahertz Large Area Survey [H-ATLAS; @EAL10] is the largest area extragalactic survey carried out by the Herschel space observatory [@PIL10] covering $\sim 610 deg^2$ with PACS \[43\] and SPIRE \[44\] instruments between 100 and 500 $\mu m$. Details of the H-ATLAS map-making, source extraction and catalogue generation can be found in @IBA10 [@PAS11; @RIG11; @VAL16; @BOU16], and @MAD20. The background sample consists of H-ATLAS sources detected in the three GAMA fields (total area of $\sim 147 deg^2$), the North Galactic Pole (NGP, $\sim 170 deg^2$) and the part of the South Galactic Pole (SGP) that overlaps with the spectroscopic foreground sample ($\sim 60 deg^2$). A photometric redshift selection of 1.2 &lt; z &lt; 4.0 has been applied to ensure no overlap in the redshift distributions of lenses and background sources, and we are thus left with $\sim66000$ ($\sim 24$ per cent of the initial sample and $z_{ph,med} = 2.20$). The redshifts estimation is described in detail in @GON17 [@BON19] and references therein. This is the same background sample used in @GON17, @BON19 and @BON20. Foreground samples ------------------ In this work we use two independent foreground samples. The first one is the same one used by @GON17, BON20 and we name it as the “$z_{spec}$ sample”. It consists of a sample extracted from the GAMA II [@DRI11; @BAL10; @BAL14; @LIS15] spectroscopic survey, with $\sim 150000$ galaxies for 0.2 &lt; $z_{spec}$ &lt; 0.8 ($z_{spec,med} = 0.28$). H-ATLAS and GAMA II surveys were carried out to maximize the common area coverage. Both surveys covered the three equatorial regions at 9, 12 and 14.5 h (referred to as G09, G12 and G15, respectively) and the SGP region was partially observed also by GAMA II. Thus, the resulting common area is of about $\sim 207 deg^2$, surveyed down to a limit of $r \simeq 19.8$ mag. This is the same foreground sample used in @GON17 and BON20. The second foreground sample is selected in the Sixteenth Data Release of the Sloan Digital Sky Survey [SDSS; @BLA17; @AHU19]. It consists of galaxies with photometric redshift between 0.2&lt; $z_{ph}$ &lt; 0.8 and photometric redshift error $z_{err}/(1+z)<1$ (*photoErrorClass*=1). SDSS have completly covered the H-ATLAS equatorial regions and the NGP one (a total area of $\sim 317 deg^2$). The sample, denominated “$z_{ph}$ sample”, comprises $\sim962000$ galaxies in total with median value of $z_{ph,med} = 0.38$. The reason to introduce this second foreground sample is to study the improvements in the final results by increasing the density of potential lenses. The higher uncertainty in the redshift estimation of the foreground photometric redshifts is not very important in the current analysis because we are using a single wide redshift bin. The normalised redshift distributions of the different samples are compared in Figure \[Fig:dNdz\_hist\]. As in @GON17, the random errors in the photometric redshifts are taken into account to estimate the redshift distributions. The main effect is to broadening the distributions beyond the selection limits. Figure \[Fig:dNdz\_hist\] clearly shows the gap in redshift between the background and the foreground sources. The same figure also highlights the different redshift distributions between the two foreground samples. Methodology {#sec:methodology} =========== Tiling area scheme ------------------ The H-ATLAS survey is divided in five different fields: three GAMA fields in the ecliptic (9h, 12h, 15h) and two in the North and South Galactic Poles (NGP, and SGP). The H-ATLAS scanning strategy produced the characteristic diamond repeated shape in most of their fields \[Fig:Tiles\]. Taking into account the available area in each field we have different possibilities to measure the cross-correlation function: - The “All” field area (blue line).- It provide the best statistics, i.e. smaller statistical uncertainties, both at small and large scales. The drawback is that we are limited to 4-5 fields to minimize the cosmic variance. - The “Tile” area (red line).- This is the straightforward shape to be selected taking into account the observational strategy. The area of each tile,  16 sq. deg, should be large enough to avoid a bias in the large scale measurements (normally limited to angular separation below 2 deg). In order to maintain a regular shape for the tiles, a small overlap among such regions is needed, typically lower than 20% of the tiles’ area. The advantage of this area scheme is in the fact that it provides around 24 different tiles, that should help to diminish the cosmic variance. - The “mini-Tile” area (magenta line).- It is built by dividing the tiles in four equal area “mini-Tiles” (each of  2x2 sq. deg). This area scheme typically provides around 96 different tiles. However, the maximum distance allowed by such area scheme is close to the cosmological scales that we want to measure. This was the area scheme used in BON20. Each tiling area scheme has its own strong and weak points and can be affected by different types of large scale biases. Therefore, we perform a detailed analysis in order to compare the measurements from the different tiling area schemes and derive a robust estimation of the cross-correlation function, in particular at the cosmological angular scales. ![Examples of the different area selection to measure the cross-correlation function for the G09 H-ATLAS field. The “All” field area is shown in blue ( 56 sq. deg). The “tile” selection is shown in red ( 4x4 sq. deg.) and the “mini-Tile” one in magenta ( 2x2 sq. deg.) []{data-label="Fig:Tiles"}](PLOTS/tile_schema.png){width="\columnwidth"} Angular cross-correlation function estimation {#sec:cross_corr} --------------------------------------------- As described in detail in [@GON17], BON20, we used a modified version of the [@LAN93] estimator [@HER01]: $$\label{eq:wx} w_x(\theta)=\frac{\rm{D}_1\rm{D}_2-\rm{D}_1\rm{R}_2-\rm{D}_2\rm{R}_1+\rm{R}_1\rm{R}_2}{\rm{R}_1\rm{R}_2}$$ where $\rm{D}_1\rm{D}_2$, $\rm{D}_1\rm{R}_2$, $\rm{D}_2\rm{R}_1$ and $\rm{R}_1\rm{R}_2$ are the normalized data1-data2, data1-random2, data2-random1 and random1-random2 pair counts for a given separation $\theta$. For each selected area, we compute the angular cross-correlation function and the statistical error (averaging between 10 different realizations using different random catalogues each time). Each final measurement corresponds to the mean value of the cross-correlation functions estimated in each individual selected area for a given angular separation bin. The uncertainties correspond to the standard error of the mean, i.e. $\sigma_\mu=\sigma/\sqrt{n}$ with $\sigma$ the standard deviation of the population and $n$ the number of independent areas (each selected region can be assumed as statistically independent due to the small overlap). {width="\columnwidth"} {width="\columnwidth"} Halo Model ---------- As described in detail in the previous related works [@GON17; @BON19], BON20 we adopt the halo model formalism proposed by @COO02 in order to interpret a foreground-background source cross-correlation signal. An halo is defined as spherical regions whose mean over-density with respect to the background at any redshift is given by its virial value, which is estimated following [@WEI03] assuming a flat $\Lambda$CDM model. We used the traditional [@NAV96] density profile with the concentration parameter given in [@BUL01]. The cross-correlation between the foreground and background sources is linked to the low redshift galaxy-mass correlation through the weak gravitational lensing effect. The foreground galaxy sample traces the mass density field that causes the weak lensing, affecting the number counts of the background galaxy sample through Magnification Bias. Following mainly [@COO02] [see @GON17 for details], we compute the correlation between the foreground and background sources adopting the standard Limber [@LIM53] and flat-sky approximations [see e.g. @KIL17 and references therein]. It can be estimated as: $$\begin{split} w_{fb}=2(\beta -1)\int^{z_s}_0 \frac{dz}{\chi^2(z)}\frac{dN_f}{dz}W^{lens}(z) \\ \int_{0}^{\infty}\frac{ldl}{2\pi}P_{gal-dm}(l/\chi^2(z),z)J_0(l\theta) \end{split}$$ where $$W^{lens}(z)=\frac{3}{2}\frac{H_0^2}{c^2}E^2(z)\int_z^{z_s} dz' \frac{\chi(z)\chi(z'-z)}{\chi(z')}\frac{dN_b}{dz'}$$ being $E(z)=\sqrt{\Omega_m(1+z)^3+\Omega_{\Lambda}}$, $dN_b/dz$ and $dN_f/dz$ as the *unit-normalised* background and foreground redshift distribution and $z_s$ the source redshift. $\chi(z)$ is the comoving distance to redshift z. The logarithmic slope of the background sources number counts is assumed $\beta=3$ ($N(S) = N_0 S^{-\beta}$) as in previous works [@LAP11; @LAP12; @CAI13; @BIA15; @BIA16; @GON17; @BON19]. Small variations of its value are almost completely compensated by small changes in the $M_{min}$ parameter. As the Halo Occupation Distribution (HOD) we adopted the three parameters @ZHE05 model: all halos above a minimum mass $M_{min}$ host a galaxy at their centre, while any remaining galaxy is classified as satellite. Satellites are distributed proportionally to the halo mass profile and halos host them when their mass exceeds the $M_1$ mass. Finally, the number of satellites is a power-law function of halo mass with $\alpha$ as the exponent, $N_{sat}(M)=(\frac{M}{M_1}) ^\alpha$. Therefore, $M_\text{min}$, $M_1$ and $\alpha$ are the astrophysical free-parameters of the model. Estimation of parameters {#sec:par_estimate} ------------------------ To estimate the different set of parameters, we performed a Markov chain Monte Carlo (MCMC) using the open source [*emcee*]{} software package [@EMCEE]. It is an MIT licensed pure-Python implementation of [@GOO10] Affine Invariant MCMC Ensemble sampler. For each run, we generated at least 90000 posterior samples to ensure a good statistical sampling after convergence. In the cross-correlation function analysis, we took into account both the astrophysical HOD parameters, and the cosmological parameters. The astrophysical parameters to be estimated are $M_{min}$, $M_1$ and $\alpha$. The cosmological parameters we want to constrain are $\Omega_m$, $\sigma_8$ and $h=H_0/100$. With the current samples, we do not have the statistical power to constrain $\Omega_B$, $\Omega_{\Lambda}$ and $n_s$ in our analysis. As we assume a flat universe, $\Omega_{\Lambda}$ is simply: $\Omega_{\Lambda}=1-\Omega_m$. For the the other two cosmological parameters, we keep them fixed to the *Planck* most recent results, $\Omega_B=0.0486$ and $n_s=0.9667$ (see @PLA18_VI). A traditional Gaussian likelihood function was used during this work. It should be noted that only the cross-correlation data in the weak lensing regime ($\theta \ge 0.2$ arcmin) are being taken into account for the fit since we are in the weak lensing approximation [see @BON19 for a detailed discussion]. In general, we used the same flat priors for all the different analyses. They are based on the ones used in BON20. As for the astrophysical parameters we chose: \[12.0-13.5\] for $\log M_{min}$, \[13.0-15.5\] for $\log M_1$ and \[0.5-1.5\] for $\alpha$. And for the cosmological parameters: \[0.1-0.8\] for $\Omega_m$, \[0.6-1.2\] for $\sigma_8$ and \[0.5-1.0\] for $h$. Large Scale Biases {#sec:LS_bias} ================== The cross-correlation function measurements using the different Tiling area schemes are compared in Figure \[Fig:xcorr\_data\]. The left panel shows the measurements before any correction is applied. While all the different measurements agree almost perfectly within the uncertainties at small scales, there is a widespread variation of estimated values for angular separations above $\sim 10$ arcmin. But the cosmological parameters affect mainly those angular scales (see BON20 appendix figures). Therefore, we need to understand the causes that produce such high variation on our observations at those large angular scales before attempting any robust cosmological analysis. It is well known that the distribution of galaxies in the Universe is not perfectly homogeneous. Therefore, in a field with a limited area, the number of detected galaxies will be somewhat higher or lower than the mean value obtained considering large enough areas. If this variation is not taken into account when building the random catalogues for a particular field, it will affect the DR and RR related terms in equation \[eq:wx\] and the estimated correlation could be stronger or weaker than the intrinsic value (see e.g. @ADE05 for a detailed discussion on this topic). To this respect, there are mainly two different biases that can affect the cross-correlation measurements at large scales: the integral constraint [IC; @ROC99] and the surface density variation [SD; @BLA02]. Integral constraint (IC) ------------------------ When many fields are averaged, the overall effect of the large scale fluctuations tends to make the observed correlation weaker mainly at the largest observed scales. This means that the estimated cross-correlation function is biased low by a constant, the IC: $w_{x\_ideal}(\theta) = w_{x}(\theta) + IC$. Although there are possible theoretical approaches to estimate the IC for a particular scanning strategy (see e.g. @ADE05), it is commonly estimated numerically using the RR counts: $$IC=\frac{\sum_i{\rm{R}_1\rm{R}_2(\theta_i)w_{x\_ideal}(\theta_i)}}{\sum_i{\rm{R}_1\rm{R}_2(\theta_i)}}.$$ As a first approximation of $w_{x\_ideal}(\theta_i)$, we assumed a power-law model, $w_{x\_ideal}(\theta_i)= A \theta^\gamma$. In order to be as much independent as possible of the exact value of the cosmological parameters (that mainly affect the largest angular scales), we estimated the best fit parameters for the power-law using only the observed cross-correlation function below 20 arcmin ($A=10 ^{-1.54}$ arcmin and $\gamma=-0.89$). With the estimated power-law, the derived IC value for the “mini-Tiles” area was $9\times10^{-4}$. We verified that choosing a smaller angular separation upper limit or using different data set did not affect the derived IC value. Moreover, assuming the best fit model of BON20 that can be considered biased low due to the fact that neglected the IC correction, the IC derived was again the same value. Therefore, we can conclude that the “mini-Tiles” estimated cross-correlation functions at the largest scales (&gt;20 arcmin) are biased low but can be safely corrected by adding an $IC = 9\times10^{-4}$. Anyway, as discussed in section \[sec:flat\_priors\], this correction does not introduce any substantial difference with respect to the BON20 results on cosmological parameters. On the other hand, the estimated IC for the the “Tiles” area is $IC = 5\times10^{-4}$, considering both the power-law fit and the BON20 best fit model. As expected, the correction is smaller than in the “mini-Tiles” case taking into account the larger area of the “Tiles”. The IC in the “Tiles” case affects marginally only the measurements above $\sim 40$ arcmin. Considering the large uncertainties at those angular scales, it can be almost considered a negligible correction for the $z_{spec}$ sample measured using the “Tiles” area. However, in the seek of precision, we decided to apply it in any case. On the other hand, the IC results are completely negligible in the case of using the “all” field area scheme, as expected. Surface density variations -------------------------- The results using the “Tiles” area differ for the $z_{spec}$ and the $z_{ph}$ samples. This difference remains after the IC correction because it is the same for both cases. Moreover, the discrepancy is even stronger in the “All” scenario case (since the “All” measurements are almost the same between both samples, we are focusing only in the $z_{ph}$ for simplicity). This is a clear indication that an additional large scale bias is affecting the measurements when larger areas are considered. The fact that the $z_{ph}$ sample is more affected is probably related to the much higher density of sources in this sample. If there is an additional variation of the source density of the foreground or the background sample that is not taken into account when building the random catalogues, it can produce a spurious enhancement of the measured correlation. As explained by @BLA02, the number of close pairs depended on the local surface density while the random pairs are related to the global average surface density. Then, systematic fluctuations produce $DD>RR$ that means a higher correlation (e.g. consider just the simplest estimator of the auto-correlation: $w(\theta)=DD/RR-1$). Therefore, if present, the surface density variation produces the opposite effect with respect to the IC, that is what we are observing with the $z_{ph}$ sample. ### Instrumental Noise variation For the background sample, there is a well known surface density variation related to the instrumental noise due to the scanning strategy (see Figure \[Fig:SD\_maps\], top panel). The overlap between the “Tiles” reduces the instrumental noise that allows fainter SMGs to be detected with respect the rest of the field. For the auto-correlation analysis it was demonstrated that the potential effect can be considered negligible [@AMV19]. Moreover, our results indicate that the relatively low surface density of the $z_{spec}$ sample makes this effect also negligible. In other words, the number of additional pairs due to the fainter background sources in those areas is not relevant enough to affect the measurements for the $z_{spec}$ sample. However, the much higher surface density of the $z_{ph}$ sample could produce a relevant enough enhancement of background-foreground pairs in those regions and, therefore, inducing a large scale surface density variation for the “Tiles” and “All” area schemes (we can consider the “mini-Tile” measurements simply dominated by the IC correction and neglect this other type of large scale bias even for the $z_{ph}$ sample). In order to correct the instrumental noise surface density bias, we adopted the same procedure to generate random catalogues used in @AMV19 for the auto-correlation analysis of the SMGs. First, a flux was chosen randomly from the flux densities of our background sample. Then the simulated galaxy is situated in a random position on the field. At this position the local noise was estimated as the instrumental noise and the confusion noise [see Table 3 of @VAL16 for the GAMA fields]. The estimated local noise is used to introduce a random Gaussian perturbation in the flux density. Finally, the simulated galaxy was kept in the sample if its flux density was greater than four times the local noise, the same detection limit used to produce the official H-ATLAS catalogue. This process was repeated for each random galaxy until the completion of the random catalogue. These newly generated random catalogues correspond only to the background sample, i.e. it was only applied to build the $R_1$ random catalogues (used to estimate the $D_2R_1$ and $R_1R_2$ terms). When the instrumental noise variation is considered, the cross-correlation functions showed a small correction toward lower values at the largest angular scales (not shown individually in Figure \[Fig:xcorr\_data\]). Although this result confirms that this bias is not negligible, it also highlights that it is not enough to explain the stronger correlation observed in the “Tiles” scheme for the $z_{ph}$ sample and the “All” one for both samples. Therefore, we studied additional sources of surface density variations in the foreground samples. ![Top panel: Example of the instrumental noise variation in the G09 field due to the scanning strategy. Bottom panel: Example of the surface density variation for the $z_{ph}$ sample in the G15 filed after been filtered using a gaussian kernel with a standard deviation of 180 arcmin.[]{data-label="Fig:SD_maps"}](PLOTS/ND_map "fig:"){width="\columnwidth"}\ ![Top panel: Example of the instrumental noise variation in the G09 field due to the scanning strategy. Bottom panel: Example of the surface density variation for the $z_{ph}$ sample in the G15 filed after been filtered using a gaussian kernel with a standard deviation of 180 arcmin.[]{data-label="Fig:SD_maps"}](PLOTS/SD_map "fig:"){width="\columnwidth"} ### Surface density variation of the foreground samples. There are different causes of surface density variations in large area galaxy surveys, such as scanning strategy, sensitivity variation with time and foreground contamination. Moreover, the sample selection can amplify or reduce these variations, for example a region where the conditions for spectroscopic observations are different from the mean field ones. The detailed correction of these possible variations is complicated and requires a deep knowledge of the particular details of the instrument and the pipeline used for the production of the catalogue. For the purpose of this work we adopted a simple approach to investigate the existence and correction of surface density variations in the foreground samples. As we can only observe a discrepancy at the largest angular scales, we decided to focus just on this range. First, we created a surface density map by adding $+1$ to the pixel value at the position of each galaxy on the sample. Then we smoothed the map using a Gaussian kernel with a certain standard deviation (see discussion later in this section). Next, we apply the H-ATLAS survey masks (so that we can neglect border effects due to the smoothing step). These surface density maps are then used to generate the Random catalogues, $R_2$, for the foreground samples (used to estimate $D_1R_2$ and $R_1R_2$ terms in equation \[eq:wx\]). The bottom panel in Figure \[Fig:SD\_maps\] shows an example of smoothed surface density map built using the $z_{ph}$ sample, with a standard deviation of 180 arcmin, for the G15 field. As expected, the overall density map at those angular scales is almost homogeneous. However, there are some variations that might be biasing our measurements: the source density in the second “Tile” from the left is higher than the fourth one. However, the exact value to be used as the Gaussian kernel dispersion is an unknown quantity. Using values smaller than 180 arcmin, the resulting density map starts to mimic the two-halo correlation of the foreground data. This means that the obtained $R_2$ contain part of the real auto-correlation and will remove part of this power from the estimated cross-correlation. For this reason and considering that the cross-correlation function decreases steeply for $\theta \sim 100$ arcmin, we can set a Gaussian dispersion of &gt; 150 arcmin as a lower limit. On the other hand, for dispersion values above 180 arcmin, the surface density variation along the area becomes almost negligible in the derived $R_2$. Therefore, we can considered a dispersion of &lt; 200 – 220 arcmin as an upper limit. Overall, we decided to proceed using a dispersion of 180 arcmin as a representative value, but taking into account that it is arbitrarily chosen. At the same time, given the uncertainties of the measurements at the relevant angular scales, small variations around the chosen deviation value became only a second order effect in our large scale measurements. When both surface density variations are taken into account to generate the random catalogues the large scale bias observed in the “Tiles” scheme for the $z_{ph}$ sample or the “All” field area one for both samples disappear. The right panel of Figure \[Fig:xcorr\_data\] shows the estimated cross-correlation functions using different tiling area schemes for the two samples after all the large scale bias corrections. The difference between the mean values at each angular scale is much smaller than the uncertainties. Considering this good agreement, we are confident that the measurements can be considered robust in all the angular scales commonly used for the cosmological analysis. As a final summary, to minimise the number of corrections applied to the data, we recommend to apply just the IC correction to the “mini-Tile” measurements for both samples and to the “Tile” one in the $z_{spec}$ case. In the other cases, the surface density correction is the most relevant one to be considered. Cosmological constraints {#sec:results} ======================== Once the cross-correlation measurements are corrected for the different large scale biases discussed in the previous section, we focus our analysis in their application to the estimate of some relevant parameters as done in BON20: the astrophysical parameters ($M_{min}$, $M_1$ and $\alpha$) and the cosmological ones ($\Omega_m$, $\sigma_8$ and $h$). The higher number of independent smaller sky areas allows to minimise the error contribution given by the cosmic variance resulting in smaller uncertainties. For this reason and considering the almost perfect agreement between the “All” tiling scheme and the “Tiles” ones for both samples, we decided to focus just on the second case in order to simplify the discussion. Therefore, we focus on just four cases (all of them corrected for the relevant large scale biases): “mini-Tiles” and “Tiles” tiling schemes for both samples ($z_{spec}$ and $z_{ph}$). . \[Tab:zspec\] [c c c c c c c c]{} Param & Priors & &\ & $\mathcal{U}$\[a,b\] & $\mu$ & $\sigma$ & peak & $\mu$ & $\sigma$ & peak\ & & $\pm 68 CL$ & $ $ & & $\pm 68 CL$ & &\ \ $\log(M_{min}/M_\odot)$ & \[12.0, 14.0\] & $12.57_{- 0.17}^{+ 0.23}$ & 0.20& 12.61 & $12.61_{- 0.15}^{+ 0.19}$ & 0.18 & 12.56\ \ $\log(M_1/M_\odot)$ & \[12.5, 15.5\] & $14.26_{- 0.38}^{+ 1.24}$ & 0.78& 15.03 & $14.37_{- 0.37}^{+ 1.13}$ & 0.74 & 14.71\ \ $\alpha$ & \[0.5, 1.5\] & – & – & – & – & – & –\ \ $\Omega_m$ & \[0.1, 0.8\] & $ 0.45_{- 0.21}^{+ 0.13}$ & 0.16& 0.38 & $ 0.42_{- 0.24}^{+ 0.14}$ & 0.18 & 0.31\ \ $\sigma_8$ & \[0.6, 1.2\] & $ 0.84_{- 0.18}^{+ 0.11}$ & 0.14& 0.83 & $ 0.82_{- 0.20}^{+ 0.08}$ & 0.14 & 0.75\ \ $h$ & \[0.5, 1.0\] & – & – & – & – & – & –\ \ [c c c c c c c c]{} Param & Priors & &\ & $\mathcal{U}$\[a,b\] & $\mu$ & $\sigma$ & peak & $\mu$ & $\sigma$ & peak\ & & $\pm 68 CL$ & $ $ & & $\pm 68 CL$ & &\ \ $\log(M_{min}/M_\odot)$ & \[12.0, 14.0\] & $12.60_{- 0.13}^{+ 0.20}$ & 0.18& 12.67 & $12.61_{- 0.13}^{+ 0.20}$ & 0.17 & 12.66\ \ $\log(M_1/M_\odot)$ & \[12.5, 15.5\] & $13.81_{- 1.09}^{+ 0.53}$ & 0.76& 13.60 & $13.95_{- 0.95}^{+ 0.74}$ & 0.76 & 13.74\ \ $\alpha$ & \[0.5, 1.5\] & $0.96_{-0.46}^{+0.15}$ & 0.27 & 0.77 & $0.96_{-0.46}^{+0.15}$ & 0.28 & 0.73\ \ $\Omega_m$ & \[0.1, 0.8\] & $ 0.46_{- 0.18}^{+ 0.11}$ & 0.14& 0.38 & $ 0.46_{- 0.19}^{+ 0.12}$ & 0.15 & 0.39\ \ $\sigma_8$ & \[0.6, 1.2\] & $ 0.99_{- 0.11}^{+ 0.12}$ & 0.11& 0.98 & $ 0.98_{- 0.10}^{+ 0.16}$ & 0.12 & 1.00\ \ $h$ & \[0.5, 1.0\] & $ 0.71_{- 0.21}^{+ 0.06}$ & 0.14 & 0.50 & – & – & –\ \ ![Comparison of the derived posterior distributions for the constrained parameters using the four data sets: $\log M_{min}$ (top left panel), $\log M_1$ (top right), $\Omega_m$ (bottom left) and $\sigma_8$ (bottom right). []{data-label="Fig:Comp_1D"}](PLOTS/comp_1D_new){width="\columnwidth"} Flat priors {#sec:flat_priors} ----------- The main results of this work are derived imposing flat priors as described in section \[sec:par\_estimate\]. The full set of posterior distributions can be found in the Appendix \[sec:corner\_plots\]. Figures \[Fig:zspec\_corner\] and \[Fig:zph\_corner\] compare the results derived from the different tiling schemes for the same sample. Moreover, the main statistical quantities that describe the posterior distributions are summarised in Table \[Tab:zspec\] and \[Tab:zph\] for $z_{spec}$ and $z_{ph}$ samples, respectively. The model prediction using the best fit values for both samples using the “mini-Tile” scheme are shown in the right panel of Figure \[Fig:xcorr\_data\] (black solid and green dashed lines, respectively). As in BON20 both $\alpha$ and $h$ are not well constrained. For this reason the comparison will focus on the rest of the parameters (see Figure \[Fig:Comp\_1D\]). All the cases provide similar constraints for $M_{min}$ and $\Omega_m$. On the first one, all of them agree to a mean value of $\log(M_{min}/M_\odot) \simeq 12.6 \pm 0.2$ at 68% CL. This value is very similar with the one found by BON20, $\log(M_{min}/M_\odot)= 12.53^{+0.29}_{-0.16}$. With respect the BON20 results, the introduction of the IC correction did not affect the estimated value of this well constrained parameter. In the case of $\Omega_m$, the new results moved the mean, $\sim0.45$, toward lower, more traditional values. This indicates that the large scale corrections helped to increase slightly the recovered values at the largest angular scales and to reduce their uncertainties. As a consequence, the highest $\Omega_m$ values become less probable based on our current measurements. However, similar lower limits as in BON20 are confirmed, e.g. &gt;0.22 for the $z_{spec}$ cases. On the other hand, the results for $\log M_1$ and $\sigma_8$ are different depending on the sample used. However, the results based on the same sample but using different Tile schemes are consistent between them. For $M_1$, using the $z_{spec}$ sample, we find a preference for $\log(M_1/M_\odot) \geq 13.8$ but only at 68% CL, whereas it shows a clear peak around $\log(M_1/M_\odot) \sim 13.6-13.7$, using the $z_{ph}$ one. In both cases these results are consistent with the BON20 ones. In a similar way, $\sigma_8$ mean estimated value moves from $\sim 0.8$, obtained with the $z_{spec}$ sample, to $\sim 1.0$ using the $z_{ph}$ one. Therefore, with the $z_{spec}$ sample, the same as in BON20, we obtain similar $\sigma_8$ constraints, but not confirmed by the $z_{ph}$ ones. Taking into account that the measurements of the cross-correlation function are almost the same between both samples (see again right panel of Figure \[Fig:xcorr\_data\]), this discrepancy in some of the recovered parameters can only be related to the fact that both samples have different redshift distributions. In fact, @GON17 perform a tomographic analysis of the cross-correlation function using four different redshift bins, bewteen $0.1<z<0.8$, and study the evolution of the same HOD parameters. While the $M_1$ parameter remains almost constant with redhsift, there is a clear evolution of an increasing $M_{min}$ values with redshift. The results of $\alpha$ are inconclusive as it is unconstrained in most of the redshift bins. By using a single wide redshift bin, we are deriving an average of the astrophysical parameters weigthed by the sample redshift distribution. Therefore, by analysing samples with different redshift distributions, it is expected to estimate different astrophysical parameter values, at least for those showing an evolution with redshift as $M_{min}$. Gaussian priors for the unconstrained parameters ------------------------------------------------ As discussed in the previous section, there are two parameters that remain unconstrained with the current data sets: $\alpha$ and $h$. In this section, we study the potential improvements on the results by assuming external constraints on these two parameters. This additional information is introduced in the MCMC as Gaussian priors. For all the analysis in this section we used only the $z_{spec}$ sample with the “mini-Tile” scheme. In the case of $\alpha$ we adopted a normal distribution with mean 1.0 and a dispersion of 0.1 (very similar to the Gaussian priors also used in BON20). The results are summarize in Table \[Tab:Galpha\] and the derived posterior distribution are shown in Figure \[Fig:zspec\_corner\_Galpha\]. In general, adopting a Gaussian prior for the $\alpha$ parameters produces almost no variation with respect to the default case. Only the most related parameters, $\log M_1$ and $\sigma_8$ move slightly toward lower values with a reduction on their dispersion of $\sim9$ and $\sim 21$ %, respectively. For the Hubble constant, we adopted the two popular values given by the local estimation, $74.03\pm 1.42$ km/s/Mpc [@RIE19], and the CMB one, $67.4\pm0.5$ km/s/Mpc [@PLA18_VIII]. The results obtained in these two cases are summarized in Table \[Tab:h\_high\], while the derived posterior distributions are compared in Figure \[Fig:zspec\_corner\_htest\]. The only relevant variation with respect to the default case is that the $\sigma_8$ distribution moves again slightly toward lower values with a reduction on their dispersion of $\sim 29$ %. When comparing between both $h$ priors cases, the results are almost identical. However, as also indicated in BON20, higher values of $h$ seem to perform slightly better: the $\Omega_m$ posterior distribution becomes thinner and moves towards lower, more traditional, values. However, the current uncertainties do not allow us to derive stronger conclusions on this particular topic. Overall, adopting more restrictive priors on the unconstrained parameters does not improve remarkably the results in general. The parameter that seems to benefit more from the reduction of uncertainty in both cases is $\sigma_8$. This is probably due to the fact that it is the parameter that mostly depends on the intermediate angular scales and, therefore, it is the one mostly affected by changes induced both by the smallest scales ($\alpha$’s main influence) and by the largest scales ($h$’s main influence), see appendix in BON20. . \[Tab:Galpha\] [c c c c c]{} Params & Priors & $\mu$ & $\sigma$ & peak\ & & $\pm 68 CL$ & $ $ &\ \ $\log(M_{min}/M_\odot)$ & $\mathcal{U}$\[12.0, 14.0\] & $12.53_{- 0.04}^{+ 0.16}$ & 0.21& 12.59\ \ $\log(M_1/M_\odot)$ & $\mathcal{U}$\[12.5, 15.5\] & $14.31_{- 0.38}^{+ 0.47}$ & 0.71& 14.32\ \ $\alpha$ & $\mathcal{N}$\[1.0, 0.1\] & $0.99_{-0.05}^{+ 0.06}$ & 0.10 & 1.00\ \ $\Omega_m$ & $\mathcal{U}$\[0.1, 0.8\] & $ 0.46_{- 0.16}^{+ 0.01}$ & 0.15& 0.37\ \ $\sigma_8$ & $\mathcal{U}$\[0.6, 1.2\] & $ 0.76_{- 0.16}^{+ 0.01}$ & 0.10& 0.64\ \ $h$ & $\mathcal{U}$\[0.5, 1.0\] & $ 0.75_{- 0.09}^{+ 0.09}$ & 0.14 & 0.66\ \ [c c c c c c c c c]{} Param & &\ & Priors & $\mu$ & $\sigma$ & peak & Priors & $\mu$ & $\sigma$ & peak\ & & $\pm 68 CL$ & $ $ & & & $\pm 68 CL$ & &\ \ $\log(M_{min}/M_\odot)$ & $\mathcal{U}$\[12.0, 14.0\] & $12.54_{- 0.05}^{+ 0.13}$ & 0.19& 12.58 & $\mathcal{U}$\[12.0, 14.0\] & $12.57_{- 0.06}^{+ 0.12}$ & 0.19& 12.61\ \ $\log(M_1/M_\odot)$ & $\mathcal{U}$\[12.5, 15.5\] & $14.29_{- 0.01}^{+ 1.02}$ & 0.76& 14.83 & $\mathcal{U}$\[12.5, 15.5\] & $14.29_{- 0.01}^{+ 1.12}$ & 0.77& 14.93\ \ $\alpha$ & $\mathcal{U}$\[0.5, 1.5\] & – & – & – & $\mathcal{U}$\[0.5, 1.5\] & – & – & –\ \ $\Omega_m$ & $\mathcal{U}$\[0.1, 0.8\] & $ 0.44_{- 0.15}^{+ 0.01}$ & 0.15& 0.35 & $\mathcal{U}$\[0.1, 0.8\] & $ 0.49_{- 0.15}^{+ 0.02}$ & 0.15& 0.41\ \ $\sigma_8$ & $\mathcal{U}$\[0.6, 1.2\] & $ 0.75_{- 0.11}^{+ 0.01}$ & 0.10& 0.69 & $\mathcal{U}$\[0.6, 1.2\] & $ 0.76_{- 0.15}^{+ 0.01}$ & 0.10& 0.68\ \ $h$ & $\mathcal{N}$\[0.74, 0.014\] & $ 0.74_{- 0.01}^{+ 0.01}$ & 0.014 & 0.74 & $\mathcal{N}$\[0.67, 0.005\] & $ 0.67_{- 0.003}^{+ 0.002}$ & 0.005 & 0.67\ \ Combining both data sets {#sec:tomo} ------------------------ {width="\textwidth"} [c c c c]{} Params & $\mu$ & $\sigma$ & peak\ & $\pm 68 CL$ & $ $ &\ \ $\log(M_{min}/M_\odot)$ $z_{spec}$ & $12.48_{- 0.16}^{+ 0.21}$ & 0.18& 12.51\ \ $\log(M_1/M_\odot)$ $z_{spec}$ & $14.37_{- 0.36}^{+ 1.13}$ & 0.74& 15.5\ \ $\alpha$ $z_{spec}$& – & – & –\ \ $\log(M_{min}/M_\odot)$ $z_{ph}$ & $12.60_{- 0.12}^{+ 0.21}$ & 0.19& 12.67\ \ $\log(M_1/M_\odot)$ $z_{ph}$ & $13.69_{- 1.03}^{+ 0.46}$ & 0.71& 13.49\ \ $\alpha$ $z_{ph}$ & $0.97_{- 0.44}^{+ 0.45}$ & 0.27 & 0.88\ \ $\Omega_m$ & $ 0.42_{- 0.14}^{+ 0.08}$ & 0.12& 0.37\ \ $\sigma_8$ & $ 0.81_{- 0.09}^{+ 0.09}$ & 0.09& 0.81\ \ $h$ & $ 0.72_{- 0.22}^{+ 0.09}$ & 0.14 & 0.5\ \ The $z_{ph}$ sample has much better statistics with respect to the $z_{spec}$ one, but we do not see a relevant improvement in the obtained constraints. In addition, even if the measured cross-correlation function is almost the same, each sample provides different results in some of the studied parameters. This is probably linked to the different redshift functions. On this respect, @GON17 tomographic analysis of the cross-correlation function show a strong evolution with redshift at least for the $\log M_{min}$ parameter. As explained before, by using a single wide redshift bin, the derived astrophysical parameters are the average of the evolving values measured by @GON17 weighted by the particular sample redshift distribution. As we saw, the different averaged astrophysical parameter values between the two samples is affecting also the recovered values of some of the cosmological parameters. In particular $\sigma_8$ changes from 0.84 for the $z_{spec}$ sample to 0.99 for the $z_{ph}$ one. A proper tomographic analysis is beyond the scope of this paper, but we can try a simple but interesting analysis: taking into account that the results from both samples are independent and have different redshift distributions, we can try to constraint the cosmological parameters using both samples at the same time. We perform a joint analysis allowing different astrophysical parameters constraints for each sample but keeping the same cosmological parameters. Therefore, we run an additional MCMC analysis but this time with nine parameters to be constrained (three astrophysical ones for each sample and three common cosmological ones). We used for both samples the “mini-Tile” scheme as it needs the simplest large scale bias correction. The results are summarize in Table \[Tab:tomo\] and the derived posterior distributions for the nine parameters are shown in Figure \[Fig:zspec\_corner\_tomo\]. Regarding the astrophysical parameters (see Figure \[Fig:Comb\_astro\]) the main changes of the combined analysis with respect to the individual ones are the following. Imposing a common cosmological parameters values seems not to affect the $\log M_{min}$ constraints for the $z_{ph}$ sample but it produces a shift toward slightly lower mean values for the $z_{spec}$ one (from 12.57 to 12.48). This is probably due to the more peaked redshift distribution of the $z_{ph}$ sample. In the case of the $M_{1}$ parameter, there is only a small reduction in the parameter uncertainty. Finally, there is no improvement in the $\alpha$ parameter, that remains unconstrained. As expected, the most relevant changes are within the cosmological parameters. While for the $\Omega_m$ the estimated posterior distribution is simply less skewed toward high values (from an associated gaussian standard deviation of 0.16 to 0.12), the $\sigma_8$ parameter is the most affected. Its posterior distribution become almost gaussian with a mean value of $\sigma_8=0.81_{- 0.09}^{+ 0.09}$ and a standard deviation of 0.09. However, for the Hubble constant the results give only an upper limit very similar to the one derived from the $z_{ph}$ sample alone ($h<0.8$ at 68% CL). A more detailed discussion on the $\Omega_m$ and $\sigma_8$ results will be presented in the next subsection. Comparison with other results ----------------------------- {width="\columnwidth"} {width="\columnwidth"} The weak gravitational lensing results available in the literature are usually related with a different and complementary observable, the shear. In this section we compare with measurements by cosmic shear of galaxies, focusing on the most constraining, and the CMB lensing by *Planck* [@PLA18_VIII]. In particular the results from the following surveys (with different redshift ranges and affected by different systematic effects) are being taken into account for the comparison: the Canada-France-Hawaii Telescope Lensing Survey presented in CFHTLenS [@JOU17], the Kilo Degree Survey and VIKING based on $450\,$deg$^2$ data [KV450, @HIL20], the first-year lensing data from the Dark Energy Survey (DES, [@TRO18]) and the Subaru Hyper Suprime-Cam first-year data [HSC, @HAM20]. For the comparison, we use publicly released MCMC results. Moreover, the different results we are comparing with have different priors. Since we are not interested in an in-depth comparison, we do not adjust them to our fiducial set-up. In particular, we compare the constraints in the $\Omega_m$ - $\sigma_8$ plane: cosmic shear measures the combination $\sigma_8 \Omega_m^{0.5}$ and CMB lensing the $\sigma_8 \Omega_m^{0.25}$ one. Such combinations highlight degeneracy directions, shown in the marginalised posterior contours ($68\%$ and $95\%\,$C.L.) in Figure \[Fig:External\_OmgM\_sgm8\] for the data-sets described above. To have a direct comparison with literature, the contours of these plots (0.68 and 0.95) are different from those used in the corner plots of this work (0.393 and 0.865, corresponding to the relevant 1-sigma and 2-sigma levels in the 1D histograms in the upper part of the same corner plots). We also show *Planck* CMB temperature and polarisation angular power spectra (dark blue) that, although in certain agreement with the HSC (cyan) and DES (green) constraints, presents the tension issues with the CFHTLenS (red) and KV450 (orange) data. The relevant cosmological constraints derived in this paper are shown in Figure \[Fig:External\_OmgM\_sgm8\] for both samples, $z_{spec}$ and $z_{ph}$, using the “mini-Tiles” scheme. The left panel shows the results from the analysis of each sample individually (grey filled contours for the $z_{ph}$ sample and black dashed curves for the $z_{spec}$ one) while the right panel shows the results from the combination of both samples as described in section \[sec:tomo\]. With respect to the previous BON20 constraints, by analysing each sample individually, the correction of the large scale bias has shifted the constraints on the $\Omega_m$ parameter toward lower values, more in agreement with the rest of the results from other studies. However, even when combining the two data-sets, the Hubble constant remains unconstrained. There is only a mild preference for the lowest values allowed by the flat prior, which is analogous to the one we found from the $z_{ph}$ sample alone. As displayed in Figure \[Fig:External\_OmgM\_sgm8\], it is very relevant to underline that when both samples are analyzed together, the constraints in the $\Omega_m$-$\sigma_8$ plane becomes much more restrictive: $\Omega_m= 0.42_{- 0.14}^{+ 0.08}$ and $\sigma_8= 0.81_{- 0.09}^{+ 0.09}$. It can also be noted, that their almost perpendicular direction with respect to the other lensing results can help to break the typical degeneracy. In any case, the constraints derived in this work confirm the main conclusions from BON20. Finally, we note that the data here discussed cannot be used to place useful constraints on the Hubble constant yet. Conclusions {#sec:conclusion} =========== As discussed in detail in BON20 (see their Figure A.1) the cosmological parameters depend mainly on the largest angular separation measurements. Therefore, the large scale biases can affect the cosmological constraint derived from the analysis of the magnification bias through the cross-correlation function. In this work, we study and correct the main large scale biases that affect our samples in order to product a robust estimation of the cross-correlation function. The result is a remarkable agreement among the different cross-correlation measurements, calculated independently of the used Tiling scheme or foreground samples. Then we analyse these results to estimate cosmological constraints after correcting the different large scale biases. We get minor improvements with respect to the BON20 results, mainly confirming their conclusions: a lower bound on $\Omega_m > 0.22$ at $95\%$ C.L. and an upper bound $\sigma_8 < 0.97$ at $95\%$ C.L. (results from the $z_{spec}$ sample using the “mini-Tile” scheme). Therefore, the large scale biases are a systematic that need to be corrected in order to derive robust and consistent results between different foreground samples or Tiling schemes, but does not help much to improve the precision of the derived constraints. In addition, we compare the estimates derived using two different and independent foreground samples: one consisting of foreground galaxies with spectroscopic redshifts, the $z_{spec}$ sample, and another one with only photometric redshifts, the $z_{ph}$ one. Analysing only one single broad redshift bin, we conclude that the higher errors of the photometric redhsifts do not have a relevant role in our outcomes. The $z_{ph}$ sample here considered has $\sim 6$ times more sources than the $z_{spec}$ one. Its better surface density makes it more sensitive to some large scale biases but helps to reduce the uncertainty in the measured cross-correlation function at intermediate and small angular scales. On the other hand, our current results show that the uncertainty is still dominated by the cosmic variance rather than by the surface density of the specific foreground sample at the largest angular scales. However, the constraints obtained making use of the $z_{ph}$ sample, which provides a more accurate cross-correlation measurements, are generally consistent with those derived using the $z_{spec}$ ones, with similar uncertainties. Moreover, adopting gaussian priors for the unconstrained parameters (i.e. $\alpha$ and the Hubble constant, similarly to BON20) does not improve much the results. Therefore, we are probably reaching the accuracy limit of the cosmological constraints that can be achieved with the analysis of a single redshift bin. Increasing the total area in order to decrease further the cosmic variance is probably an interesting improvement to be considered in the future. Although the measured cross-correlation function is almost the same between both foregrounds samples, we find different constraints for $\log M_1$ and $\sigma_8$ parameters. This is caused by the different redshift distributions between both samples. With a single wide redshift bin, the derived astrophysical parameters, that evolve with time as shown in the tomographic analysis of the cross-correlation function by @GON17, are averaged quantities weighted by the specific redshift distribution of the selected sample. Therefore, taking into account that the measurements of the cross-correlation function from both foreground samples are independent, we make use of the different redshift distributions to perform a simplified tomographic analysis combining both samples into a single MCMC run. We jointly performed the estimation of the cosmological parameters for both samples, but allowed different values of the astrophysical parameters for each sample. In this way, the effect of having different redshift distributions is included in the astrophysical parameters allowing us to determine with higher precision the cosmological parameters. In fact, the improvements on the $\Omega_m$-$\sigma_8$ plane are evident in the right panel of Figure \[Fig:External\_OmgM\_sgm8\]. The cosmological constraints obtained with this independent technique are starting to become competitive with respect to the other lensing results and its particular characteristics make it an interesting possibility in breaking the usual $\Omega_m$-$\sigma_8$ degeneracy. As a general conclusion, we showed that we are probably reaching the limits of the constraints than can be derived using just a single redshift bin, although there are still some ways to improve the results. However, the most promising advances with the study of the SMGs magnification bias are probably going to be obtained by performing a more complex tomographic analysis. JGN, MMC, LB, FA and LT acknowledge the PGC 2018 project PGC2018-101948-B-I00 (MICINN/FEDER). LB and JGN also acknowledge PAPI-19-EMERG-11 (Universidad de Oviedo). MM is supported by the program for young researchers “Rita Levi Montalcini" year 2015. A.L. acknowledges support from PRIN MIUR 2017 prot. 20173ML3WW002, ‘Opening the ALMA window on the cosmic evolution of gas, stars and supermassive black holes’, the MIUR grant ‘Finanziamento annuale individuale attivitá base di ricerca’, and the EU H2020-MSCA-ITN-2019 Project 860744 ‘BiD4BEST: Big Data applications for Black hole Evolution STudies’. We deeply acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support. In particular the projects “SIS19\_lapi”, “SIS20\_lapi” in the framework “Convenzione triennale SISSA-CINECA”.\ The Herschel-ATLAS is a project with Herschel, which is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. The H-ATLAS web- site is http://www.h-atlas.org. GAMA is a joint European- Australasian project based around a spectroscopic campaign using the Anglo- Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programs including GALEX MIS, VST KIDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA web- site is: http://www.gama-survey.org/.\ Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.\ In this work, we made extensive use of `GetDist` [@GETDIST], a Python package for analysing and plotting MC samples. In addition, this research has made use of the python packages `ipython` [@ipython], `matplotlib` [@matplotlib] and `Scipy` [@scipy] Posterior distributions of the MCMC results {#sec:corner_plots} =========================================== Posterior distributions for the different analyses discussed during the article. The contours for all these plots are set to 0.393 and 0.865. Notice that the relevant 1-sigma and 2-sigma levels for a 2D histogram of samples is 39.3% and 86.5% not 68% and 95%. Otherwise, there is not a direct comparison with the 1D histograms above the contours. {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"}

ArXiv

--- abstract: 'Recently N. Nitsure showed that for a coherent sheaf ${{\mathcal F}}$ on a noetherian scheme the automorphism functor ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable if and only if ${{\mathcal F}}$ is locally free. Here we remove the noetherian hypothesis and show that the same result holds for the endomorphism functor ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ even if one asks for representability by an algebraic space.' author: - Niko Naumann title: 'Representability of ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ and ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$' --- [MSC2000: 14A25]{} Statement of results {#s1} ==================== {#s11} Let $X$ be a scheme and ${{\mathcal F}}$ a quasi-coherent ${{\mathcal O}}_X$-module of finite presentation. We are interested in the representability of the following two functors on the category of $X$-schemes: $$\begin{aligned} {\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}} (X') & := & {\mathrm{Aut}\,}_{{{\mathcal O}}_{X'}} (f^* {{\mathcal F}}) \\ {\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}} (X') & := & {\mathrm{End}\,}_{{{\mathcal O}}_{X'}} (f^* {{\mathcal F}})\end{aligned}$$ for $f : X' \to X$ an $X$-scheme. The result is as follows: \[thm11\] Let $X$ be a scheme and ${{\mathcal F}}$ a quasi-coherent ${{\mathcal O}}_X$-module of finite presentation. Then the following are equivalent: 1. ${{\mathcal F}}$ is locally free. 2. ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by a scheme. 3. ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is representable by a scheme. If $X$ is locally noetherian, these conditions are also equivalent to the following: 1. ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by an algebraic space. 2. ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is representable by an algebraic space. {#s12} The equivalence of 1) and 2) in theorem \[thm11\] in case $X$ is noetherian is the main result of [@N]. Our proof follows the ideas of [*loc.cit.*]{} closely. The main steps are contained in the following two lemmas: \[t1\] Let $A$ be a local ring and $M$ a finitely presented $A$-module which is [*not*]{} free. Then there is a local homomorphism $A \to B$ such that $$M \otimes_A B \cong B^n \oplus (B/b)^m \; ,$$ for some $0 \neq b \in B , b^2 = 0$ and $m \ge 1,n \ge 0$.\ If $A$ is noetherian, $B$ can be chosen to be artin. We observe that in the last statement of the lemma the noetherian hypothesis is indispensable: let $(B , {{\mathfrak{m}}})$ be a local ring such that there is $0 \neq b \in \bigcap_{n \ge 1} {{\mathfrak{m}}}^n$. Clearly $(b^2) \subsetneq (b)$, so after dividing out $(b^2)$ one gets a ring $B$ as in the lemma but for any local homomorphism $f : B \to C$ with $C$ [*noetherian*]{} one clearly has $f (b) = 0$. \[t2\] Let $S$ be a scheme and $S_0 \subseteq S$ a closed subscheme defined by a nilpotent ideal sheaf. Assume $X$ is a flat $S$-scheme and $f : X \to Y$ is an $S$-morphism such that $f \times {\mathrm{id}}_{S_0}$ is an isomorphism. Then $f$ is an isomorphism. {#s13} In order to treat the representability of ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ we will use the following observation: \[t3\] Under the assumptions of 1.1 the obvious natural transformation of (set-valued) functors ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}} \to {\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is relatively representable by an open immersion. For completeness we also include a proof of the next lemma which is essentially lemma 5 of [@N] and shows the relative representability of a “parabolic” sub-group functor:\ Let $X$ be a scheme and $$\label{eq:1} 0 \longrightarrow {{\mathcal F}}' \longrightarrow {{\mathcal F}}\longrightarrow {{\mathcal F}}'' \longrightarrow 0$$ a short exact sequence of quasi-coherent ${{\mathcal O}}_X$-modules with ${{\mathcal F}}'$ finitely presented and ${{\mathcal F}}''$ locally free. For any morphism $f : Y \to X$, the sequence $f^* ((\ref{eq:1}))$ is exact because ${{\mathcal F}}''$ is in particular ${{\mathcal O}}_X$-flat and it makes sense to consider $$P (Y) := \{ \alpha \in {\mathrm{Aut}\,}_{{{\mathcal O}}_Y} (f^* {{\mathcal F}}) {\, | \,}\alpha (f^* {{\mathcal F}}') \subseteq f^* {{\mathcal F}}' \} \subseteq {\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}} (Y) \; .$$ \[t4\] In the above situation, the natural transformation $P \hookrightarrow {\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is relatively representable by a closed immersion. For basic facts about (relative) representability we refer to [@BLR], 7.6. Proofs ====== {#s21} In this subsection we dispense with the easy implications of theorem \[thm11\], the assumptions and notations of which we now assume:\ As ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ and ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ are clearly Zariski sheaves the problem of representing them is Zariski local on $X$, i.e. we can assume that $X$ is affine and ${{\mathcal F}}$ corresponds to a free module of finite rank. In this case, representability of both ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ and ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is obvious; we have proved the implications 1) $\Rightarrow$ 2) and 1) $\Rightarrow$ 2’). Finally, the implications 2) $\Rightarrow$ 3) and 2’) $\Rightarrow$ 3’) are trivial. {#s22} [**Proof of lemma \[t1\]:**]{} Let $(A , {{\mathfrak{m}}})$ be a local ring and $M$ a finitely presented $A$-module which is not free. We will find the required local homomorphism $A \to B$ as a suitable quotient of $A$:\ Let $$\label{eq:2} A^m \xrightarrow{\alpha} A^n \xrightarrow{\beta} M \rightarrow 0$$ be a minimal presentation of $M$, i.e. $n = \dim_k (M / {{\mathfrak{m}}}M)$ where $k := A / {{\mathfrak{m}}}$ is the residue field of $A$. Then $M$ is free if and only if $\alpha = 0$: clearly $\alpha = 0$ is sufficient for freeness of $M$ and conversely, if $M$ is free, it is necessarily so of rank $n$, hence $\beta$ is a surjective endomorphism of $A^n$ which must be an isomorphism by a standard application of Nakayama’s lemma, c.f. [@M], thm. 2.4., hence $\alpha = 0$.\ For any $J \subseteq {{\mathfrak{m}}}$, (\[eq:2\]) $\otimes_A A / J$ is a minimal presentation of the $A / J$-module $M / JM$. If we denote by $I \subseteq A$ the ideal generated by the coefficients of any matrix representation of $\alpha$ and note that the minimality of (\[eq:2\]) implies $I\subseteq {{\mathfrak{m}}}$ we find that $M / JM$ is $A / J$-free if and only if $\alpha\otimes id_{A/J}=0$ if and only if $I \subseteq J$. As $M$ is not $A$-free we have $I \neq 0$ and as $I$ is finitely generated we get ${{\mathfrak{m}}}I \subsetneq I$, again by Nakayama’s lemma. By Zorn’s lemma, using again that $I$ is finitely generated, there is an ideal $J$ with ${{\mathfrak{m}}}I \subseteq J \subsetneq I$ and which is maximal subject to these conditions (indeed, any ascending chain of such ideals admits its union as an upper bound because I is finitely generated). We claim that $B := A / J$ is as required:\ By the maximality of $J$ the ideal ${\overline{I}}:= I / J$ is non-zero principal: ${\overline{I}}= (b) , 0 \neq b \in B$ and we neccessarily have $b^2 = 0$: if not, we would have $b \in (b^2)$, i.e. $b = xb^2$ or $b (1-xb) = 0$ for some $x \in B$. As $b \in {\overline{{{\mathfrak{m}}}}}:= {{\mathfrak{m}}}/ J$, the maximal ideal of $B , 1-xb$ was a unit of $B$, so we would have $b = 0$. We now show that $M\otimes_{A} B$ has the desired structure: any coefficient $\alpha_{ij}$ of a matrix representation of $\alpha \otimes {\mathrm{id}}_B$ is of the form $\alpha_{ij} = bu_{ij} , u_{ij} \in B$. As by construction ${\overline{{{\mathfrak{m}}}}}b = 0$ we see that if $\alpha_{ij} \neq 0$, then $u_{ij} \in B^*$. We get a matrix equation $(\alpha_{ij}) = b (u_{ij})$ and $(u_{ij})$ can be chosen with $u_{ij} = 0$ or $u_{ij} \in B^*$, all $i,j$. Then the usual Gau[ß]{}-algorithm can be applied to $(u_{ij})$, showing that indeed $M \otimes_A B \cong B^n \oplus (B / b)^m$ for some $m,n\geq 0$. As, by construction, $M \otimes_A B$ is not $B$-free, we finally see that $m \ge 1$. If $A$ is noetherian we can start the construction of $B$ by first dividing out a suitable high power of ${{\mathfrak{m}}}$: Indeed, if $M / {{\mathfrak{m}}}^n M$ was free for all $n \ge 1$ we would have $I \subseteq \bigcap_{n \ge 1} {{\mathfrak{m}}}^n = (0)$. Then the ring $B$ we obtain in the above construction is noetherian local with ${\overline{{{\mathfrak{m}}}}}$ nilpotent, hence zero-dimensional, i.e. $B$ is artin local. [**Proof of lemma \[t2\]:**]{} We can assume that the ideal sheaf ${{\mathcal I}}$ of $S_0 \subseteq S$ satisfies ${{\mathcal I}}^2 = 0$. Our assertion is local on $S, X$ and $Y$ and thus reduces to the following:\ Given a ring $k$ and an ideal $I \subseteq k$ of square zero, if $f : A \to B$ is a morphism of $k$-algebras with $B$ $k$-flat and such that $f \otimes_k {\mathrm{id}}_{k / I}$ is an isomorphism, then $f$ is an isomorphism:\ 1) $f$ is surjective: any $b \in B$ can be written $$b = f (a) + \sum_j\alpha_jb_j'\; \mbox{ ,some} \; \alpha_j \in I , b'_j \in B , a \in A \; .$$ Applying this to the $b'_j$ we get (for some $\alpha_{ij} \in I , b''_{ij} \in B , a_j \in A)$: $$b = f (a) + \sum_j\alpha_j(f(a_j)+\sum_{ij}\alpha_{ij}b_{ij}'')=f(a+\sum_j\alpha_j a_j )\; .$$ 2) $f$ is injective: For $K := $ker$ (f)$ the $k$-flatness of $B$ implies $K / IK = 0$ and the same argument as in 1) shows that $K = 0$. [**Proof of 2) $\Rightarrow$ 1) in theorem \[thm11\]:**]{} Under the notations of \[s11\] we assume that ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by a scheme and, by contradiction, that ${{\mathcal F}}$ is not locally free. Note that the assumption on representability is stable under base-change $Y\rightarrow X$. So, base-changing to a suitable local ring of $X$, we can assume $X = {\mathrm{Spec}\,}(A)$ with $A$ local and ${{\mathcal F}}$ corresponding to a finitely presented $A$-module $M$ which is not free. According to lemma \[t1\] we can assume $M \cong A^n \oplus (A / a)^m$ for some $0 \neq a \in A$ with $a^2 = 0$ and $m \ge 1$. Let $G \to S:= {\mathrm{Spec}\,}(A)$ be the group-scheme representing ${\underline{{\mathrm{Aut}\,}}}_M$ and put $S_0 := {\mathrm{Spec}\,}(A / a)$. The sub-functor $G' \hookrightarrow G$ of automorphisms preserving (base-changes of) the direct summand $(A / a)^m$ is represented by a closed sub-group scheme (still to be denoted $G'$) according to lemma \[t4\]. Let $P \subseteq {\mathrm{Gl}\,}_{n+m , S}$ denote the standard parabolic sub-group of automorphisms preserving the rank $m$ direct summand. $P$ is flat over $S$, as can be seen over ${\mathrm{Spec}\,}({{\mathbb{Z}}})$. There is a morphism of $S$-groups $f : P \to G'$ which on points is given by sending $\left( \begin{smallmatrix} \alpha & 0 \\ \beta & \gamma \end{smallmatrix} \right)$ to $\left( \begin{smallmatrix} \alpha & 0 \\ \pi \beta & \overline{\gamma} \end{smallmatrix} \right)$, where $\alpha \in {\mathrm{Aut}\,}_A (A^n) , \gamma \in {\mathrm{Aut}\,}_A (A^m) , \beta : A^n \to A^m$ and $\overline{\gamma} \in {\mathrm{Aut}\,}_{A/a} ((A / a)^m)$ denotes the reduction of $\gamma$ and $\pi : A^m \to (A / a)^m$ is the natural map. This “point-wise” description of $f$ is immediately checked to be functorial and a homomorphism and hence does indeed define a morphism of $S$-groups. Obviously, $f \times {\mathrm{id}}_{S_0}$ is an isomorphism, hence so is $f$ by lemma \[t2\]. This is however a contradiction, because $f (S) : P (S) \to G' (S)$ is not injective, as $f (S) ({\mathrm{id}}_{A^n} \oplus (1-a) {\mathrm{id}}_{A^m}) = 1$ and $a \neq 0 , m \ge 1$. {#s23} [**Proof of lemma \[t3\]:**]{} Given a scheme $X$, a quasi-coherent ${{\mathcal O}}_X$-module ${{\mathcal F}}$ of finite presentation and some $\varphi \in {\mathrm{End}\,}_{{{\mathcal O}}_X} ({{\mathcal F}})$ we have to show that there is an open sub-scheme $X_0 \subseteq X$ such for all $f : Y \to X , f^* (\varphi) \in {\mathrm{Aut}\,}_{{{\mathcal O}}_Y} (f^* {{\mathcal F}}) \subseteq {\mathrm{End}\,}_{{{\mathcal O}}_Y} (f^* {{\mathcal F}})$ if and only if $f$ factors through $X_0$. Consider ${{\cal G}}:= {\mathrm{coker}\,}(\varphi)$ and the exact sequence of ${{\mathcal O}}_X$-modules $$\label{eq:3} {{\mathcal F}}\xrightarrow{\varphi} {{\mathcal F}}\xrightarrow{} {{\cal G}}\xrightarrow{} 0 \; .$$ We claim that $f^* (\varphi)$ is an automorphism if and only if $f^* ({{\cal G}}) = 0$: as $f^* ((\ref{eq:3}))$ is exact, necessity is obvious. If, conversely, $f^* ({{\cal G}}) = 0$ then for any $y \in Y$ $f^* (\varphi)_y$ is a surjective endomorphism of the finitely generated ${{\mathcal O}}_{Y,y}$-module ${{\mathcal F}}_y$, hence is an isomorphism, hence so is $f^* (\varphi)$. So the sought for $X_0 \subseteq X$ is the complement of the support of ${{\cal G}}$ which is open, because ${{\cal G}}$ is finitely presented. [**Proof of lemma \[t4\]:**]{} Given a scheme $X$ and a short exact sequence $0 \to {{\mathcal F}}' \to {{\mathcal F}}\to {{\mathcal F}}'' \to 0$ of quasi-coherent ${{\mathcal O}}_X$-modules with ${{\mathcal F}}'$ finitely presented and ${{\mathcal F}}''$ locally free and some $\alpha \in {\mathrm{Aut}\,}_{{{\mathcal O}}_X} ({{\mathcal F}})$, we have to show the representability by a closed sub-scheme of $X$ of the following functor on $X$-schemes: $$F (Y \xrightarrow{f} X) := \left\{ \begin{array}{ccl} * & , & f^* (\alpha) (f^* {{\mathcal F}}') \subseteq f^* {{\mathcal F}}' \\ \emptyset &, & \mbox{otherwise} \; . \end{array} \right.$$ Clearly, $F$ is a Zariski sheaf, so the problem is local on $X$, i.e. we can assume that $X = {\mathrm{Spec}\,}(A)$ is affine, ${{\mathcal F}}''$ corresponds to some $A^n$, ${{\mathcal F}}'$ corresponds to some $A$-module $M$ for which there is a presentation $A^a \to A^b \to M \to 0$ and ${{\mathcal F}}$ corresponds to some $A$-module $N$. The exact sequence $0 \to {{\mathcal F}}' \to {{\mathcal F}}\to {{\mathcal F}}'' \to 0$ then becomes an exact sequence $0 \to M \xrightarrow{\iota} N \xrightarrow{\pi} A^n \to 0$ of $A$-modules and we are given some $\alpha \in {\mathrm{Aut}\,}_A (N)$. Consider $\nu := \pi \alpha \iota$. As all the above sequences are exact after [*any*]{} base-change, we have $F (Y \xrightarrow{f} X) \neq \emptyset \iff f^* (\nu) = 0$. We have a diagram (defining $\psi$): $$\xymatrix{ A^a \ar[r] & A^b \ar[r] \ar[dr]^{\psi} & M \ar[r] \ar[d]^{\nu} & 0 \\ & & A^n & }$$ which is exact after any base-change, hence $f^* (\nu) = 0 \iff f^* (\psi) = 0$, for any $f : Y \to X$. So the closed sub-scheme of $X$ we are looking for is the one defined by the ideal of $A$ generated by the coefficients of any matrix representation of $\psi$. [**Proof of 2’) $\Rightarrow$ 1) in theorem \[thm11\]:**]{} Under the notations of \[s11\] we assume that ${\underline{{\mathrm{End}\,}}}_{{{\mathcal F}}}$ is representable by a scheme. Then so is ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ by lemma \[t3\], hence ${{\mathcal F}}$ is locally free by what has been shown in \[s22\]. {#s24} \ [**Proof of 3) $\Rightarrow$ 1) and 3’) $\Rightarrow$ 1) in theorem \[thm11\]:**]{} Under the notations of 1.1 we assume that $X$ is locally noetherian and that either 3) or 3’) holds as well as, by contradiction, that ${{\mathcal F}}$ is not locally free. By lemma \[t3\] we know in either case that ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by an algebraic space. Using the last assertion of lemma \[t1\] we can assume $X = {\mathrm{Spec}\,}(A)$ with $A$ artin local. Then ${\underline{{\mathrm{Aut}\,}}}_{{{\mathcal F}}}$ is representable by a scheme according to [@K], p. 25, 7) contradicting what we proved in \[s22\]. [**Acknowledgements.**]{} I would like to thank H. Frommer and M. Volkov for interesting discussions and G. Weckermann for excellent type-setting. [9999]{} S. Bosch, W. Lütkebohmert, M. Raynauld, Néron Models, Ergebnisse der Mathematik, 3. Folge, Band 21, Springer, Heidelberg 1990. D. Knutson, Algebraic Spaces, Springer LNM 203. H. Matsumura, Commutative ring theory, Cambridge studies in advanced mathematics 8, 1997. N. Nitsure, Representability of ${\mathrm{Gl}\,}_E$, arXiv:math.AG/0204047. [Mathematisches Institut der WWU Münster\ Einsteinstr. 62\ 48149 Münster\ Germany\ e-mail: [email protected]]{}

ArXiv

--- abstract: 'We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen’s theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. Along the way, we introduce an appropriate generalisation of LOCC operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von Neumann algebras. As an application of our result in the setting of ${\ensuremath{\mathrm{II}_1}}$-factors, we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of Vidal, thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we show that trace vectors play the role of maximally entangled states for general ${\ensuremath{\mathrm{II}_1}}$-factors. Examples are drawn from infinite spin chains, quasi-free representations of the CAR, and discretised versions of the CCR.' address: - 'School of Mathematics & Statistics, Carleton University, Ottawa, ON, Canada H1S 5B6' - 'Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1 & Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada N2L 3G1' - 'School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland' - 'Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom, and School of Mathematical Sciences, Nankai University, 300071 Tianjin, China' author: - Jason Crann - 'David W. Kribs' - 'Rupert H. Levene' - 'Ivan G. Todorov' date: 18 April 2019 title: State Convertibility in the von Neumann Algebra Framework --- Introduction {#s_bc} ============ Quantum entanglement is a central notion in quantum information theory and a key resource in the applications that are driving efforts to develop quantum technologies. While there is a growing depth of understanding of the concept and its many potential uses, the theory of quantum entanglement remains an active, challenging, and fundamental area of investigation in quantum information theory with, of particular note here, relatively little progress having been made in the general infinite-dimensional and von Neumann algebra settings. The mathematical theory that provides the foundation for these efforts rests in many ways on an understanding of how entanglement between quantum states can be transformed through various tasks and processes. A very natural and central question is to ask whether a given state, entangled between multiple parties, can be transformed by certain restricted classes of quantum operations to other types of entangled states, with restrictions determined by theoretical or physical limitations. This is a basic question that is relevant, for instance, in the development of any quantum communication scheme, realisations of quantum algorithms, physical implementations of quantum networks, etc. As the simplest starting point in this subject, consider the scenario in which two parties $A$ and $B$ each have the ability to implement all local quantum operations, described mathematically by completely positive and trace-preserving maps on the algebras of bounded linear operators on their respective system Hilbert spaces $H_A$, $H_B$, but such that the parties are limited in that they can only communicate with each other using classical communication. An initial core problem then is to start with an entangled state $\psi \in H_A \otimes H_B$ shared by the parties, and to determine what are the possible entangled states that $\psi$ can be converted to through local operations and classical communication (LOCC) between them. This question has a neat matrix theoretic solution in the finite-dimensional case, known as Nielsen’s Theorem [@nielsen]. For every pure state $\psi \in H_A \otimes H_B$, let $\rho_\psi = \operatorname{tr}_B (\psi\psi^*)$ be the (in general, mixed) state on $H_A$ found by applying the partial trace map over $H_B$ to the projection $\psi\psi^*$ with range the one-dimensional space of scalar multiples of $\psi$. The eigenvalues of $\rho_\psi$ (including multiplicities) form a probability distribution and can be arranged in non-increasing order, thus giving rise to a real vector $\lambda_\psi$. Nielsen’s Theorem states that $\psi$ can be converted into another state $\phi$ by LOCC between $A$ and $B$ if and only if $\lambda_\psi$ is majorised by $\lambda_\phi$, that is, all partial sums of values from $\lambda_\psi$ (respecting the non-increasing indexing) are bounded above by the corresponding partial sums of values from $\lambda_\phi$. Recall that $\rho_\psi$ is a pure (rank one) state if and only if ${\psi}$ is separable. Obviously if we start with a separable state ${\psi}$, then we can only transform it to another separable state via LOCC, and this case is easily captured by the theorem with $\lambda_\psi=(1,0,\ldots ,0)$. On the other hand, given any separable state ${\phi}$, any arbitrary state ${\psi}$ can be transformed to it via LOCC, in particular by making use of local depolarising maps on the individual systems. The power of Nielsen’s theorem lies in the fact that it gives a matrix and spectral theoretic description of which entangled states are attainable through LOCC when we start with a given entangled state. The theorem has far-reaching implications and applications throughout finite-dimensional quantum information theory; indeed, it is one of the most important and widely used results in the entire field. While the primary focus of research in quantum information has been on challenges and applications in the finite-dimensional and qubit setting for more than two decades now, ultimately general quantum mechanics is an infinite-dimensional theory that is rooted in the theory of von Neumann algebras [@vN]. Thus, one can reasonably expect that continued long-term progress in quantum information theory and its connections within theoretical physics will depend at least partly on the successful extension of central results in the field to the infinite-dimensional and general von Neumann algebra settings, with new peculiarities and connections uncovered along the way. This is clearly a desirable goal, and there has been a recent reemergence of activity in this direction, including quantum error correction and privacy (e.g. [@bkk2; @cklt]), entropy theory (e.g. [@bfs; @hiaif1; @hiaif2; @longo; @lx]), Bell inequalities (e.g. [@jungep]), the Connes embedding conjecture (e.g. [@musath; @slofstra; @dpp]), and entanglement in quantum field theory (e.g. [@hs] and the references therein). In this paper, we make new progress in this direction, establishing, as our main result, a generalisation of Nielsen’s Theorem to the context of von Neumann algebras, and more specifically for bipartite quantum systems modelled by commuting semi-finite von Neumann algebras, say ${\mathcal{A}}$ and ${\mathcal{B}}$ [@haagerup2; @t1; @t2]. We note that the case where ${\mathcal{A}}$ and ${\mathcal{B}}$ are separably acting factors of type I was considered in [@obnm]. While some parts of the theory extend in a somewhat straightforward way, there are, as one would expect, significant technical challenges to overcome, well beyond the generalisation provided in [@obnm]. En route, we introduce an appropriate generalisation of LOCC operations [@clmow] to our context. The setting for our version of Nielson’s Theorem is provided by the theory of singular numbers [@fk] and majorisation [@h; @h2] in von Neumann algebras. We build our analysis on key aspects of operator algebra theory, such as the standard form of a von Neumann algebra, the theory of completely bounded maps, the Haagerup tensor product, and dilation theory [@fk; @t2; @blecher_smith; @haagerup; @haagerup2; @h; @ksw]. We include a number of examples and applications of our results. In particular, we show that the entropy of the singular value distribution relative to the unique tracial state of a type ${\ensuremath{\mathrm{II}_1}}$-factor is an entanglement monotone in the sense of Vidal [@v; @op], thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we also show that trace vectors play the role of maximally entangled states for general ${\ensuremath{\mathrm{II}_1}}$-factors. Examples are drawn from quantum measurement theory [@vN], infinite spin chains [@keylsw; @kmsw], quasi-free representations of the CAR [@bcs; @dg; @dmm], and discretised versions of the CCR [@arv; @bkk2; @Faddeev]. This paper is organised as follows. The next section includes requisite preliminaries, focussed mainly on von Neumann algebra theory. In Section \[s\_locc\], we introduce the general notion of LOCC maps and derive some basic properties. In Section \[s:conv\], we investigate approximate convertibility of states by LOCC maps, showing that we can restrict attention to one-way convertibility. This is used in Section \[s:main\] to establish our main result on state convertibility and majorisation; this section also includes several supporting technical results. Section \[s\_ttf\] includes the aforementioned applications and examples. Other illustrative examples are presented throughout the paper. We finish with a brief outlook discussion on our results and the subject. Preliminaries ============= Let ${\mathcal{A}}$ be a von Neumann algebra. We denote by ${\mathcal{A}}_*$ its predual; thus, the elements of ${\mathcal{A}}_*$ are normal (that is, weak\* continuous) linear functionals on ${\mathcal{A}}$. If ${\mathcal{B}}$ is another von Neumann algebra with ${\mathcal{A}}\subseteq {\mathcal{B}}$, a map $\Phi : {\mathcal{B}}\to {\mathcal{B}}$ is called an ${\mathcal{A}}$-bimodule map if $\Phi(axb) = a\Phi(x)b$, for all $a,b\in {\mathcal{A}}$ and all $x\in {\mathcal{B}}$. We denote by $\operatorname{CP}_{{\mathcal{A}}}^\sigma({\mathcal{B}})$ the cone of all normal completely positive ${\mathcal{A}}$-bimodule maps on ${\mathcal{B}}$. We let $\operatorname{UCP}_{{\mathcal{A}}}^\sigma({\mathcal{B}})$ stand for the convex subset of all unital maps in $\operatorname{CP}^\sigma_{{\mathcal{A}}}({\mathcal{B}})$. We set $\operatorname{CP}^\sigma({\mathcal{B}}) = \operatorname{CP}_{{\mathbb{C}}I}^\sigma({\mathcal{B}})$ and $\operatorname{UCP}^\sigma({\mathcal{B}}) = \operatorname{UCP}_{{\mathbb{C}}I}^\sigma({\mathcal{B}})$. We call the elements of $\operatorname{UCP}^\sigma({\mathcal{B}})$ *quantum channels* on ${\mathcal{B}}$. We refer the reader to [@paulsen_book] for basics of the theory of completely positive and completely bounded maps and some standard notation. We denote by $S({\mathcal{A}})$ the convex set of all normal states of ${\mathcal{A}}$, and by $S_{{\rm ext}}({\mathcal{A}})$ the set of all pure normal states of ${\mathcal{A}}$. If $\omega\in {\mathcal{A}}_*$ and $a\in {\mathcal{M}}$, define $a\omega, \omega a\in {\mathcal{A}}_*$ by $(a\omega)(x) = \omega(xa)$ and $(\omega a)(x) = \omega(ax)$. We sometimes use the duality pairing notation ${{{ \left\langlex,\omega\right\rangle}}}:=\omega(x)$ with angled brackets (in contrast, we use rounded parentheses for inner products in Hilbert spaces); in this notation, the bimodule action just described may be written as ${{{ \left\langlex,a\omega b\right\rangle}}}={{{ \left\langlebxa,\omega\right\rangle}}}$, $a,b\in {\mathcal{A}}$. As usual, we let ${\mathcal{B}}(H)$ be the $C^*$-algebra of all bounded operators on a Hilbert space $H$, and set $M_n={{\mathcal{B}}}({{\mathbb{C}}}^n)$, the algebra of $n\times n$ matrices with complex entries. We assume throughout this paper that all Hilbert spaces under consideration are separable, sometimes mentioning this explicitly for emphasis. For an element $a\in {\mathcal{B}}(H)$, we write $\operatorname{Ad}(a)$ for the map given by $\operatorname{Ad}(a)(x) = axa^*$, where $a^*$ denotes the adjoint of $a$; clearly, $\operatorname{Ad}(a) \in \operatorname{CP}^\sigma({\mathcal{B}}(H))$. If $\nph,\psi\in H$, we let $\nph\psi^*$ be the rank one operator on $H$ given by $(\nph\psi^*)(\xi) = (\xi,\psi)\nph$ (note that our inner products are linear in the first argument). Let ${\mathcal{T}}(H)$ be the space of all trace class operators in ${\mathcal{B}}(H)$ and ${\rm tr} : {\mathcal{T}}(H)\to {\mathbb{C}}$ be the trace. We have a canonical identification ${\mathcal{T}}(H) \equiv {\mathcal{B}}(H)_*$, given by letting ${{{ \left\langleT,S\right\rangle}}} = {\rm tr}(TS)$, $T\in {\mathcal{B}}(H)$, $S\in {\mathcal{T}}(H)$. We denote by $H_1$ the set of unit vectors in a Hilbert space $H$. If $\phi\in H$, we write $\omega_\phi\in {\mathcal{B}(H)}_*$ for the (positive, normal) functional given by $\omega_\phi(x) = {{ \left(x\phi,\phi\right)}}$. Throughout the paper, we will let ${\mathcal{A}}\subseteq {\mathcal{B}(H)}$ be a von Neumann algebra, for some (separable) Hilbert space $H$, with unit $1$, projection lattice $\mathcal P({\mathcal{A}})$ and positive cone ${\mathcal{A}}^+$. We will mainly be interested in the case when ${\mathcal{A}}$ is semi-finite, equipped with a normal semi-finite faithful trace $\tau$. Let $\tilde{{\mathcal{A}}}$ be the \*-algebra of all $\tau$-measurable operators [@fk], that is, the set of all densely defined closed operators $T : {\mathcal{D}}(T)\to H$, (where ${\mathcal{D}}(T) \subseteq H$ is the domain of $T$), affiliated with ${\mathcal{A}}$, with the property that for every $\epsilon > 0$ there exists a projection $e\in {\mathcal{A}}$ such that $\tau(1-e) \leq \epsilon$ and $eH \subseteq {\mathcal{D}}(T)$. For $p\geq 1$, let $${\mathcal{A}}_p = \{a\in {\mathcal{A}} : \tau(|a|^p) < \infty\},$$ and let $L^p({\mathcal{A}},\tau)$ be the completion of ${\mathcal{A}}_p$ with respect to the norm $\|\cdot\|_p$, given by $\|a\|_p = \tau(|a|^p)^{1/p}$, $a\in {\mathcal{A}}_p$. Set $L^{\infty}({\mathcal{A}},\tau) = {\mathcal{A}}$. We will extensively use the fact that the elements of the space $L^p({\mathcal{A}},\tau)$ can be canonically identified with operators in $\tilde{{\mathcal{A}}}$ (see [@fk]). Note that $L^2({\mathcal{A}},\tau)$ is a Hilbert space, which is separable since ${\mathcal{A}}$ is separably acting, with inner product given by $${{ \left(a,b\right)}} = \tau(b^*a), \ \ \ a,b\in {\mathcal{A}}_2.$$ In fact, $L^2({\mathcal{A}},\tau)$ is the Hilbert space arising from the GNS construction applied to $\tau$. The associated (normal, faithful) \*-representation $\pi_\tau : {\mathcal{A}} \to {\mathcal{B}}(H)$ is given by $$\pi_{\tau}(a)b = ab, \ \ \ b\in {\mathcal{A}}_2, \ a\in {\mathcal{A}}.$$ We will suppress the use of the notation $\pi_{\tau}$; in this way, we will consider ${\mathcal{A}}$ as a von Neumann subalgebra of ${\mathcal{B}}(L^2({\mathcal{A}},\tau))$. We then have that ${\mathcal{A}}$ is in its standard form [@haagerup2]; we also say that ${{\mathcal{A}}}$ is standardly represented on $L^2({\mathcal{A}},\tau)$. Working in this standard representation, let ${\mathcal{A}}'$ be the commutant of ${\mathcal{A}}$, and let $J : L^2({\mathcal{A}},\tau) \to L^2({\mathcal{A}},\tau)$ be the associated conjugate linear isometry with the property that ${\mathcal{A}}' = J{\mathcal{A}}J$. Note that $J$ is the (unique) extension of the adjoint map $a\mapsto a^*$ on ${\mathcal{A}}_2$, and for $\psi\in L^2({\mathcal{A}},\tau)$, we have that $\psi^* = J\psi$, where the left hand side in the latter identity is the adjoint of the linear operator $\psi$ (by which we mean the element of $\tilde{{\mathcal{A}}}$ canonically identified with $\psi$). For $\xi,\eta\in L^2({\mathcal{A}},\tau)$, we have $$\label{eq_jaj} (J\xi,J\eta) = (\xi^*,\eta^*) = \tau(\eta\xi^*) = \tau(\xi^*\eta) = (\eta,\xi).$$ The map $\pi'_\tau : {\mathcal{A}}\rightarrow {\mathcal{B}}(L^2({\mathcal{A}},\tau))$, given by $$\pi'_\tau(a)b = ba, \ \ \ b\in {\mathcal{A}}_2, \ a\in {\mathcal{A}},$$ is a faithful anti-\*-homomorphism, satisfying $\pi'_\tau(a) = Ja^*J$, $a\in {\mathcal{A}}$ (see [@t1 Theorem V.2.22]). Let $R : {\mathcal{A}}'\rightarrow {\mathcal{A}}$ be the anti-\*-isomorphism given by $$R(a') = Ja'^*J, \ \ \ a'\in{\mathcal{A}}'.$$ We note that $L^1({\mathcal{A}},\tau)$ can be identified in a canonical way with the predual of ${\mathcal{A}}$. In fact, if $\omega\in {\mathcal{A}}_*$ then there exists a unique $\rho_{\omega}\in L^1({\mathcal{A}},\tau)$ such that $$\omega(a) = \tau(\rho_{\omega}a), \ \ \ a\in {\mathcal{A}}.$$ Here, and in the sequel, we use the fact that $L^1({\mathcal{A}},\tau)$ is an ${\mathcal{A}}$-bimodule, that is, given $\rho\in L^1({\mathcal{A}},\tau)$ and $a,b\in {\mathcal{A}}$, we have that $a\rho b$ is a well-defined $\tau$-measurable operator and belongs to $L^1({\mathcal{A}},\tau)$. Note that if $\omega\in {\mathcal{A}}_*^+$, then $\rho_{\omega}$ is a positive (in general unbounded) operator, which we call the *density operator* of $\omega$. If $x\in \tilde{{\mathcal{A}}}$ and $t > 0$, the *$t$-th singular value* $\mu_t(x)$ of $x$ is defined by letting $$\mu_t(x) = \mu_t(x;{{\mathcal{A}}},\tau)= \inf\{\|xp\| : p\in\mathcal P({\mathcal{A}}),\, \tau(1-p)\leq t\}.$$ The *singular value function of $x$*, namely $\mu(x)\colon (0,\infty)\to (0,\infty)$, $t\mapsto \mu_t(x)$, is decreasing and continuous from the right [@fk Lemma 2.5]. If $x,y\in \tilde{{\mathcal{A}}}^+$, we say that $x$ is *majorised* by $y$ if $$\int_0^s \mu_t(x)\, dt \leq \int_0^s \mu_t(y)\, dt, \ \ \ 0< s \leq \infty;$$ we write $x\prec y$ to designate the fact that $x$ is majorised by $y$ and $\tau(x) = \tau(y)$. We refer to [@h] for extensive details on majorisation of elements of $\tilde{{\mathcal{A}}}^+$ and to [@fk] for background on the theory of singular values. Let $\tau' : {\mathcal{A}}'\to {\mathbb{C}}$ be the functional given by $\tau'(a') = \tau(R(a'))$, $a'\in {\mathcal{A}}'$. Then $\tau'$ is a normal faithful semi-finite trace on ${\mathcal{A}}'$. Since ${\mathcal{A}}'\subseteq {\mathcal{B}}(L^2({\mathcal{A}},\tau))$, the elements of $L^1({\mathcal{A}}',\tau')$ can be identified with linear densely defined operators on $L^2({\mathcal{A}},\tau)$. Given a normal functional $\omega'$ on ${\mathcal{A}}'$, there exists, by the preceding discussion, a (unique) element $\rho'_{\omega'}\in L^1({\mathcal{A}}',\tau')$ such that $\omega'(b) = \tau'(\rho'_{\omega'}b)$, $b\in {\mathcal{A}}'$. The constructions described above can be performed relative to the pair $({\mathcal{A}}',\tau')$; in particular, one may define the corresponding singular values $\mu'_t(x'):=\mu_t(x';{{\mathcal{A}}}',\tau')$ associated with any $\tau'$-measurable operator $x'$, relative to $({\mathcal{A}}',\tau')$. We finish this section with two important examples of the previous notions. ${\mathcal{A}}=L^\infty(X,m)$, for a $\sigma$-finite measure space $(X,m)$. In this case, $\tau$ is integration by the measure $m$ and, for any $p\geq 1$, $L^p({\mathcal{A}},\tau)=L^p(X,m)$. In particular, the standard representation is given by the pointwise action of $L^\infty(X,m)$ on $L^2(X,m)$. For a non-negative element $f\in L^1(X,m)$, its singular value function $\mu_t(f)$ satisfies $$\mu_t(f)=\inf\left\{\strut s\geq 0\mid m(\{x\in X\mid f(x)>s\})\leq t\right\};$$ in other words, $t\mapsto \mu_t(f)$ is the non-increasing rearrangement of $f$. ${\mathcal{A}}={\mathcal{B}}(H)$ for a Hilbert space $H$. Here, the (essentially unique) normal semi-finite faithful trace $\tau$ is the canonical trace $\operatorname{tr}$. In this case, for $p\geq 1$, the space $L^p({\mathcal{A}},\tau)$ coincides with the Schatten $p$-class ${\mathcal{S}}_p(H)$. In particular, the standard representation of ${\mathcal{B}}(H)$ is given by the left multiplication action on the Hilbert–Schmidt operators ${\mathcal{S}}_2(H)$. Equivalently, fixing a unitary equivalence ${\mathcal{S}}_2(H)\cong H{\otimes}\overline{H}$ (where $\overline{H}$ is the conjugate Hilbert space of $H$), the standard representation is the canonical action of ${\mathcal{B}}(H){\otimes}1_{\overline{H}}$ on $H{\otimes}\overline{H}$. Given a positive element $\rho\in {\mathcal{T}}(H)={\mathcal{S}}_1(H)$, its singular value function $\mu_t(\rho)$ satisfies $$\mu_t(\rho)=\sum_{n=1}^\infty\lambda_n \chi_{[n-1,n)}(t),$$ where $\lambda_n$ is the $n^{th}$ largest eigenvalue of $\rho$ (including multiplicity). Local Operations and Classical Communication {#s_locc} ============================================ In this section, we introduce the class of maps that realise local operations and classical communication (LOCC) in our general setting and establish some of their properties needed in the sequel. The origins of our approach lie within the development of algebraic quantum field theory where the framework is typically encoded in two commuting \*-subalgebras of a larger $C^*$-algebra. In this paper, a bipartite quantum system is given by a Hilbert space $H$, together with a von Neumann algebra ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$, and its commutant ${\mathcal{B}}:={\mathcal{A}}'$. In the language of [@kmsw §4], this forms a simple, bipartite system satisfying Haag duality. The standard intuition comes from viewing ${\mathcal{A}}$ and ${\mathcal{B}}$ as the observable algebras of parties Alice and Bob, respectively, which have joint access to a quantum system modelled on the Hilbert space $H$. For example, when $H=H_A{\otimes}H_B$ for Hilbert spaces $H_A$ and $H_B$, then ${\mathcal{A}}={\mathcal{B}}(H_A){\otimes}1_{H_B}$ and ${\mathcal{B}} = 1_{H_A}{\otimes}{\mathcal{B}}(H_B)$ define the canonical bipartite system structure in the tensor product framework. We now examine a suitable generalisation of local operations and classical communication (LOCC) in this general bipartite setting, inspired by the approach in [@clmow] and related to, but slightly different from, the proposed notion in [@vw §5]. \[d\_locc\] Let ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$ be a von Neumann algebra on a Hilbert space $H$, and let ${{\mathcal{B}}}={{\mathcal{A}}}'$. (i) A *one-way right local map relative to ${\mathcal{A}}$* is a normal completely positive map $\Theta : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $\Theta = \Phi \circ\Psi$, where $\Phi\in \operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ and $\Psi\in \operatorname{CP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$. Similarly, a *one-way left local map relative to ${\mathcal{A}}$* is a normal completely positive map $\Theta : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $\Theta = \Phi \circ\Psi$, where $\Phi\in \operatorname{CP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ and $\Psi\in \operatorname{UCP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$. (ii) An *instrument* is a collection ${\mathcal{I}} = (\Theta_k)_{k\in {\mathbb{N}}}$ of normal completely positive maps on ${\mathcal{B}}(H)$ such that, for every $x\in {\mathcal{B}}(H)$, the series $\sum_{k=1}^{\infty} \Theta_k(x)$ converges in the weak\* topology to a limit, say $\Theta_{{\mathcal{I}}}(x)$, and the map $x\to \Theta_{{\mathcal{I}}}(x)$ is a normal unital completely positive map. In this case, we sometimes write $\sum_{k=1}^{\infty}\Theta_k$ to denote the map $\Theta_{{\mathcal{I}}}$. We will identify two instruments if they differ only by a bijective relabelling of the index set. (iii) A *one-way right instrument relative to ${\mathcal{A}}$* is an instrument ${\mathcal{I}} = (\Theta_k)_{k\in {\mathbb{N}}}$, where each of the maps $\Theta_k$ is a one-way right local map relative to ${\mathcal{A}}$. A *one-way right LOCC map relative to ${\mathcal{A}}$* is a normal completely positive map $\Theta : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $\Theta = \Theta_{{\mathcal{I}}}$, where ${\mathcal{I}}$ is a one-way right instrument relative to ${\mathcal{A}}$. Similarly, a *one-way left instrument relative to ${\mathcal{A}}$* is an instrument ${\mathcal{I}} = (\Theta_k)_{k\in {\mathbb{N}}}$, where each of the maps $\Theta_k$ is a one-way left local map relative to ${\mathcal{A}}$, and a *one-way left LOCC map relative to ${\mathcal{A}}$* is a normal completely positive map $\Theta : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $\Theta = \Theta_{{\mathcal{I}}}$, where ${\mathcal{I}}$ is a one-way left instrument relative to ${\mathcal{A}}$. (iv) An instrument $(\Gamma_j)_{j\in {\mathbb{N}}}$ is a *coarse-graining* of an instrument $(\Theta_k)_{k\in {\mathbb{N}}}$ if there is a partition ${{\mathbb{N}}}=\bigcup_{j\in {{\mathbb{N}}}}S_j$ so that $\Gamma_j=\sum_{k\in S_j}{\Theta_k}$ for $j\in {\mathbb{N}}$, where each series converges point-weak\*. (v) An instrument ${\mathcal{I}}$ is called *one-way local relative to ${\mathcal{A}}$* if ${\mathcal{I}}$ is either a one-way right instrument relative to ${\mathcal{A}}$ or a one-way left instrument relative to ${\mathcal{A}}$. We say that an instrument ${\mathcal{J}}$ is *linked* to an instrument ${\mathcal{I}} = (\Theta_k)_{k\in {\mathbb{N}}}$ if there exist one-way instruments $(\Theta_{ki})_{i\in {\mathbb{N}}}$, $k\in {\mathbb{N}}$, such that ${\mathcal{J}}$ is a coarse-graining of the instrument $(\Theta_k\circ \Theta_{ki})_{i,k\in {\mathbb{N}}}$. (vi) A map $\Theta\in \operatorname{UCP}^\sigma({\mathcal{B}}(H))$ is an *LOCC map relative to ${\mathcal{A}}$* if there exists a sequence $({\mathcal{I}}_0, \dots,{\mathcal{I}}_n)$ of instruments such that ${\mathcal{I}}_0$ is a one-way local instrument relative to ${\mathcal{A}}$, ${\mathcal{I}}_{l+1}$ is linked to ${\mathcal{I}}_l$, $l = 0,\dots,n-1$, and $\Theta = \Theta_{{\mathcal{I}}_n}$. We denote by $\operatorname{LOCC}({\mathcal{A}})$ the set of all LOCC maps relative to ${\mathcal{A}}$. We also write $\operatorname{LOCC}^r({{\mathcal{A}}})$ and $\operatorname{LOCC}^l({{\mathcal{A}}})$ for the subsets of $\operatorname{LOCC}({{\mathcal{A}}})$ consisting of the one-way right and left LOCC maps relative to ${{\mathcal{A}}}$, respectively. Thus, any $\Theta$ in $\operatorname{LOCC}^r({{\mathcal{A}}})$ is given by a point weak-\* convergent series $$\label{eq:locc-r} \Theta=\sum_{k\in {\mathbb{N}}} \Phi_k\circ \Psi_k$$ where $\Phi_k\in \operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ and $\Psi_k\in \operatorname{CP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$. \[ex\_pol\] Let $H$ be a Hilbert space, ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$ be a von Neumann algebra, and $\{a_k\mid k\in{\mathbb{N}}\} \subseteq {\mathcal{A}}$ be a countably infinite measurement system, that is, a sequence in ${\mathcal{A}}$ for which $\sum_{k=1}^\infty a_k^*a_k = 1$ in the weak\* topology. For each $k\in{\mathbb{N}}$, let $\Psi_k=\operatorname{Ad}(a_k^*)$ on ${\mathcal{B}}(H)$ and let $\Phi_k\in \operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ be a channel. Then the series $\Theta = \sum_{k=1}^\infty\Phi_k\circ\Psi_k$ defines a one-way right LOCC map relative to ${\mathcal{A}}$. Indeed, $$\sum_{k=1}^N \Phi_k\circ\Psi_k(1) = \sum_{k=1}^N \Psi_k\circ\Phi_k(1) = \sum_{k=1}^N a_k^* a_k \to_{N\to\infty} 1$$ in the weak\* topology. If $x\in {\mathcal{B}}(H)^+$ then $x\leq \|x\|1$ and hence the partial sums $\sum_{k=1}^N \Phi_k\circ\Psi_k(x)$ are dominated by $\|x\|1$. Since they form an increasing sequence, the series $\sum_{k=1}^{\infty} \Phi_k\circ\Psi_k(x)$ converges in the weak\* topology. If $x\in {\mathcal{B}}(H)$ is arbitrary then, using polarisation, we can write $x = \sum_{l=1}^4 \lambda_l x_l$, where $x_l\in {\mathcal{B}}(H)^+$ and $\lambda_l\in {\mathbb{C}}$, $l = 1,2,3,4$. It follows that the partial sums $$\sum_{k=1}^N \Phi_k\circ\Psi_k(x) = \sum_{k=1}^N \sum_{l=1}^4 \lambda_l \Phi_k\circ\Psi_k(x_l) = \sum_{l=1}^4 \sum_{k=1}^N \lambda_l \Phi_k\circ\Psi_k(x_l)$$ form a weak\* convergent sequence. The predual $\Theta_* : {\mathcal{T}}(H)\to {\mathcal{T}}(H)$ of the map $\Theta$ is given by $$\Theta_*(\rho) = \sum_{k=1}^\infty(\Phi_k)_*(a_k\rho a_k^*), \ \ \ \rho\in {\mathcal{T}}(H),$$ so one can think of $\Theta_*$ as a protocol where Alice makes a measurement corresponding to the system $\{a_k\mid k\in{\mathbb{N}}\}$, sends the result $k$ to Bob, who then applies $(\Phi_k)_*$. Below we show that any one-way right LOCC map relative to ${\mathcal{A}}$ is of this form, similar to the finite-dimensional setting. The use of measurements with countably many outcomes is natural in an infinite-dimensional context. Indeed, even the measurement of an observable modelled by a (possibly unbounded) self-adjoint operator with continuous spectrum can, within an arbitrarily small amount of error, be modelled by an observable with countably many disjoint outcomes [@vN §III.3]. \[r\_locc\] Let $\Theta\in \LOCC^r({{\mathcal{A}}})$ and let $\Phi_k$, $\Psi_k$ be as in . Then $\Phi_k(a) = a \Phi_k(1) = a$, $a\in {\mathcal{A}}$, $k\in{\mathbb{N}}$, and, in the weak\* topology we have $$\label{eq_psi1} 1 = \Theta(1) = \sum_{k=1}^\infty \Psi_k\circ \Phi_k(1) = \sum_{k=1}^{\infty} \Psi_k(1).$$ It follows that, for every positive element $x\in {\mathcal{B}}(H)$, the partial sums $\sum_{i=1}^m \Psi_k(x)$ are norm bounded; since they form an increasing sequence, they converge in the weak\* topology. Using polarisation, we conclude, as in Example \[ex\_pol\], that the sequence $\left(\sum_{i=1}^m \Psi_k(x)\right)_{m\in {\mathbb{N}}}$ converges in the weak\* topology for every $x\in {\mathcal{B}}(H)$. The locality of an LOCC operation is reflected through the bimodule structure of its implementing maps. For example, if $\Theta$ is a map of the form in , then Alice’s local operations are modelled by the maps $\Psi_k$, and Bob’s local operations by the channels $\Phi_k$. Since the $\Phi_k$ are ${\mathcal{A}}$-bimodule maps, they do not affect any of Alice’s observables and, as shown below, they admit Kraus decompositions with operators belonging to Bob’s observable algebra ${\mathcal{B}}$. A similar intuition is applied for Alice’s local operations $\Psi_k$. In the case where $H_A$ and $H_B$ are finite dimensional Hilbert spaces, $H = H_A\otimes H_B$, ${\mathcal{A}} = {\mathcal{B}}(H_A)\otimes 1$ and ${\mathcal{B}} = 1\otimes {\mathcal{B}}(H_B)$, Definition \[d\_locc\] reduces to the usual notions as described, for example, in [@clmow]. A notion of LOCC operation for general bipartite systems was proposed by Verch–Werner in [@vw §5]. There, a bipartite system is modelled by commuting unital $C^*$-subalgebras ${\mathcal{A}}$ and ${\mathcal{B}}$ of an ambient unital $C^*$-algebra ${\mathcal{C}}$. In this setting, they defined a one-way right LOCC map between bipartite systems $({\mathcal{A}}_1,{\mathcal{B}}_1, {\mathcal{C}}_1)$ and $({\mathcal{A}}_2,{\mathcal{B}}_2, {\mathcal{C}}_2)$, where ${\mathcal{A}}_i$ and ${\mathcal{B}}_i$ are commuting C\*-subalgebras of ${\mathcal{C}}_i$, $i = 1,2$, by a UCP map $\Theta:{\mathcal{C}}_1\rightarrow{\mathcal{C}}_2$, for which there exist finitely many completely positive maps $\Psi_k:{\mathcal{A}}_1\rightarrow{\mathcal{A}}_2$ and UCP maps $\Phi_k:{\mathcal{B}}_1\rightarrow{\mathcal{B}}_2$ satisfying $$\label{e:vw}\Theta(ab)=\sum_{k}\Psi_k(a)\Phi_k(b), \ \ \ a\in{\mathcal{A}}_1, \ b\in{\mathcal{B}}_1.$$ In the special case when ${\mathcal{C}}_1={\mathcal{C}}_2={\mathcal{B}}(H)$, ${\mathcal{A}}_1={\mathcal{A}}_2 =: {\mathcal{A}}$ is a von Neumann algebra, and ${\mathcal{B}}_1={\mathcal{B}}_2={\mathcal{A}}'$, our definition of a one-way right LOCC map relative to ${\mathcal{A}}$ satisfies this condition, albeit, allowing a countable summation over a classical” index $k$. This follows from the fact that any normal completely positive ${\mathcal{A}}'$-bimodule map on ${\mathcal{B}}(H)$ admits a Kraus decomposition with operators from ${\mathcal{A}}$ (see e.g. [@blecher_smith; @haagerup]), and similarly for ${\mathcal{B}}$. If, in addition, one assumes that ${\mathcal{A}}$ and ${\mathcal{B}}$ are injective factors, then by [@cs Theorem 4.2], any completely positive map $\Psi:{\mathcal{A}}\rightarrow {\mathcal{A}}$ admits a net of completely positive elementary operators $\Psi_i:{\mathcal{A}}\rightarrow{\mathcal{A}}$ (i.e., operators admitting finitely many Kraus operators from ${\mathcal{A}}$) satisfying $\norm{\Psi_i}_{cb} \leq \norm{\Psi}_{cb}$ and $\Psi_i\rightarrow\Psi$ in the point weak\* topology of $\operatorname{CB}({\mathcal{A}})$. The maps $\Psi_i$ admit canonical extensions to maps in $\operatorname{CP}^\sigma_{{\mathcal{A}}'}({\mathcal{B}}(H))$ (through their finite Kraus decompositions), so that we may approximate the (potentially non-normal) completely positive maps $\Psi_k$ occurring in by normal maps satisfying our bimodule requirements. Similar considerations hold for the maps $\Phi_k$. Hence, in the case of injective factors, one may view the proposed definition of Verch–Werner as a limit case” of ours. Note that by [@cs Remark 4.3], when ${\mathcal{A}}$ is an injective factor of type II or III, it is *not* true that every *normal* completely positive map $\Psi : {\mathcal{A}}\rightarrow{\mathcal{A}}$ extends to a *normal* completely positive map $\widetilde{\Psi}\in\operatorname{CP}^\sigma_{{\mathcal{A}}'}({\mathcal{B}}(H))$. \[p\_comp\] Let ${{\mathcal{A}}}\subseteq {\mathcal{B}(H)}$ be a von Neumann algebra on a separable Hilbert space $H$. (i) The class $\LOCC^r({{\mathcal{A}}})$ of one-way right LOCC maps is closed under finite compositions. (ii) The maps $\Psi_k$ in  can be taken of the form $\Psi_k = \operatorname{Ad}(a_k^*)$, for some $a_k\in {\mathcal{A}}$, $k\in {\mathbb{N}}$, with $\sum_{k=1}^{\infty} a_k^*a_k = 1$ in the weak\* topology. (iii) The maps $\Phi_k$ in  can be taken of the form $\Phi_k=\sum_{i=1}^\infty \operatorname{Ad}(c_{ki}^*)$, a point weak\*-convergent series, for some $c_{ki}\in {{\mathcal{A}}}'$, $k,i\in {{\mathbb{N}}}$, with $\sum_{i=1}^\infty c_{ki}^*c_{ki}=1$ in the weak\* topology for every $k\in {{\mathbb{N}}}$. Let, as before, ${\mathcal{B}} = {\mathcal{A}}'$. We first claim that if $\Phi \in \operatorname{CP}_{{\mathcal{A}}}^\sigma({\mathcal{B}}(H))$ and $\Psi \in \operatorname{CP}_{{\mathcal{B}}}^\sigma({\mathcal{B}}(H))$, then $$\label{eq_comm} \Phi \circ\Psi = \Psi \circ\Phi.$$ Indeed, since $H$ is separable, by [@haagerup] (see also [@blecher_smith]), there exists a bounded column operator $(a_{i})_{i\in {\mathbb{N}}}$ with entries ${\mathcal{A}}$, such that $$\Psi(x) = \sum_{i=1}^{\infty} a_{i}^* x a_{i}, \ \ \ x\in {\mathcal{B}}(H),$$ where the series converges in the weak\* topology. For every $x\in {\mathcal{B}}(H)$ we now have $$\Phi\circ\Psi(x) = \Phi \left(\sum_{i=1}^{\infty} a_{i}^* x a_{i}\right) = \sum_{i=1}^{\infty} \Phi(a_{i}^* x a_{i}) = \sum_{i=1}^{\infty} a_{i}^* \Phi(x) a_{i}=\Psi\circ \Phi(x),$$ showing . Let $\Theta \in \LOCC^r({{\mathcal{A}}})$, with corresponding maps $\Phi_k\in \UCP^\sigma_{{{\mathcal{A}}}}({\mathcal{B}(H)})$ and $\Psi_k\in \CP^\sigma_{{{\mathcal{B}}}}({\mathcal{B}(H)})$ for $k\in {{\mathbb{N}}}$, as in . Write $$\Psi_k(x) = \sum_{i=1}^{\infty} a_{ki}^* x a_{ki}, \ \ \ x\in {\mathcal{B}}(H),$$ for some bounded column operator $(a_{ki})_{i\in {\mathbb{N}}}$ with entries in ${\mathcal{A}}$ [@haagerup]. By , $\Psi_k(1) \leq 1$, and hence the column operator $(a_{ki})_{i\in {\mathbb{N}}}$ is contractive. Thus, $a_{ki}$ is a contraction for all $k,i\in {\mathbb{N}}$. Set $\Phi_{ki} = \Phi_k$ for all $k,i\in {\mathbb{N}}$ and note that, in the weak\* topology, we have $$\Theta(x) = \lim_{p\to\infty} \lim_{q\to\infty} \sum_{k=1}^p \sum_{i=1}^q \Phi_{ki}\circ \operatorname{Ad}(a_{ki}^*)(x), \ \ \ x\in {\mathcal{B}}(H).$$ Let $x\in {\mathcal{B}}(H)^+$ and $\rho\in {\mathcal{T}}(H)^+$. Then the double limit $$\label{eq_doub} \lim_{p\to\infty} \lim_{q\to\infty} \sum_{k=1}^p \sum_{i=1}^q {{ \left\langle\Phi_{ki}\circ \operatorname{Ad}(a_{ki}^*)(x),\rho\right\rangle}}$$ exists. Since the terms of the sequence in are positive, the limit $$\lim_{L\to\infty} \sum_{k,i=1}^L {{ \left\langle\Phi_{ki}\circ \operatorname{Ad}(a_{ki}^*)(x),\rho\right\rangle}}$$ exists. Thus, the partial sums $\sum_{k,i=1}^L \Phi_{ki}\circ \operatorname{Ad}(a_{ki}^*)(x)$ converge in the weak\* topology for every $x\in {\mathcal{B}}(H)^+$ and hence, by the polarisation identity, for every $x\in {\mathcal{B}}(H)$. It is now straightforward to see that the limit coincides with $\Theta(x)$. Note, moreover, that identity shows that $\sum_{k,i=1}^{\infty} a_{ki}^* a_{ki} = 1$ in the weak\* topology. This establishes (ii). The assertion in (iii) follows by considering the Kraus decomposition of the maps $\Phi_k$. To show (i), assume that $\Theta_i\in \LOCC^r({{\mathcal{A}}})$ and let $\Phi_k^{(i)}$, $\Psi_k^{(i)}$, $k\in {\mathbb{N}}$, be the maps as in , associated with $\Theta_i$, $i = 1,2$. By , for every $x\in {\mathcal{B}}(H)$, we have that, in the weak\* topology, $$\begin{aligned} (\Theta_1\circ \Theta_2)(x) & = & \lim_{p\to \infty} \lim_{q\to \infty} \sum_{k=1}^p \sum_{l=1}^q \left(\Phi_k^{(1)}\circ \Psi_k^{(1)} \circ \Phi_l^{(2)} \circ \Psi_l^{(2)}\right)(x)\\ & = & \lim_{p\to \infty} \lim_{q\to \infty} \sum_{k=1}^p \sum_{l=1}^q \left(\Phi_k^{(1)}\circ \Phi_l^{(2)} \circ \Psi_k^{(1)}\circ \Psi_l^{(2)}\right)(x).\end{aligned}$$ Set $\Phi_{kj} = \Phi_k^{(1)}\circ \Phi_l^{(2)}$ and $\Psi_{kj} = \Psi_k^{(1)}\circ \Psi_l^{(2)}$. An argument similar to the one in the previous paragraph now implies that $$(\Theta_1\circ \Theta_2)(x) = \sum_{k,l=1}^{\infty} (\Phi_{kj} \circ \Psi_{kj})(x), \ \ \ x\in {\mathcal{B}}(H),$$ in the weak\* topology, so $\Theta_1\circ \Theta_2\in \LOCC^r({{\mathcal{A}}})$. \[r\_mirror\] (i) The expression of one-way right LOCC maps given in Proposition \[p\_comp\](ii) reflects the notion of fine-graining of LOCC channels described in [@clmow]. \(ii) By symmetry, observations analogous to those above for one-way right LOCC maps also hold for the one-way left LOCC maps. State Convertibility via $\operatorname{LOCC}({{\mathcal{A}}})$ {#s:conv} =============================================================== Having established an appropriate generalisation of LOCC operations in the preceding section, we now define the corresponding notions of convertibility. \[d\_conv\] Let ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$ be a von Neumann algebra on a Hilbert space $H$, and let $\psi, {\varphi}\in H_1$. (i) We say that $\psi$ *is convertible to* ${\varphi}$ *via* $\operatorname{LOCC}({\mathcal{A}})$ if there exists $\Theta\in \operatorname{LOCC}({\mathcal{A}})$ such that $\Theta_*({\omega}_\psi) = {\omega}_{\varphi}$. (ii) We say that $\psi$ *is approximately convertible to* ${\varphi}$ *via* $\operatorname{LOCC}({\mathcal{A}})$ if for every $\varepsilon > 0$ there exists $\Theta\in \operatorname{LOCC}({\mathcal{A}})$ such that ${\left\Vert \Theta_*({\omega}_\psi)- {\omega}_{\varphi}\right\Vert}<{\varepsilon}$. We also make analogous definitions with $\operatorname{LOCC}^l({{\mathcal{A}}})$ and $\operatorname{LOCC}^r({{\mathcal{A}}})$ in place of $\operatorname{LOCC}({{\mathcal{A}}})$. The goal of the next few results is to show that approximate convertibility can be realised by using only one-way LOCC maps (see Corollary \[c\_oneway\]). This generalises to the commuting operator framework a result of Lo-Popescu [@lp] for finite-dimensional bipartite systems. We note that another generalisation of this theorem was established in [@obnm], which we recover from our results by taking the special case ${{\mathcal{A}}}={\mathcal{B}(H)}$ in its standard representation, for $H$ a separable Hilbert space. The essential feature of our argument, similar to the finite-dimensional case, is the symmetry induced from the standard form of a von Neumann algebra [@haagerup2], as highlighted in Section \[s\_bc\]. Let $H$ be a Hilbert space, ${\mathcal{A}}\subseteq{\mathcal{B}}(H)$ be a (semi-finite) von Neumann algebra equipped with a normal semi-finite faithful trace $\tau$, and $\psi\in H$. To avoid double subscripts, we will write $\rho_\psi\in L^1({{\mathcal{A}}},\tau)$ for the density operator $\rho_{\omega_\psi}$ arising from the restriction of the vector state ${\omega}_\psi$ to ${\mathcal{A}}$. Recall that, in the case where $H=L^2({{\mathcal{A}}},\tau)$ and ${\mathcal{A}}$ is in standard form, we write $\rho_\psi'\in L^1({{\mathcal{A}}}',\tau')$ for the density operator affiliated to ${\mathcal{A}}'$ satisfying $${{ \left(a'\psi,\psi\right)}}={\omega}_{\psi}|_{{\mathcal{A}}'}(a')=\tau'(\rho'_\psi a'), \ \ \ a'\in {\mathcal{A}}',$$ where $\tau'=\tau\circ R$ is the canonical trace on ${\mathcal{A}}'$. Recall that for $t > 0$, we write $\mu'_t(\rho'_\psi)=\mu_t(\rho'_\psi;{{\mathcal{A}}}',\tau')$ for the $t$-th singular value of $\rho'_\psi$ relative to $({{\mathcal{A}}}',\tau')$. \[l\_Schmidt\] Let $({\mathcal{A}},\tau)$ be a semi-finite von Neumann algebra, represented in its standard form on $L^2({{\mathcal{A}}},\tau)$, and let $\psi\in L^2({\mathcal{A}},\tau)$. Then $$J\rho_\psi'J=\rho_{\psi^*}{{\quad\text{and}\quad}}\mu'_t(\rho_\psi')=\mu_t(\rho_{\psi^*})=\mu_t(\rho_\psi),\quad t>0.$$ First we show that $\rho_{\psi^*}=J\rho'_\psi J$ and $\mu_t'(\rho_\psi')=\mu_t(\rho_{\psi^*})$. Recall that $\psi^*=J\psi$. For $a\in {{\mathcal{A}}}$, we therefore have $$\begin{aligned} \tau(\rho_{\psi^*}a)&={{ \left(aJ\psi,J\psi\right)}}={{ \left(Ja^*J\psi,\psi\right)}}=\tau'(\rho'_\psi Ja^*J)\\&=\tau\left(J(JaJ\rho_\psi')J\right)=\tau(J\rho_\psi'Ja).\end{aligned}$$ Hence, $$\label{eq_psiJ} \rho_{\psi^*} = J\rho'_\psi J.$$ Recall that $J$ is isometric, with $J^2=1$, and note that $R\colon {{\mathcal{A}}}'\to {{\mathcal{A}}}$ induces a trace-preserving bijection ${{\mathcal{P}}}({{\mathcal{A}}}')\to {{\mathcal{P}}}({{\mathcal{A}}})$, $p'\mapsto Jp'J$. For $t > 0$, using we obtain $$\begin{aligned} \mu'_t(\rho_\psi')&=\inf\{{\left\Vert \rho_\psi' p'\right\Vert}\colon {p'\in {{\mathcal{P}}}({{\mathcal{A}}}'),\,\tau'(1-p')\le t}\}\\ &=\inf\{{\left\Vert \rho_\psi'JpJ\right\Vert}\colon {p\in {{\mathcal{P}}}({{\mathcal{A}}}),\,\tau(1-p)\le t}\}\\ &=\inf\{{\left\Vert J\rho_\psi'Jp\right\Vert}\colon {p\in {{\mathcal{P}}}({{\mathcal{A}}}),\,\tau(1-p)\le t}\}\\ &=\inf\{{\left\Vert \rho_{\psi^*}p\right\Vert}\colon {p\in {{\mathcal{P}}}({{\mathcal{A}}}),\,\tau(1-p)\le t}\} = \mu_t(\rho_{\psi^*}).\end{aligned}$$ Now, by [@t2 Exercises IX.1.2–3], there exists a partial isometry $u\in{\mathcal{A}}$ such that $|\psi^*|=\psi u^*$, and $\psi^*=u^*|\psi^*|$. It follows that $$\psi^* = u^*\psi u^* = u^* |\psi^*| = u^* J |\psi^*| = u^* J u \psi^* = u^*JuJ\psi.$$ Since $u\in{\mathcal{A}}$, we have $\rho_{\psi*}=\rho_{u^*JuJ\psi}=u^*\cdot\rho_{JuJ\psi}\cdot u$. By [@fk Lemma 2.5(vi)], $$\label{eq_mut} \mu_t(\rho_{\psi*})\leq{\left\Vert u^*\right\Vert}{\left\Vert u\right\Vert}\mu_t(\rho_{JuJ\psi})\leq\mu_t(\rho_{JuJ\psi}),$$ for $t > 0$. But $JuJ$ is a contraction in ${\mathcal{A}}'$, so ${\omega}_{JuJ\psi}|_{{\mathcal{A}}}\leq {\omega}_{\psi}|_{{\mathcal{A}}}$, that is, $\rho_{JuJ\psi}\leq \rho_{\psi}$. By [@fk Lemma 2.5(iii)], $\mu_t(\rho_{JuJ\psi})\leq\mu_t(\rho_\psi)$, $t>0$. Thus, by , $\mu_t(\rho_{\psi^*})\leq\mu_t(\rho_{\psi})$, for $t>0$. By symmetry, we obtain equality. \[p\_lopopescu\] Let $({\mathcal{A}},\tau)$ be a semi-finite factor in its standard form on $L^2({{\mathcal{A}}},\tau)$, let ${\mathcal{B}} = {\mathcal{A}}'$ and let $\epsilon > 0$. For any $(\psi,b)\in H\times {{\mathcal{B}}}$, there exist a unitary $u\in {\mathcal{B}}$ and partial isometries $v,w\in {{\mathcal{A}}}$ so that $\psi=v^*v\psi$ and, if $z=JbJv$, then $${\left\Vert b\psi\right\Vert}={\left\Vert z\psi\right\Vert}{{\quad\text{and}\quad}}{\left\Vert b\psi-uwz\psi\right\Vert}<\epsilon.$$ Moreover, the partial isometry $v$ can be chosen independently of $b$. Let $\psi^*=v|\psi^*|$ be the polar decomposition of $\psi^*$ [@t2 Exercises IX.1.2–3]; thus, $v\in {\mathcal{A}}$ is a partial isometry with $\psi=v^*|\psi|$, and the projections $v^*v$ and $vv^*$ have ranges $\overline{{\mathcal{B}}\psi}$ and $\overline{{\mathcal{B}}|\psi|}$, respectively, so in particular, $\psi=v^*v\psi$. It follows that $$\label{eq_vpsi} v\psi=vv^*|\psi|=|\psi|=J|\psi|=Jv\psi.$$ Note that $z \in {{\mathcal{A}}}$. We have $v^*vb\psi = b v^*v\psi = b\psi$, and hence, using , $${\left\Vert b\psi\right\Vert}={\left\Vert vb\psi\right\Vert}={\left\Vert bv\psi\right\Vert}={\left\Vert bJv\psi\right\Vert}={\left\Vert JbJv\psi\right\Vert}={\left\Vert z\psi\right\Vert},$$ as desired. Note that $$\label{eq_taudash} \tau'(\rho') = \tau(J\rho'\mbox{}^{*} J), \ \ \ \rho'\in L^1({\mathcal{B}}, \tau');$$ indeed, the formula holds by the definition of $\tau'$ in the case $\rho' \in {\mathcal{B}}$, and the general case follow by approximating $\rho'$ by a sequence in ${\mathcal{B}}$ in the norm $\|\cdot\|_1$. Let $\alpha=z\psi$ and $\beta=b\psi$. For $c\in{\mathcal{B}}$, by , and , we have $$\begin{aligned} \tau(\rho_\alpha R(c)) &= (R(c)z\psi,z\psi) ={{ \left(Jc^*JJbJv\psi,JbJv\psi\right)}}={{ \left(Jc^*bv\psi,Jbv\psi\right)}}\\ &={{ \left(bv\psi,c^*bv\psi\right)}} = {{ \left(cv^*vb\psi,b\psi\right)}} ={{ \left(c \beta,\beta\right)}} =\tau'(\rho'_\beta c)\\ &=\tau(J\rho_\beta'J R(c)).\end{aligned}$$ Thus, by Lemma \[l\_Schmidt\], $\rho_\alpha=J\rho_\beta'J=\rho_{\beta^*}$ and $$\mu'_t(\rho'_{\alpha})=\mu_t(\rho_{\alpha})=\mu_t(\rho_{\beta^*})=\mu_t'(\rho'_{\beta}), \ \ \ t>0.$$ Since ${{\mathcal{A}}}$ is a factor, by [@h Theorem 3.4(1)] for every ${\varepsilon}>0$ there exists a unitary $u\in{\mathcal{B}}$ such that $${\left\Vert \rho'_{\beta}-\rho'_{u\alpha}\right\Vert}_1 = {\left\Vert \rho'_{\beta}-u \rho'_{\alpha} u^*\right\Vert}_1 < {\varepsilon}^2.$$ By the continuity of Stinespring’s representation [@ksw Theorem 1], there exist a Hilbert space $K$, a \*-homomorphism $\pi:{\mathcal{B}}\rightarrow {\mathcal{B}}(K)$ and vectors $\xi,\eta\in K$ such that ${\omega}_{\beta}|_{{\mathcal{B}}}={\omega}_\xi\circ\pi$ and ${\omega}_{u\alpha}|_{{\mathcal{B}}}={\omega}_\eta\circ\pi$, and $${\left\Vert \xi-\eta\right\Vert}\leq{\left\Vert \omega_\beta|_{{{\mathcal{B}}}}-\omega_{u\alpha}|_{{{\mathcal{B}}}}\right\Vert}^{1/2}={\left\Vert \rho'_{\beta}-\rho'_{u\alpha}\right\Vert}_1^{1/2}<{\varepsilon}.$$ By the uniqueness of Stinespring representations, there exist partial isometries $w_1 : L^2({\mathcal{A}},\tau)\rightarrow K$ and $w_2 : K\rightarrow L^2({\mathcal{A}},\tau)$ such that $w_1c=\pi(c)w_1$ and $w_2\pi(c)=cw_2$ for all $c\in {\mathcal{B}}$, $w_1u\alpha=\eta$ and $w_2\xi=\beta$. Then $w:=w_2w_1$ is a contraction in ${\mathcal{A}}$, and $${\left\Vert b\psi-uwz\psi\right\Vert}={\left\Vert \beta-wu\alpha\right\Vert} = {\left\Vert w_2\xi-w_2\eta\right\Vert} < {\varepsilon}.\qedhere$$ The following estimate is straightforward and we will make use of it multiple times. \[l\_piso\] Let $\psi\in H$, ${\left\Vert \psi\right\Vert}\leq 1$, and let $v\in{\mathcal{B}(H)}$ be a contraction. If ${\left\Vert \psi\right\Vert}-{\left\Vert v\psi\right\Vert}<{\varepsilon}$, then ${\left\Vert (1-v^*v)^{1/2}\psi\right\Vert}<\sqrt{2{\varepsilon}}$. The assumption implies $$\begin{aligned} {\left\Vert (1-v^*v)^{1/2}\psi\right\Vert}^2&={{ \left(\strut(1-v^*v)\psi,\psi\right)}}={\left\Vert \psi\right\Vert}^2-{\left\Vert v\psi\right\Vert}^2\\ &=({\left\Vert \psi\right\Vert}+{\left\Vert v\psi\right\Vert})({\left\Vert \psi\right\Vert}-{\left\Vert v\psi\right\Vert}) \leq 2({\left\Vert \psi\right\Vert}-{\left\Vert v\psi\right\Vert}) <2{\varepsilon}.\qedhere\end{aligned}$$ \[l\_dini\] Let ${\mathcal{A}}$ be a von Neumann algebra equipped with a normal faithful semi-finite trace $\tau$, and let $(\rho_k)_{k\in {\mathbb{N}}}$ be an increasing sequence of hermitian elements of $L^1({\mathcal{A}},\tau)$. If $\rho_k\to_{k\to \infty}\rho$ in the weak topology, for some $\rho\in L^1({\mathcal{A}},\tau)$, then $\rho_k\to_{k\to \infty}\rho$ in norm. We have that $\rho_k\leq \rho$; thus, $\rho - \rho_k \geq 0$ for every $k$. Thus, $$\|\rho - \rho_k\| = (\rho - \rho_k)(1) \to_{k\to\infty} 0.\qedhere$$ \[l\_ineq\] Let $H$ be a Hilbert space and $\Phi:{\mathcal{B}(H)}\rightarrow{\mathcal{B}(H)}$ be positive. Then for any self-adjoint $T\in{\mathcal{B}(H)}$ and any self-adjoint $\rho\in{\mathcal{T}}(H)$, $$|{\langle}\Phi(T),\rho{\rangle}|\leq\norm{T}{\langle}\Phi(1),|\rho|{\rangle}.$$ Let $(e_n)_{n=1}^\infty$ be orthonormal eigenvectors of $\rho$ with corresponding eigenvalues $(\lambda_n)_{n=1}^\infty\subseteq{\mathbb{R}}$. Since $-\norm{T}1\leq T\leq\norm{T}1$, by positivity of ${\omega}_{e_n}\circ\Phi$ we have $$-\norm{T}(\Phi(1)e_n,e_n)\leq(\Phi(T)e_n,e_n)\leq\norm{T}(\Phi(1)e_n,e_n),$$ so that $|(\Phi(T)e_n,e_n)|\leq\norm{T}(\Phi(1)e_n,e_n)$ for all $n\in{\mathbb{N}}$. Hence $$\begin{aligned} |{\langle}\Phi(T),\rho{\rangle}|&=\bigg|\sum_{n=1}^\infty\lambda_n(\Phi(T)e_n,e_n)\bigg|\leq\sum_{n=1}^\infty|\lambda_n||(\Phi(T)e_n,e_n)|\\ &\leq\sum_{n=1}^\infty|\lambda_n|\norm{T}(\Phi(1)e_n,e_n)=\norm{T}{\langle}\Phi(1),|\rho|{\rangle}.\qedhere\end{aligned}$$ \[th\_lopopescu\_std\] Let $({\mathcal{A}},\tau)$ be a semi-finite factor in its standard form on $H=L^2({{\mathcal{A}}},\tau)$. For any $\Theta\in\operatorname{LOCC}({\mathcal{A}})$, $\psi\in H$ and ${\varepsilon}> 0$, there exists $\Theta_{\varepsilon}\in \LOCC^r({{\mathcal{A}}})$ such that ${\left\Vert \Theta_*({\omega}_\psi)-\Theta_{{\varepsilon}*}({\omega}_\psi)\right\Vert} < {\varepsilon}$. We may assume that ${\left\Vert \psi\right\Vert}\le1$. Any LOCC map $\Theta\in\operatorname{LOCC}({\mathcal{A}})$ is of the form $\Theta=\Theta_{{\mathcal{I}}_n}$ where $({\mathcal{I}}_0,\dots,{\mathcal{I}}_n)$ is a sequence of instruments such that ${\mathcal{I}}_0$ is one-way local relative to ${\mathcal{A}}$, and ${\mathcal{I}}_{l+1}$ is linked to ${\mathcal{I}}_{l}$ for each $l=0,\dots,n-1$. Since the trivial instrument $(\operatorname{id},0,0,\dots)$ is one-way right local and any one-way left local instrument is linked to it, without loss of generality, we may suppose that ${\mathcal{I}}_0$ is a one-way right instrument. Writing ${\mathcal{I}}_n=(\Theta_k^{(n)})_{k}$, consider the following proposition $P(n)$: for every ${\varepsilon}>0$ there exists an instrument ${\mathcal{I}}_{{\varepsilon}}=(\Gamma_{k})_{k}$ which is a coarse-graining of some one-way right instrument relative to ${\mathcal{A}}$, such that $$\sum_{k=1}^\infty\norm{\Theta_{k*}^{(n)}({\omega}_\psi)-\Gamma_{k*}({\omega}_\psi)}<{\varepsilon}.$$ If $P(n)$ were true for every $n\in{\mathbb{N}}$ then the Theorem follows with $\Theta_{{\varepsilon}}=\Theta_{{\mathcal{I}}_{{\varepsilon}}}$. We therefore proceed by induction on $n$, starting with the base case $n=1$ (as the claim for $n=0$ is vacuous). Let ${\varepsilon}> 0$ and write ${\mathcal{I}}_0=(\Theta_k)_{k\in {\mathbb{N}}}$. Each $\Theta_k$ is one-way right local, so by Proposition \[p\_comp\](ii) we may write $\Theta_k=\sum_{j,l=1}^\infty \operatorname{Ad}(a_{kj}^*)\circ\operatorname{Ad}(c_{kl}^*)$, where $a_{kj}\in{\mathcal{A}}$ satisfy $\sum_{k=1}^\infty\sum_{j=1}^{\infty}a_{kj}^*a_{kj}=1$ and $c_{kl}\in{\mathcal{B}}$ satisfy $\sum_{l=1}^\infty c_{kl}^*c_{kl}=1$ for each $k\in{\mathbb{N}}$. We define $\psi_{kj}:=a_{kj}\psi\in H$, for $k,j\in {\mathbb{N}}$. Since ${\mathcal{I}}_1$ is linked to ${\mathcal{I}}_0$, the instrument ${\mathcal{I}}_1$ is a coarse-graining of an instrument of the form $(\Theta_k\circ \Theta_{ki})_{k,i}$, for a collection of one-way instruments ${\mathcal{J}}_k=(\Theta_{ki})_i$ indexed by $k\in{\mathbb{N}}$. Write $L=\{k\in{\mathbb{N}}\mid {\mathcal{J}}_k \ \textnormal{is one-way left}\}$ and $R=\{k\in{\mathbb{N}}\mid {\mathcal{J}}_k \ \textnormal{is one-way right}\}$. Suppose first that $k\in L$. By Proposition \[p\_comp\](ii) and Remark \[r\_mirror\](ii), we may assume that $\Theta_{ki}=\Psi^L_{ki}\circ\operatorname{Ad}(b_{ki}^*)$, where $\Psi^L_{ki}\in\operatorname{UCP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$, $b_{ki}\in{\mathcal{B}}$ and $\sum_{i=1}^\infty b_{ki}^*b_{ki}=1$. Hence, in the point weak\*-topology, we have $$\label{e_rl} \Theta_k\circ \Theta_{ki}={\sum_{j=1}^\infty} {\sum_{l=1}^\infty} \operatorname{Ad}((b_{ki}c_{kl}a_{kj})^*)\circ \Psi^L_{ki}.$$ Since ${{\mathcal{A}}}$ is a factor, we can apply Proposition \[p\_lopopescu\] to the pairs $(\psi_{kj},b_{ki}c_{kl})\in H\times {{\mathcal{B}}}$. We obtain unitaries $u_{kijl}\in {{\mathcal{B}}}$ and partial isometries $v_{kj},w_{kijl}\in {{\mathcal{A}}}$ so that $\psi_{kj}=v_{kj}^*v_{kj}\psi_{kj}$ and $z_{kijl}=Jb_{ki}c_{kl}Jv_{kj}\in {{\mathcal{A}}}$ satisfy ${\left\Vert b_{ki}c_{kl}\psi_{kj}\right\Vert}={\left\Vert z_{kijl}\psi_{kj}\right\Vert}$ and $$\label{e_bound1} \|b_{ki}c_{kl}\psi_{kj}-u_{kijl}w_{kijl}z_{kijl}\psi_{kj}\|<\frac{\epsilon}{2^{i+j+k+l+2}}$$ for $i,j,k,l\in {{\mathbb{N}}}$. Let $a_{kijl}=w_{kijl}z_{kijl}\in {{\mathcal{A}}}$, and observe that $${\left\Vert z_{kijl}\psi_{kj}\right\Vert}-\norm{a_{kijl}\psi_{kj}}={\left\Vert b_{ki}c_{kl}\psi_{kj}\right\Vert}-\norm{u_{kijl}w_{kijl}z_{kijl}\psi_{kj}}<\frac{{\varepsilon}}{2^{i + j + k+l+2}}.$$ Let $\widetilde{a_{kijl}} := (1-w_{kijl}^*w_{kijl})z_{kijl}\in {{\mathcal{A}}}$. By Lemma \[l\_piso\], the preceding inequality implies $$\label{eq_kij1} {\left\Vert \widetilde{a_{kijl}}\psi_{kj}\right\Vert}={\left\Vert (1-w_{kijl}^*w_{kijl}) z_{kijl}\psi_{kj}\right\Vert}<\sqrt{\frac{{\varepsilon}}{2^{i + j+k+l+1}}}.$$ Now suppose that $k\in R$. The map $\Theta_{{\mathcal{J}}_k}=\sum_{i=1}^\infty \Theta_{ki}$ is then a one-way right LOCC map relative to ${\mathcal{A}}$, with $\Theta_{ki}=\Psi^R_{ki}\circ\Phi_{ki}$, where $\Psi^R_{ki}\in\operatorname{CP}^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$ and $\Phi_{ki}\in\operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$. Then $$\label{eq_kR}\Theta_k\circ \Theta_{ki}=\sum_{j=1}^\infty\sum_{l=1}^\infty \operatorname{Ad}(a_{kj}^*)\circ\Psi^R_{ki}\circ(\operatorname{Ad}(c_{kl}^*)\circ\Phi_{ki})$$ in the point weak\*-topology. Recall that $v_{kj}$ was defined above for $k\in L$ and $j\in {\mathbb{N}}$; we define $v_{kj}:=1$ for $k\in R$ and $j\in{\mathbb{N}}$. Then $$A:=\sum_{k=1}^\infty\sum_{j=1}^\infty a_{kj}^*v_{kj}^*v_{kj}a_{kj}\leq\sum_{k=1}^\infty\sum_{j=1}^\infty a_{kj}^*a_{kj}=1;$$ thus, the operator $(1-A)^{1/2}$ is a well-defined element of ${\mathcal{A}}$. Moreover, since $a_{kj}\psi=\psi_{kj}=v_{kj}^*v_{kj}\psi_{kj}=v_{kj}^*v_{kj}a_{kj}\psi$ for all $j,k\in{\mathbb{N}}$, we have $\psi=\sum_{k=1}^\infty\sum_{j=1}^\infty a_{kj}^*a_{kj}\psi=A\psi$, showing that $\psi\in\operatorname{Ker}(1-A)$ and thus that $\psi \in \operatorname{Ker}((1-A)^{1/2})$. In particular, $$\label{eq_1-A} \left(\operatorname{Ad}((1-A)^{1/2})\right)_*(\omega_\psi) = 0.$$ Fix a probability distribution $(p_n)$ over ${\mathbb{N}}$ with $p_n > 0$ for each $n$. For $k\in L$ and $i,j,l\in {\mathbb{N}}$, define $\Gamma^L_{kijl}\in \operatorname{CP}_{{\mathcal{B}}}^\sigma({\mathcal{B}}(H))$ for $i\in{{\mathbb{N}}}$ by $$\Gamma^L_{kijl}=\left(\operatorname{Ad}(a_{kj}^*)\circ (\operatorname{Ad}(a_{kijl}^*)+\operatorname{Ad}(\widetilde{a_{kijl}}^*)) + p_kp_ip_j p_l\operatorname{Ad}((1-A)^{1/2})\right)\circ \Psi^L_{ki}.$$ Since $$a_{kijl}^*a_{kijl}+\widetilde{a_{kijl}}^*\widetilde{a_{kijl}}=z_{kijl}^*z_{kijl}=v_{kj}^*Jc_{kl}^*b_{ki}^*b_{ki}c_{kl}Jv_{kj}$$ and ${\sum_{i=1}^\infty} {\sum_{l=1}^\infty} c_{kl}^*b_{ki}^*b_{ki}c_{kl}=1$ for every $k\in L$, in the weak\* topology we have $$\begin{aligned} \sum_{k\in L}\sum_{i=1}^\infty{\sum_{j=1}^\infty}{\sum_{l=1}^\infty}\Gamma^L_{kijl}(1) &=\sum_{k\in L}\sum_{i=1}^\infty\sum_{j=1}^\infty\sum_{l=1}^\infty a_{kj}^*z_{kijl}^*z_{kijl} a_{kj}+p_kp_ip_jp_l(1-A)\\ &=\sum_{k\in L}\sum_{j=1}^\infty a_{kj}^*v_{kj}^*v_{kj}a_{kj} +p_kp_j(1-A).\end{aligned}$$ For $k\in R$, define one-way right local maps $\Gamma^R_{ki}$ by $$\Gamma^R_{ki}=\sum_{j=1}^\infty\sum_{l=1}^\infty(\operatorname{Ad}(a_{kj}^*)\circ\Psi^R_{ki}+p_kp_ip_j\operatorname{Ad}((1-A)^{1/2}))\circ(\operatorname{Ad}(c_{kl}^*)\circ\Phi_{ki}).$$ Then $$\begin{aligned} \sum_{k\in R}\sum_{i=1}^\infty\Gamma^R_{ki}(1)&=\sum_{k\in R}\sum_{i=1}^\infty\sum_{j=1}^\infty a_{kj}^*\Psi^R_{ki}(1)a_{kj}+p_kp_ip_j(1-A)\\ &=\sum_{k\in R}\sum_{j=1}^\infty a_{kj}^*a_{kj}+p_kp_j(1-A).\\\end{aligned}$$ Putting things together, we get $$\begin{aligned} &\sum_{k\in L}\sum_{i=1}^\infty{\sum_{j=1}^\infty}{\sum_{l=1}^\infty}\Gamma^L_{kijl}(1)+ \sum_{k\in R}\sum_{i=1}^\infty\Gamma^R_{ki}(1)\\ &=\sum_{k\in L}\sum_{j=1}^\infty a_{kj}^*v_{kj}^*v_{kj}a_{kj} +p_kp_j(1-A)+\sum_{k\in R}\sum_{j=1}^\infty a_{kj}^*a_{kj}+p_kp_j(1-A)\\ &=\sum_{k=1}^\infty\sum_{j=1}^\infty a_{kj}^*v_{kj}^*v_{kj}a_{kj} +p_kp_j(1-A)\\ &=A+(1-A)\\ &=1.\end{aligned}$$ It follows that the series $$\sum_{k\in L}\sum_{i=1}^\infty\sum_{j=1}^\infty \sum_{l=1}^\infty \operatorname{Ad}(u_{kijl}^*)\circ \Gamma^L_{kijl}(x)+\sum_{k\in R}\sum_{i=1}^\infty\Gamma^R_{ki}(x)$$ is convergent in the weak\* topology for every positive $x\in{\mathcal{B}}(H)$. By polarisation, it is convergent in the weak\* topology for every $x\in{\mathcal{B}}(H)$. With $$\Gamma^L_{ki}:=\sum_{j=1}^\infty \sum_{l=1}^\infty \operatorname{Ad}(u_{kijl}^*)\circ \Gamma^L_{kijl},$$ and $\Gamma_{ki}:=\Gamma^L_{ki}$ for $k\in L$, $i\in{\mathbb{N}}$, and $\Gamma_{ki}:=\Gamma^R_{ki}$ for $k\in R$, $i\in{\mathbb{N}}$, it follows that $(\Gamma_{ki})_{k,i}$ is a coarse-graining of a one-way right instrument relative to ${\mathcal{A}}$. By and , for $k\in R$ we have $\Theta_{ki*}(\Theta_{k*}({\omega}_\psi))=\Gamma_{ki*}({\omega}_\psi)$, $i\in{\mathbb{N}}$. Also, using , , and with the identity $\operatorname{Ad}(a_{kj})\omega_\psi=\omega_{\psi_{kj}}$ and Lemma \[l\_dini\] consecutively, we have $$\begin{aligned} &\sum_{k\in L}{\sum_{i=1}^\infty}{\left\Vert \Gamma_{ki*}({\omega}_\psi)-\Theta_{ki*}(\Theta_{k*}({\omega}_\psi))\right\Vert}\\ &\leq\sum_{k\in L}\sum_{i,j,l=1}^\infty{\left\Vert \Psi^L_{ki*} \circ\left(\operatorname{Ad}(u_{kijl})\circ(\operatorname{Ad}(a_{kijl})+\operatorname{Ad}(\widetilde{a_{kijl}})) -\operatorname{Ad}(b_{ki}c_{kl}) \right)\omega_{\psi_{kj}}\right\Vert}\\ &\leq\sum_{k\in L}\sum_{i,j,l=1}^\infty{\left\Vert u_{kijl}(a_{kijl}{\omega}_{\psi_k}a_{kijl}^*+\widetilde{a_{kijl}}{\omega}_{\psi_{kj}}\widetilde{a_{kijl}}^*)u_{kijl}^*-b_{ki}c_{kl}{\omega}_{\psi_{kj}}c_{kl}^*b_{ki}\right\Vert}\\ &\leq\sum_{k\in L}\sum_{i,j,l=1}^\infty{\left\Vert u_{kijl}a_{kijl}{\omega}_{\psi_{kj}}a_{kijl}^*u_{kijl}^*-b_{ki}c_{kl}{\omega}_{\psi_{kj}}c_{kl}^*b_{ki}\right\Vert}+{\left\Vert \widetilde{a_{kijl}}{\omega}_{\psi_{kj}}\widetilde{a_{kijl}}^*\right\Vert}\\ &\leq\sum_{k\in L}\sum_{i,j,l=1}^\infty{\left\Vert u_{kijl}a_{kijl}\psi_{kj}-b_{ki}c_{kl}\psi_{kj}\right\Vert}({\left\Vert a_{kijl}\psi_{kj}\right\Vert}+{\left\Vert b_{ki}c_{kl}\psi_{kj}\right\Vert})+{\left\Vert \widetilde{a_{kijl}}\psi_{kj}\right\Vert}^2\\ &\leq2\sum_{k\in L}\sum_{i,j,l=1}^\infty\frac{{\varepsilon}}{2^{k + i + j+l+1}} \leq {\varepsilon}.\end{aligned}$$Since ${\mathcal{I}}_1$ is a coarse-graining of $(\Theta_k\circ \Theta_{ki})_{k,i}$, applying the same coarse-graining to $(\Gamma_{ki})_{k,i}$ produces an instrument ${\mathcal{I}}_{{\varepsilon}}$ satisfying $P(1)$. Now, assuming $P(n)$ is true, let $\Theta$ be an LOCC map of the form $\Theta=\Theta_{{\mathcal{I}}_{n+1}}$ where $({\mathcal{I}}_0,\dots,{\mathcal{I}}_{n+1})$ is a sequence of instruments such that ${\mathcal{I}}_0$ is a one-way local right instrument relative to ${\mathcal{A}}$, and ${\mathcal{I}}_{l+1}$ is linked to ${\mathcal{I}}_{l}$ for each $l=0,\dots,n$. Then ${\mathcal{I}}_{n+1}$ is a coarse-graining of $(\Theta_k\circ \Theta_{ki})_{k,i}$, where ${\mathcal{I}}_{n}=(\Theta_k)_k$ and ${\mathcal{J}}_k=(\Theta_{ki})_i$ is a one-way instrument for each $k\in{\mathbb{N}}$. Given ${\varepsilon}>0$, by $P(n)$ there exists a coarse-graining ${\mathcal{I}}_{{\varepsilon}}=(\Gamma_{k})_{k}$ of some one-way right instrument such that $$\sum_{k=1}^\infty\norm{\Theta_{k*}({\omega}_\psi)-\Gamma_{k*}({\omega}_\psi)}<\frac{{\varepsilon}}{2}.$$ We have $\Gamma_k=\sum_{j\in S_k} \tilde \Gamma_j$ for some one-way right instrument ${\mathcal{I}}_0'=(\tilde\Gamma_j)_j$ and some partition $(S_k)_k$ of ${\mathbb{N}}$. Let $\tilde \Theta_{ji}=\Theta_{ki}$ for $j\in S_k$, $k\in {\mathbb{N}}$. Then $(\tilde \Theta_{ji})_i$ is a one-way instrument for each $j\in {\mathbb{N}}$, and $(\Gamma_k\circ \Theta_{ki})_{k,i}$ is a coarse-graining of ${\mathcal{I}}_1'=(\tilde \Gamma_j\circ \tilde \Theta_{ji})_{j,i}$, and ${\mathcal{I}}_1'$ is linked to ${\mathcal{I}}_0'$. Applying $P(1)$ to the pair $({\mathcal{I}}_0', {\mathcal{I}}_1')$, we obtain an instrument $(\hat \Gamma_{ji})_{j,i}$ which is a coarse-graining of some one-way right instrument, and satisfies $$\sum_{j,i=1}^\infty{\left\Vert \tilde \Theta_{ji*}( \tilde \Gamma_{j*}(\omega_\psi))-\hat\Gamma_{ji*}(\omega_\psi)\right\Vert}<\frac\epsilon2.$$ Setting $\Gamma_{ki}:=\sum_{j\in S_k}\hat \Gamma_{ji}$, we obtain an instrument $(\Gamma_{ki})_{k,i}$ which is a coarse-graining of a one-way right instrument, and satisfies $$\begin{aligned} {\sum_{k=1}^\infty}{\sum_{i=1}^\infty}\norm{\Theta_{ki*}(\Gamma_{k*}({\omega}_\psi))-\Gamma_{ki*}({\omega}_\psi)}&\le {\sum_{k=1}^\infty}{\sum_{i=1}^\infty}\sum_{j\in S_k}\norm{\tilde\Theta_{ji*}(\tilde \Gamma_{j*}({\omega}_\psi))-\hat\Gamma_{ji*}({\omega}_\psi)}\\&={\sum_{i=1}^\infty}{\sum_{j=1}^\infty} \norm{\tilde\Theta_{ji*}(\tilde \Gamma_{j*}({\omega}_\psi))-\hat\Gamma_{ji*}({\omega}_\psi)}<\frac{{\varepsilon}}{2}.\end{aligned}$$ For each $k,i\in{\mathbb{N}}$, $\Theta_{ki*}((\Theta_{k*}-\Gamma_{k*})({\omega}_\psi))\in{\mathcal{T}}(H)$ is self-adjoint, and attains its norm on self-adjoint operators in ${\mathcal{B}(H)}$. Hence, there exists $T_{ki}\in{\mathcal{B}(H)}$ with $T_{ki}=T_{ki}^*$ and $\norm{T_{ki}}\leq 1$ satisfying $$\norm{\Theta_{ki*}((\Theta_{k*}-\Gamma_{k*})({\omega}_\psi))}=|{\langle}\Theta_{ki}(T_{ki}),(\Theta_{k*}-\Gamma_{k*})({\omega}_\psi){\rangle}|.$$ By Lemma \[l\_ineq\], $$\begin{aligned} \norm{\Theta_{ki*}((\Theta_{k*}-\Gamma_{k*})({\omega}_\psi))}&\leq\norm{T_{ki}}{\langle}\Theta_{ki}(1),|(\Theta_{k*}-\Gamma_{k*})({\omega}_\psi)|{\rangle}\\ &\leq{\langle}\Theta_{ki}(1),|(\Theta_{k*}-\Gamma_{k*})({\omega}_\psi)|{\rangle}.\end{aligned}$$ Hence, $$\begin{aligned} \sum_{k,i=1}^\infty\norm{\Theta_{ki*}(\Theta_{k*}({\omega}_\psi))-\Theta_{ki*}(\Gamma_{k*}({\omega}_\psi))}&\leq\sum_{k=1}^\infty\sum_{i=1}^\infty{\langle}\Theta_{ki}(1),|(\Theta_{k*}-\Gamma_{k*})({\omega}_\psi)|{\rangle}\\ &=\sum_{k=1}^\infty\norm{\Theta_{k*}({\omega}_\psi)-\Gamma_{k*}({\omega}_\psi)}\\ &<\frac{{\varepsilon}}{2}.\end{aligned}$$ By the triangle inequality we obtain $${\sum_{k=1}^\infty}{\sum_{i=1}^\infty}\norm{\Theta_{ki*}(\Theta_{k*}({\omega}_\psi))-\Gamma_{ki*}({\omega}_\psi)}<{\varepsilon}.$$ Recall that ${\mathcal{I}}_{n+1}$ is a coarse-graining of $(\Theta_k\circ \Theta_{k,i})_{k,i}$. Letting ${\mathcal{I}}_\epsilon$ be the result of applying the same coarse-graining to $(\Gamma_{ki})_{k,i}$, the preceding inequality and the triangle inequality then show that ${\mathcal{I}}_\epsilon$ satisfies $P(n+1)$. The proof of Theorem \[th\_lopopescu\_std\] may seem complicated when compared to Lo and Popescu’s intuitive argument for the special case ${{\mathcal{A}}}=M_n\otimes 1$. This may be explained by our approximate version of convertibility together with the additional approximation provided by Proposition \[p\_lopopescu\], the latter not being required in the type I case. Using the representation theory of properly infinite von Neumann algebras, we now remove the standardness assumption in the previous theorem. Recall that a von Neumann algebra is $\sigma$-finite if every set of mutually orthogonal projections is at most countable, and that every von Neumann algebra ${{\mathcal{A}}}\subseteq{\mathcal{B}(H)}$ on a separable Hilbert space $H$ enjoys this property. \[t:lopopescu\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a semi-finite factor on a separable Hilbert space $H$. Given $\Theta\in\operatorname{LOCC}({\mathcal{A}})$ and $\psi\in H$, for every ${\varepsilon}>0$ there exists $\Xi \in \LOCC^r({{\mathcal{A}}})$ such that $${\left\Vert \Theta_*({\omega}_\psi)-\Xi_*({\omega}_\psi)\right\Vert} < {\varepsilon}.$$ Clearly, we may assume that $\psi\in H_1$. Let $K$ be a separable infinite-dimensional Hilbert space and fix a unit vector $\xi\in K$. Consider the factors $\widetilde{{\mathcal{A}}} :={\mathcal{A}}{\otimes}{\mathcal{B}}(K){\otimes}1_K$ and $\widetilde{{\mathcal{B}}}:=\widetilde{{\mathcal{A}}}'={\mathcal{B}}{\otimes}1_K{\otimes}{\mathcal{B}}(K)$, acting on $H{\otimes}K{\otimes}K$, and equip $\widetilde{{\mathcal{A}}}$ with the trace $\widetilde{\tau} = \tau\otimes {\rm tr} \otimes {\rm tr}$. Letting $L^{\infty}(\widetilde{{\mathcal{A}}})\subseteq{\mathcal{B}}(L^2(\widetilde{{\mathcal{A}}}))$ denote the standard representation of $\widetilde{{\mathcal{A}}}$, one sees that both $\widetilde{{\mathcal{B}}}=\widetilde{{\mathcal{A}}}'$ and $L^{\infty}(\widetilde{{\mathcal{A}}})'\cong L^{\infty}(\widetilde{{\mathcal{A}}})$ are $\sigma$-finite and properly infinite factors. Hence, by [@t1 Proposition V.3.1], the representations $(\widetilde{{\mathcal{A}}},H{\otimes}K{\otimes}K)$ and $(\widetilde{{\mathcal{A}}},L^2(\widetilde{{\mathcal{A}}}))$ are unitarily equivalent, implemented by the unitary operator $U : H{\otimes}K {\otimes}K\rightarrow L^2(\widetilde{{\mathcal{A}}})$, say. Clearly, $\operatorname{Ad}(U)\circ (\Theta{\otimes}\operatorname{id}_{{\mathcal{B}}(K{\otimes}K)})\circ \operatorname{Ad}(U^*)\in\operatorname{LOCC}(L^\infty(\widetilde{{\mathcal{A}}}))$. Since $L^\infty(\widetilde{{{\mathcal{A}}}})$ is a factor, by Theorem \[th\_lopopescu\_std\], for every ${\varepsilon}>0$ there exists a one-way right LOCC map $\widetilde{\Xi}:{\mathcal{B}}(L^2(\widetilde{{\mathcal{A}}}))\rightarrow{\mathcal{B}}(L^2(\widetilde{{\mathcal{A}}}))$ relative to $L^\infty(\widetilde{{\mathcal{A}}})$ such that $$\norm{(\operatorname{Ad}(U)\circ (\Theta{\otimes}\operatorname{id}_{{\mathcal{B}}(K{\otimes}K)})\circ \operatorname{Ad}(U^*))_*({\omega}_{U(\psi{\otimes}\xi{\otimes}\xi)})-\widetilde{\Xi}_*({\omega}_{U(\psi{\otimes}\xi{\otimes}\xi)})}<{\varepsilon}.$$ Then $\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U):{\mathcal{B}}(H{\otimes}K{\otimes}K)\rightarrow{\mathcal{B}}(H{\otimes}K{\otimes}K)$ is a one-way right LOCC map relative to $\widetilde{{\mathcal{A}}}$ satisfying $$\norm{(\Theta{\otimes}\operatorname{id}_{{\mathcal{B}}(K{\otimes}K)})_*({\omega}_{\psi{\otimes}\xi{\otimes}\xi})-(\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U))_*({\omega}_{\psi{\otimes}\xi{\otimes}\xi})}<{\varepsilon}.$$ Let $\widetilde{\Psi}_k\in \CP^\sigma_{\widetilde{{\mathcal{B}}}}({\mathcal{B}}(H{\otimes}K{\otimes}K))$ and $\widetilde{\Phi}_k\in \UCP^\sigma_{\widetilde{{\mathcal{A}}}}({\mathcal{B}}(H{\otimes}K{\otimes}K))$ satisfy $\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U)= \sum_{k=1}^\infty \widetilde{\Phi}_k\circ\widetilde{\Psi}_k$, and let ${\mathcal{E}}:{\mathcal{B}}(H{\otimes}K {\otimes}K)\rightarrow {\mathcal{B}}(H)$ denote the normal unital completely positive map given by ${\mathcal{E}}(X)=(\operatorname{id}{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)(X)$. Define $\Psi_k\in \CP^\sigma_{{\mathcal{B}}}({\mathcal{B}}(H))$ and $\Phi_k\in \UCP^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ by $$\Psi_k={\mathcal{E}}\circ \widetilde{\Psi}_k|_{{\mathcal{B}(H)}{\otimes}1_K{\otimes}1_K}\circ \iota{{\quad\text{and}\quad}}\Phi_k={\mathcal{E}}\circ \widetilde{\Phi}_k|_{{\mathcal{B}(H)}{\otimes}1_K{\otimes}1_K}\circ \iota,$$ where $\iota\colon {\mathcal{B}(H)}\to {\mathcal{B}(H)}{\otimes}1_K{\otimes}1_K$ is the embedding $\iota(T)= T{\otimes}1{\otimes}1$. Then $\Xi:=\sum_{k=1}^\infty \Phi_k\circ \Psi_k$ is a one-way right LOCC map on ${\mathcal{B}(H)}$ relative to ${\mathcal{A}}$. Moreover, letting $\sigma_{r,s}$, for $r,s\in \{2,3,4,5\}$ be the flip between terms $r$ and $s$ acting on the tensor product $H\otimes K \otimes K \otimes K \otimes K$, for every $T\in{\mathcal{B}(H)}$, we have $$\begin{aligned} &{{{ \left\langleT,\Xi_*({\omega}_\psi)\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle{\mathcal{E}}(\widetilde{\Phi}_k(T{\otimes}1{\otimes}1)){\otimes}1{\otimes}1,(\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1,({\mathcal{E}}_*{\otimes}\operatorname{id}{\otimes}\operatorname{id})((\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi))\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1,\sigma_{24}\sigma_{35}((\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi){\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1,\sigma_{35}((\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi){\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)\right\rangle}}}\\ &=\sum_{k=1}^\infty{{{ \left\langle\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1,(\widetilde{\Psi}_k)_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi){\otimes}{\omega}_\xi{\otimes}{\omega}_\xi\right\rangle}}}\\ &={{{ \left\langleT{\otimes}1{\otimes}1{\otimes}1{\otimes}1,(\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U))_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi){\otimes}{\omega}_\xi{\otimes}{\omega}_\xi\right\rangle}}}\\ &={{{ \left\langleT{\otimes}1{\otimes}1,(\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U))_*({\omega}_\psi{\otimes}{\omega}_\xi{\otimes}{\omega}_\xi)\right\rangle}}},\end{aligned}$$ where in the fourth equality we used the fact that $\widetilde{\Phi}_k(T{\otimes}1{\otimes}1){\otimes}1{\otimes}1\in {\mathcal{B}}(H){\otimes}1{\otimes}B(K){\otimes}1{\otimes}1$ is symmetric under $\sigma_{24}$ and in the fifth equality we used the fact that $(\widetilde{\Psi}_k)_*$ acts trivially on the third leg. (These facts follow from Proposition \[p\_comp\], for example.) It follows that $$\begin{aligned} &\norm{\Theta_*({\omega}_\psi)-\Xi_*({\omega}_\psi)}\\ &\le \norm{(\Theta{\otimes}\operatorname{id}_{{\mathcal{B}}(K{\otimes}K)})_*({\omega}_{\psi{\otimes}\xi{\otimes}\xi})-(\operatorname{Ad}(U^*)\circ\widetilde{\Xi}\circ\operatorname{Ad}(U))_*({\omega}_{\psi{\otimes}\xi{\otimes}\xi})}<{\varepsilon}.\;\qedhere\end{aligned}$$ By left-right symmetry, Theorem \[t:lopopescu\] immediately yields the following corollary: \[c\_oneway\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a semi-finite factor on a separable Hilbert space. For unit vectors $\psi,{\varphi}\in H$, the following are equivalent: (i) $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$; (ii) $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}^r({\mathcal{A}})$; (iii) $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}^l({\mathcal{A}})$. The Main Theorem {#s:main} ================ In this section we establish Theorem \[th\_maj\], a version of Nielsen’s theorem for bipartite systems modelled by semi-finite, $\sigma$-finite von Neumann algebras (or by standardly represented von Neumann algebras). The next group of lemmas will help justify certain technical arguments in its proof. \[l\_M\] Let $({\mathcal{A}},\tau)$ be a semi-finite von Neumann algebra. For any $\rho_1,\dots,\rho_n\in {{\mathcal{A}}}_*^+$, there exist $M_1,\dots,M_{n+1}\in {\mathcal{A}}$ with $\sum_{i=1}^{n+1}M_i^*M_i=1$ such that, for $\rho=\sum_{i=1}^n \rho_i$, we have $$M_i \rho M_i^*=\rho_i,\quad 1\le i\le n,{{\quad\text{and}\quad}}M_{n+1}\rho M_{n+1}=0.$$ We may assume that ${{\mathcal{A}}}$ is standardly represented on $H=L^2({{\mathcal{A}}},\tau)$, and identify ${{\mathcal{A}}}_*$ with $L^1({{\mathcal{A}}},\tau)$. Let ${{\mathcal{B}}}={{\mathcal{A}}}'$. Since $\rho$ is a positive element of $L^1({{\mathcal{A}}},\tau)$, the (positive, densely defined) operator $\xi:=\rho^{1/2}$ is an element of $H=L^2({{\mathcal{A}}},\tau)$. Let $p$ and $p'$ be the orthogonal projections onto $\overline{{{\mathcal{B}}}\xi}$ and $\overline{{{\mathcal{A}}}\xi}$, respectively, and note that $$p\in {{\mathcal{A}}},\quad p'\in {{\mathcal{B}}},\quad Jp'J=p{{\quad\text{and}\quad}}JpJ=p'$$ (see [@t2 Section IX.1]; for the last two equalities, $J\xi=\xi^*=\xi$ so $JpJa\xi=Jp(JaJ\xi)=J(JaJ\xi)=a\xi$, so $p'\le JpJ$ and similarly $p\le Jp'J$.) For $1\le i\le n$, we have $\rho_i\leq\rho$, that is, ${\omega}_{\rho_i^{1/2}}\leq {\omega}_\xi$. By the Radon–Nikodym theorem (see [@kr2 Proposition 7.3.5] and its proof) there exists $b_i\in {\mathcal{B}}^+$ such that $${\omega}_{\rho_i^{1/2}}(a) = {{ \left(a b_i\xi,\xi\right)}} = {{ \left(a b_i^{1/2}\xi,b_i^{1/2}\xi\right)}} = {\omega}_{b_i^{1/2} \xi}(a),\quad a\in {\mathcal{A}}.$$ By the uniqueness of the GNS representation, there exists a partial isometry $v_i\in {\mathcal{B}}$ such that for $c_i:=v_ib_i^{1/2}\in {{\mathcal{B}}}$, we have $$\label{eq_cixi} c_i \xi=\rho_i^{1/2},\quad 1\le i\le n.$$ Consider $M_1,\dots,M_{n+1}\in {{\mathcal{A}}}$, given by $$M_i:=Jc_iJp,\quad 1\le i\le n,\quad M_{n+1}:=1-p.$$ For $1\le i\le n$, using we have $$M_i\xi=Jc_iJ\xi=\xi c_i^*=(c_i\xi)^*=(\rho_i^{1/2})^*=\rho_i^{1/2},$$ so $$M_i \rho M_i^*=(M_i\xi)(M_i\xi)^*=\rho_i,\quad 1\le i\le n.$$ Similarly, $M_{n+1}\xi=(1-p)\xi=0$, so $M_{n+1}\rho M_{n+1}^*=0$. For $a,b\in {{\mathcal{A}}}$, we have $p'a\xi=a\xi$ and $p'b\xi=b\xi$; hence, for $1\le i\le n$, $$\begin{aligned} {{ \left(JM_i^*M_iJa\xi,b\xi\right)}}&={{ \left(JpJc_i^*c_iJpJa\xi,b\xi\right)}}= {{ \left(c_ip'a\xi,c_ip'b\xi\right)}} \\&={{ \left(c_ia\xi,c_ib\xi\right)}}={{ \left(ac_i\xi,bc_i\xi\right)}}={{ \left(a\rho_i^{1/2},b\rho_i^{1/2}\right)}} \\&=\tau(\rho_i^{1/2}b^*a\rho_i^{1/2})=\tau(b^*a\rho_i). \end{aligned}$$ Thus, the operator $S:=\sum_{i=1}^n M_i^*M_i$ satisfies $ {{ \left(JSJ a\xi,b\xi\right)}}=\tau(b^*a\rho)={{ \left(a\xi,b\xi\right)}}$. Hence, $p'JSJp'=p'$, so $$S = pSp = Jp'JSJp'J = Jp'J = p,$$ and $$\sum_{i=1}^{n+1}M_i^*M_i=S+M_{n+1}^*M_{n+1}=p+(1-p)=1.\qedhere$$ The version of the following lemma for the case where ${\varepsilon}= 0$ is well-known. We require the following approximate extension. \[l\_pure\] Let $H$ be a Hilbert space, ${\varphi}\in H_1$ and $(\omega_k)_{k=1}^\infty$ be a sequence in ${\mathcal{T}}(H)^+$ such that $\sum_{k=1}^\infty{\omega}_k$ converges weakly to an element in the closed unit ball of ${\mathcal{T}}(H)$. Set $\alpha_k=\langle \phi\phi^*,{\omega}_k\rangle$, $k\in {\mathbb{N}}$. If $\epsilon > 0$ and $$\label{l_1}{\left\Vert {\omega}_{\varphi}- \sum_{k=1}^\infty{\omega}_k\right\Vert}<{\varepsilon},$$ then $\sum_{k=1}^\infty{\left\Vert {\omega}_k-\alpha_k{\omega}_{\varphi}\right\Vert} < 2\sqrt{{\varepsilon}}+{\varepsilon}$. Let $\operatorname{tr}$ denote the canonical trace on ${\mathcal{T}}(H)$ and, for simplicity, write $p = {\varphi}{\varphi}^*$ and $p^{\perp} = 1 - p$. Observe that $p^\perp {\omega}_{\varphi}p^\perp =0$. Thus, by , $$\begin{aligned} \label{eq_tlines} \sum_{k=1}^\infty{\left\Vert p^\perp{\omega}_k p^\perp\right\Vert}&=\sum_{k=1}^\infty\operatorname{tr}(p^{\perp}{\omega}_kp^{\perp})=\operatorname{tr}\bigg(p^{\perp}\bigg(\sum_{k=1}^\infty{\omega}_k\bigg)p^{\perp}\bigg)\\ &={\left\Vert p^{\perp}\bigg(\sum_{k=1}^\infty{\omega}_k\bigg)p^{\perp}\right\Vert} < {\varepsilon}. \nonumber\end{aligned}$$ By the Cauchy–Schwarz inequality, for any $T\in{\mathcal{B}(H)}$, we have $$\begin{aligned} \left|p\omega_k p^{\perp}(T)\right| = \left|{\omega}_k(p^{\perp}Tp)\right| & \leq {\omega}_k\left(p^{\perp}TT^*p^{\perp}\right)^{1/2}{\omega}_k(p)^{1/2}\\ &\leq {\left\Vert T\right\Vert}{\left\Vert p^{\perp}{\omega}_kp^{\perp}\right\Vert}^{1/2}{\omega}_k(p)^{1/2}.\end{aligned}$$ Hence, ${\left\Vert p{\omega}_kp^\perp\right\Vert}\leq{\left\Vert p^{\perp}{\omega}_kp^{\perp}\right\Vert}^{1/2}{\omega}_k(p)^{1/2}$. Applying the Cauchy–Schwarz inequality once again and using , we obtain $$\begin{aligned} \sum_{k=1}^\infty{\left\Vert p{\omega}_kp^\perp\right\Vert}&\leq\sum_{k=1}^\infty {\omega}_k(p)^{1/2} {\left\Vert p^{\perp}{\omega}_kp^{\perp}\right\Vert}^{1/2}\\ &\leq \bigg(\sum_{k=1}^\infty{\omega}_k(p)\bigg)^{1/2} \bigg(\sum_{k=1}^\infty{\left\Vert p^{\perp}{\omega}_kp^{\perp}\right\Vert}\bigg)^{1/2} < \sqrt{{\varepsilon}}.\end{aligned}$$ Since ${\left\Vert p^\perp {\omega}_k p \right\Vert}={\left\Vert (p{\omega}_k p^\perp)^*\right\Vert}={\left\Vert p{\omega}_k p^\perp\right\Vert}$, we have $\sum_{k=1}^\infty {\left\Vert p^\perp {\omega}_k p \right\Vert}< \sqrt{{\varepsilon}}$. Decomposing ${\omega}_k=p{\omega}_kp+p{\omega}_kp^{\perp}+p^\perp{\omega}_k p+p^\perp{\omega}_k p^\perp$, and noting $p{\omega}_k p={\omega}_k(p){\omega}_{\varphi}=\alpha_k {\omega}_{\varphi}$, we see that $$\begin{aligned} \sum_{k=1}^\infty{\left\Vert {\omega}_k-\alpha_k{\omega}_{\varphi}\right\Vert}&\leq\sum_{k=1}^\infty \Big( {\left\Vert p{\omega}_kp^\perp\right\Vert}+{\left\Vert p^\perp {\omega}_k p \right\Vert}+{\left\Vert p^\perp{\omega}_k p^\perp\right\Vert}\Big) < 2\sqrt{{\varepsilon}}+{\varepsilon}.\qedhere\end{aligned}$$ We are now in a position to prove the main result of the paper. It provides a version of Nielsen’s theorem for bipartite systems without any explicit (spatial) tensor product structure. \[th\_maj\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a semi-finite factor on a separable Hilbert space $H$. For unit vectors $\psi,{\varphi}\in H$, the following are equivalent: \(i)  $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$; \(ii) $\rho_{\psi}\prec \rho_{{\varphi}}$. $(i)\Rightarrow(ii)$ Let ${\varepsilon}> 0$ and $\delta > 0$ be such that $2(\delta+\sqrt{\delta}) < \tfrac12{\varepsilon}$. By Corollary \[c\_oneway\], there exists a one-way right LOCC map $\Theta$ relative to ${\mathcal{A}}$ such that $$\label{eq_nor}{\left\Vert {\omega}_{\varphi}-\Theta_*({\omega}_\psi)\right\Vert}<\delta.$$ By Proposition \[p\_comp\](ii), we may write $$\Theta(x) = \sum_{k=1}^\infty \Phi_k(a_k^*xa_k), \ \ x\in{\mathcal{B}(H)},$$ for some $\Phi_k\in \operatorname{UCP}^\sigma_{{\mathcal{A}}}({\mathcal{B}}(H))$ and $a_k\in {\mathcal{A}}$, $k\in{\mathbb{N}}$, with $$\label{eq_akak} \sum_{k=1}^{\infty} a_k^* a_k = 1,$$ where the series converge in the weak\* topology. Let $$\omega_k=\Phi_{k*}(a_k\omega_\psi a_k^*) = a_k\Phi_{k^*}(\omega_\psi) a_k^*\in {\mathcal{T}}(H)^+;$$ then the series $\sum_{k=1}^\infty \omega_k$ is weakly convergent to $\Theta_*(\omega_\psi)$. Let us write $\alpha_k:={{ \left\langle\phi\phi^*,\omega_k\right\rangle}}$, $k\in {\mathbb{N}}$. Since ${{ \left\langle\phi\phi^*,\omega_\phi\right\rangle}}=1$, the bound  implies $$\label{eq_alphak} \sum_{k=1}^\infty \alpha_k={{ \left\langle\phi\phi^*,\Theta_*(\omega_\psi)\right\rangle}}\in (1-\delta,1].$$ By Lemma \[l\_pure\], $\sum_{k=1}^\infty{\left\Vert \omega_k -\alpha_k{\omega}_{\varphi}\right\Vert}<2\sqrt{\delta}+\delta$. Taking restrictions to ${\mathcal{A}}$, and using the fact that $\Phi_k|_{{\mathcal{A}}}$ coincides with the identity map, we obtain $$\label{eq_no1} \sum_{k=1}^\infty{\left\Vert a_k\rho_\psi a_k^*-\alpha_k\rho_{\varphi}\right\Vert}_1 < 2 \sqrt{\delta} + \delta.$$ For each $a\in{\mathcal{A}}$, by we have $$\begin{aligned} \sum_{k=1}^l &{{{ \left\langlea,\rho_\psi^{1/2}a_k^*a_k\rho_\psi^{1/2}\right\rangle}}} = \sum_{k=1}^l \tau(a\rho_\psi^{1/2}a_k^*a_k\rho_\psi^{1/2}) = \sum_{k=1}^l \tau(a_k^*a_k\rho_\psi^{1/2}a\rho_\psi^{1/2})\\ &= \sum_{k=1}^l {{{ \left\langlea_k^*a_k,\rho_\psi^{1/2}a\rho_\psi^{1/2}\right\rangle}}} \to_{l\to\infty} {{{ \left\langle1,\rho_\psi^{1/2}a\rho_\psi^{1/2}\right\rangle}}} ={{{ \left\langlea,\rho_\psi\right\rangle}}}.\end{aligned}$$ Hence, the series $\sum_{k=1}^\infty\rho_\psi^{1/2}a_k^*a_k\rho_\psi^{1/2}$ converges weakly to $\rho_\psi$. By Lemma \[l\_dini\], the convergence is in norm. Choose $L\in {\mathbb{N}}$ so that $$\label{eq_norrho} {\left\Vert \rho_\psi-\sum_{k=1}^L \rho_\psi^{1/2} a_k^*a_k\rho_\psi^{1/2}\right\Vert}_1<\tfrac12\epsilon.$$ By the right polar decomposition, there exists a partial isometry $v_k\in{\mathcal{A}}$ such that $a_k\rho_\psi^{1/2} = (a_k\rho_\psi a_k^*)^{1/2}v_k^*$, $k\in {\mathbb{N}}$. Writing $\alpha_0 = 1 - \sum_{k=1}^\infty\alpha_k$, we have by that $\alpha_0 <\delta$. Setting $v_0 = 1$ and using and , we see that $$\begin{aligned} {\left\Vert \rho_\psi-\sum_{k=0}^L\alpha_kv_k\rho_{\varphi}v_k^*\right\Vert}_1 &< \delta+{\left\Vert \sum_{k=1}^L \rho_\psi^{1/2}a_k^*a_k \rho_\psi^{1/2}-\alpha_kv_k\rho_\phi v_k^*\right\Vert}_1+\tfrac12\epsilon\\ &= \delta+{\left\Vert \sum_{k=1}^L v_k(a_k \rho_\psi a_k^*-\alpha_k\rho_\phi) v_k^*\right\Vert}_1+\tfrac12\epsilon\\ &\le \delta+\sum_{k=1}^L {\left\Vert a_k\rho_\psi a_k^*-\alpha_k\rho_\phi\right\Vert}_1+\tfrac12\epsilon \\ & < 2(\delta+\sqrt \delta)+\tfrac12\epsilon<{\varepsilon}.\end{aligned}$$ Since ${\varepsilon}>0$ was arbitrary, it follows from [@h Theorem 2.5(3)] that $\rho_\psi\prec\rho_{\varphi}$. $(ii)\Rightarrow(i)$ Suppose $\rho_\psi \prec \rho_{\varphi}$, and fix ${\varepsilon}>0$. Pick $\delta>0$ such that $4\sqrt{\delta}<{\varepsilon}$. Since ${{\mathcal{A}}}$ is a factor, by [@h Theorem 2.5], there exist a family $(u_i)_{i=1}^n$ of unitary operators in ${\mathcal{A}}$ and a probability distribution $(p_i)_{i=1}^n$, such that, if $\widetilde{\rho_\psi}=\sum_{i=1}^n p_i u_i\rho_{\varphi}u_i^*$, then $\norm{\rho_\psi-\widetilde{\rho_\psi}}_1 < \delta$. Set $m=n+1$. By Lemma \[l\_M\], there exist $M_1,\dots,M_m\in {\mathcal{A}}$ with $\sum_{i=1}^m M_i^*M_i=1$, such that $$\label{e_M} M_i\widetilde{\rho_\psi}M_i^*= p_i\rho_{\varphi}\text{ for $1\le i\le n$,}{{\quad\text{and}\quad}}M_m\widetilde{\rho_\psi}M_m^*= 0.$$ Let $e_1,\dots,e_m$ be the standard basis of ${\mathbb{C}}^m$, and consider the UCP maps $\Psi,\Phi:{\mathcal{B}}(H{\otimes}{\mathbb{C}}^m)\rightarrow{\mathcal{B}}(H)$ given by $$\Psi(T) = \sum_{i=1}^mM_i^*(\operatorname{id}{\otimes}{\omega}_{e_i})(T)M_i{{\quad\text{and}\quad}}\Phi(T)=\sum_{i=1}^n p_i(\operatorname{id}{\otimes}{\omega}_{e_i})(T)$$ for $T\in {\mathcal{B}}(H{\otimes}{\mathbb{C}}^m)$. We have that $$\label{e_pre} \Psi_*(\rho)=\sum_{i=1}^m M_i\rho M_i^*{\otimes}e_ie_i^*{{\quad\text{and}\quad}}\Phi_*(\rho)=\sum_{i=1}^n p_i\rho{\otimes}e_ie_i^*,$$ for $\rho\in {\mathcal{T}}(H)$. Letting $V,W: H\rightarrow H{\otimes}{\mathbb{C}}^m{\otimes}{\mathbb{C}}^m$ be the isometries given by $$V\eta = \sum_{i=1}^mM_i\eta{\otimes}e_i{\otimes}e_i{{\quad\text{and}\quad}}W\eta=\sum_{i=1}^n \sqrt p_i\eta{\otimes}e_i{\otimes}e_i,\quad\eta\in H,$$ we have Stinespring representations $$\label{st_VW}\Psi(T)=V^*(T{\otimes}1)V {{\quad\text{and}\quad}}\Phi(T) = W^*(T{\otimes}1)W,\quad T\in {\mathcal{B}}(H{\otimes}{\mathbb{C}}^m).$$ Consider the states $\nu_\psi,\nu_\phi\colon {\mathcal{A}}{\otimes}M_m\to {\mathbb{C}}$ given by $$\nu_\psi=\omega_\psi\circ \Psi|_{{\mathcal{A}}{\otimes}M_m}{{\quad\text{and}\quad}}\nu_\phi=\omega_\phi\circ \Phi|_{{\mathcal{A}}{\otimes}M_m}.$$ By  and , we have $$\begin{aligned} {\left\Vert \nu_\psi-\nu_\phi\right\Vert}_{cb}&={\left\Vert \nu_\psi-\nu_\phi\right\Vert}\\ &={\left\Vert (\Psi|_{{\mathcal{A}}{\otimes}M_m})_*(\rho_\psi)-(\Phi|_{{\mathcal{A}}{\otimes}M_m})_*(\rho_{\varphi})\right\Vert}\\ &={\left\Vert \sum_{i=1}^mM_i\rho_\psi M_i^* {\otimes}e_ie_i^*- M_i\widetilde{\rho_\psi} M_i^*{\otimes}e_ie_i^*\right\Vert}\\ &={\left\Vert (\Psi|_{{\mathcal{A}}{\otimes}M_m})_*(\rho_\psi-\widetilde{\rho_\psi})\right\Vert} < \delta.\end{aligned}$$ Let $\theta\colon {\mathcal{A}}\otimes M_m\to {\mathcal{B}} (H\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^m)$ be the $*$-homomorphism given by $\theta(X)=X\otimes 1$, $X\in {\mathcal{A}}\otimes M_m$. By , the maps $\nu_\psi$ and $\nu_\phi$ have Stinespring representations $$\nu_\psi=\omega_{V\psi}\circ \theta{{\quad\text{and}\quad}}\nu_\phi=\omega_{W\phi}\circ \theta.$$ By the continuity of the Stinespring representation [@ksw Theorem 1] there exist a Hilbert space $K$, a $*$-homomorphism $\pi:{\mathcal{A}}{\otimes}M_m\rightarrow {\mathcal{B}}(K)$, and vectors $\eta_1,\eta_2\in K$ yielding Stinespring representations $$\nu_\psi={\omega}_{\eta_1}\circ \pi{{\quad\text{and}\quad}}\nu_\phi={\omega}_{\eta_2}\circ\pi$$ with $$\label{eq_sqrd} {\left\Vert \eta_1-\eta_2\right\Vert} < \sqrt{\delta}.$$ By the uniqueness of Stinespring representations, there exist partial isometries $U_1:H{\otimes}{\mathbb{C}}^m{\otimes}{\mathbb{C}}^m\rightarrow K$ and $U_2:K\rightarrow H{\otimes}{\mathbb{C}}^m{\otimes}{\mathbb{C}}^m$ satisfying $U_1V\psi=\eta_1$, $U_2\eta_2=W{\varphi}$, $$U_1(X{\otimes}1)=\pi(X)U_1 \quad \mbox{ and } \quad U_2\pi(X) = (X{\otimes}1)U_2,\quad X\in {\mathcal{A}}{\otimes}M_m.$$ Let ${\mathcal{B}}={\mathcal{A}}'$. The preceding relations imply that the contraction $U:=U_2U_1$ satisfies $$U\in ({\mathcal{A}}{\otimes}M_m{\otimes}1)'={\mathcal{A}}'{\otimes}1{\otimes}M_m={\mathcal{B}}{\otimes}1{\otimes}M_m;$$ moreover, by , $$\label{eq_uvw} {\left\Vert UV\psi-W{\varphi}\right\Vert} < \sqrt{\delta}.$$ Since $U \in {\mathcal{B}}{\otimes}1{\otimes}M_m$, we have $U = \sum_{k,l=1}^mb_{kl}\otimes 1\otimes e_ke_l^*$ for some $b_{kl}\in{\mathcal{B}}$. Set $b_i = b_{ii}$, $1\leq i\leq m$. Then $b_i$ is a contraction in ${\mathcal{B}}$. Let $\Phi_i\in\operatorname{UCP}_{{\mathcal{A}}}^\sigma({\mathcal{B}(H)})$ be the channel given by $$\Phi_i(x) = b_i^*xb_i + (1 - b_i^*b_i)^{1/2}x(1 - b_i^*b_i)^{1/2}, \ \ \ x\in{\mathcal{B}(H)},$$ and define $\Theta\in\operatorname{LOCC}({\mathcal{A}})$ by $$\Theta(x) = \sum_{i=1}^m \Phi_i(M_i^*x M_i), \ \ \ x\in{\mathcal{B}(H)}.$$ We claim that ${\left\Vert \Theta_*({\omega}_\psi)-{\omega}_{\varphi}\right\Vert} < {\varepsilon}$, which will complete the proof. To see this, let $P : {\mathbb{C}}^m{\otimes}{\mathbb{C}}^m\rightarrow{\mathbb{C}}^m{\otimes}{\mathbb{C}}^m$ denote the orthogonal projection onto $\operatorname{span}\{e_i{\otimes}e_i\mid 1\leq i\leq m\}$, and consider the contraction $$\tilde{U} = (1{\otimes}P)U \in {\mathcal{B}}(H\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^m).$$ A calculation shows that $$\label{tUV} \tilde U V\psi = \sum_{i=1}^m b_i M_i\psi\otimes e_i\otimes e_i.$$ Since $W\phi$ lies in the range of $1{\otimes}P$, the bound  implies $$\label{eq_wdelta} {\left\Vert \tilde UV\psi-W{\varphi}\right\Vert} = {\left\Vert (1{\otimes}P)\left(UV\psi-W{\varphi}\right)\right\Vert} < \sqrt{\delta},$$ and so $$\begin{aligned} \label{eq_anewo} {\left\Vert \omega_{\tilde U V\psi}-\omega_{W\phi}\right\Vert}\le 2{\left\Vert \tilde U V\psi-W\phi\right\Vert}< 2 \sqrt{\delta}.\end{aligned}$$ Observe that for $x\in {\mathcal{B}(H)}$, equation  yields $$\begin{aligned} {{{ \left\langlex,\left(\sum_{i=1}^m\omega_{b_iM_i\psi}\right)-\omega_\phi\right\rangle}}}={{{ \left\langlex{\otimes}1{\otimes}1,\omega_{\tilde U V\psi}-\omega_{W\phi}\right\rangle}}}\end{aligned}$$ so, in particular, using , we have $$\begin{aligned} \label{eq_mipsi} {\left\Vert \sum_{i=1}^m b_iM_i\omega_\psi M_i^*b_i^*-\omega_\phi\right\Vert} &= {\left\Vert \left(\sum_{i=1}^m \omega_{b_iM_i\psi}\right)-\omega_{\phi}\right\Vert} \\&\leq {\left\Vert \omega_{\tilde UV\psi}-\omega_{W\phi}\right\Vert}<2\sqrt\delta. \nonumber\end{aligned}$$ Since $V$ and $W$ are isometries, we have $\|V\psi\|=1=\|W\phi\|$. Using , we thus have $$\|V\psi\| - \|\tilde{U}V\psi\| = \|W{\varphi}\| - \|\tilde{U}V\psi\| < \sqrt{\delta}.$$ By Lemma \[l\_piso\], $$\begin{aligned} \sum_{i=1}^m{\left\Vert (1-b_i^*b_i)^{1/2}M_i\psi\right\Vert}^2 &= \sum_{i=1}^m \|M_i\psi\|^2-\|b_iM_i\psi\|^2 =\|V\psi\|^2-\|\tilde UV\psi\|^2\\ &={\left\Vert (1-\tilde{U}^*\tilde{U})^{1/2}V\psi\right\Vert}^2 < 2\sqrt{\delta}.\end{aligned}$$ Thus, using , we have $$\begin{aligned} & {\left\Vert \Theta_*({\omega}_\psi)-{\omega}_{\varphi}\right\Vert}\\ &= {\left\Vert \sum_{i=1}^mb_iM_i{\omega}_\psi M_i^*b_i^*+(1-b_i^*b_i)^{1/2}M_i{\omega}_\psi M_i^*(1-b_i^*b_i)^{1/2}-{\omega}_{\varphi}\right\Vert}\\ &\leq {\left\Vert \sum_{i=1}^mb_iM_i{\omega}_\psi M_i^*b_i^*-{\omega}_{\varphi}\right\Vert} +\sum_{i=1}^m{\left\Vert (1-b_i^*b_i)^{1/2}M_i{\omega}_\psi M_i^*(1-b_i^*b_i)^{1/2}\right\Vert}\\ &<2\sqrt\delta+\sum_{i=1}^m{\left\Vert (1-b_i^*b_i)^{1/2}M_i\psi\right\Vert}^2 < 4\sqrt{\delta} < {\varepsilon}.\qedhere\end{aligned}$$ The question of convertibility for purely infinite factors becomes trivial. For instance, if ${\mathcal{A}}$ is a factor of type $\mathrm{III}_1$ with separable predual then, by [@t2 Theorem XII.5.12], for all normal states ${\omega}_1,{\omega}_2$ on ${\mathcal{A}}$, we have $$\inf\{\norm{u^*{\omega}_1 u - {\omega}_2}\mid u\in {\mathcal{U}}({\mathcal{A}})\}=0.$$ Hence, given states $\psi,{\varphi}$ in the representation space $H$ of ${\mathcal{A}}$, for every ${\varepsilon}>0$ there exists a unitary $u\in{\mathcal{A}}$ such that $\norm{u^*{\omega}_\psi|_{{\mathcal{A}}} u - {\omega}_{\varphi}|_{{\mathcal{A}}}}<{\varepsilon}$. Appealing to continuity and uniqueness of Stinespring representations (as in Proposition \[p\_lopopescu\]) one can build a channel $\Theta\in\operatorname{LOCC}({\mathcal{A}})$ for which $$\norm{\Theta_*({\omega}_\psi)-{\omega}_{\varphi}} < 2\sqrt{{\varepsilon}}.$$ Hence, $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$ and vice-versa. It is natural to ask if the statement of Theorem \[th\_maj\] holds in the case of general semi-finite von Neumann algebras. Such an extension would require a treatment of integral decompositions of normal completely positive maps, and is left for a further study. Here we only include an illustration involving a typical non-factor case. Let ${\mathcal{D}}$ be a maximal abelian selfadjoint algebra with separable predual, acting on a Hilbert space $H$. We may assume, without loss of generality, that $(X,\mu)$ is a probability measure space such that $H = L^2(X,\mu)$ and ${\mathcal{D}} = \{M_a : a\in L^{\infty}(X,\mu)\}$, where, for $a\in L^{\infty}(X,\mu)$, we have let $M_a\in {\mathcal{B}}(H)$ be the operator of multiplication by $a$. We equip ${\mathcal{D}}$ with the trace $\tau$ given by $\tau(M_a) = \int_{X} a\, d\mu$. Note that, since ${\mathcal{D}} = {\mathcal{D}}'$, we have $\operatorname{LOCC}^r({\mathcal{D}})=\operatorname{LOCC}^l({\mathcal{D}})=\operatorname{LOCC}({\mathcal{D}})$, and these sets consist of all unital positive Schur multipliers relative to $(X,\mu)$, that is, the maps $\Phi : {\mathcal{B}}(H)\to {\mathcal{B}}(H)$ of the form $$\label{eq_schur} \Phi(T) = \sum_{i=1}^{\infty} M_{a_i}^* T M_{a_i}, \ \ \ T\in {\mathcal{B}}(H),$$ where $a_i\in L^\infty(X,\mu)$, $i\in {\mathbb{N}}$ and $$\sum_{i=1}^{\infty} |a_i(s)|^2 = 1 \ \ \mbox{ for almost all } s\in X.$$ Let $\psi,\nph\in H$. We claim that the following are equivalent: - there exists $\Phi\in \operatorname{LOCC}({\mathcal{D}})$ such that $\Phi_*({\omega}_\psi) = {\omega}_\nph$; - $\psi$ is approximately convertible to $\nph$ via $\LOCC({{\mathcal{D}}})$; - $|\psi| = |\nph|$ almost everywhere. Indeed, the implication (i)$\Rightarrow$(ii) is trivial. Assuming (ii), fix $\epsilon > 0$ and let $\Phi\in \operatorname{LOCC}({\mathcal{D}})$ be such that $\|\omega_{\nph} - \omega_{\psi} \circ \Phi\| < \epsilon$. Writing $\Phi$ in the form , we have $$\begin{aligned} \sup_{\|c\|_{\infty} \leq 1} \left| \int_X c (|\nph|^2 - |\psi|^2) \,d\mu \right| & = & \sup_{\|c\|_{\infty} \leq 1} \left| \int_X c (|\nph|^2 - \left(\sum_{i=1}^{\infty}|a_i|^2\right) |\psi|^2) \,d\mu \right|\\ & = & \sup_{\|c\|_{\infty} \leq 1} |{\omega}_\nph(M_c) - {\omega}_\psi(\Phi(M_c))|\\ & \leq & \|{\omega}_\nph - {\omega}_\psi \circ \Phi\| < \epsilon.\end{aligned}$$ Thus, $\||\nph|^2 - |\psi|^2\|_1 < \epsilon$. Hence $|\nph|^2 = |\psi|^2$ in $L^1(X,\mu)$, and (iii) follows. Finally, assuming (iii), let $\theta : X\to{\mathbb{C}}$ be a unimodular function such that $\nph = \theta \psi$, and let $\Phi : {\mathcal{B}}(H) \to {\mathcal{B}}(H)$ be the map given by $\Phi(T) = M_{\theta}^* T M_{\theta}$. Then $\Phi\in \operatorname{LOCC}({\mathcal{D}})$ and $\Phi_*({\omega}_\psi) = {\omega}_\nph$. Trace Vectors and Entanglement in ${\ensuremath{\mathrm{II}_1}}$-factors {#s_ttf} ======================================================================== This section is dedicated to some examples and applications of our convertibility result from Section \[s:main\]. In its first part, we consider a generalisation of maximally entangled vectors to the commuting von Neumann algebra setting, while in its second part we show that entropy of states, relative to the trace, is an entanglement monotone in the sense of [@v]. Trace vectors ------------- Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a finite factor on a Hilbert space. A unit vector $\psi\in H$ is said to be a *trace vector* for ${\mathcal{A}}$ if ${\omega}_{\psi}|_{{\mathcal{A}}}=\tau$, the unique (normal) tracial state on ${\mathcal{A}}$. \[r\_1A\] Since $\omega_{\psi}|_{{{\mathcal{A}}}}(a)=\tau(\rho_\psi a)$ for $a\in {{\mathcal{A}}}$, we see that $\psi$ is a trace vector if and only if $\rho_\psi=1_{{\mathcal{A}}}$. It follows from Nielsen’s theorem [@nielsen] that the maximally entangled state $\psi=\frac{1}{\sqrt{n}}\sum_{i=1}^ne_n{\otimes}e_n\in{\mathbb{C}}^n{\otimes}{\mathbb{C}}^n$ is LOCC-convertible (that is, convertible via $\operatorname{LOCC}(M_n\otimes 1)$) to any other state ${\varphi}\in{\mathbb{C}}^n{\otimes}{\mathbb{C}}^n$. Notice that ${\omega}_\psi|_{M_n{\otimes}1}=\frac{1}{n}\operatorname{tr}$, the normalised trace on $M_n$. Hence, $\psi$ is a trace vector for $M_n{\otimes}1_n\subseteq {\mathcal{B}}({\mathbb{C}}^n{\otimes}{\mathbb{C}}^n)$. The next proposition shows that trace vectors play the role of maximally entangled states relative to ${\ensuremath{\mathrm{II}_1}}$-factors, and provides additional evidence for viewing maximal entanglement through the lens of tracial states [@keylsw §V.A]. \[II1\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a ${\ensuremath{\mathrm{II}_1}}$-factor on a separable Hilbert space $H$. If $\psi\in H$ is a trace vector for ${\mathcal{A}}$, then $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$ for any ${\varphi}\in H_1$. Conversely, if there exists a trace vector $\psi_0\in H$ for ${\mathcal{A}}$, and $\psi\in H_1$ is approximately convertible to any ${\varphi}\in H_1$ via $\operatorname{LOCC}({\mathcal{A}})$, then $\psi$ is a trace vector for ${\mathcal{A}}$. Suppose that $\psi$ is a trace vector for ${\mathcal{A}}$. By Remark \[r\_1A\], $\rho_\psi = 1_{{\mathcal{A}}}$. The map on ${{\mathcal{A}}}$, given by $a\mapsto \tau(a)1_{{\mathcal{A}}}$, is doubly stochastic (i.e., it is positive, normal, unital and trace-preserving) and its extension to $L^1({{\mathcal{A}}},\tau)$ maps $\rho_\phi$ to $1_{{\mathcal{A}}}=\rho_\psi$, since $\tau(\rho_\phi)=\omega_\phi(1_{{\mathcal{A}}})=(\phi,\phi)=1$. It follows from [@h Theorem 4.5] that $\rho_\psi\prec\rho_{\varphi}$, hence, $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$ by Theorem \[th\_maj\]. For the converse statement, suppose $\psi_0\in H_1$ is a trace vector for ${\mathcal{A}}$, and that $\psi\in H_1$ is approximately convertible to every ${\varphi}\in H_1$ via $\operatorname{LOCC}({\mathcal{A}})$. Then $\psi$ is approximately convertible to $\psi_0$ and vice-versa. By Theorem \[th\_maj\] and Remark \[r\_1A\], $\rho_\psi\prec 1_{{\mathcal{A}}}$ and $1_{{\mathcal{A}}} \prec\rho_\psi$, that is, $\rho_\psi$ and $1_{{\mathcal{A}}}$ are spectrally equivalent in the sense of [@h §3]. By [@h Theorem 3.4(2)], for every ${\varepsilon}>0$, there exists a unitary $u\in{\mathcal{A}}$ such that $$\norm{\rho_\psi - 1_{{\mathcal{A}}}}_1 = \norm{\rho_\psi-u\cdot\rho_{\psi_0}\cdot u^*}_1 < {\varepsilon}.$$ Since ${\varepsilon}> 0$ was arbitrary we have $\rho_\psi = 1_{{\mathcal{A}}}$; by Remark \[r\_1A\], $\psi$ is a trace vector for ${\mathcal{A}}$. Amongst ${\ensuremath{\mathrm{II}_1}}$-factors, the hyperfinite (i.e., approximately finite dimensional) ${\ensuremath{\mathrm{II}_1}}$-factor is best suited for applications in mathematical physics. In that context, it typically appears through an infinite tensor product construction, an algebra of canonical commutation/anti-commutation relations, or an irrational rotation algebra. We now present examples of maximally entangled states relative to the hyperfinite ${\ensuremath{\mathrm{II}_1}}$-factor in each of the three aforementioned manifestations. This example is based on [@kmsw §4.2]. Consider an infinite spin chain consisting of infinitely many qubits arranged on a one-dimensional lattice, say ${\mathbb{Z}}$. The underlying $C^*$-algebra of the system is given by the infinite tensor product $A=\bigotimes_{{\mathbb{Z}}} M_2$, that is, the inductive limit of the system $A_F=\bigotimes_{n\in F} M_2$, with canonical inclusion maps, where $F$ ranges through the finite subsets of ${\mathbb{Z}}$. For $n\in{\mathbb{Z}}$, let $\psi_n$ be the maximally entangled state on $A_{\{-n,n+1\}}$. Then ${\omega}=\bigotimes_{n\in{\mathbb{Z}}}\psi_n \psi_n^*$ defines a state on $A$. Let ${\mathcal{A}}=\pi_{\omega}(A_{(-\infty,0)})''\subseteq{\mathcal{B}}(H_{\omega})$ be the von Neumann algebra generated by the left half-chain in the cyclic GNS-representation $(H_{\omega},\pi_{\omega},\psi_{\omega})$ of ${\omega}$. Then ${\mathcal{B}}:={\mathcal{A}}'=\pi_{\omega}(A_{[0,\infty)})''$ is the von Neumann algebra generated by the right half-chain. Both ${\mathcal{A}}$ and ${\mathcal{B}}$ are ${\ensuremath{\mathrm{II}_1}}$-factors, and by construction it follows that $\psi_{\omega}$ is a trace vector for ${\mathcal{A}}$. Thus, by Proposition \[II1\], $\psi_{\omega}$ is approximately convertible to any state ${\varphi}\in H_{\omega}$ via $\operatorname{LOCC}({\mathcal{A}})$. Naturally, one may view $\psi_{\omega}$ as a state representing infinitely many pairs of entangled qubits. \[eg\_fock\] Let $K$ be a real Hilbert space and $H=K\oplus iK$ its complexification. Let ${\mathcal{F}}_a(H)$ denote the anti-symmetric Fock space over $H$, given by $${\mathcal{F}}_a(H)=\bigoplus_{n\geq 0}\wedge^n H,$$ where $\wedge^n H$ is the anti-symmetric subspace of $H^n:=\bigotimes_{k=1}^n H$ for $n\ge1$ and $\wedge^0:={\mathbb{C}}$. For $\psi\in H$, let $a(\psi)^*$ and $a(\psi)$ denote the Fock creation and annihilation operators, namely the bounded [@de] linear maps ${\mathcal{F}}_a(H)\to {\mathcal{F}}_a(H)$ given by $$a(\psi)^*{\varphi}=\sqrt{n+1}P_a^{n+1}(\psi{\otimes}{\varphi}), \ \ a(\psi){\varphi}=\sqrt{n}P_a^n(\psi^*{\otimes}\operatorname{id}){\varphi},$$ where $n\geq 1$, ${\varphi}\in\wedge^n H$ and $P_a^n:H^n\rightarrow\wedge^n H$ is the canonical projection. Let $S\in {{\mathcal{B}}}({\mathcal{F}}_a(H))$ denote the parity operator defined by $S=\bigoplus_{n\geq0}(-1)^{{\otimes}n}$. Letting $B(\psi):=a(\psi)^*+a(\psi)$ represent the corresponding (self-adjoint) Fermionic field operators, it follows that ${\mathcal{A}}:=\{B(\psi)\mid\psi\in K\}''$ is a ${\ensuremath{\mathrm{II}_1}}$-factor associated to a real-wave representation of the canonical anti-commutation relations [@dg §13] whose commutant satisfies ${\mathcal{B}}:={\mathcal{A}}'=\{S B(i\psi)\mid \psi\in K\}''$. It is known that the vacuum vector ${\Omega}=(1,0,0,\ldots)\in{\mathcal{F}}_a(H)$ is a quasi-free trace vector for ${\mathcal{A}}$, with $$\label{e:vac}(B(\psi)B({\varphi}){\Omega},{\Omega})=(\psi,{\varphi}), \ \ \ \psi,{\varphi}\in K.$$ More generally, given an anti-symmetric tensor $c\in H\wedge H$ (seen as a Hilbert-Schmidt operator from $\overline{H}$ to $H$), the Fermionic Gaussian vector associated with $c$ is given by $${\Omega}_c=\det(1+c^*c)^{-\frac{1}{4}}e^{-\frac{1}{2}a^*(c)}{\Omega},$$ where $\det(\cdot)$ is the Fredholm determinant and $a^*(c)$ is the two particle creation operator defined by $$a^*(c)\psi=\sqrt{(n+2)(n+1)}P_a^{n+2}(c{\otimes}\psi), \ \ \ \psi\in\wedge^n H, \ n\geq 1.$$ Such vectors occur in the Hartree–Fock–Bogoliubov method for approximating Fermionic systems (see e.g. [@dmm §4]), which is related to the Bardeen–Cooper–Schrieffer theory of superconductivity [@bcs]. For every $c$ there exists an orthogonal transformation $O_c$ on $K$, and a unitary $U_c$ on ${\mathcal{F}}_a(H)$ satisfying ${\Omega}_c=U_c^*{\Omega}$ and $U_cB(\psi)U_c^*=B(O_c\psi)$, $\psi\in K$ [@dg]. Thus, ${\omega}_{{\Omega}_c}|_{{\mathcal{A}}}={\omega}_{{\Omega}}\circ\operatorname{Ad}(U_c)$, and it follows from that ${\Omega}_c$ is also a trace vector for ${\mathcal{A}}$. Thus, by Proposition \[II1\], any of the Fermionic Gaussian vectors ${\Omega}_c$ may be converted into any Fock state ${\varphi}\in{\mathcal{F}}_a(H)$ by means of local operations and classical communication relative to the real-wave representation ${\mathcal{A}}$ of the CAR. In particular, the vectors $\Omega_c$ display properties of maximal entanglement relative to ${\mathcal{A}}$ and its commutant. We present one more instance of Proposition \[II1\], based on the example from [@bkk2 §7], which in turn was partly motivated by [@Faddeev]. This example is a particular realization of the irrational rotation algebra and is related to discretised CCR relations, whose relevance to numerical analysis of quantum systems was advocated by Arveson [@arv]. Suppose Alice and Bob have access to a quantum system represented by the Hilbert space $L^2({\mathbb{R}})$. Let $q$ and $p$ denote the self-adjoint operators corresponding to position and momentum: $$q\psi(x)=x\psi(x), \ \ \ p\psi(x)=i\frac{d}{dx}\psi(x),$$ where $\psi$ belongs to a common dense domain for $q$ and $p$. Suppose that Alice can measure periodic functions of position and momentum, with periods $t_q$ and $t_p$, respectively. Such functions are given (respectively) by integer powers of the unitary operators $$U:=e^{i\omega_q q} {{\quad\text{and}\quad}}V:=e^{i\omega_p p},$$ where, following [@bkk2], we let ${\omega}_q:=\frac{2\pi}{t_q}$ and ${\omega}_p:=\frac{2\pi}{t_p}$. The operators $U$ and $V$ satisfy $$UV=e^{2\pi i\theta}VU,$$ where $\theta:=\frac{{\omega}_q{\omega}_p}{2\pi}$. In what follows, we assume that ${\omega}_q{\omega}_p>4\pi$ and that $\theta$ is irrational. The algebra describing Alice’s measurement statistics is the von Neumann subalgebra ${\mathcal{A}}$ of ${\mathcal{B}}(L^2({\mathbb{R}}))$ generated by $U$ and $V$, and is known to be a type ${\ensuremath{\mathrm{II}_1}}$-factor. The $C^*$-algebra generated by $U$ and $V$ is known as the irrational rotation algebra corresponding to $\theta$. The von Neumann algebra describing Bob’s measurement statistics, ${\mathcal{B}} = {\mathcal{A}}'$, is generated by $$U':=e^{i\frac{{\omega}_q}{\theta}q}, {{\quad\text{and}\quad}}V':=e^{i\frac{{\omega}_p}{\theta}p},$$ and is also a type ${\ensuremath{\mathrm{II}_1}}$-factor. Let $\psi=\frac{1}{\sqrt{2t_q}}\chi_{[-t_q,t_q]}\in L^2({\mathbb{R}})$. If $x\in[-t_q,t_q]$ and $m\in{\mathbb{Z}}$ then, since ${\omega}_p>\frac{4\pi}{{\omega}_q}=2t_q$, it follows that $x+m{\omega}_p\in[-t_q,t_q]$ if and only if $m=0$. Hence, for all $n,m\in{\mathbb{Z}}$ we have $$\begin{aligned} (U^nV^m\psi,\psi)&=\frac{1}{2t_q}\int_{{\mathbb{R}}}e^{in\frac{2\pi}{t_q} x}\chi_{[-t_q,t_q]}(x+m{\omega}_p)\chi_{[-t_q,t_q]}(x) \ dx\\ &=\delta_{m,0}\frac{1}{2t_q}\int_{-t_q}^{t_q}e^{in\frac{2\pi}{t_q}x} \ dx\\ &=\delta_{m,0}\delta_{n,0}1.\end{aligned}$$ By [@d Corollary VI.1.2], ${\omega}_\psi|_{{\mathcal{A}}}$ is the unique normal tracial state $\tau$ on ${\mathcal{A}}$. By Proposition \[II1\], we have that $\psi$ is approximately convertible to any unit vector ${\varphi}\in L^2({\mathbb{R}})$ via $\operatorname{LOCC}({\mathcal{A}})$. One can think of $\psi$ as representing the state of a particle whose position is uniformly distributed over the interval $[-t_q,t_q]$. This uniformity is playing the role of maximal entanglement relative to the bipartitite system $({\mathcal{A}}, {\mathcal{B}})$. Entanglement monotones ---------------------- The practical importance of quantifying the degree of entanglement present in a given state cannot be overestimated. In the standard finite-dimensional tensor product framework, this quantification is studied through notions of entanglement measures. Since entanglement at its very core is a form of non-local quantum correlation, any reasonable entanglement measure ought to be monotonic with respect to local operations and classical communication. The term entanglement monotone has since emerged for such a measure, and it was argued by Vidal [@v] that monotonicity under LOCC is the *only* natural requirement for measures of entanglement. As an application of our main result, we show that the entropy of the singular value distribution satisfies this requirement for pure states relative to ${\ensuremath{\mathrm{II}_1}}$-factors, thus yielding an entanglement monotone. Let ${\mathcal{A}}$ be a ${\ensuremath{\mathrm{II}_1}}$-factor with unique tracial state $\tau$. Given a normal state $\rho\in{\mathcal{S}}({\mathcal{A}})$, we define the *entropy of $\rho$ relative to $\tau$* by $$H_\tau(\rho):=H(\mu(\rho))=-\int_0^1\mu_t(\rho)\log(\mu_t(\rho)) \ dt.$$ By splitting the entropy function $\eta(x)=-x\log(x)$ into $\chi_{[0,1]}\eta+\chi_{(1,\infty)}\eta$, and applying [@fk Remark 3.3] to the non-negative Borel functions $\chi_{[0,1]}\eta$ and $-\chi_{(1,\infty)}\eta$, it follows that $$H_\tau(\rho)=\tau(\eta(\rho))=-\tau(\rho\log(\rho))=-S(\rho,\tau),$$ whenever $|H_\tau(\rho)|<\infty$, where $S(\cdot,\cdot)$ is the relative entropy between normal states of ${\mathcal{A}}$ (see e.g. [@op §5]). As such, we see that $H_\tau(\rho)\leq 0$ and $H_\tau(\rho)=0$ if and only if $\rho=\tau$. In particular, if ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ and $\psi\in H_1$, then $H_\tau(\rho_\psi)=0$ if and only if $\psi$ is a trace vector for ${\mathcal{A}}$. \[p:monotone\] Let ${\mathcal{A}}\subseteq{\mathcal{B}(H)}$ be a ${\ensuremath{\mathrm{II}_1}}$-factor with tracial state $\tau$. The function $$H_\tau:{\mathcal{S}}({\mathcal{A}})\ni \rho\mapsto H_\tau(\rho)\in[-\infty,0]$$ is non-increasing under approximate convertibility by $\operatorname{LOCC}({\mathcal{A}})$, when restricted to states of the form $\rho_\psi$, $\psi\in H_1$. First note that for any state $\rho\in{\mathcal{S}}({\mathcal{A}})$, $$H_\tau(\rho)=-S(\rho,\tau)=-S(\mu(\rho),\chi_{[0,1]}),$$ where $S(\mu(\rho),\chi_{[0,1]})$ is the relative entropy of the density $\mu(\rho)\colon t\mapsto \mu_t(\rho)$ on $[0,1]$ with respect to the uniform distribution. Given $\psi,{\varphi}\in H_1$, if $\psi$ is approximately convertible to ${\varphi}$ via $\operatorname{LOCC}({\mathcal{A}})$ then by Theorem \[th\_maj\] we have $\rho_{\psi}\prec \rho_{{\varphi}}$, meaning that $\mu_t(\rho_\psi)\prec\mu_t(\rho_{\varphi})$ as probability densities on $[0,1]$. By [@veh Theorem 10], it follows that $$S(\mu_t(\rho_\psi),\chi_{[0,1]})\leq S(\mu_t(\rho_{\varphi}),\chi_{[0,1]}).$$ Hence, $H_\tau(\rho_\psi)\geq H_\tau(\rho_{\varphi})$. In the proof of Proposition \[p:monotone\] we could instead appeal to [@h Theorem 4.7(1)] for the connection between majorisation and double stochasticity together with monotonicity of the relative entropy [@op Theorem 5.3]. Recall that any density matrix $\rho\in M_n$ satisfies $$S(\rho)=-S(\rho,\tau)+\log(n),$$ where $S(\cdot)$ is the von Neumann entropy and $\tau=\frac{1}{n}\operatorname{tr}$ is the maximally mixed state (the $\log(n)$ factor would disappear if we used the unnormalised trace $\operatorname{tr}$). It is known that the restriction of any entanglement monotone to pure states $\psi\in {\mathbb{C}}^n{\otimes}{\mathbb{C}}^n$ is a concave function of the reduced density $\rho_\psi=(\operatorname{id}{\otimes}\operatorname{tr})(|\psi{\rangle}{\langle}\psi|)$ [@v Theorem 3]. The entanglement monotone $-S(\rho_\psi,\tau)$ is equivalent (up to the translational factor $\log(n)$) to the common choice of $S(\rho_\psi)$, and is the finite-dimensional analogue of our proposed monotone above. Note that $-S(\rho_\psi,\tau)\in[-\log(n),0]$, with the largest value of 0 occurring for maximally entangled $\psi$, and the lowest value of $-\log(n)$ occurring for separable $\psi$. Outlook ======= Several natural lines of investigation arise from this work. First, we intend to study the generalisation of our main result to the non-factor setting in connection with [@h2], as mentioned at the end of Section \[s:main\]. This could be useful for the study of entanglement in hybrid systems [@kuper; @DevShor; @bkk0; @bkk1]. Second, a rectangular version of our main theorem, describing convertibility between states $\psi\in H$ and ${\varphi}\in K$, with respect to distinct bipartite systems $({\mathcal{A}}_1,{\mathcal{B}}_1)$ in ${\mathcal{B}(H)}$ and $({\mathcal{A}}_2,{\mathcal{B}}_2)$ in ${\mathcal{B}}(K)$, would be desirable. Among other things, this could have applications to the structure of quantum correlation matrices and values of certain non-local games [@psstw]. We also plan to explore notions of distillability/dilution of entanglement for general bipartite systems in connection with this and previous work [@vw; @kmsw]. In that direction it would be interesting to explore uniqueness of entanglement monotones in the asymptotic regime for pure states, analogous to the finite-dimensional setting [@pr]. Finally, we observe that while many of the von Neumann algebras of interest in algebraic quantum field theory are of type III [@haag], and hence not semi-finite, it is conceivable that semi-finite von Neumann algebras may be relevant in the discretisation of space-time via tensor networks—a current area of research—analogous to those arising from discretisations of the canonical commutation relations [@slawny]. [*Acknowledgements.*]{} J.C. was partially supported by the NSERC Discovery Grant RGPIN-2017-06275. D.W.K. was partly supported by NSERC Discovery Grant 400160 and by a Royal Society grant allowing research visits to Queen’s University Belfast. R.H.L. was partly supported by UCD Seed Funding. [99]{} , [*Discretized CCR algebras*]{}, [J. Operator Thy. 26 (1991), no. 2, 225–239]{}. , [*Theory of superconductivity*]{}, [Phys. Rev. 108 (1957), 1175]{}. , [*Generalization of quantum error correction via the Heisenberg picture*]{}, [Phys. Rev. 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ArXiv

--- abstract: 'The Bethe-Salpeter equation for ground state of two fermions exchanging a gauge boson presents divergences in the momentum transverse, even in the ladder aproximation projected in light-front. Gauge theories with light-front gauge also present the difficulty associated to the instantaneous term of the propagator of a system composed by fermions bosons-exchange interaction. We used a prescription that allowed an apropriate description of the singularity in the propagator of the gauge boson in the light-front.' author: - | B.M.Pimentel$^{a}$, J.H.O.Sales$^{a}$ and Tobias Frederico$^{b}$\ $^{a}$Instituto de Física Teórica-UNESP, 01405-900 São Paulo, Brazil.\ $^{b}$Instituto Tecnológico de Aeronaútica, CTA, 12228-900\ São José dos Campos, Brazil. title: '**Gauge field divergences in the light-front**' --- Light-Front Dynamics: Definition ================================ Beginning from Dirac’s idea [@dirac] of representing the dynamics of the quantum system at ligth-front times $x^{+}=t+z$, we derive the Green’s function from the covariant propagator that evolutes the system from one light-front hyper-surface to another one. The light-front Green’s function is the probability amplitude for an initial state at $x^{+}=0$ do evolvey to a final state in the Fock-state at some $x^{+}$, where the evolution operator is defined by the light-front Hamiltonian [@Kogut] The Scalar Field Propagator =========================== The Feynman propagator for the scalar field is $$S(x^{\mu })=\int \frac{d^{4}k}{\left( 2\pi \right) ^{4}}\frac{ie^{-ik^{\mu }x_{\mu }}}{k^{2}-m^{2}+i\varepsilon }. \label{1}$$ and in terms of light-front variables [@jhs2002], we have $$S(x^{+})=\frac{1}{2}\int \frac{dk^{-}dk^{+}dk^{\perp }}{\left( 2\pi \right) }\frac{ie^{\frac{-i}{2}k^{-}x^{+}}}{k^{+}\left( k^{-}-\frac{k_{\perp }^{2}+m^{2}-i\varepsilon }{k^{+}}\right) }. \label{2}$$ The Fourier transform of the single boson state propagator to the in the light-front time is giver by: $$\widetilde{S}(k^{-})=\int dk^{+}dk^{\perp }\frac{i}{k^{+}\left( k^{-}-\frac{k_{\perp }^{2}+m^{2}-i\varepsilon }{k^{+}}\right) }. \label{3}$$ Fermion Field ============= Let $S_{\text{F}}$ denote fermion field propagator in covariant theory $$S_{\text{F}}(x^{\mu })=\int \frac{d^{4}k}{\left( 2\pi \right) ^{4}}\frac{i(\rlap\slash k_{\text{on}}+m)}{k^{2}-m^{2}+i\varepsilon }e^{-ik^{\mu }x_{\mu }}, \label{4}$$ where $\rlap\slash k_{\text{on}}=\frac{1}{2}\gamma ^{+}\frac{(k^{\perp })^{2}+m^{2}}{k^{+2}}+\frac{1}{2}\gamma ^{-}k^{+}-\gamma ^{\perp }k^{\perp }$. Using light-front variables in Eq.(\[4\]), we have $$S_{\text{F}}(x^{+})=\frac{i}{2}\int \frac{dk^{-}dk^{+}dk^{\perp }}{\left( 2\pi \right) }\left[ \frac{\rlap\slash k_{on}+m}{k^{+}\left( k^{-}-k_{on}^{-}+\frac{i\varepsilon }{k^{+}}\right) }+\frac{\gamma ^{+}}{2k^{+}}\right] e^{\frac{-i}{2}k^{-}x^{+}}. \label{5}$$ We note that for the fermion field, light-front propagator differs from the Feynmam propagator by an instantaneous propagator. Gauge Boson Propagator ====================== Let $S^{\mu \nu }$gauge propagator, $$S^{\mu \nu }(x^{\mu })=\int \frac{d^{4}k}{\left( 2\pi \right) ^{4}}\frac{ie^{-ik^{\mu }x_{\mu }}}{k^{2}+i\varepsilon }\left[ \frac{-nkg^{\mu \nu }+n^{\mu }k^{\nu }+n^{\nu }k^{\mu }}{nk}\right] , \label{6}$$ where we choose the light-front gauge $A^{+}=0$, $n^{\mu }=(1,0,0,-1)$ and the metric tensor is given from [@Kogut]$.$ The light-front components (\[6\]) can be as written $S^{+-}=S^{-+}=S^{++}=S^{+\perp }=0$ and $$S^{--}=4\frac{ik^{-}}{k^{+}(k^{2}+i\varepsilon )},\text{ }S^{-\perp }=S^{\perp -}=2\frac{ik^{\perp }}{k^{+}(k^{2}+i\varepsilon )},\text{ }S^{\perp \perp }=-1\frac{i}{k^{2}+i\varepsilon } \label{7a}$$ Interaction in First Order ========================== We consider the fermion-antifermion system in the light-front with one-gauge boson exchange ($A^{+}=0$), for which the interaction Lagrangian density is given by $$\mathcal{L}_{I}=g\overline{\Psi }_{1}\gamma _{\mu }A^{\mu }\Psi _{1}+g\overline{\Psi }_{2}\gamma _{\nu }A^{\nu }\Psi _{2}. \label{8}$$ The fermion corresponds to the field $\Psi $ with rest masses $m$ and the exchanged gauge boson to the field $A^{\mu }$ with mass $\mu =0.$ The coupling constant is $g.$ The perturbative correction to the two-body propagator which comes from the  exchange of one intermediate virtual boson, is $$\begin{aligned} \Delta S_{g^{2}}(x^{+}) &=&\left( ig\right) ^{2}\int d\overline{x}_{1}^{+}d\overline{x}_{2}^{+}S_{k^{\prime }}(x^{+}-\overline{x}_{1}^{+})(\gamma _{\mu })S_{k}(\overline{x}_{1}^{+}) \label{9} \\ &&S^{\mu \nu }(\overline{x}_{2}^{+}-\overline{x}_{1}^{+})S_{p}(x^{+}-\overline{x}_{2})(\gamma _{\nu })S_{p^{\prime }}(\overline{x}_{2}^{+}). \notag\end{aligned}$$ The intermediate boson propagates between the time interval $\overline{x}_{2}^{+}-\overline{x}_{1}^{+}.$ The labels in the particle propagators $k$ and $p$ indicates initial and $k^{\prime }$ and $p^{\prime }$ final states. Performing the Fourier transform from $x^{+}$ to $P^{-}$ and for the total kinematical momentum $P^{+}$, which we choose positive, and $P^{\perp }$. The double integration in $k^{-}$ is performed analytically in Eq.(\[10a\]), $$\begin{aligned} \Delta S_{g^{2}}(P^{-}) &=&\frac{-\left( ig\right) ^{2}i}{(4\pi )^{2}}\int \frac{dk^{-}dk^{\prime ^{-}}}{k^{+}k^{\prime ^{+}}(P^{+}-k^{\prime +})(P^{+}-k^{+})} \\ &&\left\{ \frac{\rlap\slash k_{on}^{\prime }+m}{\left( k^{\prime -}-k_{on}^{\prime -}+\frac{i\varepsilon }{k^{\prime +}}\right) }\right. \begin{array}{c} \gamma _{-} \end{array} \frac{\rlap\slash k_{on}+m}{\left( k^{-}-k_{on}^{-}+\frac{i\varepsilon }{k^{+}}\right) } \\ &&\frac{4\left( k^{-}-k^{\prime -}\right) }{(q^{+})^{2}\left( k^{-}-k^{\prime -}-q_{on}^{-}+\frac{i\varepsilon }{q^{+}}\right) }\frac{\rlap\slash p_{on}^{\prime }+m}{\left( p^{\prime -}-p_{on}^{\prime -}+\frac{i\varepsilon }{p^{\prime +}}\right) } \\ && \begin{array}{c} \gamma _{-} \end{array} \frac{\rlap\slash p_{on}+m}{\left( p^{-}-p_{on}^{-}+\frac{i\varepsilon }{p^{+}}\right) }+\end{aligned}$$ $$\begin{aligned} &&+\frac{\rlap\slash k_{on}^{\prime }+m}{\left( k^{\prime -}-k_{on}^{\prime -}+\frac{i\varepsilon }{k^{\prime +}}\right) } \notag \\ && \begin{array}{c} \gamma _{-} \end{array} \frac{\rlap\slash k_{on}+m}{\left( k^{-}-k_{on}^{-}+\frac{i\varepsilon }{k^{+}}\right) }\frac{2\left( k^{\perp }-k^{\prime \perp }\right) }{(q^{+})^{2}\left( k^{-}-k^{\prime -}-q_{on}^{-}+\frac{i\varepsilon }{q^{+}}\right) } \notag \\ &&\frac{\rlap\slash p_{on}^{\prime }+m}{\left( p^{\prime -}-p_{on}^{\prime -}+\frac{i\varepsilon }{p^{\prime +}}\right) } \begin{array}{c} \gamma _{\perp } \end{array} \frac{\rlap\slash p_{on}+m}{\left( p^{-}-p_{on}^{-}+\frac{i\varepsilon }{p^{+}}\right) }+\left[ \gamma _{\perp }\rightarrow \gamma _{-}\right] + \notag \\ &&\frac{\rlap\slash k_{on}^{\prime }+m}{\left( k^{\prime -}-k_{on}^{\prime -}+\frac{i\varepsilon }{p^{\prime +}}\right) } \begin{array}{c} \gamma _{\perp } \end{array} \frac{\rlap\slash k_{on}+m}{\left( k^{-}-k_{on}^{-}+\frac{i\varepsilon }{p^{+}}\right) } \notag \\ &&\frac{(-1)}{(q^{+})^{2}\left( k^{-}-k^{\prime -}-q_{on}^{-}+\frac{i\varepsilon }{q^{+}}\right) } \label{10a} \\ &&\frac{\rlap\slash p_{on}^{\prime }+m}{\left( p^{\prime -}-p_{on}^{\prime -}+\frac{i\varepsilon }{p^{\prime +}}\right) } \begin{array}{c} \gamma _{\perp } \end{array} \left. \frac{\rlap\slash p_{on}+m}{\left( p^{-}-p_{on}^{-}+\frac{i\varepsilon }{p^{+}}\right) }\right\} , \notag\end{aligned}$$ Conclusion ========== From equation (\[10a\]), we verified the existence of singularity in the components $(--)$ and $(-\perp )$ in the coordinate $q^{+}=k^{+}-k^{\prime +}.$ We hoped to remove those singularity using the technique of displacement $\left( \delta ^{+}\right) $ of the pole of the phase space in $q^{+}$ [@jhs97]. T.F and B.M. Pimentel thank to CNPq for partial support. J.H.O. Sales is supported by FAPESP/Brazil pos-doctoral fellowship. We acknowledge discussions with J.F.Libonati and J.Messias. [9]{} P. A. M. Dirac, Rev. Mod. Phys. **21**, 392 (1949). A.Harindranath, *Light-Front Quantization and Non-Perturbative QCD*, editors J.P.Vary and F.Wölz, International Institute of Theoretical and Applied Physics (1997). S.D.Glasek and K.G.Wilson Phys.Rev.**D49** (1994) 4214. J.H.O. Sales, T.Frederico, B.V. Carlson and P.U.Sauer, Phys. Rev. **C63**:064003(2001). J.H.O. Sales, T.Frederico, B.V. Carlson and P.U.Sauer, Phys. Rev. **C61**:044003(2000). J.P.B.C.de Melo, J.H.O.Sales, T.Frederico and P.U.Sauer, Nucl. Phys.**A631** (1998) 574c.

ArXiv

--- abstract: 'We here discuss the emergence of Quasi Stationary States (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian Mean Field (HMF) model, numerical simulations are performed based on both the original $N$-body setting and the continuum Vlasov model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously demonstrates that the Vlasov-wave system provides the correct framework to address the study of QSS. Further, analytical calculations based on Lynden-Bell’s theory of violent relaxation are shown to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical solutions are shown to be affected by small scale fluctuations, a finding that points to the need for novel schemes able to account for particles correlations.' author: - | Andrea Antoniazzi$^{1}$[^1], Francesco Califano$^ {2}$[^2], Duccio Fanelli$^{1,3}$[^3], Stefano Ruffo$^{1}$[^4] title: 'Exploring the thermodynamic limit of Hamiltonian models: convergence to the Vlasov equation.' --- The Vlasov equation constitutes a universal theoretical framework and plays a role of paramount importance in many branches of applied and fundamental physics. Structure formation in the universe is for instance a rich and fascinating problem of classical physics: The fossile radiation that permeates the cosmos is a relic of microfluctuation in the matter created by the Big Bang, and such a small perturbation is believed to have evolved via gravitational instability to the pronounced agglomerations that we see nowdays on the galaxy cluster scale. Within this scenario, gravity is hence the engine of growth and the Vlasov equation governs the dynamics of the non baryonic “dark matter" [@peebles]. Furthermore, the continuous Vlasov description is the reference model for several space and laboratory plasma applications, including many interesting regimes, among which the interpretation of coherent electrostatic structures observed in plasmas far from thermodynamic equilibrium. The Vlasov equation is obtained as the mean–field limit of the $N$–body Liouville equation, assuming that each particle interacts with an average field generated by all plasma particles (i.e. the mean electromagnetic field determined by the Poisson or Maxwell equations where the charge and current densities are calculated from the particle distribution function) while inter–particle correlations are completely neglected. Numerical simulations are presently one of the most powerful resource to address the study of the Vlasov equation. In the plasma context, the Lagrangian Particle-In-Cell approach is by far the most popular, while Eulerian Vlasov codes are particularly suited for analyzing specific model problems, due to the associated low noise level which is secured even in the non–linear regime [@mangeney]. However, any numerical scheme designed to integrate the continuous Vlasov system involves a discretization over a finite mesh. This is indeed an unavoidable step which in turn affects numerical accuracy. A numerical (diffusive and dispersive) characteristic length is in fact introduced being at best comparable with the grid mesh size: as soon as the latter matches the typical length scale relative to the (dynamically generated) fluctuations a violation of the continuous Hamiltonian character of the equations occurs (see Refs. [@califano]). It is important to emphasize that even if such [*non Vlasov*]{} effects are strongly localized (in phase space), the induced large scale topological changes will eventually affect the system globally. Therefore, aiming at clarifying the problem of the validity of Vlasov numerical models, it is crucial to compare a continuous Vlasov, but numerically discretized, approach to a homologous N-body model. Vlasov equation has been also invoked as a reference model in many interesting one dimensional problems, and recurrently applied to the study of wave-particles interacting systems. The Hamiltonian Mean Field (HMF) model [@antoni-95], describing the coupled motion of $N$ rotators, is in particular assimilated to a Vlasov dynamics in the thermodynamic limit on the basis of rigorous results [@BraunHepp]. The HMF model has been historically introduced as representing gravitational and charged sheet models and is quite extensively analyzed as a paradigmatic representative of the broader class of systems with long-range interactions [@Houches02]. A peculiar feature of the HMF model, shared also by other long-range interacting systems, is the presence of [*Quasi Stationary States*]{} (QSS). During time evolution, the system gets trapped in such states, which are characterized by non Gaussian velocity distributions, before relaxing to the final Boltzmann-Gibbs equilibrium [@ruffo_rapisarda]. An attempt has been made [@rapisarda_tsallis] to interpret the emergence of QSSs by invoking Tsallis statistics [@Tsallis]. This approach has been later on criticized in [@Yamaguchi], where QSSs were shown to correspond to stationary stable solutions of the Vlasov equation, for a particular choice of the initial condition. More recently, an approximate analytical theory, based on the Vlasov equation, which derives the QSSs of the HMF model using a maximum entropy principle, was developed in  [@antoniazziPRL]. This theory is inspired by the pioneering work of Lynden-Bell  [@LyndenBell68] and relies on previous work on 2D turbulence by Chavanis [@chava2D]. However, the underlying Vlasov ansatz has not been directly examined and it is recently being debated [@EPN]. In this Letter, we shall discuss numerical simulations of the continuous Vlasov model, the kinetic counterpart of the discrete HMF model. By comparing these results to both direct N-body simulations and analytical predictions, we shall reach the following conclusions: (i) the Vlasov formulation is indeed ruling the dynamics of the QSS; (ii) the proposed analytical treatment of the Vlasov equation is surprisingly accurate, despite the approximations involved in the derivation; (iii) Vlasov simulations are to be handled with extreme caution when exploring specific regions of the parameters space. The HMF model is characterized by the following Hamiltonian $$\label{eq:ham} H = \frac{1}{2} \sum_{j=1}^N p_j^2 + \frac{1}{2 N} \sum_{i,j=1}^N \left[1 - \cos(\theta_j-\theta_i) \right]$$ where $\theta_j$ represents the orientation of the $j$-th rotor and $p_j$ is its conjugate momentum. To monitor the evolution of the system, it is customary to introduce the magnetization, a macroscopic order parameter defined as $M=|{\mathbf M}|=|\sum {\mathbf m_i}| /N$, where ${\mathbf m_i}=(\cos \theta_i,\sin \theta_i)$ stands for the microscopic magnetization vector. As previously reported [@antoni-95], after an initial transient, the system gets trapped into Quasi-Stationary States (QSSs), i.e. non-equilibrium dynamical regimes whose lifetime diverges when increasing the number of particles $N$. Importantly, when performing the mean-field limit ($N \rightarrow \infty$) [*before*]{} the infinite time limit, the system cannot relax towards Boltzmann–Gibbs equilibrium and remains permanently confined in the intermediate QSSs. As mentioned above, this phenomenology is widely observed for systems with long-range interactions, including galaxy dynamics [@Padmanabhan], free electron lasers [@Barre], 2D electron plasmas [@kawahara]. In the $N \to \infty$ limit the discrete HMF dynamics reduces to the Vlasov equation $$\partial f / \partial t + p \, \partial f / \partial \theta \,\, - (dV / d \theta ) \, \partial f / \partial p = 0 \, , %\frac{\partial f}{\partial t} + p\frac{\partial f}{\partial \theta} - %\frac{d V}{d \theta} \frac{\partial f}{\partial p}=0\quad , \label{eq:VlasovHMF}$$ where $f(\theta,p,t)$ is the microscopic one-particle distribution function and $$\begin{aligned} V(\theta)[f] &=& 1 - M_x[f] \cos(\theta) - M_y[f] \sin(\theta) ~, \\ M_x[f] &=& \int_{-\pi}^{\pi} \int_{-\infty}^{\infty} f(\theta,p,t) \, \cos{\theta} {\mathrm d}\theta {\mathrm d}p\quad , \\ M_y[f] &=& \int_{-\pi}^{\pi} \int_{\infty}^{\infty} f(\theta,p,t) \, \sin{\theta}{\mathrm d}\theta {\mathrm d}p\quad . \label{eq:pot_magn}\end{aligned}$$ The specific energy $h[f]=\int \int (p^2/{2}) f(\theta,p,t) {\mathrm d}\theta {\mathrm d}p - ({M_x^2+M_y^2 - 1})/{2}$ and momentum $P[f]=\int \int p f(\theta,p,t) {\mathrm d}\theta {\mathrm d}p$ functionals are conserved quantities. Homogeneous states are characterized by $M=0$, while non-homogeneous states correspond to $M \ne 0$. Rigorous mathematical results [@BraunHepp] demonstrate that, indeed, the Vlasov framework applies in the continuum description of mean-field type models. This observation corroborates the claim that any theoretical attempt to characterize the QSSs should resort to the above Vlasov based interpretative picture. Despite this, the QSS non-Gaussian velocity distributions have been [*fitted*]{} [@rapisarda_tsallis] using Tsallis’ $q$–exponentials, and the Vlasov formalism assumed valid [*only*]{} for the limiting case of homogeneous initial conditions [@EPN]. In a recent paper [@antoniazziPRL], the aforementioned velocity distribution functions were instead reproduced with an analytical expression derived from the Vlasov scenario, with no adjustable parameters and for a large class of initial conditions, including inhomogeneous ones. The key idea dates back to the seminal work by Lynden-Bell [@LyndenBell68] (see also [@Chavanis06], [@Michel94]) and consists in coarse-graining the microscopic one-particle distribution function $f(\theta,p,t)$ by introducing a local average in phase space. It is then possible to associate an entropy to the coarse-grained distribution $\bar{f}$: The corresponding statistical equilibrium is hence determined by maximizing such an entropy, while imposing the conservation of the Vlasov dynamical invariants, namely energy, momentum and norm of the distribution. We shall here limit our discussion to the case of an initial single particle distribution which takes only two distinct values: $f_0=1/(4 \Delta_{\theta} \Delta_{p})$, if the angles (velocities) lie within an interval centered around zero and of half-width $\Delta_{\theta}$ ($\Delta_{p}$), and zero otherwise. This choice corresponds to the so-called “water-bag" distribution which is fully specified by energy $h[f]=e$, momentum $P[f]=\sigma$ and the initial magnetization ${\mathbf M_0}=(M_{x0}, M_{y0})$. The maximum entropy calculation is then performed analytically [@antoniazziPRL] and results in the following form of the QSS distribution $$\label{eq:barf} \bar{f}(\theta,p)= f_0\frac{e^{-\beta (p^2/2 - M_y[\bar{f}]\sin\theta - M_x[\bar{f}]\cos\theta)-\lambda p-\mu}} {1+e^{-\beta (p^2/2 - M_y[\bar{f}]\sin\theta - M_x[\bar{f}]\cos\theta)-\lambda p-\mu}}$$ where $\beta/f_0$, $\lambda/f_0$ and $\mu/f_0$ are rescaled Lagrange multipliers, respectively associated to the energy, momentum and normalization. Inserting expression (\[eq:barf\]) into the above constraints and recalling the definition of $M_x[\bar{f}]$, $M_y[\bar{f}]$, one obtains an implicit system which can be solved numerically to determine the Lagrange multipliers and the expected magnetization in the QSS. Note that the distribution (\[eq:barf\]) differs from the usual Boltzmann-Gibbs expression because of the “fermionic” denominator. Numerically computed velocity distributions have been compared in [@antoniazziPRL] to the above theoretical predictions (where no free parameter is used), obtaining an overall good agreement. However, the central part of the distributions is modulated by the presence of two symmetric bumps, which are the signature of a collective dynamical phenomenon [@antoniazziPRL]. The presence of these bumps is not explained by our theory. Such discrepancies has been recently claimed to be an indirect proof of the fact that the Vlasov model holds only approximately true. We shall here demonstrate that this claim is not correct and that the deviations between theory and numerical observation are uniquely due to the approximations built in the Lynden-Bell approach. A detailed analysis of the Lynden-Bell equilibrium (\[eq:barf\]) in the parameter plane $(M_{0},e)$ enabled us to unravel a rich phenomenology, including out of equilibrium phase transitions between homogeneous ($M_{QSS}=0$) and non-homogeneous ($M_{QSS} \ne 0$) QSS states. Second and first order transition lines are found that separate homogeneous and non homogeneous states and merge into a tricritical point approximately located in $(M_{0},e)=(0.2,0.61)$. When the transition is second order two extrema of the Lynden-Bell entropy are identified in the inhomogeneous phase: the solution $M=0$ corresponds to a saddle point, being therefore unstable; the global maximum is instead associated to $M \neq 0$, which represents the equilibrium predicted by the theory. This argument is important for what will be discussed in the following. Let us now turn to direct simulations, with the aim of testing the above scenario, and focus first on the kinetic model (\[eq:VlasovHMF\])–(\[eq:pot\_magn\]). The algorithm solves the Vlasov equation in phase space and uses the so-called “splitting scheme", a widely adopted strategy in numerical fluid dynamics. Such a scheme, pioneered by Cheng and Knorr [@Cheng], was first applied to the study of the Vlasov-Poisson equations in the electrostatic limit and then employed for a wide spectrum of problems [@califano]. For different values of the pair $(M_{0},e)$, which sets the widths of the initial water-bag profile, we performed a direct integration of the Vlasov system (\[eq:VlasovHMF\])–(\[eq:pot\_magn\]). After a transient, magnetization is shown to eventually attain a constant value, which corresponds to the QSS value observed in the HMF, discrete, framework. The asymptotic magnetizations are hence recorded when varying the initial condition. Results (stars) are reported in figure \[fig1\](a) where $M_{QSS}$ is plotted as function of $e$. A comparison is drawn with the predictions of our theory (solid line) and with the outcome of N-body simulation (squares) based on the Hamiltonian (\[eq:ham\]), finding an excellent agreement. This observation enables us to conclude that (i) the Vlasov equation governs the HMF dynamics for $N \to \infty$ [*both*]{} in the homogeneous and non homogeneous case; (ii) Lynden-Bell’s violent relaxation theory allows for reliable predictions, including the transition from magnetized to non-magnetized states. Deviations from the theory are detected near the transition. This fact has a natural explaination and raises a number of fundamental questions related to the use of Vlasov simulations. As confirmed by the inspection of figure \[fig1\](b), close to the transition point, the entropy $S$ of the Lynden-Bell coarse-grained distribution takes almost the same value when evaluated on the global maximum (solid line) or on the saddle point (dashed line). The entropy is hence substantially flat in this region, which in turn implies that there exists an extended basin of states accessible to the system. This interpretation is further validated by the inset of figure \[fig1\](a), where we show the probability distribution of $M_{QSS}$ computed via N-body simulation. The bell-shaped profile presents a clear peak, approximately close to the value predicted by our theory. Quite remarkably, the system can converge to final magnetizations which are sensibly different from the expected value. Simulations based on the Vlasov code running at different resolutions (grid points) confirmed this scenario, highlighting a similar degree of variability. These findings point to the fact that in specific regions of the parameter space, Vlasov numerics needs to be carefully analyzed (see also Ref. [@Elskens]). Importantly, it is becoming nowadays crucial to step towards an “extended«« Vlasov theoretical model which enables to account for discreteness effects, by incorporating at least two particles correlations interaction term. ![Panel (a): The magnetization in the QSS is plotted as function of energy, $e$, at $M_0=0.24$. The solid line refers to the Lynden-Bell inspired theory. Stars (resp. squares) stand for Vlasov (resp. N-body) simulations. Inset: Probability distribution of $M_{QSS}$ computed via N-body simulation (the solid line is a Gaussian fit). Panel (b): Entropy $S$ at the stationary points, as function of energy, $e$: magnetized solution (solid line) and non–magnetized one (dashed line).[]{data-label="fig1"}](fig1.eps){width="7cm"} ![Phase space snapshots for $(M_{0},e)=(0.5,0.69)$.[]{data-label="fig2"}](fig2.ps){width="7cm"} Qualitatively, one can track the evolution of the system in phase space, both for the homogeneous and non homogeneous cases. Results of the Vlasov integration are displayed in figure \[fig2\] for $(M_{0},e)=(0.5,0.69)$, where the system is shown to evolve towards a non magnetized QSS. The initial water-bag distribution splits into two large resonances, which persist asymptotically: the latter acquire constant opposite velocities which are maintained during the subsequent time evolution, in agreement with the findings of [@antoniazziPRL]. The two bumps are therefore an emergent property of the model, which is correctly reproduced by the Vlasov dynamics. For larger values of the initial magnetization ($M_{0}>0.89$), while keeping $e=0.69$, the system evolves towards an asymptotic magnetized state, in agreement with the theory. In this case several resonances are rapidly developed which eventually coaelesce giving rise to complex patterns in phase space. More quantitatively, one can compare the velocity distributions resulting from, respectively, Vlasov and N-body simulations. The curves are diplayed in figure \[fig3\] (a),(b),(c) for various choices of the initial conditions in the magnetized region. The agreement is excellent, thus reinforcing our former conclusion about the validity of the Vlasov model. Finally, let us stress that, when $e=0.69$, the two solutions (resp. magnetized and non magnetized) [@antoniazziPRL] are associated to a practically indistinguishible entropy level (see figure \[fig3\] (d)). As previously discussed, the system explores an almost flat entropy landscape and can be therefore be stuck in local traps, because of finite size effects. A pronounced variability of the measured $M_{QSS}$ is therefore to be expected. ![Symbols: velocity distributions computed via N-body simulations. Solid line: velocity distributions obtained through a direct integration of the Vlasov equation. Here $e=0.69$ and $M_0=0.3$ (a), $M_0=0.5$ (b), $M_0=0.7$ (c). Panel (d): Entropy at the stationary points as a function of the initial magnetization: the solid line refers to the global maximum, while the dotted line to the saddle point.[]{data-label="fig3"}](fig3.eps){width="7cm"} In this Letter, we have analyzed the emergence of QSS, a universal feature that occurs in systems with long-range interactions, for the specific case of the HMF model. By comparing numerical simulations and analytical predictions, we have been able to unambiguously demonstrate that the Vlasov model provides an accurate framework to address the study of the QSS. Working within the Vlasov context one can develop a fully predictive theoretical approach, which is completely justified from first principles. Finally, and most important, results of conventional Vlasov codes are to be critically scrutinized, especially in specific regions of parameters space close to transitions from homogeneous to non homogeneous states. We acknowledge financial support from the PRIN05-MIUR project [*Dynamics and thermodynamics of systems with long-range interactions*]{}. [99]{} P.J. Peebles, *The Large-scale structure of the Universe*, Princeton, NJ: Princeton Universeity Press (1980). A. Mangeney et al., J. Comp. Physics [**179**]{}, 495 (2002). L. Galeotti et al., Phys. Rev. Lett. [**95**]{}, 015002 (2005); F. Califano et al., Phys. Plasmas [**13**]{}, 082102 (2006). M. Antoni et al., Phys. Rev. E **52**, 2361 (1995). W. Braun et al., Comm. Math. Phys. **56**, 101 (1977). T. Dauxois et al., Lect. Not. Phys. [**602**]{}, Springer (2002). V. Latora et al. Phys. Rev. Lett. **80**, 692 (1998). V. Latora et al. Phys. Rev. E **64** 056134 (2001). C. Tsallis, J. Stat. Phys. [**52**]{}, 479 (1988). Y.Y. Yamaguchi et al. Physica A, [**337**]{}, 36 (2004). A. Antoniazzi et al., Phys. Rev. E **75** 011112 (2007); P.H. Chavanis Eur. Phys. J. B [**53**]{}, 487 (2006). D. Lynden-Bell et al., Mon. Not. R. Astron. Soc. [**138**]{}, 495 (1968). P.H. Chavanis, Ph. D Thesis, ENS Lyon (1996). A. Rapisarda et al., Europhysics News, [**36**]{}, 202 (2005); F. Bouchet et al., Europhysics News, [**37**]{}, 2, 9-10 (2006). T. Padmanabhan, Phys. Rep. [**188**]{}, 285 (1990). J. Barr[é]{} et al., Phys. Rev E [**69**]{}, 045501(R) (2004). R. Kawahara and H. Nakanishi, cond-mat/0611694. P. H. Chavanis, [Physica A]{} **359**, 177 (2006). J. Michel et al., Comm. Math. 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ArXiv

--- abstract: | A [*rack*]{} is a set equipped with a bijective, self-right-distributive binary operation, and a [*quandle*]{} is a rack which satisfies an idempotency condition. In this paper, we introduce a new definition of modules over a rack or quandle, and show that this definition includes the one studied by Etingof and Graña [@etingof/grana:orc] and the more general one given by Andruskiewitsch and Graña [@andr/grana:pointed-hopf]. We further show that this definition coincides with the appropriate specialisation of the definition developed by Beck [@beck:thesis], and hence that these objects form a suitable category of coefficient objects in which to develop homology and cohomology theories for racks and quandles. We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, Elhamdadi, Kamada and Saito [@carter/elhamdadi/saito:twisted; @carter/kamada/saito:diag] as special cases. address: | Mathematics Institute\ University of Warwick\ Coventry\ CV4 7AL\ United Kingdom author: - Nicholas Jackson title: Extensions of racks and quandles --- Introduction ============ A [*rack*]{} (or [*wrack*]{}) is a set $X$ equipped with a self-right-distributive binary operation (often written as exponentiation) satisfying the following two axioms: 1. For every $a,b \in X$ there is a unique $c \in X$ such that $c^b = a$. 2. For every $a,b,c \in X$, the [*rack identity*]{} holds: $$a^{bc} = a^{cb^c}$$ In the first of these axioms, the unique element $c$ is often denoted $a^{\overline{b}}$, although $\overline{b}$ should not itself be regarded as an element of the rack. Association of exponents should be understood to follow the usual conventions for exponential notation. In particular, the expressions $a^{bc}$ and $a^{cb^c}$ should be interpreted as $(a^b)^c$ and $(a^c)^{(b^c)}$ respectively. A rack which, in addition, satisfies the following idempotency criterion is said to be a [*quandle*]{}. 1. For every $a \in X$, $a^a = a$. There is an obvious notion of a [*homomorphism*]{} of racks: a function $f{\colon}X {\rightarrow}Y$ such that $f(a^b) = f(a)^{f(b)}$ for all $a,b \in X$. We may thus form the categories ${\mathsf{Rack}}$ and ${\mathsf{Quandle}}$. For any element $x \in X$ the map $\pi_x{\colon}a \mapsto a^x$ is a bijection. The subgroup of ${\operatorname{Sym}}X$ generated by $\{ \pi_x : x \in X \}$ is the [*operator group*]{} of $X$, denoted ${\operatorname{Op}}X$. This assignment is not functorial since there is not generally a well-defined group homomorphism ${\operatorname{Op}}f{\colon}{\operatorname{Op}}X {\rightarrow}{\operatorname{Op}}Y$ corresponding to an arbitrary rack homomorphism $f{\colon}X {\rightarrow}Y$. The group ${\operatorname{Op}}X$ acts on the rack $X$, and divides it into [*orbits*]{}. Two elements $x,y \in X$ are then said to be in the same orbit (denoted $x \sim y$ or $x \in [y]$) if there is a (not necessarily unique) word $w \in {\operatorname{Op}}X$ such that $y = x^w$. A rack with a single orbit is said to be [*transitive*]{}. The set of orbits of $X$ is denoted ${\operatorname{Orb}}X$. Given any group $G$, we may form the [*conjugation rack*]{} ${\operatorname{Conj}}G$ of $G$ by taking the underlying set of $G$ and defining the rack operation to be conjugation within the group, so $g^h := h^{-1}gh$ for all $g,h \in G$. This process determines a functor ${\operatorname{Conj}}{\colon}{\mathsf{Group}} {\rightarrow}{\mathsf{Rack}}$ which has a left adjoint, the [*associated group*]{} functor ${\operatorname{As}}{\colon}{\mathsf{Rack}} {\rightarrow}{\mathsf{Group}}$. For a given rack $X$, the associated group ${\operatorname{As}}X$ is the free group on the elements of $X$ modulo the relations $$a^b = b^{-1}ab$$ for all $a,b \in X$. Racks were first studied by Conway and Wraith [@conway/wraith:wracks] and later (under the name ‘automorphic sets’) by Brieskorn [@brieskorn:automorphic], while quandles were introduced by Joyce [@joyce:knot-quandle]. A detailed exposition may be found in the paper by Fenn and Rourke [@fenn/rourke:racks-links]. A [*trunk*]{} ${\mathsf{T}}$ is an object analogous to a category, and consists of a class of [*objects*]{} and, for each ordered pair $(A,B)$ of objects, a set ${\operatorname{Hom}}_{{\mathsf{T}}}(A,B)$ of [*morphisms*]{}. In addition, ${\mathsf{T}}$ has a number of [*preferred squares*]{} $$\bfig\square[A`B`C`D;f`g`h`k]\efig$$ of morphisms, a concept analogous to that of composition in a category. Morphism composition need not be associative, although it is in all the cases discussed in this paper, and particularly when the trunk in question is also a category. Given two arbitrary trunks ${\mathsf{S}}$ and ${\mathsf{T}}$, a [*trunk map*]{} or [*functor*]{} $F{\colon}{\mathsf{S}}{\rightarrow}{\mathsf{T}}$ is a map which assigns to every object $A$ of ${\mathsf{S}}$ an object $F(A)$ of ${\mathsf{T}}$, and to every morphism $f{\colon}A {\rightarrow}B$ of ${\mathsf{S}}$ a morphism $F(f){\colon}F(A) {\rightarrow}F(B)$ of ${\mathsf{T}}$ such that preferred squares are preserved: $$\bfig\square/>`>`>`>/<500,500>[F(A)`F(B)`F(C)`F(D);f_*`g_*`h_*`k_*]\efig$$ For any category ${\mathsf{C}}$ there is a well-defined trunk ${\operatorname{Trunk}({\mathsf{C}})}$ which has the same objects and morphisms as ${\mathsf{C}}$, and whose preferred squares are the commutative diagrams in ${\mathsf{C}}$. In particular, we will consider the case ${\operatorname{Trunk}({\mathsf{Ab}})}$, which we will denote ${\mathsf{Ab}}$ where there is no ambiguity. Trunks were first introduced and studied by Fenn, Rourke and Sanderson [@fenn/rourke/sanderson:trunks]. In this paper, we study extensions of racks and quandles in more generality than before, in the process describing a new, generalised notion of a module over a rack or quandle, which is shown to coincide with the general definition of a module devised by Beck [@beck:thesis]. Abelian groups ${\operatorname{Ext}}(X,{\mathcal{A}})$ and ${\operatorname{Ext}}_Q(X,{\mathcal{A}})$ are defined and shown to classify (respectively) Abelian rack and quandle extensions and to be generalisations of all known existing ${\operatorname{Ext}}$ groups for racks and quandles. This paper contains part of my doctoral thesis [@jackson:thesis]. I am grateful to my supervisor Colin Rourke, and to Alan Robinson, Ronald Brown, and Simona Paoli for many interesting discussions and much helpful advice over the past few years. I also thank the referees for their kind comments and helpful suggestions. Modules ======= Given a rack $X$ we define a trunk ${\mathsf{T}}(X)$ as follows: let ${\mathsf{T}}(X)$ have one object for each element $x \in X$, and for each ordered pair $(x,y)$ of elements of $X$, a morphism $\alpha_{x,y}{\colon}x {\rightarrow}x^y$ and a morphism $\beta_{y,x}{\colon}y {\rightarrow}y^x$ such that the squares $$\bfig\square/>`>`>`>/<1000,500>[x`x^y`x^z`x^{yz} = x^{zy^z}; \alpha_{x,y}`\alpha_{x,z}`\alpha_{x^y,z}`\alpha_{x^z,y^z}]\efig \qquad \bfig\square/>`>`>`>/<1000,500>[y`x^y`y^z`x^{yz} = x^{zy^z}; \beta_{y,x}`\alpha_{y,z}`\alpha_{x^y,z}`\beta_{y^z,x^z}]\efig$$ are preferred for all $x,y,z \in X$. Thus a trunk map $A{\colon}{\mathsf{T}}(X){\rightarrow}{\mathsf{Ab}}$, as defined in the previous section, determines Abelian groups $A_x$, and Abelian group homomorphisms $\phi_{x,y}{\colon}A_x {\rightarrow}A_{x^y}$ and $\psi_{y,x}{\colon}A_y {\rightarrow}A_{x^y}$, such that $$\begin{aligned} \phi_{x^y,z}\phi_{x,y} & = & \phi_{x^z,y^z}\phi_{x,z} \\ \mbox{and}\qquad \phi_{x^y,z}\psi_{y,x} & = & \psi_{y^z,x^z}\phi_{y,z}\end{aligned}$$ for all $x,y,z \in X$. It will occasionally be convenient to denote such a trunk map by a triple $(A,\phi,\psi)$. Rack modules ------------ Let $X$ be an arbitrary rack. Then a [*rack module*]{} over $X$ (or an [*$X$–module*]{}) is a trunk map ${\mathcal{A}} = (A,\phi,\psi){\colon}{\mathsf{T}}(X) {\rightarrow}{\mathsf{Ab}}$ such that each $\phi_{x,y}{\colon}A_x {\cong}A_{x^y}$ is an isomorphism, and $$\label{eqn:rmod} \psi_{z,x^y}(a) = \phi_{x^z,y^z}\psi_{z,x}(a) + \psi_{y^z,x^z}\psi_{z,y}(a)$$ for all $a \in A_z$ and $x,y,z \in X$. If $x,y$ lie in the same orbit of $X$ then this implies that $A_x {\cong}A_y$ (although the isomorphism is not necessarily unique). For racks with more than one orbit it follows that if $x \not\sim y$ then $A_x$ need not be isomorphic to $A_y$. Rack modules where the constituent groups are nevertheless all isomorphic are said to be [*homogeneous*]{}, and those where this is not the case are said to be [*heterogeneous*]{}. It is clear that modules over transitive racks must be homogeneous. An $X$–module ${\mathcal{A}}$ of the form $(A,{\operatorname{Id}},0)$ (so that $\phi_{x,y} = {\operatorname{Id}}{\colon}A_x {\rightarrow}A_{x^y}$ and $\psi_{y,x}$ is the zero map $A_y {\rightarrow}A_{x^y}$) is said to be [*trivial*]{}. \[exm:abgroup\] [Any Abelian group $A$ may be considered as a homogeneous trivial $X$–module ${\mathcal{A}}$, for any rack $X$, by setting $A_x = A, \phi_{x,y} = {\operatorname{Id}}_A$, and $\psi_{y,x} = 0_A$ for all $x,y \in X$.]{} \[exm:asx\] [Let $X$ be a rack, and let $A$ be an Abelian group equipped with an action of ${\operatorname{As}}X$. Then $A$ may be considered as a homogeneous $X$–module ${\mathcal{A}} = (A,\phi,\psi)$ by setting $A_x = A$, and defining $\phi_{x,y}(a) = a \cdot x$ and $\psi_{y,x}(a) = 0$ for all $a \in A$ and $x,y \in X$.]{} In particular, Etingof and Graña [@etingof/grana:orc] study a cohomology theory for racks, with ${\operatorname{As}}X$–modules as coefficient objects. \[exm:andr/grana\] In [@andr/grana:pointed-hopf], Andruskiewitsch and Graña define an [*$X$–module*]{} to be an Abelian group $A$ equipped with a family $\eta = \{\eta_{x,y} : x,y \in X\}$ of automorphisms of $A$ and another family $\tau = \{\tau_{x,y} : x,y \in X\}$ of endomorphisms of $A$ such that (after slight notational changes): $$\begin{aligned} \eta_{x^y,z}\eta_{x,y} & = & \eta_{x^z,y^z}\eta_{x,z} \\ \eta_{x^y,z}\tau_{y,x} & = & \tau_{y^z,x^z}\eta_{y,z} \\ \tau_{z,x^y} & = & \eta_{x^z,y^z}\tau_{z,x} + \tau_{y^z,x^z}\tau_{z,y}\end{aligned}$$ This may readily be seen to be a homogeneous $X$–module in the context of the current discussion. As a concrete example, let $X$ be $C_3=\{0,1,2\}$, the cyclic rack with three elements. This has rack structure given by $x^y = x+1 \pmod{3}$ for all $x,y \in X$. Let $A=\mathbb{Z}_5$ and define: $$\begin{aligned} &\eta_{x,y}{\colon}A {\rightarrow}A;\quad n \mapsto 2n\pmod{5}\\ &\tau_{y,x}{\colon}A {\rightarrow}A;\quad n \mapsto 4n\pmod{5}\end{aligned}$$ Then this satisfies Andruskiewitsch and Graña’s definition of a $C_3$–module, and (by setting $A_0=A_1=A_2=A=\mathbb{Z}_5$) is also a homogeneous $C_3$–module in the context of the current discussion. \[exm:alexander\] [Let $h = \{ h_i : i \in {\operatorname{Orb}}X \}$ be a family of Laurent polynomials in one variable $t$, one for each orbit of the rack $X$, and let $n = \{ n_i : i \in {\operatorname{Orb}}X \}$ be a set of positive integers, also one for each orbit. Then we may construct a (possibly heterogeneous) $X$–module ${\mathcal{A}} = (A,\phi,\psi)$ by setting $A_x = \mathbb{Z}_{n_{[x]}}[t,t^{-1}]/h_{[x]}(t)$, $\phi_{x,y}{\colon}a \mapsto ta$, and $\psi_{y,x}{\colon}b \mapsto (1-t)b$ for all $x,y \in X$, $a \in A_x$ and $b \in A_y$. The case where $A_x = \mathbb{Z}[t,t^{-1}]/h_{[x]}(t)$ for all $x$ in some orbit(s) of $X$ is also an $X$–module.]{} \[exm:dihedral\] [Let $n = \{ n_i : i \in {\operatorname{Orb}}X \}$ be a set of positive integers, one for each orbit of $X$. Then let ${\mathcal{D}} = (D,\phi,\psi)$ denote the (possibly heterogeneous) $X$–module where $D_x = \mathbb{Z}_{n_{[x]}}$, $\phi_{x,y}(a) = -a$, and $\psi_{y,x}(b) = 2b$ for all $x,y \in X$, $a \in A_x$ and $b \in A_y$. This module is isomorphic to the Alexander module where $h_i(t) = (1+t)$ for all $i \in {\operatorname{Orb}}X$. The case where $A_x = \mathbb{Z}$ for all $x$ in some orbit(s) of $X$, is also an $X$–module. The [*$n$th homogeneous dihedral $X$–module*]{} (where all the $n_i$ are equal to $n$) is denoted ${\mathcal{D}}_n$. The case where $D_x = \mathbb{Z}$ for all $x \in X$ is the [*infinite homogeneous dihedral $X$–module*]{} ${\mathcal{D}}_\infty$.]{} Given two $X$–modules ${\mathcal{A}} = (A,\phi,\psi)$ and ${\mathcal{B}} = (B,\chi,\omega)$, a [*homomorphism*]{} of $X$–modules, or an [*$X$–map*]{}, is a natural transformation $f{\colon}{\mathcal{A}} {\rightarrow}{\mathcal{B}}$ of trunk maps, that is, a collection $f = \{ f_x{\colon}A_x {\rightarrow}B_x : x \in X \}$ of Abelian group homomorphisms such that $$\begin{aligned} \phi_{x,y}f_x & = & f_{x^y}\phi_{x,y} \\ \mbox{and}\qquad\psi_{y,x}f_y & = & f_{x^y}\psi_{y,x}\end{aligned}$$ for all $x,y \in X$. We may thus form the category ${\mathsf{RMod}}_X$ whose objects are $X$–modules, and whose morphisms are $X$–maps. In his doctoral thesis [@beck:thesis], Beck gives a general definition of a ‘module’ in an arbitrary category. Given a category ${\mathsf{C}}$, and an object $X$ of ${\mathsf{C}}$, a [*Beck module*]{} over $X$ is an Abelian group object in the slice category ${\mathsf{C}}/X$. For any group $G$, the category ${\mathsf{Ab}}({\mathsf{Group}}/G)$, for example, is equivalent to the category of $G$–modules. Similar results hold for Lie algebras, associative algebras and commutative rings. The primary aim of this section is to demonstrate a categorical equivalence between the rack modules just defined, and the Beck modules in the category ${\mathsf{Rack}}$. For an arbitrary rack $X$ and an $X$–module ${\mathcal{A}} = (A,\phi,\psi)$, we define the [*semidirect product*]{} of ${\mathcal{A}}$ and $X$ to be the set $${\mathcal{A}} \rtimes X = \{ (a,x) : x \in X, a \in A_x \}$$ with rack operation given by $$(a,x)^{(b,y)} := \left(\phi_{x,y}(a) + \psi_{y,x}(b), x^y\right).$$ \[thm:semidirect-rack\] \[prp:semidirect-rack\] For any rack $X$ and $X$–module ${\mathcal{A}} = (A,\phi,\psi)$, the semidirect product ${\mathcal{A}} \rtimes X$ is a rack. For any three elements $(a,x), (b,y), (c,z) \in {\mathcal{A}} \rtimes X$, $$\begin{aligned} (a,x)^{(b,y)(c,z)}\! &= (\phi_{x,y}(a) + \psi_{y,x}(b),x^y)^{(c,z)} \\ &= (\phi_{x^y,z}\phi_{x,y}(a) + \phi_{x^y,z}\psi_{y,x}(b) + \psi_{z,x^y}(c),x^{yz}) \\ &= (\phi_{x^z,y^z}\phi_{x,z}(a) + \psi_{y^z,x^z}\phi_{y,z}(b) + \phi_{x^z,y^z}\psi_{z,x}(c) + \psi_{y^z,x^z}\psi_{z,y}(c), x^{zy^z}) \\ &= (\phi_{x,z}(a) + \psi_{z,x}(c),x^z)^{(\phi_{y,z}(b) + \psi_{z,y}(c),y^z)} \\ &= (a,x)^{(c,z)(b,y)^{(c,z)}}.\end{aligned}$$ Furthermore, for any two elements $(a,x), (b,y) \in {\mathcal{A}} \rtimes X$, there is a unique element $$(c,z) = (a,x)^{\overline{(b,y)}} = (\phi_{z,y}^{-1}(a - \psi_{y,z}(b)),x^{\overline{y}}) \in {\mathcal{A}} \rtimes X$$ such that $(c,z)^{(b,y)} = (a,x)$. Hence ${\mathcal{A}} \rtimes X$ satisfies the rack axioms. \[thm:rmod-beck\] For any rack $X$, the category ${\mathsf{RMod}}_X$ of $X$–modules is equivalent to the category ${\mathsf{Ab}}({\mathsf{Rack}}/X)$ of Abelian group objects over $X$. Given an $X$-module ${\mathcal{A}} = (A,\phi,\psi)$, let $T{\mathcal{A}}$ be the object $p{\colon}{\mathcal{A}} \rtimes X {\twoheadrightarrow}X$ in the slice category ${\mathsf{Rack}}/X$, where $p$ is defined as projection onto the second coordinate. Given an $X$–map $f{\colon}{\mathcal{A}}{\rightarrow}{\mathcal{B}}$, we obtain a slice morphism $Tf:{\mathcal{A}} \rtimes X {\rightarrow}{\mathcal{B}} \rtimes X$ defined by $T(f)(a,x) = (f_x(a),x)$ for all $a \in A_x$ and $x \in X$. This is functorial since, for any $X$–module homomorphism $g{\colon}{\mathcal{B}}{\rightarrow}{\mathcal{C}}$, $$\begin{aligned} T(fg)(a,x) &= ((fg)_x(a),x) \\ &= (f_xg_x(a),x) \\ &= T(f)(g_x(a),x) \\ &= T(f)T(g)(a,x)\end{aligned}$$ for all $a \in A_x$ and $x \in X$. We thus have a functor $T{\colon}{\mathsf{RMod}}_X{\rightarrow}{\mathsf{Rack}}/X$. Our aim is to show firstly that the image of $T$ is the subcategory ${\mathsf{Ab}}({\mathsf{Rack}}/X)$, and secondly that $T$ has a well-defined inverse. To show the first, that $T{\mathcal{A}}$ has a canonical structure as an Abelian group object, we must construct an appropriate section, and suitable multiplication and inverse morphisms. Let: $$\begin{aligned} & r{\colon}{\mathcal{A}} \rtimes X {\rightarrow}{\mathcal{A}} \rtimes X ; && (a,x) \mapsto (-a,x) \\ & m{\colon}({\mathcal{A}} \rtimes X)\times_X({\mathcal{A}} \rtimes X) {\rightarrow}{\mathcal{A}} \rtimes X ; && ((a_1,x),(a_2,x)) \mapsto (a_1+a_2,x) \\ & s{\colon}X {\rightarrow}{\mathcal{A}} \rtimes X ; && x \mapsto (0,x)\end{aligned}$$ The maps $r$ and $m$ both compose appropriately with the projection map $p$: $$\begin{aligned} &p(a,x) = x = p(-a,x) = p(r(a,x)) \\ &p(a_1,x) = p(a_2,x) = x = p(a_1+a_2,x) = p(m((a_1,x),(a_2,x))\end{aligned}$$ Furthermore, $ps = {\operatorname{Id}}_X$. Also $$\begin{aligned} m(m((a_1,x),(a_2,x)),(a_3,x)) &= m((a_1+a_2,x),(a_3,x)) \\ &= (a_1+a_2+a_3,x) \\ &= m((a_1,x),(a_2+a_3,x)) \\ &= m((a_1,x),m((a_2,x),(a_3,x))),\\ m(s(x),(a,x)) &= m((0,x),(a,x)) \\ &= (a,x) \\ &= m((a,x),(0,x)) \\ &= m((a,x),s(x)), \\ m((a_1,x),(a_2,x)) &= (a_1+a_2,x) \\ &= (a_2+a_1,x) \\ &= m((a_2,x),(a_1,x)),\end{aligned}$$ $$\begin{aligned} \mbox{and}\qquad m(r(a,x),(a,x)) &= m((-a,x),(a,x)) \\ &= (0,x) \\ &= m((a,x),(-a,x)) \\ &= m((a,x),r(a,x)),\end{aligned}$$ so $T{\mathcal{A}}$ is an Abelian group object in ${\mathsf{Rack}}/X$. Now, given an Abelian group object $p{\colon}R {\rightarrow}X$ in ${\mathsf{Rack}}/X$, with multiplication map $\mu$, inverse map $\nu$, and section $\sigma$, let $R_x$ be the preimage $p^{-1}(x)$ for each $x \in X$. Each of the $R_x$ has a canonical Abelian group structure defined in terms of the maps $\mu, \nu$, and $\sigma$: $\sigma(x)$ is the identity in $R_x$, and for any $u,v \in R_x$ let $u+v := \mu(u,v)$ and $-u := \nu(u)$. That the preimage $R_x$ is closed under addition and inversion follows immediately from the fact that $\mu$ and $\nu$ are rack homomorphisms over $X$. Next, we define maps $$\rho_{x,y}{\colon}R_x {\rightarrow}R_{x^y},\ \mbox{given by}\ u \mapsto u^{\sigma(y)},$$ for all $x,y \in X$ and $u \in R_x$. These are Abelian group homomorphisms, since $\rho_{x,y}\sigma(x) = \sigma(x)^{\sigma(y)} = \sigma(x^y)$ (which is the identity in $R_{x^y}$) and, for any $u_1,u_2 \in R_x$, $$\begin{aligned} \rho_{x,y}(u_1+u_2) &= \mu(u_1,u_2)^{\sigma(y)} \\ &= \mu(u_1,u_2)^{\mu(\sigma(y),\sigma(y))} \\ &= \mu(u_1^{\sigma(y)},u_2^{\sigma(y)}) \\ &= \rho_{x,y}(u_1) + \rho_{x,y}(u_2).\end{aligned}$$ It is also an isomorphism, since exponentiation by a fixed element of a rack is a bijection. Furthermore, for any $x,y,z \in X$ and any $u \in R_x$ $$\begin{aligned} \rho_{x^y,z}\rho_{x,y}(u) &= u^{\sigma(y)\sigma(z)} \\ &= u^{\sigma(z)\sigma(y)^{\sigma(z)}} \\ &= u^{\sigma(z)\sigma(y^z)} \\ &= \rho_{x^z,y^z}\rho_{x,z}(u).\end{aligned}$$ Now we define maps $$\lambda_{y,x}{\colon}R_y {\rightarrow}R_{x^y},\qquad \mbox{given by}\ v \mapsto \sigma(x)^v,$$ for all $x,y \in X$ and $v \in R_y$. These are also Abelian group homomorphisms since $$\lambda_{y,x}\sigma(y) = \sigma(x)^{\sigma(y)} = \sigma(x^y)$$ (which is the identity in $R_{x^y}$) and, for any $v_1,v_2 \in R_y$, $$\begin{aligned} \lambda_{y,x}(v_1+v_2) &= \sigma(x)^{\mu(v_1,v_2)} \\ &= \mu(\sigma(x),\sigma(x))^{\mu(v_1,v_2)} \\ &= \mu(\sigma(x)^{v_1},\sigma(x)^{v_2}) \\ &= \lambda_{y,x}(v_1) + \lambda_{y,x}(v_2).\end{aligned}$$ Also, for any $x,y,z \in X$, $v \in R_y$ and $w \in R_z$ $$\begin{aligned} \rho_{x^y,z}\lambda_{y,x}(v) &= \sigma(x)^{v\sigma(z)} \\ &= \sigma(x)^{\sigma(z)v^{\sigma(z)}} \\ &= \sigma(x^z)^{v^{\sigma(z)}} \\ &= \lambda_{y^z,x^z}\rho_{y,z}(v) \\ \mbox{and}\qquad\lambda_{z,x^y}(w) &= \sigma(x^y)^w \\ &= \sigma(x)^{\sigma(y)w} \\ &= \sigma(x)^{w\sigma(y)^w} \\ &= \mu(\sigma(x),\sigma(x))^{\mu(\sigma(z),w) \mu(\sigma(y),\sigma(y))^{\mu(w,\sigma(z))}} \\ &= \mu\left(\sigma(x)^{\sigma(z)\sigma(y)^w}, \sigma(x)^{w\sigma(y)^{\sigma(z)}}\right) \\ &= \sigma(x)^{\sigma(z)\sigma(y)^w} + \sigma(x)^{w\sigma(y)^{\sigma(z)}} \\ &= \sigma(x^z)^{\sigma(y)^w} + \sigma(x)^{w\sigma(y^z)} \\ &= \lambda_{y^z,x^z}\lambda_{z,y}(w) + \rho_{x^z,y^z}\lambda_{z,x}(w).\end{aligned}$$ Thus an Abelian group object $R {\rightarrow}X$ in ${\mathsf{Rack}}/X$ determines a unique rack module ${\mathcal{R}} = (R,\rho,\lambda)$ over $X$. For any two such Abelian group objects $p_1{\colon}R_1 {\rightarrow}X$ and $p_2{\colon}R_2 {\rightarrow}X$, together with a rack homomorphism $f_1{\colon}R_1 {\rightarrow}R_2$ over $X$, we may construct two $X$–modules ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ as described above, and an $X$–map $g_1{\colon}{\mathcal{R}}_1{\rightarrow}{\mathcal{R}}_2$ by setting $(g_1)_x(u) = f_1(u)$ for all $u \in (R_1)_x$ and $x \in X$. It may be seen that $(g_1)_x:(R_1)_x {\rightarrow}(R_2)_x$ since $f_1$ is a rack homomorphism over $X$. It may also be seen that $g_1$ is a natural transformation of trunk maps ${\mathsf{T}}(X) {\rightarrow}{\mathsf{Ab}}$ since $$\begin{aligned} (g_1)_{x^y}((\rho_1)_{x,y}(u)) &= f_1((\rho_1)_{x,y}(u)) \\ &= f_1(u^{\sigma_1(y)}) \\ &= f_1(u)^{f_1\sigma_1(y)} \\ &= f_1(u)^{\sigma_2(y)} \\ &= (\rho_2)_{x,y}(g_1)_x(u) \\ \mbox{and}\qquad(g_1)_{x^y}(\lambda_1)_{y,x}(v) &= f_1((\lambda_1)_{y,x}(v)) \\ &= f_1(\sigma_1(x)^v) \\ &= f_1\sigma_1(x)^{f_1(v)} \\ &= \sigma_2(x)^{f_1(v)} \\ &= (\lambda_2)_{y,x}(g_1)_x(v)\end{aligned}$$ for all $u \in R_x, v \in R_y$, and $x,y \in X$. Given a third Abelian group object $p_3{\colon}R_3 {\rightarrow}X$ together with another slice morphism $f_2{\colon}R_2 {\rightarrow}R_3$, we may construct another $X$–module ${\mathcal{R}}_3$ and $X$–map $g_2{\colon}{\mathcal{R}}_2 {\rightarrow}{\mathcal{R}}_3$. From the composition $f_2f_1$ we may similarly construct a unique $X$–map $g{\colon}{\mathcal{R}}_1{\rightarrow}{\mathcal{R}}_3$. Then $$g_x(u) = (f_2f_1)(u) = (g_2)_x(f_1(u)) = (g_2)_x(g_1)_x(u).$$ Hence this construction determines a functor $S{\colon}{\mathsf{Ab}}({\mathsf{Rack}}/X){\rightarrow}{\mathsf{RMod}}_X$, which is the inverse of the functor $T{\colon}{\mathsf{RMod}}_X{\rightarrow}{\mathsf{Ab}}({\mathsf{Rack}}/X)$ described earlier. \[cor:rmod-abelian\] \[thm:rmod-abelian\] The category ${\mathsf{RMod}}_X$ is Abelian. The category ${\mathsf{RMod}}_X$ is additive, as for any $X$–modules ${\mathcal{A}}$ and ${\mathcal{B}}$, the set ${\operatorname{Hom}}_{{\mathsf{RMod}}_X}({\mathcal{A}},{\mathcal{B}})$ has an Abelian group structure given by $(f+g)_x(a) = f_x(a) + g_x(a)$ for all $f,g{\colon}{\mathcal{A}}{\rightarrow}{\mathcal{B}}$, all $x \in X$ and all $a \in A_x$. Furthermore, composition of $X$–maps distributes over this addition operation. The $X$–module with trivial orbit groups and structure homomorphisms is the zero object in ${\mathsf{RMod}}_X$, and for any two $X$–modules ${\mathcal{A}} = (A,\alpha,{\varepsilon})$ and ${\mathcal{B}} = (B,\beta,\zeta)$, the Cartesian product ${\mathcal{A}}\times{\mathcal{B}} = (A \times B, \alpha\times\beta, {\varepsilon}\times\zeta)$ is also an $X$–module. Given an $X$–map $f{\colon}{\mathcal{B}} = (B,\beta,\zeta) {\rightarrow}{\mathcal{C}} = (C,\gamma,\eta)$ let ${\mathcal{A}} = (A,\alpha,{\varepsilon})$ such that $A_x = \{a \in B_x : f_x(a) = 0\}$, with $\alpha_{x,y} = \beta_{x,y}|_{A_x}$ and ${\varepsilon}_{y,x} = \zeta_{y,x}|_{A_y}$. Then ${\mathcal{A}}$ is a submodule of ${\mathcal{B}}$ and the inclusion $\iota{\colon}{\mathcal{A}}\hookrightarrow{\mathcal{B}}$ is the (categorical) kernel of $f$. Now define ${\mathcal{D}} = (D,\delta,\xi)$ where $D_x = C_x/{\operatorname{im}}f_x$, and $\delta_{x,y} =\gamma_{x,y}+{\operatorname{im}}f_x$ and $\xi_{y,x} = \eta_{y,x} + {\operatorname{im}}f_y$. Then ${\mathcal{D}}$ is a quotient of ${\mathcal{C}}$ and the canonical projection map $\pi{\colon}{\mathcal{C}}{\rightarrow}{\mathcal{D}}$ is the (categorical) cokernel of $f$. Let $\mu{\colon}{\mathcal{H}}{\rightarrow}{\mathcal{K}}$ be an $X$–monomorphism. Then the inclusion $\iota{\colon}{\operatorname{im}}\mu{\rightarrow}{\mathcal{K}}$ is a kernel of the quotient map $\pi{\colon}{\mathcal{K}}{\rightarrow}{\mathcal{K}}/{\operatorname{im}}\mu$. Since $\mu$ is injective, $\mu'{\colon}{\mathcal{H}}{\cong}{\operatorname{im}}\mu$ where $\mu'_x(a) = \mu_x(a)$ for all $x \in X$ and $a \in H_x$. But since kernels are unique up to composition with an isomorphism, and since $\mu = \iota\mu'$, it follows that $\mu$ is the kernel of its cokernel, the canonical quotient map $\pi$. Let $\nu{\colon}{\mathcal{H}}{\rightarrow}{\mathcal{K}}$ be an $X$–epimorphism. Then the inclusion map $\iota{\colon}\ker\nu\hookrightarrow{\mathcal{H}}$ is a kernel of $\nu$. Given another $X$–map $\kappa{\colon}{\mathcal{H}}{\rightarrow}{\mathcal{L}}$ such that $\kappa\iota = 0$, then $\ker\nu\subseteq\ker\kappa$ so that $\nu(a) = \nu(b)$ implies that $\kappa(a)=\kappa(b)$. But since $\nu$ is surjective we can define an $X$–map $\theta{\colon}{\mathcal{K}}{\rightarrow}{\mathcal{L}}$ by $\theta_x\nu_x(a) = \kappa_x(a)$ for all $a \in H_x$ and $x \in X$. Then $\theta\nu = \kappa$ and so $\nu$ is a cokernel of $\iota$. So, every $X$–map has a kernel and a cokernel, every monic $X$–map is the kernel of its cokernel, and every epic $X$–map is the cokernel of its kernel, and hence ${\mathsf{RMod}}_X$ is an Abelian category. These results justify the use of the term ‘rack module’ to describe the objects under consideration, and show that ${\mathsf{RMod}}_X$ is an appropriate category in which to develop homology theories for racks. Papers currently in preparation will investigate new homology theories for racks, based on the derived functor approach of Cartan and Eilenberg [@cartan/eilenberg:homalg] and the cotriple construction of Barr and Beck [@barr/beck:standard-constructions]. We now introduce a notational convenience which may serve to simplify matters in future. Let $X$ be a rack, ${\mathcal{A}} = (A,\phi,\psi)$ an $X$–module, and $w = y_1 y_2 \ldots y_n$ a word in ${\operatorname{As}}X$. Then we may denote the composition $$\phi_{x^{y_1 \ldots y_{n-1}},y_n} \phi_{x^{y_1 \ldots y_{n-2}},y_{n-1}} \ldots \phi_{x,y_1}$$ by $\phi_{x,w} = \phi_{x,y_1 \ldots y_n}$. This shorthand is well-defined as the following lemma shows: \[lem:asword\] \[thm:asword\] If $y_1 \ldots y_n$ and $z_1 \ldots z_m$ are two different representative words for the same element $w \in {\operatorname{As}}X$, then the compositions $$\begin{aligned} &\phi_{x^{y_1 \ldots y_{n-1}},y_n} \phi_{x^{y_1 \ldots y_{n-2}},y_{n-1}} \ldots \phi_{x,y_1} \\ \mbox{and}\qquad &\phi_{x^{z_1 \ldots z_{m-1}},z_m} \phi_{x^{z_1 \ldots z_{m-2}},z_{m-1}} \ldots \phi_{x,z_1}\end{aligned}$$ are equal, for all $x \in X$. Furthermore, $\phi_{x,1} = {\operatorname{Id}}_{A_x}$, where $1$ denotes the identity in ${\operatorname{As}}X$. Let $T{\colon}{\mathsf{RMod}}_X{\rightarrow}{\mathsf{Ab}}({\mathsf{Rack}}/X)$ be the functor constructed in the proof of Theorem \[thm:rmod-beck\], and recall that $R_x = T({\mathcal{A}})_x$ has an Abelian group structure. For any $x,y \in X$, the homomorphism $T(\phi_{x,y}){\colon}R_x {\rightarrow}R_{x^y}$ maps $u \mapsto u^{\sigma(y)}$, where $\sigma$ is the section of $T{\mathcal{A}}$. Then for any $u \in R_x$ $$\begin{aligned} T(\phi_{x^{y_1 \ldots y_{n-1}},y_n} \phi_{x^{y_1 \ldots y_{n-2}},y_{n-1}} \ldots &\phi_{x,y_1})(u) \\ &= u^{\sigma(y_1)\ldots\sigma(y_n)} \\ &= u^{\sigma(y_1 \ldots y_n)} \\ &= u^{\sigma(z_1 \ldots z_m)} \\ &= u^{\sigma(z_1)\ldots\sigma(z_m)} \\ &= T(\phi_{x^{z_1 \ldots z_{m-1}},z_m} \phi_{x^{z_1 \ldots z_{m-2}},z_{m-1}} \ldots \phi_{x,z_1})(u)\end{aligned}$$ where the equality in the second and third lines follows from the functoriality of the associated group. The final statement follows from the observation $$T(\phi_{x,1})(u) = u^1 = u = T({\operatorname{Id}}_{A_x})(u).$$ Hence this notation is well-defined. Quandle modules --------------- We now study the specialisation of rack modules to the subcategory ${\mathsf{Quandle}}$. A [*quandle module*]{} is a rack module ${\mathcal{A}} = (A,\phi,\psi)$ which satisfies the additional criterion $$\label{eqn:qmod} \psi_{x,x}(a) + \phi_{x,x}(a) = a$$ for all $a \in A_x$ and $x \in X$. Where the context is clear, we may refer to such objects as [*$X$–modules*]{}. There is an obvious notion of a [*homomorphism*]{} (or, in the absence of ambiguity, an [*$X$–map*]{}) of quandle modules, and thus we may form the category ${\mathsf{QMod}}_X$ of quandle modules over $X$. Similarly to example \[exm:andr/grana\], Andruskiewitsch and Graña’s definition of quandle modules coincides with the definition of a homogeneous quandle module in the sense of the current discussion. Examples \[exm:abgroup\], \[exm:alexander\], and \[exm:dihedral\] of the previous subsection, are also quandle modules. Example \[exm:asx\] is not, but the variant obtained by setting $\psi_{y,x} = {\operatorname{Id}}_{A} - \phi_{x,y}$, for all $x,y \in X$, is. \[exm:andr/grana-qmod\] [For an arbitrary quandle $X$, Andruskiewitsch and Graña [@andr/grana:pointed-hopf] further define a [*quandle $X$–module*]{} to be a rack module (as in example \[exm:andr/grana\]) which satisfies the additional condition $$\eta_{x,x} + \tau_{x,x} = {\operatorname{Id}}_A$$ for all $x \in X$. This may be seen to be a homogeneous quandle $X$–module in the context of the current discussion.]{} Given a quandle $X$ and a quandle $X$–module ${\mathcal{A}}$, the semidirect product ${\mathcal{A}} \rtimes X$ has the same definition as before. \[thm:semidirect-quandle\] \[prp:semidirect-quandle\] If $X$ is a quandle and ${\mathcal{A}} = (A,\phi,\psi)$ a quandle module over $X$, the semidirect product ${\mathcal{A}} \rtimes X$ is a quandle. By proposition \[prp:semidirect-rack\], ${\mathcal{A}} \rtimes X$ is a rack, so we need only verify the quandle axiom. For any element $(a,x) \in {\mathcal{A}} \rtimes X$, $$(a,x)^{(a,x)} = (\phi_{x,x}(a) + \psi_{x,x}(a),x^x) = (a,x)$$ and so ${\mathcal{A}} \rtimes X$ is a quandle. These objects coincide with the Beck modules in the category ${\mathsf{Quandle}}$. \[thm:qmod-beck\] For any quandle $X$, there is an equivalence of categories $${\mathsf{QMod}}_X {\cong}{\mathsf{Ab}}({\mathsf{Quandle}}/X)$$ As in the proof of Theorem \[thm:rmod-beck\], we identify the quandle module ${\mathcal{A}} = (A,\phi,\psi)$ with ${\mathcal{A}} \rtimes X {\rightarrow}X$ in the slice category ${\mathsf{Quandle}}/X$. Proposition \[prp:semidirect-quandle\] ensures that this object is indeed a quandle over $X$, and hence we obtain a well-defined functor $T{\colon}{\mathsf{QMod}}_X{\rightarrow}{\mathsf{Ab}}({\mathsf{Quandle}}/X)$. Conversely, suppose that $R{\rightarrow}X$ is an Abelian group object in ${\mathsf{Quandle}}/X$, with multiplication map $\mu$, inverse map $\nu$, and section $\sigma$. As before, we may construct a rack module ${\mathcal{R}} = (R,\rho,\lambda)$ over $X$. It remains only to show that this module satisfies the additional criterion (\[eqn:qmod\]) for it to be a quandle module over $X$. But $$\begin{aligned} \lambda_{x,x}(a) + \rho_{x,x}(a) &= \mu(\sigma(x)^a,a^{\sigma(x)}) \\ &= \mu(\sigma(x),a)^{\mu(a,\sigma(x))} \\ &= \mu(\sigma(x),a)^{\mu(\sigma(x),a)} \\ &= \mu(\sigma(x)^{\sigma(x)},a^a) = a\end{aligned}$$ and so ${\mathcal{R}}$ is indeed a quandle $X$–module. The category ${\mathsf{QMod}}_X$ is Abelian. This proof is exactly the same as the proof of Theorem \[cor:rmod-abelian\]. Analogously to the previous subsection, we may conclude that our use of the term ‘quandle module’ is justified, and that the category ${\mathsf{QMod}}_X$ is a suitable environment in which to study the homology and cohomology of quandles. Abelian extensions ================== Having characterised suitable module categories, we may now study extensions of racks and quandles by these objects. Rack extensions have been studied before, in particular by Ryder [@ryder:thesis] under the name ‘expansions’; the constructs which she dubs ‘extensions’ are in some sense racks formed by disjoint unions, whereby the original rack becomes a subrack of the ‘extended’ rack. Ryder’s notion of rack expansions is somewhat more general than the extensions studied here, as she investigates arbitrary congruences (equivalently, rack epimorphisms onto a quotient rack) whereas we will only examine certain classes of such objects. Abelian extensions of racks --------------------------- An [*extension*]{} of a rack $X$ by an $X$–module ${\mathcal{A}} = (A,\phi,\psi)$ consists of a rack $E$ together with an epimorphism $f{\colon}E {\twoheadrightarrow}X$ inducing a partition $E = \bigcup_{x \in X} E_x$ (where $E_x$ is the preimage $f^{-1}(x)$), and for each $x \in X$ a left $A_x$–action on $E_x$ satisfying the following three conditions: 1. The $A_x$–action on $E_x$ is simply transitive, which is to say that for any $u,v \in E_x$ there is a unique $a \in A_x$ such that $a \cdot u = v$. 2. For any $u \in E_x$, $a \in A_x$, and $v \in E_y$, $(a \cdot u)^v = \phi_{x,y}(a) \cdot (u^v)$. 3. For any $u \in E_y$, $b \in A_y$, and $v \in E_y$, $u^{(b \cdot v)} = \psi_{y,x}(b) \cdot (u^v)$. Two extensions $f_1{\colon}E_1 {\twoheadrightarrow}X$ and $f_2{\colon}E_2 {\twoheadrightarrow}X$ by the same $X$–module ${\mathcal{A}}$ are [*equivalent*]{} if there exists a rack isomorphism (an [*equivalence*]{}) $\theta{\colon}E_1 {\rightarrow}E_2$ which respects the projection maps and the group actions: 1. $f_2\theta(u) = f_1(u)$ for all $u \in E_1$ 2. $\theta(a \cdot u) = \theta(a) \cdot u$ for all $u \in E_x$, $a \in A_x$ and $x \in X$. Let $f{\colon}E {\twoheadrightarrow}X$ be an extension of $X$ by ${\mathcal{A}}$. Then a [*section*]{} of $E$ is a function (not necessarily a rack homomorphism) $s{\colon}X {\rightarrow}E$ such that $fs = {\operatorname{Id}}_X$. Since the $A_x$ act simply transitively on the $E_x$, there is a unique $x \in X$ and a unique $a \in A_x$ such that a given element $u \in E_x$ can be written as $u = a \cdot s(x)$. Since $f$ is a homomorphism, it follows that $s(x)^{s(y)} \in E_{x^y}$ and so there is a unique $\sigma_{x,y} \in A_{x^y}$ such that $s(x)^{s(y)} = \sigma_{x,y} \cdot s(x^y) $. The set $\sigma = \{ \sigma_{x,y} : x,y \in X \}$ is the [*factor set*]{} of the extension $E$ [*relative to*]{} the section $s$, and may be regarded as an obstruction to $s$ being a rack homomorphism. It follows that, for all $x,y \in X$, $a \in A_x$, and $b \in A_y$ $$\begin{aligned} (a \cdot s(x))^{(b \cdot s(y))} &= \phi_{x,y}(a) \cdot s(x)^{(b \cdot s(y))} \\ &= (\psi_{y,x}(b) + \phi_{x,y}(a)) \cdot s(x)^{s(y)} \\ &= (\psi_{y,x}(b) + \phi_{x,y}(a) + \sigma_{x,y}) \cdot s(x^y) \end{aligned}$$ Thus the rack structure on $E$ is determined completely by the factor set $\sigma$. The next result gives necessary and sufficient conditions on factor sets of arbitrary rack extensions. \[thm:rack-ext1\] \[prp:rack-ext1\] Let $X$ be a rack, and ${\mathcal{A}} = (A,\phi,\psi)$ be an $X$–module. Let $\sigma = \{ \sigma_{x,y} \in A_{x^y} : x,y \in X \}$ be a collection of group elements. Let $E[{\mathcal{A}},\sigma]$ be the set $\{ (a,x) : a \in A_x, x \in X \}$ with rack operation $$(a,x)^{(b,y)} = (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b), x^y)$$ for all $a \in A_x$, $b \in A_y$, and $x,y \in X$. Then $E[{\mathcal{A}},\sigma]$ is an extension of $X$ by ${\mathcal{A}}$ with factor set $\sigma$ if $$\label{eqn:rack-ext1} \sigma_{x^y,z} + \phi_{x^y,z}(\sigma_{x,y}) = \phi_{x^z,y^z}(\sigma_{x,z}) + \sigma_{x^z,y^z} + \psi_{y^z,x^z}(\sigma_{y,z})$$ for all $x,y,z \in X$. Conversely, if $E$ is an extension of $X$ by ${\mathcal{A}}$ with factor set $\sigma$ then (\[eqn:rack-ext1\]) holds, and $E$ is equivalent to $E[{\mathcal{A}},\sigma]$. To prove the first part, we require that $E[{\mathcal{A}},\sigma]$ satisfy the rack axioms. Given $(a,x), (b,y) \in E[{\mathcal{A}},\sigma]$, there is a unique $(c,z) \in E[{\mathcal{A}},\sigma]$ such that $(c,z)^{(b,y)} = (a,x)$, given by $$(c,z) = (\phi_{x,y}^{-1}(a - \sigma_{x,y} - \psi_{y,x}(b)),x^{\overline{y}})$$ Also, for any $(a,x), (b,y), (c,z) \in E[{\mathcal{A}},\sigma]$, $$\begin{aligned} (a,x)^{(b,y)(c,z)} &= (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b),x^y)^{(c,z)} \\ &= (\phi_{x^y,z}\phi_{x,y}(a) + \phi_{x^y,z}(\sigma_{x,y}) + \phi_{x^y,z}\psi_{y,x}(b) + \sigma_{x^y,z} + \psi_{z,x^y}(c), x^{yz})\end{aligned}$$ and $$\begin{aligned} (a,x)^{(c,z)(b,y)^{(c,z)}} &= (\phi_{x,z}(a) + \sigma_{x,z} + \psi_{z,x}(c),x^z)^{(\phi_{y,z}(b) + \sigma_{y,z} + \psi_{z,y}(c),y^z)} \\ &= (\phi_{x^z,y^z}\phi_{x,z}(a) + \phi_{x^z,y^z}(\sigma_{x,z}) + \phi_{x^z,y^z}\psi_{z,x}(c) + \sigma_{x^z,y^z} \\ &\qquad+ \psi_{y^z,x^z}\phi_{y,z}(b) + \psi_{y^z,x^z}(\sigma_{y,z}) + \psi_{y^z,x^z}\psi_{z,y}(c), x^{zy^z})\end{aligned}$$ are equal if (\[eqn:rack-ext1\]) holds, and so $E[{\mathcal{A}},\sigma]$ is a rack. Now define $f{\colon}E[{\mathcal{A}},\sigma] {\twoheadrightarrow}X$ to be projection onto the second coordinate, and let $A_x$ act on $E[{\mathcal{A}},\sigma]_x = f^{-1}(x)$ by $a_1 \cdot (a_2,x) := (a_1 + a_2,x)$ for each $a_1,a_2 \in A_x$ and all $x \in X$. These actions are simply transitive and satisfy the requirements $$\begin{aligned} (a_1 \cdot (a_2,x))^{(b,y)} = (a_1+a_2, x)^{(b,y)} &= (\phi_{x,y}(a_1+a_2) + \sigma_{x,y} + \psi_{y,x}(b),x^y) \\ &= (\phi_{x,y}(a_1) + \phi_{x,y}(a_2) + \sigma_{x,y} + \psi_{y,x}(b),x^y) \\ &= \phi_{x,y}(a_1) \cdot (\phi_{x,y}(a_2) + \sigma_{x,y} + \psi_{y,x}(b),x^y) \\ &= \phi_{x,y}(a_1) \cdot (a_2,x)^{(b,y)}\\ \mbox{and}\qquad(a,x)^{b_1 \cdot (b_2,y)} &= (a,x)^{(b_1+b_2,y)} \\ &= (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b_1+b_2),x^y) \\ &= (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b_1) + \psi_{y,x}(b_2),x^y) \\ &= \psi_{y,x}(b_1) \cdot (\phi_{x,y}(a) + \sigma_{x,y} + \psi_{y,x}(b_2),x^y) \\ &= \psi_{y,x}(b_1) \cdot (a,x)^{(b_2,y)}\end{aligned}$$ so $E[{\mathcal{A}},\sigma]$ is an extension of $X$ by ${\mathcal{A}}$. Now define $s{\colon}X {\twoheadrightarrow}E[{\mathcal{A}},\sigma]$ by $s(x) = (0,x)$ for all $x \in X$. This is clearly a section of this extension. Also, $$s(x)^{s(y)} = (0,x)^{(0,y)} = (\sigma_{x,y},x^y) = \sigma_{x,y} \cdot s(x^y)$$ so $\sigma$ is the factor set of this extension relative to the section $s$. Conversely, let $f{\colon}E {\twoheadrightarrow}X$ be an extension of $X$ by a given $X$–module ${\mathcal{A}}$, with factor set $\sigma$ relative to some extension $s{\colon}X {\rightarrow}E$. By the simple transitivity of the $A_x$–action on the $E_x = f^{-1}(x)$, the map $\theta{\colon}(a,x) \mapsto a \cdot s(x)$ is an isomorphism $E[{\mathcal{A}},\sigma] {\cong}E$. Since $E$ is a rack, the earlier part of the proof shows that (\[eqn:rack-ext1\]) holds, and so $E[{\mathcal{A}},\sigma]$ is another extension of $X$ by ${\mathcal{A}}$. Furthermore, $\theta$ respects the projection maps onto $X$, and $$\theta(a_1 \cdot (a_2,x)) = \theta(a_1+a_2,x) = (a_1+a_2) \cdot s(x) = a_1 \cdot (a_2 \cdot s(x)) = a_1 \cdot \theta(a_2,x)$$ so $\theta$ is an equivalence of extensions. Andruskiewitsch and Graña [@andr/grana:pointed-hopf] introduce the notion of an extension by a [*dynamical cocycle*]{}. Given an arbitrary rack $X$ and a non-empty set $S$, we select a function $\alpha{\colon}X \times X {\rightarrow}{\operatorname{Hom}}_{\mathsf{Set}}(S \times S,S)$ (which determines, for each ordered pair $x,y \in X$, a function $\alpha_{x,y}{\colon}S \times S {\rightarrow}S$) satisfying the criteria 1. $\alpha_{x,y}(s,-)$ is a bijection on $S$ 2. $\alpha_{x^y,z}(s,\alpha_{x,y}(t,u)) = \alpha_{x^z,y^z}(\alpha_{x,z}(s,t),\alpha_{x,y}(s,u))$ for all $x,y,z \in X$ and $s,t,u \in S$. Then we may define a rack structure on the set $X \times S$ by defining $(x,s)^{(y,t)} = (x^y,\alpha_{x,y}(s,t))$. This rack, denoted $X \times_\alpha S$, is the extension of $X$ by $\alpha$. In the case where $S$ is an Abelian group, and $\alpha_{x,y}(s,t) = \phi_{x,y}(s) + \sigma_{x,y} + \psi_{y,s}(t)$ for some suitably-chosen Abelian group homomorphisms $\phi_{x,y},\psi_{y,x}{\colon}S {\rightarrow}S$, and family $\sigma = \{ \sigma_{x,y} \in S : x,y \in X \}$ of elements of $S$, then this is equivalent to the construction $E[{\mathcal{A}},\sigma]$ just discussed, for a homogeneous $X$–module ${\mathcal{A}} = (A,\phi,\psi)$. \[thm:rack-ext2\] \[prp:rack-ext2\] Let $\sigma$ and $\tau$ be factor sets corresponding to extensions of a rack $X$ by an $X$–module ${\mathcal{A}}$. Then the following are equivalent: 1. $E[{\mathcal{A}},\sigma]$ and $E[{\mathcal{A}},\tau]$ are equivalent extensions of $X$ by ${\mathcal{A}}$ 2. there exists a family $\upsilon = \{ \upsilon_x \in A_x : x \in X \}$ such that $$\label{eqn:rack-ext2} \tau_{x,y} = \sigma_{x,y} + \phi_{x,y}(\upsilon_x) + \psi_{y,x}(\upsilon_y) - \upsilon_{x^y}$$ for $x,y \in X$. 3. $\sigma$ and $\tau$ are factor sets of the same extension of $X$ by ${\mathcal{A}}$, relative to different sections. Let $\theta{\colon}E[{\mathcal{A}},\tau] {\cong}E[{\mathcal{A}},\sigma]$ be the hypothesised equivalence. Then it follows that $\theta(0,x) = (\upsilon_x,x)$ for some $\upsilon_x \in A_x$ and, furthermore, $$\theta(a,x) = \theta(a \cdot (0,x)) = a \cdot \theta(0,x) = a \cdot (\upsilon_x,x) = (a + \upsilon_x,x)$$ for all $a \in A_x$, since $\theta$ preserves the $A_x$–actions. Then $$\begin{aligned} &\theta\bigl((a,x)^{(b,y)}\bigr) = (\phi_{x,y}(a) + \psi_{y,x}(b) + \tau_{x,y} + \upsilon_{x^y}, x^y) \\ \mbox{and}\qquad&\theta(a,x)^{\theta(b,y)} = (a + \upsilon_x,x)^{(b+\upsilon_y,y)} = (\phi_{x,y}(a+\upsilon_x) + \psi_{y,x}(b+\upsilon_y) + \sigma_{x,y},x^y)\end{aligned}$$ which are equal since $\theta$ is a rack homomorphism, and so (\[eqn:rack-ext2\]) holds. This argument is reversible, showing the equivalence of the first two statements. Now, given such an equivalence $\theta$, define a section $s{\colon}X {\rightarrow}E[{\mathcal{A}},\tau]$ by $x \mapsto (\upsilon_x,x)$. Then the above argument also shows that $$\begin{aligned} &s(x)^{b \cdot s(y)} = (\upsilon_x,x)^{(\upsilon_y+b,y)} = (\sigma_{x,y} + \psi_{y,x}(b)) \cdot s(x^y) \\ \mbox{and}\qquad &(a \cdot s(x))^{s(y)} = (\upsilon_x+a,x)^{(\upsilon_y,y)} = (\sigma_{x,y} + \phi_{x,y}(a)) \cdot s(x^y)\end{aligned}$$ so $\sigma$ is the factor set of $E[{\mathcal{A}},\tau]$ relative to the section $s$. This property holds for any extension equivalent to $E[{\mathcal{A}},\tau]$. Conversely, if $\sigma$ and $\tau$ are factor sets of some extension $E$ of $X$ by ${\mathcal{A}}$ relative to different sections $s,t{\colon}X {\rightarrow}E$ then $s(x) = \upsilon_x \cdot t(x)$ for some $\upsilon_x \in A_x$, and so the first and third conditions are equivalent. The following corollary justifies the earlier assertion that the factor set is in some sense the obstruction to a section being a rack homomorphism. \[thm:rack-ext3\] \[cor:rack-ext3\] For an extension $f{\colon}E {\twoheadrightarrow}X$ by an $X$–module ${\mathcal{A}} = (A,\phi,\psi)$, the following statements are equivalent: 1. There exists a rack homomorphism $s{\colon}X {\rightarrow}E$ such that $fs = {\operatorname{Id}}_X$ 2. Relative to some section, the factor set of $E {\twoheadrightarrow}X$ is trivial 3. Relative to any section there exists, for the factor set $\sigma$ of $E {\twoheadrightarrow}X$, a family $\upsilon = \{\upsilon_x \in A_x : x \in X \}$ such that for all $x,y \in X$ $$\label{eqn:rack-ext3} \sigma_{x,y} = \phi_{x,y}(\upsilon_x) - \upsilon_{x^y} + \psi_{y,x}(\upsilon_y)$$ Extensions of this type are said to be [*split*]{}. We are now able to classify rack extensions: \[thm:rack-ext4\] Let $X$ be a rack and ${\mathcal{A}} = (A,\phi,\psi)$ an $X$–module. Then there is an Abelian group ${\operatorname{Ext}}(X,{\mathcal{A}})$ whose elements are in bijective correspondence with extensions of $X$ by ${\mathcal{A}}$. Let the set $Z(X,{\mathcal{A}})$ consist of extensions of $X$ by ${\mathcal{A}}$. As shown above, these are determined by factor sets $\sigma$ satisfying (\[eqn:rack-ext1\]). Defining an addition operation by $(\sigma+\tau)_{x,y} := \sigma_{x,y} + \tau_{x,y}$ gives this an Abelian group structure with the trivial factor set as identity. A routine calculation confirms that the set $B(X,{\mathcal{A}})$ of split extensions (equivalently, factor sets satisfying (\[eqn:rack-ext3\])) forms an Abelian subgroup of $Z(X,{\mathcal{A}})$, and so we may define ${\operatorname{Ext}}(X,{\mathcal{A}}) := Z(X,{\mathcal{A}})/B(X,{\mathcal{A}})$. In the case where ${\mathcal{A}}$ is a trivial homogeneous $X$–module (equivalently, an Abelian group $A$) the group ${\operatorname{Ext}}(X,{\mathcal{A}})$ coincides with $H^2(BX;A)$, the second cohomology group of the rack space of $X$ as defined by Fenn, Rourke and Sanderson [@fenn/rourke/sanderson:trunks]. Abelian extensions of quandles ------------------------------ We now turn our attention to the case where $X$ is a quandle. Extensions of $X$ by a quandle $X$–module ${\mathcal{A}}$ and their corresponding factor sets are defined in an analogous manner. \[thm:quandle-ext1\] \[prp:quandle-ext1\] Let $X$ be a quandle and ${\mathcal{A}} = (A,\phi,\psi)$ be a quandle module over $X$. Then extensions $f{\colon}E {\twoheadrightarrow}X$ such that $E$ is also a quandle are in bijective correspondence with factor sets $\sigma$ satisfying hypothesis (\[eqn:rack-ext1\]) of proposition \[thm:rack-ext1\] together with the additional criterion $$\label{eqn:quandle-ext1} \sigma_{x,x} = 0$$ for all $x \in X$. Following the reasoning of proposition \[thm:rack-ext1\], for $E$ to be a quandle is equivalent to the requirement that $$(a,x)^{(a,x)} = (\phi_{x,x}(a) + \sigma_{x,x} + \psi_{x,x}(a),x^x) = (a,x)$$ for all $x \in X$ and $a \in A_x$. Since ${\mathcal{A}}$ is a quandle module, this is equivalent to the requirement that (\[eqn:quandle-ext1\]) holds. We may now classify quandle extensions of $X$ by ${\mathcal{A}}$: \[thm:quandle-ext2\] For any quandle $X$ and quandle $X$–module ${\mathcal{A}}$, there is an Abelian group ${\operatorname{Ext}}_Q(X,{\mathcal{A}})$ whose elements are in bijective correspondence with quandle extensions of $X$ by ${\mathcal{A}}$. We proceed similarly to the proof of Theorem \[thm:rack-ext4\]. Let $Z_Q(X,{\mathcal{A}})$ be the subgroup of $Z(X,{\mathcal{A}})$ consisting of factor sets satisfying the criterion (\[eqn:quandle-ext1\]), and let $B_Q(X,{\mathcal{A}}) = B(X,{\mathcal{A}})$. Then we define ${\operatorname{Ext}}_Q(X,{\mathcal{A}}) = Z_Q(X,{\mathcal{A}})/B_Q(X,{\mathcal{A}})$. In the case where ${\mathcal{A}}$ is trivial homogeneous (and hence equivalent to an Abelian group $A$), extensions of $X$ by ${\mathcal{A}}$ correspond to Abelian quandle extensions, in the sense of Carter, Saito and Kamada [@carter/kamada/saito:diag] and so ${\operatorname{Ext}}_Q(X,{\mathcal{A}}) = H^2_Q(X;A)$. If the module ${\mathcal{A}}$ is a homogeneous Alexander module as defined in example \[exm:alexander\], then extensions of $X$ by ${\mathcal{A}}$ are exactly the twisted quandle extensions described by Carter, Saito and Elhamdadi [@carter/elhamdadi/saito:twisted], and so ${\operatorname{Ext}}_Q(X,{\mathcal{A}}) = H^2_{TQ}(X;A)$. [99]{} , [Matías Graña]{}, *From racks to pointed Hopf algebras*, Advances in Mathematics **178** (2003) 177–243 , [Jonathan Beck]{}, *Homology and standard constructions*, from: “Seminar on Triples and Categorical Homology Theory”, volume 80 of Lecture Notes in Mathematics, Springer–Verlag (1969) 245–335 , *Triples, algebras and cohomology*, PhD thesis, Columbia University (1967). Republished as: Reprints in Theory and Applications of Categories **2** (2003) 1–59 , *Automorphic sets and singularities*, Contemporary Mathematics **78** (1988) 45–115 , [Samuel Eilenberg]{}, *Homological Algebra*, Princeton University Press (1999) , [Mohamed Elhamdadi]{}, [Masahico Saito]{}, *Twisted quandle homology theory and cocycle knot invariants*, Algebraic and Geometric Topology **2** (2002) 95–135 , [Seiichi Kamada]{}, [Masahico Saito]{}, *Diagrammatic computations for quandles and cocycle knot invariants*, from: “Diagrammatic morphisms and applications (San Francisco, CA, 2000)”, Contemporary Mathematics **318** (2003) 51–74 , [Gavin Wraith]{}, unpublished correspondence (1959) , [Matías Graña]{}, *On rack cohomology*, Journal of Pure and Applied Algebra **177** (2003) 49–59 , [Colin Rourke]{}, *Racks and links in codimension 2*, Journal of Knot Theory and its Ramifications **1** (1992) 343–406 , [Colin Rourke]{}, [Brian Sanderson]{}, *Trunks and classifying spaces*, Applied Categorical Structures **3** (1995) 321–356 , *Homological algebra of racks and quandles*, PhD thesis, Mathematics Institute, University of Warwick (2004) , *A classifying invariant of knots: the knot quandle*, Journal of Pure and Applied Algebra **23** (1982) 37–65 , *The structure of racks*, PhD thesis, Mathematics Institute, University of Warwick (1993)

ArXiv

--- abstract: 'In this paper, we explore propagation of energy flux in the future Poincaré patch of de Sitter spacetime. We present two results. First, we compute the flux integral of energy using the symplectic current density of the covariant phase space approach on hypersurfaces of constant radial physical distance. Using this computation we show that in the tt-projection, the integrand in the energy flux expression on the cosmological horizon is same as that on the future null infinity. This suggests that propagation of energy flux in de Sitter spacetime is sharp. Second, we relate our energy flux expression in tt-projection to a previously obtained expression using the Isaacson stress-tensor approach.' --- **** Sk Jahanur Hoque$^1$ and Amitabh Virmani$^{1,2,3, }$[^1] $^1$Chennai Mathematical Institute, H1 SIPCOT IT Park,\ Kelambakkam, Tamil Nadu, India 603103\ $^2$Institute of Physics, Sachivalaya Marg,\ Bhubaneswar, Odisha, India 751005\ $^3$Homi Bhabha National Institute, Training School Complex,\ Anushakti Nagar, Mumbai 400085, India\ \ $ \, $\ Introduction ============ The era of gravitational wave astronomy has begun [@Abbott:2016blz; @TheLIGOScientific:2017qsa; @GBM:2017lvd; @Coulter]. It is now all the more important that our theoretical understanding be at par with the impressive experimental developments that have gone into the discovery of gravitational waves. There are several theoretical aspects that are potentially important in relation to generation and propagation of gravitational waves but have not been fully explored. One such aspect is the effect of the positive cosmological constant on the propagation of gravitational waves. The discovery of the accelerated expansion of the universe from distant supernovae and cosmic microwave background surveys have shown that around 68% of the energy density of the universe is dark energy. While at a fundamental level dark energy is poorly understood, the positive cosmological constant is the simplest explanation of it. From the theoretical point of view, positive cosmological constant posses numerous challenges in relation to study of gravitational waves. In a recent series of papers Ashtekar, Bonga, and Kesavan [@ABKI; @ABKII; @ABKIIIPRL; @ABKIII] have systematically initiated the study of gravitational waves focusing on the numerous effects that the presence of a positive cosmological constant brings. Subsequently, several authors have contributed to the development of the subject [@DHI; @DHII; @Bishop:2015kay; @Bonga; @JA]. The primary aim of this work is to expand on some of these studies, in particular on some aspects of [@DHI; @DHII], and to clarify their relation to [@ABKII; @ABKIII]. In comparison to Minkowski spacetime there are several effects that the positive cosmological constant brings on the propagation of linearised gravitational field. For a detailed discussion of these points, we refer the reader to [@ABKII; @ABKIII]; here we wish to focus on two points especially. First, while wavelengths of linear waves remain constant in flat space, they increase in de Sitter spacetime as the universe undergoes de Sitter expansion. So much so that in the asymptotic region of interest, the wavelengths diverge. Naively, this seems to invalidate the geometrical optics approximation commonly used in the gravitational waves literature. Secondly, due to the curvature of the background spacetime, the linear gravitational field satisfies a *massive* wave equation, i.e., propagation of waves in de Sitter spacetime is not on the light cone. Due to backscattering from the background curvature, in general, there is a tail term. Partial understanding of these effects is already available. Our study expands on that knowledge. Firstly, although in the asymptotic region of interest, wavelengths diverge, reference [@DHII] made precise how the geometrical optics approximation is still useful. They arrived at an effective stress tensor for gravitational waves following the original work of Isaacson [@Isaacson:1968zza; @Isaacson:1967zz]. An aim of this paper is to re-obtain appropriate version of those expressions from the covariant phase space approach, thus clarifying their relation to [@ABKII; @ABKIII]. The second aim of the paper is to make precise the notion of the “sharp" propagation of energy flux in de Sitter spacetime, i.e., to understand in what sense the tail term mentioned above does not matter for radiated energy flux. The rest of the paper is organized as follows. We start with a brief review of linearised gravity on de Sitter spacetime in section \[sec:linear\] and write various identities involving derivatives of the radiative field that we need in later sections. In section \[sec:current\_and\_energy\] we compute the symplectic current density for linearised gravity on de Sitter spacetime and write a general expression for the energy flux through a hypersurface $\Sigma$. Since symplectic current density is conserved, it allows us to compute energy flux through any hypersurface. In section \[sec:tt\] we use the general expression obtained in section \[sec:current\_and\_energy\] to compute the flux integrals on hypersurfaces of constant radial physical distance. These hypersurfaces allow us to interpolate between the cosmological horizon and the future null infinity. We show that in the tt-projection, the integrand in the energy flux expression on the cosmological horizon is same as that on the future null infinity. This suggests that the propagation of energy flux in de Sitter spacetime is sharp. We also relate our energy flux expression to the previously obtained expression of reference [@DHII]. This section constitutes the main results of our work. We close with a discussion in section \[sec:disc\]. ![The full square is the Penrose diagram of global de Sitter spacetime, with each point representing a 2-sphere. In this paper we exclusively work in the future Poincaré patch of de Sitter spacetime — the upper triangular region (red triangle) of this diagram. Blue lines denote hypersurfaces of constant radial physical distance. These hypersurfaces are generated by the time-translational (dilatation) Killing vector. On these hypersurfaces, $\tau$ is a Killing parameter that runs from $-\infty$ to $\infty$. The dotted lines are lines of constant retarded time. Green line is the worldline of the radiating source. []{data-label="PoincarePatch"}](GlobalConformalChart.pdf){width="70.00000%"} Linearised gravity and various identities involving radiative field {#sec:linear} =================================================================== Linearised gravity on de Sitter spacetime ----------------------------------------- We are interested in linearised gravity over de Sitter background. We exclusively work in the future Poincaré patch of de Sitter spacetime. The background de Sitter metric in the Poincaré patch is ds\^2 = |g\_ dx\^dx\^= a\^[2]{} ( - d\^2 + dx\^2), \[background\] a=-(H)\^[-1]{}, H=, where $\Lambda$ is the positive cosmological constant. Linearised perturbations over the background are written as g\_ = |g\_ + \_. Coordinates $x_i$, with $i = 1,2,3$, ranges from $(-\infty, \infty)$, whereas coordinate $ \eta$ takes values in the range $(-\infty, 0)$, with $\eta = 0$ at the future null infinity $\mathcal{I}^+$. The future infinity $\mathcal{I}^+$ is a spacelike surface, see figure \[PoincarePatch\]. For the background metric the Christoffel symbol is \^\_[c]{} = - ( \^0\_\^\_+ \^0\_c\^\_+ \^\_0 \_[c]{}). Using this useful expressions for the d’Alembertian and for various other derivative operators can be written, see e.g. [@deVega:1998ia; @DHI]. In terms of the trace reversed combination $\hat{\gamma}_{\alpha\beta}:=\gamma_{\alpha\beta}-\frac{1}{2}\bar{g} _{\alpha\beta} \ (\bar{g}^{\mu\nu}\gamma_{\mu\nu})$, the linearised Einstein equations take the form, $$\frac{1}{2}\left[ - \overline{\Box} \hat{\gamma}_{\mu\nu} + \left\{ \overline{\nabla}_{\mu}B_{\nu} + \overline{\nabla}_{\nu}B_{\mu} - \bar{g}_{\mu\nu}(\overline{\nabla}^{\alpha}B_{\alpha})\right\}\right] + \frac{\Lambda}{3}\left[\hat{\gamma}_{\mu\nu} - \hat{\gamma}\bar{g}_{\mu\nu}\right] ~ = ~ 8\pi G T_{\mu\nu} \label{LinEqn} $$ where $B_{\mu} := \overline{\nabla}_{\alpha}\hat{\gamma}^{\alpha}_{~\mu}$ and $\overline{\nabla}_{\alpha}$ is the metric compatible covariant derivative with respect to the background metric $\bar g_{\a \b}$. As is well known in the literature [@deVega:1998ia; @DHI; @ABKIII], these equations written in terms of a rescaled variables leads to a great deal of simplification. We define, \_:=a\^[-2]{}\_, and using the gauge condition [@deVega:1998ia], \^\_ + (2 \_[0]{} + \_\^0 \_\^[ ]{} ) = 0, \[ChiGauge\] equation becomes, - 16 G T\_ = \_ + \_0\_ - (\_\^0\_\^0\_\^[ ]{} + \_\^0\_[0]{} + \_\^0\_[0]{} ) \[LinEqnChi\], where $\Box$ is simply the d’Alembertian with respect to Minkowski metric in cartesian coordinates, $\Box = - \partial_\eta^2 + \partial_i^2$. In terms of variables $ \hat{\chi} :=\chi_{00}+\chi_{~i}^{i}, ~\chi_{0i},~ \chi_{ij} $ equation (\[LinEqnChi\]) decomposes into three decoupled equations () &=& - , \[decoupled1\]\ ()&=&- , \[decoupled2\]\ \_[ij]{} + \_0 \_[ij]{} &=& - 16G T\_[ij]{}, \[decoupled3\] where $\hat{T}:=T_{00}+T_{~i}^{i}$. Under a linearised diffeomorphism $\xi^{\mu}$, $\chi_{\mu\nu}$ transforms as, \_ = (\_\_ + \_\_ - \_\^\_) - \_\_0, where \_ := a\^[-2]{} \_ = \_\^. \[ChiGaugeTrans\] A small calculation then shows that the gauge condition (\[ChiGauge\]) is preserved under transformations generated by vector fields $\xi^{\mu}$ — the residual gauge transformations — satisfying, $$\Box\underline{\xi}_{\mu} + \frac{2}{\eta}\partial_0\underline{\xi}_{\mu} - \frac{2}{\eta^2}\delta_{\mu}^0\underline{\xi}_0= 0. \label{ResidualChiGauge}$$ Under these residual gauge transformations equation is also invariant. We can exhaust the residual gauge freedom as follows. We note that $\delta{\hat{\chi}}$ satisfies, $$\begin{aligned} \Box~{\big(\delta{\hat{\chi}}\big)}&=& 4~\bigg[\Box~\bigg(\partial_0 \underline{\xi}_0- \frac{\underline{\xi}_0}{\eta}\bigg)\bigg] \\ &=&-\frac{2}{\eta}\partial_0~ \big(\delta{\hat{\chi}}\big)+\frac{2}{\eta^{2}}~\big(\delta{\hat{\chi}}\big) \eea where in going from the first step to the second step we have used \eqref{ResidualChiGauge}. This form of the equation implies that, \be \Box~\bigg(\frac{\delta{\hat{\chi}}}{\eta}\bigg)=0, \ee i.e., $\delta{\hat{\chi}}$ satisfies the wave equation \eqref{decoupled1} outside the source. Therefore, using an appropriate residual gauge transformation we can set $ \hat{\chi}=0$ outside the source. Similarly $\chi_{0i}$ can be set to zero outside the source \cite{deVega:1998ia}. Gauge condition (\ref{ChiGauge}) then implies $\partial^0\chi_{00} = 0$. Choosing $\chi_{00}$ to be zero at some initial $\eta = $ constant hypersurface we can take $\chi_{00} = 0$ everywhere. Doing so, gauge condition (\ref{ChiGauge}) becomes \be \partial^{i}\chi_{ij} = \chi^{i}_{~i} = 0. \label{TTconditions} \ee \subsection{TT-gauge vs tt-projection} With conditions \eqref{TTconditions} imposed there are no further gauge transformations allowed. Thus, transverse and traceless (TT) solutions are fully gauge fixed. Therefore, away from the source it suffices to focus on equation (\ref{decoupled3}). In general, solutions of this inhomogeneous equation do not satisfy the TT conditions. However, any spatial rank-2 symmetric tensor can be decomposed into its irreducible components as, \begin{equation} \chi_{ij}=\frac{1}{3}\delta_{ij}\delta^{kl}\chi_{kl}+(\partial_{i}\partial_{j}- \frac{1}{3}\delta_{ij} \nabla^{2})B+\partial_{i}B_{j}^{\mathrm{T}}+\partial_{j}B_{i}^{\mathrm{T}}+\chi_{ij}^{\mathrm{TT}}, \end{equation} where $\chi_{ij}^{\mathrm{TT}} $ refers to the transverse-traceless part of the field $\chi_{ij}$, i.e., it satisfies, \be \partial^{i} \chi_{ij}^{\mathrm{TT}}=\delta^{ij} \chi_{ij}^{\mathrm{TT}}=0. \ee The vector $B_{i}^{\mathrm{T}}$ is transverse, $\partial^i B_i^{\mathrm{T}} = 0.$ In this decomposition only $ \chi_{ij}^{\mathrm{TT}}$ is the gauge invariant piece. Hence, $\chi_{ij}^{\mathrm{TT}}$ is best regarded as the physical component of the field $\chi_{ij}$. Given a tensor $\chi_{ij}$, in general it is highly non-trivial to extract $\chi_{ij}^{\mathrm{TT}}$; see \cite{Bonga} for an explicit example. In the context of gravitational waves, another conceptually distinct notion of transverse-traceless tensors is often used in the literature. This notion is operationally simpler but inequivalent to the above notion. Here one `extracts' the `transverse-traceless' part of a rank-two tensor simply by defining an algebraic projection operator, \begin{align} P_i^{~j} &= \delta_i^{~j} - \hat{x}_i\hat{x}^{j}, & \Lambda_{ij}^{~~kl} &= \frac{1}{2}(P_i^{~k}P_j^{~l} + P_i^{~l}P_j^{~k} - P_{ij}P^{kl}), \end{align} where $\hat x^i = x^i/r$ with $r= \sqrt{x^i x_i}$. In order to distinguish it from the the above notion, we use the notation $\chi_{ij}^{\mathrm{tt}}$, \be \chi_{ij}^{\mathrm{tt}} := \Lambda_{ij}^{~~kl}\chi_{kl} \label{ttProjection} \ee For a detailed discussion of the differences between these two notions see \cite{Ashtekar:2017ydh, Ashtekar:2017wgq}. For asymptotically flat space-times the two notions match only at null infinity $\mathcal{I}^+$ \cite{Ashtekar:2017wgq, Bonga}. The tt-projection is well tailored to the $1/r$ expansion commonly used for asymptotically flat spacetimes. The global structure of de Sitter spacetime is very different from Minkowski spacetime. Expansion in powers of $1/r$ is not a useful tool to analyse asymptotically de Sitter spacetimes. In particular, the radial tt-projection is not a valid operation to extract the transverse-traceless part of a rank-2 tensor on the full $\mathcal{I}^+$. The TT-tensor is the correct notion of transverse traceless tensors. However, if one restricts oneself to large radial distances away from the source, one may expect that the tt-projection also gives useful answers. In fact, it appears to work better than expected. In the context of the power radiated by a spatially compact circular binary system analysed in \cite{Bonga,JA} the difference does not seem to matter. The tt-projection being algebraic allows us to do various non-trivial computations which seem difficult to perform otherwise. In particular, this simplicity allows us to gain a physical understanding of the propagation of gravitational waves in de Sitter spacetime. In this paper we mostly restrict ourselves to tt-projection, with the understanding that our results need to be generalised to TT gauge. A detailed study of this we leave for future research. \subsection{Derivatives of radiative field} \label{Identity} In order to compute energy flux through different slices, we need various derivatives of radiative field $\chi_{ij}$. In this subsection we establish those identities. The expression for radiative $\chi_{ij}$ we use was obtained in references \cite{ABKIII, DHI}, \be \chi_{ij} (\eta, r) = 4 G\frac{\eta}{r (\eta - r)} \int d^3 x' T_{ij}(\eta - r, x') + 4 G \int_{-\infty}^{\eta - r} d\eta' \frac{1}{\eta'{}^2}\int d^3 x' T_{ij}(\eta', x'), \ee where $T_{ij}$ is the source energy-momentum tensor. In arriving at this expression, Green's functions for the differential operator in \eqref{decoupled3} is used together with the approximation \be \eta - | \vec x - \vec x'| \approx \eta - | \vec x| = \eta - r, \ee in order to pull the factor of $\frac{1}{r(\eta - r)}$ out from the integral. The integral of the stress tensor can be expressed in terms of the mass and pressure quadrupole moments $Q_{ij}$ and $\overline{Q} _{ij}$ at the retarded time $\eta_{\mathrm{ret}}:=\eta - r$ \cite{ABKIII, DHI}, \be \int d^3 x' T_{ij}(\eta - r, x') = \frac{1}{2 a(\eta_{\mathrm{ret}})} \left( \ddot{Q}_{ij} + 2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2 {\overline{Q}} _{ij} \right) (\eta_{\mathrm{ret}}), \label{moments} \ee where dots denote Lie derivatives with respect to time-translation (dilatation) Killing vector \be T^{\mu} \partial_\mu = - H ( \eta \partial_\eta + r \partial_r). \label{time_translation} \ee The mass and pressure quadrupole moments $Q_{ij}$ and $\overline{Q} _{ij}$ are defined as an integrals over the source at some fixed time $\eta$, \bea Q_{ij} (\eta)= \int \ a^3 (\eta) T_{00} (\eta, x) x_i x_j d^3 x, \eea \bea \overline{Q}_{ij} (\eta)= \int \ a^3 (\eta) \delta^{kl}T_{kl}(\eta, x) x_i x_j d^3 x. \eea Using these expressions, we get the identities \be \partial_\eta \chi_{ij} (\eta, x) = 4G\frac{\eta}{(\eta - r)r} \partial_\eta \left[ \int d^3 x' T_{ij}(\eta - r, x') \right] =: \frac{2G\eta}{(\eta - r)r} R_{ij} (\eta_{\mathrm{ret}}). \ee where \be R_{ij}(\eta_{\mathrm{ret}}) ~ = ~\bigg[\dddot{Q}_{ij} + 3H\ddot{Q}_{ij} + 2H^2\dot{Q}_{ij} + H\ddot{\overline{Q}}_{ij} + 3H^2\dot{\overline{Q}}_{ij} + 2H^3\overline{Q}_{ij}\bigg](\eta_{\mathrm{ret}}) . \ee Similarly, \bea \partial_r \chi_{ij} &=& - \partial_\eta \chi_{ij} - \frac{4}{r^2}\int d^3 x' T_{ij} (\eta - r, x') \label{useful_identity_DH} \\ &=&-\frac{2G\eta}{r (\eta - r)} R_{ij}(\eta_{\mathrm{ret}})+ 2H G \ \frac{(\eta -r)}{r^{2}}\left( \ddot{Q}_{ij} + 2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2 \overline{Q} _{ij} \right)(\eta_{\mathrm{ret}}). \eea As a result \bea (T \cdot \partial) \chi_{ij} &=& - H (\eta \partial_\eta + r \partial_r ) \chi_{ij} \\ &=&-\frac{2G H \eta}{r} R_{ij} (\eta_{\mathrm{ret}}) -2GH^{2} \bigg(\frac{\eta - r }{r}\bigg)\left( \ddot{Q}_{ij} + 2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2 \overline{Q}_{ij} \right) (\eta_{\mathrm{ret}}). \label{T_dot_partial_chi} \eea For later convenience we also define \be A_{ij}=\ddot{Q}_{ij}+2H\dot{Q}_{ij}+H\dot{\overline{Q}}_{ij}+2H^{2} \overline{Q}_{ij}. \label{A_def} \ee This quantity is interesting as it satisfies the relations \be R_{ij}=\dot{A}_{ij}+HA_{ij}= (T \cdot \partial) A_{ij}-HA_{ij}, \label{R_A_old} \ee which we will need later. On the future cosmological horizon of the source defined by \be \cH^+: \qquad \eta + r =0, \ee equation \eqref{moments} simplifies to, \be \left[\int d^3 x' T_{ij}(\eta - r, x')\right]\Bigg{|}_{\cH^+} = (H r)\left( \ddot{Q}_{ij} + 2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2 \overline{Q}_{ij} \right) (\eta_{\mathrm{ret}}), \ee and equation \eqref{T_dot_partial_chi} simplifies to, \be (T \cdot \partial) \chi_{ij} \Big{|}_{\cH^+} = 2G H R_{ij} (\eta_{\mathrm{ret}}) + 4G H^2 \left( \ddot{Q}_{ij} + 2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2 \overline{Q}_{ij} \right) (\eta_{\mathrm{ret}}). \ee \section{Symplectic current density and energy flux} \label{sec:current_and_energy} We are interested in computing energy flux through any Cauchy surface and more generally through other surfaces. Perhaps the most convenient way to do this is via the covariant phase space approach. For linearised gravity, the covariant phase space can be taken to be simply the space of solutions $\gamma_{ab}$ of the linearised Einstein's equations together with appropriate gauge conditions \cite{ABKII}. A standard procedure \cite{ABR, LeeWald} then gives a symplectic structure. When restricted to cosmological slices, the symplectic structure was computed and used in \cite{ABKII, ABKIII}. In this work we are interested in other slices. In our discussion below we focus on the symplectic current density and its integrals, rather than on the careful construction of the phase space itself. The phase space construction is somewhat subtle \cite{ABKII} due to certain divergences as the future null infinity $\mathcal{I}^+$ is approached. Some of our intermediate expressions below are formally divergent as the future null infinity is approached, however, our final answers are all finite and have a well defined limit at $\mathcal{I}^+$. We start with an expression of symplectic current of linearised Einstein gravity with a cosmological constant, which we can evaluate on different slices. A convenient form is \cite{Hollands:2005wt}, \be \omega^\a =\frac{1}{32\pi G} \ P^{ \a \b \c \d \e \f } \left( \delta_1 g_{ \b \c } \overline{\nabla}_{ \d } \delta_2 g_{\e \f } - \delta_2 g_{\b \c } \overline{\nabla}_{\d} \delta_1 g_{\e \f }\right), \label{omega_exp_1} \ee where \be P^{\a \b \c \d \e \f} = \bar g^{\a \e} \bar g^{\f \b} \bar g^{\c \d} - \frac{1}{2} \bar g^{\a \d} \bar g^{\b \e} \bar g^{\f \c} - \frac{1}{2} \bar g^{\a \b} \bar g^{\c \d} \bar g^{\e \f} - \frac{1}{2} \bar g^{\b \c} \bar g^{\a \e} \bar g^{\f \d} + \frac{1}{2} \bar g^{\b \c} \bar g^{\a \d} \bar g^{\e \f}. \ee We use the notation \bea \delta_1 g_{\a \b} &=& \gamma_{\a \b}, \\ \delta_2 g_{\a \b} &=& \widetilde \gamma_{\a \b}, \eea where $\gamma_{\a \b}$ and $\widetilde \gamma_{\a \b}$ are fully gauge fixed physical solutions of the (homogeneous) linearised Einstein equations. We take them to satisfy Lorentz and radiation gauge, \be \overline{\nabla}^\a {\gamma}_{\a\b} = 0, \qquad \gamma_{0\a} = 0, \qquad \bar{g}^{\a\b} \gamma_{\a\b} = 0. \label{gauge_gamma} \ee These gauge conditions are the same as \eqref{TTconditions}. Since $\gamma_{\a\b}$ and $\widetilde \gamma_{\a\b}$ are both traceless, the last three terms in $P^{\a \b \c \d \e \f}$ do not contribute to the symplectic current $\omega^\a$. We effectively have \be P^{\a \b \c \d \e \f} = \bar g^{\a \e} \bar g^{\f \b} \bar g^{\c \d} - \frac{1}{2} \bar g^{\a \d} \bar g^{\b \e} \bar g^{\f \c}. \label{P_simple} \ee Expanding out the covariant derivatives in \eqref{omega_exp_1} in terms of the Christoffel symbols we get a simplified expression, \bea \omega^\a &=& \frac{1}{32\pi G} \ P^{\a \b \c \d \e \f} \gamma_{\b \c} \left(\partial_\d \widetilde \gamma_{\e \f} - \overline{\Gamma}^\m_{\d \e} \widetilde \gamma_{\m \f} - \overline{\Gamma}^\m_{\d \f} \widetilde \gamma_{\e \m} \right) - (1 \leftrightarrow 2), \eea with $P^{\a \b \c \d \e \f}$ given in \eqref{P_simple}. \subsubsection*{Time component} Using the simplified expressions above, the time component of the symplectic current is \be \omega^\eta = \frac{1}{64\pi G} (H^2 \eta^2) \left(\gamma^{\b \c} \partial_\eta \widetilde \gamma_{\b \c} -\widetilde \gamma^{\b \c} \partial_\eta \gamma_{\b \c} \right). \ee We note that due to the gauge conditions \eqref{gauge_gamma}, $\gamma_{\a \b}$ has only spatial components. In terms of the rescaled field $\gamma_{ij} = a^2 \chi_{ij}$, we have \be \label{SympCur1} \omega^\eta = \frac{1}{64 \pi G} (H^2 \eta^2) \left(\chi^{ij} \partial_\eta \widetilde \chi_{ij} -\widetilde \chi_{ij} \partial_\eta \chi_{ij}\right). \ee This expression matches with the corresponding expression in reference \cite{ABKII}. In such expressions TT superscript on $\chi_{ij}$ is implicit. \subsubsection*{Space components} A similar calculation gives \be \label{SympCur2} \omega^i = \frac{1}{32\pi G}a^{-2} \delta^{ij} \left\{ \chi^{lm} \partial_m \widetilde \chi_{jl} - \frac{1}{2} \chi^{lm} \partial_j \widetilde \chi_{lm} - \widetilde \chi^{lm} \partial_m \chi_{jl} + \frac{1}{2} \widetilde \chi^{lm} \partial_j \chi_{lm} \right\}. \ee \subsubsection*{Energy flux} From general results on the covariant phase space approach \cite{ABR, LeeWald, ABKII}, it follows that the energy flux (Hamiltonian for time-translation symmetry $T$) is given as \be E_T (\gamma) = -\int_{\Sigma} \omega^\alpha (\gamma, \pounds_T \gamma) \left(n_{\alpha} \sqrt{h_\Sigma}d^3 \x\right), \label{ET} \ee where $h_\Sigma$ is the determinant of the induced metric on the slice $\Sigma$ with coordinates $\xi^i$ and $n^\alpha$ is the future directed normal vector to the slice $\Sigma$. In this expression we have evaluated the symplectic current density with $\widetilde \gamma_{\a\b} = \pounds_T \gamma_{\a\b}$. For use in equations \eqref{SympCur1} and \eqref{SympCur2}, we need to evaluate $\widetilde \chi_{ij} = a^{-2} \widetilde \gamma_{ij} = a^{-2} \pounds_T \gamma_{ij}$. This quantity is computed to be \bea \widetilde \chi_{ij} = a^{-2} \ \pounds_T (a^2 \chi_{ij}) = (T \cdot \partial) \chi_{ij}. \eea As a result we have the following components of the current $j^\a := \omega^\alpha (\gamma, \pounds_T \gamma)$ for computing the energy flux, \bea j^\eta &=& \frac{1}{64\pi G} \ (H^2 \eta^2) \left(\chi^{ij} \partial_\eta \left[ (T \cdot \partial) \chi_{ij} \right] - (T \cdot \partial) \chi_{ij} \partial_\eta \chi^{ij}\right), \label{EFI} \\ j^i &=& \frac{1}{32\pi G} \ (H^2 \eta^2) \delta^{ik} \left\{ \chi^{lm} \partial_m \left[ (T \cdot \partial) \chi_{kl} \right] - \frac{1}{2} \chi^{lm} \partial_k \left[ (T \cdot \partial) \chi_{lm} \right] \right. \nn \\ & &\qquad \qquad \qquad \qquad \left. - \left[ (T \cdot \partial) \chi^{lm} \right] \partial_m \chi_{kl} + \frac{1}{2} \left[ (T \cdot \partial) \chi^{lm} \right] \partial_k \chi_{lm} \right\}.\label{EFII} \eea Since $j^\a$ is conserved, we can use it to compute flux across any hypersurface. In this paper we will restrict ourselves to hypersurfaces generated by the time-translation Killing vector $T$. Near the future null infinity $\mathcal{I}^+$ these hypersurfaces are spacelike. Inside the cosmological horizon $\mathcal{H}^+$ these hypersurfaces are timelike. See figure~\ref{PoincarePatch}. \section{Energy flux in tt-projection} \label{sec:tt} In this section we compute the energy flux across hypersurfaces generated by the time-translation Killing vector $T$. We exclusively work with tt-projection. We start by observing some useful properties of the tt-projection, \begin{eqnarray} \partial_{\eta}(\chi_{ij}^{\mathrm{tt}}(\eta, r)) & = & (\partial_{\eta}\chi_{ij}(\eta, r))^{\mathrm{tt}}, \label{tt_commute1} \\ \partial_{r}(\chi_{ij}^{\mathrm{tt}}(\eta, r)) &=& (\partial_{r}\chi_{ij}(\eta, r))^{\mathrm{tt}}, \label{tt_commute2} \eea i.e., tt-projection commutes with $\partial_\eta$ and $\partial_r$. Moreover, \be \partial_m(\chi_{ij}^{\mathrm{tt}}(\eta, r)) = (\partial_{m}\Lambda_{ij}^{~~kl})\chi_{kl}(\eta, r) + \hat{x}_m(\partial_r\chi_{ij}(\eta, r))^{\mathrm{tt}} , \ee as a result we have, \bea \partial^j(\chi_{ij}^{\mathrm{tt}}(\eta, r)) & = & \hat{x}^j\Lambda_{ij}^{~~kl}\partial_r\chi_{ij}(\eta, r) + (\partial^{j}\Lambda_{ij}^{~~kl})\chi_{kl}(\eta, r) \nn \\ &=& \mathcal{O}(r^{-1}), \label{Transversality} \eea where we used, \bea \partial_{m}\Lambda_{ij}^{~~kl} & = & - \frac{1}{r}\left[\hat{x}_i\Lambda_{mj}^{~~~kl} + \hat{x}_j\Lambda_{mi}^{~~~kl} + \hat{x}^k\Lambda_{ijm}^{~~~~l} + \hat{x}^l\Lambda_{ijm}^{~~~~k}\right] = \mathcal{O}(r^{-1}) \ . \label{ProjectorDerivative} \end{aligned}$$ The traceless-ness of $\chi_{ij}^{\mathrm{tt}}$ is manifest, but $\chi_{ij}^{\mathrm{tt}}$ satisfies the spatial transversality condition to $\mathcal{O}(r^{-1})$ only, cf. . Energy flux across hypersurfaces of constant radial physical distance {#energy_flux} --------------------------------------------------------------------- Hypersurfaces of constant radial physical distance can be defined as, \_: := a () r = - = . These hypersurfaces are generated by the time-translation Killing vector $T$, cf. , T = T\^\_= - H (\_+ r \_r).Let $\tau$ be the Killing parameter along the integral curves of the Killing vector $T$ satisfying, $$\begin{aligned} \frac{d\eta}{d\tau}&=-H\eta,& \frac{d x^i}{d\tau}&=-H x^i, & \label{KillingParameter}\end{aligned}$$ then, coordinates on $\Sigma_\rho$ can be taken to be $\tau, \theta,$ and $\phi$. The induced metric on $\Sigma_\rho$ is, h\_[ab]{} = (H\^2\^2 - 1, \^2, \^2 \^2). This metric is of Lorentzian signature for $H\rho < 1$ (inside the cosmological horizon), is degenerate for $H\rho = 1$ (the cosmological horizon), and is of Euclidean signature for $H\rho > 1$ (outside the cosmological horizon); see figure \[PoincarePatch\]. In this subsection we work with the timelike and spacelike cases; the case of the null cosmological horizon is considered in the next subsection (section \[cosmological\_horizon\]). The volume element for the non-null cases is =  \^2 , and the unit normal is n\_ =  a | H\^2\^2-1|\^[-1/2]{}(H, x\_i/r). Here $\epsilon = +1$ for time-like hypersurfaces $H\rho < 1$, and $-1$ for space-like hypersurfaces $H\rho > 1$. Therefore, the infinitesimal volume element vector field is [@Poisson] d \_= n\_  d\^3 = a\^[3]{} r\^2  (H, ) d d d . The hypersurface integral for energy flux is then written as, \[flux\_physical\_radius\] E\_T = -\_[\_]{}d \_ j\^ & = & -\_[-]{}\^[+]{}d\_[S\^2]{}d  r\^[2]{} a\^[3]{} (H j\^+ ), where $\tau$ is the Killing parameter defined in . Using , this expression can be rewritten as, E\_T =- \_[\_ ]{} a\^[4]{} ( j\^+ j\^r ) d\^3 x. \[omega\_rho\] In this expression, both terms diverge as $\eta \rightarrow 0$, or as $\rho \to \infty$. It is easily seen from and that $j^{\eta}$ term diverges as $\eta^{-2} $ while $j^r$ term diverges as $\eta^{-1}$. This situation is similar to $E_T$ evaluated on constant $\eta$ slices in [@ABKII]. We will see below that, as in [@ABKII], the divergent pieces turn out to be total derivative. ### $j^{\eta}$ contribution {#jeta-contribution .unnumbered} Let us first look at the $j^{\eta}$ part of integral , we call it $E_T^{(1)}$, E\_T\^[(1)]{} &=&-\_[-]{}\^[+]{}d\_[S\^2]{}d  r\^[2]{} a\^[3]{} (H j\^)\ &=& - \_[-]{}\^[+]{} d \_[S\^2]{} d  r\^[2]{} a\^[3]{}  H a\^[-2]{}\ &=& H\^2 \^[3]{} \_[-]{}\^[+]{} d \_[S\^2]{} d\ &=&-H\^2 \^[3]{} {\_[-]{}\^[+]{} d \_[S\^2]{} d - \_[-]{}\^[+]{} d d },\ \[omega\_eta\_final\] where we have done the following manipulations. In the first step we have substituted . In the second step we have used the property that $T \cdot \partial = \frac{d}{d\tau}$ and $ \partial_{\eta}[\frac{d}{d\tau}]=\frac{d}{d\tau}[\partial_{\eta}]-H \partial_{\eta}$. In the third step we have done integrations by part with respect to the Killing parameter $\tau$ and have made use of equation . This integration by parts is valid because $\frac{d} {d\tau}$ is tangential to $\Sigma_\rho$. The second term in expression is a total derivative. This integral is zero for the following reasons. On timelike $\rho$ = constant hypersurfaces, $ \tau=+\infty$ corresponds to future timelike infinity $i^+ $ and $\tau=-\infty$ corresponds to past timelike infinity $i^- $. Assuming that the source is static at the boundary points [@ABKIII; @DHI], i.e., $ \frac{dQ_{ij}}{d\tau}\big|_{\tau=\pm\infty}=0$ and $\frac{d\overline{Q} _{ij}}{d\tau}\big|_{\tau= \pm\infty}=0$, $\chi_{ij}$ vanishes at $i^+$ and $i^-$. Hence the end point contributions in the integral vanish for timelike hypersurfaces. On spacelike $\rho$ = constant hypersurfaces, $\tau=+\infty$ corresponds to future timelike infinity $i^+ $ and $\tau=-\infty$ corresponds to spatial infinity $i^0$; see figure \[PoincarePatch\]. $\chi_{ij}$ vanishes at $i^0$ due to no incoming radiation boundary conditions at $\eta = - \infty$. Hence the end point contributions in the integral also vanish for spacelike hypersurfaces. ### $j^{i}$ contributions {#ji-contributions .unnumbered} Let us now look at the $j^{i}$ part of integral . We call this piece $E_T^{(2)}$. Upon substituting we get four terms. We separate the contributions of these terms based on their derivative structures. Two of these terms are, $E_T^{(2, I)}$, E\_T\^[(2, I)]{}&=&-\_[-]{}\^[+]{} d\_[S\^2]{} d  r\^[2]{} a\^[3]{} a\^[-2]{} { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }\ &=& - \_[-]{}\^[+]{} d\_[S\^2]{} d [x\^k]{} { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }\ &=&- d\^[3]{}x   { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }. Since we are working with tt-projection, we have for the integrand, $$\begin{aligned} &{x^k}\left\{ \chi^{lm}_{\mathrm{tt}}~ \partial_m \left[ (T \cdot \partial) \chi_{kl}^{\mathrm{tt}} \right] - \left[ (T \cdot \partial) \chi^{lm}_{\mathrm{tt}} \right] \partial_m \chi_{kl} ^{\mathrm{tt}} \right\} \\ &= {x^k}\left\{ \chi^{lm}_{\mathrm{tt}}~ \partial_m \left[ (T \cdot \partial) (\Lambda^{ij}_{kl}~\chi_{ij}) \right] -\left[ (T \cdot \partial) \chi^{lm}_{\mathrm{tt}} \ \partial_{m}(\Lambda^{ij}_{kl} \chi_{ij}) \right] \right\} \nn \\ &=r \ \hat{x}^k \left\{ \chi^{lm}_{\mathrm{tt}}\bigg[(\partial_{m}\Lambda^{ij}_{kl}) (T \cdot \partial) \chi_{ij} +\Lambda^{ij}_{kl}~\partial_{m}(T \cdot \partial)\chi_{ij}\bigg] -(T \cdot \partial) \chi_{lm}^{\mathrm{tt}} \bigg[ \Lambda^{ij}_{kl} \ (\partial_{m}\chi_{ij}) +\chi_{ij}(\partial_{m} \Lambda^{ij}_{kl}) \bigg] \right\} \nn \\ &= r \ \chi^{lm}_{\mathrm{tt}} \left\{-\frac{1}{r}\Lambda_{lm}^{ij} (T \cdot \partial) \chi_{ij}\right\}- r \ (T \cdot \partial) \chi_{lm}^{\mathrm{tt}} \left\{ -\frac{1}{r}\Lambda^{ij}_{lm} \ \chi_{ij}\right\} \nn \\ &=- \chi^{lm}_{\mathrm{tt}} (T \cdot \partial)\chi_{lm}^{\mathrm{tt}}+\chi^{lm}_{\mathrm{tt}} (T \cdot \partial)\chi_{lm}^{\mathrm{tt}} \nn\\ &= 0, \end{aligned}$$ i.e., these two terms cancel each other. The remaining terms $E_T^{(2,II)}$ in the $j^i$ integral are, E\_T\^[(2,II)]{} &= & \_[-]{}\^[+]{} d \_[S\^2]{} d  r\^[2]{} a\^[3]{} a\^[-2]{}  \ &=& - H\^2 \_[-]{}\^[+]{} d \_[S\^2]{} d\ &=& H\^2 { \_[-]{}\^[+]{} d \_[S\^2]{} d  \_[r]{} \^[ij]{} - \_[-]{}\^[+]{} d \_[S\^2]{} d },\ \[omega\_i\_final\] where in arriving at these expressions we have done manipulations similar to ones done above. In the first step we have used the property that $T \cdot \partial = \frac{d}{d\tau}$ and $\partial_{r}[\frac{d}{d\tau}]=\frac{d} {d\tau}[\partial_{r}]-H\partial_{r}$. In the second step we have done integrations by part with respect to the Killing parameter $\tau$ and have made use of equation . The second term in expression is a total derivative. On $\rho$ = constant surfaces, $\tau=+\infty$ corresponds to the boundary point $i^+$. For $\rho$ = constant timelike (spacelike) surfaces, $\tau=-\infty$ corresponds to $i^-(i^0)$, see figure  \[PoincarePatch\]. At all these points the field $\chi_{ij}$ vanishes. Hence contributions from the total derivative term are zero in . ### Adding the two contributions {#adding-the-two-contributions .unnumbered} The non zero contributions from $j^{\eta}$ and $j^{i}$ to the flux integral are E\_T = \^[ik]{}\^[jl]{}. \[Flux\_Phys\] At this stage we can use various identities from section \[Identity\] to get, E\_T &=& \_[S\^2]{} d\_[-]{}\^ d  \^[ik]{}\^[jl]{}\ & & , \[energy\_main\_sec\] where we recall that $A_{ij}$ is defined in A\_[ij]{}=\_[ij]{}+2H\_[ij]{}+H\_[ij]{}+2H\^[2]{} \_[ij]{}, and it satisfies identities R\_[ij]{}=\_[ij]{}+HA\_[ij]{}=A\_[ij]{}-HA\_[ij]{}. \[R\_A\] In arriving at expression we have used the fact that $\partial_{r}$ and $\partial_{\eta}$ commute with the tt-projection, cf. equations –. We also note that the operation of tt-projection commutes with the dot operation. Interestingly, all the other terms except the $RR$ term in expression combine into a total derivative. Substituting $R_{ij}$ in terms of $A_{ij}$ in the other terms in we get, \^[ik]{}\^[jl]{} =\^[ik]{}\^[jl]{}, which is a total derivative on $\Sigma_\rho$. Like in the previous subsection, contributions from this total derivatives terms vanish. This is so because $A_{ij}$ vanishes due to the staticity assumption of the source at the boundary points. Hence, these terms do not contribute to the energy flux. Note that, formally several of these total derivative terms do not have a good limit as $\rho \rightarrow \infty$, reflecting the fact that the hypersurface integral of the symplectic current density itself does not have a good limit on $\mathcal{I}^{+}.$ However, the divergent terms turn out to be total derivatives, as in [@ABKII]. A final expression is therefore, E\_T = \_[S\^2]{} d\_[-]{}\^ d  \^[ik]{}\^[jl]{}. \[energy\_flux\_final\] Flux integral on cosmological horizon {#cosmological_horizon} ------------------------------------- The analog of the above computation can also be done on the cosmological horizon. The cosmological horizon is a null surface at, \^+ : + r = 0. The fact that it is a null surface brings about some non-trivial changes to the computation of subsection \[energy\_flux\], which we highlight below. On the cosmological horizon $\sqrt{h} = H^{-2} \sin\theta, $ and we fix the normalisation of the normal vector as, n\_ = - |H|\^[-1]{}(1, x\_i/r) , so that $n^{\mu} = T^{\mu}$ at $\cH^+$. The flux integral is therefore, E\_[T]{}=-\_[\^+]{} d\_ j\^ &=&- \^[+]{}\_[-]{} \_[S\^[2]{}]{} (j\^+). \[energy\_main\_H\] ### $j^{\eta}$ contribution {#jeta-contribution-1 .unnumbered} The $j^{\eta}$ terms in integral are E\_T\^[(1)]{}&=&\^[+]{}\_[-]{} d \_[S\^2]{} d      (\^[ij]{} \_- (T ) \_[ij]{} \_\^[ij]{}). Following the step similar to the previous subsection, this contribution becomes, E\_T\^[(1)]{}&=& \^\_[-]{} d \_[S\^2]{} d  r \^[ik]{} \^[jl]{}. ### $j^{i}$ contributions {#ji-contributions-1 .unnumbered} The $j^{i}$ part of integral again has two types of terms. The terms with the derivative structure of the form \[tt\_zero\] -\^[+]{}\_[-]{} d\_[S\^2]{} d   x\^k { \^[lm]{} \_m - \_m \_[kl]{} } . cancel with each other like in the previous subsection. The remaining terms in the integral become, E\_T\^[(2)]{} &=&-\_[-]{}\^[+]{} d\_[S\^2]{} d   { \^[lm]{} \_r - \_r \_[lm]{} }\ &=& d\_[S\^2]{} d   \_r \_[lm]{}\ &=& d\_[S\^2]{} d  \^[ik]{}\^[jl]{}. ### Adding the two contributions {#adding-the-two-contributions-1 .unnumbered} The energy flux across $\mathcal{H}^+$ is, E\_T = \_[-]{}\^ d\_[S\^2]{} d \^[ik]{}\^[jl]{}. \[temp\] Now using the identities from section \[Identity\] and substituting $H\rho=1$ on the cosmological horizon, the energy flux expression becomes, $$\begin{aligned} E_T = \frac{G}{8\pi}\int_{S^2} d\Omega \int_{-\infty}^{\infty} d\tau \bigg[ R_{ij}^{\mathrm{tt}}R_{kl}^{\mathrm{tt}} +4H A_{ij}^{\mathrm{tt}}R_{kl}^{\mathrm{tt}}+4 H^{2} A_{ij}^{\mathrm{tt}}A_{kl}^{\mathrm{tt}}\bigg] \delta^{ik}\delta^{jl}. \end{aligned}$$ Again terms other than the $RR$ term combine into a total derivative. Using , we note that, 4H \_[-]{}\^ d\^[ik]{}\^[jl]{} = 2 H \_[-]{}\^ d= 0. Hence, the energy flux across $\mathcal{H}^+$ is simply, E\_T = \_[-]{}\^ d\_[S\^2]{} d  R\_[ij]{}\^ R\_[kl]{}\^ \^[ik]{}\^[jl]{}. \[EF\_CH\] Sharp propagation of energy --------------------------- The integrands in integrals and are exactly the same. In particular, the integrand is independent of $\rho$. Hence the power radiated P = = \_[S\^2]{} d  R\_[ij]{}\^ R\_[kl]{}\^ \^[ik]{}\^[jl]{} \[power\] is independent of $\rho$. The power is a function of retarded time alone. Along the outgoing null rays, retarded time is constant, see figure \[PoincarePatch\]. Specifically, the power can be computed at a cross-section of the cosmological horizon or at a cross-section of the future null infinity. As long as the cross-sections are on the same retarded time the two expressions are identical. This is the sense in which propagation of energy flux is sharp in de Sitter spacetime. See also [@DHII; @Bonga] for related comments. Comparison with the stress-tensor approach of [@DHII] ------------------------------------------------------ Reference [@DHII] also obtained an expression for the energy flux across hypersurfaces of constant radial physical distance. It uses the Isaacson stress-tensor approach. To compare our energy flux expression to theirs, we first use = (T ) = - H (\_+ r \_r), and then expand out the resulting expression to get, E\_T = - d\_[S\^2]{} d H\^[2]{}\^[2]{}r   { \_ \_[ij]{}\_\_[kl]{}+\_[r]{}\_[ij]{}\_[r]{}\_[kl]{} -   \_[r]{}\_[ij]{}\_\_[kl]{} } \^[ik]{}\^[jl]{}. \[energy\_flux\_DH\] This expression matches with that of [@DHII] (equations (52) and (53)), modulo the ‘averaging’. The averaging is part of the Isaacson stress-tensor approach. Our analysis differs from [@DHII] in another technical aspect. In reference [@DHII], to obtain energy flux in the form of equation from , the approximation \_ \_[ij]{} -\_[r]{}\_[ij]{} \[approximation\] was used. From the computation of section \[energy\_flux\], we note that this approximation is not needed. The terms it ignores combine into a total derivative. Discussion {#sec:disc} ========== We have explored propagation of energy flux in the future Poincaré patch of de Sitter spacetime. We computed energy flux integral on hypersurfaces of constant radial physical distance. We showed that in the tt-projection, the integrand in the energy flux expression on the cosmological horizon is same as that on the other hypersurfaces of constant physical radial distance. This strongly suggests that the energy flux propagates sharply in de Sitter spacetime. We also related our flux expression to a previously obtained expression of [@DHII], where a Isaacson stress-tensor approach was used. Our work can be extended in several directions. Perhaps the most pressing extension is to generalise our computations in TT-gauge and clarify their relation to [@ABKII; @ABKIII]. To systematically study this problem, it will be useful to carefully define the covariant phase space using hypersurfaces of constant radial physical distance. Such an approach offers advantages over [@ABKII; @ABKIII], as in this foliation, slices near future null infinity do not intersect source’s worldvolume. Hence the covariant phase space based on homogeneous solutions of Einstein’s equations is better defined. It can perhaps also be useful to compute the electric and magnetic parts of the Weyl tensors adapted to $\rho =$ constant slicing and write the flux expression in terms of these tensors. We hope to return to some of these problems in our future work. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank Ghanashyam Date and Alok Laddha for discussions. We are grateful to Ghanashyam Date for carefully reading a version of the manuscript, and for his detailed comments. This research is supported in part by the DST Max-Planck partner group project “Quantum Black Holes” between CMI, Chennai and AEI, Golm. Addendum: Energy flux in TT gauge {#app:TT} ================================= In this appendix we evaluate expression in TT-gauge. Although we are not able to match our final answer to that of [@ABKIII], the computations involved are sufficiently interesting to include this discussion as an appendix. This appendix is not included in the journal version of the paper. For ease of reference we write the energy flux expression again, E\_T = -\_[\_]{}d \_ j\^ & = & -\_[-]{}\^[+]{}d\_[S\^2]{}d  r\^[2]{} a\^[3]{} (H j\^+ ), \[flux\_physical\_radius\_TT\] where recall that $\tau$ is the Killing parameter defined in . ### $j^{\eta}$ contribution {#jeta-contribution-2 .unnumbered} Computation of the $j^{\eta}$ part of integral is identical to the corresponding computation presented in section \[energy\_flux\]. A final answer is E\_T\^[(1)]{} &=&-  H\^2 \^[3]{} \_[-]{}\^[+]{} d \_[S\^2]{} d  \^[ik]{} \^[jl]{}. ### $j^{i}$ contributions {#ji-contributions-2 .unnumbered} Let us first look at the $j^{i}$ part of integral . We call this piece $E_T^{(2)}$. Upon substituting we get four terms. We separate the contributions of these terms based on their derivative structures. Two of these terms are, $E_T^{(2, I)}$, E\_T\^[(2, I)]{}&=& \_[-]{}\^[+]{} d\_[S\^2]{} d  r\^[2]{} a\^[3]{} a\^[-2]{} { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }\ &=& \_[-]{}\^[+]{} d\_[S\^2]{} d [x\^k]{} { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }\ &=& d\^[3]{}x   { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }. Upon an integration by parts we get, E\_T\^[(2, I)]{}&=&\ &=&0, i.e., these two terms exactly cancel each other in TT gauge since $\partial^{m} \chi_{lm}^{\mathrm{TT}}=0$. For the remaining terms in the $j^i$ integral, computation is identical to the corresponding computation presented in section \[energy\_flux\]. A final answer is E\_T\^[(2,II)]{} &=&  H\^2 \_[-]{}\^[+]{} d \_[S\^2]{} d  \_[r]{} \_[ij]{}\^ \^[ik]{} \^[jl]{}. ### Adding the two contributions {#adding-the-two-contributions-2 .unnumbered} A final expression for the energy flux in TT gauge is, E\_T &=& H\^[2]{}{d\_[S\^2]{} d (r\_\_[kl]{}\^+\_[r]{}\_[kl]{}\^)} \^[ik]{}\^[jl]{}\ &=& H\^[2]{}{d\_[S\^2]{} d\^ \^}\^[ik]{}\^[jl]{}, \[TT\_energy\]\ where we have used the fact that $\partial_\eta$ and $r\partial_r$ commute with the TT operation. Although we do not have a clear interpretation of , neither a detailed understanding of its relation of [@ABKIII], we make the following (possibly interesting/useful) observation. Under the integral sign, we can first evaluate the expressions at $\rho$ = constant surface and then take its TT part[^2]. Thought of it in this way, it appears appropriate to pull out factors of $\rho$ from bracketed expressions in . Then, we can express energy flux as, E\_T= d\_[S\^2]{} d\^[ik]{}\^[jl]{}, as the remaining three terms in energy expressions can be written as a total derivative, (1+H)\^[2]{} (A\_[ij]{}\^A\_[kl]{} \^) \^[ik]{}\^[jl]{}. [99]{} B. P. Abbott [*et al.*]{}, “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett.  [**116**]{}, no. 6, 061102 (2016) doi:10.1103/PhysRevLett.116.061102 \[arXiv:1602.03837 \[gr-qc\]\]. B. P. Abbott [*et al.*]{}, “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett.  [**119**]{}, no. 16, 161101 (2017) doi:10.1103/PhysRevLett.119.161101 \[arXiv:1710.05832 \[gr-qc\]\]. B. P. Abbott [*et al.*]{}, “Multi-messenger Observations of a Binary Neutron Star Merger,” Astrophys. J.  [**848**]{}, no. 2, L12 (2017) doi:10.3847/2041-8213/aa91c9 \[arXiv:1710.05833 \[astro-ph.HE\]\]. 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Aggarwal, “Quadrupolar power radiation by a binary system in de Sitter Background,” \[arXiv:1710.01357\] R. A. Isaacson, “Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Ef fective Stress Tensor,” Phys. Rev.  [**166**]{}, 1272 (1968). doi:10.1103/PhysRev.166.1272 R. A. Isaacson, “Gravitational Radiation in the Limit of High Frequency. I. The Linear Approximation and Geometrical Optics,” Phys. Rev.  [**166**]{}, 1263 (1967). doi:10.1103/PhysRev.166.1263 H. J. de Vega, J. Ramirez and N. G. Sanchez, “Generation of gravitational waves by generic sources in de Sitter space-time,” Phys. Rev. D [**60**]{}, 044007 (1999) doi:10.1103/PhysRevD.60.044007 \[astro-ph/9812465\]. A. Ashtekar and B. Bonga, “On a basic conceptual confusion in gravitational radiation theory,” Class. Quant. Grav.  [**34**]{}, no. 20, 20LT01 (2017) doi:10.1088/1361-6382/aa88e2 \[arXiv:1707.07729 \[gr-qc\]\]. 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ArXiv

--- abstract: 'We show that within the inverse seesaw mechanism for generating neutrino masses minimal supergravity is more likely to have a sneutrino as the lightest superparticle than the conventional neutralino. We also demonstrate that such schemes naturally reconcile the small neutrino masses with the correct relic sneutrino dark matter abundance and accessible direct detection rates in nuclear recoil experiments.' author: - 'C. Arina' - 'F. Bazzocchi' - 'N. Fornengo' - 'J. C. Romao' - 'J. W. F. Valle' title: Minimal supergravity sneutrino dark matter and inverse seesaw neutrino masses --- Introduction ============ Over the last fifteen years we have had solid experimental evidence for neutrino masses and oscillations [@Maltoni:2004ei], providing the first evidence for physics beyond the Standard Model. On the other hand, cosmological studies clearly show that a large fraction of the mass of the Universe in dark and must be non–baryonic. The generation of neutrino masses may provide new insight on the nature of the dark matter [@Berezinsky:1993fm]. In this Letter we show that in a minimal supergravity (mSUGRA) scheme where the smallness of neutrino masses is accounted for within the inverse seesaw mechanism the lightest supersymmetric particle is likely to be represented by the corresponding neutrino superpartner (sneutrino), instead of the lightest neutralino. This opens a new window for the mSUGRA scenario. Here we consider the implications of the model for the dark matter issue. We demonstrate that such a model naturally reconciles the small neutrino masses with the correct relic abundance of sneutrino dark matter and experimentally accessible direct detection rates. Minimal SUGRA inverse seesaw model {#Model} ================================== Let us add to the Minimal Supersymmetric Standard Model (MSSM) three sequential pairs of 1 singlet neutrino superfields ${\widehat \nu^c}_i$ and $\widehat{S}_i$ ($i$ is the generation index), with the following superpotential terms [@mohapatra:1986bd; @Deppisch:2004fa], $${\cal W} = {\cal W_{\rm MSSM}} +\varepsilon_{ab}\, h_{\nu}^{ij}\widehat L_i^a\widehat \nu^c_j\widehat H_u^b + M_{R}^{ij}\widehat \nu^c_i\widehat S_j +\frac{1}{2}\mu_S^{ij} \widehat S_i \widehat S_j \label{eq:Wsuppot}$$ where ${\cal W_{\rm MSSM}}$ is the usual MSSM superpotential. In the limit $\mu_S^{ij} \to 0$ there are exactly conserved lepton numbers assigned as $(1,-1,1)$ [@mohapatra:1986bd; @Deppisch:2004fa] for $\nu$, $\nu^{c}$ and $S$, respectively. The extra singlet superfields induce new terms in the soft–breaking Lagrangian: $$\begin{aligned} \mathcal{-L}_{\rm soft} &=& \mathcal{-L}_{\rm soft}^{\rm MSSM} + \tilde{\nu}^c_i\ \mathbf{M^2_{\nu^c}}_{ij} \tilde{\nu}^c_j + \tilde{S}_i\, \mathbf{M^2_{S}}_{ij} \tilde{S}_j \\ & & + \varepsilon_{ab}\, A_{h_{\nu}}^{ij} \tilde{L}_i^a \tilde{\nu}^c_j H_u^b + B_{M_{R}}^{ij} \tilde{\nu}^c_i \tilde{S}_j +\frac{1}{2} B_{\hat{\mu_S}}^{ij} \tilde{S}_i \tilde{S}_j \nonumber \label{eq:soft}\end{aligned}$$ where $\mathcal{L}_{\rm soft}^{\rm MSSM}$ is the MSSM SUSY–breaking Lagrangian. Small neutrino masses are generated through the inverse seesaw mechanism [@mohapatra:1986bd; @Deppisch:2004fa; @Nunokawa:2007qh]: the effective neutrino mass matrix $m^{\rm eff}_{\nu}$ is obtained by the following relation: $$\label{eq:1} m^{\rm eff}_{\nu}= -v_u^2 h_{\nu} \left(M_R^T\right)^{-1} {\mu_S} M_R^{-1} h_{\nu}^T = \left(U^T\right)^{-1} m_{\mu}^{\rm diag}\ U^{-1}$$ where $h_\nu$ defines the Yukawa matrix and $v_u$ is the $H_u$ vacuum expectation value. The smallness of the neutrino mass is ascribed to the smallness of the $\mu_S$ parameter, rather than the largeness of the Majorana–type mass matrix $M_R$, as required in the standard seesaw mechanism [@Nunokawa:2007qh]. In this way light (eV scale or smaller) neutrino masses allow for a sizeable magnitude for the Dirac–type mass $m_D=v_u h_\nu$ and a TeV–scale mass for the right-handed neutrinos, features which have been shown to produce an interesting sneutrino dark matter phenomenology [@Arina:2007tm]. The main feature of our model is that the nature of the dark matter candidate, its mass and couplings all arise from the same sector responsible for the generation of neutrino masses. In order to illustrate the mechanism we consider the simplest one-generation case, for simplicity. In this case where the sneutrino mass matrix reads $$\begin{aligned} \mathcal{M}^2 = \begin{pmatrix} \mathcal{M}^2_+ & \mathbf{0}\cr \mathbf{0} & \mathcal{M}^2_-\cr \end{pmatrix} $$ where the two sub–matrices $\mathcal{M_\pm}^2$ are: $$\begin{aligned} \label{eq:snumatrix} \mathcal{M_{\pm}}^2 = \begin{pmatrix} m^2_L+\frac{1}{2} m^2_Z \cos 2\beta+m^2_D & \pm (A_{h_{\nu}}v_u-\mu m_D {\rm cotg} \beta) & m_D M_R\cr \pm (A_{h_{\nu}}v_u-\mu m_D {\rm cotg}\beta) & m^2_{\nu^c}+M_R^2+m^2_D & \mu_S M_R \pm B_{M_R}\cr m_D M_R & \mu_S M_R \pm B_{M_R} & m^2_S+\mu^2_S+M^2_R\pm B_{\mu_S} \end{pmatrix} $$ in the CP eigenstates basis: $\Phi^{\dag} = (\snu_{+}^\ast \,\tilde{\nu}_{+}^{c\ast} \, \tilde{S}_+^\ast \,\, \snu_-^\ast \, \tilde{\nu}_-^{c\ast} \, \tilde{S}_-^\ast)$. Once diagonalized, the lightest of the six mass eigenstates is our dark matter candidate and it is stable by $R$–parity conservation. A novel supersymmetric spectrum =============================== ![Supersymmetric particle spectrum in the standard MSUGRA scheme \[panel (a)\] and in the inverse seesaw mSUGRA model \[panel (b)\] with parameters chosen as: $m_0= 358$ GeV, $m_{1/2}= 692$ GeV, $A_0 = 0$, $\tan\beta=35$ and sign $\mu >0$. The sneutrino sector has the additional parameter $B_{\mu_S}$, fixed at 10 GeV$^2$. The squark sector is not shown. []{data-label="fig:spectrum"}](tower2_new.eps){width="\columnwidth"} ![The $m_{0}-m_{1/2}$ plane for $\tan\beta=35$, $A_0=0$ and $\mu>0$. The red and yellow areas denote the set of supersymmetric parameters where the sneutrino is the LSP in inverse seesaw models (notice that it includes all the yellow region where the $\tilde{\tau}$ is the LSP in the standard mSUGRA case). The white region has the neutralino as LSP in both standard and modified mSUGRA. For the sneutrino LSP region, the additional parameters are: $B_{\mu_S}= 10 {\, {\rm GeV}}^2$, $M_R=500 {\, {\rm GeV}}$, $m_D=50 {\, {\rm GeV}}$ and $\mu_S=1$ eV. The blue region is excluded (see text). []{data-label="fig:m0mh12"}](mhalfm0_new.eps){width="\columnwidth"} Let us now consider the model within a minimal SUGRA scenario. In the absence of the singlet neutrino superfields, the mSUGRA framework predicts the lightest supersymmetric particle (LSP) to be either a stau or a neutralino, and only the latter case represents a viable dark matter candidate. In most of the mSUGRA parameter space, however, the neutralino relic abundance turns out to exceed the WMAP bound [@Komatsu:2008hk] and hence the cosmologically acceptable regions of parameter space are quite restricted. In contrast, when the singlet neutrino superfields are added, a combination of sneutrinos emerges quite naturally as the LSP. Indeed, we have computed the resulting supersymmetric particle spectrum and couplings by adapting the SPheno code [@Porod:2003um] so as to include the additional singlet superfields. An illustrative example of how the minimal supergravity particle spectrum is modified by the presence of such states is given in Fig. \[fig:spectrum\]. This figure shows explicitly how a sneutrino LSP is in fact realized. A more general analysis in the mSUGRA parameter space is shown in Fig. \[fig:m0mh12\]: the dark (blue) shaded area is excluded either by experimental bounds on supersymmetry and Higgs boson searches, or because it does not lead to electroweak symmetry breaking, while the (light) yellow region refers to stau LSP in the conventional (unextended) mSUGRA case. As expected, in all of the remaining region of the plane, the neutralino is the LSP in the standard mSUGRA case. The new phenomenological possibility which opens up thanks to the presence of the singlet neutrino superfields where the sneutrino is the LSP corresponds to the full mid-gray (red) and light (yellow) areas. In what follows we demonstrate that in this region of parameter space such a sneutrino reproduces the right amount of dark matter and is not excluded by direct detection experiments. Sneutrino LSP as Dark Matter ============================ ![Sneutrino relic abundance $\Omega h^2$ as a function of the LSP sneutrino mass $m_1$, for a full scan of the supersymmetric parameter space: $100 {\, {\rm GeV}}< m_0 < 3 {\, {\rm TeV}}$, $100 {\, {\rm GeV}}< m_{1/2}< 3 {\, {\rm TeV}}$, $1 {\, {\rm GeV}}^2 <B_\mu < 100 {\, {\rm GeV}}^2$, $A_0=0$, $3 < \tan\beta<50$, $10^{-9} {\, {\rm GeV}}<\mu_S<10^{-6} {\, {\rm GeV}}$. The yellow band delimits the WMAP [@Komatsu:2008hk] cold dark matter interval at 3 $\sigma$ of C.L.: $0.104 \leq \Omega_{\rm{CDM}} h^2 \leq 0.124$.[]{data-label="fig:omega"}](relic_inv_wmap5.eps){width="\columnwidth"} ![Sneutrino–nucleon scattering cross section $\xi \sigma^{\rm (scalar)}_{\rm nucleon}$ vs. the sneutrino relic abundance $\Omega h^{2}$, for the same scan of the supersymmetric parameter space given in Fig. \[fig:omega\]. The horizontal \[light blue\] band denotes the current sensitivity of direct detection experiments; the vertical \[yellow\] band delimits the 3 $\sigma$ C.L. WMAP cold dark matter range [@Komatsu:2008hk].[]{data-label="fig:direct"}](relic_direct1.eps){width="\columnwidth"} The novelty of the spectrum implied by mSUGRA implemented with the inverse seesaw mechanism is that it may lead to a bosonic dark matter candidate, the lightest sneutrino $\tilde{\nu}_1$, instead of the fermionic neutralino. To understand the physics it suffices for us to consider the simple one sneutrino generation case [^1]. The relic density of the sneutrino candidate is shown in Fig. \[fig:omega\]. The lightest mass eigenstate is also a CP eigenstate and coannihilates with the NLSP, a corresponding heavier opposite–CP sneutrino eigenstate. We notice that this situation provides a nice realization of inelastic dark matter, a case where the dark matter possesses a suppressed scattering with the nucleon, relevant for the direct detection scattering cross section, shown in Fig. \[fig:direct\]. From Fig. \[fig:omega\] we see that a large fraction of the sneutrino configuration are compatible with the WMAP cold dark matter range, and therefore represents viable sneutrino dark matter models. Fig. \[fig:direct\] in addition shows that direct detection experiments do not exclude this possibility: instead, a large fraction of configurations are actually compatible and under exploration by current direct dark matter detection experiments. This fact is partly possible because of the inelasticity characteristics we have mentioned above, which reduces the direct detection cross section to acceptable levels [@Arina:2007tm]. We stress that all models reported in Figs. \[fig:omega\] and \[fig:direct\] have the inverse seesaw-induced neutrino masses consistent with current experimental observations for natural values of its relevant parameters. We also note that the lepton–number violating parameter $B_{\mu_S}$, which determines the lightest mass sneutrino eigenstate and its couplings, also has an impact on the neutrino sector, since it can induce one-loop corrections to the neutrino mass itself (for details, see Ref. [@Arina:2007tm] and references therein). These corrections must be small, in order not to go into conflict with the bounds on neutrino masses, and this in turn implies that the mass splitting between the sneutrino LSP and sneutrino NLSP is small (less than MeV or so) [@Arina:2007tm], implying the inelasticity of the sneutrino scattering with nuclei [@Arina:2007tm]. The parameter $\mu_s$ therefore plays a key role in controlling the neutrino mass generation, the sneutrino relic abundance and the direct detection cross section. In conclusion, in this Letter we have presented an mSUGRA scenario in which neutrino masses and dark matter arise from the same sector of the theory. Over large portions of the parameter space the model successfully accommodates light neutrino masses and sneutrinos dark matter with the correct relic abundance indicated by WMAP as well as direct detection rates searches consistent with current dark matter searches. The neutrino mass is generated by means of an inverse seesaw mechanism, while in a large region of parameters the dark matter is represented by sneutrinos. The small superpotential mass parameter $\mu_S$ controls most of the successfull phenomenology of both the neutrino and sneutrino sector. In the absence of $\mu_S$ neutrinos become massless, Eq. (\[eq:1\]). The bilinear superpotential term $\mu_S^{ij} \widehat S_i \widehat S_j$ could arise in a spontaneous way in a scheme with an additional lepton-number-carrying singlet superfield $\sigma$, implying the existence of a majoron [@gonzalez-garcia:1988rw]. In this case, the dominant decays of the Higgs boson are likely to be into a pair of majorons [@Joshipura:1992hp]. Such invisible mode would be “seen” experimentally as missing momentum, but the corresponding signal did not show up in the LEP data [@Abdallah:2003ry]. Although hard to catch at the LHC such decays would provide a clean signal in a future ILC facility. Similarly, the standard bilinear superpotential term $\mu H_u H_d$ present in the minimal supergravity model could also be substituted by a trilinear, in a NMSSM-like scheme [@CerdenoMunoz:2008]. Note that our proposed scheme may also have important implications for supersymmetric particle searches at the LHC, due to modified particle spectra and decay chains. Additional experimental signatures could be associated with the (quasi-Dirac) neutral heavy leptons formed by $\nu^c$ and $S$, whose couplings and masses are already restricted by LEP searches [@Dittmar:1989yg; @Abreu:1996pa]. [*Acknowledgements.*]{} We warmly thank M. Hirsch for stimulating discussions. This work was supported by MEC grant FPA2005-01269, by EC Contracts RTN network MRTN-CT-2004-503369 and ILIAS/N6 RII3-CT-2004-506222, by FCT grant POCI/FP/81919/2007 and by research grants funded jointly by the Italian Ministry of Research and by the Istituto Nazionale di Fisica Nucleare (INFN) within the [*Astroparticle Physics Project*]{}. [10]{} For an updated review of neutrino oscillations see archive version of M. Maltoni, T. Schwetz, M. A. Tortola and J. W. F. Valle, New J. Phys. [**6**]{}, 122 (2004), \[hep-ph/0405172\]. V. Berezinsky and J. W. F. Valle, Phys. Lett. [**B 318**]{} (1993) 360; E. K. Akhmedov, Z. G. Berezhiani, R. N. Mohapatra and G. Senjanovic,Phys. Lett. B [**299**]{} (1993) 90 \[hep-ph/9209285\]; M. Lattanzi and J. W. F. Valle, Phys. Rev. Lett. [**99**]{}, 121301 (2007), \[0705.2406\]; F. Bazzocchi, M. Lattanzi, S. Riemer-Sorensen and J. W. F. Valle, 0805.2372. R. N. Mohapatra and J. W. F. Valle, Phys. Rev. [**D34**]{}, 1642 (1986). F. Deppisch and J. W. F. Valle, Phys. Rev. [**D72**]{}, 036001 (2005), \[hep-ph/0406040\]. H. Nunokawa, S. J. Parke and J. W. F. Valle, Prog. Part. Nucl. Phys. [**60**]{}, 338 (2008), \[arXiv:0710.0554 \[hep-ph\]\], this review gives an updated discussion of the standard seesaw mechanism and its variants. C. Arina and N. Fornengo, JHEP [**11**]{}, 029 (2007), \[arXiv:0709.4477\]. WMAP collaboration, E. Komatsu [*et al.*]{}, arXiv:0803.0547. W. Porod, Comput. Phys. Commun. [**153**]{}, 275 (2003), \[hep-ph/0301101\]. M. C. Gonzalez-Garcia and J. W. F. Valle, Phys. Lett. [**B216**]{}, 360 (1989). A. S. Joshipura and J. W. F. Valle, Nucl. Phys. [**B397**]{}, 105 (1993). See, e. g. DELPHI collaboration, J. Abdallah [*et al.*]{}, Eur. Phys. J. [**C32**]{}, 475 (2004), \[hep-ex/0401022\]. D. Cerdeno, talk at DSU08, Cairo, June 2008. M. Dittmar, A. Santamaria, M. C. Gonzalez-Garcia and J. W. F. Valle, Nucl. Phys. [**B332**]{}, 1 (1990). See, e. g. DELPHI collaboration, P. Abreu [*et al.*]{}, Z. Phys. [**C74**]{}, 57 (1997). [^1]: We adopt the same approximation used in the relic density calculation within the standard minimal mSUGRA model, which we have checked holds in our case as well.

ArXiv

--- abstract: 'An Archimedean copula is characterised by its generator. This is a real function whose inverse behaves as a survival function. We propose a semiparametric generator based on a quadratic spline. This is achieved by modelling the first derivative of a hazard rate function, in a survival analysis context, as a piecewise constant function. Convexity of our semiparametric generator is obtained by imposing some simple constraints. The induced semiparametric Archimedean copula produces Kendall’s tau association measure that covers the whole range $(-1,1)$. Inference on the model is done under a Bayesian approach and for some prior specifications we are able to perform an independence test. Properties of the model are illustrated with a simulation study as well as with a real dataset.' author: - | [Ricardo Hoyos & Luis Nieto-Barajas]{}\ [*Department of Statistics, ITAM, Mexico*]{}\ \ title: '**A Bayesian semiparametric Archimedean copula**' --- [*Keywords*]{}: Archimedean copula, Bayes nonparametrics, piecewise constant, survival analysis, quadratic spline. [*AMS Classification*]{}: 60E05 $\cdot$ 62G05 $\cdot$ 62N86. Introduction {#sec:intro} ============ Let $\varphi(\cdot)$ be a continuous, strictly decreasing function from $[0,1]$ to $[0,\infty)$ such that $\varphi(1)=0$. Let $\varphi^{-1}(\cdot)$ be the inverse or the pseudo-inverse of $\varphi$, where the latter is defined as zero for $t>\varphi(0)$. If $\varphi(0)=\infty$ the generator is called strict. An Archimedean copula $C(u,v)$ with generator $\varphi$ is a function from $[0,1]^2$ to $[0,1]$ defined as $$\label{eq:arqc} C(u,v)=\varphi^{-1}\left(\varphi(u)+\varphi(v)\right).$$ A further requirement for to be well defined is that $\varphi$ must be convex [e.g. @nelsen:06]. There are many properties that characterize Archimedean copulas, for instance, they are symmetric, associative and their diagonal section $C(u,u)$ is always less than $u$ for all $u\in(0,1)$. Generators $\varphi(\cdot)$ are usually parametric families defined by a single parameter. Most of them are summarised in @nelsen:06 [Table 4.1]. Association measures induced by Archimedean copulas are a function of the generator. For instance, Kendall’s tau becomes $$\label{eq:ktau} \kappa_\tau=1+4\int_0^1\frac{\varphi(t)}{\varphi^{\prime}(t^+)}\d t,$$ where $\varphi^{\prime}(t^+)$ denotes right derivative of $\varphi$ at $t$. In this work we propose a Bayesian semiparametric generator defined through a quadratic spline. Within a survival analysis context, we model the first derivative of a hazard rate function with a piecewise constant function. The hazard rate and the cumulative hazard functions become linear and quadratic continue functions, respectively. The induced survival function is used as an inverse generator for an Archimedean copula. Convexity constrains are properly addressed and inference on the model is done under a Bayesian approach. Other studies on semiparametric generators for Archimedean copulas can be found in where their model is based on an empirical Kendall’s process. A new approach and extensions of this latter methodology can be found in . In the model arises from the one-to-one correspondence between an Archimedean generator and a distribution function of a nonnegative random variable. In particular they use a mixture of Pólya trees as a prior for the corresponding distribution function under a Bayesian nonparametric approach. In a work more related to ours, use the relationship between quantile functions and Archimedean generators to define a semiparametric generator by supplementing a parametric generator with $n+1$ dependence parameters. Differing to their work, our model is not based on any parametric generator and the Kendall’s tau can take values on the whole interval $(-1,1)$. The contents of the rest of the paper is as follows. In Section \[sec:model\] we present our proposal and characterise its properties. In Section \[sec:post\] we provide details of how to make posterior inference under a Bayesian approach. In Section \[sec:illust\] we illustrate the performance of our model with a simulation study as well as with a real data set. We conclude with some remarks in Section \[sec:concl\]. Model {#sec:model} ===== To define our proposal we realise that $\varphi^{-1}$ is a decreasing function from $[0,\infty)$ to $[0,1]$, so it behaves as a survival function, in a failure time data analysis context . The idea is to propose a semi/non parametric form for the inverse generator $\varphi^{-1}$ by using survival analysis ideas. For that we recall some basic definitions. Let $h(t)$ be a nonnegative function with domain in $[0,\infty)$ such that $H(t)=\int_0^t h(s)\d s\to\infty$ as $t\to\infty$. Then $S(t)=\exp\{-H(t)\}$ is a decreasing function from $[0,\infty)$ to $[0,1]$, so it behaves like an inverse generator $\varphi^{-1}(t)$. In a survival analysis context, functions $h(\cdot)$, $H(\cdot)$ and $S(\cdot)$ are the hazard rate, cumulative hazard and survival functions, respectively. In particular, if $h(t)=\theta$, i.e. constant for all $t$, then $S(t)=e^{-\theta t}$. If we take $\varphi(t)^{-1}=e^{-\theta t}$, then $\varphi(t)=-(\log t)/\theta$. Using we obtain that the resulting copula $C(u,v)=uv$ is the independence copula, and what is interesting, is that it does not depend on $\theta$. Using these ideas we construct a semiparametric generator in the following way. We first consider a partition of size $K$ of the positive real line, with interval limits given by $0=\tau_0<\tau_1<\cdots<\tau_K=\infty$. Then, we define the first derivative of the hazard rate, as a piecewise constant function of the form $$\label{eq:hp} h'(t)=\sum_{k=1}^K \theta_k I(\tau_{k-1}<t\leq\tau_k),$$ where $\theta_K\equiv 0$. We recover the hazard rate function as $h(t)=\int_0^t h'(s)\d s+\theta_0$, where $h(0)=\theta_0>0$ is an initial condition. Using , the hazard rate becomes a piecewise linear function of the form $$\label{eq:h} h(t)=\sum_{k=1}^K \left(A_k+\theta_k t\right) I(\tau_{k-1}<t\leq\tau_k),$$ where $A_1=\theta_0$ and $A_k=\theta_0+\sum_{j=1}^{k-1}(\theta_j-\theta_{j+1})\tau_j$, for $k=2,\ldots,K$. Integrating now the hazard function , the cumulative hazard is a piecewise quadratic function given by $$\label{eq:H} H(t)=\sum_{k=1}^K \left(B_k+A_k t+\frac{\theta_k}{2} t^2\right) I(\tau_{k-1}<t\leq\tau_k),$$ where $B_1=0$ and $B_k=\sum_{j=2}^k(\theta_j-\theta_{j-1})\tau_{j-1}^2/2$, for $k=2,\ldots,K$. We therefore define a semiparametric inverse generator as the induced survival function, which can be written as $$\label{eq:phiinv} \varphi^{-1}(t)=\exp\{-H(t)\},$$ where $H(t)$ is given in . After doing some algebra, we can invert this function to obtain an expression for the generator $$\begin{aligned} \nonumber \varphi(t)=\sum_{k=1}^{K}&\left(\left[\operatorname{sgn}(\theta_k)\left\{\frac{2}{\theta_k}\left(\frac{A_k^2}{2\theta_k}-B_k-\log(t)\right)\right\}^{1/2}-\frac{A_k}{\theta_k}\right]I(\theta_k\neq 0)\right.\\ \nonumber &\left.\hspace{5mm}-\frac{B_k+\log(t)}{A_k}I(\theta_k=0)\right) I\left(\varphi^{-1}(\tau_k)\leq t< \varphi^{-1}(\tau_{k-1})\right). \\ \label{eq:phi}\end{aligned}$$ The value $K$ controls the flexibility of the generator, and thus of the copula. If $K=1$, the induced Archimedean copula is the independent copula, whereas for larger $K$, the generator, and the induced copula, become more nonparametric. Potentially $K$ could be infinite. We now discuss some properties of our semiparametric generator. \[prop:1\] Consider the semiparametric inverse generator $\varphi^{-1}(t)$, given in , and the corresponding generator $\varphi(t)$, given in , and assume that $\{\theta_k\}$ are such that $\theta_0>0$, $\theta_K=0$ and satisfy conditions (C1) and (C2) given by 1. $A_k+\theta_k t\geq 0$, for $t\in(\tau_{k-1},\tau_k]$ and for all $k=1,\ldots,K$. 2. $(A_k + \theta_k t)^2 > \theta_k$, for $t\in(\tau_{k-1},\tau_k]$ and for all $k=1,\ldots,K$. Then, 1. $\varphi^{-1}(t)$ and $\varphi(t)$ are continuous functions of $t$, 2. $\varphi^{-1}(t)$ is a convex function, 3. $\varphi(t)$ a strict generator. For (i) we know that $h'(t)$, as in , is a piecewise constant discontinuous function, however, function $h(t)$, as in , is continuous. To see this, for each $k=1,\ldots,K$, the limit from the left is $\lim_{t\to\tau_k^{-}} h(t) = \lim_{t \to \tau_k^{-}} A_k + \theta_k t = A_k + \theta_k \tau_k$, and the limit from the right becomes $\lim_{t \to \tau_k^{+}} h(t) = \lim_{t \to \tau_k^{+}} A_{k+1} + \theta_{k+1} t = A_{k+1} + \theta_{k+1} \tau_k$. Since $A_{k+1} = A_{k} + (\theta_{k} - \theta_{k+1})\tau_{k}$, then both limits coincide. For (ii) we take the second derivative of $\varphi^{-1}(t)$ which becomes $\varphi^{-1(\prime \prime)}(t) = \{h(t)\}^2 \exp\{-H(t)\} - h^{\prime}(t)\exp\{-H(t)\}$, this is positive if and only if $\{h(t)\}^2-h'(t)>0$. For this to happen we require condition $(C2)$. For (iii), $\varphi^{-1}(t)$ must be a proper survival function, that is, $h(t)$ must be nonnegative, which is achieved by imposing condition $(C1)$. Furthermore, we need $\lim_{t\to\infty}\varphi^{-1}(t)=0$, which is equivalent to prove that $\lim_{t\to\infty}H(t)=\lim_{t\to\infty}\left(B_K+A_K t+\theta_K t^2/2\right)=\infty$. This is true since $B_K$ is a finite constant, $A_K>0$ and $\theta_K=0$, so the linear part goes to infinity when $t\to\infty$. To see the kind of association induced by our proposal, we computed the Kendall’s tau using expression with generator . This is given in the following result. \[prop:kt\] The Kendall’s tau obtained by the Archimedean copula with semiparametric generator is given by $$\kappa_\tau = -1+2\sum_{k=1}^K A_k \int_{\tau_{k-1}}^{\tau_k}\exp\left(-2B_k-2A_k t-\theta_k t^2\right)\d t.$$ Moreover, this $\kappa_\tau\in(-1,1)$. Rewriting expression in terms of the inversed generator we obtain $\kappa_\tau=1-4\int_0^\infty t\{\varphi^{-1(\prime)}(t)\}^2\d t$. Computing the derivative we get $\varphi^{-1(\prime)}(t)=-\sum_{k=1}^K(A_k+\theta_k t)\times$ $\exp\{-(B_k + A_k t+\theta_k t/2)\}I(\tau_{k-1}<t\leq\tau_k)$. Doing the integral we obtain the expression. To obtain the range of possible values of $\kappa_\tau$ it is easier to re-write $\kappa_\tau$ in terms of $h(t)$ and $H(t)$. This becomes $\kappa_{\tau} = -2 \int_0^{\infty} t h^{\prime}(t)\exp\{-H(t)\}\,dt$. Here it is straightforward to see that the sign of $\kappa_\tau$ is determined by the sign of $h'(t)$, therefore $h'(t)>0$ for all $t$ implies $-1 < \kappa_{\tau} < 0$ and $h'(t)\le 0$ implies $0 \le\kappa_{\tau}<1$. The expression for $\kappa_\tau$ tells us that the concordance induced by our semiparametric copula is a function of both, the parameters $\{\theta_k\}$, as well as of the partition limits $\{\tau_k\}$. It depends on a definite integral and can be evaluated numerically. What is more important is that $\kappa_\tau$ covers the whole range from $-1$ to $1$, showing that our proposal is very flexible. To illustrate the flexibility of our model we define a partition of the positive real line of size $K=10$, such that $\tau_k=-\log(1-k/10)$ for $k=0,1,\ldots,10$. We consider two scenarios for the values of the parameters $\{\theta_k\}$. The first scenario is defined by $\theta_k<0$ for all $k$, whereas the second scenario contains $\theta_k>0$ for all $k$. Conditions $(C1)$ and $(C2)$ were satisfied in both cases. Figure \[fig:ilust1\] contains functions $h'(t)$, $H(t)$ and $\varphi^{-1}(t)$ for two different scenarios, the solid (blue) line corresponds to the first scenario and the dotted (red) line to the second scenario. In the first case the corresponding hazard function (middle panel) is decreasing, whereas for the second case the hazard function is increasing. The induced concordance values are $\kappa_\tau=0.368$ and $\kappa_\tau=-0.202$, respectively. As a second example, we consider a partition of size $K=50$, such that $\tau_k = -\log(1 - k/50)$ for $k=0,1,\ldots,50$. We consider three different scenarios for the parameters $\{\theta^{(i)}_k\}$ with $i=1,2,3$, respectively. In the first scenario we assume $\theta_1^{(1)}\sim \un(-1,1)$, in the second $\theta_1^{(2)}\sim \un(-50,0)$ and in the third $\theta_1^{(3)}\sim \un(0,1)$. Posteriorly, we define sequentially $\theta_k^{(i)}\sim \un(a_k^{(i)},b_k^{(i)})$ with $a_k^{(i)}$ and $b_k^{(i)}$ constants such that constrains $(C1)$ and $(C2)$ are satisfied, for $k=2,\ldots,K-1$ and $i=1,2,3$. We repeated sampling from these distributions a total of 5,000 times, and for each repetition we computed $\kappa_\tau$. The induced histogram densities for the three scenarios are presented in Figure \[fig:simkt\]. For the first scenario, the values of $\kappa_\tau$ range from $-0.3$ to $0.4$, showing that our model can capture both negative and positive concordance measures. For the second scenario, the values of $\kappa_\tau$ are all positive and the distribution is right skewed, and for the third scenario the values of $\kappa_\tau$ are all negative showing a left skewed distribution. According to [@nelsen:06], new generators can be defined if we apply a scale transformation of the form $\phi^{-1}(t)=\varphi^{-1}(\alpha t)$ iff $\phi(t)=\varphi(t)/\alpha$, for $\alpha>0$, where $\phi(t)$ becomes a new Archimedean copula generator. More recently, realised that the new generator $\phi(t)$ induces exactly the same copula as that obtained with $\varphi(t)$. To see this we have that $C_\phi(u,v)=\phi^{-1}(\phi(u)+\phi(v))=\varphi^{-1}\left(\alpha\left\{\frac{1}{\alpha}\varphi(u)+\frac{1}{\alpha}\varphi(v)\right\}\right)=C_\varphi(u,v)$. In other words, an Archimedean copula generator is not unique. Moreover, in terms of the hazard rate functions, $h_\phi(t)$ and $h_\varphi(t)$, induced by generators $\phi$ and $\varphi$, respectively, the relationship becomes $h_\phi(t)=\alpha h_\varphi(\alpha t)$. In order to make our semiparametric generator identifiable, without loss of generality, we impose the new constraint 1. $\theta_0=1$. This constraint is equivalent to impose $h(0)=1$ in definition . Posterior inference {#sec:post} =================== The copula density $f_C(u,v)$, of an Archimedean copula, can be obtained by taking the second crossed derivatives with respect to $u$ and $v$ in expression . In terms of the generator and its inverse this density becomes $$\label{eq:cdensity} f_C(u,v)=\varphi^{-1(\prime\prime)}\left(\varphi(u)+\varphi(v)\right) \varphi^{(\prime)}(u)\varphi^{(\prime)}(v),$$ where the single and double primes denote first and second derivatives, respectively, and are given by $$\varphi^{-1(\prime\prime)}(t)=\sum_{k=1}^K \left\{(A_k+\theta_k t)^2-\theta_k\right\}\exp\left\{-\left(B_k+A_k t+\frac{\theta_k}{2}t^2\right)\right\}I(\tau_{k-1}<t\leq\tau_K)$$ and $$\varphi^{(\prime)}(t)=-\sum_{k=1}^{K}\frac{1}{t}\left(-2\theta_k B_k+A_k^2-2\theta_k\log(t)\right)^{-1/2}I\left(\varphi^{-1}(\tau_k)\leq t< \varphi^{-1}(\tau_{k-1})\right).$$ Let $(U_i,V_i)$, $i=1,\ldots,n$ be a bivariate sample of size $n$ from $f_C(u,v)$ defined in . With this we can construct the likelihood for ${\boldsymbol{\theta}}=(\theta_0,\theta_1,\ldots,\theta_K)$ as $\mbox{lik}({\boldsymbol{\theta}}\mid\bu,\bv)=\prod_{i=1}^n f_C(u_i,v_i\mid{\boldsymbol{\theta}})$, where we have made explicit the dependence on ${\boldsymbol{\theta}}$ in the notation of the copula density. Recall that the parameters must satisfy several conditions, $(C1)$ and $(C2)$ given in Proposition \[prop:1\], $(C3)$ to make our generator unique, and $\theta_K=0$. We assume a prior distribution for the $\theta_k$’s of the form $$\label{eq:prior} f(\theta_k)=\pi_0 I(\theta_k=0)+(1-\pi_0)\mbox{N}(\theta_k\mid \mu_0,\sigma^2_0),$$ independently for $k=1,\ldots,K-1$. Note that we explicitly allow the $\theta_k$’s, for $k=1,\ldots,K-1$ to be zero with positive probability $\pi_0$. This prior choice is useful to define an independence test. Specifically, the hypothesis $H_0:U \mbox{ and } V$ independent is equivalent to $H_0:\theta_1=\cdots=\theta_{K-1}=0$. To perform the test we can compute the posterior probability of $H_0$ and its complement and make the decision, say via Bayes factors . The posterior distribution of ${\boldsymbol{\theta}}$ is simply given by the product of expressions and , up to a proportionality constant. It is somehow easier to characterize the posterior distribution by implementing a Gibbs sampler and sampling from the conditional posterior distributions $$\label{eq:postc} f(\theta_k\mid {\boldsymbol{\theta}}_{-k},\data)\propto\mbox{lik}({\boldsymbol{\theta}}\mid\bu,\bv)f(\theta_k),$$ for $k=1,\ldots,K-1$. However, sampling from conditional distributions is not trivial, we therefore propose a Metropolis-Hastings step [@tierney:94] by sampling $\theta_k^*$ at iteration $(r+1)$ from a random walk proposal distribution $$q(\theta_k\mid{\boldsymbol{\theta}}_{-k},\theta_k^{(r)})=\pi_1 I(\theta_k=0)+(1-\pi_1)\un(\theta_k\mid \max\{a_k,\theta_k^{(r)}-\delta c_k\},\min\{b_k,\theta_k^{(r)}+\delta c_k\})$$ where the interval $(a_k,b_k)$ represents the conditional support of $\theta_k$, $c_k=b_k-a_k$ is its length, with $a_k = \max_{k \le j \le K-1} \left\{\left(\sqrt{\theta_{j+1}}I(\theta_{j+1}\geq 0) -\theta_0 - \sum_{i=1,i \ne k}^{j}(\tau_i - \tau_{i-1})\theta_i\right)/(\tau_{k} - \tau_{k-1})\right\}$, for $k=1,\ldots,K-1$, $b_k = \left(\theta_0 + \sum_{j=1}^{k-1} (\tau_j - \tau_{j-1})\theta_j \right)^2$, for $k=2,\ldots,K-1$, and $b_1 = 1$. The justification of these bounds obeys the inclusion of constraints $(C1)$ and $(C2)$ and their derivations are given in Appendix \[sec:appendix\]. The parameters $\pi_1$ and $\delta$ are tuning parameters that control de acceptance rate. Therefore, at iteration $r+1$ we accept $\theta_k^*$ with probability $$\alpha\left(\theta_k^*,\theta_k^{(r)}\right)=\min\left\{1\,,\;\frac{f(\theta_k^*\mid{\boldsymbol{\theta}}_{-k},\data)\,q(\theta_k^{(r)}\mid{\boldsymbol{\theta}}_{-k},\theta_k^{*})}{f(\theta_k^{(r)}\mid{\boldsymbol{\theta}}_{-k},\data)\,q(\theta_k^*\mid{\boldsymbol{\theta}}_{-k},\theta_k^{(r)})}\right\}.$$ Numerical studies {#sec:illust} ================= We illustrate the performance of our model in two ways, through a simulation study, and with a real data set. To define the partition $\{\tau_k\}$ of the positive real line we consider a Log-$\alpha$ partition defined by $\tau_k = - \alpha \log(1-k/K)$ for $k = 0,\ldots,K-1$, with $\alpha>0$. This partition is the result of transforming a uniform partition in the interval $[0,1]$ via a convex function. In particular we inspired ourselves in the generator of the product copula. Larger values of $\alpha$ increase the spread of the partition along the positive real line. Simulation study ---------------- We generated simulated data from four parametric Archimedean copulas, namely the product, Clayton, Ali-Mikhail-Haq (AMH) and Gumbel copulas. Their features are summarised in Table \[tab:parcopulas\], where we include the parameter space, the generator, the inverse generator, an indicator whether the copula is strict or not and the induced $h(t)$ function obtained through inversion of relationship . For each parametric copula we took a sample of size $n=200$. To specify the copulas we took $\theta\in\{-0.8,1\}$ for the Clayton copula, $\theta\in\{-0.7,0.7\}$ for the AMH copula, and $\theta=1.4$ for the Gumbel copula. For the partition size we compared $K\in\{10,20\}$ and tried values $\alpha \in \{0.3,0.5,0.9,1,2,\ldots,10\}$. For the prior distributions we took $\pi_0=0, \mu_0=-1$ and $\sigma_0^2=10$. We implemented a MH step with-in the Gibbs sampler where the proposal distributions were specified by $\pi_1=0$ and $\delta=0.25$. The acceptance rate attained with these specifications are around 30%, which according to are optimal for random walks. Finally, the chains were ran for 20,000 iterations with a burn-in of 2,000 and keeping one of every 5$^{th}$ iteration to produce posterior estimates. To assess goodness of fit (GOF) we computed several statistics. The logarithm of the pseudo marginal likelihood (LPML), originally suggested by , to assess the fitting of the model to the data. The supremum norm, defined by $\sup_{t}|\varphi^{-1}(t)-\widehat{\varphi}^{-1}(t)|$ to assess the discrepancy between our posterior estimate (posterior mean) $\widehat{\varphi}^{-1}(t)$ from the true inverse generator $\varphi^{-1}(t)$. We also computed the Kendall’s tau coefficient and compare the posterior point and 95% interval estimates with the true value. These values are shown in Tables \[tab:prod\] to \[tab:gumbel14\]. Although we fitted our model with all values of $\alpha$ mentioned above, we only show results for some of them in the tables. Note that, due to the nonunicity of an Archimedean generator, an equivalent constraint to $(C3)$ has to be imposed to the parametric generators that we are comparing to. That is we set $h(0)=1$ for the product, Clayton and AMH copulas, and $h(\epsilon)=1$ for the Gumbel copula, for say $\epsilon=0.01$. The difference in the latter case is because, for a Gumbel copula, $h(t)\to\infty$ when $t\to 0$. These conditions are already included in the parametrisation used in Table \[tab:parcopulas\]. For the product copula the GOF measures are presented in Table \[tab:prod\]. With exception of the partition Log-$3$ for $K=30$, for all settings considered, the true $\kappa_\tau$ lies inside the 95% credible intervals. The LPML chooses the model with Log-$1$ partitions of size $K=10$, and corresponds to the second smallest value of the supremum norm. Posterior estimates of functions $h(t)$ and $\varphi^{-1}(t)$ are shown in Figure \[fig:prod\]. In both cases the true function lies inside the 95% credible intervals. For the Clayton copula we have two choices of $\theta$, $-0.8$ and $1$. The first choice, $\theta=-0.8$, corresponds to a generator that is not strict, that is, $\varphi^{-1}(t)>0$ for $t\in[0,5/4]$, and $\varphi^{-1}(t)=0$ for $t>5/4$. This is an interesting challenge because our model defined only strict generators. The settings with smallest supremum norm, Log-$0.5$ with $K=10$, produces the 95% credible interval for $\kappa_\tau$ closest to the true value, however it does not achieve the largest LPML. The inconsistency of the GOF measures might be due to the non strictness feature of the true generator. Moreover, if we look at the graphs of the posterior estimates of $h(t)$ and $\varphi^{-1}(t)$ (Figure \[fig:clay\_08\]), for larger values of $t$ the true functions lie outside of our posterior estimates. For $\theta=1$, the best model is obtained with a Log-$6$ partition of size $10$. In this case, posterior estimates of functions $h(t)$ and $\varphi^{-1}(t)$ with the best fitting (Figure \[fig:clay1\]), contain the true functions. For the AMH copula we have two values of $\theta$, $-0.7$ and $0.7$. The best fitting consistently chosen by the three GOF criteria is obtained with a Log-$6$ and Log-$1$ partitions of size $K=10$, respectively for the two values of $\theta$. Posterior estimates of functions $h(t)$ and $\varphi^{-1}(t)$ with the best fitting are shown in Figures \[fig:amh\_07\] and \[fig:amh07\], respectively. In all cases the true functions lie within the 95% credible intervals. For the Gumbel copula with $\theta=1.4$ we have an interesting behaviour. The true $h(t)$ function has the feature that $h(0)=\infty$. This represents a challenge for our model since we have imposed the constrain $(C3)$ which is equivalent to $h(0)=1$. The highest LPML value is obtained with a Log-$7$ partition of size $K=10$, however the posterior 95% credible interval for $\kappa_\tau$ does not contain the true value. On the other hand, the second best value of LPML is obtained with an Log-$3$ partition of size $10$, and in this case the 95% credible interval for $\kappa_\tau$ does contain the true value. We select this latter as the best fitting. Posterior estimates of functions $h(t)$ and $\varphi^{-1}(t)$ are shown in Figure \[fig:gumbel14\]. Recalling that the true hazard function goes asymptotically to infinity when $t\to 0$, therefore, for values close to zero the true $h(t)$ lies outside our posterior credible intervals, something similar happens in the estimates of the inverse generator. Apart from this, our posterior estimates are very good for $t>\epsilon$. An important learning from the previous examples is that increasing the partition size doe not necessarily implies a better fitting. Real data analysis ------------------ In public health it is important to study the factors that determine the birth weight of a child. Low birth weight is associated with high perinatal mortality and morbility . We study the dependence structure between the age of a mother ($X$) and the weight of her child ($Y$), and concentrated on mothers of 35 years old and above. The dataset was obtained from the General Hospital of Mexico through the opendata platform that can be accessed at https://datos.gob.mx/busca/dataset/perfiles-metabolicos-neonatales/resource/4ab603eb-b73a-498f-8c56-0dc6d21930e8. It contains $n = 208$ records of the neonatal metabolic profile of male babies registered in the year 2017 in Mexico City. The marginal distributions for variables $U$ and $V$, induced by copula , are uniform. In practice, copulas are used to model the dependence for any pair of random variables regardless of their marginal distributions. Let $X$ and $Y$ be two random variables with marginal cumulative distributions $F(x)$ and $G(y)$ respectively. Then the joint cumulative distribution function for $(X,Y)$ is obtained as [@sklar:59], $H(x,y)=C(F^{-1}(x),G^{-1}(y))$, where $C$ is given in . Since we are just interested in modelling the dependence between $X$ and $Y$, it is common in practice to transform the original data, $(X_i,Y_i)$, $i=1,\ldots,n$, to the unit interval via a modified rank transformation [@Deheuvels:79] in the following way. Let $\bX'=(X_1,\ldots,X_n)$ and $\bY'=(Y_1,\ldots,Y_n)$ then $U_i=\mbox{rank}(i,\bX)/n$ and $V_i=\mbox{rank}(i,\bY)/n$ are the transformed data, where $\mbox{rank}(i,\bX)=k$ iff $X_i=X_{(k)}$ for $i,k=1,\ldots,n$. This is based on the probability integral transform using the empirical cumulative distribution function of each coordinate. In Figure \[fig:realdata\] we show a dispersion diagram of the original data (left panel) and the rank transformed data (right panel). To avoid problems due to ties in the original data, we fist include a perturbation to the data by adding a uniform random variable $\un(0,0.01)$ to each coordinate. The sample Kendall’s tau value for the transformed data is $\tilde{\kappa}_{\tau} = -0.1122$. We fitted our model to the transformed data with the following specifications. To define the partitions we took values $\alpha \in \{0.3,0.5,0.9,1,2,\ldots,10\}$ with sizes $K\in\{10,20\}$. For the prior we took $\pi_0=0$, $\mu_0=-1$ and $\sigma_0^2=10$. The MCMC specifications were the same as those used for the simulated data. The GOF measures computed were the LPML and the posterior estimates (point and 95% credible interval) of $\kappa_\tau$. The results are reported in Table \[tab:realdata\]. The best fitting model according to LPML is that obtained with a partition of size $K=10$ and Log-$10$. The sample concordance $\tilde{\kappa}_\tau$ is included in our posterior 95% credible interval estimate $\kappa_\tau\in(-0.213,-0.098)$. The estimated hazard rate function $h(t)$ and the inverse generator $\varphi^{-1}(t)$, with the best fitting model, are included in Figure \[fig:realdata\]. The solid thick line corresponds to the point estimates and the solid thin lines to the 95% credible intervals. For a visual comparison, the blue dotted line corresponds to the function of the independence (product) copula. This confirms that there is a negative (weak) dependence between the age of the mother and the birth weight of the child. The older the mother, the less weight of the child. This finding could potentially help the policy makers to focus campaigns to help the awareness of future mothers. As mentioned in Section \[sec:model\], we can use our model to undertake an independence test. For that we choose the prior distribution for the $\theta_k$’s, as in , such that the prior probability of $H_0:\theta_1=\cdots=\theta_{K-1}=0$ is around $1/2$, in other words, we want $\P(H_0)=\pi_0^{K-1}=1/2$. Particularly, for a partition of size $K=10$ we need to specify $\pi_0=0.9258$. In order to get a point of mass proposal in the Metropolis-Hasting step we consider $\pi_1 = 0.3$. We re-ran our model using these values with the other specifications left unchanged. The posterior probability of $H_0$ becomes $\P(H_0\mid\data)= 0.53$, which leads to an inconclusive test. This might be explained by the fact that although the association is negative, it is very close to zero. To calibrate our independence test, we consider the simulated data from previous section of two copulas, the product copula and the Clayton copula with $\theta=-0.8$. In these cases the best fitting was obtained with a partition of size $K=10$, so we chose the same prior values as for the real data to perform an independence test. Posterior probabilities of $H_0$ are $0.81$ and $0$, respectively, showing that for independent data the posterior probability is a lot larger that 0.5, whereas for clearly dependent data the posterior probability of independence is zero. Concluding remarks {#sec:concl} ================== We have proposed a semiparametric Archimedean copula that is flexible enough to capture the behaviour of several families of parametric arquimedean copulas. Our model is capable of modelling positive and negative dependence. The number of parameters in the model to produce a good estimation of the dependence in the data should not be extremely high. For most of the examples considered here 10 parameters are enough. Defining an appropriate partition to analyse real data sets is not trivial. We suggest to try different values of $\alpha$ in a wide range and compare using a GOF criteria like the LPML we used here. In the exposition and in examples considered here, we concentrated on bivariate copulas, however extensions to more than two dimensions is also possible, say $C(u_1,\ldots,u_m)=\varphi^{-1}\left(\varphi(u_1)+\cdots+\varphi(u_m)\right)$. Performance of our semiparametric copula in this multivariate setting is worth studying. Our model is motivated by semiparametric proposals for survival analysis functions and appropriately modified to satisfy the properties of an Archimedean generator. The semiparametric generator presented here turned out to be based on quadratic splines, however, alternative proposals are possible. Instead of starting with a piecewise function for the derivative of a hazard rate, we could start by defining a piecewise constant function for the hazard rate itself. That is $h(t)=\sum_{k=1}^K\theta_k I(\tau_{k-1}<t\leq\tau_k)$ with $\theta_k>0$, and $\{\tau_k\}$ a partition of the positive real line. In this case the cumulative hazard function becomes $H(t)=\sum_{j=1}^{k}\theta_j\Delta_j+\theta_{k}(t-\tau_{k-1})$, for $t\in(\tau_{k-1},\tau_{k}]$, with $\Delta_j=\tau_{j}-\tau_{j-1}$. The inverse generator is then a linear spline of the form $$\nonumber \varphi^{-1}(t)=\exp\left\{-\sum_{k=1}^K\theta_k w_k(t)\right\},$$ with $$w_k(t)=\left\{\begin{array}{ll} \Delta_k, & t>\tau_k \\ t-\tau_{k-1} & t\in(\tau_{k-1},\tau_k] \\ 0 & \mbox{otherwise} \end{array}\right.$$ and the corresponding Archimedean generator has the form $$\nonumber \varphi(t)=\sum_{k=1}^K \left\{\tau_{k-1}-\frac{1}{\theta_{k}}(\log t+\vartheta_{k-1})\right\}I(\vartheta_{k-1}<-\log t\leq\vartheta_k),$$ with $\vartheta_k=\sum_{j=1}^{k}\theta_j\Delta_j$. To ensure convexity of the generator we further require $\theta_1\geq\theta_2\geq\cdots\geq\theta_K$. Furthermore, the Kendall’s tau has a simpler expression $$\nonumber \kappa_\tau=1+\sum_{k=1}^K\left\{e^{-2\vartheta_k}(1+2\theta_k\tau_k)-e^{-2\vartheta_{k-1}}(1+2\theta_k\tau_{k-1})\right\}.$$ However, it can be shown that this expression for the Kendall’s tau only allows positive values, constraining the possible associations captured by the model. Acknowledgements {#acknowledgements .unnumbered} ================ This research was supported by *Asociación Mexicana de Cultura, A.C.* Derivation of posterior conditional support of $\theta_k$ {#sec:appendix} ========================================================= In order to satisfy constraint $(C1)$, we consider first the case $\theta_k\leq 0$. Therefore\ $\min_{t \in (\tau_{k-1},\tau_k]}A_k + \theta_k t = A_k + \theta_k \tau_{k}$. This implies the following constraint for $\theta_k$, $$\theta_k \geq \max_{k \le j \le K-1} \left\{- \left( \theta_0 + \sum_{i=1,i \ne k}^{j}(\tau_i - \tau_{i-1})\theta_i\right)/(\tau_{k} - \tau_{k-1})\right\},$$ for $k = 1,\ldots,K-1$, where we define the empty sum as zero. On the order hand, if $\theta_k > 0$ we have $\min_{t \in (\tau_{k-1},\tau_k]}A_k + \theta_k t = A_k + \theta_k \tau_{k-1},$ and we get, from condition $(C2)$, the following restriction $$\theta_k < \left(\theta_0 + \sum_{i=1}^{k-1} (\tau_i - \tau_{i-1})\theta_i \right)^2.$$ This defines the upper bound $b_k$, for $k = 2,\ldots,K-1$, and $b_1=1$. Because the term $\theta_k$ appears on the right side of the previous inequality for $j=k+1,\ldots,K-1$, we need to consider the following restriction $$\theta_k > \left.\left( \sqrt{\theta_j} - \theta_0 - \sum_{i=1,i \ne k}^{j-1}(\tau_i - \tau_{i-1})\theta_i\right)\right/(\tau_{k} - \tau_{k-1})$$ if $\theta_j \geq 0$. Combining this with the constraint when $\theta_k\leq 0$ above, we get the lower bound $a_k$ for $k=1,\ldots,K-1$. [99]{} Deheuvels, P. (1979). La fonction de dépendance empirique et ses proprietés. Un test non paramétrique d’indépendance. [*Roy. Belg. Bull. Cl. Sci.*]{}, [**65**]{}, (5), 274–292. Di Bernardino, E. and Rullière, D. (2013). On certain transformation of Archimedean copulas: Application to non-parametric estimation of their generators. *Dependence Modeling* [**1**]{}, 1–36. Geisser, S. and Eddy, W. F. (1979). A predictive approach to model selection. *Journal of the American Statistical Association* **74**, 153–160. Genest, C., Nešlehová J. and Ziegel J. (2011). Inference in multivariate Archimedean copula models. *TEST* **20**: 223. https://doi.org/10.1007/s11749-011-0250-6. Genest, C. and Rivest, L. (1993). Statistical inference procedures for bivariate Archimedean copulas . *Journal of the American Statistical Association*, **88**, 1034-1043. Guillote, S. and Perron, J. (2015). Inference on Archimedean copulas using mixtures of Pólya trees. *Journal of Statistical Planning and Inference*, **166**, 2-13. Kass, R.E. and Raftery, A.E. (1995). Bayes Factors. *Journal of the American Statistical Association* **90**, 773–795. Klein, J.P. and Moeschberger, M.L. (2003). [*Survival analysis*]{}. Springer, New York. Nelsen, R.B. (2006). [*An introduction to copulas*]{}. Springer, New York. Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter. *The Annals of Mathematical Statistics* , **42** (4), 1285-1295. Nieto-Barajas, L.E. and Walker, S.G. (2002). Markov beta and gamma processes for modeling hazard rates. *Scandinavian Journal of Statistics* **29**, 413–424. Robert, C. P. and Casella, G. (2010). [*Introducing Monte Carlo methods with R*]{}. Springer, New York. Sklar, M. (1959). Fonctions de répartition á $n$ dimensions et leurs marges. Université Paris 8. Smith, A. and Roberts, G. (1993). Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods. [*Journal of the Royal Statistical Society, Series B*]{} [**55**]{}, 3-–23. Stevens-Simon, C. and Orleans, M. (2001). Low-birthweight prevention programs: The enigma of failure. *Birth* **26**, 184–191. Tierney, L. (1994). Markov chains for exploring posterior distributions. [*Annals of Statistics*]{} [**22**]{}, 1701-–1722. Vanderhende, F. and Lampert, P. (2005). Local dependence estimation using semiparametric Archimedean copulas. [*The Canadian Journal of Statistics*]{} [**33**]{}, 377–388. ![Functions $h'(t)$ (first panel), $h(t)$ (second panel) and $\varphi^{-1}(t)$ (third panel) for two scenarios of $\{\theta_k\}$. All negative values (solid line), and all positive values (dotted line).[]{data-label="fig:ilust1"}](Grafica_Derivadah.pdf "fig:"){width="5.5cm" height="5.cm"} ![Functions $h'(t)$ (first panel), $h(t)$ (second panel) and $\varphi^{-1}(t)$ (third panel) for two scenarios of $\{\theta_k\}$. All negative values (solid line), and all positive values (dotted line).[]{data-label="fig:ilust1"}](Grafica_h.pdf "fig:"){width="5.5cm" height="5.cm"} ![Functions $h'(t)$ (first panel), $h(t)$ (second panel) and $\varphi^{-1}(t)$ (third panel) for two scenarios of $\{\theta_k\}$. All negative values (solid line), and all positive values (dotted line).[]{data-label="fig:ilust1"}](Grafica_GeneradorInverso.pdf "fig:"){width="5.5cm" height="5.cm"} \[fig:2\] ![Prior distributions of Kendall’s tau under three different scenarios.[]{data-label="fig:simkt"}](HistTauKendall_v1.pdf "fig:"){width="5.5cm" height="5.cm"} ![Prior distributions of Kendall’s tau under three different scenarios.[]{data-label="fig:simkt"}](HistTauKendall_thetanegative.pdf "fig:"){width="5.5cm" height="5.cm"} ![Prior distributions of Kendall’s tau under three different scenarios.[]{data-label="fig:simkt"}](HistTauKendall_thetapositive.pdf "fig:"){width="5.5cm" height="5.cm"} ![ \[fig:prod\]](Grafh_Producto.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:prod\]](GrafInvGen_Producto.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:clay\_08\]](Grafh_Clayton-08.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:clay\_08\]](GrafInvGen_Clayton-08.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:clay1\]](Grafh_Clayton.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:clay1\]](GrafInvGen_Clayton.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:amh07\]](Grafh_AMH07.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:amh07\]](GrafInvGen_AMH07.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:amh\_07\]](Grafh_AMH-07.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:amh\_07\]](GrafInvGen_AMH-07.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:gumbel14\]](Grafh_Gumbel14.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:gumbel14\]](GrafInvGen_Gumbel14.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:realdata\]](MuestraOriginal.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:realdata\]](MuestraTransformada.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:perealdata\]](Grafh_Datos.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:perealdata\]](GrafInvGen_Datos.pdf "fig:"){width="6.5cm" height="6.0cm"} Copula $\Theta$ $\varphi(t)$ $\varphi^{-1}(t)$ Estrict? h(t) --------- --------------- ------------------------------------------------------------------------- -------------------------------------------------------------------------------- -------------------- -------------------------------------------------------------- Product - $ - \log(t)$ $e^{-t}$ Yes $1$ Clayton $[-1,\infty)$ $\frac{1}{\theta}(t^{- \theta}-1)$ $ (1+ \theta t)^{-1/\theta}$ If $\theta \geq 0$ $\frac{1}{1 + \theta t}$ AMH $[-1,1)$ $\frac{1}{1-\theta}\log \left( \frac{1 - \theta + \theta t}{t} \right)$ $ \frac{1- \theta}{e^{(1-\theta)t} - \theta}$ Yes $\frac{(1-\theta)e^{(1-\theta)t}}{e^{(1-\theta)t} - \theta}$ Gumbel $[1,\infty)$ $ \epsilon\left(\frac{-\log(t)}{\epsilon\theta}\right)^{\theta}$ $\exp\left\{-\epsilon\theta\left(\frac{t}{\epsilon}\right)^{1/\theta}\right\}$ Yes $\left(\frac{\epsilon}{t}\right)^{1- 1/\theta}$ : Summary of some parametric Archimedean copulas parametrised such that $h(0)=1$, for the first three copulas, and $h(\epsilon)=1$, for the Gumbel copula.[]{data-label="tab:parcopulas"}

ArXiv

ArXiv

--- abstract: 'We present an analysis of the photometry and spectroscopy of the host galaxy of [*Swift*]{}-detected GRB080517. From our optical spectroscopy, we identify a redshift of $z=0.089\pm0.003$, based on strong emission lines, making this a rare example of a very local, low luminosity, long gamma ray burst. The galaxy is detected in the radio with a flux density of $S_{4.5\,GHz}=$0.22$\pm$0.04mJy - one of relatively few known GRB hosts with a securely measured radio flux. Both optical emission lines and a strong detection at 22$\mu$m suggest that the host galaxy is forming stars rapidly, with an inferred star formation rate $\sim16$M$_\odot$yr$^{-1}$ and a high dust obscuration (E$(B-V)>1$, based on sight-lines to the nebular emission regions). The presence of a companion galaxy within a projected distance of 25kpc, and almost identical in redshift, suggests that star formation may have been triggered by galaxy-galaxy interaction. However, fitting of the remarkably flat spectral energy distribution from the ultraviolet through to the infrared suggests that an older, 500Myr post-starburst stellar population is present along with the ongoing star formation. We conclude that the host galaxy of GRB080517 is a valuable addition to the still very small sample of well-studied local gamma-ray burst hosts.' author: - | Elizabeth R. Stanway$^{1}$[^1], Andrew J. Levan$^{1}$, Nial Tanvir$^{2}$, Klaas Wiersema$^{2}$, Alexander van der Horst$^3$, Carole G. Mundell$^4$, Cristiano Guidorzi$^5$\ $^{1}$Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK\ $^{2}$Department of Physics and Astronomy, University of Leicester, University Road, Leicester LE1 7RH, UK\ $^{3}$Anton Pannekoek Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands\ $^{4}$Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK\ $^{5}$Department of Physics and Earth Sciences, University of Ferrara, via Saragat 1, I-44122, Ferrara, Italy\ date: 'Accepted 2014 October 28. Received 2014 October 23; in original form 2014 September 19' title: 'GRB 080517: A local, low luminosity GRB in a dusty galaxy at z=0.09' --- \[firstpage\] gamma-ray burst:individual:080517 – galaxies:star formation – galaxies:structure – galaxies: distances and redshifts Introduction ============ Long Gamma Ray Bursts (GRBs) are intense, relativistically beamed, bursts of radiation, likely emitted during the collapse of a massive star at the end of its life [@2006ApJ...637..914W]. As well as constraining the end stages of the evolution for massive stars, they also mark out star formation in the distant Universe, in galaxies often too small to observe directly through their stellar emission or molecular gas [e.g. @2012ApJ...754...46T]. However, extrapolating from the detection of a single stellar event (the burst) to their wider environment, and the contribution of their hosts to the volume averaged cosmic star formation rate [e.g. @2012ApJ...744...95R], is challenging. Doing so relies on a good understanding of the stellar populations and physical conditions that give rise to GRB events. This understanding has improved significantly over recent years. A number of studies now constrain the stellar properties of typical GRB hosts [e.g. @2009ApJ...691..182S; @2010MNRAS.tmp..479S; @2012ApJ...756..187H], their radio properties [e.g. @2012ApJ...755...85M; @2010MNRAS.409L..74S; @radiopaper; @2014arXiv1407.4456P] and behaviour in the far-infrared [@2014arXiv1402.4006H; @2014arXiv1406.2599S]. However these studies have also demonstrated diversity within the population. GRB host galaxies range from low mass, metal poor galaxies forming stars at a moderate rate [e.g. @2010AJ....140.1557L], to more massive moderately dusty but not extreme (SMG-like) starbursts such as the ‘dark’ burst population [@2013ApJ...778..128P; @2013ApJ...778..172P]. The challenge of understanding these sources has been complicated by the high redshifts at which they typically occur. The long GRB redshift distribution peaks beyond $z=1$ [@2012ApJ...752...62J], tracing both the rise in the volume-averaged star formation rate and the decrease in typical metallicity - which may favour the formation of GRB progenitors [see e.g. @2012ApJ...744...95R and references therein]; local examples which can be studied in detail are rare. Of long duration ($>$2s) bursts in the official [*Swift Space Telescope*]{} GRB catalogue table[^2], only three are listed as having $z<0.1$. A few other (pre-[*Swift*]{}) bursts are also known at low redshifts [e.g. GRB980425 at $z=0.009$ @1998Natur.395..670G], but were detected by instruments with quite different systematics and tend to be unusual systems. One of the most recent studies, which exploited ALMA data, identified the host of GRB980425 as a dwarf system with low dust content and suggested that this is typical of GRB hosts as a whole . However each low redshift host investigated in detail has informed our understanding of the population as a whole and proven to differ from the others [e.g. @2011ApJ...741...58W; @2011MNRAS.411.2792S]. Low redshift bursts include several which are sub-luminous, such as GRBs090825 and 031203 [@1998Natur.395..670G; @2004ApJ...609L...5M; @2004Natur.430..648S], and others such as GRBs060505 and 060614 that were long bursts without associated supernovae [@2006Natur.444.1047F; @2006Natur.444.1050D]. Cross-correlation with local galaxy surveys (at $z<0.037$) has suggested that some low redshift GRBs in the existing burst catalogues have yet to be identified as such [@2007MNRAS.382L..21C] and hence opportunities to study their properties in detail have been missed. Given the very small sample, and the variation within it, it is important that we continue to follow up the hosts of low redshift bursts and do not allow a few examples to skew our perception of the population. We have acquired new evidence suggesting that a previously overlooked burst, GRB080517, and its host galaxy might prove a valuable addition to the study of local gamma ray bursts. The WISE all-sky survey [@2010AJ....140.1868W], publically released in 2012, maps the sky at 3-22$\mu$m. While the observations are relatively shallow and most GRB hosts remain undetected or confused, we have identified the host of GRB080517 as anomalous. Not only is an infrared-bright source clearly detected coincident with the burst location, but it has a sharply rising spectrum and is extremely luminous in the 22$\mu$m W4 band, suggesting that it is a rather dusty galaxy, and likely at low redshift. In this paper, we present new photometry and spectroscopy of the host of GRB080517, identifying its redshift as $z=0.09$. Compiling archival data, we consider the spectral energy distribution (SED) of the host galaxy, and also its larger scale environment, evaluating the source as a low redshift example of a dusty GRB host galaxy. In section \[sec:initial\] we discuss the initial identification of this GRB and its properties. In section \[sec:data\] we present new data on the host galaxy of this source. We present our optical photometry and spectroscopy of the GRB host and a neighbouring companion in section \[sec:spec\] and report a detection of the GRB host at radio frequencies in section \[sec:radio\]. In section \[sec:reassess\] we reassess the initial burst properties and its early evolution in the light of our new redshift information. In section \[sec:sed\] we compile new and archival photometry to secure an analysis of the spectral energy distribution, and in section \[sec:sfr\] report constraints on the host galaxy’s star formation rate. In section \[sec:disc\] we discuss the properties of the host galaxy in the context of other galaxy populations before presenting our conclusions in section \[sec:conc\]. Throughout, magnitudes are presented in the AB system [@1983ApJ...266..713O] and fluxes in $\mu$Jy unless otherwise specified. Where necessary, we use a standard cosmology with $h_0$=70kms$^{-1}$Mpc$^{-1}$, $\Omega_M=0.3$ and $\Omega_\Lambda$=0.7. Initial Observations {#sec:initial} ==================== GRB080517 triggered the [*Swift*]{} Burst Alert Telescope (BAT) at 21:22:51 UT on 17th May 2008 as a flare with a measured T$_{90}$ (i.e. period during which 90% of the burst energy was detected) of 65$\pm$27s, classifying the event as a long GRB. The X-ray Telescope (XRT) identified a fading, uncatalogued point source and the presence of a known optical source was noted within the X-Ray error circle. The final enhanced XRT position, with uncertainty $1\farcs5$, was 06h 48m 58.03s +50° 44′ 07.7′′ (J2000), coincident with the optical source [@2008GCN..7742....1P]. The Galactic longitude and latitude ($l=$165.369, $b=$20.301) correspond to a sight-line with moderate dust extinction (A$_V$=0.25) from our own galaxy [@2011ApJ...737..103S]. Early observations with the Liverpool Telescope, starting 11 minutes after the BAT trigger, did not detect an optical transient outside of the known source [@2008GCN..7743....1S] and no further optical follow-up was undertaken - in part due to the difficult proximity (within 50$^\circ$) of the Sun at the time the burst triggered [*Swift*]{}. Both the lack of an optical afterglow and analysis of the BAT spectrum suggested that the source might lie at high redshifts [@2008GCN..7748....1M; @2011ApJ...731..103X], but constraints on the X-ray spectrum precluded a high redshift fit to the data [@2008GCN..7742....1P]. Association with the known, bright optical source would suggest a lower redshift for the burst, but it was not clear whether this was the host galaxy or a star in chance alignment. While the afterglow was not detected in the optical, the $\gamma$-ray and X-ray emission was also relatively weak, with an early time flux at 0.3-10keV of 2.52$^{+1.20}_{-0.75}\times10^{-10}$ergcm$^{-2}$s$^{-1}$, measured in a 10s exposure at a mean photon arrival time of T$_0$+133s [based on analysis from @2009MNRAS.397.1177E]. In the absence of a redshift for the host, the time-averaged X-ray analysis also suggested the presence of an excess neutral hydrogen column density of $3.0^{+2.1}_{-1.8} \times 10^{21}$ cm$^{-2}$ above the Galactic value of $1.09\times10^{21}$cm$^{-2}$ [where these are 90% confidence intervals in analysis from the UK Swift Science Data Centre, @2009MNRAS.397.1177E]. This represents an excess in the X-ray inferred hydrogen column at the $\sim$3$\sigma$ level. [*Swift*]{} observations ended approximately 20 hours after the initial trigger. Initial observations for this source were therefore ambiguous, with different elements of the data either suggesting a high redshift solution (non-detection of the optical transient, BAT spectrum) or appearing to preclude it (optical source association, X-ray spectrum), and the excess extinction seen in the afterglow implying the presence of dust either in the host galaxy or along the line of sight. However the burst’s location, within 50$^\circ$ of the Sun at the time the burst went off, precluded further early time studies, and the burst has largely been ignored since. [*Swift*]{} has not observed this location at any other time. Given the presence of a relatively bright, $r_{AB}=17.73$, catalogued source within the [*Swift*]{} XRT error circle, an obvious question arises: what is the probability that this is a chance alignment rather than a genuine host galaxy identification? Two main factors contribute to this determination. The surface density of galaxies observable at a given magnitude will depend both on the properties of the galaxy population with redshift, and with galactic latitude (which will govern the fraction of the sky affected by foregrounds and crowding). To evaluate this, we have studied the galaxy population in regions of the Sloan Digital Sky Survey Data Release 10 [SDSS DR10, @2014ApJS..211...17A] at comparable Galactic latitude (within $\sim$5$^\circ$) and offset by 30-50$^\circ$ in Galactic longitude. The population identified by the SDSS photometric pipeline as galaxies were selected in ten regions, each with a diameter of 1$^\circ$, and their surface density evaluated as a function of $r'$-band magnitude. As figure \[fig:coin\] illustrates, the surface density of galaxies comparable to the proposed host of GRB080517 is low, with $0.028\pm0.006$ galaxies typically found per square arcminute. Assuming that the SDSS photometric classification is accurate, the probability of finding a galaxy of this brightness within 3arcseconds (see figure \[fig:whtim\]) of a given X-ray location is just 0.007%. Taking into account the 604 long bursts with X-ray localisations in the [*Swift Space Telescope*]{} GRB catalogue table, we would expect 0.04 chance alignments amongst the entire long GRB population. A further constraint arises from the nature of the GRB itself. GRB080517 was a long burst, believed to be associated with a core collapse progenitor, and so likely to be associated with recent or ongoing star formation. If we consider only those galaxies in SDSS with the flat optical colours associated with ongoing star formation, i.e. $|r'-i'|<0.5$, the surface density of galaxies drops still further, to just $0.021\pm0.006$ galaxies per square arc minute, and a predicted 0.03 chance alignments in the entire GRB sample. As will be discussed below, the possible host galaxy of GRB080517 is strongly star forming, and lies within 3 arcseconds of the burst location. Thus we propose its identification as the burst host. ![The surface density of galaxies brighter than a given $r'$-band magnitude, at comparable galactic latitude to GRB080517, based on photometric classification in the Sloan Digital Sky Survey. The solid line shows the surface density of all galaxies, with the standard deviation measured across ten 1$^\circ$-diameter fields. The dashed line shows the lower surface density of relatively blue galaxies likely to be star-forming. The dotted vertical line indicates the magnitude of the proposed host of GRB080517. \[fig:coin\]](coincidence_prob.ps){width="0.99\columnwidth"} Follow-Up Data {#sec:data} ============== WHT Imaging {#sec:whtim} ----------- We targetted the host of GRB080517 on the night of 2014 Feb 25 (i.e. 6 years post-burst) using the auxiliary-port camera, ACAM, on the William Herschel Telescope (WHT). Photometric imaging was acquired in the Sloan $g$, $r$ and $i$ bands, with an integration time of 180s in each band. Observations were carried out as part of programme W/2014/9 (PI: Levan) and photometric data were reduced and calibrated using standard [IRAF]{} procedures. As figure \[fig:whtim\] demonstrates, the 1.2arcsecond seeing was sufficient to determine the morphology of both the host galaxy and a near neighbour, separated from it by 16 arcseconds. While the GRB host shows a relatively smooth, relaxed morphology, it is resolved in our imaging with a measured gaussian FWHM of 2.1arcseconds. Deconvolution with the seeing, as measured from unresolved sources in the image, yields an estimated intrinsic size of 1.6arcseconds, or 2.7kpc at $z=0.09$ (see next section). ![The structure of the compact GRB host galaxy (lower left) and its near-neighbour (upper right) in the Sloan-$i$ band from our WHT imaging. The neighbour clearly has two cores within a more diffuse galaxy, and is likely to be undergoing a major merger. Both sources are at the same redshift (see section \[sec:spec\]), the scale bar indicates physical distance at this redshift. The 1.5arcsecond 90% confidence error circle from the [*Swift*]{} XRT detection of the burst is indicated in red. \[fig:whtim\]](targets.ps){width="0.99\columnwidth"} ![The radial surface brightness profile of the GRB host galaxy in the Sloan-$i$ band from our WHT imaging. Sersic profiles have been convolved with the seeing and overplotted for comparison. Given the large error bars - due to compact morphology relative to the pixel scale and seeing - a range of Sersic parameters provide a reasonable fit to the data. Normalising the profiles close to the centre suggests a Sersic index of $n\sim1.0-2.0$ may provide the best description of this galaxy’s light profile. The gaussian seeing is shown as a solid line for comparison. \[fig:sersic\]](plot_sersic2.ps){width="0.99\columnwidth"} ![The allowed regions of parameter space for a Sersic light profile, quantified by $\chi^2$-fitting against the data. Small effective radii ($<2''$) and low Sersic indices ($n\sim1-2$) are favoured by the data, but there are substantial degeneracies between these parameters. Contours are shown at 1, 2 and 3$\sigma$ confidence levels. \[fig:sersic2\]](fit_sersic2.ps){width="0.99\columnwidth"} While we are unable to distinguish clumpiness on sub-kiloparsec scales, the host galaxy is sufficiently resolved in this new imaging to place constraints on its radial light profile, although such constraints are necessarily limited by the relatively large (0.253 arcsecond) pixels relative to the seeing. In figures \[fig:sersic\] and \[fig:sersic2\] we compare the radially averaged light profile of the galaxy with Sersic profiles [see @2005PASA...22..118G for definitions and discussion], which have been convolved with the seeing in the image. It is clear that a de Vaucouleurs law ($n=4$), such as describes local giant elliptical galaxies would predict far too steep a light profile. Allowing the effective radius and Sersic parameter to vary simultaneously, the best fit to the data is found for $n=1.5\pm1.0$ and $R_e=1.7\pm0.8''$. WHT Spectroscopy {#sec:spec} ---------------- We also obtained spectroscopic data from ACAM on the same night, using the V400 grating and a total integration time of $4\times600$s, producing a spectrum spanning 4000-9000Å with a spectral resolution measured from unblended arc lines of $\sim$18Å($\sim$1000kms$^{-1}$). Both photometric and spectroscopic data were reduced and calibrated using standard [IRAF]{} procedures. Absolute flux and wavelength calibration were achieved through observations of a standard star field and arc lamps immediately preceding the science data. The slit was oriented at a position angle of 50$^\circ$, so as to pass through the centres both of the GRB host and the bright neighbour, separated from it by 16$\arcsec$ measured along the 1$\farcs5$ slit. Both the GRB host galaxy and its neighbour are detected at high signal to noise in our spectroscopy. The latter clearly shows two components, A and B. Of these, component A is the stronger continuum source, while component B appears to show relatively stronger line emission (see figure \[fig:2dspec\]). In table \[tab:lines\] we provide the relative emission line strengths of each source (also shown graphically in figure \[fig:specall\]). Line equivalent widths are presented in the observed frame. We make no adjustment for slit losses since it is difficult to reconstruct precisely where on the object the 1.5 arcsec wide slit was placed, and harder still to estimate whether line ratios in the regions of the galaxy outside the slit are comparable to those in observed regions. The measured redshift for the host galaxy is $z=0.089\pm0.003$ and for the neighbour $z=0.091\pm0.003$ (for both components). The uncertainty, estimated by cross correlation against a template starburst spectrum, comprises instrumental resolution effects, the effects of blending on many of the lines and uncertainty due to small shifts in velocity offset between different emission lines. While we adopt the cross-correlation redshifts and conservative associated uncertainties for our analysis, we also consider the redshift derived from the observed wavelength of individual emission lines. Fitting gaussian profiles to the unblended H$\beta$ and \[OIII\] and the strong, but somewhat blended H$\alpha$ lines, we derive redshifts $z=0.0903\pm0.0006$, $0.0925\pm0.0006$ and $0.0930\pm0.0003$ for the GRB host and components A and B of the companion respectively, where the error now represents the scatter between individual line centroids on each source rather than including other uncertainties. These imply velocity offsets of $\Delta v = 150\pm155$kms$^{-1}$ (i.e. no significant offset) between the two components of the companion and $\Delta v = 576\pm155$kms$^{-1}$ between the host and the companion. This velocity offset places the galaxy pair just outside (although within one standard deviation of) the criteria used to select galaxy pairs in the SDSS by @2008AJ....135.1877E, who placed a cutoff for their sample at $\Delta v = 500$kms$^{-1}$. Those authors recognise however that this cutoff requires a trade-off between contamination and completeness, with genuine pairs observed out to $\Delta v \sim 600$kms$^{-1}$ separations [@2008AJ....135.1877E; @2000ApJ...536..153P]. @2008AJ....135.1877E identified an enhancement in star formation rate for pairs with projected separations $<30-40$kpc, a criterion easily satisifed by the companion in this case (16$''$ represents $\sim27$kpc at this redshift), suggesting that the star formation observed in both host and companion is likely influenced by their proximity. In figures \[fig:specha\] and \[fig:spechb\], we present the spectral regions in the GRB host galaxy associated with line ratios used to classify an ionising spectrum (see section \[sec:disc\]). At this spectral resolution, H$_\alpha$ is blended with the \[N[II]{}\] doublet, and a fit to the three lines must be obtained simultaneously in order to measure their line strengths. With the exception of close doublets (i.e. \[O[II]{}\], \[S[II]{}\]) the other lines in the spectrum are all comparitively unblended, and all lines are consistent with being essentially unresolved at the instrumental resolution. {width="1.97\columnwidth"} Line Host Neighbour A B -------------- ----------------- ---------------- ---------------- O[II]{}3726 61.9 $\pm$ 8.0 O[II]{}3729 31.0 $\pm$ 4.0 H$\gamma$ 6.5 $\pm$ 0.4 H$\beta$ 16.8 $\pm$ 1.0 3.2 $\pm$ 1.3 22 $\pm$ 9 O[III]{}4959 5.8 $\pm$ 0.3 4.1 $\pm$ 1.7 27 $\pm$ 11 O[III]{}5007 17.1 $\pm$ 1.0 12.4 $\pm$ 5.0 77 $\pm$ 32 He I 5875 2.8 $\pm$ 0.1 O[I]{}6300 6.1 $\pm$ 0.2 N[II]{}6548 10.3 $\pm$ 0.2 1.6 $\pm$ 0.1 4.1 $\pm$ 0.4 H$\alpha$ 103.1 $\pm$ 2.4 31.8 $\pm$ 1.5 116 $\pm$ 12 N[II]{}6583 31.5 $\pm$ 0.7 4.9 $\pm$ 0.2 12.5 $\pm$ 1.2 S[II]{}6716 13.3 $\pm$ 0.5 S[II]{}6730 13.0 $\pm$ 0.5 : Line strengths measured for the target objects. All measures are given as observed-frame equivalent widths in Angstroms. Measurement of weak lines is not attempted in the fainter neighbour, and it is impossible to isolate the two components in the \[O[II]{}\] line.\[tab:lines\] {width="1.97\columnwidth"} {width="1.97\columnwidth"} ![The spectral region containing H$\alpha$ and the \[N[II]{}\] doublet. All three lines are consistent with the being unresolved at the instrumental FWHM. The relative strength of the doublet lines is consistent with that predicted from the electron transition probabilities. \[fig:specha\]](spec_080517_ha.ps){width="0.98\columnwidth"} ![The spectral region containing H$\beta$ and the \[O[III]{}\] doublet.\[fig:spechb\]](spec_080517_hb.ps){width="0.98\columnwidth"} The relative intensity of the nebular emission in hydrogen Balmer lines allows us to make an estimate of the dust extinction in the actively star forming region of the GRB host galaxy. The flux in H$\alpha$ is expected to scale relative to H$\beta$ by a ratio determined by the temperature and electron density of the emitting region. Standard assumptions for these parameters in HII regions (T=10,000K, low density limit) yields the widely applied expected ratio of 2.87 [see @2006agna.book.....O]. In figure \[fig:specratio\], we scale the line fluxes from the GRB host galaxy accordingly, such that, for Case B reionisation, we would expect each line to have the same relative intensity as H$\beta$. It is clear that the Balmer series shows a decrement in the later lines, most likely attributable to dust. Given a Calzetti-like dust law [$R_V=4.05$, @2000ApJ...533..682C], the measured H$\alpha$/H$\beta$ ratio implies an extinction of flux from the nebular emission region of the GRB host galaxy of E($B-V$)=1.2. Of course, uncertainty arises in whether the HII region parameters and extinction law adopted are indeed appropriate for GRB hosts galaxies. @2011MNRAS.414.2793W explored a grid of temperatures and dust extinction laws for fitting the H[I]{} Balmer series in example spectra and suggested that in the case of the GRB060218 host a higher temperature and steeper extinction law (T=15,000K, $R_V=4.5$) might be appropriate, while the host of GRB100316D is best fit with T$\sim7500$K and $R_V=3.5$. The Calzetti dust law lies between that inferred from these two examples, as does our adopted temperature. More detailed spectroscopy (with fainter Balmer lines, and ideally also the He II recombination series) would be required to reach a tighter constraint. ![A comparison of the Balmer series emission lines in the GRB host galaxy, scaled by the appropriate line ratios for Case B H[I]{} recombination (T=$10^4$K, low density, Osterbrock & Ferland 2006), such that in the absence of dust, all peaks would be expected to match H$\beta$ in intensity. The line intensities are offset from zero for clarity. Even accounting for blending with the \[N[II]{}\] doublet, H$\alpha$ still shows a relative excess, consistent with the blue-wards lines being attenuated by a dusty line of sight. \[fig:specratio\]](spec_080517_hratios2.ps){width="0.98\columnwidth"} We also obtain a tentative spectroscopic redshift for an unrelated galaxy falling on the long slit. The galaxy, located at RA and Declination 06$^h$48$^m$59.756$^2$ +50$^\circ$44$'$23.50$''$ (J2000), lies at $z=0.56$, based on identification of an emission feature as the \[O[II]{}\] 3727Å doublet. Radio Observations {#sec:radio} ------------------ The low redshift confirmed for this GRB host makes it an ideal candidate for radio observation. The majority of radio observations of GRB hosts to date have resulted in non-detections, implying star formation rates that do not significantly exceed their UV-optical estimates [e.g. @2012ApJ...755...85M; @2010MNRAS.409L..74S; @radiopaper]. However, some fraction of GRB hosts appear to be luminous in the submillimetre-radio [@2003ApJ...588...99B; @2004MNRAS.352.1073T], particularly amongst those that show evidence for strong dust extinction [i.e. dark bursts, @2013ApJ...778..172P; @2014arXiv1407.4456P]. Radio observations of the GRB080517 host galaxy were performed with the Westerbork Synthesis Radio Telescope (WSRT) at 4.8GHz on 2014 May 2 and May 3 UT, i.e. almost 6 years after the gamma-ray trigger. We used the Multi Frequency Front Ends [@tan1991] in combination with the IVC+DZB back end in continuum mode, with a bandwidth of 8x20 MHz. Gain and phase calibrations were performed with the calibrator 3C147. The data were analyzed using the Multichannel Image Reconstruction Image Analysis and Display [[MIRIAD]{}; @1995ASPC...77..433S] package. Both observations resulted in a detection of a source at the position of GRB080517, with consistent flux densities. We have measured the flux density in an image of the combined data set as $S_{4.8\,GHz}=$0.22$\pm$0.04mJy. The detection is consistent with a point source, in observations with a synthesised beam of $14.2\times5.3''$ as shown in figure \[fig:radim\]. No significant detection is made of the neighbour galaxy - somewhat surprisingly given its high inferred star formation rate (based on H$\alpha$ emission, see section \[sec:disc\]). ![The 4.8GHz radio flux measured at the WSRT (contours), overlaying the compact GRB host galaxy (lower left) and its near-neighbour (upper right) in the Sloan-$r$ band (greyscale). The contours indicate levels of zero flux (dotted) and +2, 3, 4 and 5$\sigma$. There are no signals below $-2$$\sigma$ in this region. The burst location is indicated with a cross. \[fig:radim\]](radio_targets.ps){width="0.95\columnwidth"} Reassessing GRB080517 {#sec:reassess} ===================== Given the identification of a redshift for the host galaxy of GRB080517, we are able to reassess the properties of the burst and its immediate afterglow in the context of an accurate distance (and thus luminosity) measurement, allowing for more meaningful comparison with the rest of the GRB population. Burst properties {#sec:grb} ---------------- At $z=0.09$ the inferred isotropic equivalent energy of GRB080517 is only $E_{iso} = (1.03 \pm 0.21) \times 10^{49}$ ergs, while its 10 hour X-ray luminosity is $L_X \sim 10^{42}$ ergs s$^{-1}$. Both of these values lie orders of magnitude below the expectations for most GRBs, which have characteristic values of $E_{iso} \sim 10^{52-54}$ ergs [@2008ApJ...680..531K; @2009ApJ...693.1484C] and $L_X \sim 10^{45-47}$ ergs s$^{-1}$ [@2006ApJ...642..389N]. These properties mark GRB080517 as a member of the observationally rare population of low luminosity GRBs (LLGRBs). Only a handful of such low luminosity events have been identified in the past decade, all of which have been relatively local (given the difficulty in observing low luminosity bursts beyond $z\sim0.1$). These include the well studied GRB-supernova pairs GRB980425/SN 1998bw [@1998Natur.395..670G], GRB031203/SN 2003lw [@2004ApJ...609L...5M; @2004Natur.430..648S], GRB 060218/SN 2006aj [@2006Natur.442.1011P] and GRB100316D/SN 2010bh [@2011MNRAS.411.2792S; @2011ApJ...740...41C], and the enigmatic GRBs 060505 and 060614, where associated SNe have been ruled out to deep limits, and whose origin remains mysterious [@2006Natur.444.1047F; @2006Natur.444.1050D]. They may be low luminosity events akin to those above, but where the SNe is absent [e.g. @2006Natur.444.1053G], or they could be GRBs with a similar physical origin to the short-GRBs [most likely NS-NS mergers based on recent observations, @2006Natur.444.1044G; @2013Natur.500..547T], in which case their luminosities would be more typical of their population. GRB 080517 adds a further example to these local, low luminosity events. Its highly star forming host galaxy (as discussed later) is perhaps most in keeping with the expectations of long-GRBs, although its high stellar mass and metallicity would be unusual at such low redshift [@2013ApJ...774..119G]. The prompt emission from GRB080517, as reported by the [ *Swift*]{}/BAT instrument, shows a single ‘fast rise, exponential decay’ [FRED @1995PASP..107.1145F] lightcurve, albeit at low signal to noise. In this respect, its profile is similar to that of low luminosity GRBs031203 and 980425 [@2007ApJ...654..385K], although the profile is not unusual amongst GRBs more generally [@1996ApJ...473..998F]. Interestingly, within the low luminosity population there appears to be a good deal of internal diversity. The [*Swift*]{}-identified low luminosity events to date – GRB060218 [@2006Natur.442.1008C] and GRB100316D [@2011MNRAS.411.2792S] – appear to be of extremely long duration (2000s in the case of GRB060218) with extremely smooth light curves. They are also very soft events in which the X-ray emission exceeds that in the $\gamma-$ray (so called X-ray Flashes). In contrast, the pre-[*Swift*]{} events (GRB980425 and GRB031203) appear to be much closer in prompt properties to classical GRBs, exhibiting shorter durations (tens of seconds) and relatively hard $\gamma$-ray spectra. Although there has been some suggestions that GRB 031203 was a softer X-ray event, integrated over longer time periods, this is based on inferences from its X-ray echo [@2004ApJ...603L...5V], and favour a soft component arising after the initial burst [@2004ApJ...605L.101W]. In retrospect this is most likely an X-ray flare, as are commonly seen in [*Swift*]{} X-ray afterglows [@2006ApJ...636..967W; @2006ApJ...642..389N]. Hence we consider the $E_p$ measured via [*INTEGRAL*]{} as likely indicative of the true burst $E_p$ [@2004Natur.430..646S]. In this case, both GRB 980425 and GRB 031203 lie well away from the correlation between the peak of the $\nu F_{\nu}$ spectrum ($E_p$) and $E_{iso}$ . GRB080517 appears to have much more in common with these pre-Swift events, with a $T_{90} = 65\pm27$s and a hard photon index of $\Gamma \sim 1.5$. While $E_p$ is difficult to directly constrain with the limited BAT bandpass, the Bayesian method of @2007ApJ...663..407B suggests that $E_p > 55$ keV, making GRB080517 a significant outlier in the $E_p$ – $E_{iso}$ relation, with a similar location to GRB980425 and GRB031203 (see figure \[fig:amati\_rel\]). Its recovery, some 6 years after the initial detection implies that other, similar, low luminosity events may be present within the [*Swift*]{} catalog, since a significant number of bursts have not been followed in depth due to observational constraints. However, GRB080517 was unusual in having a bright catalogued source within its error box - a relatively rare occurrence. In this context, it should be noted that the host of GRB080517 is relatively luminous for a GRB host. By contrast, the hosts of GRB980425 and GRB060218 would have had observed magnitudes of $r\approx$20 and $r\approx$22.5 at $z=0.1$ and so would not be readily cataloged in DSS and similar survey observations, suggesting that the presence of a catalogued source is not necessarily a good indicator of event frequency. ![The location of GRB080517 on the $E_p$ – $E_{iso}$ (Amati) relation, given its redshift of $z=0.09$. Black points indicate long GRBs, while those in grey are the short GRB population. GRB080517 and previously identified low luminosity bursts are labelled. The burst lies in an unusual region of parameter space, well below the commonly seen relation for GRBs, placing it in the class of low redshift, low luminosity bursts. \[fig:amati\_rel\]](plot_amati.ps){width="1.05\columnwidth"} Afterglow reanalysis {#sec:afterglow} -------------------- Making use of data analysis tools available from the UK Swift Data Centre[^3], specifying the burst redshift as matching that determined for the host, we have reanalysed the Swift XRT afterglow spectrum and early time series data. Allowing for absorption at the host redshift only slightly modifies the hydrogen column density required to fit the burst X-ray spectrum observed by [*Swift*]{}. The effect on the late time spectrum (mean photon arrival time = T0+25863, where T0 is the [*Swift*]{} trigger time) is negligible, not modifying the required intrinsic absorption from the N$_H=6^{+4}_{-5}\times10^{21}$cm$^{-2}$ in excess of the estimated Galactic absorption estimated with the absorber at $z=0$ (where the errors on N$_H$ from Swift are 90% confidence rather than 1$\sigma$ intervals). The early time PC-mode data, with a mean photon arrival of T0+9559s, yields a lower (but consistent) estimated intrinsic absorption (N$_H=3.4^{+2.4}_{-2.0}\times10^{21}$cm$^{-2}$), and a photon index of 1.9$\pm$0.4. Optical/ultraviolet imaging was also obtained by the [*Swift*]{}/UVOT instrument, from first acquiring the field to the end of observations at T0+19hrs. Data were observed in 6 bands ($V$, $B$, $U$, $UVW1$, $UVM2$ and $UVW2$), and photometric imaging was obtained in each band at intervals throughout this early period. As figure \[fig:uvotearly\] demonstrates, these is little evidence of temporal variation in five of the six bands. Each is consistent with a constant flux. There is a hint of declining flux in the last observations taken in the reddest, $V$, band but the substantial uncertainties on this data preclude a firm identification of afterglow flux. No other band shows a comparable decline during the observation interval. ![Early time optical and ultraviolet photometry from [*Swift*]{}/UVOT. The flux density in the $B$ and $U$-band and the three ultraviolet wavebands shows little evidence of variation. There is a hint of declining flux in the $V$-band, but given the large photometric errors in this band, any decline is difficult to constrain with any accuracy. \[fig:uvotearly\]](lightcurve_plot2.ps){width="0.99\columnwidth"} ![Comparison of averaged early time optical and ultraviolet photometry from [*Swift*]{}/UVOT (red, crosses), with the late time observations of the host from other sources (see section \[sec:sed\]). The UVOT observations are averaged from T0+0hr to T0+19hr. The late time observations are taken at several years post-burst. Nonetheless, the UVOT observations are consistent (within the photometric errors) with the host galaxy data, suggesting that any afterglow was below the UVOT detection limit. \[fig:uvotcomp\]](comp_early_phot.ps){width="0.99\columnwidth"} In figure \[fig:uvotcomp\], we compare this early time UVOT data, now integrated across the 19 hour observation assuming no temporal variation, with the late time host galaxy data described in sections \[sec:whtim\] and \[sec:sed\]. While observations in $U$ and $B$ are not available at late times, the measured flux in the early time integrated $V$ band and near-ultraviolet bands are consistent with that in late time observations of the host galaxy. With the exception of possible variation the $V$-band data, no afterglow is detected within the photometric errors, suggesting that any optical supernova was at or below the UVOT detection limit. Taking the 1$\sigma$ upper limit on the early time photometry, and subtracting off the late time galaxy flux (see below), we constrain the optical afterglow to $F_\lambda<2\times10^{-17}$ergss$^{-1}$Å$^{-1}$ at T0+4000s, measured at 5500Å. The [*Swift*]{} detected X-ray flux at the same epoch (T0$\sim5227^{+1471}_{-1016}$s), was $(3.1\pm0.8)\times10^{-13}$ergss$^{-1}$cm$^{-2}$ in the 0.3-10keV band. Comparing these yields a limit on X-ray to optical ratio, $\beta_{OX}<1.0$. Finally, we have reexamined the early time data from the Liverpool Telescope (LT) observations. Smith et al (2008, GCN7743) reported limits for non-detection of an optical transient, based on the assumption that it was not coincident with the known source. LT observations commenced at 21:34:05 UT, 674s after the burst trigger. The data comprised imaging of the field in SDSS-$r'$, $i'$ and $z'$ bands, with 120s individual exposures, the last of which ended at T0+2290s. We confirm that there is no evidence for an early-time excess due to afterglow flux in this source, either in subtractions against our late-time WHT imaging or in pairwise subtraction of early-time exposures. These data provide a relatively weak constraint, but at an early time, enabling us to limit the flux at T0+900s to $F_\lambda<2\times10^{-16}$ergss$^{-1}$Å at 7500Å. Comparing to the [*Swift*]{}-detected X-ray flux at the same epoch, we determine an identical limit to that from the [*Swift*]{} optical data alone – $\beta_{OX}<1.0$. These limits are formally too weak to satisfy the $\beta_{\rm{OX}}$ criterion that is applied to select dark bursts [@2004ApJ...617L..21J; @2009ApJ...699.1087V]. Thus the non-detection of the afterglow is consistent with either ‘dark’ or ‘normal’ interpretation. However, we note that this burst shows the high column and red, dusty host more common amongst the dark population [e.g. @2014arXiv1402.4006H; @2013ApJ...778..128P]. Additional $V$-band time series data for this target exists in the second data release of the Catalina Real-time Transient Survey [CRTS, @2009ApJ...696..870D]. Both the GRB host and neighbour are well detected. Unfortunately the burst itself occured during a hiatus in CRTS observing, with no data available until 140 days after the GRB trigger (likely due to Sun avoidance). However, as figure \[fig:catalina\] demonstrates, the time series data of the GRB host shows no strong evidence for variability (although a few photometric points are outliers), and allows a precise measurement of the optical magnitude of the host galaxy, $V_{AB}=17.60\pm0.08$. We also investigate the late time optical afterglow, and find no statistically significant evidence for an excess over the host galaxy flux at T+160 days. ![Optical time-series data from the Catalina Real-time Transient Survey for the host galaxy of GRB 0801517. There is no evidence for strong variability in the host galaxy, allowing the time series data to be combined to determine a precise magnitude for the object. In the lower left panel we consider the data closest to the burst data (MJD=54603) in detail, averaging the data points on each night on which the host was observed. While the first three data points lie above the mean measured magnitude, they are within one standard deviation of the time-series mean (shown with shaded region).\[fig:catalina\]](plot_catalina2.ps){width="0.99\columnwidth"} The Spectral Energy Distribution {#sec:sed} ================================ Our WHT optical imaging in the SDSS $g$, $r$, and $i$ bands (described in section \[sec:whtim\]) is supplemented by extensive archival data on these relatively bright sources. In addition to the $V$-band data from the Catalina Real-time Transient Survey described in section \[sec:afterglow\], we compile archival data in the ultraviolet from the [*GALEX*]{} [GR6, @2003SPIE.4854..336M] survey and in the near-infrared from the Two Micron All-Sky Survey [2MASS, @2006AJ....131.1163S] as well as the [*Wide-Field Infrared Survey Explorer*]{} [WISE, @2010AJ....140.1868W]. Both the GRB host and its merging neighbour are detected in the majority of bands from the near-ultraviolet (NUV, 0.15$\mu$m) through to the mid-infrared (W4, 22$\mu$m). While flux from the neighbour is clearly dominated by two principle components in our WHT imaging, these are blended in the remaining data, and we do not attempt to gauge the relative contribution of the two components. Photometry was measured on the images at the source locations, and checked against catalog magnitudes where these were available. For the W3 and W4 bands (at 12 and 22$\mu$m) where the host and its neighbour are blended in the imaging data, we use magnitudes derived from the ‘ALLWISE’ catalogue values (corrected to AB magnitudes) rather than attempting an independent deblending of the two sources. As noted in the introduction, it is the exceptional brightness of this target in the W3 and W4 bands that initially motivated the follow-up observations described here. While the sources are blended in the W4 band, the light distribution of the host galaxy is distorted, such that it is likely that both the GRB host galaxy and its near neighbour are luminous at 22$\mu$m. The multi-wavelength photometry of both the GRB host and its neighbour are given in table \[tab:phot\] and figure \[fig:allbands\] presents snapshots of the host and neighbour in imaging across the full wavelength range. At $z=0.09$, the $g$-band absolute magnitude is M$_g=-20.12\pm0.05$ (comparable to that of the Milky Way). Band $\lambda_\mathrm{cen}$ / Å Source GRB Host Neighbour ------- ---------------------------- ----------- ------------------ ------------------ $FUV$ 1540 GALEX 20.84 $\pm$ 0.20 20.92 $\pm$ 0.22 $NUV$ 2316 GALEX 20.42 $\pm$ 0.11 20.74 $\pm$ 0.16 $g $ 4660 This work 18.03 $\pm$ 0.05 18.42 $\pm$ 0.06 $V $ 5500 CRTS 17.60 $\pm$ 0.09 18.14 $\pm$ 0.12 $r $ 6140 This work 17.73 $\pm$ 0.02 18.18 $\pm$ 0.05 $i $ 7565 This work 17.46 $\pm$ 0.01 18.22 $\pm$ 0.02 $J $ 12400 2MASS 17.37 $\pm$ 0.13 17.91 $\pm$ 0.21 $H $ 16600 2MASS 17.22 $\pm$ 0.15 19.56 $\pm$ 0.95 $Ks$ 21600 2MASS 17.50 $\pm$ 0.20 17.91 $\pm$ 0.26 $W1$ 33500 WISE 17.33 $\pm$ 0.04 18.49 $\pm$ 0.05 $W2$ 46000 WISE 17.57 $\pm$ 0.05 19.05 $\pm$ 0.12 $W3$ 120000 WISE 15.09 $\pm$ 0.05 17.12 $\pm$ 0.32 $W4$ 220000 WISE 13.68 $\pm$ 0.11 14.78 $\pm$ 0.23 Radio 6cm WSRT 0.22$\pm$0.04mJy : Observed photometry measured from broadband observations of the GRB host and its neighbouring galaxy. All magnitudes are given in the AB system. Note that the neighbouring galaxy is undetected in the 2MASS $H$-band and barely detected in $Ks$. Radio fluxes are described in section \[sec:radio\]. In the 12 and 22$\mu$m bands we make use of WISE catalog data rather than attempting to deblend the two sources independently.\[tab:phot\] {width="1.5\columnwidth"} We fit the spectral energy distribution (SED) of the host galaxy using a template fitting approach, minimising the $\chi^2$ parameter to determine the best fit age, mass, star formation history and dust extinction. While we could constrain the dust parameter using the extinction derived from the hydrogen Balmer lines, we allow it to vary, recognising that regions contributing to the continuum flux at long wavelengths and those contibuting nebular emission flux in the optical may well differ in their extinction properties. We use as templates the Binary Population and Spectral Synthesis ([BPASS]{}) stellar population models of @2012MNRAS.419..479E [@2009MNRAS.400.1019E] which include a prescription for the nebular emission excited by the stellar continuum. The [BPASS]{} models consider the instantaneous-burst and constant star formation rate cases. In both cases, we modify the templates using the @2000ApJ...533..682C dust extinction law. This was derived for local infrared-luminous galaxies with active star formation, and would appear appropriate in this case, given the bright infrared fluxes measured. The photometry shows a challenging combination of a very flat optical spectrum (which implies little extinction and potentially even a non-stellar continuum) and evidence for strong dust emission in the infrared (see figure \[fig:bpass\]). The [BPASS]{} population synthesis models include a treatment of stellar evolution pathways through binary evolution. For young stellar populations, this treatment results in a relatively blue UV-optical continuum at a given age (and hence larger energy budget for heating dust) compared to models which neglect such pathways. To model the re-emission of thermal photons at longer wavelengths, we adopt the energy-balance prescription of @2008MNRAS.388.1595D, and re-emit the energy lost from the UV-optical as a combination of black body and PAH emission components. For the latter we scale the composite mid-infrared spectrum determined for star forming galaxies by @2007ApJ...656..770S. The W3 and W4 bands were excluded from the fitting procedure, in order to assess whether the observed UV-optical continuum were able to correctly predict the mid-infrared flux, or whether an additional, heavily dust-extincted component was required. The best fitting single SED model to the host photometry is not one dominated by nebular emission or continuous star formation, but rather a post-starburst template, observed 500Myr after the initial starburst, as shown in figure \[fig:bpass\]. The derived stellar mass is log$_{10}$(M$_\ast$/M$_\odot$)=9.58$^{+0.12}_{-0.16}$, and a relatively low extinction of A$_V$=0.16$\pm$0.02 is required for the dominant stellar component. This model reproduces the GRB host galaxy’s ultraviolet and infrared continua, while underestimating its optical flux by $\sim25-50$% .We note that fitting instead with the [@2005MNRAS.362..799M] stellar population synthesis models returns similar parameters, albeit with a relativity poor fit to the data. The mass, low extinction and age of the dominant stellar component imply that, for $z=0.09$, the GRB host represents a relatively young, slightly sub-L$^\ast$ galaxy. As section \[sec:spec\] demonstrated, there is a substantial contribution to the optical from line emission, which may account for some part of this discrepency. In the $r$-band in particular, line emission likely contributes a minimum of 12% of the broadband flux. To address this, we also allow an additional component of continuous ongoing star formation with moderate dust extinction (as seen in the Balmer series), while cautioning that this may be overfitting the limited data. We find that the star forming component required for the best fit combination contributes a mere 0.1% of the stellar mass of the system. Including such a component improves the fit in the optical and at 12 and 22$\mu$m, but causes the flux in the $K_S$ band and 4.6$\mu$m W2 band to be somewhat overestimated (again, by a factor of $\sim$50%). Since the latter lies at the transition between the stellar and dust components of the template, it is possible that this transition is not correctly addressed in the modelling, and potentially that a steeper spectral index is required in the infrared PAH emission region than is seen in the IR-luminous galaxy composite used [@2007ApJ...656..770S]. Explanations for the comparatively low flux measured in the 2MASS $K_S$ band, and the non-detection of the neighbour in the $H$-band (see figure \[fig:allbands\]), are less clear. It is likely that deeper near-infrared photometry is required to address this issue. We note that the stellar population templates fail to entirely reproduce the high fluxes seen in the 22$\mu$m W4 band, implying that at least some fraction of the stellar emission in this system is heavily extincted and undetectable in the UV-optical. Further observations at millimetre/submillimetre wavelengths will also be required to properly constrain this emission region. ![The best fitting model templates to the host galaxy photometry. The pale cyan line indicates the best fitting single template: a mature system observed 0.5Gyr after an initial starburst. In red we show the best fit derived when a component of ongoing star formation (at age 1Myr, contributing just 0.1% of the stellar mass) is allowed in addition to the dominant model.\[fig:bpass\]](plot_best_bpass.ps){width="1.05\columnwidth"} Star Formation in the host of GRB080517 {#sec:sfr} ======================================= The host of GRB080517 is an actively star forming galaxy at $z=0.09$. The evidence for ongoing star formation is overwhelming, based on the presence of i) strong H$\alpha$ (and other Balmer) emission lines, ii) GALEX FUV and NUV flux, iii) 22$\mu$m emission and iv) a 4.8GHz radio detection, not to mention the initial selection through detection of a core-collapse gamma-ray burst. In table \[tab:sfrs\] we compare the star formation rates (SFRs) derived from these different proxies. In all cases, the star formation rate conversion used is subject to significant systematic uncertainty, but those at 0-22$\mu$m are derived primarily for young ($<100$Myr) stellar populations with continuous star formation, while that for the radio continuum is based primarily on resolved measurements of nearby star forming galaxies. No attempt is made to correct for dust extinction, which may well be affecting different indicators differently, and so these values are effectively lower limits on the total star formation rate. Unsurprisingly, the near-ultraviolet continuum (which is most affected by the presence of dust extinction) gives the lowest estimate of 0.43$\pm$0.07M$_\odot$yr$^{-1}$. In the @2000ApJ...533..682C extinction paradigm, the continuum is subject to 0.44 times the extinction in the nebular component, or E($B-V$)=0.53. This corresponds to a measured 2300Å flux only 2% of its intrinsic value. In fact, the emission in the GALEX band appears to be associated with the mature stellar population (see section \[sec:sed\]) rather than the heavily-extincted star forming component. The star formation rates derived from the optical H$\alpha$ emission line and the 22$\mu$m continuum (measured in the W4 band) are consistent at the 1$\sigma$ level, each estimating rates around 16M$_\odot$yr$^{-1}$. Interestingly, this may suggest that the 1.5arcsecond wide slit used for spectroscopy may have captured the majority of the flux from active star formation in the GRB host galaxy, despite its relatively large ($\sim$2arcsecond full-width at half-maximum) light distribution. Curiously, the final star formation rate indicator, that derived from the radio continuum at 4.5GHz produces a relatively low estimate at 7.6$\pm$1.4M$_\odot$yr$^{-1}$, only about half that determined from the previous two measures, using the conversion rate determined by @2011ApJ...737...67M, and extrapolating to 1.4GHz using a radio spectral slope $\alpha=0.75$. An alternate conversion factor [@2002ApJ...568...88Y] yields a similar but still lower estimate (4M$_\odot$yr$^{-1}$). The synthesised beam of the WSRT at 4.8GHz is insufficient to have resolved out a significant fraction of the flux in this source, and the flux density is measured at better than 5$\sigma$, making it likely that this is a genuine decrement. Gigahertz frequency radio continuum in star forming galaxies arises primarily from non-thermal synchrotron emission, generated by the electrons accelerated by supernovae and their remnants. This introduces a time delay between the onset of star formation and the establishment of a radio continuum, the length of which will depend on the mass distribution, metallicity and other properties of the stellar population. As a result, the radio continuum flux density associated with ongoing star formation rises rapidly with age of the star forming population before stabilising at $>100$Myr . If, then, the young 1Myr starburst suggested by the BPASS models presents a true picture of the ongoing star formation in this source, it is possible that a strong radio continuum has yet to become established and a radio flux - SFR conversion factor up to an order of magnitude higher might be appropriate. Future observations at further radio/submillimeter frequencies, and a measurement of the radio spectral slope, may help to constrain the effect of star formation history on this estimate. Proxy SFR/M$_\odot$yr$^{-1}$ Conversion factor ------------------------- ------------------------ ---------------------- NUV continuum 0.43$\pm$0.07 Hao et al. (2011) H$\alpha$ line emission 15.5$\pm$0.4 Hao et al. (2011) 22$\mu$m continuum 16.5$\pm$1.5 Lee et al (2013) 4.8GHz continuum 7.6$\pm$1.4 Murphy et al. (2011) : Star formation rates derived from different proxies observed for the host of GRB080517. In the radio we apply the conversion factor derived at 1.4GHz, assuming a radio spectral slope of 0.75. \[tab:sfrs\] The Host and Environment of GRB080517 {#sec:disc} ===================================== The host of GRB080517 appears to be a compact, smooth galaxy in the local Universe. Given its UV-optical photometry, there would be little reason to expect significant ongoing star formation. Nonetheless, as the previous section describes, there is substantial evidence for an ongoing, young and fairly dusty (based on H$\alpha$/H$\beta$) starburst, which likely dominates emission at $>10\mu$m. In this context, the presence of a near neighbour (separated by only $\sim$25kpc at $z=0.09$) is intriguing. The two galaxies have comparable masses (the neighbour is just $\sim$0.5 magnitudes fainter than the GRB host and similar in colour), and any interaction between them will constitute a major incident in the history of both galaxies. The gravitational forces caused by a near fly-by in the recent past could well have been sufficient to trigger the starburst detected in both sources - a starburst somewhat obscured by dust. It is also notable that the cores of both galaxies (including both components of the neighbour) and the GRB X-ray error circle lie along a common axis. Further modelling of the dynamics of this system, supported by integral field spectroscopy, and a search for low surface-brightness distortions in the morphology of both galaxies, will be necessary to confirm this picture, but the existing evidence is suggestive. Long-duration GRBs are typically associated with the peak light in their host galaxies, and with recent star formation [e.g @2010MNRAS.tmp..479S]. However, the broadband continuum emission of the host of GRB080517 is dominated by a much older stellar population. If, then, we hypothesise that the GRB is associated with the recent episode of star formation in this system, we are left with the conclusion that it occured in a dusty region (E($B-V$)=1.2) undergoing an intense star burst (SFR$\sim$16M$_\odot$yr$^{-1}$). Such dust extinction is consistent with the excess neutral hydrogen column inferred from the X-ray afterglow, and potentially with the failure of early-time optical observations to identify an optical transient distinguishable from the host galaxy. As we have already discussed, it is impossible to determine whether GRB080517 would indeed have met the ‘dark’ burst criterion if deeper observations were available. It is, however, possible to consider whether its host lies in a similar region of parameter space to known dark bursts. @2013ApJ...778..128P recently presented a systematic analysis of the dark GRB host galaxy population, examining both their afterglow properties and those derived from fitting of the host galaxy spectral energy distribution. As figures \[fig:perley1\] and \[fig:perley2\] demonstrate, the inferred characteristics of the host of GRB080517 lie comfortably within the distribution of ‘dark’ burst hosts in terms of afterglow spectral index, hydrogen column density, stellar mass and inferred star formation rate, although the last is higher than those in dark bursts at $z<0.5$, and more akin to those observed at higher redshifts [@2013ApJ...778..128P]. ![The X-ray afterglow properties of GRB080517 compared to those of ‘dark’ bursts as given by @2013ApJ...778..128P. GRB080517 (bold, red symbol) has an afterglow spectral index, and shows an excess neutral hydrogen column density above that in our own Galaxy, which is consistent with those of the dark GRB population, and follows the same correlation. \[fig:perley1\]](plot_perley1.ps){width="0.99\columnwidth"} ![The host mass of GRB080517, and star formation rate inferred from H$\alpha$ line emission, compared to those of ‘dark’ bursts as given by @2013ApJ...778..128P. GRB080517 (bold, red symbol) has an inferred stellar mass (based on SED fitting) comparable to those of the dark burst population, and follows the same mass-star formation rate trend.\[fig:perley2\]](plot_perley2.ps){width="0.99\columnwidth"} Whether or not GRB080517 is indeed a local example of a ‘dark’ host, one advantage it offers is the opportunity to study its optical spectrum in a detail challenging for higher redshift GRB hosts of either dark or normal types. In figures \[fig:bpt\] and \[fig:metal\] we compare its optical emission line ratios to those of local emission-line galaxies from the SDSS [@2004MNRAS.351.1151B]. The GRB host has line ratios consistent with a Solar or slightly super-Solar metallicity, and is within the range of scatter of the SDSS sample, although well above the relationship between the R$_{23}$ and \[O[ III]{}\]/\[O[II]{}\] at a given metallicity determined by . This is consistent with the results from the less metal-sensitive SED fitting procedure described in section \[sec:sed\], in which 0.5-1.0 Solar metallicity templates were narrowly preferred over those with significantly lower metal enrichment. While far from unique , this places the host of GRB080517 towards the upper end of the metallicity distribution for GRB hosts. Interestingly, at least two other high metallicity bursts, GRB020819 [@2010ApJ...712L..26L] and GRB080607 [@2009ApJ...691L..27P], are dark bursts. The BPT diagram [@1981PASP...93....5B] is an established indicator of starburst versus AGN character, since the different ionisation parameters arising from the two classes have a strong effect on optical emission line ratios and particularly the ratio of \[N[II]{}\] to H$\alpha$. As figure \[fig:bpt\] demonstrates, the two components of the neighbouring source are broadly consistent with a star-formation driven spectrum, although both lie above the local mean in \[O[III]{}\]/H$\beta$ [a trait also often seen in high redshift star forming galaxies, e.g. @2014ApJ...785..153M; @2014arXiv1408.4122S]. While component A has measured line ratios consistent with a ‘composite’ spectrum, large errors on the measured values permit a purely star forming spectrum. The line ratios of the GRB host galaxy are intriguing, placing it too in the region of the parameter space usually described as ‘composite’, suggesting that there might plausibly be a contribution to the ionising spectrum from an AGN. If so, this would be a surprise, since gamma ray bursts have not previously been associated with active galaxies, but may help to explain the excess flux seen in the $22\mu$m band where a hot AGN would be expected to make a contribution to PAH emission. Unfortunately [*Swift*]{} did not track the burst beyond 20 hours after the trigger, at which point the X-ray afterglow was still fading. However the measured flux at this late epoch provides a firm upper limit on possible X-ray flux from the host galaxy of 4.3$^{+2.8}_{-2.0}\times10^{-14}$ergscm$^2$s$^{-1}$ in the [ *Swift*]{} XRT 0.3-10keV band. The luminosity of any hypothetical AGN at $z=0.09$ is therefore constrained to an $L_X<8.6\times10^{41}$ergss$^{-1}$, placing it at the very low end of the AGN luminosity distribution [e.g. @2008ApJ...679..118S; @2010ApJ...716..348B]. As noted in section \[sec:afterglow\], there is also little evidence for any optical variability in the host galaxy, either before or after the gamma ray burst, as might be expected of a galaxy with a strong AGN contibution. While the centre of the host galaxy lies outside the 90% confidence interval on the X-ray location of the GRB (based on the refined XRT analysis), the two locations are consistent at the 2$\sigma$ uncertainty level. It is therefore not impossible that the gamma-ray burst resulted from activity in the galactic nucleus. A rare class of gamma ray flares are known to result from a sudden accretion event due to the tidal disruption of stars around supermassive black holes [@2011Sci...333..203B; @2012ApJ...753...77C Brown et al submitted]. The burst of accretion in such sources launches a relativistic jet, and would result in a short-lived burst of AGN activity from otherwise quiescent galactic nuclei. In this context, the plausible association of GRB080517 with a very low luminosity AGN at $\sim$6 years post burst merits further investigation and monitoring of this system. Further observations will be required to place firmer constraints on the presence or absence of an AGN at late times and any late time variability. We note that if there is no AGN contribution, then the optical emission line ratios in the host imply star formation with a steep ultraviolet spectrum, causing a higher ionisation parameter than is typical at low redshifts, and perhaps strengthening the suggestion that this is a very young, intense starburst (as suggested by the [BPASS]{} stellar population models). ![The emission line strengths of the GRB host galaxy and its neighbour (components A and B) placed on the classic BPT diagram. All three sources lie above the locus of star forming galaxies measured in the SDSS, although the neighbour remains consistent with a star forming origin. The dashed lines indicate the classification criteria of @2003MNRAS.346.1055K. The region between the dashed lines is described as a ‘composite’ region and may indicate contributions from both star formation and AGN activity. The background density plot shows the distribution of galaxies in the SDSS [@2004MNRAS.351.1151B]. Interestingly, the GRB host lies in the ‘composite’ region of the parameter space, suggesting that it may have an AGN component in addition to ongoing star formation. \[fig:bpt\]](bpt.ps){width="0.99\columnwidth"} ![Metallicity-sensitive optical line ratios for the GRB host galaxy. The well known R$_{23}$ index is plotted against the ratio of oxygen lines in order to break the degeneracy in metallicity in the former. The majority of SDSS sources (greyscale) are relatively local and high in metallicity [@2004MNRAS.351.1151B]. The solid line shows the metallicity parameterisation of . The GRB host galaxy lies well above the typical SDSS galaxy in R$_{23}$ but within the distribution of local sources. We note that the effects of correcting for differential dust extinction on the lines is to move it further from the SDSS relation. Its measured line ratios are consistent with a slightly super-Solar metallicity.\[fig:metal\]](metal.ps){width="0.99\columnwidth"} Conclusions {#sec:conc} =========== In this paper we have presented an analysis of new and archival data for the host galaxy of GRB080517. Our main conclusions can be summarised as follows: 1. GRB080517 is a rare, low luminosity, long gamma ray burst. 2. Our WHT spectroscopy reveals that the host galaxy of GRB080517 is a strong optical line emitter lying at $z=0.09$. 3. The morphology of the GRB host appears to be smooth and compact with a half light radius, deconvolved with the seeing, of 2.7kpc. Its light distribution is consistent with a Sersic index of n=$1.5\pm1.0$. 4. The strong optical emission line ratios in the GRB host are consistent with a composite AGN+starburst spectrum at Solar or super-Solar metallicity, and the ratio of Balmer lines suggests the nebular emission is subject to an extinction E($B-V$)=1.2. 5. The spectral energy distribution of the galaxy in the UV-optical is broadly reproduced by a post-starburst template at an age of 500Myr, with a relatively small component of ongoing star formation ($<$1% of the stellar mass). However no template considered provides a good match to all features of the SED, and in particular to the high fluxes measued at $>10\mu$m, suggesting that multiple components with different spectral energy distributions may contribute to the broadband flux. 6. Star formation rate estimates for the GRB host range from 0.43M$_\odot$yr$^{-1}$ to 16.5M$_\odot$yr$^{-1}$, based on different indicators. The low rate estimated from the ultraviolet continuum likely arises due to strong dust extinction in the star forming regions. Estimates from the H$\alpha$ line and $22\mu$m are consistent at 15.5$\pm$0.4 and 16.5$\pm$1.5M$_\odot$yr$^{-1}$ 7. We detect radio emission from the host galaxy with a flux density, $S_{4.8\,GHz}=$0.22$\pm$0.04mJy. This corresponds to a star formation rate of 7.6$\pm$1.4M$_\odot$yr$^{-1}$. 8. The high ionisation parameter seen in the optical line ratios, low radio flux and SED fitting are all consistent with a very young ($<$100Myr) star formation episode. 9. The host galaxy has a close companion within 25kpc in projected distance and lying at the same redshift. The companion shows distorted morphology, including two cores which appear to be undergoing a merger. The proximity of these galaxies may indicate that the GRB progenitor formed in an ongoing starburst triggered by gravitational interaction. 10. While the burst afterglow was too faint to tightly constrain the X-ray to optical flux ratio, its properties and those of its host galaxy are consistent with those of the ‘dark’ GRB population. The host galaxy’s properties and wider environment suggest that the role of galaxy-galaxy interaction in triggering bursts in relatively massive, metal rich galaxies needs to be considered more carefully. We aim to investigate this field further, obtaining stronger X-ray constraints on the presence of AGN activity, high resolution imaging, and also further radio continuum measurements of the host’s dust-obscured star formation. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the anonymous referee of this paper for helpful comments and suggestions. ERS and AJL acknowledge funding from the UK Science and Technology Facilities Council under the Warwick Astrophysics consolidated grant ST/L000733/1. ERS thanks Dr Elmé Breedt for useful discussions and recommending the CRTS. AJvdH acknowledges the support of the European Research Council Advanced Investigator Grant no. 247295 (PI: R.A.M.J. Wijers). CGM acknowledges support from the Royal Society, the Wolfson Foundation and STFC. Optical data were obtained from the William Herschel Telescope. The WHT and its override programme are operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. We also make use of data from the Liverpool Telescope which is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council. Radio data were obtained from WSRT. The WSRT is operated by ASTRON (Netherlands Institute for Radio Astronomy) with support from the Netherlands foundation for Scientific Research. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester [@2009MNRAS.397.1177E]. We also made use of Ned Wright’s very useful cosmology calculator [@2006PASP..118.1711W]. Based in part on public data from GALEX GR6. The Galaxy Evolution Explorer (GALEX) satellite is a NASA mission led by the California Institute of Technology. This publication also makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This publication further makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Data is also derived from the Catalina Real-time Transient Survey. The CSS survey is funded by the National Aeronautics and Space Administration under Grant No. NNG05GF22G issued through the Science Mission Directorate Near-Earth Objects Observations Program. The CRTS survey is supported by the U.S. National Science Foundation under grants AST-0909182 and AST-1313422. We make use of SDSS-III data. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. 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ArXiv

--- abstract: | The spectrum of $^9$He was studied by means of the $^8$He($d$,$p$)$^9$He reaction at a lab energy of 25 MeV/n and small center of mass (c.m.) angles. Energy and angular correlations were obtained for the $^9$He decay products by complete kinematical reconstruction. The data do not show narrow states at $\sim $1.3 and $\sim $2.4 MeV reported before for $^9$He. The lowest resonant state of $^9$He is found at about 2 MeV with a width of $\sim $2 MeV and is identified as $1/2^-$. The observed angular correlation pattern is uniquely explained by the interference of the $1/2^-$ resonance with a virtual state $1/2^+$ (limit on the scattering length is obtained as $a > -20$ fm), and with the $5/2^+$ resonance at energy $\geq 4.2$ MeV. author: - 'M.S. Golovkov' - 'L.V. Grigorenko' - 'A.S. Fomichev' - 'A.V. Gorshkov' - 'V.A. Gorshkov' - 'S.A. Krupko' - 'Yu.Ts. Oganessian' - 'A.M. Rodin' - 'S.I. Sidorchuk' - 'R.S. Slepnev' - 'S.V. Stepantsov' - 'G.M. Ter-Akopian' - 'R. Wolski' - 'A.A. Korsheninnikov' - 'E.Yu. Nikolskii' - 'V.A. Kuzmin' - 'B.G. Novatskii' - 'D.N. Stepanov' - 'S. Fortier' - 'P. Roussel-Chomaz' - 'W. Mittig' date: '. [File: he9-11.tex ]{}' title: 'New insight into the low-energy $^9$He spectrum' --- *Introduction.* — Since the first observation of $^9$He in the experiment [@set87] it was studied in relatively small number of works compared to the neighbouring exotic neutron dripline nuclei. This can be connected, on one hand, to the facts that technical difficulty of the precision measurements rapidly grows with move away from the stability line. On the other hand, already in the first experiment (pion double charge exchange on the $^9$Be nucleus [@set87]) several narrow resonances were observed above the $^8$He+$n$ threshold. This observation was confirmed in Ref. [@boh99], where the $^9$Be($^{14}$C,$^{14}$O)$^9$He reaction was used, and now the experimental situation with the low-energy spectrum of $^9$He is considered to be well established. A new rise of interest to $^9$He was connected with the question of a possible $2s$ state location in the framework of the shell inversion problem in nuclei with large neutron excess. The recent experiment [@che01] was focused on the search for the virtual state in $^9$He. An upper limit on the scattering length $a<-10$ fm was established in this work. The properties of states in $^{9}$He were inferred in [@gol03] basing on studies of isobaric partners in $^{9}$Li. The available results are summarized in Table \[tab:exp\]. Interpretation of the $^9$He spectrum as provided in [@set87; @boh99] faces certain difficulties which were not unnoticed (e.g. Ref. [@bar04]). Indeed, the ground $1/2^-$ state is expected to be single particle state with width estimated as $0.8-1.3$ MeV at $E = 1.27$ MeV for typical channel radii $3 - 6$ fm. This requires spectroscopic factor $S\sim 0.1$ which contradicts single particle character of the state. F. Barker in Ref. [@bar04] concludes on this point that “some configuration mixing in either the $^{9}$He($1/2^-$) or $^{8}$He($0^+$) state or both is possible, but is unlikely to be large enough to reduce the calculated width to the experimental value”. The next, presumably $3/2^-$ state, should be a complicated particle-hole excitation as $p_{3/2}$ subshell is occupied. However, a much larger spectroscopic factor $S\sim 0.3-0.4$ is required for its widths found in a range $2.0-2.6$ MeV. ![Experimental setup, angles, and momenta.[]{data-label="fig:setup"}](setup){width="44.00000%"} Having in mind the mentioned problematic issues we decided to study the $^9$He in the “classical” one-neutron transfer ($d$,$p$) reaction well populating single particle states. In contrast with the previous works complete kinematics studies were foreseen to reveal the low-energy $s$-wave mode. Following the experimental concept of [@gol04a; @gol05b], where correlation studies of $^5$H continuum were accomplished by means of the $^3$H($t$,$p$) transfer reaction, this work was performed in the so called “zero geometry”. ---------- ------------ ---------- ---------- ---------- ---------- ------------ ---------- 1/2$^+$ Ref. $a$ (fm) $E$ $\Gamma$ $E$ $\Gamma$ $E$ $\Gamma$ [@set87] 1.13(10) small 2.3 small 4.9 [@boh99] 1.27(10) 0.1(0.6) 2.42(10) 0.7(2) 4.3 small [@che01] $< \! -10$ [@gol03] 1.1 2.2 4.0 Our $> \! -20$ 2.0(0.2) 2 $\geq 4.2$ $> 1$ ---------- ------------ ---------- ---------- ---------- ---------- ------------ ---------- : Experimental positions of states in $^9$He relative to the $^8$He+$n$ threshold (energies and widths are given in MeV). \[tab:exp\] {width="86.00000%"} *Experiment.* — The experiment was done at the U-400M cyclotron of the Flerov Laboratory of Nuclear Reactions, JINR (Dubna, Russia). A 34 MeV/nucleon $^{11}$B primary beam delivered by the cyclotron hitted a 370 mg/cm$^2$ Be production target. The modified ACCULINNA fragment separator [@rod97] was used to produce a $^8$He secondary beam with a typical intensity of $2\times 10^4$ s$^{-1}$. The beam was focused on a cryogenic target [@yuk03] filled with deuterium at 1020 mPa pressure and cooled down to 25 K. The 4 mm thick target cell was supplied with 6 $\mu$m stainless steel windows, 30 mm in diameter. Experimental setup and kinematical diagram for the $^{2}$H($^8$He,$p$)$^9$He reaction are shown in Fig.\[fig:setup\]. Slow protons escaping from the target in the backward direction hitted on an annular 300 $\mu$m silicon detector with an active area of the outer and inner diameters of 82 mm and 32 mm, respectively, and a 28 mm central hole. The detector was installed 100 mm upstream of the target. It was segmented in 16 rings on one side and 16 sectors on the other side providing a good position resolution. The detection threshold for the protons ($\sim $1.2 MeV) corresponded to a $\sim$5.5 MeV cutoff in the missing mass of $^9$He. We did not use here particle identification because, due to the kinematical constraints of the $^8$He+$^2$H collisions, only protons can be emitted in the backward direction. The main cause of the background was due to evaporation protons originating from the interaction of $^8$He beam with the material of target windows. This background was almost completely suppressed by the coincidence with $^8$He. The detection of such coincidences fixed the complete kinematics for the experiment. Energy-momentum conservation was used for cleaning the spectra. Finally the comparison with an empty target run has shown that only $\sim 2 \%$ events can be treated as a background. The $^8$He nuclei resulting from the $^9$He decay, focused in narrow angular cone relative to the beam direction, were detected by a Si-CsI telescope mounted in air just behind the exit window of the scattering chamber. The 82 mm diameter exit window was closed by a 125 $\mu$m capton foil. The Si-CsI telescope consisted of two 1 mm thick silicon detectors and 16 CsI crystals with photodiode readouts. The $6 \times 6$ cm Si detectors were segmented in 32 strips both in horizontal and vertical directions, providing position resolution and particle identification by the $\Delta E$-$E$ method (together with following thick CsI detector). Sixteen $1.5 \times 2 \times 2$ cm CsI crystals were arranged as a $4 \times 4$ wall just behind the Si detectors. A $ 2 \%$ energy resolution of the CsI detectors allowed particle identification even for $Z=1$ nuclei. The distance between the target and the telescope (50 cm) was sufficient to provide a good efficiency for the detection of the $^8$He nuclei in coincidence with protons in the whole range of accessible $^9$He energies. To eliminate signals in the proton telescope coming from the beam halo a veto detector was installed upstream the proton telescope. The energy of the $^8$He beam in the middle of the target was $\sim $25 MeV/nucleon. Energy spread of the beam, angular divergence, and position spread on the target were $8.5 \%$, $0.23^{\circ}$, and about 5.4 mm respectively. A set of beam detectors was installed upstream of the veto (not shown in the Fig. \[fig:setup\]). The beam energy was measured by two time-of-flight plastic scintillators with a 785 cm base. The overall time resolution was 0.8 ns. Beam tracking was made by two multiwire proportional chambers installed 26 and 80 cm upstream of the target. Each chamber had two perpendicular planes of wires with a 1.25 mm pitch. Energy resolution was estimated by Monte-Carlo (MC) code taking into account all experimental details. It was found to be 0.8 MeV (FWHM) for the $^9$He missing mass. {width="90.00000%"}\ {width="85.00000%"}\ {width="85.00000%"}\ {width="85.00000%"} *Qualitative considerations.* — Data obtained in the experiment are shown in Figs. \[fig:expth\] and \[fig:distrib\]. The total number of counts corresponds to cross section of $\sim$7 mb/sr. This value is consistent with the direct one-neutron transfer reaction mechanism at forward angles. The narrow states known from the literature do not show up in the data. Instead, we get two broad peaks at about 2 and 4.5 MeV. Near the threshold the $^9$He spectrum exhibits behavior (although distorted by the finite energy resolution of the experiment) which is more consistent with $s$-wave ($\sigma \sim \sqrt{E}$) rather than $p$-wave ($\sigma \sim E^{3/2}$). This is an indication for a possible virtual state in $^9$He. An important feature of the data is a prominent forward-backward asymmetry with $^8$He flying preferably in the backward direction in the $^{9}$He c.m. system. This is not feasible if the narrow (means long-living) states of the $^9$He are formed. To describe such an asymmetry the interference of opposite parity states is unavoidable. As far as asymmetry is observed even at very low energy, the $s$-$p$ interference is compulsory. Such an interference can provide only a very smooth distribution described by the first order polynomial \[Eq. (\[eq:sigma-full\]) and Fig. \[fig:distrib\], $E<2.2$ MeV\]. Since above 3 MeV the character of the distribution changes to a higher polynomial, but asymmetry does not disappear, the $p$-$d$ interference is also needed. This defines the minimal set of states as $s$, $p$, and $d$. *Theoretical model.* — In the zero geometry approach the resonant states of interest are identified by the observation of recoil particle (here proton) at zero (in reality small, $3^{\circ} \leq \theta_{p}(\text{c.m.})\leq 7^{\circ}$) angle. This means that the angular momentum transferred to the studied system should have zero projection on the axis of the momentum transfer. As a result we get a complete (strong) alignment for states with $J>1/2$ in the produced system. In the $^9$He case only the magnetic substates with $M= \pm 1/2$ should be populated for $J^{\pi}=5/2^+$, $3/2^-$, and $3/2^+$ states. This strongly reduces possible ambiguity in the analysis of correlation patterns. E.g. in the case of zero spin particles the zero geometry experiments give very clean pictures with angular distributions described by pure Legendre polynomial $|P^0_l|^2$. The situation in the case of nonzero spin particles involved is more complicated (see detailed discussion in Ref. [@gol05b]), and diverse correlation patterns are possible. We have found that the observed experimental picture (Figs. \[fig:expth\], \[fig:distrib\]) can be well explained in a simple model involving only three low-lying states: $1/2^+$, $1/2^-$, and $5/2^+$. The inelastic cross in the DWBA ansatz is written as $$\begin{aligned} \frac{d\sigma(\Omega_{^9\text{He}})}{dE \,d \Omega_{^8\text{He}}} \sim \frac{v_{f}}{v_{i}}\,\sqrt{E} \, \sum \nolimits_{MM_S} \left| \sum \nolimits_J\langle \Psi^{JMM_S}_f \left| V \right| \Psi_i \rangle \right|^2 \label{eq:sigma}\\ % = \frac{v_f}{v_i} \sum _{MM_S} \sum _{JJ'} \sum _{M'_lM_l}\! \rho_{JM}^{J'M} C^{J'M'}_{l'M'_lSM_S} C^{JM}_{lM_lSM_S} \, Y^{\ast}_{lM'_l} Y_{lM_l}\,. \nonumber\end{aligned}$$ For the density matrix the generic symmetries are $$\rho_{JM}^{J'M'}= \left(\rho^{JM}_{J'M'}\right)^* \quad ; \qquad \rho_{JM}^{JM'}= (-)^{M+M'}\rho^{J-M}_{J-M'} \; ,$$ and properties specific to coordinate choice (spirality representation) and setup (zero geometry) are $$\rho_{JM}^{J'M'} \sim \; \delta_{M,M'}(\delta_{M,1/2}+\delta_{M,-1/2}) \; .$$ For the density matrix parametrization we use the following model for the transition matrix. The wave function (WF) $\Psi_f$ is calculated in the $l$-dependent square well (with depth parameters $V_l$). The well radius is taken $r_0=3$ fm, which is consistent with typical R-matrix phenomenology $1.4 A^{1/3}$. The energy dependence of the velocities $v_i$, $v_f$ (in the incoming $^8$He-$d$ and outgoing $^9$He-$p$ channels) and WF $\Psi_i$ is neglected for our range of $^9$He energies. The term $ V \left| \Psi_i \right. \rangle $, describing the reaction mechanism, is approximated by radial $\theta$-function: $$V \left| \Psi_i \right. \rangle \rightarrow C_l \;r^{-1}\,\theta(r_0-r) \; [Y_{l}(\hat{r}) \otimes \chi_{S}]_{JM} \; ,$$ where $C_l$ is (complex) coefficient defined by the reaction mechanism. For $\bigl|\rho_{J\pm 1/2}^{J'\pm 1/2}\bigr|$ denoted as $A_{l'l}$ the cross section as a function of energy $E$ and $x=\cos(\theta_{^8\text{He}})$ is $$\begin{aligned} \frac{d\sigma(\Omega_{^9\text{He}})}{dE \;d x} \sim \frac{1}{\sqrt{E}} \left[ \rule{0pt}{12pt} 4 A_{00} + 4 A_{11} + 3(1-2x^2+5x^4) A_{22} \right. \nonumber \\ % + \left. 8 \, x \cos(\phi_{10})A_{10} + 4 \sqrt{3} \, x(5x^2 - 3) \cos(\phi_{12}) A_{12} \right] \, . \, \label{eq:sigma-full} \\ % A_{l'l}=|A_{l'}| |A_{l}|\; , \;\;A_l= C_l N_l(E)\,e^{i\delta_l(E)}\int_0^{r_0}dr j_l(q_lr)\; , \nonumber \\ % q_l=\sqrt{2M(E-V_l)}\; , \;\; \phi_{l'l}(E) = \phi^{(0)}_{l'l} + \delta_{l'}(E) - \delta_{l}(E)\, , \nonumber %\end{aligned}$$ where $N_l$ is defined by matching condition on the well boundary for internal function $j_l(q_lr)$. The three coefficients $C_l$ give rise to the two phases $\phi^{(0)}_{10}$ and $\phi^{(0)}_{12}$. Positions and widths of the states are fixed by the three parameters $V_l$. Their relative contributions to the inclusive energy spectrum (Fig. \[fig:distrib\]a–c) are fixed by the three parameters $|C_l|$. The phase $\phi^{(0)}_{10}$ is fixed by the angular distributions at low energy, where the contribution of the $d$-wave resonance is small. After that the phase $\phi^{(0)}_{12}$ was varied to fit the angular distributions at higher energies (Fig. \[fig:distrib\], $E>2.2$ MeV). So, the model does not have redundant parameters and the ambiguity of the theoretical interpretation is defined by the quality of the data. Set $|C_0|^2$ $|C_1|^2$ $V_0$ $V_1$ $V_2$ $\phi^{(0)}_{10}$ $\phi^{(0)}_{12}$ ----- ----------- ----------- ---------- --------- --------- ------------------- ------------------- 1 0.26 0.35 $-4.0$ $-20.7$ $-43.4$ $0.80 \pi$ $-0.03 \pi$ 2 0.03 0.52 $-4.0$ $-20.7$ $-43.4$ $0.85 \pi$ $-0.02 \pi$ 3 0.12 0.43 $-5.817$ $-20.7$ $-43.4$ $1.00 \pi$ $-0.03 \pi$ : Parameters of theoretical model used in the work. $V_i$ values are in MeV, weight coefficients for different states are normalized to unity $\sum |C_i|^2=1$. \[tab:th\] We have found that the weight and interaction strength for the $1/2^+$ state can be varied in a relatively broad range, still providing a good description of data. The results of MC simulations of the experiment with different $s$-wave contributions are shown in Fig. \[fig:distrib\]a–c. The parameter sets of the model are given in Table \[tab:th\]; sets 1 and 2 correspond to small scattering length ($a=-4$ fm) and different weights of $s$-wave (largest and lowest possible), set 3 has $a=-25$ fm and largest possible weight of $s$-wave. It can be seen that the agreement with the data deteriorates when the population of the $s$-wave continuum falls, say, below $15-25 \%$ of the $p$-wave. On the other hand the large negative scattering length has a drastic effect below 0.5 MeV. The energy resolution and the quality of the measured angular distributions are not sufficient to draw solid conclusions about the exact properties of the $s$-wave contribution. The situations with the large contribution of the $s$-wave cross section but with moderate scattering length (say $a > -20$ fm) seem to be more plausible. Measurements with better resolution are required to refine the properties of the $1/2^+$ continuum. Position of the $d$-wave resonance is not well defined in our analysis of data due to the efficiency fall in the high-energy side of the spectrum. This can be well seen from the comparison of theoretical inputs and MC results in Fig.\[fig:distrib\]a–c. The lower limit for the resonance energy of 4.2 MeV is in a good agreement with the value 4.0 MeV found in [@gol03]. A broader energy range measured for $^9$He is needed to resolve the $5/2^+$ state completely and to make the angular distribution analysis more restrictive. *Discussion.* — It should be noted that the interference of any other combination of $s$- $p$- $d$-wave states [*can not*]{} lead to the required forward-backward asymmetry in the whole energy range. The correlation terms \[square brackets in Eq. (\[eq:sigma-full\])\] are $$\begin{aligned} % \left[\rule{0pt}{9pt}\ldots \right] &= &2 A_{00} + 2 A_{11} + (1+3x^2) A_{22} + 4 x \cos(\phi_{10}) A_{10} \nonumber \\ % & +& 2 \sqrt{2} (3x^2 - 1) \cos(\phi_{20}) A_{20}\, , \nonumber \\ % \left[ \rule{0pt}{9pt}\ldots \right] &=& 4 A_{00} + 2 (1+3x^2) A_{11} + 3(1-2x^2+5x^4) A_{22} \,, \nonumber \\ % \left[ \rule{0pt}{9pt} \ldots \right] &= &2 A_{00} + (1+3x^2) (A_{11} + A_{22}) + 2 \, x (9x^2-5) \nonumber \\ % & \times & \cos(\phi_{12})A_{12} + 2 \sqrt{2} (3x^2 - 1) \cos(\phi_{20}) A_{20} \, , \nonumber %\end{aligned}$$ for $\{s_{1/2},p_{1/2},d_{3/2} \}$, $\{s_{1/2},p_{3/2},d_{5/2} \}$, and $\{s_{1/2},p_{3/2},$ $d_{3/2} \}$ sets of states respectively. The asymmetric term ($\sim x)$ is present here either for $s$-$p$ interference only or for $p$-$d$ only or for neither. The angular distributions in energy bins (Fig. \[fig:distrib\]) provide a good indication that a [*narrow*]{} $p_{1/2}$ state is not populated in the reaction. Fig. \[fig:qualit\] shows qualitatively what happens with angular distribution if there is a narrow resonance. The phase shift changes across the narrow resonance to a value close to $\pi$ and the character of angular distribution should change drastically within this energy range. No trend of this kind is observed in Fig. \[fig:distrib\]. Phase shift for the [*broad*]{} $1/2^-$ state changes slowly and hardly achieves $\pi /2$ in our calculations. This allows to explain the smooth behaviour of asymmetry up to 3 MeV. ![Schematic illustration of possible behavior of angular distributions (shown by inserts for angles $\theta_{^8\text{He}}$ from $0^{\circ}$ to $180^{\circ}$) due to $s_{1/2}$-$p_{12}$ interference around a narrow resonance for different phases $\phi_{10}^{(0)}$.[]{data-label="fig:qualit"}](qualit){width="45.00000%"} The existing experimental data can be regarded as not contradicting our results. In Refs. [@set87; @boh99] the narrow states were observed with low statistics (15–40 events/state). If the narrow states really exist they should be observable even at such a low counting rates. However, simulations show that if the cross section behavior is smooth, “statistically driven” narrow structures are quite probable in such a situation. A look at the data of Ref. [@gol03] also shows that states (which should have analogues in $^9$He) at 2.2 MeV and 4.0 MeV are absolutely evident in the data. However, the presence of a narrow 1.1 MeV state is more likely not to contradict the data, rather than necessarily follow from these. The idea that only the $1/2^-$ resonance state can be found in the low energy region not only looks natural, but also finds support in the recent theoretical studies. In Ref. [@vol05], which deals with the whole chain of helium isotopes in continuum shell model, the $1/2^-$ state is located at 1.6 MeV above the $^8$He+$n$ threshold and the width is $\sim 0.6$ MeV indicating the dominant single particle component in the WF. The $3/2^-$ state is predicted to be at 6.6 MeV and relatively narrow ($\sim 2.5$ MeV), what is natural for complicated particle-hole excitations. *Conclusions.* — We would like to emphasize the following results of our study. \(i) Our data show two broad overlapping peaks (at 2 and at 4.5 MeV) in the $^9$He spectrum. Statistics obtained in our experiment is about factor of 10 higher than in the previous works [@set87; @boh99]. Our resolution is sufficient to resolve narrow low-lying states. Even if a narrow $p_{1/2}$ is not resolved, the rapid change of phase around resonance energy should produce the change in the forward-backward asymmetry, which is also not seen in the data. \(ii) An essential contribution of the $s$-wave $1/2^+$ state is evident from the data. It is manifested in two ways: (a) Large forward-backward asymmetry at $E \leq 3$ MeV and (b) accumulation of counts around the threshold, which should not take place for typical cross section behavior for higher $l$-values. A limit $a > -20$ fm is obtained for the scattering length of this state. \(iii) The proposed spin assignment $\{s_{1/2},p_{1/2},d_{5/2} \}$ is unique, as no other reasonable set of low-lying states can provide the observed correlation pattern. \(iv) The experimental data are well described in a simple single-particle potential model, involving only basic theoretical assumptions about the reaction mechanism and the low-energy spectrum of $^9$He. This supports the idea that $^8$He (having closed $p_{3/2}$ subshell) presents a “good” core in the $^{9}$He structure. *Acknowledgments.* — This work was supported by the Russian Foundation for Basic Research grants 02-02-16550, 02-02-16174, 05-02-16404, and 05-02-17535 by the INTAS grantS 03-51-4496 and 03-54-6545. LVG acknowledge the financial support from the Royal Swedish Academy of Science and Russian Ministry of Industry and Science grant NS-1885.2003.2. [99]{} K. Seth, *et al.*, Phys. Rev. Lett.  **58**, 1930 (1987). H.G. Bohlen, *et al.*, Prog. Part. Nucl. Phys. **42**, 17 (1999). L. Chen, *et al.*, Phys. Lett.  **B505**, 21 (2001). G.V. Rogachev, *et al.*, Phys. Rev. C **67**, 041603R (2003). F.C. Barker, Nucl. Phys. **A741**, 42 (2004). M. Golovkov, *et al.*, Phys. Rev. Lett. **93**, 262501 (2004). M. Golovkov, *et al.*, Phys. Rev. C **72**, 064612 (2005). A.M. Rodin, *et al.*, Nucl. Instr. Meth. **A** 391, 228 (1997). A.A. Yukhimchuk, *et al.*, Nucl. Instrum. Meth.  **A** 513, 439 (2003). A. Volya and V. Zelevinsky, Phys. Rev. Lett. **94**, 052501 (2005).

ArXiv

--- abstract: | Identification schemes are interactive protocols typically involving two parties, a *prover*, who wants to provide evidence of his or her identity and a *verifier*, who checks the provided evidence and decide whether it comes or not from the intended prover. In this paper, we comment on a recent proposal for quantum identity authentication from Zawadzki [@Zawadzki19], and give a concrete attack upholding theoretical impossibility results from Lo [@Lo97] and Buhrman et al. [@Buhrman12]. More precisely, we show that using a simple strategy an adversary may indeed obtain non-negligible information on the shared identification secret. While the security of a quantum identity authentication scheme is not formally defined in [@Zawadzki19], it is clear that such a definition should somehow imply that an external entity may gain no information on the shared identification scheme (even if he actively participates injecting messages in a protocol execution, which is not assumed in our attack strategy). author: - 'Carlos E. González-Guillén[^1]' - 'María Isabel González Vasco[^2]' - 'Floyd Johnson[^3]' - 'Ángel L. Pérez del Pozo[^4]' bibliography: - 'QIA.bib' title: Concerning Quantum Identity Authentication Without Entanglement --- Introduction ============ One of the major goals of cryptography is authentication in different flavours, namely, providing guarantees that certain interaction is actually involving certain parties from a designated presumed set of users. In the two party scenario, cryptographic constructions towards this goal are called *identity authentication schemes*, and have been extensively studied in classical cryptography. The advent of quantum computers spells the possible end for many of these protocols however. Since Wiesner proposed using quantum mechanics in cryptography in the 1970’s multiple directions using this concept have undergone serious research. One major role quantum mechanics has played in cryptography is the development of quantum key distribution (QKD) where two parties can securely share a one time pad using quantum mechanics, for example the seminal protocol BB84 [@BB84]. One drawback most of these protocols share is the need for authentication, which is traditionally done over an authenticated classical channel. Classically, there are different ways of defining so-called *identification schemes*, for mutual authentication of peers, mainly depending on whether the involved parties share some secret information (such as a password) or should rely on different (often certified) keys provided by a trusted third party. In the quantum scenario, different identification protocols have been introduced following the first approach, e.g., assuming that two parties may obtain authentication evidence from the common knowledge of a shared secret. These kind of constructions, often called *quantum identity authentication schemes* (or just *quantum identification schemes*), are thus closely related to protocols for *quantum equality tests* and *quantum private comparison*. All these constructions are concrete examples of two-party computations with asymmetric output, i.e. allowing only one of the two parties involved to learn the result of a computation on two inputs. Without posing restrictions on an adversary it was shown by Lo in [@Lo97] and and Buhrman et al. in [@Buhrman12] that these constructions are impossible, even in a quantum setting. As a consequence, constructions for generic unrestricted adversaries in the quantum setting are doomed to failure. All in all, the necessity for authentication in QKD has led to many authors considering approaches which are strictly quantum in nature, such as those in [@Penghao16; @Zeng00; @Huang11] which are based off entanglement or more recently [@Zawadzki19; @Hong17] which do not rely on entanglement. These are known as *quantum identity authentication* (QIA) protocols. For protocols such as BB84 that do not rely on entanglement it would be more appealing to not rely on entanglement for entity authentication purposes. [*Our Contribution.*]{} Recently, an original work about authentication without entanglement by Hong et. al. in [@Hong17] was improved by Zawadzki using tools from classical cryptography in [@Zawadzki19]. We start this contribution by summarizing in section \[sec:impossibility\] the impossibility results from Lo [@Lo97] and Buhrman et al. [@Buhrman12], concerning generic quantum two party protocols. Further, we present and discuss the Zawadzki protocol in section \[sec:zawadzki\_protocol\] and show how it succumbs under a simple attack, which we outline in section \[sec:attack\]. Our attack evidences the practical implications of the proven impossibility of identification schemes as conceived in Zawadki’s design, and thus we stress that fundamental changes in the original proposal, beyond preventing our attack, would be needed in order to derive a secure identification scheme. Quantum Equality Tests are Impossible {#sec:impossibility} ===================================== A *one sided equality test* is a cryptographic protocol in which one party, Alice, convinces another, Bob, that they share a common key by revealing nothing to either party except equality (or inequality) to Bob. Formally we define a key space $K$ and a function $F:K^2\to \{0,1\}$ which checks for equality. Let $i\in K$ be Alice’s key and $j\in K$ be Bob’s key. The goals of a one sided equality test are as follows: 1\) $F(i,j)=1$ if and only if $i=j$. 2\) Alice learns nothing about $j$ nor about $F(i,j)$. 3\) Bob learns $F(i,j)$ with certainty. If $F(i,j)=0$ then Bob learns nothing about $i$ except $i\neq j$. The above is a specific case of a one-sided two-party secure computation protocol as described in [@Lo97]. In this work, a very general result is proven indicating that any protocol realising a one-sided two party secure computation task is impossible, even in a quantum setting. In particular, Lo shows in [@Lo97] that if a protocol satisfies 1) and 2) then Bob can know the output of $F(i,j)$ for any $j$. Furthermore, the one sided equality test with some small relaxations on points 1) and 3) is also proven impossible. Hence, any one-sided QIA protocol which validates identities using equality tests by use of quantum mechanics is impossible without imposing restrictions on an adversary. Note that the above argument says nothing about protocols with built in adversarial assumptions such as those presented in [@Damgard14; @Bouman13]. Further, note that many of the QIA schemes end up with a round where Bob accepts or rejects, which makes Alice aware of the success or failure of the protocol. Indeed, those schemes can be straightforwardly turned into one-sided equality tests by suppressing Bob’s final message announcing the result. Hence, they are clearly insecure against a dishonest Bob. However, note that if any such protocol can be modified so that Alice may obtain information on the identification output at some point before the last protocol round, it is unclear how Lo’s impossibility result would apply. However, if they are built upon equality tests we can get impossibility from another well know result by Buhrman el al.[@Buhrman12]. Certainly, two-sided QIA schemes, in which both Alice and Bob learn the result of the protocol, are a particular case of two-sided two-party computations. It is shown in [@Buhrman12] that a correct quantum protocol for a classical two-sided two-party computation that is secure against one of the parties is completely insecure against the other. For equality tests, if one of the parties, say Alice, learns nothing else than $F(i,j)$, the other party, Bob, will indeed be able to compute $F(i,j)$ for all possible inputs $j$. Thus, any two-sided QIA protocol which validates identities using equality tests is also impossible without imposing further restrictions on the adversary. QIA without Entanglement {#sec:zawadzki_protocol} ======================== Here we will outline the protocol proposed in [@Zawadzki19] with some minor modifications, discussed afterword. Suppose Alice and Bob have keys $k_a$ and $k_b$ respectively. Bob wishes to verify that $k_b=k_a$ without leaking any information about $k_b$ or $k_a$. Bob randomly generates a nonce $r$ from a designated domain and generates a universal hash function $H:\{0,1\}^N\to \{0,1\}^{2d}$. This hash function may be chosen by Bob or sampled at random, in the below description we sample from a space of universal hash functions with image $\{0,1\}^{2d}$ called $\mathbb{H}$. Bob sends Alice $r$ and $H$. Alice then calculates the value $h_a=H(r||k_a)$. Alice then acts on pairs in $h_a$ with an embedding function $Q:\{0,1\}^2\to \CC ^2$. This function $Q$ uses the first of the two binary values to determine the measurement basis (horizontal/vertical or diagonal/antidiagonal) and the second to determine the specific qubit in $\{|0\rangle, |1\rangle , |+\rangle, |-\rangle\}$. For example, $Q(0,0)=|0\rangle$ and $Q(1,0)=|+\rangle$. Applying $Q$ to the pairs of bits in $h_a$ Alice prepares and sends $d$ qubits to Bob over the quantum channel. Upon reception, Bob computes $h_b=H(r||k_b)$. Note that if $k_a=k_b$ then $h_a=h_b$. Using the first bit of each pair Bob measures the quantum states and insures he obtains the correct qubit according to the second bit of the pair. If Bob measures something that disagrees with the even bits of $h_b$ then Bob rejects Alice’s challenge. If after measuring all qubits Bob has not yet rejected Alice’s challenge then he accepts her challenge. Changes made to the protocol are as follows: (1) Bob generates $r$ and $H$, this is done to thwart a simple attack discussed later; (2) the hash function changes between trials, this has no impact on the security of the protocol due to the public nature of the hash in both instances; and finally (3) here we assume for simplicity that Alice and Bob obtain the same nonce $r$ with certainty, using classical error correction techniques one can be relatively certain both parties obtain the same nonce. See below for a schematic overview of the protocol. $\quad$\ Figure 1. The protocol presented in [@Zawadzki19] The reason we force Bob to generate the randomness instead of Alice is that an adversary with unbound quantum memory may impersonate Bob but not make a measurement. Suppose an adversary does not know the key but requests Alice to identify herself. If Alice generates and sends $r,H$ with the string of states $|\varphi _i \rangle$ then the adversary may record $r,H$ and hold in memory, but not measure, the qubits. At a later time an honest participant may ask the adversary to identify themselves, in this case the adversary may send $r,H$ and the qubits in memory. Thus, the adversary correctly forges an authentication. Note that as we have presented the algorithm an adversary may still make this impersonation by waiting between Alice and Bob then passing the information between the two. The difference is as long as Bob generates the nonce then this attack must only be done while Alice and Bob are both online, whereas if Alice generates and sends the nonce then an adversary may hold the states for as long as is technologically feasible. The proposed protocol is claimed to be leakage resistant when considering an adversary measuring in a random basis. The reasoning behind this is that unless an adversary, Eve, correctly guesses the correct basis for each round, she will obtain different values for at least one of the bits of the hash. Now suppose an adversary is capable of computing preimages of hash functions through brute force with unbounded classical computational power or through dictionary attacks with unbounded classical memory. In this case it is unlikely that there will exist a $k_e\in K$ such that $H(r||k_e)$ matches what Eve measured. In the event there does exist such a $k_e$ then with overwhelming probability $k_e\neq k_a=k_b$ and Eve will not be able to falsify authentication of Alice or Bob. Unfortunately, the proposed protocol is claimed to be exactly a two-sided equality test with possible, though unlikely, relaxation of $F(i,j)=1$ if and only if $i=j$ (in this case $i$ is $k_a$ and $j$ is $k_b$). We know such a protocol has necessary leakage and due to the non-interactive nature of Bob we know that $k_b$ has no leakage, thus we know there must exist some leakage on $k_a$. Although Eve may not be able to determine any exact bit of $k_a$ she may drastically reduce the number of possible options for $k_a$ and hence construct a proper subset of $K$ such that the true value for $k_a$ is contained in this subset. An attack exemplifying this phenomenon is described in the next section. A Key Space Reduction Attack on QIA without Entanglement (our contribution) {#sec:attack} =========================================================================== Before discussing the specific attack, let $B$ be a set of orthogonal bases in ${\mathbb{C}}^2$ and consider the following fact. If a quantum state is prepared in a basis $b\in B$ with value $v\in \{0,1\}$, then an adversary may always remove one possible combination of $b$ and $v$ with a single measurement. Upon measuring in basis $b'\in B$ an adversary obtains $v'\in \{0,1\}$. The adversary is then certain the original pair $(b,v)$ was not $(b',1\bigoplus v')$, as when measured in the basis $b$ the qubit prepared by $b$ and $v$ will yield $v$ with certainty. Note that the adversary cannot say with certainty how the qubit was prepared, but they can always remove one possible option. Suppose now that instead of sampling at random for $b$ and $v$, the qubit is prepared using a private key $k\in K$ and a set of public parameters $p$, namely $b=b(k,p)$ and $v=v(k,p)$. An adversary once again measures in basis $b'\in B$ (chosen or taken at random) to obtain $v'\in \{0,1\}$, they may then determine a basis/value pair in which the qubit was not prepared. Because the adversary is assumed to be computationally unbounded they may then compute $b(\hat{k},p)$ and $v(\hat{k},p)$ for all $\hat{k}\in K$. Whenever these computations output the impossible pair $k',v'$ the adversary becomes aware that $\hat{k}\neq k$, hence reducing the key space. The extent to which the key space is reduced depends on the number of basis in $B$. If the distribution of basis choices in $B$ is low entropy the attack may be accomplished as described while if $B$ is high entropy then a probabilistic version decreases the space of likely keys. The assumption that the adversary is computationally unbounded may be lifted if $k$ is low entropy (for he can then indeed test all possible values for $k$ — given there are only a polynomial set of candidates), however assuming a computationally bounded adversary immediately removes unconditional security as an end goal. Let us now apply this key space reduction to the QIA protocol proposed in [@Zawadzki19], in this case the private key is $k$ and the public parameters are $r$ and $H$. Suppose an Eve has no a priori knowledge of the key except its existence in $K$. After receiving $r$ and $H$ over the classical channel she measures all qubits $|\varphi _i \rangle$ received from Alice in the horizontal/vertical basis and records the outputs as $M$. In the case where Eve is utilizing man-in-the-middle she is done, if she is impersonating Bob she accepts or rejects the protocol. After the protocol finishes the adversary may then compute $h_{\hat{k}}=H(r||\hat{k})$ for all $\hat{k}\in K$. Suppose the first qubit Eve measured in $M$ was $|0\rangle$. She now examines the first two bits of each $h_{\hat{k}}$, those that begin 00, 10, or 11 are all possible of obtaining the qubit $|0\rangle$ after measurement. The first of these three tuples will yield $|0\rangle$ with certainty and the later two with a probability of 0.5. The final tuple 01 however is not possible as that would imply that the qubit started in the state $|1\rangle$ and measured in $|0\rangle$. Thus, Eve knows that any $\hat{k}$ such that $h_{\hat{k}}$ begins 01 is not the key. The hash function is assumed to be independent and identically distributed so this removes approximately $\frac{1}{4}$ of all possible keys. Repeat this process for all qubits. After completion of all hash and check operations the adversary has obtained a subset of the key space which contains the key, hence causing information leakage. Specifically, the adversary knows the key is in subset $S$ defined by $$S=\{s\in K: h_{s_{2i}}=M_{i} \text{ and } h_{s_{2i-1}}=0\ \forall i\leq d\}.$$ Note that the true key $k\in S$ and $|S|\approx (\frac{3}{4})^d |K|$. This attack may be repeated using other choice of bases (i.e.- not always selecting the horizontal/vertical basis) and utilizing the same approach these different bases will likely yield different subsets of $K$. The key is in the intersection of all these subsets, decreasing the possible key space further. Note that after applying this attack the advantage of adversary may be negligible yet if $(\frac{3}{4})^d |K|$ is still not negligible in the security parameter. Parameters are not listed in [@Zawadzki19] however it does not seem unreasonable that $d$ is sufficiently large compared to $|K|$, otherwise a false positive for authentication is more likely. Other QIA protocols =================== It is worth pointing out that the attack described in section \[sec:attack\] also applies to the protocol by Hong et al. [@Hong17], which Zawadzki [@Zawadzki19] modifies. In more detail, the protocol in [@Hong17] is similar to Zawadzki’s, but do not use a hash function. Instead, whenever Alice transmits the qubits sequentially and, before sending each qubit, she randomly decides if she is going to use *security mode* or *authentication mode*. In the first case, she sends a decoy state while in the second one, a qubit encoding two bits of the authentication string is sent, similarly to [@Zawadzki19]. After Bob’s reception, Alice announces which mode she just has used. Therefore an adversary using the same strategy described in our attack in section \[sec:attack\] and collecting the information obtained whenever Alice announces authentication mode, will be able to shrink the size of the key space in the same way we have previously stated. On the other hand, other quantum identification protocols proposed in the literature are not vulnerable to our attack neither contradict the impossibility result mentioned in section \[sec:impossibility\]. For instance, some of them [@Penghao16; @Zeng00; @Yang13] are aided by the presence of a trusted third party, therefore not being real two-party protocols. Another type of protocols, such as [@Mihara02; @Shi01; @Zhang00], make use of an entangled quantum state shared between both parties. In [@Mihara02] the users, in addition, share a bitstring used as a password; both parties measures their part of the entangled state to produce a one time key that one of the users XORs with the password and sends the result to the other who checks for consistency. The downside of this approach is that, to repeat the identification process, the parties need to be provided again with new entangled states. In [@Shi01; @Zhang00] the users do not share any classical secret, they just use the entangled state to identify themselves. Conclusion ========== The protocol given by Zawadzki in [@Zawadzki19] may be secure against hash preimage attacks when attempting to find an exact match, however when considering impossible results from quantum measurements we see some hashed key values are not possible. Proverbially, the forest may be secure but each of the trees reveals enough information to reconstruct the possible forests. By eliminating approximately one quarter of the key options from each qubit we see that by measuring all the individual qubits in a random basis does in fact reveal a great deal about the key. This attack has no concern on quantum memory though relies heavily on classical computation power. Hence, unlike [@Damgard14; @Bouman13] where the authors consider a bounded quantum storage model, the only way to make this protocol secure without greatly changing its construction is to constrict an adversaries computational power. The attack proposed here is general in the sense of QIA protocols in the prepare and measure setup, thus any future protocol of this type must consider possible key space reduction attacks. Regardless of the method it is known that any identification protocol which poses no bounds on the adversary will inevitably fail due to results of Lo and Buhrman et al. For this reason we advise that any future attempts at identification schemes consider, and clearly communicate, their assumptions and objectives. [**[Acknowledgements:]{}**]{} This research was sponsored in part by the NATO Science for Peace and Security Programme under grant G5448, in part by Spanish MINECO under grants MTM2016-77213-R and MTM2017-88385-P, and in part by Programa Propio de I+D+i of the Universidad Politécnica de Madrid. [^1]: [email protected] [^2]: [email protected] [^3]: [email protected] [^4]: [email protected]

ArXiv

--- abstract: 'We extend the classic cake-cutting problem to a situation in which the “cake” is divided among families. Each piece of cake is owned and used simultaneously by all members of the family. A typical example of such a cake is land. We examine three ways to assess the fairness of such a division, based on the classic no-envy criterion: (a) Average envy-freeness means that for each family, the average value of its share (averaged over all family members) is weakly larger than the average value of any other share; (b) Unanimous envy-freeness means that in each family, each member values the family’s share weakly more than any other share; (c) Democratic envy-freeness means that in each family, at least half the members value the family’s share weakly more than any other share. We study each of these definitions from both an existential and a computational perspective.' author: - 'Erel Segal-Halevi' - Shmuel Nitzan bibliography: - '../erelsegal-halevi.bib' title: 'Envy-Free Cake-Cutting among Families' --- Introduction ============ Fair division of land and other resources among agents with different preferences has been an important issue since Biblical times. Today it is an active area of research in the interface of computer science [@Robertson1998CakeCutting; @Procaccia2015Cake] and economics [@Moulin2004Fair]. Its applications range from politics [@Brams1996Fair; @Brams2007Mathematics] to multi-agent systems [@Chevaleyre2006Issues]. In most fair division problems, the goods are divided to individual agents, and the fairness of a division is assessed based on the *valuation* of each agent. A common fairness criterion is envy-freeness: a division is called **envy-free** if each agent values his own share at least as much as any other share. However, in real life, goods are often owned and used by groups. As an example, consider a land-estate inherited by $k$ families, or a nature reserve that should be divided among $k$ states. The land should be divided to $k$ pieces, one piece per group. Each group’s share is then used by *all* members of the group together. The land-plot allotted to a family is inhabited by the entire family. The share of the nature-reserve alloted to a state becomes a national park open to all citizens of that state. In economic terms, the alloted piece becomes a “club good” [@Buchanan1965Economic]. The happiness of each group member depends on his/her valuation of the entire share of the group. But, in each group there are different people with different valuations. The same division can be considered envy-free by some family members and not envy-free by other members of the same family. How, then, should the fairness of the division be assessed? One option that comes to mind is to aggregate the valuations in each family to a single *family valuation* (also known as: *collective welfare function*). Following the utilitarian tradition [@Bentham1789Introduction], the family-valuation can be defined as the sum or (equivalently) the arithmetic average of the valuations of all family members. We call a division **[average-envy-free]{}** if, for each family, the average valuation of the family’s share (averaged over all family members) is weakly larger than the average valuation of any share allocated to another family. This definition makes sense in situations in which the numeric values of the agents’ valuations are meaningful and they are all measured in the same units, e.g. in dollars (see chapter 3 of [@Moulin2004Fair] for some real-life examples of such situations). In such cases, it may be possible to transfer value between members of a group after the division is done. Members who are more satisfied with the division can somehow compensate the less satisfied members, such that finally each member enjoys the average value. A second option is to require that no member of any family feels any envy. We call a division **[unanimous-envy-free]{}** if every agent values his family’s share at least as much as the share of any other family. The advantage of this definition is that it does not depend on the units in which the valuation functions are measured. It is applicable even when the valuations are only abstract representations of ordinal preferences. A disadvantage of [unanimous-envy-freeness]{} is that it may be difficult to attain in practice, especially when the “families” are large. As citizens in democratic states, we know that it is next to impossible to attain unanimity on even the most trivial issue. Therefore, it is not realistic to expect that all citizens agree that a certain division is envy-free. We call a division **[democratic-envy-free]{}** if at least half the citizens in each family value their family’s share at least as much as the share of any other family. This definition makes sense when land is divided between states with a democratic regime. After a division is proposed, each state conducts a referendum in which each citizen approves the division if he/she feels that the division is envy-free. The division is implemented only if, in every state, at least half of its members approve it. Of the three definitions presented above, [unanimous-envy-free]{} is clearly the strongest: it implies both [average-envy-free]{} and [democratic-envy-free]{}. The other two definitions do not imply each other, as shown below. [r]{}[3cm]{} Alice 6 3 0 0 --------- --- --- --- --- -- -- Bob 5 4 0 0 Charlie 1 8 0 0 David 0 0 6 3 Eva 0 0 6 3 Frankie 0 0 0 9 Example ------- Consider a land-estate consisting of four districts. It has to be divided between two families: (1) Alice+Bob+Charlie and (2) David+Eva+Frankie. The valuation of each member to each district is shown in the table to the right. If the two leftmost districts are given to family 1 and the two rightmost districts are given to family 2, then the division is *[unanimous-envy-free]{}*, since each member of each family feels that his family’s share is better than the other family’s share. It is also, of course, [average-envy-free]{} and [democratic-envy-free]{}. If only the single leftmost district is given to family 1 and the other three districts are given to family 2, then the division is still *[democratic-envy-free]{}*, since Alice and Bob feel that their family received a better share than the other family. However, Charlie does not feel that way, so the division is not [unanimous-envy-free]{}. Moreover, the division is not [average-envy-free]{} since the average valuation of family 1 in its own share is (6+5+1)/3=4, while the average valuation of family 1 in the other share is (3+4+8)/3=5. If the three leftmost districts are given to family 1 and only the rightmost district is given to family 2, then the division is *[average-envy-free]{}*, since the average valuation of family 2 in its share is (3+3+9)/3=5 while its average valuation in the other share is (6+6+0)/3=4. However, it is not [unanimous-envy-free]{} and not even [democratic-envy-free]{}, since David and Eva feel that the share given to the other family is more valuable. Two challenges arise once the fairness criterion is selected. First, determine whether there always exists a division satisfying this criterion. In case the answer is yes, determine whether there exists a protocol for achieving such a division. Cake-cutting protocols are traditionally characterized by two factors: the number of *connectivity components* in the final division, and the number of *queries* required to achieve the division. We now briefly explain each of these factors, as they are relevant to our results. Number of connectivity components --------------------------------- Ideally, we would like to allocate to each family a single, connected piece. This requirement is especially meaningful when the divided resource is land, since a contiguous piece of land is much easier to use than a collection of disconnected patches. However, a division with connected pieces is not always possible. In fact, many countries have a disconnected territory. A striking example is the India-Bangladesh border. According to Wikipedia,[^1] “Within the main body of Bangladesh were 102 enclaves of Indian territory, which in turn contained 21 Bangladeshi counter-enclaves, one of which contained an Indian counter-counter-enclave... within the Indian mainland were 71 Bangladeshi enclaves, containing 3 Indian counter-enclaves”. Another example is Baarle-Hertog - a Belgian municipality made of 24 separate parcels of land, most of which are exclaves in the Netherlands.[^2] In case a division with connected pieces is not possible, it is desirable to minimize the number of *connectivity components* (hence: *components*) in the division. Minimizing the number of components is a common requirement in the cake-cutting literature. It is common to assume that the cake is a 1-dimensional interval. In this case, the components are sub-intervals and their number is one plus the number of *cuts*. Hence, the number of components is minimized by minimizing the number of cuts [@Robertson1995Approximating; @Webb1997How; @Shishido1999MarkChooseCut; @Barbanel2004Cake; @Barbanel2014TwoPerson]. In a realistic, 3-dimensional world, the additional dimensions can be used to connect the components, e.g, by bridges or tunnels. Still, it is desirable to minimize the number of components in the original division in order to reduce the number of required bridges/tunnels. The goal of minimizing the number of components is also pursued in real-life politics. Going back to India and Bangladesh, after many years of negotiations they finally started to exchange most of their enclaves during the years 2015-2016. This is expected to reduce the number of components from 200 to a more reasonable number. Number of queries \[sub:Number-of-queries\] ------------------------------------------- The most common model for cake-cutting protocols is the *query model*, formally defined by Robertson and Webb [@Robertson1998CakeCutting]. Intuitively, a cake-cutting protocol uses two types of queries: a *mark query* (also called *cut query*) asks an agent “where would you cut the cake such that the value of the resulting piece is X?” and an *eval query* asks an agent “how much is piece Y worth for you?”. This model has been used to prove results about the *query complexity* of cake-cutting protocols, e.g. [@Even1984Note; @Edmonds2011Cake]. Interestingly, some cake-cutting problems cannot be solved with a finite number of queries. For example, with 3 or more agents, an envy-free division cannot be attained by a finite protocol when the pieces must be connected [@Stromquist2008Envyfree]. Results ------- Our results regarding the three fairness definitions are summarized in the following theorems. In all theorems, $k\geq 2$ is the number of families and $n$ is the total number of agents in all families. In the impossibility results, it is implicitly assumed that at least one family contains at least 2 members (which implies $n>k$).[^3] A property of cake partitions is called *feasible* if for every $k$ families and $n$ agents there exists an allocation satisfying this property. Otherwise, the property is called *infeasible*. (**[average-envy-freeness]{}**) \(a) [average-envy-freeness]{} with connected pieces is feasible. \(b) [average-envy-freeness]{}, with either connected or disconnected pieces, cannot be found by a finite protocol. (**[unanimous-envy-freeness]{}**) \(a) [unanimous-envy-freeness]{} with connected pieces is infeasible. Moreover, at least $n$ components might be required for a [unanimous-envy-free]{} division. \(b) [unanimous-envy-freeness]{} with disconnected pieces is feasible. Moreover, the number of required components is at most $1+(k-1)(n-1) = O(n k)$. \(c) A [unanimous-envy-free]{} division cannot be found by a finite protocol. (**[democratic-envy-freeness]{}**) \(a) When $k\geq3$, [democratic-envy-freeness]{} with connected pieces is infeasible. Moreover, at least $n\cdot\frac{k/2-1}{k-1}=\Omega(n)$ components might be required for a [democratic-envy-free]{} division. \(b) [democratic-envy-freeness]{} with disconnected pieces is always feasible. Moreover, the number of required components is at most $1+(k-1)(n/2-1)$. \(c) When there are $k=2$ families, [democratic-envy-freeness]{} with connected pieces is feasible. Moreover, there is a finite protocol for finding a [democratic-envy-free]{} division using at most $n$ queries. The results are summarized in the table below. For $k=2$, the results are tight: by all fairness definitions, we know that a fair division exists with the smallest possible number of connectivity components. For $k>2$, the results are not tight. As an illustration of the currently unsolved gaps, the table includes numeric values for $k=3$ and $k=4$. [|&gt;p[2.5cm]{}|c||c|c|c|]{} & & & **Finite**\ & & **Lower bound** & **Existence** & **Procedure**\ Average (Sec.\[sec:AM-fairness\]) & Any & $k$ & $k$ (connected; optimal) & No\ & 2 & $n$ & $n$ (optimal) & No\ Unanimous & 3 & $n$ & $2n-1$ & No\ (Sec. \[sec:unan-fairness\]) & 4 & $n$ & $3n-2$ & No\ & Any & $n$ & $(k-1)\cdot(n-1)+1$ & No\ & 2 & 2 & 2 (connected; optimal) & Yes\ Democratic & 3 & $n/4$ & $n-1$ & ?\ (Sec. \[sec:dem-fairness\]) & 4 & $n/3$ & $3n/2-2$ & ?\ & Any & $n\cdot\frac{k/2-1}{k-1}$ & $(k-1)\cdot(n/2-1)+1$ & ?\ Model and Notation {#sec:Model-and-Notation} ================== The cake to be divided is $C$. As in many cake-cutting papers, $C$ is assumed to be the unit interval $[0,1]$. The total number of agents in *all* families is $n$. Every agent $i\in\{1,...,n\}$ has a personal value function $V_i$, defined on the Borel subsets of $C$. The $V_i$ are assumed to be *absolutely continuous with respect to the length measure* (or simply *continuous*). This implies that all singular points have a value of 0 to all agents (a property often termed *non-atomicity*). Additionally, the value functions are assumed to be *additive* - the value of a union of two disjoint pieces is the sum of the values of the pieces. The continuity and additivity assumptions are common to most cake-cutting papers. [^4] The number of families is $k$. The families are called $F_j$, $j\in\{1,...,k\}$. The number of members in family $F_j$ is $n_j$. Each agent is a member of exactly one family, so $n=\sum_{j=1}^{k}n_j$. An *allocation* is a vector of $k$ pieces, $X=(X_1,\dots,X_k)$, one piece per family, such that the $X_i$ are pairwise-disjoint and $\cup_{i}{X_{i}} = C$. Ideally, we would like that each piece be connected (i.e, an interval). If this is not possible, then each piece should be a finite union of intervals, where the total number of components (intervals) should be as small as possible. In the division procedures presented here, it is assumed that all agents act according to their true value functions and not strategically. Designing cake-cutting mechanisms that take agents’ strategies into account is a challenging task even for individual agents [@Branzei2015Dictatorship] and we leave it to future work. Average fairness {#sec:AM-fairness} ================ In this section we prove Theorem 1. We assume that the value functions of the agents are additive, and define the following family-valuations: $$W_j(X_j)=\frac{\sum_{i\in F_j}V_i(X_j)}{n_j}\,\,\,\,\,\,\,\,\,\,\,\,\,\textrm{{for}\,}\,j\in\{1,...,k\}.$$ A division is called **[average-envy-free]{}** if $\forall j,j': W_j(X_j)\geq W_j(X_{j'})$. Existence --------- Given any $n$ additive value functions $V_i$, $i=1,...,n$ and any grouping of the agents to $k$ families, there exists an [average-envy-free]{} division in which each family receives a connected piece (the total number of components is $k$). Given any $n$ additive value functions $V_i$, the $k$ family-valuations $W_j$ defined above are also additive. Hence, the classic cake-cutting results are applicable: each of the $k$ families in our problem can be treated as an individual agent in the classic solution. For example, Simmons’ protocol [@Su1999Rental] implies the existence of an [average-envy-free]{} division in which each family receives a connected piece. Non-existence of finite protocols --------------------------------- While the existence results from classic cake-cutting are applicable in our setting, their *query complexity* is not preserved. The simplest cake-cutting protocol - cut-and-choose - finds a connected envy-free division between two individuals using only two queries (one cuts, the other chooses). However, with two *families* instead of two individuals, any finite number of queries might be insufficient, regardless of the number of components. \[lemma:infinity-of-average\]For $k\geq2$ families, if at least one family contains at least 2 members, then an [average-envy-free]{} division (with connected or disconnected pieces) cannot be found by a finite protocol. We prove that the lemma is true even in the simplest case in which there are two families. Suppose one family, e.g. $F_1$, has two members: $F_1=\{1,2\}$, so their family valuation is: $W_1 = (V_1+V_2)/2$. Also suppose that the valuation of each member of $F_2$ is $W_1$, so that their family valuation is $W_2\equiv W_1$. For convenience, assume here that the agents’ valuations are normalized such that the value of the entire cake is 1. Finding an [average-envy-free]{} division now amounts to finding a piece $X\subseteq C$ such that $W_1(X)=W_2(X)=1/2$. This is equivalent to finding a piece $X\subseteq C$ such that $V_1(X)+V_2(X)=1$. Hence, our lemma reduces to the following lemma: \[lemma:neg-average\]There is no finite protocol that, given two agents with value measures $V_1$ and $V_2$ satisfying $V_1(C)=V_2(C)=1$, always finds a piece $X\subseteq C$ such that $V_1(X)+V_2(X)=1$. Define an *average piece* as a piece $X\subseteq C$ such that: $V_1(X)+V_2(X)=1$. We now prove that finding an average piece might require an infinite number of queries. Each *eval* or *mark* query involves two points in $[0,1]$: in an eval query, both points are determined by the protocol; in a mark query, one point is replied by the agent. Call these points the “known points” and include the endpoints 0 and 1 in the set of known points. Let $P_{m}$ be the set of known points before step $m$. Initially $P_{1}=\{0,1\}$. Each query potentially increases $P_{m}$ by at most two points. For example, after a $mark(0.1,v)$ query with a reply of $0.2$, $P_{m}=\{0,0.1,0.2,1\}$. The protocol can be conceptually divided into steps, such that in each step, one point is added to $P_{m}$. Hence, for every $m\geq1$, $|P_{m}|=m+1$. If the protocol returns a result at step $m$, this result must be a collection of intervals whose endpoints are in $P_{m}$, since the values of subsets with different endpoints are not known to the protocol. Let $I_{m}$ be the set of $m$ intervals whose endpoints are nearby points in $P_{m}$. If the protocol returns at step $m$, the result must be a subset of $I_{m}$. Let $V_{m}=\{V_{1}(X)+V_{2}(X)|X\subseteq I_{m}\}$. I.e, $V_{m}$ is the set of all values of pieces that can be returned by the protocol at step $m$. Note that $V_{m}$ is finite and $|V_{m}|\leq2^{m}$. The protocol can return an average piece at step $m$, if and only if $1\in V_{m}$. We now prove that this cannot be guaranteed in a finite protocol. The proof is by induction on $m$. For $m=1$, $V_{m}=\{0,2\}$ so $1\notin V_{m}$. $V_{1}$ is illustrated by the dots: (210,20) (0,10)(200,10) (0,15) (0,5)[0]{} (100,5)[1]{} (200,15)(200,5)[2]{} Suppose the claim is true for $m$. Then $V_{m}$ is a set of values that does not contain 1, e.g: (210,20) (0,10)(200,10) (0,15) (0,5)[0]{} (30,15) (50,15) (110,15) (160,15) (190,15) (195,15) (100,5)[1]{} (200,15)(200,5)[2]{} At step $m$, the protocol adds to $P_{m}$ a new point $q$, such that $P_{m+1}=P_{m}\cup\{q\}$. This replaces an interval in $I_{m}$ with two smaller intervals, e.g. if $q\in[p,r]$ and $[p,r]\in I_{m}$, then $I_{m+1}=I_{m}\setminus\{[p,r]\}\cup\{[p,q],[q,r]\}$. Let $v_{p}:=V_{1}([p,q])+V_{2}([p,q])$ and $v_{r}:=V_{1}([q,r])+V_{2}([q,r])$. Then: $$\begin{aligned} V_{m+1}\subset V_{m}\cup\{v-v_{p}|v\in V_{m}\}\cup\{v+v_{p}|v\in V_{m}\}\cup\{v-v_{r}|v\in V_{m}\}\cup\{v+v_{r}|v\in V_{m}\}\end{aligned}$$ I.e, the new possible values are the previous possible values, plus or minus the values of the new intervals (since the new intervals can be added or subtracted from any piece). Suppose step $m$ is a mark query sent to agent 1. This means that the protocol can control the values $V_{1}([p,q]),\,V_{1}([q,r])$. It cannot, however, control the values $V_{2}([p,q]),\,V_{2}([q,r])$ since the query can be sent to only one agent at a time. Similarly, if the query is sent to agent 2 then the protocol can control $V_{2}([p,q]),\,V_{2}([q,r])$ but not $V_{1}([p,q]),\,V_{1}([q,r])$. In both cases, the protocol cannot control $v_{p},v_{r}$. This means that for every protocol, there can be value measures such that the new values added to $V_{m}$ do not include 1, so $1\notin V_{m+1}$: (210,20) (0,10)(200,10) (0,15) (0,5)[0]{} (26,15) (30,15) (34,15) (46,15) (50,15) (54,15) (106,15) (110,15) (114,15) (156,15) (160,15) (164,15) (190,15) (195,15) (100,5)[1]{} (200,15)(200,5)[2]{} To conclude: after any finite number $m$ of steps, the set of possible piece values $V_{m}$ is finite, and an adversary can select the value measures such that it does not contain 1. Hence finding an average piece cannot be guaranteed. #### Note: A similar idea was used by [@Robertson1998CakeCutting] to prove that it is impossible to find an exact division with a finite number of queries. Unanimous fairness {#sec:unan-fairness} ================== This section proves Theorem 2. A division is called **[unanimous-envy-free]{}** if: $$\begin{aligned} \forall j,j'=1,...,k: \forall i\in F_j: V_i(X_j)\geq V_i(X_{j'})\end{aligned}$$ We denote by [UnanimousEnvyFree]{}$(n,k)$ the problem of finding a [unanimous-envy-free]{} division when there are $n$ agents grouped in $k$ families. We relate this problem to the classic cake-cutting problem of finding an *exact division*: [Exact]{}$(N,K)$ is the following problem. Given $N$ agents and an integer $K$, find a division of the cake to $K$ pieces, such that each of the $N$ agents assigns exactly the same value to all pieces: $$\begin{aligned} \forall j,j'=1,...,K:\,\,\forall i=1,...,N:\,\,V_{i}(X_{j})=V_{i}(X_{j'}).\end{aligned}$$ Alon [@Alon1987Splitting] proved that for every $N$ and $K$, [Exact]{}$(N,K)$ has a solution with at most $N(K-1)+1$ components. He also showed that this number is the smallest that can be guaranteed. We now use these results in our setting. To this end, we show a two-way reduction between the problem of [unanimous-envy-free]{} division and the problem of exact division. \[lemma:exact-implies-unprop\] For each $n,k$, a solution to [Exact]{} $(n-1,k)$ implies a solution to [UnanimousEnvyFree]{} $(n,k)$ for any grouping of the $n$ agents to $k$ families. Suppose we are given an instance of [UnanimousEnvyFree]{}$(n,k)$, i.e, $n$ agents in $k$ families. Select $n-1$ agents arbitrarily. Use [Exact]{}$(n-1,k)$ to find a partition of the cake to $k$ pieces, such that each of the $n-1$ agents is indifferent between these $k$ pieces. Ask the $n$-th agent to choose his favorite piece. Give that piece to the family of the $n$-th agent. Give the other $k-1$ pieces arbitrarily to the remaining $k-1$ families. The division is [unanimous-envy-free]{}. Combining this lemma with the result of [@Alon1987Splitting] immediately implies the following upper bound on the number of required components: \[cor:minprop-general\] Given $n$ agents in $k$ families, there exists a [unanimous-envy-free]{} division with at most $(k-1)\cdot(n-1)+1$ components. \[lemma:unprop-implies-exact\] For each $N,K$, a solution to [UnanimousEnvyFree]{} $(N(K-1)+1,\,K)$ implies a solution to [Exact]{} $(N,K)$. Given an instance of [Exact]{}$(N,K)$ ($N$ agents and a number $K$ of required pieces), create $K$ families. In each of the first $K-1$ families, put a copy of each of the $N$ agents. In the $K$-th family, put a single agent whose value measure is the average of the given $N$ value measures: $$V^{*}=\frac{1}{N}\sum_{i=1}^{N}V_{i}.$$ The total number of agents in all $K$ families is $N(K-1)+1$. Use [UnanimousEnvyFree]{} $(N(K-1)+1,\,K)$ to find a [unanimous-envy-free]{} division, X. By the pigeonhole principle, for each agent $i$ in family $j$: $V_i(X_j) \geq 1/K$. By construction, each of the first $K-1$ families has a copy of agent $i$. Hence, all $N$ agents values each of the first $K-1$ pieces as at least $1/K$ and: $$\forall i=1,...,N:\,\,\,\,\,\,\sum_{j=1}^{K-1}V_{i}(X_{j})\geq\frac{K-1}{K}.$$ Hence, by additivity, every agent values the $K$-th piece as at most $1/K$: $$\forall i=1,...,N:\,\,\,\,\,\,V_{i}(X_{K})\leq1/K.$$ The piece $X_{K}$ is given to the agent with value measure $V^{*}$, so again by the pigeonhole principle: $V^{*}(X_{K})\geq1/K$. By construction, $V^{*}(X_{K})$ is the average of the $V_{i}(X_{K})$. Hence, necessarily: $$\forall i=1,...,N:\,\,\,\,\,\,V_{i}(X_{K})=1/K.$$ Again by additivity: $$\forall i=1,...,N:\,\,\,\,\,\,\sum_{j=1}^{K-1}V_{i}(X_{j})=\frac{K-1}{K}.$$ Hence, necessarily: $$\forall i=1,...,N,\,\,\,\,\,\forall j=1,...,K-1:\,\,\,\,\,\,V_{i}(X_{j})=1/K.$$ So we have found an exact division and solved [Exact]{}$(N,K)$ as required. @Alon1987Splitting proved that for every $N$ and $K$, an [Exact]{}$(N,K)$ division might require at least $N(K-1)+1$ components. Combining this result with the above lemma implies the following negative result: \[cor:neg-unprop\] For every $N,K$, let $n=N(K-1)+1$. A [unanimous-envy-free]{} division for $n$ agents in $K$ families might require at least $n$ components. This corollary implies that, in particular, [unanimous-envy-freeness]{} with connected pieces is infeasible. This impossibility result is generalized in Lemma \[thm:neg-majority\]. Infinite procedures and approximations -------------------------------------- It is impossible to solve [Exact]{}$(N,K)$ by a finite protocol whenever $N\geq2$ and $K\geq2$ [@Robertson1998CakeCutting pp. 103-104]. By Lemma \[lemma:unprop-implies-exact\], this implies: [UnanimousEnvyFree]{} cannot be solved by a finite protocol whenever $n>k$.[^5] However, there is an approximation procedure that converges to an exact division of a cake to $k=2$ pieces [@Simmons2003Consensushalving]. By Lemma \[lemma:exact-implies-unprop\], this procedure can be used to find an approximate [unanimous-envy-free]{} division for two families. Democratic fairness {#sec:dem-fairness} =================== In this section we prove Theorem 3. A division X is called **[democratic-envy-free]{}** if for all $j,j'=1,...,k$, for at least half the members $i\in F_j$: $$\begin{aligned} V_i(X_j)\geq V_i(X_{j'})\end{aligned}$$ Existence and number of components ---------------------------------- Given a specific allocation of cake to families, define a *positive* agent as an agent that values his family’s share as more than 0. Note that this is a much weaker requirement than proportionality. Define a *zero* agent as a non-positive agent. \[thm:neg-majority\] Assume there are $n=mk$ agents, divided into $k$ families with $m$ members in each family. To guarantee that at least $q$ members in each family are positive, the total number of components might have to be at least: $$\begin{aligned} k\cdot\frac{kq-m}{k-1} \end{aligned}$$ Number the families by $j=0,...,k-1$ and the members in each family by $i=0,...,m-1$. Assume that the cake is the 1-dimensional interval $[0,mk]$. In each family $j$, each member $i$ wants only the following interval: $(ik+j,\,ik+j+1)$. Thus there is no overlap between desired pieces of different members. The table below illustrates the construction for $k=2,\,m=3$. The families are {Alice,Bob,Charlie} and {David,Eva,Frankie}: Alice 1 0 0 0 0 0 --------- --- --- --- --- --- --- Bob 0 0 1 0 0 0 Charlie 0 0 0 0 1 0 David 0 1 0 0 0 0 Eva 0 0 0 1 0 0 Frankie 0 0 0 0 0 1 Suppose the piece $X_j$ (the piece given to family $j$) is made of $l\geq1$ components. We can make $l$ members of $F_j$ positive using $l$ intervals of positive length inside their desired areas. However, if $q>l$, we also have to make the remaining $q-l$ members positive. For this, we have to extend $q-l$ intervals to length $k$. Each such extension totally covers the desired area of one member in each of the other families. Overall, each family creates $q-l$ zero members in each of the other families. The number of zero members in each family is thus $(k-1)(q-l)$. Adding the $q$ members which must be positive in each family, we get the following necessary condition: $(k-1)(q-l)+q\leq m$. This is equivalent to: $$\begin{aligned} l\geq\frac{kq-m}{k-1}. \end{aligned}$$ The total number of components is $k\cdot l$, which is at least the expression stated in the Lemma. In a [unanimous-envy-free]{} division, all members in each family must be positive. Taking $q=m$ gives $l\geq m$ and the number of components is at least $km=n$, which is the bound of Corollary \[cor:neg-unprop\]. In a [democratic-envy-free]{} division, at least half the members in each family must be positive. Taking $q=m/2$ gives: \[cor:neg-majority\] In a [democratic-envy-free]{} division with $n$ agents grouped into $k$ families, the number of components might have to be at least $$\begin{aligned} n\cdot\frac{k/2-1}{k-1}.\end{aligned}$$ Note that when $k=2$, the lower bound of Corollary \[cor:neg-majority\] is 0. Indeed, for two families there always exist [democratic-envy-free]{} divisions with connected pieces. This is proved in the next subsection. Division procedure ------------------ Algorithm \[alg:majprop\] describes a procedure that achieves a [democratic-envy-free]{} division for two families. For each family, a location $M_j$ is calculated such that, if the cake is cut at $M_j$, half the members value the interval $[0,M_j]$ as at least $1/2$ and the other half value the interval $[M_j,1]$ as at least $1/2$. Then, the cake is cut between the two family medians, and each family receives the piece containing its own median. By construction, at least half the members in each family value their family’s share as at least 1/2, so the division is [democratic-envy-free]{}. The division has only 2 components (each family receives a connected piece). In contrast to the impossibility results of the previous sections, this protocol is *finite*. In fact, it requires only $n$ mark queries (one query per agent). INPUT: - A cake, which is assumed to be the unit interval $[0,1]$. - $n$ agents, all of whom value the cake as 1. - A grouping of the agents to $2$ families, $F_1,F_2$.\ OUTPUT: A [democratic-envy-free]{} division of the cake to $2$ pieces.\ ALGORITHM: - Each agent $i=1,...,n$ marks an $x_{i}\in[0,1]$ such that $V_i([0,x_{i}])=V_i([x_{i},1])=1/2$. - For each family $j=1,2$, find the median of its members’ marks: $M_{j}=\textrm{{median}}_{i\in F_j}x_{i}$. Find the median of the family medians: $M^{*}=(M_1+M_2)/2$. - If $M_1<M_2$ then give $[0,M^*]$ to $F_1$ and $[M^*,1]$ to $F_2$.\ Otherwise give $[0,M^*]$ to $F_2$ and $[M^*,1]$ to $F_1$. The above procedure does not work for more than 2 families. Currently, all we have is the following existence result, which is a trivial outcome of the existence results of Section \[sec:unan-fairness\]. Apply these results with $n/2$ instead of $n$: select half of the members in each family arbitrarily, then find a division which is [unanimous-envy-free]{} for them while ignoring all other members. Hence: \[cor:mjef-general\] Given $n$ agents in $k$ families, there exists a [democratic-envy-free]{} division with at most $(k-1)\cdot(n/2-1)+1$ components. It is an interesting open question whether a [democratic-envy-free]{} division for 3 or more families (with disconnected pieces) can be found by a finite protocol. Alternatives ============ Instead of envy-freeness, it is possible to use *proportionality* as the basic fairness criterion. Proportionality means that, when there are $k$ families, each family receives at least $1/k$ of its total cake value. Then, **[average-proportionality]{}** means that the average value of each family in its allocated share (averaged over all family members) is at least $1/k$ of the average value of the entire cake; **[unanimous-proportionality]{}** means that every agent values its family’s share as at least $1/k$ of the total; **[democratic-proportionality]{}** is defined analogously. When the valuations are additive, envy-freeness implies proportionality; when there are only two families, proportionality implies envy-freeness. Theorem 1 (Section \[sec:AM-fairness\]) holds as-is for [average-proportionality]{}. In Theorems 2 and 3, the number of components in the positive results can be improved from $O(n k)$ to $O(n \log k)$, using a recursive halving technique. See [@SegalHalevi2016FamilyProportional] for details. The above criteria assume that all families have equal entitlements. This makes sense, for example, when $k$ siblings inherit their parents’ estate. While an heir will probably like to take his family’s preferences into account when selecting a share, each heir is entitled to exactly $1/k$ of the estate regardless of the size of his/her family. In general, each family may have a different entitlement. The entitlement of a family may depend on its size but may also depend on other factors. For example, when two states jointly discover a new island, they will probably want to divide the island between them in proportion to their investments and not in proportion their population. This generalized problem can be solved by applying results in cake-cutting with unequal entitlements. In particular, Stromquist and Woodall [@Stromquist1985Sets] prove that, for every fraction $r\in[0,1]$, it is possible to cut a piece of cake made of at most $n$ intervals, which each agent values as exactly $r$ of the total cake value. This result can be used to generalize our theorems to families with different entitlements. See [@SegalHalevi2016FamilyProportional] for details. One could consider the following alternative fairness criterion: an allocation is *individually-proportional* if the allocation $X=(X_{1},\ldots,X_{k})$ admits a refinement $Y=(Y_{1},\ldots,Y_{n})$, where for each family $F_{j}$, $\cup_{i\in F_{j}}Y_{i}=X_{j}$, such that for each agent $i$, $V_{i}(Y_{i})\geq1/n$. Individually-proportional allocations always exist and can be found by using any classic proportional cake-cutting procedure on the individual agents, disregarding their families. The number of components is at most $n$. Individual-proportionality makes sense if, after the land is divided among the families, each family intends to further divide its share among its members. However, often this is not the case. When an inherited land-estate is divided between two families, the members of each family intend to live and use their entire share together, rather than dividing it among them. Therefore, the happiness of each family member depends on the entire value of his family’s share, rather than on the value of a potential private share he would get in a hypothetic sub-division. Related Work {#sec:related} ============ There are numerous papers about fair division in general and fair cake-cutting in particular. We mentioned some of them in the introduction. Here we survey some work that is more closely related to family-based fairness. Group-envy-freeness and on-the-fly coalitions --------------------------------------------- [@Berliant1992Fair; @Husseinov2011Theory] study the concept of *group-envy-free* cake-cutting. Their model is the standard cake-cutting model in which the cake is divided among *individuals* (and not among families as in our model). They define a group-envy-free division as a division in which no coalition of individuals can take the pieces allocated to another coalition with the same number of individuals and re-divide the pieces among its members such that all members are weakly better-off. Coalitions are also studied by [@DallAglio2009Cooperation; @DallAglio2012Finding]. In our setting, the families are pre-determined and the agents do not form coalitions on-the-fly. In an alternative model, in which agents *are* allowed to form coalitions based on their preferences, the family-cake-cutting problem becomes easier. For instance, it is easy to achieve a [unanimous-proportional]{} division with connected pieces between two coalitions: ask each agent to mark its median line, find the median of all medians, then divide the agents to two coalitions according to whether their median line is to the left or to the right of the median-of-medians. Fair division with public goods ------------------------------- In our setting, the piece given to each family is considered a “public good” in this specific family. The existence of fair allocations of homogeneous goods when some of the goods are public has been studied e.g. by @Diamantaras1992Equity [@Diamantaras1994Generalization; @Diamantaras1996Set; @Guth2002NonDiscriminatory]. In these studies, each good is either private (consumed by a single agent) or public (consumed by all agents). In the present paper, each piece of land is consumed by all agents in a single family - a situation not captured by existing public-good models. Family preferences in matching markets -------------------------------------- Besides land division, family preferences are important in matching markets, too. For example, when matching doctors to hospitals, usually a husband and a wife want to be matched to the same hospital. This issue poses a substantial challenge to stable-matching mechanisms [@Klaus2005Stable; @Klaus2007Paths; @Kojima2013Matching]. Fairness in group decisions --------------------------- The notion of fairness between groups has been studied empirically in the context of the well-known *ultimatum game*. In the standard version of this game, an individual agent (the *proposer*) suggests a division of a sum of money to another individual (the *responder*), which can either approve or reject it. In the group version, either the proposer or the responder or both are groups of agents. The groups have to decide together what division to propose and whether to accept a proposed division. Experiments by [@Robert1997Group; @Bornstein1998Individual] show that, in general, groups tend to act more rationally by proposing and accepting divisions which are less fair. [@Messick1997Ultimatum] studies the effect of different group decision rules while [@Santos2015Evolutionary] uses a threshold decision rule which is a generalized version of our majority rule (an allocation is accepted if at least $M$ agents in the responder group vote to accept it). These studies are only tangentially relevant to the present paper, since they deal with a much simpler division problem in which the divided good is *homogeneous* (money) rather than heterogeneous (cake/land). Non-additive utilities ---------------------- As explained in Sections \[sec:unan-fairness\] and \[sec:dem-fairness\], the difficulty with [unanimous-envy-freeness]{} and [democratic-envy-freeness]{} is that the associated family-valuation functions are not additive. It is interesting to compare our work to other works on cake-cutting with non-additive valuations. [@Berliant1992Fair; @Maccheroni2003How; @DallAglio2005Fair] focus on sub-additive, or concave, valuations, in which the sum of the values of the parts is *more* than the value of the whole. These works are not applicable to the family-cake-cutting problem, because the family-valuations are not necessarily sub-additive - the sum of values of the parts might be less than the value of the whole. [@Sagara2005Equity; @DallAglio2009Disputed; @Husseinov2013Existence] consider general non-additive value functions. They provide pure existence proofs and do not say much about the nature of the resulting divisions (e.g, the number of connectivity components), which we believe is important in practical division applications. [@Su1999Rental] presents a protocol for envy-free division with connected pieces which does not assume additivity of valuations. However, when the valuations are non-additive, there are no guarantees about the value per agent. In particular, with non-additive valuations, the resulting division is not necessarily proportional. [@Mirchandani2013Superadditivity] suggests a division protocol for non-additive valuations using non-linear programming. However, the protocol is practical only when the cake is a collection of a small number of *homogeneous* components, where the only thing that matters is what fraction of each component is allocated to each agent. Our model is the standard, general model where the cake is a single *heterogeneous* good. Finally, [@Berliant2004Foundation; @Caragiannis2011Towards; @SegalHalevi2015EnvyFree] study specific non-additive value functions which are motivated by geometric considerations (location, size and shape). The present paper contributes to this line of work by studying specific non-additive value functions which are motivated by a different consideration: handling the different preferences of family members. A possible future research topic is to find fair division rules that handle these considerations simultaneously, as both of them are important for fair division of land. Conclusions =========== One practical conclusion that can be drawn from our results concerns the selection of fairness criterion. When $n$ (the total number of agents) is sufficiently small, it is reasonable to use unanimous-fairness, which guarantees that all agents are satisfied with their family’s share. However, when $n$ is large, as is the case when dividing land between states, insisting on unanimous-fairness might result in each country having an absurdly fractioned territory. In this case, democratic-fairness is a more reasonable choice. This is particularly true when there are only two states, since in this case democratic-fairness can be achieved with connected pieces and in finite time, which is impossible with the other fairness criteria. Acknowledgements ================ This research was funded in part by the following institutions: The Doctoral Fellowships of Excellence Program at Bar-Ilan University, the Mordechai and Monique Katz Graduate Fellowship Program, and the Israel Science Fund grant 1083/13. We are grateful to Yonatan Aumann, Avinatan Hassidim, Noga Alon, Christian Klamler, Ulle Endriss and Neill Clift for helpful discussions. This paper started with a discussion in the MathOverflow website.[^6] We are grateful to the members that participated in the discussion: Pietro Majer, Tony Huynh and Manfred Weis. Other members of the StackExchange network contributed useful answers and ideas: Alex Ravsky, Andrew D. Hwang, BKay, Christian Elsholtz, Daniel Fischer, David K, D.W, Hurkyl, Ittay Weiss, Kittsil, Michael Albanese, Raphael Reitzig, Real, babou, Dömötör Pálvölgyi (domotorp), Ian Turton (iant) and ivancho. [^1]: Wikipedia page “IndiaBangladesh enclaves”. [^2]: Wikipedia page “Baarle-Hertog”. Many other examples are listed in Wikipedia page “List of enclaves and exclaves”. We are grateful to Ian Turton for the references. [^3]: Without this assumption, the problem is equivalent to cake-cutting among individuals. [^4]: A third common assumption is that the valuations are normalized such that for every agent $i$: $V_i(C)=1$. The results of the present paper do not rely on this assumption. [^5]: Because $N(K-1)+1>K$ iff $(N-1)(K-1)>0$ iff $N\geq2$ and $K\geq2$. [^6]: http://mathoverflow.net/questions/203060/fair-cake-cutting-between-groups

ArXiv

--- abstract: 'The luminous $z=0.286$ quasar [HE0450–2958]{} is interacting with a companion galaxy at 6.5 kpc distance and the whole system radiates in the infrared at the level of an ultraluminous infrared galaxy (ULIRG). A so far undetected host galaxy triggered the hypothesis of a mostly “naked” black hole (BH) ejected from the companion by three-body interaction. We present new HST/NICMOS 1.6$\mu$m imaging data at 01 resolution and VLT/VISIR 11.3$\mu$m images at 035 resolution that are for the first time resolving the system in the near- and mid-infrared. We combine these data with existing optical HST and CO maps. (i) At 1.6$\mu$m we find an extension N-E of the quasar nucleus that is likely a part of the host galaxy, though not its main body. If true, a combination with upper limits on a main body co-centered with the quasar brackets the host galaxy luminosity to within a factor of $\sim$4 and places [HE0450–2958]{} directly onto the $M_\mathrm{BH}-M_\mathrm{bulge}$-relation for nearby galaxies. (ii) A dust-free line of sight to the quasar suggests a low dust obscuration of the host galaxy, but the formal upper limit for star formation lies at 60 M$_\odot$/yr. [HE0450–2958]{} is consistent with lying at the high-luminosity end of Narrow-Line Seyfert 1 Galaxies, and more exotic explanations like a “naked quasar” are unlikely. (iii) All 11.3$\mu$m radiation in the system is emitted by the quasar nucleus. It has warm ULIRG-strength IR emission powered by black hole accretion and is radiating at super-Eddington rate, $L/L_\mathrm{Edd}=6.2^{+3.8}_{-1.8}$, or 12 $M_\odot$/year. (iv) The companion galaxy is covered in optically thick dust and is not a collisional ring galaxy. It emits in the far infrared at ULIRG strength, powered by Arp220-like star formation (strong starburst-like). An M82-like SED is ruled out. (v) With its black hole accretion rate [HE0450–2958]{} produces not enough new stars to maintain its position on the $M_\mathrm{BH}-M_\mathrm{bulge}$-relation, and star formation and black hole accretion are spatially disjoint. This relation can either only be maintained averaging over a longer timescale ($\la$500 Myr) and/or the bulge has to grow by redistribution of preexisting stars. (vi) Systems similar to [HE0450–2958]{} with spatially disjoint ULIRG-strength star formation and quasar activity might be common at high redshifts but at $z<0.43$ we only find $<$4% (3/77) candidates for a similar configuration.' author: - 'Knud Jahnke, David Elbaz, Eric Pantin, Asmus Böhm$^,$, Lutz Wisotzki, Geraldine Letawe, Virginie Chantry, Pierre-Olivier Lagage' title: | The QSO HE0450–2958: Scantily dressed or heavily robed?\ A normal quasar as part of an unusual ULIRG. --- Introduction ============ In the current framework of galaxy evolution, galaxies and black holes are intimately coupled in their formation and evolution. The masses of galactic bulges and their central black holes (BHs) in the local Universe follow a tight relation [e.g. @haer04] with only 0.3 dex scatter. Currently it is not clear how this relation comes about and if and how it evolved over the last 13 Gyrs, but basically all semi-analytic models now include feedback from active galactic nuclei (AGN) as a key ingredient to acquire consensus with observations [e.g. @hopk06c; @some08]. In these models it is assumed that black hole growth by accretion and energetic re-emission from the ignited AGN back into the galaxy can form a self regulating feedback chain. This feedback loop can potentially regulate or possibly also truncate star formation and in this process create and maintain the red/blue color–magnitude bimodality of galaxies. In this light, any galaxy with an abnormal deviation from the $M_\mathrm{BH}$–$M_\mathrm{bulge}$-relation will be an important laboratory for understanding the coupling mechanisms of black hole and bulge growth. It will set observational limits for these models, and constrain the time-lines and required physics involved. Since the early work by @bahc94 [@bahc95b] on QSO host galaxies with the [*Hubble Space Telescope (HST)*]{} and the subsequently resolved dispute about putatively “naked” QSOs [@mcle95a], no cases for QSOs without surrounding host galaxies were found – when detection limits were correctly interpreted. Only recently the QSO [HE0450–2958]{} renewed the discussion, when @maga05 made a case for a 6$\times$ too faint upper limit of the host galaxy of [HE0450–2958]{} with respect to the $M_\mathrm{BH}$–$M_\mathrm{bulge}$-relation. In light of a number of competing explanations for this, the nature of the [HE0450–2958]{} system needs to be settled. The QSO [HE0450–2958]{} (a.k.a. IRAS 04505–2958) at a redshift of $z=0.286$ was discovered by @low88 as a warm IRAS source. [HE0450–2958]{} is a radio-quiet quasar, with a distorted companion galaxy at 15 (=6.5 kpc) distance at the same redshift, likely in direct interaction with the QSO [@cana01]. The combined system shows an infrared luminosity of an ultraluminous infrared galaxy (ULIRG, $L_\mathrm{IR}>10^{12}$ L$_\odot$). [HE0450–2958]{} was observed with the [Hubble Space Telescope (HST)]{} and its WFPC2 camera [@boyc96] in F702W (=$R$ band) and ACS camera [@maga05] in F606W (=$V$ band), both observations did not allow to detect a host galaxy centered on the quasar position within their limits (Figure \[fig:allwave\], left column). @maga05 estimated an expected host galaxy brightness if [HE0450–2958]{} was a normal QSO system that obeyed the $M_\mathrm{BH}$–$M_\mathrm{bulge}$-relation in the local Universe and given a BH mass estimate or luminosity of the QSO. They concluded that the ACS F606W detection limits were six times fainter than the expected value for the host galaxy, which qualified [HE0450–2958]{} to be very unusual. {width="\textwidth"} @maga05 sparked a flurry of subsequent papers to explain the undetected host galaxy to black hole relation. Over time three different alternative explanations have been put forward and were substantiated: 1. [HE0450–2958]{} is a normal QSO nucleus, but with a massive black hole residing in an under-massive host galaxy. The system is lying substantially off the local $M_\mathrm{BH}$–$M_\mathrm{bulge}$-relation; the host galaxy possibly hides just below the F606W detection limit [@maga05]. 2. The host galaxy is actually absent, [HE0450–2958]{} is a truly “naked” QSO, by means of a black hole ejection event in a gravitational three body interaction or gravitational recoil following the merger of [HE0450–2958]{} with the companion galaxy [@hoff06; @haeh06; @bonn07]. 3. The original black hole mass estimate was too high [@merr06; @kim07; @leta07] and is in fact $\sim$10 times lower. With comparably narrow ($\sim$1500 km/s FWHM) broad QSO emission lines the QSO could be the high-luminosity analog of the class of narrow-line Seyfert 1 galaxies (NLSy1). The host galaxy could be normal for the black hole mass and be absolutely consistent with the ACS upper limits. In this article we present new data initially motivated by the still undetected host galaxy and by the possibility that the host galaxy might be obscured by substantial amounts of dust. We want to investigate the overall cool and warm dust properties of the system, using new near infrared (NIR) and mid infrared (MIR) images. The F606W ACS band is strongly susceptible to dust attenuation, and dust could have prevented the detection of the host galaxy in the optical. With new NIR data we look at a substantially more transparent wavelength. At the same time the new infrared data is meant to localize the source(s) of the ULIRG emission. Three components are candidates for this: The AGN nucleus, the host galaxy, and the companion galaxy. Our NIR data allow to trace star formation and the MIR image traces the hot dust in the system. We present the new data and interpret it in the view of the so far collected knowledge from X-ray to radio-wavelengths that was built up since the article of @maga05. Throughout we will use Vega zero-points and a cosmology of $h=H_0/(100\mathrm{km s^{-1} Mpc^{-1}})=0.7$, $\Omega_M=0.3$, and $\Omega_\Lambda=0.7$, corresponding to a distance modulus of 40.84 for $z=0.286$ and linear scales of 4.312 kpc/. The IR angle ============ Up to now the only existing infrared observations on [HE0450–2958]{} were from the 2MASS survey in the near infrared $J$, $H$, and $K$ bands at $\sim$4resolution, and in the MIR from the IRAS mission [@grij87; @low88] at 12, 25, 60, and 100 $\mu$m with about 4 resolution. Both surveys do not resolve the different individual components of the system (QSO, companion galaxy, foreground star). De Grijp et al. ([-@grij87]) noted that the [HE0450–2958]{} system is showing the MIR/FIR luminosities of a ULIRG system, but it was not clear which components of the system are responsible for this emission due to the coarse IRAS resolution. We want to localize the dust emission in two ways: (a) A direct observation of the hot dust component at 8.9$\mu$m (rest-frame) with the VISIR imager at the ESO VLT. (b) A localization of dust in general by combining new HST near infrared and the existing ACS optical data. For this purpose we obtained HST NIC2 imaging in the rest-frame $J$-band at $\sim$1.3$\mu$m. VISIR 11.3$\mu$m imaging data ----------------------------- In the near and mid infrared the [HE0450–2958]{} system clearly has a spectral energy distribution (SED) that is composed of more than a single component: In Figure \[fig:iras\_sed\] we model the IRAS and VISIR flux densities with a composite SED of a quasar plus a star forming component. For the quasar we test the median and 68 percentile reddest quasar SED from @elvi94, and for the star forming component an Arp220-like starbursting SED, but we also tried a medium star formation M82 SED, both from @elba02. The median quasar SED plus Arp220 can reproduce the data at all wavlengths, except at observed 25$\mu$m, where it leaves a small mismatch. The 68 percentile reddest SED on the other hand creates a perfect match also there. For both cases the flux predicted for the companion galaxy at 11.3$\mu$m lies below the detection limit as observed. Milder, M82-like star formation can be ruled out on the same basis, as it predicts a detection of the companion also at 11.3$\mu$m – both with the information from the mid infrared, as well as when extrapolating the observed $H$-band flux. @papa08 match a simple model of two black-body emission curves to the four IRAS points, yielding a cool dust component heated by star formation and a warm dust component which can be attributed to intense AGN emission (see Section \[sec:discussion\_ulirg\]). While it is not possible to spatially resolve the system at FIR wavelengths with current telescopes, we aim for the highest wavelength where this is currently possible, in order to localize the warm emission component and test whether this comes solely from the (optically visible) QSO or from extra sources. ![\[fig:iras\_sed\] The SED of [HE0450–2958]{} in the mid-infrared: Shown are the IRAS flux density measurements from @grij87 [*(open circles)*]{}, our VISIR data point [*(filled circle)*]{} and upper limit on the companion galaxy [*(arrow)*]{} and overlaid composite AGN plus starburst SEDs [*(lines)*]{}. For the quasar nucleus we use the median [*(green dashed line)*]{} and 68 percentile reddest SEDs [*(black dashed line)*]{} from @elvi94, the starburst [*(red solid line)*]{} is a model for Arp220 by @elba02. The median quasar plus Arp220 SED [*(green solid line)*]{} can explain the data except for a slightly too low value at the observed 25$\mu$m point, but with the 68 percentile SED [*(black solid line)*]{} the match is perfect. The predicted flux of the companion galaxy where the star formation of the system is located [*(bar)*]{} lies below our detection limit, consistent with the data. Milder star formation templates as e.g. M82 can be ruled out, since they predict too high fluxes for the companion – also from the observed $H$-band data – which should be visible in the VISIR image. ](fig2.eps){width="\columnwidth"} The observations were performed using VISIR, the ESO/VLT mid-infrared imager and spectrograph mounted on unit 3 of the VLT (Melipal). VISIR gives a pixel size of 0075 and a total field-of-view of 192. The diffraction limited resolution is 035 FWHM. Standard “chopping and nodding” mid-infrared observational technique was used to supress the background dominating at these wavelength. All the observations were interlaced with standard star observations of HD 29085 (4.45 Jy) and HD 41047 (7.21 Jy). The estimated sensitivity was 4 mJy/10$\sigma$/1h. Imaging data were obtained on the 12th of December 2005 in service observing mode, through the PAH2 filter centered on 11.3 $\mu$m having a half-band width of 0.6 $\mu$m. Weather conditions were very good, optical seeing was below 1, and the object was observed always at an airmass of 1.15, which resulted in a diffraction limited image of 035 resolution. Chopping/nodding parameters were 8/8and 0.25 Hz/0.033 Hz. The total time spent on-source was 1623 s. The data were reduced using a dedicated pipeline written in IDL, which does the chopping/nodding correction and removes the spurious stripes due to detector instabilities [@pant07]. The reduced data were finally flux-calibrated using the two reference stars as photometric calibrators. The error on the photometry due to variations of the atmospheric transmission are estimated to be less than 2% (3$\sigma$). NICMOS $H$-band imaging data ---------------------------- The ACS $V$-band is too blue to penetrate any substantial amount of dust. With the scenario of a dust enshrouded host galaxy in mind, we acquired new HST NICMOS data (NIC2 with 0075 plate scale) in the F160W $H$-band (program \#10797, cycle 15) to reduce the dust attenuation by a factor of 3.5 in magnitude space. A total of 5204s integration on target was forcedly split into two observation attempts due to telescope problems, and carried out in July 2006 and 2007. These yielded two sets of data with 2602 s integration each, but slightly different orientations. In order to minimize chromatic effects, we also observed a point spread function (PSF) calibrator star (EIS J033259.33–274638.5) with the SED-characteristics over the F160W filter bandpass similar to a mean QSO template. We do not know the actual SED of [HE0450–2958]{} itself, as no NIR imaging or spectroscopic data of the system with high enough spatial resolution exist to date. As the stellar type yielding the likely most similar PSF we found K4III, by comparing the PSFs predicted by the TinyTim package [@kris03]. The only cataloged stars faint enough to not immediately saturate were observed by the ESO Imaging Survey [EIS, @groe02] located in the E-CDFS, and had to be observed at 6 months distance in time to [HE0450–2958]{}. Since we also want to minimize the PSF variation due to differences in observing strategy, we applied the same dither patterns for both [HE0450–2958]{} and the PSF star. Due to the absolute pointing accuracy of HST the centroid location of the star relative to the chip is shifted about 15 pixels (11) from the QSO centroid towards the companion galaxy. Data reduction and combination of the individual frames were carried out using a mix of STScI pipeline data products, pyraf, and our own procedures in MIDAS and Fortran. The resulting image is shown together with the analysis in Figure \[fig:dataimages\]a. Two parts of the team analyzed the combined images in complementary ways, by decomposition of the components using two-dimensional modeling and by image deconvolution. ### Uncertainty in the PSF {#sec:psfuncertainty} In order to detect a putative faint host galaxy underneath the bright QSO nucleus we require a precise knowledge of the PSF. The PSF will vary spatially, with the energy distribution in the filter as well as temporally, with a changing effective focus of the telescope due to changing thermal history. We opt for a double approach: First, we observe the separate PSF star with the properties described in the last section (and see Fig. \[fig:dataimages\]b). Secondly, we also have the foreground star available that is located at 18 distance from the QSO to the north-west. It is classified as a G star [@low89]. Its on-chip distance to the QSO will leave only room for small spatial variations, but its SED in the $H$-band likely will not perfectly match the SED of the QSO. It is difficult to assess the PSF uncertainty at the position of the QSO. In principle we have a combined effect of color, spatial, and temporal variation, but only one bit of information: the difference between the foreground star and the PSF star. We thus model the expected difference in the shape of these two stars with TinyTim and then compare their actual observed shapes. This shows that the foreground star should be slightly narrower than the observed PSF star, which is consistent with PSF star’s later, redder spectral type and an increase of PSF width with wavelength. We observe this effect also in the data, however somewhat stronger. A temporal variation can thus not be separated and ruled out. In any case we conclude that the PSF star is wider and thus will yield more conservative (=fainter) estimates for a QSO host galaxy, while in case of a non-detection the foreground star will yield brighter upper limits. For two-dimensional modeling of the system we use [galfit]{} [@peng02]. In order to quantify the PSF uncertainty for this process, we first let [ galfit]{} fit a single point source, represented by the PSF star, to the foreground star. In this process we use an error map created from the data itself and we add the sky as a free parameter. We minimize the influence of the nearby QSO on the foreground star by first fitting the former with a single point source as well, removing its modeled contribution, and mask out the remaining residuals starting at 09 from the star. The PSF created in this way is shown in Figure \[fig:dataimages\]c. This image is fed into the modeling process of the PSF star, or later the QSO/host/companion system. The residual flux in this process is of the order of 3% of the total, inside the 05 radius aperture where most apparent residuals are located, the absolute value of the residuals in the same region is 14%. This means that it will be generally impossible to detect any host galaxy of less than 3% of the total flux of the QSO, and it will even be difficult to isolate a somewhat brighter smooth galaxy in the non-smooth residuals. This level of residuals is consistent with experience from the HST ACS camera, where we find that due to PSF uncertainties 5% of the total flux are approximate detection limits for faint host galaxies [@jahn04b Jahnke et al. in prep.]. Including the structured PSF residuals we will only consider a host galaxy component as significant if it has clearly more than 3–5% of residual flux inside an 05 radius of the QSO, or that shows up as a non co-centric structure above the noise outside this region. In absolute magnitudes and related to the QSO these limits correspond to the following: inside an 05 radius of the QSO we can hide a galaxy co-centric with the QSO of at least $M_H\sim-24.7$ (for the 3% case) or $M_H\sim-25.2$ (for 5%). Results ======= VISIR ----- We detect a single unresolved point source in the VISIR field-of-view with a flux density 62.5 mJy at observed 11.3$\mu$m (Figure \[fig:allwave\], right column). This compares to 69.3 mJy in the IRAS 12 $\mu$m channel. There is no second source detected in the field down to a point source sensitivity of at least 3 mJy at the 5$\sigma$ level. Extended sources of the visual size of the companion galaxy have a 5$\sigma$ detection limit of 5.5 mJy. With only one source in the total 192 VISIR field three optical sources have in principle to be considered as potential counterparts: The QSO nucleus, the companion galaxy, and the foreground star. However, the star is a G spectral type and can thus be safely ruled out. We find that the initial position of the MIR point source as recorded in the VISIR image header comes to lie between the QSO and the companion, somewhat closer to the QSO. To clarify this we conducted an analysis of the pointing accuracy of VISIR testing the astrometry of a number of reference stars observed with VISIR at different epochs. The two results are: (1) In all cases the offset between targeted and effective RA,Dec is less than 1 rms, but (2) there is a systematic offset of 0.15s in RA recorded in the fits header, so the true positions need to be corrected by –0.15s in RA. This correction places the MIR point source exactly onto the locus of the QSO in the HST ACS images. It is thus clearly the QSO nucleus that is responsible for all of the 11.3 $\mu$m emission. NICMOS ------ ### Host galaxy {#results:host} To extract information on the host galaxy, we use three different methods to remove the flux contribution from the QSO nucleus. First, we make a model-independent test for obvious extended emission: In a simple peak subtraction we remove a PSF from the QSO, scaled to the total flux inside two pixels radius around the QSO center. This is a robust approach that is independent of specific model assumptions and quite insensitive to the noise distribution in the image [@jahn04b]. As a result, the peak subtracted image shows no obvious extended residual, i.e. host galaxy, centered on the QSO, when using the PSF star as PSF. As a second step we use on the one hand [galfit]{} to model the 2-dimensional light distribution of the [HE0450–2958]{} system and decompose it into different morphological components. On the other hand we use the MCS deconvolution method [@maga98] to mathematically deconvolve the system to a well defined and narrower PSF. The procedure we follow is based on the one described in @chan07. For [galfit]{} we use the two empirical PSFs, for MCS deconvolution we construct a number of combinations of empirical PSF and TinyTim models including very red dust-like SED components. While these two approaches are complementary in method, their results agree as can be seen in Figure \[fig:dataimages\]: The inner part of the QSO inside of 05 radius is consistent with a point source within the PSF uncertainties, but there is extra flux present outside of this radius. The structure of the PSF removal or deconvolution residuals points to a substantial mismatch between shape of the QSO nucleus and the separately observed PSF star, but also to too simple models of TinyTim. In order to remove obvious residual PSF structure a very red SED needs to be assumed, which at this point can not be discriminated from a marginally resolved red component on top of the AGN point source. However, in light of the non-average properties of this QSO, a mean QSO SED is also not expected. In the following we present our results in more detail and focus on the [galfit]{} results, since it allows a more direct estimate of the significance of detected structures. A comparison of the original and point source-removed images in the optical and NIR, and the MIR image are shown in Figure \[fig:allwave\]. We use [galfit]{} to perform a number of different model fits. In all of them the companion star and QSO nucleus are described by a pure point source, while the companion galaxy is fit with one or two Sérsic[^1] components with free axis ratio, or left unmodelled. We also attempt to add another Sérsic component for the putative host galaxy. We always leave the Sérsic parameter $n$ free, although the companion galaxy is too complex and the putative host galaxy too faint for $n$ to be interpreted physically. With the PSF star used as PSF [galfit]{} finds a result consistent with the peak subtraction. A positive residual of $H\sim17.7$ inside a $\sim$1 radius aperture has a flux below 2% of the 13.05 mag of the QSO itself (see Figure \[fig:dataimages\]e+f). Even though we choose an aperture larger than in our calculation in Section \[sec:psfuncertainty\], we receive a value far below our significance limit, so no significant co-centered host galaxy is seen in this way. If we use the foreground star as PSF (Fig. \[fig:dataimages\]g) we find – as expected – a residual flux that is slightly higher than before, and consistent values for two different approaches: First, for a pure PSF fit to the QSO location, integrating the flux of the residual within a 1 radius aperture around the QSO, except along the SE–NW-axis where we expect residual flux from foreground star and companion galaxy. Secondly, we get a similar flux for a fitted additional host galaxy Sérsic component. These two approaches yield a magnitude of $H$$\sim$15.8 and 16.2, respectively, for the host, $\sim$1.5mag brighter than for the PSF star fit. $H$$\sim$16 corresponds to $\sim$6% of the 13.05 mag of the QSO nucleus. Again the QSO residual shows substantial structure as reported in Section \[sec:psfuncertainty\]. It consists of nested rings of positive and negative flux, typical signs of a close but different width between the PSF we use and the actual one. The bulk of structure is contained in the innermost 05 radius and contains 2/3 of the residual flux. The remaining residual of 2% of the total flux outside this radius is again insignificant, and no main body of the host galaxy co-centered with the quasar is found which satisfies our significance criterion. Going back to the PSF residuals that we quantified earlier on, we detect no co-centered host galaxy at a level above 3% of the flux of the quasar nucleus, corresponding to an upper limit of $H=16.9$. {width="\textwidth"} However, after removal of the point source, a feature becomes clearer, what we dub the “NE-extension”. This faint structure extends from the QSO to the N–E, and it can be traced starting at the edge of the strong PSF residuals at 06 (2.5 kpc) N–E of the nucleus (Figures \[fig:dataimages\] and \[fig:ne-extension\]). Some signs of it are already visible in the optical, when going back to the the F606W image [@maga05 see also Fig. \[fig:allwave\]], but it is much more pronounced in the new $H$-band data compared to the $V$-band. The NE-extension is possibly part of a tidal arm similar to the arm towards the south of the companion, already described by @cana01, but our $H$-band image shows it to be clearly disjoint from the companion galaxy. Due to its proximity it is very likely associated with the QSO, even though it is clearly not its main body. It is unlikely that the NE-extension is just a gas cloud with star formation induced by the radio jet in the system, since it lies at least 50$^\circ$ from the jet direction [@feai07]. It is also unlikely a chance superposition of a gas cloud with emission line gas, as seen by @leta08, since the observed $H$-band does not contain any strong enough line. The NE-extension contains non-negligible flux far above the noise of the background and is unaffected by QSO residuals and independent of the PSF used. We estimate its brightness at $H=18.8$ using an aperture encompassing all visible extension outside the QSO nucleus residual. The same region in the ACS image has $V=21.6$, so $(V-H)=2.8$ ![\[fig:ne-extension\] A slight zoom into the inner region of [HE0450–2958]{} to show the newly found “NE-extension” of the QSO. We removed the star and the QSO using the PSF star as PSF. An extension to the N–E is visible (marked with [*red brackets*]{}) at a distance of 06–15 that is clearly not due to PSF residuals – a very similar result is seen when using the narrower foreground star as PSF (Fig. \[fig:dataimages\]e), or MCS deconvolution (Fig. \[fig:dataimages\]d). This structure is disjoint from the companion galaxy so very likely belongs to the QSO host galaxy itself. The estimated brightness is $H=18.8$. The image size is 45 on the side. ](fig4.eps){width="\columnwidth"} In summary, we detect no significant host galaxy that is co-centered with the QSO. We conclude this from the size and shape of the residuals underneath the QSO in comparison to the “PSF star minus foreground star” subtraction residuals we discussed above. The NE-extension, however, that can be seen outside of the residuals of the QSO nucleus, is a real and significant emission structure – and it is very likely associated with the main part of the host galaxy. ### Companion galaxy In the ACS $V$-band the companion galaxy located 15 to the S–E appears clumpy, with several bright knots as well as lower surface brightness in the center. @cana01 even call the companion a “collisional ring galaxy”. With the NICMOS $H$-band we get a substantially different picture. The galaxy at $H=15.2$ is still asymmetric, with tidal extensions, but contrary to the visible wavelengths it is smooth and shows a pronounced center: clear signs for substantial dust, distributed not smoothly but unevenly and clumpily, with concentration towards the center that only shows up in the optical (Fig. \[fig:companion\]). The complexity of the companion is manifested in that there is no good description with neither one or two Sersic components, when the azimuthal shape is restricted to ellipses. The Sersic index of the companion is around $n=2$ for a single Sersic component, and $n<1$ if two Sersic components are used. Taken at face value, both cases point to a more disk- than bulge-like companion, but a substantial fraction in flux is containted in the non-symmetric distorted part of the companion – and this should be the main description of the companion. More complex descriptions were put forward, with either a proposed additional faint AGN hosted by the companion galaxy [@leta09], explosive quasar outflows [@lipa09], or quasar-induced star formation [@elba09]. ![\[fig:companion\] Zoom on the companion galaxy. As a difference to the $I$-band (Fig. \[fig:allwave\]) the galaxy has a pronounced peak of emission and no ring. The light in the optical is obviously attenuated by dust, very strongly in the center where the dust is optically thick, less in the outer regions. Image size is 3 on the side. ](fig5.eps){width="\columnwidth"} Discussion ========== Where is the ULIRG? {#sec:discussion_ulirg} ------------------- There was substantial confusion about the source for the ULIRG-strength IRAS MIR and FIR emission in the literature. From the uncorrected \[OII\] line flux a star formation rate (SFR) of only 1 M$_\odot$/yr can be inferred [@kim07]. @maga05 still assign the ULIRG emission to the companion galaxy due to its Balmer decrement which yields non-negligible dust extinction, while @kim07 note that the corrected SFR would still be below 10 M$_\odot$/yr. This number is in strong disagreement with a SFR up to $\sim$800 M$_\odot$/yr inferred from total IR-luminosity or 370 M$_\odot$/yr from CO [@papa08]. The new NICMOS images show that the stars in the companion galaxy are not distributed in a ring, but smoothly (Fig. \[fig:companion\]) and that an optically thick dust creates the ring-like structure in the optical ACS images (Fig. \[fig:allwave\]a+b). This means that at optical wavelengths only information from the less extincted outer regions of the galaxy as well as the surface of the strongly extincted central regions is seen. UV-based SFRs must therefore dramatically underestimate the true SFRs when corrected with dust extinction estimated from (also optical wavelength) Balmer-decrements. The actual scale of the uncertainty in $A_V$, the optical extinction correction, can be estimated by comparing $A_V$ estimates from Balmer lines and Paschen/Bracket lines in other ULIRGs. @dann05 studied five ULIRGS for which they estimated $A_V$ both from H$_\alpha$/H$_\beta$ as well as from Pa$\alpha$/Br$\gamma$. NIR-derived values for A$_V$ were in every case significantly larger, ranging from factors of $\sim$1.16 to $\sim$10 (mean 4.0) times higher. As this factor does not scale in any way with the optical $A_V$ estimate, but only with the NIR estimate, we can not determine a correction for [HE0450–2958]{}. When starting out with the redshifted \[OII\]-line at $\lambda$3727 and $A_\mathrm{3727}=A_V\times1.57$ like @kim07, and the correction factors from @dann05, a huge range of possible star formation rates arises. An average of the two [*lowest*]{} correction values from @dann05 of 1.16 and 1.65 means $A_V\sim 2.1$, $A_\mathrm{3727}\sim3.3$ or corrected SFRs of 21 M$_\odot$/yr. Using their mean correction factor of $\sim$4 would lead to $A_\mathrm{3727}\sim9.4$ or $>$5000 M$_\odot$/yr. So already a number below the mean correction ($A_V\sim3$) would make these numbers consistent with FIR-emission based SFR estimates. This directly shows that optical/UV line-emission based SFRs as used by @kim07 can not at all be used to constrain the true SFR of ULIRGs and does not provide an argument against strong star formation in the companion. @papa08 approximate the IRAS IR SED with a 2-component black-body model and find a cool component $T_\mathrm{dust}^\mathrm{cool}=47$K, dust mass $M_\mathrm{dust}^\mathrm{cool}\sim10^8$ M$_\odot$ and $L_\mathrm{FIR} \sim 2.1\times10^{12}$ L$_\odot$, and a warm component with $T_\mathrm{dust}^\mathrm{warm}=184$K, $M_\mathrm{dust}^\mathrm{warm}\sim5\times10^4$ M$_\odot$ and $L_\mathrm{MIR} \sim 2.6\times10^{12}$ L$_\odot$. We can now for the first time spatially localize the warm component from the detection of the single 11.3$\mu$m point source with VISIR to be coincident with the position of the QSO nucleus. Since the measured flux density is consistent with a warm component having the previously known 12$\mu$m IRAS flux density, we conclude that the QSO nucleus itself already is a ULIRG-level emitter, but with a warmer component compared to star formation. For localizing star formation in the system, there are two recent new datasets available, radio data from @feai07 and the CO maps by @papa08. While the radio maps do not set strong constraints when trying to exploit the radio–FIR relation to assign a location for the FIR emission, the CO data are more powerful: at least the bulk, possibly all of molecular gas and thus star formation activity is located in the companion galaxy. We can add two further constraints from our NICMOS and VISIR images. Both the mid infrared SED of the system (Fig. \[fig:iras\_sed\]) as well as an extrapolation from the $H$-band are consistent with an Arp220-like star formation, while ruling out milder, M82-like conditions. In the latter case the companion would have to be visible in our observed 11.3$\mu$m image, but it is absent (Fig. \[fig:allwave\]). Together with the dense and clumpy dust geometry of the companion when comparing optical and NIR morphology, it becomes clear that the companion is responsible for most, if not all, of the 370 M$_\odot$/yr star formation. If we follow the 5:1 CO detection significance for the companion given by @papa08, this means that as a minimum 5/6=83% of CO are located in the companion and thus also $\ge83$% of the star formation and FIR emission. This number converts to an integrated IR luminosity of $L_\mathrm{FIR} \ge 1.75\times10^{12}$ L$_\odot$, so the companion also qualifies as a ULIRG. While the presence of very strong star formation in the companion is clear now, its trigger is a priori not so clear. The most probably solution is merger induced SF, so the system would be a classical ULIRG – just with a non-standard geometry – but there is room for a radio jet induced effect as well. One of the lobes of the jets from the QSO is located directly at the companion position. If and how much this contributes to star formation in the companion still needs to be quantified. Host galaxy detection {#sec:hostgalaxydetection} --------------------- With the companion identified as the main star-former, we get limits from the CO that less than 1/6 of the total cool dust is located within the putative host galaxy. Thus 1/6 of the FIR-inferred SFR by @papa08 of $\mathrm{SFR} = 1.76\cdot 10^{10} (\mathrm{L_{IR}/L_\odot})$ M$_\odot$/yr correspond to an upper limit of 62 M$_\odot$/yr. This leaves room for a non-negligible amount of SF in the host galaxy, but is also an upper limit[^2]. If we assume the host galaxy to have a mix of old and young stellar population as we find for other QSO host galaxies at these redshifts [@jahn04a; @leta07], we can convert this to an expected $H$-band flux. If the host galaxy had the same population mix as @cana01 modelled for the companion galaxy[^3] – 95.5% of a 10 Gyr old population with 5 Gyr e-folding SFR timescale plus 4.5% of a 128 Myr young population –, this SFR upper limit would translate to an expected NIR magnitude of 1.75 mag fainter than the companion or $H\ge16.95$. The combined color and $K$-correction term is $V-H_\mathrm{z=0.285} = 1.66$, and changes by only about $\pm$0.3mag for a pure old (10 Gyr) or young (100 Myr) population. So they are rather insensitive to the exact choice of stellar population. However, this limit will get brighter if the host galaxy contained less dust – by about 0.3 mag per magnitude decrease in $A_V$. With that in mind, this limit is not more stringent than the limit from NICMOS itself: No significant main host galaxy body is found after PSF removal (Section \[results:host\]) and so an upper limit from the NIR decomposition of $H=16.9$ applies for a host galaxy co-centered with the quasar nucleus. We therefore conclude that the current upper limit from NICMOS lies at around $H\sim16.9$. This is consistent with the CO/FIR limits. How do these numbers relate to the current upper limit for a co-centered host galaxy from the optical HST data? We convert our $H$-band limit to absolute $V$-band magnitudes with again the assumption of the host galaxy having the same stellar population mix as the companion. In the conversion to $M_V$ we assume two different values for dust extinction, (a) $A_V=0$, motivated by the nearly dust-free line of sight to the QSO nucleus, and (b) a moderate $A_V=1$ (corresponding to $A_{H\mathrm{(z=0.285)}}\sim0.29$). This yields host-galaxy upper limits of $M_V>-21.25$ and $>-22.55$, for the cases (a) and (b) respectively. If we convert the @maga05 upper limits to our $h=0.7$ cosmology and assume the same stellar population and dust properties, we receive $M_V>-20.6$ and $>-21.6$, respectively. We note here that this corresponds to a detection limit of only 1.5% of the total quasar flux in the optical. This factor of two is owed to the better determined PSF in the ACS images. This allows @maga05 to set somewhat stricter upper limits for a nucleus co-centered host galaxy component, particularly if a low dust extinction is present. Concerning lower limits to the host galaxy, the NE-extension (Figure \[fig:ne-extension\]) is a structure of real emission that can be traced towards the QSO from $\sim$15 to a radius of 06, where the region of substantial PSF residuals begins. We can not say for sure whether it continues further inward from this position. Signs of this structure are visible in the ACS $V$-band (see Figure \[fig:allwave\], left column) but it is not clear whether the more compact region only $\sim$02 N–E of the nucleus in ACS image is real or an artefact of the deconvolution process. We measured the $(V-H)$-color to be 2.8 outside this region, which is consistent with a stellar population of intermediate age. In the dust-free case this color corresponds to a $\sim$2.1 Gyr old single stellar population [@bruz03 solar metallicity], for $A_V=1.0$ to an age of 800 Myr. This is consistent with stellar material from a host galaxy, e.g. tidally ejected disk stars. We conclude that with its spatial detachment from the companion galaxy this NE-extension is likely a part of the host galaxy, possibly as a tidal extension, but its vicinity to the QSO makes other interpretations less likely. With this interpretation, we receive an $H\le18.8$ [*lower*]{} limit for the host, corresponding to $M_V<-20.4$ ($A_V=0$) or $<-20.7$ ($A_V=1$). If we include this off-center emission to the upper limit of a co-centered host galaxy, we obtain a total host galaxy upper limit of $M_V>-21.2$ and $-22.0$. We thus bracket the host galaxy luminosity in the $V$-band by 0.8 and 1.3 mag or factors of $\sim2$ and $\sim3.5$, respectively. Formally, the CO detection significance and NICMOS give the same limit on a star formation rate of up to $\sim60$ M$_\odot$/yr. If we take into account the stricter ACS $V$-band limits of $M_V>-20.6$ and $>-21.6$, depending on dust cases (a) and (b), these are fainter by 1.3 and 0.6mag than the CO predicted magnitures. Inversely, these reduce the upper limits on star formation to 18 and 35 M$_\odot$/yr, respectively. Beyond $A_V=2$mag the CO and NICMOS limits again become the most stringent. This means that we can not rule out dust obscuration in the host galaxy. At the same time the dust-free line of sight to the quasar nucleus is a strong argument against large amounts of dust, unless a very special geometrical configuration is invoked, while the warm ULIRG emission from the QSO points to dust in the very central few 100 pc. Only better CO limits or a detection of the host galaxy in the NIR will be able to finally resolve this matter. Black hole mass, galaxy luminosity, and the NLSy1 angle ------------------------------------------------------- Black hole mass estimates for [HE0450–2958]{} vary significantly through the literature. The original 8$\times$10$^8$ $M_\odot$ [@maga05] were revised later to a substantially lower value of 4$\times$10$^7$ [@leta07]. Both values are virial estimates based on H$\beta$ width, but while narrow and broad components were separately measured in the former study, the FWHM of the whole line was used in the latter. This revised value is consistent with the independent virial estimate of 6–9$\times$10$^7$ by @merr06, and even with an estimate from X-ray variability, $2^{+7}_{-1.3}$$\times$10$^7$ [@zhou07]. Since the virial estimates agree now, we will adopt the range 4–9$\times$10$^7$ $M_\odot$ for the black hole mass. @merr06 noted the rather narrow broad emission lines of [HE0450–2958]{} and suggested that it should actually be viewed not as a standard QSO but as a higher-$L$ analog of local NLSy1s. If we compare [HE0450–2958]{} with estimates from the literature [@grup04; @ohta07], we find that [HE0450–2958]{} is consistent with the high black hole mass end of the known NLSy1 distribution and does not need to constitute a new “higher-$L$ NLSy1 analog” class of its own. But is it consistent regarding other properties as well? Morphologically, NLSy1 are mostly spirals, often barred, mostly not strongly disturbed [@ohta07]. Since galaxies have increasing bulge mass with increasing black hole mass it is not clear which structural properties to expect and if a merging system like this is consistent with the properties of the local, lower mass NLSy1 population. There is even a debate on how different NLSy1 actually are from normal Seyferts. Recent studies show smaller BH mass differences between normal broad-line Sy1 and NLSy1 when using line dispersions instead of FWHM [@wats07], although a difference might remain. If galaxies with potentially core outflow-affected lines are considered separately, NLSy1 share the same $M_\mathrm{BH}-\sigma_\mathrm{bulge}$-relation with BLSy1, but their accretion rates are confirmed as lying often close to the Eddington limit [@komo07]. If we compute the [HE0450–2958]{} accretion rate – as derived from the $V$-band absolute magnitude of the quasar nucleus ($M_V=-25.75$, recomputed from the HST/ACS data with updated AGN color and $K$-correction) and a bolometric correction of $BC_V\sim8$ [@marc04; @elvi94] – in relation to its Eddington accretion rate, we obtain from $M_\mathrm{BH}=6.5\pm2.5\times10^7$ $M_\odot$ a super-Eddington accretion rate of $L/L_\mathrm{Edd}=6.2^{+3.8}_{-1.8}$. This is consistent with high Eddington ratios observed for NLSy1 [@warn04; @math05a]. ![\[fig:m\_m\] $M_\mathrm{BH}$–$L_\mathrm{bulge}$-relation for inactive galaxies in the local Universe as presented by @tund07, with data from @haer04, @shan04 and @mclu04 [*(black lozenges and lines)*]{}. Overplotted are the upper limits for the host galaxy of [HE0450–2958]{} for the dust-free case by @maga05 from $V$-band imaging [*(small blue arrow)*]{} and with an $A_V=1$ added [*(small red arrow)*]{}, with their original black hole estimate, converted to our cosmology. The [*blue and red rectangles*]{} show the range for black hole mass estimates and our new lower limits for the (total) galaxy luminosity from NICMOS and new upper limits based on the (still better constrained) optical HST data. Note: Here we combined the off-center flux lower limit (NICMOS $H$-band) with the upper limit for a co-centered host galaxy (ACS $V$-band) for a total upper limit. The arrows to the bottom right show the conversion of our $L_\mathrm{galaxy}$ limits to $L_\mathrm{bulge}$ limits for bulge-to-disk ratios of 1:2 and 1:4. Both the dust-free as well as the $A_V=1$ dust case show a galaxy that is absolutely consistent with the black hole mass, even if the bulge-to-disk ratio is accounted for. ](fig6.eps){width="\columnwidth"} With the new data and an explicit assumption/interpretation that the NE-extension is indeed associated with the host galaxy, we can for the first time present a black hole mass for [HE0450–2958]{} and bracketing limits for its host galaxy luminosity. We can thus place [HE0450–2958]{} on the $M_\mathrm{BH}$–$L_\mathrm{bulge}$-relation of active and inactive galaxies, with more than just an upper limit for galaxy luminosity. In Figure \[fig:m\_m\] we show data from @haer04 and others, as collected by @tund07. We overplotted the limits on [HE0450–2958]{} for the two assumptions of dust attenuation strength (Sec. \[sec:hostgalaxydetection\]). This shows that even when applying a sensible conversion factor of 1 to 1/4 (up to 1.5 mag) to convert from total to bulge luminosity, the host of [HE0450–2958]{} will be a perfectly normal galaxy in this parameter space, with a luminosity around the knee of the galaxy luminosity function, $L\sim L^*$. Contrary to the claim by @maga05 it does not deviate substantially from the local $M_\mathrm{BH}$–$L_\mathrm{bulge}$-relation for normal inactive local massive galaxies, mainly due to the revised mass estimate for the black hole. However, this also means that [HE0450–2958]{} does not show a $M_\mathrm{BH}$/$L_\mathrm{bulge}$ different from local broad-line AGN, consistent with being a NLSy1-analog if the @komo07 result is taken as a base. With the normal $M_\mathrm{BH}$/$L_\mathrm{bulge}$-ratio and the fact that we can now rule out huge amounts of obscuring dust around the QSO nucleus, the most likely explanation for the evasive host galaxy is indeed a high $L/L_\mathrm{Edd}$ accretion rate system – a NLSy1 at the high mass end of the normal NLSy1 population. With the current evidence Occam’s Razor favors this explanation over more exotic scenarios as the ejection of the QSO’s black hole in a 3-body interaction or a gravitational recoil event involving the companion galaxy [e.g. @hoff06; @haeh06; @merr06; @bonn07]. However, these scenarios are formally not ruled out even if the upper limit can be pushed down by another $\sim$5 magnitudes. All evidence combined is consistent with a system of a QSO with ULIRG-size IR emission, residing in an $L^*$ host galaxy that is in the process of colliding with a substantially more luminous and possibly more massive companion ULIR-galaxy[^4]. Much deeper high-resolution NIR imaging with a well controlled PSF are the best way to finally find and trace the here predicted host galaxy (bulge) component of [HE0450–2958]{} co-centered with the QSO nucleus and to estimate its luminosity and mass directly. Black hole – galaxy coevolution ------------------------------- Given the black hole mass and Eddington ratio the accretion rate of the BH is 1.4 M$_\odot$/yr. At the same time @papa08 derive a star formation rate from CO of 370 M$_\odot$/yr, predominantly in the companion galaxy. Applying a correction factor of 0.5 for mass returned to the interstellar matter by stellar winds, the stellar mass growth of the whole [HE0450–2958]{} system from star formation is 185 M$_\odot$/yr. The ratio of black hole accretion and stellar mass growth is then 12/185=6.5%, which is substantially higher than the $M_\mathrm{BH}$/$M_\mathrm{bulge}$ relation for local galaxies of 0.14% [@haer04]. We can conclude the following: If activity timescales are identical for star formation and BH accretion, this system grows in black hole mass much more rapidly than the bulge is required to grow to keep the system on the $M_\mathrm{BH}$/$M_\mathrm{bulge}$ relation. This is not possible, since the star formation is taking place in the companion and not the host galaxy. So in any case a potential maintainance of the relation for this system, if actually true, needs to be seen as an integral over more than several 10$^8$ yrs. On the other hand, a gas consumption timescale of 9.5$\times$10$^7$ yrs – if we divide the H$_2$ masses and SF rates derived by @papa08 and account for 50% mass recycling – is possibly longer than the luminous quasar accretion phase. This would add to the requirement, that processes like the tidal forces of the galaxy interaction redistribute mass, adding stars to the bulge of the host galaxy. These were to the larger extent already preexisting in the host galaxies disk or the companion before the interaction and not created only now. The “coevolution” of the host galaxy and its black hole in [HE0450–2958]{} is clearly a two-part process: the build-up of stellar mass and the build-up of black hole and bulge mass. The former will take place on timescales of $>$1 Gyr through star formation, the latter two can “coevolve” if seen as an average over timescales of longer than the BH accretion lifetime, and a few dynamical timescales for redistribution of stellar orbits of, say, $<$500 Myrs. How many [HE0450–2958]{}s are there? ------------------------------------ [HE0450–2958]{} is an unusual object. AGN in ULIRGs are common, but AGN right next to ULIRGs are not, particularly not luminous QSOs with inconspicuous host galaxies next to extreme starformers. So is [HE0450–2958]{} one of a kind or was it just the scarceness of IR imaging with 1 resolution and high-resolution CO maps that prevents us from finding similar objects en masse? In the higher redshift Universe there was a recent report of a very similar system [@youn08]. LH850.02 at $z=3.3$ is the brightest submm galaxy in the Lockman hole. Using the Submillimeter Array, the authors find two components of which one is a ULIRG with intense star formation, while the other component likely harbors an AGN. At $z>2$ however, objects like this might be quite common, since merging rates and gas reservoirs were much larger than today. If there existed a substantial number of similar systems at low redshifts, this would allow to study mechanisms of the high-redshift Universe at much lower distances. We try to estimate the frequency of such systems in the local Universe using the three morphologically best studied samples of quasars at $0.05\la z<0.43$. We deliberately use optically selected quasars only, as they have no bias with respect to frequency of merger signatures or extreme SFRs as IR-selected samples have by construction. In this way statements about the general population are possible. @jahn04a investigated a volume-limited and complete sample of 19 luminous QSOs out to $z=0.2$. While at least five of these QSOs are seen in intermediate and late stages of major mergers, only one, HE1254–0934, is a likely ULIRG[^5], as determined from its IRAS fluxes. It is also among the most distorted systems, with a companion at $\sim$1 distance from the QSO nucleus. The companion is more luminous than the host galaxy, and shows a substantial tidal tail. It looks remarkably similar to [HE0450–2958]{}. The two other samples are not volume-limited samples, so the selection function is unclear – except that these quasars stem from either optical or radio surveys, but not the IR. @floy04 studied the morphologies of two intermediate- and high-luminosity samples of ten radio-quiet and seven radio-loud quasars at $0.29<z<0.43$, using HST-imaging data. Only one of their 17 quasars shows a distorted geometry similar to [HE0450–2958]{}(1237–040 at $z=0.371$) but there exists no information about the total IR emission or star-formation rates. The IRAS flux limits of 200mJy is equivalent to upper limits of $L_\mathrm{ir}\sim6\times10^{12}$ $L_\odot$ at $z=0.37$. ULIRG-strength emission for 1237–040 could have gone unnoticed by IRAS. A recent study by @kim08b determined the morphologies of 45 HST-archived quasars at $z<0.35$. It has one object in common with @floy04 and three objects with @jahn04a. Of their sample, three other objects (HE0354–5500, PG1613+658, PKS2349–01) are clearly merging with a nearby companion, and are likely ULIRGs as judged from their IRAS fluxes. However, only in the case of HE0354–5500 the quasar and companion are still well separated and their envelopes have not yet merged into a common halo. The two other cases are in a very late merger state and star-formation will likely occur all over the system. This adds up to only $\le$3/77 QSOs to possibly be [HE0450–2958]{}-like in the three samples combined. At $\le 4$% such systems are indeed rare in the local Universe. These three quasars however should be investigated in more detail. It needs to be tested how strong their star-formation actually is, where in the system it is localized, and if the separated companion is in any way connected to the AGN-fuelling. If a similar situation as for [HE0450–2958]{}is found, the result can set strong constraints on the ULIRG–AGN evolutionary scenario [@sand96] and the creation mechanisms of AGN at high redshifts. It can contribute to answering the question whether SF-ULIRG activity in AGN systems is an indicator of a specific mechanism of AGN fuelling. Or, if these are just the most gas-rich merger-triggered AGN systems at the top end of SFRs, with a continuous sequence towards less gas-rich merger-triggered AGN systems. The merging–AGN fuelling mechanism could be identical from ULIRGs down to the Seyfert regime, where at some point secular mechanisms become more dominant. Lower SFR systems could just be the consequence of lower gas mass, but this might only mildly impact on the – much smaller – AGN fuelling rate. Conclusions =========== With new NIR and MIR images to spatially resolve the [HE0450–2958]{} system, and in the light of previously existing data, we find: 1. The companion galaxy is covered in optically thick and unevenly distributed dust. This makes it appear as a collisional ring galaxy in the optical, but intrinsically it is smooth and has smooth NIR emission increasing towards a pronounced center. The star formation in the companion is similar to the strong starburst Arp220, while softer M82-like star formation is ruled out. This can reconcile the SFR estimates from the optical and FIR. The companion is a star-formation powered ULIRG. 2. Our MIR image confirms a single warm dust point source at the location of the QSO nucleus. This supports a two component dust SED with the warm component fully associated with the QSO nucleus, which is an AGN-powered ULIRG. 3. A dust-free line of sight to the quasar nucleus is evidence that the host galaxy is not obscured by large amounts of dust. However, the ULIRG-strength warm IR emission by the nucleus and the upper limit on star formation in the host galaxy of substantial 60 M$_\odot$/yr leave room for dust. 4. With $H\ge16.9$ the current NICMOS images do not set stronger upper limits on the host galaxy of [HE0450–2958]{}. The $V$-band, $H$-band, and CO-constraints give $M_V\ge-21.2$ to $M_V\ge-22.0$ depending on the assumed dust masses. 5. Flux in the NE-extension of $H=18.8$ is likely associated with the QSO’s host galaxy. It corresponds to a first lower limit of $M_V<-20.4$ for the host galaxy. With a black hole of $\sim6.5\pm 2.5 \times10^7$ M$_\odot$, an accreting rate of 12 M$_\odot$/yr equal to super-Eddington accretion, $L/L_\mathrm{Edd}=6.2^{+3.8}_{-1.8}$, the host galaxy is consistent with the $M_\mathrm{BH}$–$M_\mathrm{bulge}$-relation for normal galaxies. It is also consistent with [HE0450–2958]{} being a NLSy1 at the high end of the known black hole mass distribution. The reason for the high accretion rate is unclear but could be connected to [HE0450–2958]{} being in an early stage of merging with its gas-rich companion. A more exotic explanation for the system is currently not required by any data, but can in the end only be ruled out with much deeper, high-resolution NIR images to find the main body and bulge of the host galaxy. 6. If host galaxy and black hole in [HE0450–2958]{} are co-evolving according to the local $M_\mathrm{BH}$–$M_\mathrm{bulge}$ relation, it has to occur over longer timescales ($\le$500 Myr) and/or the mass growth for the bulge is predominantly not caused by the current star formation in the system, but by redistribution of preexisting stars. 7. A constellation as in the [HE0450–2958]{} system with separate locations of QSO nucleus and strongly star forming ULIRG companion might be common at $z>2$ where gas masses and merger rates were higher, but at a fraction of $\le$4% it is extremely rare in the local Universe. The authors would like to thank E. F. Bell, A. Martínez Sansigre, H.Dannerbauer, E. Schinnerer, K. Meisenheimer, F. Courbin, P. Magain and H.-R. Klöckner for very fruitful discussions and helpful pointers. Based on observations made with ESO Telescopes at the Paranal Observatory under programme ID 276.B-5011. Also based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program \#10797. This research has made use of the NASA/IPAC Extragalactic Database (NED). KJ acknowledges support through the Emmy Noether Programme of the German Science Foundation (DFG) with grant number JA 1114/3-1. AB is funded by the Deutsches Zentrum für Luft- und Raumfahrt (DLR) under grant 50 OR 0404. VC, Research Fellow, thanks Belgian Funds for Scientific Research. This work was also supported by PRODEX experiment arrangement 90312 (ESA and PPS Science Policy, Belgium). [*Facilities:*]{} , . 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ArXiv

--- abstract: | In this paper, we design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed method has several appealing features including the support of general meshes and high-order, unconditional inf-sup stability, and orders of convergence that match those obtained for Leray–Lions scalar problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau–Yasuda models. Numerical examples complete the exposition.\ **Keywords:** Hybrid High-Order methods, non-Newtonian fluids, power-law, Carreau–Yasuda law, discrete Korn inequality author: - 'Michele Botti[^1]' - 'Daniel Castanon Quiroz [^2]' - 'Daniele A. Di Pietro [^3]' - 'André Harnist [^4]' bibliography: - 'pstokes.bib' title: 'A Hybrid High-Order method for creeping flows of non-Newtonian fluids' --- Introduction {#sec:introduction} ============ In this paper, we design and analyze a Hybrid High-Order (HHO) discretization method for the steady motion of a non-Newtonian, incompressible fluid in the Stokes approximation of small velocities. Notable applications include ice sheet dynamics [@Isaac.Stadler.ea:15], mantle convection [@Schubert.Turcotte.ea:01], chemical engineering [@Ko.Pustejovska.ea:18], and biological fluids rheology [@Lai.Kuei.ea:78; @Galdi.Rannacher.ea:18]. We focus on fluids with shear-rate-dependent viscosity, whose behavior is characterized by a nonlinear strain rate-shear stress function. Physical interpretations and discussions of non-Newtonian fluid models can be found, e.g., in [@Bird.Armstrong.ea:87; @Malek.Rajagopal.ea:95]. Typical examples that are frequently used in the applications include the power-law and Carreau–Yasuda model. The earliest investigations of fluids with shear-dependent viscosities date back to the pioneering work of Ladyzhenskaya [@Ladyzhenskaya:69]. For a detailed mathematical study of the well-posedness and regularity of the continuous problem, see also [@Malek.Rajagopal:05; @Ruzicka.Diening:07; @Diening.Ettwein:08; @Beirao-da-Veiga:09; @Berselli.Ruzicka:20] and references therein. Early results on the numerical analysis of non-Newtonian fluid flow problems were given in [@Sandri:93; @Barrett.Liu:94; @Glowinski.Rappaz:03]. Later, these results were improved in [@Belenki.Berselli.ea:12] and [@Hirn:13] by proving error estimates that are optimal for fluids with shear thinning behavior (described by a power law exponent ${r}\le 2$). In [@Belenki.Berselli.ea:12], the authors considered a conforming inf-sup stable finite element discretization, while in [@Hirn:13] a low-order scheme with local projection stabilization was proposed. In both works, the use of Orlicz functions is instrumental to unify the treatment of the shear thinning and shear thickening cases (also called pseudoplastic and dilatant, respectively; cf. Example \[ex:Carreau–Yasuda\]). More recently, a finite element method based on a four-field formulation of the nonlinear Stokes equations has been analyzed in [@Sandri:14]. Other notable contributions on the numerical approximation of generalized Stokes problems include [@Diening.Kreuzer.ea:13; @Isaac.Stadler.ea:15; @Kreuzer.Suli:16; @Ko.Suli:18]. The main issues to be accounted for in the numerical solution of non-Newtonian fluid flow problems are the presence of local features emerging from the nonlinear strain rate-shear stress relation, the incompressibility condition leading to indefinite systems, the roughly varying model coefficients, and, possibly, complex geometries requiring unstructured and highly-adapted meshes. The HHO method provides several advantages to deal with the complex nature of the problem, such as the support of general polygonal or polyhedral meshes, the possibility to select the approximation order, and unconditional inf-sup stability. Moreover, HHO schemes can be efficiently implemented thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem encountered, e.g., when solving the nonlinear system by the Newton method. Hybrid High-Order methods have been successfully applied to the simulation of incompressible flows of Newtonian fluids governed by the Stokes [@Aghili.Boyaval.ea:15] and Navier–Stokes equations [@Di-Pietro.Krell:18; @Botti.Di-Pietro.ea:19*1], possibly driven by large irrotational volumetric forces [@Di-Pietro.Ern.ea:16; @Castanon-Quiroz.Di-Pietro:20]. Works related to the problem of creeping flows of non-Newtonian fluids are [@Botti.Di-Pietro.ea:17] and [@Di-Pietro.Droniou:17; @Di-Pietro.Droniou:17*1], respectively dealing with nonlinear elasticity and Leray–Lions problems. Going from nonlinear coercive elliptic equations to the nonlinear Stokes system involves additional difficulties arising from the pressure and the divergence constraint. Finally, we mention, in passing, that HHO methods are members of a wider family of polytopal methods that also includes, e.g., Virtual Element methods; cf., e.g., [@Beirao-da-Veiga.Lovadina.ea:17; @Beirao-da-Veiga.Lovadina.ea:18] for their application to Newtonian incompressible flows. The HHO discretization presented in this paper is inspired by the previously mentioned works. It hinges on discontinuous polynomial unknowns on the mesh and on its skeleton, from which discrete differential operators are reconstructed. The reconstruction operators are then used to define a consistency term inspired by the weak formulation of the creeping flow problem and a cleverly designed stabilization term penalizing boundary residuals. We carry out a complete analysis of the proposed method. In particular, under general assumptions on the strain rate-shear stress function, we derive error estimates for the velocity and pressure approximations. The energy-norm error estimate for the velocity given in Theorem \[thm:error.estimate\] is optimal in the sense that it yields the same convergence orders established in [@Di-Pietro.Droniou:17*1 Theorem 7] for the scalar Leray–Lions elliptic problem. A key tool in our analysis is provided by Lemma \[lem:discrete.korn.inequality\], in which we prove a generalization of the discrete Korn inequality of [@Botti.Di-Pietro.ea:19 Lemma 1] to the non-Hilbertian case. The other main contributions are a novel formulation of the requirements on the strain rate-shear stress function allowing a unified treatment of pseudoplastic and dilatant fluids and the identification of a set of general assumptions on the nonlinear stabilization function ensuring the desired consistency properties along with the well-posedness of the discrete problem. The rest of the paper is organized as follows. In Section \[sec:continuous.setting\] we introduce the strong and weak formulations of the nonlinear Stokes problem and present the assumptions on the strain rate-shear stress function. The construction of the HHO discretization is given in Section \[sec:discrete.setting\] by defining the discrete counterparts of the viscous and coupling terms. Section \[sec:discrete.setting\] also contains the proof of the discrete Korn inequality and the discussion on the nonlinear stabilization function. Section \[sec:well-posedness\] establishes the well-posedness of the discrete problem by proving the Hölder continuity and the strong-monotonicity of the viscous term, as well as the inf-sup stability of the pressure-velocity coupling. In Section \[sec:error.estimate\], we show the consistency of the discrete viscous function and coupling bilinear form. These results are then used to prove the error estimate. In Section \[sec:num.res\], we investigate the performance of the method by performing a convergence test with analytical solution on various families of refined meshes. Finally, in Appendix \[sec:properties.stress\] we provide a sufficient condition for the strain rate-shear stress law to fulfil the assumptions presented in Section \[sec:continuous.setting\]. Continuous setting {#sec:continuous.setting} ================== Let $\Omega \subset {\mathbb{R}}^d$, $d\in\{2,3\}$, denote a bounded, connected, polyhedral open set with Lipschitz boundary $\partial\Omega$. We consider a possibly non-Newtonian fluid occupying $\Omega$ and subjected to a volumetric force field $\b f : \Omega \to {\mathbb{R}}^d$. Its flow is governed by the generalized Stokes problem, which consists in finding the velocity field $\b u : \Omega \to {\mathbb{R}}^d$ and the pressure field $p : \Omega \to {\mathbb{R}}$ such that \[eq:stokes.continuous\] $$\begin{aligned} {2} -{\b\nabla{\cdot}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u) + {\b\nabla}p &= \b f &\qquad& \mbox{ in } \Omega, \label{eq:stokes.continuous:momentum} \\ {\b\nabla{\cdot}}\b u &= 0 &\qquad& \mbox{ in } \Omega, \label{eq:stokes.continuous:mass} \\ \b u &= \b 0 &\qquad& \mbox{ on } \partial \Omega, \label{eq:stokes.continuous:bc} \\ \int_\Omega p(\b x){\,\mathrm{d}}\b x &= 0, \label{eq:stokes.continuous:closure} \end{aligned}$$ where ${\b\nabla{\cdot}}$ denotes the divergence operator applied to vector fields, ${\b{\nabla}_{\mathrm{s}}}$ is the symmetric part of the gradient operator ${\b\nabla}$ applied to vector fields, and, denoting by ${{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ the set of square, symmetric, real-valued $d\times d$ matrices, ${\b\sigma}: \Omega \times {{\mathbb{R}}^{d \times d}_{\mathrm{s}}} \to {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ is the strain rate-shear stress law. In what follows, we formulate assumptions on ${\b\sigma}$ that encompass common models for non-Newtonian fluids and state a weak formulation for problem that will be used as a starting point for its discretization. Strain rate-shear stress law {#sec:strain.rate.shear.stress.law} ---------------------------- We define the Frobenius inner product such that, for all $\b\tau= (\tau_{ij})_{1 \le i,j \le d}$ and $\b\eta= (\eta_{ij})_{1 \le i,j \le d}$ in ${{\mathbb{R}}^{d \times d}}$, $\b\tau : \b\eta \coloneqq \sum_{i,j=1}^d \tau_{ij}\eta_{ij}$, and we denote by $|\b\tau|_{d \times d}\coloneqq \sqrt{\b\tau : \b\tau}$ the corresponding norm. \[ass:stress\] Let a real number ${r}\in (1,+\infty)$ be fixed, denote by ${r}' \coloneqq \frac{{r}}{{r}-1} \in (1,+\infty)$ the conjugate exponent of ${r}$, and define the singular exponent of ${r}$ by $$\label{eq:sing} {r^{\circ}}\coloneq \min({r},2) \in (1,2].$$ The strain rate-shear stress law satisfies \[eq:ass:sigma\] $$\begin{gathered} {\b\sigma}(\b x,\b 0) = \b 0 \text{ for almost every } \b x \in \Omega,\label{eq:ass-stress:0} \\ {\b\sigma}: \Omega \times {{\mathbb{R}}^{d \times d}_{\mathrm{s}}} \to {{\mathbb{R}}^{d \times d}_{\mathrm{s}}} \text{ is measurable}.\label{eq:ass-stress:power-framed} \end{gathered}$$ Moreover, there exist real numbers $\sigma_{\mathrm{de}} \in [0,+\infty)$ and $\sigma_{\mathrm{hc}},\sigma_{\mathrm{sm}} \in (0,+\infty)$ such that, for all $\b\tau,\b\eta \in {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ and almost every $\b x \in \Omega$, we have the Hölder continuity property \[eq:power-framed:s.holder.continuity.strong.monotonicity\] $$\begin{aligned} \left| {\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta) \right|_{d\times d} &\le \sigma_{\mathrm{hc}} \left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}+|\b\eta|_{d\times d}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| \b\tau-\b\eta |_{d\times d}^{{r^{\circ}}-1},\label{eq:power-framed:s.holder.continuity} \end{aligned}$$ and the strong monotonicity property $$\begin{aligned} \left({\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\right):(\b\tau-\b\eta) \left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}+|\b\eta|_{d\times d}^{r}\right)^\frac{2-{r^{\circ}}}{{r}} \ge \sigma_{\mathrm{sm}}|\b\tau-\b\eta|_{d\times d}^{{r}+2-{r^{\circ}}}.\label{eq:power-framed:s.strong.monotonicity} \end{aligned}$$ Some remarks are in order. Assumption can be relaxed by taking ${\b\sigma}(\cdot,\b 0) \in L^{{r}'}(\Omega,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$. This modification requires only minor changes in the analysis, not detailed for the sake of conciseness. Inequalities – can be proved starting from the following assumptions, which correspond to the conditions characterizing an ${r}$-power-framed function: For all $\b\tau,\b\eta \in {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ with $\b\tau \neq \b\eta$ and almost every $\b x \in \Omega$, $$\begin{aligned} |{\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)|_{d \times d} &\le \sigma_{\mathrm{hc}} \left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d \times d}^{r}+|\b\eta|_{d \times d}^{r}\right)^\frac{{r}-2}{{r}}| \b\tau-\b\eta |_{d \times d}, \\ \left({\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\right):\left(\b\tau-\b\eta\right) &\ge \sigma_{\mathrm{sm}}\left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d \times d}^{r}+|\b\eta|_{d \times d}^{r}\right)^\frac{{r}-2}{{r}}|\b\tau-\b\eta|_{d \times d}^{2}. \end{aligned}$$ These relations are reminiscent of the ones used in [@Di-Pietro.Droniou:17*1] in the context of scalar Leray–Lions problems. The advantage of assumptions -, expressed in terms of the singular index ${r^{\circ}}$, is that they enable a unified treatment of the cases ${r}< 2$ and ${r}\ge 2$ in the proofs of Lemma \[lem:ah:holder.continuity.strong.monotonicity\], Theorem \[thm:well-posedness\], Lemma \[lem:consistency:ah\], and Theorem \[thm:error.estimate\] below. Inequalities and give $$\label{eq:power-framed:constants.bound} \sigma_{\mathrm{sm}} \leq \sigma_{\mathrm{hc}}.$$ Indeed, let $\b\tau \in {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ be such that $|\b\tau|_{d\times d} > 0$. Using the strong monotonicity (with $\b \eta = \b 0$), the Cauchy–Schwarz inequality, and the Hölder continuity (again with $\b \eta = \b 0$), we infer that $$\begin{aligned} \sigma_{\mathrm{sm}}\left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}\right)^\frac{{r^{\circ}}-2}{{r}}|\b\tau|_{d\times d}^{{r}+2-{r^{\circ}}} &\leq {\b\sigma}(\cdot,\b\tau):\b\tau\leq |{\b\sigma}(\cdot,\b\tau)|_{d\times d}|\b\tau|_{d\times d}\leq \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| \b\tau|_{d\times d}^{{r^{\circ}}} \end{aligned}$$ almost everywhere in $\Omega$. Hence, $\frac{\sigma_{\mathrm{sm}}}{\sigma_{\mathrm{hc}}} \le \left(\frac{\sigma_{\mathrm{de}}^{r}+|\b\tau|_{d\times d}^{r}}{| \b\tau|_{d\times d}^{r}}\right)^\frac{|{r}-2|}{{r}}$. Letting $|\b\tau|_{d\times d} \to +\infty$ gives . \[ex:Carreau–Yasuda\] $(\mu,\delta,a,{r})$-Carreau–Yasuda fluids, introduced in [@Yasuda.Armstrong.Cohen:81] and later generalized in [@Hirn:13 Eq. (1.2)], are fluids for which it holds, for almost every $\b x\in\Omega$ and all $\b\tau \in {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$, $$\label{eq:Carreau--Yasuda} {\b\sigma}(\b x,\b\tau) = \mu(\b x)\left(\delta^{a(\b x)}+|\b\tau|_{d \times d}^{a(\b x)}\right)^\frac{{r}-2}{a(\b x)}\b\tau,$$ where $\mu : \Omega \to [\mu_-,\mu_+]$ is a measurable function with $\mu_-,\mu_+ \in (0,+\infty)$ corresponding to the local flow consistency index, $\delta \in [0,+\infty)$ is the degeneracy parameter, $a : \Omega \to [a_-,a_+]$ is a measurable function with $a_-,a_+ \in (0,+\infty)$ expressing the local transition flow behavior index, and ${r}\in (1,+\infty)$ is the flow behavior index. The Carreau–Yasuda law is a generalization of the Carreau law (corresponding to $a_- = a_+ = 2$) that takes into account the different local levels of flow behavior in the fluid. The degenerate case $\delta=0$ corresponds to the power-law model. Non-Newtonian fluids described by constitutive laws with a $(\mu,\delta,a,{r})$-structure exhibit a different behavior according to the value of ${r}$. If ${r}> 2$, then the fluid shows shear thickening behavior and is called *dilatant*. Examples of dilatant fluids are wet sand and oobleck. The case ${r}< 2$, on the other hand, corresponds to *pseudoplastic* fluids having shear thinning behavior, such as blood. Finally, if ${r}= 2$, then the fluid is Newtonian and becomes the classical (linear) Stokes problem. We show in Appendix \[sec:properties.stress\] that the strain rate-shear stress law is an ${r}$-power-framed function with $\sigma_{\mathrm{de}} = \delta$, $$\sigma_{\mathrm{hc}} = \begin{cases} \frac{\mu_+}{{r}-1}2^{\left[-\left(\frac{1}{a_+}-\frac{1}{{r}}\right)^\ominus-1\right]({r}-2)+\frac{1}{r}} & \text{if } {r}< 2, \\ \mu_+({r}-1)2^{\left(\frac{1}{a_-}-\frac{1}{{r}}\right)^\oplus({r}-2)} & \text{if } {r}\ge 2, \end{cases} \quad\text{and}\quad \sigma_{\mathrm{sm}} = \begin{cases} \mu_-({r}-1)2^{\left(\frac{1}{a_-}-\frac{1}{{r}}\right)^\oplus({r}-2)} & \text{if } {r}\le 2, \\ \frac{\mu_-}{{r}-1}2^{\left[-\left(\frac{1}{a_+}-\frac{1}{{r}}\right)^\ominus-1\right]({r}-2)-1} & \text{if } {r}> 2, \end{cases}$$ where $\xi^\oplus\coloneq\max(0,\xi)$ and $\xi^\ominus\coloneq-\min(0,\xi)$ denote, respectively, the positive and negative parts of a real number $\xi$. As a consequence, it matches Assumption \[ass:stress\]. Weak formulation {#sec:weak.formulation} ---------------- From this point on, we omit both the integration variable and the measure from integrals, as they can be in all cases inferred from the context. We define the following velocity and pressure spaces embedding, respectively, the homogeneous boundary condition for the velocity and the zero-average constraint for the pressure: $$\b U \coloneqq \left\{\b v \in W^{1,{r}}(\Omega,{\mathbb{R}}^d)\ : \ \b v{\ \!\!_{|_{\partial\Omega}}} = \b 0 \right\}, \qquad P \coloneqq L^{{r}'}_0(\Omega,{\mathbb{R}}) \coloneqq \left\{q \in L^{{r}'}(\Omega,{\mathbb{R}})\ : \ \textstyle\int_\Omega q = 0 \right\}.$$ Assuming $\b f \in L^{{r}'}(\Omega,{\mathbb{R}}^d)$, the weak formulation of problem reads: Find $(\b u,p) \in \b U \times P$ such that \[eq:stokes.weak\] $$\begin{aligned} {2} a(\b u,\b v)+b(\b v,p) &= \displaystyle\int_\Omega \b f \cdot \b v &\qquad \forall \b v \in \b U,\label{eq:stokes.weak:momentum} \\ -b(\b u,q) &= 0 &\qquad \forall q \in P, \label{eq:stokes.weak:mass} \end{aligned}$$ where the function $a : \b U \times \b U \to {\mathbb{R}}$ and the bilinear form $b : \b U \times L^{{r}'}(\Omega,{\mathbb{R}}) \to {\mathbb{R}}$ are defined such that, for all $\b v,\b w \in \b U$ and all $q \in L^{{r}'}(\Omega,{\mathbb{R}})$, $$\label{eq:a.b} a(\b w,\b v) \coloneqq \displaystyle\int_\Omega {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) : {\b{\nabla}_{\mathrm{s}}}\b v,\qquad b(\b v,q) \coloneqq -\displaystyle\int_\Omega ({\b\nabla{\cdot}}\b v) q.$$ The test space in can be extended to $L^{{r}'}(\Omega,{\mathbb{R}})$ since, for all $\b v \in \b U$, the divergence theorem and the fact that $\b v{\ \!\!_{|_{\partial\Omega}}} = \b 0$ yield $b(\b v,1) = -\int_\Omega {\b\nabla{\cdot}}\b v = - \int_{\partial\Omega} \b v \cdot \b n_{\partial \Omega} = 0$, with $\b n_{\partial \Omega}$ denoting the unit vector normal to $\partial\Omega$ and pointing out of $\Omega$. \[rem:a-priori\] It can be checked that, under Assumption \[ass:stress\], the continuous problem admits a unique solution $(\b u,p) \in \b U \times P$; see, e.g., [@Hirn:13 Section 2.4], where slightly stronger assumptions are considered. For future use, we also note the following a priori bound on the velocity: $$\label{eq:continuous.solution:bounds:uh} |\b u|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)} \lesssim \left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}-1}+\left(\sigma_{\mathrm{de}}^{2-{r^{\circ}}}\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}+1-{r^{\circ}}}.$$ To prove , use the strong-monotonicity of ${\b\sigma}$, sum written for $\b v=\b u$ to written for $q=p$, and use the Hölder and Korn inequalities to write $$\begin{aligned} \sigma_{\mathrm{sm}}\left( \sigma_{\mathrm{de}}^{r}+ \|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{{r^{\circ}}-2}{{r}} \|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r}+2-{r^{\circ}}} &\lesssim a(\b u,\b u) = \displaystyle\int_\Omega \b f \cdot \b u \lesssim \| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}, \end{aligned}$$ that is, $$\label{eq:continuous.solution:bounds:uh:1} \mathcal{N}\coloneqq \left( \sigma_{\mathrm{de}}^{r}+ \|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{{r^{\circ}}-2}{{r}} \|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r}+1-{r^{\circ}}} \lesssim \sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}.$$ Observing that $\|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r}+1-{r^{\circ}}} \lesssim \max\left(\|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})},\sigma_{\mathrm{de}}\right)^{2-{r^{\circ}}}\mathcal{N} $, we obtain, enumerating the cases for the maximum and summing the corresponding bounds, $\|{\b{\nabla}_{\mathrm{s}}}\b u\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})} \lesssim \mathcal{N}^\frac{1}{{r}-1}+(\sigma_{\mathrm{de}}^{2-{r^{\circ}}}\mathcal{N})^\frac{1}{{r}+1-{r^{\circ}}} $. Combining this inequality with gives . Discrete setting {#sec:discrete.setting} ================ In this section, we recall the notion of polyhedral mesh along with the definitions and properties of $L^2$-orthogonal projectors on local and broken polynomial spaces. Then, after introducing the spaces of discrete unknowns for the velocity and the pressure, we prove a discrete Korn inequality, define the discrete counterparts of the function $a$ and of the bilinear form $b$, and formulate the HHO scheme. Mesh and notation for inequalities up to a multiplicative constant ------------------------------------------------------------------ We define a mesh as a couple $\mathcal M_h\coloneq({\mathcal{T}}_h,{\mathcal{F}}_h)$, where ${\mathcal{T}}_h$ is a finite collection of polyhedral elements $T$ such that $h=\max_{T\in{\mathcal{T}}_h}h_T$ with $h_T$ denoting the diameter of $T$, while ${\mathcal{F}}_h$ is a finite collection of planar faces $F$ with diameter $h_F$. Notice that, here and in what follows, we use the three-dimensional nomenclature also when $d=2$, i.e., we speak of polyhedra and faces rather than polygons and edges. It is assumed henceforth that the mesh $\mathcal M_h$ matches the geometrical requirements detailed in [@Di-Pietro.Droniou:20 Definition 1.7]. In order to have the boundedness property for the interpolator, we additionally assume that the mesh elements are star-shaped with respect to every point of a ball of radius uniformly comparable to the element diameter; see [@Di-Pietro.Droniou:20 Lemma 7.12] for the Hilbertian case. Boundary faces lying on $\partial\Omega$ and internal faces contained in $\Omega$ are collected in the sets ${\mathcal{F}}_h^{\rm b}$ and ${\mathcal{F}}_h^{\rm i}$, respectively. For every mesh element $T\in{\mathcal{T}}_h$, we denote by ${\mathcal{F}}_T$ the subset of ${\mathcal{F}}_h$ containing the faces that lie on the boundary $\partial T$ of $T$. For every face $F \in {\mathcal{F}}_h$, we denote by ${\mathcal{T}}_F$ the subset of ${\mathcal{T}}_h$ containing the one (if $F\in{\mathcal{F}}_h^{\rm b}$) or two (if $F\in{\mathcal{F}}_h^{\rm i}$) elements on whose boundary $F$ lies. For each mesh element $T\in{\mathcal{T}}_h$ and face $F\in{\mathcal{F}}_T$, $\b n_{TF}$ denotes the (constant) unit vector normal to $F$ pointing out of $T$. Our focus is on the $h$-convergence analysis, so we consider a sequence of refined meshes that is regular in the sense of [@Di-Pietro.Droniou:20 Definition 1.9] with regularity parameter uniformly bounded away from zero. The mesh regularity assumption implies, in particular, that the diameter of a mesh element and those of its faces are comparable uniformly in $h$ and that the number of faces of one element is bounded above by an integer independent of $h$. To avoid the proliferation of generic constants, we write henceforth $a\lesssim b$ (resp., $a\gtrsim b$) for the inequality $a\le Cb$ (resp., $a\ge Cb$) with real number $C>0$ independent of $h$, of the constants $\sigma_{\mathrm{de}},\sigma_{\mathrm{hc}},\sigma_{\mathrm{sm}}$ in Assumption \[ass:stress\], and, for local inequalities, of the mesh element or face on which the inequality holds. We also write $a\simeq b$ to mean $a\lesssim b$ and $b\lesssim a$. The dependencies of the hidden constants are further specified when needed. Projectors and broken spaces ---------------------------- Given $X \in {\mathcal{T}}_h \cup {\mathcal{F}}_h$ and $l \in {\mathbb{N}}$, we denote by ${\mathbb{P}}^l(X,{\mathbb{R}})$ the space spanned by the restriction to $X$ of scalar-valued, $d$-variate polynomials of total degree $\le l$. The local $L^2$-orthogonal projector ${\pi_{X}^{l}} : L^{1}(X,{\mathbb{R}}) \to {\mathbb{P}}^l(X,{\mathbb{R}})$ is defined such that, for all $v \in L^{1}(X,{\mathbb{R}})$, $$\label{eq:proj} \displaystyle\int_X ({\pi_{X}^{l}} v-v) w = 0 \qquad \forall w \in {\mathbb{P}}^{l}(X,{\mathbb{R}}).$$ When applied to vector-valued fields in $L^1(X,{\mathbb{R}}^d)$ (resp., tensor-valued fields in $L^1(X,{{\mathbb{R}}^{d \times d}})$), the $L^2$-orthogonal projector mapping on ${\mathbb{P}}^l(X,{\mathbb{R}}^d)$ (resp., ${\mathbb{P}}^l(X,{{\mathbb{R}}^{d \times d}})$) acts component-wise and is denoted in boldface font. Let $T\in{\mathcal{T}}_h$, $n\in[0,l+1]$ and $m\in[0,n]$. The following $(n,{r},m)$-approximation properties of ${\pi_{T}^{l}}$ hold: For any $v\in W^{n,{r}}(T,{\mathbb{R}})$, \[eq:proj:app\] $$\label{eq:proj:app:T} |v-{\pi_{T}^{l}}v|_{W^{m,{r}}(T,{\mathbb{R}})} \lesssim h_T^{n-m}|v|_{W^{n,{r}}(T,{\mathbb{R}})}.$$ The above property will also be used in what follows with ${r}$ replaced by its conjugate exponent ${r}'$. If, additionally, $n\ge 1$, we have the following $(n,{r}')$-trace approximation property: $$\label{eq:proj:app:F} \|v-{\pi_{T}^{l}}v\|_{L^{{r}'}(\partial T,{\mathbb{R}})}\lesssim h_T^{n-\frac{1}{{r}'}}|v|_{W^{n,{r}'}(T,{\mathbb{R}})}.$$ The hidden constants in are independent of $h$ and $T$, but possibly depend on $d$, the mesh regularity parameter, $l$, $n$, and ${r}$. The approximation properties are proved for integer $n$ and $m$ in [@Di-Pietro.Droniou:17 Appendix A.2] (see also [@Di-Pietro.Droniou:20 Theorem 1.45]), and can be extended to non-integer vales using standard interpolation techniques (see, e.g., [@Lions.Magenes:72 Theorem 5.1]). At the global level, for a given integer $l\ge 0$, we define the broken polynomial space ${\mathbb{P}}^l({\mathcal{T}}_h,{\mathbb{R}})$ spanned by functions in $L^1(\Omega,{\mathbb{R}})$ whose restriction to each mesh element $T\in{\mathcal{T}}_h$ lies in ${\mathbb{P}}^l(T,{\mathbb{R}})$, and we define the global $L^2$-orthogonal projector ${\pi_{h}^{l}} : L^{1}(\Omega,{\mathbb{R}}) \to {\mathbb{P}}^l({\mathcal{T}}_h,{\mathbb{R}})$ such that, for all $v \in L^{1}(\Omega,{\mathbb{R}})$ and all $T \in {\mathcal{T}}_h$, $$({\pi_{h}^{l}} v){\ \!\!_{|_{T}}} \coloneq {\pi_{T}^{l}} v{\ \!\!_{|_{T}}}.$$ Broken polynomial spaces are subspaces of the broken Sobolev spaces $$W^{n,{r}}({\mathcal{T}}_h,{\mathbb{R}})\coloneq\left\{ v\in L^{r}(\Omega,{\mathbb{R}})\ : \ v{\ \!\!_{|_{T}}}\in W^{n,{r}}(T,{\mathbb{R}})\quad\forall T\in{\mathcal{T}}_h\right\}.$$ We define the broken gradient operator ${\b\nabla_h}: W^{1,1}({\mathcal{T}}_h,{\mathbb{R}}) \rightarrow L^1(\Omega,{\mathbb{R}}^d)$ such that, for all $v \in W^{1,1}({\mathcal{T}}_h,{\mathbb{R}})$ and all $T \in {\mathcal{T}}_h$, $({\b\nabla_h}v){\ \!\!_{|_{T}}} \coloneq {\b\nabla}v{\ \!\!_{|_{T}}}$. We define similarly the broken gradient acting on vector fields along with its symmetric part ${\b{\nabla}_{{\mathrm{s}},h}}$, as well as the broken divergence operator ${\b\nabla_h \cdot}$ acting on tensor fields. The global $L^2$-orthogonal projector ${\b{\pi}_{h}^{l}}$ mapping vector-valued fields in $L^1(\Omega,{\mathbb{R}}^d)$ (resp., tensor-valued fields in $L^1(\Omega,{{\mathbb{R}}^{d \times d}})$) on ${\mathbb{P}}^l({\mathcal{T}}_h,{\mathbb{R}}^d)$ (resp., ${\mathbb{P}}^l({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})$) is obtained applying ${\pi_{h}^{l}}$ component-wise. Discrete spaces and norms ------------------------- Let an integer $k\ge 1$ be fixed. The HHO space of discrete velocity unknowns is $${{{\und{\b{U}}}}_{h}^{k}} \coloneqq \left\{ {\und{\b{v}}}_h \coloneq ((\b v_T)_{T \in {\mathcal{T}}_h},(\b v_F)_{F\in {\mathcal{F}}_h}) \ : \ \b v_T \in {\mathbb{P}}^k(T,{\mathbb{R}}^d)\ \ \forall T \in {\mathcal{T}}_h\ \mbox{ and }\ \b v_F \in {\mathbb{P}}^k(F,{\mathbb{R}}^d)\ \ \forall F \in {\mathcal{F}}_h \right\}.$$ The interpolation operator ${{\und{\b{I}}}_{h}^{k}} : W^{1,1}(\Omega,{\mathbb{R}}^d) \to {{{\und{\b{U}}}}_{h}^{k}} $ maps a function $\b v \in W^{1,1}(\Omega,{\mathbb{R}}^d)$ on the vector of discrete unknowns ${{\und{\b{I}}}_{h}^{k}}\b v$ defined as follows: $${{\und{\b{I}}}_{h}^{k}} \b v \coloneqq (({\b{\pi}_{T}^{k}} \b v{\ \!\!_{|_{T}}})_{T \in {\mathcal{T}}_h},({\b{\pi}_{F}^{k}} \b v{\ \!\!_{|_{F}}})_{F \in {\mathcal{F}}_h}).$$ For all $T \in {\mathcal{T}}_h$, we denote by ${{{\und{\b{U}}}}_{T}^{k}}$ and ${{\und{\b{I}}}_{T}^{k}}$ the restrictions of ${{\und{\b{I}}}_{h}^{k}}$ and ${{{\und{\b{U}}}}_{h}^{k}}$ to $T$, respectively and, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, we let ${\und{\b{v}}}_T \coloneqq (\b v_T,(\b v_F)_{F\in {\mathcal{F}}_T}) \in {{{\und{\b{U}}}}_{T}^{k}}$ denote the vector collecting the discrete unknowns attached to $T$ and its faces. Furthermore, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, we define the broken polynomial field $\b v_h\in{\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}^d)$ obtained patching element unknowns, that is, $$(\b v_h){\ \!\!_{|_{T}}} \coloneqq \b v_T\qquad\forall T \in {\mathcal{T}}_h.$$ We define on ${{{\und{\b{U}}}}_{h}^{k}}$ the $W^{1,{r}}(\Omega,{\mathbb{R}}^d)$-like seminorm $\| {\cdot} \|_{{\boldsymbol{\varepsilon}},{r},h}$ such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, \[eq:norm.epsilon.r\] $$\begin{gathered} \label{eq:norm.epsilon.r.h} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} \coloneqq \left(\displaystyle\sum_{T \in {\mathcal{T}}_h}\| {\und{\b{v}}}_T \|_{{\boldsymbol{\varepsilon}},{r},T}^{r}\right)^\frac{1}{{r}} \\\label{eq:norm.epsilon.r.T} \text{with $\| {\und{\b{v}}}_T \|_{{\boldsymbol{\varepsilon}},{r},T} \coloneqq \left(\| {\b{\nabla}_{\mathrm{s}}}\b v_T \|^{r}_{L^{r}(T,{{\mathbb{R}}^{d \times d}})} + \displaystyle\sum_{F \in {\mathcal{F}}_T} h_F^{1-{r}} \| \b v_F - \b v_T\|^{r}_{L^{r}(F,{\mathbb{R}}^d)}\right)^\frac{1}{{r}}$ for all $T \in {\mathcal{T}}_h$. } \end{gathered}$$ The following boundedness property for ${{\und{\b{I}}}_{T}^{k}}$ can be proved adapting the arguments of [@Di-Pietro.Droniou:20 Proposition 6.24] and requires the star-shaped assumption on the mesh elements: For all $T \in {\mathcal{T}}_h$ and all $\b v \in W^{1,{r}}(T,{\mathbb{R}}^d)$, $$\label{eq:I:boundedness} \|{{\und{\b{I}}}_{T}^{k}} \b v\|_{{\boldsymbol{\varepsilon}},{r},T} \lesssim | \b v |_{W^{1,{r}}(T,{\mathbb{R}}^d)},$$ where the hidden constant depends only on $d$, the mesh regularity parameter, ${r}$, and $k$. The discrete velocity and pressure are sought in the following spaces, which embed, respectively, the homogeneous boundary condition for the velocity and the zero-average constraint for the pressure: $${{{\und{\b{U}}}}_{h,0}^{k}} \coloneqq \left\{ {\und{\b{v}}}_h = ((\b v_T)_{T \in {\mathcal{T}}_h},(\b v_F)_{F\in {\mathcal{F}}_h}) \in {{{\und{\b{U}}}}_{h}^{k}} \ : \ \b v_F = \b 0 \quad \forall F \in {{\mathcal{F}}_h^{\mathrm{b}}}\right\},\quad {P_{h}^{k}} \coloneqq {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}})\cap P.$$ By the discrete Korn inequality proved in Lemma \[lem:discrete.korn.inequality\] below, $\| {\cdot} \|_{{\boldsymbol{\varepsilon}},{r},h}$ is a norm on ${{{\und{\b{U}}}}_{h,0}^{k}}$ (the proof is obtained reasoning as in [@Di-Pietro.Droniou:20 Corollary 2.16]). Discrete Korn inequality {#sec:discrete.korn.inequality} ------------------------ We prove in this section a discrete counterpart of the following Korn inequality (see [@Geymonat.Suquet:86 Theorem 1]): For all $\b v \in\b U$. $$\label{eq:Korn} \|\b v\|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)} \lesssim \|{\b{\nabla}_{\mathrm{s}}}\b v\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}.$$ We start by recalling a few preliminary results. The first concerns inequalities between sums of powers, and will be often used in what follows without necessarily recalling this fact explicitly each time. Let an integer $n\ge 1$ and a real number $m \in (0,+\infty)$ be given. Then, for all $a_1,\ldots,a_n \in (0,+\infty) $, we have $$\label{eq:sum-power} n^{-(m-1)^\ominus}\sum_{i=1}^n a_i^m \le \left(\sum_{i=1}^n a_i\right)^m \le n^{(m-1)^\oplus}\sum_{i=1}^n a_i^m.$$ If $m=1$, then holds with the equal sign. If $m < 1$, [@Ursell:59 Eqs. (5) and (3)] with $\alpha = 1$ and $\beta = m$ give $ n^{m-1}\sum_{i=1}^n a_i^m \le \left(\sum_{i=1}^n a_i\right)^m \le \sum_{i=1}^n a_i^m. $ If, on the other hand, $m > 1$, [@Ursell:59 Eqs. (3) and (5)] with $\alpha = m$ and $\beta = 1$ give $ \sum_{i=1}^n a_i^m \le \left(\sum_{i=1}^n a_i\right)^m \le n^{m-1}\sum_{i=1}^n a_i^m. $ Gathering the above cases yields . The second preliminary result concerns the node-averaging interpolator. Let $\mathfrak T_h$ be a matching simplicial submesh of $\mathcal M_h$ in the sense of [@Di-Pietro.Droniou:20 Definition 1.8]. The node-averaging operator ${\b{I}_{{\mathrm{av}},h}^{k}} : {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}^d) \to {\mathbb{P}}^k(\mathfrak T_h,{\mathbb{R}}^d) \cap W^{1,{r}}(\Omega,{\mathbb{R}}^d)$ is such that, for all $\b v_h \in {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}^d)$ and all Lagrange node $V$ of $\mathfrak T_h$, denoting by $\mathfrak T_V$ the set of simplices sharing $V$, $$({\b{I}_{{\mathrm{av}},h}^{k}}\b v_h)(V) \coloneq \begin{cases} \frac{1}{\rm{card}(\mathfrak T_V)}\sum_{\b\tau\in\mathfrak T_V} \b v_h{\ \!\!_{|_{\b\tau}}}(V) & \text{if }\ V \in \Omega, \\ \b 0 & \text{if }\ V \in \partial\Omega. \end{cases}$$ For all $F \in {{\mathcal{F}}_h^{\mathrm{i}}}$, denote by $T_1,T_2\in{\mathcal{T}}_h$ the elements sharing $F$, taken in an arbitrary but fixed order. We define the jump operator such that, for any function $\b v\in W^{1,1}({\mathcal{T}}_h,{\mathbb{R}}^d)$, $[\b v]_F \coloneq (\b v{\ \!\!_{|_{T_1}}}){\ \!\!_{|_{F}}}-(\b v{\ \!\!_{|_{T_2}}}){\ \!\!_{|_{F}}}$. This definition is extended to boundary faces $F\in{{\mathcal{F}}_h^{\mathrm{b}}}$ by setting $[\b v]_F \coloneq \b v{\ \!\!_{|_{F}}}$. For all $\b v_h \in {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}^d)$, it holds $$\label{eq:Iav:bound} |\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\lesssim \sum_{F\in{\mathcal{F}}_h} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)}.$$ Combining [@Di-Pietro.Droniou:20 Eq. (4.13)] (which corresponds to for ${r}=2$) with the local Lebesgue embeddings of [@Di-Pietro.Droniou:20 Lemma 1.25] (see also [@Di-Pietro.Droniou:17 Lemma 5.1]) gives, for any $T\in{\mathcal{T}}_h$, $$\label{eq:Iav:bound:0} \|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h\|_{L^{r}(T,{\mathbb{R}}^d)}^{r}\lesssim \sum_{F\in{\mathcal{F}}_{\mathcal V,T}}h_F \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)},$$ where ${\mathcal{F}}_{\mathcal V,T}$ collects the faces whose closure has non-empty intersection with $\overline{T}$. Using the local inverse inequality of [@Di-Pietro.Droniou:20 Lemma 1.28] (see also [@Di-Pietro.Droniou:17 Eq. (A.1)]) we can write $$\begin{aligned} |\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}&\lesssim \sum_{T\in{\mathcal{T}}_h} h_T^{-{r}}\|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h\|_{L^{r}(T,{\mathbb{R}}^d)}^{r}\\ &\lesssim \sum_{T\in{\mathcal{T}}_h}\sum_{F\in{\mathcal{F}}_{\mathcal V,T}} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)} \\ &\lesssim \sum_{F\in{\mathcal{F}}_h} \sum_{T\in{\mathcal{T}}_{\mathcal V,F}} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)} \\ &\le\max_{F\in{\mathcal{F}}_h}\operatorname{card}({\mathcal{T}}_{\mathcal V,F}) \sum_{F\in{\mathcal{F}}_h} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)}, \end{aligned}$$ where we have used the fact that $h_T^{-{r}} \le h_F^{-{r}}$ along with inequality to pass to the second line, and we have exchanged the sums after setting ${\mathcal{T}}_{\mathcal V,F} \coloneqq \big\{T \in {\mathcal{T}}_h : \overline F \cap \overline T \neq \emptyset\big\}$ for all $F \in {\mathcal{F}}_h$ to pass to the third line. Observing that $\max_{F\in{\mathcal{F}}_h}\operatorname{card}({\mathcal{T}}_{\mathcal V,F}) \lesssim 1$ (since, for any $F\in{\mathcal{F}}_h$, $\operatorname{card}({\mathcal{T}}_{\mathcal V,F})$ is bounded by the left-hand side of [@Di-Pietro.Droniou:20 Eq. (4.23)] written for any $T\in{\mathcal{T}}_h$ to which $F$ belongs), follows. (Discrete Korn inequality)\[lem:discrete.korn.inequality\] We have, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, $$\label{eq:discrete.Korn} \| \b v_h \|_{L^{r}(\Omega,{\mathbb{R}}^d)}^{r}+|\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\lesssim \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}.$$ Let ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$. Using a triangle inequality followed by , we can write $$\begin{aligned} |\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}&\lesssim |{\b{I}_{{\mathrm{av}},h}^{k}} \b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}+|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\\ &\lesssim \|{\b{\nabla}_{\mathrm{s}}}({\b{I}_{{\mathrm{av}},h}^{k}} \b v_h) \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\\ &\lesssim \|{\b{\nabla}_{{\mathrm{s}},h}}\b v_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+|\b v_h-{\b{I}_{{\mathrm{av}},h}^{k}}\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\\ &\lesssim \|{\b{\nabla}_{{\mathrm{s}},h}}\b v_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\sum_{F\in{\mathcal{F}}_h} h_F^{1-{r}} \|[\b v_h]_F\|^{r}_{L^{r}(F,{\mathbb{R}}^d)}, \end{aligned}$$ where we have used the continuous Korn inequality to pass to the second line, we have inserted $\pm{\b{\nabla}_{{\mathrm{s}},h}}\b v_h$ into the first norm and used a triangle inequality followed by to pass to the third line, and we have invoked the bound to conclude. Observing that, for any $F\in{\mathcal{F}}_h$, $|[\b v_h]_F| \leq \sum_{T\in{\mathcal{T}}_F}|\b v_F-\b v_T|$ by a triangle inequality, and using , we can continue writing $$|\b v_h|_{W^{1,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{r}\lesssim \|{\b{\nabla}_{{\mathrm{s}},h}}\b v_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\sum_{F\in{\mathcal{F}}_h}\sum_{T\in{\mathcal{T}}_F}h_F^{1-{r}} \| \b v_F - \b v_T\|^{r}_{L^{r}(F,{\mathbb{R}}^d)} = \|{\und{\b{v}}}_h\|_{{\boldsymbol{\varepsilon}},{r},h}^{r},$$ where we have exchanged the sums over faces and elements and recalled definition to conclude. This proves the bound for the second term in the left-hand side of . Combining this result with the global discrete Sobolev embeddings of [@Di-Pietro.Droniou:17 Proposition 5.4] yields the bound for the first term in . Viscous term ------------ ### Local symmetric gradient reconstruction For all $T \in {\mathcal{T}}_h$, we define the local symmetric gradient reconstruction ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} : {{{\und{\b{U}}}}_{T}^{k}} \to {\mathbb{P}}^{k}(T,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$ such that, for all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:G} \displaystyle\int_T {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} {\und{\b{v}}}_T : \b\tau = \int_T {\b{\nabla}_{\mathrm{s}}}\b v_T : \b\tau + \sum_{F \in {\mathcal{F}}_T} \int_F (\b v_F-\b v_T)\cdot (\b\tau \b n_{TF})\qquad \forall \b\tau \in {\mathbb{P}}^{k}(T,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}}).$$ This symmetric gradient reconstruction, originally introduced in [@Botti.Di-Pietro.ea:17 Section 4.2], is designed so that the following relation holds (see, e.g., [@Botti.Di-Pietro.ea:17*1 Proposition 5] or [@Di-Pietro.Droniou:20 Section 7.2.5]): For all $\b v\in W^{1,1}(T,{\mathbb{R}}^d)$, $$\label{eq:G:proj} {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} ({{\und{\b{I}}}_{T}^{k}} \b v) = {\b{\pi}_{T}^{k}}({\b{\nabla}_{\mathrm{s}}}\b v).$$ The global symmetric gradient reconstruction ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} : {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{P}}^{k}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$ is obtained patching the local contributions, that is, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:Gh} ({\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{v}}}_h){\ \!\!_{|_{T}}} \coloneq {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} {\und{\b{v}}}_T \qquad \forall T\in{\mathcal{T}}_h.$$ ### Discrete viscous function The discrete counterpart of the function $a$ defined by is the function ${\mathrm{a}}_h : {{{\und{\b{U}}}}_{h}^{k}} \times {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{R}}$ such that, for all ${\und{\b{v}}}_h,{\und{\b{w}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:ah} {\mathrm{a}}_h({\und{\b{w}}}_h, {\und{\b{v}}}_h) \coloneqq \displaystyle\int_\Omega {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{w}}}_h): {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{v}}}_h +\gamma {\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h).$$ In the above definition, recalling , $\gamma$ is a stabilization parameter such that $$\label{eq:gamma} \gamma \in [\sigma_{\mathrm{sm}},\sigma_{\mathrm{hc}}],$$ while the stabilization function ${\mathrm{s}}_h : {{{\und{\b{U}}}}_{h}^{k}} \times {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{R}}$ is such that, for all ${\und{\b{v}}}_h,{\und{\b{w}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:sh} {\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h) \coloneqq \displaystyle\sum_{T \in {\mathcal{T}}_h}{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{v}}}_T),$$ where the local contributions are assumed to satisfy the following assumption. \[ass:sT\] For all $T \in {\mathcal{T}}_h$, the local stabilization function ${\mathrm{s}}_T:{{{\und{\b{U}}}}_{T}^{k}}\times{{{\und{\b{U}}}}_{T}^{k}}\to{\mathbb{R}}$ is linear in its second argument and satisfies the following properties, with hidden constants independent of both $h$ and $T$: \[eq:ass:sT\] 1. \[ass:sT:stability.boundedness\] *Stability and boundedness.* Recalling the definition of the local $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},T}$-seminorm, for all ${\und{\b{v}}}_T\in{{{\und{\b{U}}}}_{T}^{k}}$ it holds: $$\label{eq:sT:stability.boundedness} \| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}}{\und{\b{v}}}_T \|_{L^{r}(T,{{\mathbb{R}}^{d \times d}})}^{r}+ {\mathrm{s}}_T({\und{\b{v}}}_T,{\und{\b{v}}}_T) \simeq \|{\und{\b{v}}}_T\|_{{\boldsymbol{\varepsilon}},{r},T}^{r}.$$ 2. *Polynomial consistency.*\[ass:sT:polynomial.consistency\] For all $\b w\in{\mathbb{P}}^{k+1}(T,{\mathbb{R}}^d)$ and all ${\und{\b{v}}}_T\in{{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:sT:polynomial.consistency} {\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{\und{\b{v}}}_T) = 0.$$ 3. *Hölder continuity.*\[ass:sT:holder-continuity\] For all ${\und{\b{u}}}_T, {\und{\b{v}}}_T, {\und{\b{w}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, it holds, setting ${\und{\b{e}}}_T\coloneq{\und{\b{u}}}_T - {\und{\b{w}}}_T$, $$\label{eq:sT:holder-continuity} \hspace{-0.5cm} \left|{\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{v}}}_T)-{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{v}}}_T)\right| \lesssim \left({\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{u}}}_T)+{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{w}}}_T)\right)^\frac{{r}-{r^{\circ}}}{{r}}{\mathrm{s}}_T({\und{\b{e}}}_T,{\und{\b{e}}}_T)^\frac{{r^{\circ}}-1}{{r}}{\mathrm{s}}_T({\und{\b{v}}}_T,{\und{\b{v}}}_T)^\frac{1}{{r}}.$$ 4. *Strong monotonicity.*\[ass:sT:strong-monotonicity\] For all ${\und{\b{u}}}_T, {\und{\b{w}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$ , it holds, setting again ${\und{\b{e}}}_T\coloneq{\und{\b{u}}}_T - {\und{\b{w}}}_T$, $$\label{eq:sT:strong-monotonicity} \left({\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{e}}}_T) - {\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{e}}}_T)\right)\left( {\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{u}}}_T)+{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{w}}}_T)\right)^\frac{2-{r^{\circ}}}{{r}} \gtrsim {\mathrm{s}}_T({\und{\b{e}}}_T,{\und{\b{e}}}_T)^\frac{{r}+2-{r^{\circ}}}{{r}}.$$ If ${r}=2$, ${\mathrm{s}}_T$ can be any symmetric bilinear form satisfying [[(S\[ass:sT:stability.boundedness\])]{}]{}–[[(S\[ass:sT:polynomial.consistency\])]{}]{}. Indeed, property [[(S\[ass:sT:holder-continuity\])]{}]{} coincides in this case with the Cauchy–Schwarz inequality, while, by linearity of ${\mathrm{s}}_T$, property [[(S\[ass:sT:strong-monotonicity\])]{}]{} holds with the equal sign. \[lem:sT:consist\] For any $T\in{\mathcal{T}}_h$ and any ${\mathrm{s}}_T$ satisfying Assumption \[ass:sT\], it holds, for all $\b w \in W^{k+2,{r}}(T,{\mathbb{R}}^d)$ and all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:sT:consist} |{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{\und{\b{v}}}_T)| \lesssim h_T^{(k+1)({r^{\circ}}-1)}| \b w |_{W^{1,{r}}(T,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w|_{W^{k+2,{r}}(T,{\mathbb{R}}^d)}^{{r^{\circ}}-1}\| {\und{\b{v}}}_T \|_{{\boldsymbol{\varepsilon}},{r},T},$$ where the hidden constant is independent of $h$, $T$, and $\b w$. The proof adapts the arguments of [@Di-Pietro.Droniou:20 Propositon 2.14]. Using the polynomial consistency property [[(S\[ass:sT:polynomial.consistency\])]{}]{}, we can write $$\begin{aligned} |{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{\und{\b{v}}}_T)| &= |{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{\und{\b{v}}}_T)-{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}}({\b{\pi}_{T}^{k+1}} \b w),{\und{\b{v}}}_T)| \\ &\lesssim {\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}} \b w,{{\und{\b{I}}}_{T}^{k}} \b w)^\frac{{r}-{r^{\circ}}}{{r}}{\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}}(\b w-{\b{\pi}_{T}^{k+1}} \b w),{{\und{\b{I}}}_{T}^{k}}(\b w-{\b{\pi}_{T}^{k+1}} \b w))^\frac{{r^{\circ}}-1}{{r}}{\mathrm{s}}_T({\und{\b{v}}}_T,{\und{\b{v}}}_T)^\frac{1}{{r}} \\ &\lesssim \|{{\und{\b{I}}}_{T}^{k}} \b w\|_{{\boldsymbol{\varepsilon}},{r},T}^{{r}-{r^{\circ}}}\|{{\und{\b{I}}}_{T}^{k}}(\b w-{\b{\pi}_{T}^{k+1}} \b w)\|_{{\boldsymbol{\varepsilon}},{r},T}^{{r^{\circ}}-1}\|{\und{\b{v}}}_T\|_{{\boldsymbol{\varepsilon}},{r},T}\\ &\lesssim | \b w |_{W^{1,{r}}(T,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w-{\b{\pi}_{T}^{k+1}} \b w|_{W^{1,{r}}(T,{\mathbb{R}}^d)}^{{r^{\circ}}-1}\|{\und{\b{v}}}_T\|_{{\boldsymbol{\varepsilon}},{r},T}\\ &\lesssim h_T^{(k+1)({r^{\circ}}-1)}| \b w |_{W^{1,{r}}(T,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w|_{W^{k+2,{r}}(T,{\mathbb{R}}^d)}^{{r^{\circ}}-1}\|{\und{\b{v}}}_T\|_{{\boldsymbol{\varepsilon}},{r},T}, \end{aligned}$$ where we have used the Hölder continuity [[(S\[ass:sT:holder-continuity\])]{}]{} and observed that, by the consistency property [[(S\[ass:sT:polynomial.consistency\])]{}]{}, ${\mathrm{s}}_T({{\und{\b{I}}}_{T}^{k}}({\b{\pi}_{T}^{k+1}} \b w),{{\und{\b{I}}}_{T}^{k}}({\b{\pi}_{T}^{k+1}} \b w))=0$ to pass to the second line, we have used the boundedness property [[(S\[ass:sT:stability.boundedness\])]{}]{} to pass to the third line, the boundedness of ${{\und{\b{I}}}_{T}^{k}}$ to pass to the fourth line, and the $(k+2,{r},1)$-approximation property of ${\b{\pi}_{T}^{k+1}}$ to conclude. In what follows, we will need generalized versions of the continuous and discrete Hölder inequalities, recalled hereafter for the sake of convenience. Let $X \subset {\mathbb{R}}^d$ be measurable, $n \in {\mathbb{N}}^*$, and let $t,p_1,\ldots,p_n \in (0,+\infty\rbrack$ be such that $\sum_{i=1}^n\frac{1}{p_i} = \frac{1}{t}$. The continuous $(t;p_1,\ldots,p_n)$-Hölder inequality reads: For any $(f_1,\ldots,f_n) \in \bigtimes_{i=1}^n L^{p_i}(X,{\mathbb{R}})$, $$\label{eq:holder} \left\| \prod_{i=1}^n f_i \right\|_{L^t(X,{\mathbb{R}})} \le\ \prod_{i=1}^n\| f_i \|_{L^{p_i}(X,{\mathbb{R}})}.$$ Let $m \in {\mathbb{N}}^*$. For all $f : \{1,\ldots,m\} \to {\mathbb{R}}$ and all $q \in [1,+\infty)$, setting $\|f\|_q \coloneqq \left(\sum_{i=1}^m |f(i)|^q\right)^\frac{1}{q}$, and $\|f\|_\infty \coloneqq \max_{1 \le i \le m}|f(i)|$, the discrete $(t;p_1,\ldots,p_n)$-Hölder inequality reads: For any $f_1,\ldots,f_n : \{1,\ldots,m\} \to {\mathbb{R}}$, $$\label{eq:discrete.holder} \left\| \prod_{i=1}^n f_i \right\|_{t} \le\ \prod_{i=1}^n\| f_i \|_{p_i}.$$ Let ${\mathrm{s}}_h$ be given by with, for all $T\in{\mathcal{T}}_h$, ${\mathrm{s}}_T$ satisfying Assumption \[ass:sT\]. Then it holds, for all ${\und{\b{v}}}_h \in{{{\und{\b{U}}}}_{h}^{k}}$, \[eq:sh:properties\] $$\label{eq:sh:stability.boundedness} \| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+ {\mathrm{s}}_h({\und{\b{v}}}_h,{\und{\b{v}}}_h) \simeq \|{\und{\b{v}}}_h\|_{{\boldsymbol{\varepsilon}},{r},h}^{r}.$$ Furthermore, for all ${\und{\b{u}}}_h, {\und{\b{v}}}_h,{\und{\b{w}}}_h\in{{{\und{\b{U}}}}_{h}^{k}}$ it holds, setting ${\und{\b{e}}}_h\coloneq{\und{\b{u}}}_h - {\und{\b{w}}}_h$, $$\begin{gathered} \label{eq:sh:holder-continuity} \left|{\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)-{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h)\right| \lesssim \left({\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{u}}}_h)+{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{w}}}_h)\right)^\frac{{r}-{r^{\circ}}}{{r}}{\mathrm{s}}_h({\und{\b{e}}}_h,{\und{\b{e}}}_h)^\frac{{r^{\circ}}-1}{{r}}{\mathrm{s}}_h({\und{\b{v}}}_h,{\und{\b{v}}}_h)^\frac{1}{{r}}, \\\label{eq:sh:strong-monotonicity} \left( {\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h) - {\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{e}}}_h) \right)\left( {\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{u}}}_h)+{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{w}}}_h) \right)^\frac{2-{r^{\circ}}}{{r}} \gtrsim {\mathrm{s}}_h({\und{\b{e}}}_h,{\und{\b{e}}}_h)^\frac{{r}+2-{r^{\circ}}}{{r}} . \end{gathered}$$ Finally, for any $\b w \in \b U\cap W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)$, it holds $$\label{eq:sh:consist} \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1}{\mathrm{s}}_h({{\und{\b{I}}}_{h}^{k}} \b w,{\und{\b{v}}}_h) \lesssim h^{(k+1)({r^{\circ}}-1)}| \b w |_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1}.$$ Above, the hidden constants are independent of $h$ and of the arguments of ${\mathrm{s}}_h$. For the sake of conciseness, we only sketch the proof and leave the details to the reader. Summing over $T \in {\mathcal{T}}_h$ immediately yields . The Hölder continuity property follows applying to the quantity in the left-hand side triangle inequalities, using , and concluding with a discrete $(1;\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1},{r})$-Hölder inequality. Moving to , starting from $|{\mathrm{s}}_h({\und{\b{e}}}_h,{\und{\b{e}}}_h)|$, we use and apply a discrete $(1;\frac{{r}+2-{r^{\circ}}}{2-{r^{\circ}}},\frac{{r}+2-{r^{\circ}}}{{r}})$-Hölder inequality to conclude. Finally, to prove we start from ${\mathrm{s}}_h({\und{\b{I}}}_h^k\b w,{\und{\b{v}}}_h)$, expand this quantity according to , use, for all $T \in {\mathcal{T}}_h$, the local consistency property together with $h_T \le h$, invoke the discrete $(1;\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1},{r})$-Hölder inequality, and pass to the supremum to conclude. ### An example of viscous stabilization function Taking inspiration from the scalar case (cf., e.g., [@Di-Pietro.Droniou:17 Eq. (4.11c)]), a local stabilization function that matches Assumption \[ass:sT\] can be obtained setting, for all ${\und{\b{v}}}_T,{\und{\b{w}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:sT} {\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{v}}}_T) \coloneqq \int_{\partial T} |{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T \cdot {\b{\Delta}^{k}_{\partial T}} {\und{\b{v}}}_T,$$ where, denoting by ${\mathbb{P}}^k({\mathcal{F}}_T,{\mathbb{R}}^d)$ the space of vector-valued broken polynomials of total degree $\le k$ on ${\mathcal{F}}_T$, the boundary residual operator ${\b{\Delta}^{k}_{\partial T}} : {{{\und{\b{U}}}}_{T}^{k}} \to {\mathbb{P}}^k({\mathcal{F}}_T,{\mathbb{R}}^d)$ is such that, for all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$({\b{\Delta}^{k}_{\partial T}} {\und{\b{v}}}_T){\ \!\!_{|_{F}}} \coloneq h_F^{-\frac1{{r}'}}\left( {\b{\pi}_{F}^{k}}({\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T-\b v_F)-{\b{\pi}_{T}^{k}}({\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T-\b v_T) \right) \qquad\forall F\in{\mathcal{F}}_T,$$ with velocity reconstruction ${\b{{\mathrm{r}}}^{k+1}_{T}} : {{{\und{\b{U}}}}_{T}^{k}} \to {\mathbb{P}}^{k+1}(T,{\mathbb{R}}^d)$ such that $$\begin{gathered} \displaystyle\int_T ({\b{\nabla}_{\mathrm{s}}}{\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T - {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}} {\und{\b{v}}}_T) : {\b{\nabla}_{\mathrm{s}}}\b w = 0\qquad\forall\b w \in {\mathbb{P}}^{k+1}(T,{\mathbb{R}}^d), \\ \text{ $\displaystyle\int_T {\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T = \int_T \b v_T$, and $\int_T {\b{\nabla}_{\mathrm{ss}}}{\b{{\mathrm{r}}}^{k+1}_{T}} {\und{\b{v}}}_T = \frac{1}{2}\sum_{F \in {\mathcal{F}}_T} \int_F ({\b v_F \otimes \b n_{TF}} - {\b n_{TF} \otimes \b v_F})$. }\end{gathered}$$ Above, ${\b{\nabla}_{\mathrm{ss}}}$ denotes the skew-symmetric part of the gradient operator ${\b\nabla}$ applied to vector fields and ${ \otimes }$ is the tensor product such that, for all $\b x = (x_i)_{1 \le i \le d}$ and $\b y = (y_i)_{1 \le i \le d}$ in ${\mathbb{R}}^d$, ${\b x \otimes \b y} \coloneq (x_i y_j)_{1 \le i,j \le d} \in {{\mathbb{R}}^{d \times d}}$. \[lem:sT\] The local stabilization function defined by satisfies Assumption \[ass:sT\]. The proof of [[(S\[ass:sT:stability.boundedness\])]{}]{} for ${r}=2$ is given in [@Botti.Di-Pietro.ea:17 Eq. (25)]. The result can be generalized to ${r}\neq 2$ using the same arguments of [@Di-Pietro.Droniou:17 Lemma 5.2]. Property [[(S\[ass:sT:polynomial.consistency\])]{}]{} is an immediate consequence of the fact that ${\b{\Delta}^{k}_{\partial T}}({{\und{\b{I}}}_{T}^{k}}\b w) = \b 0$ for any $\b w\in{\mathbb{P}}^{k+1}(T,{\mathbb{R}}^d)$, which can be proved reasoning as in [@Di-Pietro.Droniou:20 Proposition 2.6]. Let us prove [[(S\[ass:sT:holder-continuity\])]{}]{}. First, we remark that, since the function $\alpha \mapsto \alpha^{{r}-2}$ verifies the conditions in , we can apply Theorem \[thm:1d.power-framed\] to infer that the function ${\mathbb{R}}^d\ni\b x \mapsto |\b x|^{{r}-2}\b x$ satisfies for all $\b x,\b y \in {\mathbb{R}}^d$, \[eq:rho.holder.strong\] $$\begin{gathered} \big| |\b x|^{{r}-2}\b x-|\b y|^{{r}-2}\b y \big| \lesssim \big( |\b x|^{r}+|\b y|^{r}\big)^\frac{{r}-{r^{\circ}}}{{r}}|\b x-\b y|^{{r^{\circ}}-1},\label{eq:rho:s.holder.continuity} \\ \big( |\b x|^{{r}-2}\b x-|\b y|^{{r}-2}\b y \big) \cdot (\b x-\b y) \big( |\b x|^{r}+|\b y|^{r}\big)^\frac{2-{r^{\circ}}}{{r}} \gtrsim |\b x-\b y|^{{r}+2-{r^{\circ}}}.\label{eq:rho:s.strong.monotonicity} \end{gathered}$$ Recalling , we can write $$\begin{aligned} \left|{\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{v}}}_T)-{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{v}}}_T)\right| &\leq \int_{\partial T} \left||{\b{\Delta}^{k}_{\partial T}} {\und{\b{u}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}} {\und{\b{u}}}_T-|{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T\right|| {\b{\Delta}^{k}_{\partial T}} {\und{\b{v}}}_T|\\ &\lesssim \int_{\partial T} \left(|{\b{\Delta}^{k}_{\partial T}} {\und{\b{u}}}_T|^{r}+|{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T|^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| {\b{\Delta}^{k}_{\partial T}} {\und{\b{e}}}_T|^{{r^{\circ}}-1}| {\b{\Delta}^{k}_{\partial T}} {\und{\b{v}}}_T|\\ &\le \left({\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{u}}}_T)+{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{w}}}_T)\right)^\frac{{r}-{r^{\circ}}}{{r}}{\mathrm{s}}_T({\und{\b{e}}}_T,{\und{\b{e}}}_T)^\frac{{r^{\circ}}-1}{{r}}{\mathrm{s}}_T({\und{\b{v}}}_T,{\und{\b{v}}}_T)^\frac{1}{{r}}, \end{aligned}$$ where we have used to pass to the second line and the $(1;\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1},{r})$-Hölder inequality to conclude. Moving to [[(S\[ass:sT:strong-monotonicity\])]{}]{}, and the $(1;\frac{{r}+2-{r^{\circ}}}{2-{r^{\circ}}},\frac{{r}+2-{r^{\circ}}}{{r}})$-Hölder inequality yield $$\begin{aligned} &{\mathrm{s}}_T({\und{\b{e}}}_T,{\und{\b{e}}}_T) \\ &\quad = \int_{\partial T} |{\b{\Delta}^{k}_{\partial T}}{\und{\b{u}}}_T-{\b{\Delta}^{k}_{\partial T}}{\und{\b{w}}}_T|^{r}\\ &\quad \lesssim \int_{\partial T} \left(|{\b{\Delta}^{k}_{\partial T}} {\und{\b{u}}}_T|^{r}+|{\b{\Delta}^{k}_{\partial T}} {\und{\b{w}}}_T|^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}\left[ \left(|{\b{\Delta}^{k}_{\partial T}}{\und{\b{u}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}}{\und{\b{u}}}_T-|{\b{\Delta}^{k}_{\partial T}}{\und{\b{w}}}_T|^{{r}-2}{\b{\Delta}^{k}_{\partial T}}{\und{\b{w}}}_T\right)\cdot {\b{\Delta}^{k}_{\partial T}}{\und{\b{e}}}_T \right]^\frac{{r}}{{r}+2-{r^{\circ}}}\\ &\quad \le \left( {\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{u}}}_T)+{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{w}}}_T)\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}\left({\mathrm{s}}_T({\und{\b{u}}}_T,{\und{\b{e}}}_T)-{\mathrm{s}}_T({\und{\b{w}}}_T,{\und{\b{e}}}_T)\right)^\frac{{r}}{{r}+2-{r^{\circ}}}.\qedhere \end{aligned}$$ Pressure-velocity coupling -------------------------- For all $T \in {\mathcal{T}}_h$, we define the local divergence reconstruction ${{\mathrm{D}}^{k}_{T}} : {{{\und{\b{U}}}}_{T}^{k}} \to {\mathbb{P}}^{k}(T,{\mathbb{R}})$ by setting, for all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, ${{\mathrm{D}}^{k}_{T}}{\und{\b{v}}}_T \coloneq {\mathrm{tr}}({\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}}{\und{\b{v}}}_T)$. We have the following characterization of ${{\mathrm{D}}^{k}_{T}}$: For all ${\und{\b{v}}}_T \in {{{\und{\b{U}}}}_{T}^{k}}$, $$\label{eq:D} \int_T {{\mathrm{D}}^{k}_{T}} {\und{\b{v}}}_T~ q = \int_T ({\b\nabla{\cdot}}\b v_T)~q + \sum_{F \in {\mathcal{F}}_T} \int_F (\b v_F-\b v_T)\cdot \b n_{TF}~q \qquad \forall q \in {\mathbb{P}}^{k}(T,{\mathbb{R}}),$$ as can be checked writing for $\b\tau = q\mathrm{I}_d$. Taking the trace of , it is inferred that, for all $T\in{\mathcal{T}}_h$ and all $\b v\in W^{1,1}(T,{\mathbb{R}}^d)$, ${{\mathrm{D}}^{k}_{T}} ({{\und{\b{I}}}_{T}^{k}} \b v) = {\pi_{T}^{k}}({\b\nabla{\cdot}}\b v)$. The pressure-velocity coupling is realized by the bilinear form ${\mathrm{b}}_h : {{{\und{\b{U}}}}_{h}^{k}} \times {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}) \to {\mathbb{R}}$ such that, for all $({\und{\b{v}}}_h,q_h) \in {{{\und{\b{U}}}}_{h}^{k}} \times {\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}})$, setting $q_T\coloneq (q_h){\ \!\!_{|_{T}}}$ for all $T\in{\mathcal{T}}_h$, $$\label{eq:bh} {\mathrm{b}}_h({\und{\b{v}}}_h,q_h) \coloneqq -\sum_{T \in {\mathcal{T}}_h}\int_T {{\mathrm{D}}^{k}_{T}} {\und{\b{v}}}_T~ q_T.$$ Discrete problem ---------------- The discrete problem reads: Find $({\und{\b{u}}}_h,p_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ such that \[eq:stokes.discrete\] $$\begin{aligned} {2} \label{eq:stokes.discrete:momentum} {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h) + {\mathrm{b}}_h({\und{\b{v}}}_h,p_h) &= \displaystyle\int_\Omega \b f \cdot \b v_h &\qquad& \forall {\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}, \\ \label{eq:stokes.discrete:mass} -{\mathrm{b}}_h({\und{\b{u}}}_h,q_h) &= 0 &\qquad& \forall q_h \in {P_{h}^{k}}. \end{aligned}$$ Before proceding, some remarks are in order. The space of test functions in can be extended to ${\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}})$ since, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, the divergence theorem together with the fact that $\b v_F = \b 0$ for all $F \in {{\mathcal{F}}_h^{\mathrm{b}}}$ and $\sum_{T\in {\mathcal{T}}_F} \int_F \b v_F \cdot \b n_{TF} = 0$ for all $F \in {{\mathcal{F}}_h^{\mathrm{i}}}$, yield $${\mathrm{b}}_h({\und{\b{v}}}_h,1) = -\sum_{T \in {\mathcal{T}}_h} \sum_{F \in {\mathcal{F}}_T} \int_F \b v_F \cdot \b n_{TF} = -\sum_{F \in {{\mathcal{F}}_h^{\mathrm{i}}}}\sum_{T \in {\mathcal{T}}_F} \int_F \b v_F \cdot \b n_{TF} = 0.$$ When solving the system of nonlinear algebraic equations corresponding to by a first-order (e.g., Newton) algorithm, all element-based velocity unknowns and all but one pressure unknown per element can be locally eliminated at each iteration by computing the corresponding Schur complement element-wise. As all the computations are local, this procedure is an embarrassingly parallel task which can fully benefit from multi-thread and multi-processor architectures. This implementation strategy has been described for the linear Stokes problem in [@Di-Pietro.Ern.ea:16 Section 6.2]. After further eliminating the boundary unknowns by strongly enforcing the boundary condition , we end up solving, at each iteration of the nonlinear solver, a linear system of size $d{\mathrm{card}}({{\mathcal{F}}_h^{\mathrm{i}}}){k+d-1\choose d-1} + {\mathrm{card}}({\mathcal{T}}_h)$. Well-posedness {#sec:well-posedness} ============== In this section, after studying the stability properties of the viscous function ${\mathrm{a}}_h$ and of the velocity-pressure coupling bilinear form ${\mathrm{b}}_h$, we prove the well-posedness of problem . Hölder continuity and strong monotonicity of the viscous function ----------------------------------------------------------------- \[lem:ah:holder.continuity.strong.monotonicity\] For all ${\und{\b{u}}}_h, {\und{\b{v}}}_h, {\und{\b{w}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, setting ${\und{\b{e}}}_h\coloneq{\und{\b{u}}}_h - {\und{\b{w}}}_h$, it holds \[eq:ah:holder.continuity.strong.monotonicity\] $$\begin{gathered} \label{eq:ah:holder.continuity} \left| {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)-{\mathrm{a}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h) \right| \lesssim \sigma_{\mathrm{hc}}\left( \sigma_{\mathrm{de}}^{r}+ \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \\\label{eq:ah:strong.monotonicity} \left({\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{w}}}_h,{\und{\b{e}}}_h)\right)\left( \sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}} \gtrsim \sigma_{\mathrm{sm}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}}. \end{gathered}$$ \(i) *Hölder continuity.* Denote by $|\Omega|_d$ the measure of $\Omega$. Using a Cauchy–Schwarz inequality followed by the Hölder continuity of ${\b\sigma}$, we can write $$\label{C2:rge2:1} \begin{aligned} &\left|\int_\Omega \big( {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h)-{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h) \big):{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h\right| \\ &\quad \le \sigma_{\mathrm{hc}} \int_\Omega \left(\sigma_{\mathrm{de}}^{r}+|{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h|_{d\times d}^{r}+|{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h|_{d\times d}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{e}}}_h|_{d\times d}^{{r^{\circ}}-1} |{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h|_{d \times d} \\ &\quad \lesssim \sigma_{\mathrm{hc}}\left(|\Omega|_d\sigma_{\mathrm{de}}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}} \\ & \qquad \times \| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{e}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r^{\circ}}-1}\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}\\ &\quad \lesssim \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+ \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where we have used the $(1;\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1},{r})$-Hölder inequality in the second bound and the global seminorm equivalence together with the fact that $|\Omega|_d\lesssim 1$ (since $\Omega$ is bounded) to conclude. For the stabilization term, combining the Hölder continuity of ${\mathrm{s}}_h$ and the seminorm equivalence readily gives $$\label{C2:rge2:2} \left| {\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)-{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h) \right| \lesssim \left(\sigma_{\mathrm{de}}^{r}+ \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h},$$ where we have additionally noticed that $\sigma_{\mathrm{de}}^{r}\ge 0$ to add this term to the quantity inside parentheses. Using the definition of ${\mathrm{a}}_h$, a triangle inequality followed by and , and recalling that $\gamma \le \sigma_{\mathrm{hc}}$ (cf. ), follows.\ (ii) *Strong monotonicity.* Using the strong monotonicity of ${\b\sigma}$ and the $(1;\frac{{r}+2-{r^{\circ}}}{2-{r^{\circ}}},\frac{{r}+2-{r^{\circ}}}{{r}})$-Hölder inequality , we get $$\label{eq:ah:sm:1} \hspace{-0.2cm}\begin{aligned} &\sigma_{\mathrm{sm}}^\frac{{r}}{{r}+2-{r^{\circ}}}\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{e}}}_h\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\\ &\leq \int_\Omega \left(\sigma_{\mathrm{de}}^{r}+|{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h|_{d\times d}^{r}+|{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h|_{d\times d}^{r}\right)^{\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}}\left) \left({\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h)-{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h)\right):{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{e}}}_h \right)^\frac{{r}}{{r}+2-{r^{\circ}}}\\ &\lesssim \left(\sigma_{\mathrm{de}}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{u}}}_h\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{w}}}_h\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}} \\ &\qquad \times \left( \int_\Omega \left( {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h)-{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h) \right):{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{e}}}_h \right)^\frac{{r}}{{r}+2-{r^{\circ}}}\\ &\lesssim \left(\sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}} \left( \int_\Omega \left( {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{u}}}_h)-{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{w}}}_h) \right):{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{e}}}_h \right)^\frac{{r}}{{r}+2-{r^{\circ}}}, \end{aligned}$$ where the conclusion follows from the global seminorm equivalence . Additionally, using the strong monotonicity of ${\mathrm{s}}_h$ together with the fact that $\sigma_{\mathrm{sm}} \le \gamma$ (cf. ) and invoking again the seminorm equivalence , we readily obtain $$\label{eq:ah:sm:2} \sigma_{\mathrm{sm}}^\frac{{r}}{{r}+2-{r^{\circ}}}{\mathrm{s}}_h({\und{\b{e}}}_h,{\und{\b{e}}}_h) \lesssim \left(\sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}\left( \gamma{\mathrm{s}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-\gamma{\mathrm{s}}_h({\und{\b{w}}}_h,{\und{\b{e}}}_h) \right)^\frac{{r}}{{r}+2-{r^{\circ}}}.$$ Finally, combining again the norm equivalence with and , and using yields $$\begin{aligned} \sigma_{\mathrm{sm}}^\frac{{r}}{{r}+2-{r^{\circ}}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\lesssim &\left( \sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}+2-{r^{\circ}}}\left( {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{w}}}_h,{\und{\b{e}}}_h) \right)^\frac{{r}}{{r}+2-{r^{\circ}}}. \end{aligned}$$ Raising this inequality to the power $\frac{{r}-2-{r^{\circ}}}{{r}}$ yields . Stability of the pressure-velocity coupling ------------------------------------------- \[lem:bh:inf-sup\] It holds, for all $q_h \in {P_{h}^{k}}$, $$\label{eq:bh:inf-sup} \| q_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} \lesssim \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}, \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} {\mathrm{b}}_h({\und{\b{v}}}_h,q_h),$$ with hidden constant depending only on $d$, $k$, ${r}$, $\Omega$, and the mesh regularity parameter. The proof follows the classical Fortin argument (cf., e.g., [@Boffi.Brezzi.ea:13 Section 8.4]), adapted here to the non-Hilbertian setting: we first prove that ${{\und{\b{I}}}_{h}^{k}}$ is a Fortin operator, then combine this fact with the continuous inf-sup condition.\ (i) *Fortin operator.* We need to prove that the following properties hold for any $\b v\in W^{1,{r}}(\Omega,{\mathbb{R}}^d)$: \[eq:fortin\] $$\begin{gathered} \| {{\und{\b{I}}}_{h}^{k}} \b v\|_{{\boldsymbol{\varepsilon}},{r},h} \lesssim | \b v |_{W^{1,{r}}(\Omega,{\mathbb{R}}^d),}\label{eq:fortin:boundedness} \\ {\mathrm{b}}_h({{\und{\b{I}}}_{h}^{k}} \b v,q_h) = b(\b v,q_h)\qquad\forall q_h\in{\mathbb{P}}^k({\mathcal{T}}_h,{\mathbb{R}}).\label{eq:fortin:consistency} \end{gathered}$$ Property is obtained by raising both sides of to the power ${r}$, summing over $T \in {\mathcal{T}}_h$, then taking the $r$th root of the resulting inequality. The proof of is given, e.g., in [@Di-Pietro.Droniou:20 Lemma 8.12].\ (ii) *Inf-sup condition on ${\mathrm{b}}_h$.* Let $q_h \in {P_{h}^{k}}$ and set $c_h \coloneqq \int_\Omega |q_h|^{{r}'-2}q_h$. Using a triangle inequality, the Hölder inequality, and the fact that $|\Omega|_d\lesssim 1$, we get $$\label{eq:inf-sup:qT} \| |q_h|^{{r}'-2}q_h-c_h \|_{L^{{r}}(\Omega,{\mathbb{R}})} \le \| q_h\|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1}+|c_h||\Omega|_d^\frac{1}{{r}} \le \left(1+|\Omega|_d\right)\| q_h\|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1} \lesssim \| q_h\|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1},$$ where we have used the fact that $|c_h|\le\| q_h\|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1} |\Omega|_d^{\frac1{{r}'}}$ along with $\frac1{r}+\frac1{{r}'}=1$ in the second bound and the fact that $|\Omega|_d\lesssim 1$ to conclude. Since $q_h \in L^{{r}'}(\Omega,{\mathbb{R}})$, bound implies that $|q_h|^{{r}'-2}q_h-c_h \in L^{{r}}_0(\Omega,{\mathbb{R}}) \coloneq \left\{ q \in L^{r}(\Omega,{\mathbb{R}}) : \int_\Omega q = 0 \right\}$ by construction. Thus, using the surjectivity of the continuous divergence operator ${\b\nabla{\cdot}}: \b U \to L^{{r}}_0(\Omega,{\mathbb{R}})$, (c.f. [@Duran.Muschietti.ea:10] and also [@Bogovski:79 Theorem 1]), we infer that there exists $\b v_{q_h} \in \b U$ such that $$\label{eq:velocity.lifting} -{\b\nabla{\cdot}}\b v_{q_h} = |q_h|^{{r}'-2}q_h-c_h \quad {\mathrm{and}} \quad | \b v_{q_h}|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)} \lesssim \| |q_h|^{{r}'-2}q_h-c_h \|_{L^{r}(\Omega,{\mathbb{R}})}.$$ Denote by $\$$ the supremum in . Using the fact that $q_h$ has zero mean value over $\Omega$, the equality in together with the definition of $b$, and the second Fortin property , we have $$\|q_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'} {=} \int_\Omega \big(|q_h|^{{r}'-2}q_h-c_h\big) q_h = b(\b v_{q_h},q_h) = {\mathrm{b}}_h({{\und{\b{I}}}_{h}^{k}}\b v_{q_h},q_h) \le \$ \| {{\und{\b{I}}}_{h}^{k}} \b v_{q_h} \|_{{\boldsymbol{\varepsilon}},{r},h} \lesssim \$ \| q_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})}^{{r}'-1},$$ where, to conclude, we have used followed by and . Simplifying yields . Well-posedness {#well-posedness} -------------- We are now ready to prove the main result of this section. \[thm:well-posedness\] There exists a unique solution $({\und{\b{u}}}_h,p_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ to the discrete problem . Additionally, the following a priori bounds hold: \[eq:discrete.solution:bounds\] $$\begin{aligned} \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} &\lesssim \left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}-1}+\left(\sigma_{\mathrm{de}}^{2-{r^{\circ}}}\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}+1-{r^{\circ}}}, \label{eq:discrete.solution:bounds:uh}\\ \| p_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}+\sigma_{\mathrm{de}}^{|{r}-2|({r^{\circ}}-1)}\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{{r^{\circ}}-1}{{r}+1-{r^{\circ}}}\right). \label{eq:discrete.solution:bounds:ph} \end{aligned}$$ \(i) *Existence.* Denote by ${P_{h}^{k,*}}$ the dual space of ${P_{h}^{k}}$ and let $B_h : {{{\und{\b{U}}}}_{h,0}^{k}} \to {P_{h}^{k,*}}$ be such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, $$\langle B_h{\und{\b{v}}}_h, q_h \rangle \coloneqq -{\mathrm{b}}_h({\und{\b{v}}}_h,q_h) \qquad \forall q_h \in {P_{h}^{k}}.$$ Here and in what follows, $\langle{\cdot},{\cdot}\rangle$ denotes the appropriate duality pairing as inferred from its arguments. Define the following subspace of ${{{\und{\b{U}}}}_{h,0}^{k}}$ spanned by vectors of discrete unknowns with zero discrete divergence: $$\label{eq:Whk} {\und{\b{W}}}_h^k \coloneq \operatorname{Ker}(B_h) = \left\{ {\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}} : {\mathrm{b}}_h({\und{\b{v}}}_h,q_h) = 0 \quad \forall q_h \in {P_{h}^{k}} \right\},$$ and consider the following problem: Find ${\und{\b{u}}}_h\in{\und{\b{W}}}_h^k$ such that $$\label{eq:existence:auxiliary} {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h) = \int_\Omega \b f\cdot \b v_h\qquad\forall {\und{\b{v}}}_h\in{\und{\b{W}}}_h^k.$$ Existence of a solution to this problem for a fixed $h$ can be proved adapting the arguments of [@Di-Pietro.Droniou:17 Theorem 4.5]. Specifically, equip ${\und{\b{W}}}_h^k$ with an inner product $(\cdot,\cdot)_{\b W,h}$ (which need not be further specified), denote by $\|{\cdot}\|_{\b W,h}$ the induced norm, and let $\b\Phi_h:{\und{\b{W}}}_h^k\to {\und{\b{W}}}_h^k$ be such that, for all ${\und{\b{w}}}_h\in{\und{\b{W}}}_h^k$, $(\b\Phi_h({\und{\b{w}}}_h),{\und{\b{v}}}_h)_{\b W,h} = {\mathrm{a}}_h({\und{\b{w}}}_h,{\und{\b{v}}}_h)$ for all ${\und{\b{v}}}_h\in{\und{\b{W}}}_h^k$. The strong monotonicity of ${\mathrm{a}}_h$ yields, for any ${\und{\b{v}}}_h\in{\und{\b{W}}}_h^k$ such that $\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} \ge \sigma_{\mathrm{de}}$, $$(\b\Phi_h({\und{\b{v}}}_h),{\und{\b{v}}}_h)_{\b W,h}\ge \sigma_{\mathrm{sm}} (\sigma_{\mathrm{de}}^{r}+ \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r})^\frac{{r^{\circ}}-2}{{r}} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}} \gtrsim \sigma_{\mathrm{sm}}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\ge C^{r}\sigma_{\mathrm{sm}}\| {\und{\b{v}}}_h\|_{\b W,h}^{r},$$ where $C$ denotes the constant (possibly depending on $h$) in the equivalence of the norms $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},h}$ and $\|{\cdot}\|_{\b W,h}$ (which holds since ${\und{\b{W}}}_h^k$ is finite-dimensional). This shows that $\b\Phi_h$ is coercive hence, by [@Deimling:85 Theorem 3.3], surjective. Let now ${\und{\b{w}}}_h\in{\und{\b{W}}}_h^k$ be such that $({\und{\b{w}}}_h,{\und{\b{v}}}_h)_{\b W,h}=\int_\Omega \b f\cdot \b v_h$ for all ${\und{\b{v}}}_h\in{\und{\b{W}}}_h^k$. By the surjectivity of $\b\Phi_h$, there exists ${\und{\b{u}}}_h\in {\und{\b{W}}}_h^k$ such that $\b\Phi_h({\und{\b{u}}}_h)={\und{\b{w}}}_h$ which, by definition of ${\und{\b{w}}}_h$ and $\b\Phi_h$, is a solution to the discrete problem . The proof of existence now continues as in the linear case; see, e.g., [@Boffi.Brezzi.ea:13 Theorem 4.2.1]. Denote by ${{{\und{\b{U}}}}_{h,0}^{k,*}}$ the dual space of ${{{\und{\b{U}}}}_{h,0}^{k}}$ and consider the linear mapping $\ell_h\in{{{\und{\b{U}}}}_{h,0}^{k,*}}$ such that, for all ${\und{\b{v}}}_h\in{{{\und{\b{U}}}}_{h,0}^{k}}$, $$\langle\ell_h,{\und{\b{v}}}_h\rangle\coloneq \int_\Omega \b f\cdot \b v_h - {\mathrm{a}}_h({\und{\b{u}}}_h, {\und{\b{v}}}_h).$$ Thanks to , $\ell_h$ vanishes identically for every ${\und{\b{v}}}_h\in{\und{\b{W}}}_h^k$, that is to say, $\ell_h$ lies in the polar space of ${\und{\b{W}}}_h^k$ which, denoting by $B_h^*:{P_{h}^{k}}\to{{{\und{\b{U}}}}_{h,0}^{k,*}}$ the adjoint operator of $B_h$, coincides in our case with $\operatorname{Im}(B_h^*)$ (see, e.g., [@Boffi.Brezzi.ea:13 Theorem 4.14]). Hence, $\ell_h\in\operatorname{Im}(B_h^*)$, and there exists therefore a $p_h\in {P_{h}^{k}}$ such that $B_h^* p_h = \ell_h$. This means that, for all ${\und{\b{v}}}_h\in{{{\und{\b{U}}}}_{h,0}^{k}}$, $${\mathrm{b}}_h({\und{\b{v}}}_h,p_h) = \langle B_h^* p_h,{\und{\b{v}}}_h\rangle = \langle\ell_h,{\und{\b{v}}}_h\rangle = \int_\Omega \b f\cdot \b v_h - {\mathrm{a}}_h({\und{\b{u}}}_h, {\und{\b{v}}}_h),$$ i.e., the $({\und{\b{u}}}_h,p_h)$ satisfies the discrete momentum equation . On the other hand, since ${\und{\b{u}}}_h\in{\und{\b{W}}}_h^k$, we also have, by the definition of ${\und{\b{W}}}_h^k$, $ {\mathrm{b}}_h({\und{\b{u}}}_h,q_h) = 0$ for all $q_h\in {P_{h}^{k}}$, which shows that the discrete mass equation is also verified. In conclusion, $({\und{\b{u}}}_h,p_h)\in{{{\und{\b{U}}}}_{h,0}^{k}}\times {P_{h}^{k}}$ solves .\ (ii) *Uniqueness.* We start by proving uniqueness for the velocity. Let $({\und{\b{u}}}_h,p_h),({\und{\b{u}}}'_h,p'_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ be two solutions of . Making ${\und{\b{v}}}_h = {\und{\b{u}}}_h - {\und{\b{u}}}'_h$ in written first for $({\und{\b{u}}}_h, p_h)$ then for $({\und{\b{u}}}'_h, p'_h)$, then taking the difference and observing that ${\mathrm{b}}_h({\und{\b{u}}}_h-{\und{\b{u}}}'_h,p_h)={\mathrm{b}}_h({\und{\b{u}}}_h - {\und{\b{u}}}'_h,p'_h)=0$ by , we infer that $${\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{u}}}_h - {\und{\b{u}}}'_h)-{\mathrm{a}}_h({\und{\b{u}}}'_h,{\und{\b{u}}}_h - {\und{\b{u}}}'_h) = 0.$$ Thus, the strong monotonicity of ${\mathrm{a}}_h$ yields $\| {\und{\b{u}}}_h - {\und{\b{u}}}'_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 0$, which implies ${\und{\b{u}}}_h = {\und{\b{u}}}'_h$ since $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},h}$ is a norm on ${{{\und{\b{U}}}}_{h,0}^{k}}$. Moreover, using the inf-sup stability of ${\mathrm{b}}_h$ and written first for ${\und{\b{u}}}_h$ then for ${\und{\b{u}}}_h'$, we get $$\begin{aligned} \| p_h-p'_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} {\mathrm{b}}_h({\und{\b{v}}}_h,p_h-p'_h) \\ &= \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \left({\mathrm{a}}_h({\und{\b{u}}}'_h,{\und{\b{v}}}_h)-{\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)\right) = 0, \end{aligned}$$ hence $p_h=p'_h$.\ (iii) *A priori estimates.* Using the strong monotonicity of ${\mathrm{a}}_h$ (with ${\und{\b{w}}}_h = {\und{\b{0}}}$), equation together with , and the Hölder inequality together with the discrete Korn inequality , we obtain $$\label{eq:well-posedness:0} \begin{aligned} \sigma_{\mathrm{sm}}\big( \sigma_{\mathrm{de}}^{r}+ \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\big)^\frac{{r^{\circ}}-2}{{r}} \| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}} &\lesssim {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{u}}}_h) = \displaystyle\int_\Omega \b f \cdot \b u_h \lesssim \| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}. \end{aligned}$$ We then conclude as in the continuous case to infer (see Remark \[rem:a-priori\]). To prove the bound on the pressure, we use the inf-sup stability of ${\mathrm{b}}_h$ to write $$\begin{aligned} \| p_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}, \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} {\mathrm{b}}_h({\und{\b{v}}}_h,p_h) \\ &= \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}, \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \left(\displaystyle\int_\Omega \b f \cdot \b v_h - {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h) \right) \\ &\lesssim \| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)} +\sigma_{\mathrm{hc}}(\sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r})^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1} \\ &\lesssim \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}+\sigma_{\mathrm{de}}^{|{r}-2|({r^{\circ}}-1)}\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{{r^{\circ}}-1}{{r}+1-{r^{\circ}}}\right), \end{aligned}$$ where we have used the discrete momentum equation to pass to the second line, the Hölder and discrete Korn inequalities together with the Hölder continuity of ${\mathrm{a}}_h$ to pass to the third line, and the a priori bound on the velocity together with $\frac{\sigma_{\mathrm{hc}}}{\sigma_{\mathrm{sm}}}\ge 1$ (see ) to conclude. Error estimate {#sec:error.estimate} ============== In this section, after studying the consistency of the viscous and pressure-velocity coupling terms, we prove an energy error estimate. Consistency of the viscous function {#sec:consistency:viscous.function} ----------------------------------- \[lem:consistency:ah\] Let $\b w \in \b U \cap W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)$ be such that ${\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \in W^{1,{r}'}(\Omega,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}}) \cap W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$. Define the viscous consistency error linear form $\mathcal E_{{\mathrm{a}},h}(\b w;\cdot) : {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{R}}$ such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:Eah} \mathcal E_{{\mathrm{a}},h}(\b w;{\und{\b{v}}}_h) \coloneqq \int_\Omega ({\b\nabla{\cdot}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)) \cdot \b v_h + {\mathrm{a}}_h({{\und{\b{I}}}_{h}^{k}} \b w,{\und{\b{v}}}_h).$$ Then, under Assumptions \[ass:stress\] and \[ass:sT\], we have $$\begin{gathered} \label{eq:consistency:ah} \sup_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \mathcal E_{{\mathrm{a}},h}(\b w;{\und{\b{v}}}_h) \lesssim h^{(k+1)({r^{\circ}}-1)}\bigg[ \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+ |\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1} \\ +|{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})} \bigg]. \end{gathered}$$ Let ${\und{\b{\hat w}}}_h \coloneqq {{\und{\b{I}}}_{h}^{k}} \b w$ and ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$. Expanding ${\mathrm{a}}_h$ according to its definition in the expression of $\mathcal E_{{\mathrm{a}},h}$, inserting $\pm\left( \int_\Omega{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w): {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h + \int_\Omega{\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) : {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h \right)$, and rearranging, we obtain $$\begin{gathered} \label{eq:consistency:ah:EJ} \mathcal E_{{\mathrm{a}},h}(\b w;{\und{\b{v}}}_h) =\!\! \underbrace{ \int_\Omega ({\b\nabla{\cdot}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)) \cdot \b v_h {+} \int_\Omega {\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\! :\! {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h }_{\mathcal T_1} +\! \underbrace{ \int_\Omega\! \left( {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) - {\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\right)\! :\! {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h }_{\mathcal T_2} \\ {+} \underbrace{ \int_\Omega\!\left( {\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h) - {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\right)\!:\! {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h }_{\mathcal T_3} + \underbrace{\vphantom{\int_\Omega}\gamma {\mathrm{s}}_h({\und{\b{\hat w}}}_h,{\und{\b{v}}}_h)}_{\mathcal T_4}. \end{gathered}$$ We proceed to estimate the terms in the right-hand side. For the first term, we start by noticing that $$\label{eq:consistency:ah:null} \sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T} \int_F \b v_F \cdot\left({\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\b n_{TF}\right) = 0$$ as a consequence of the continuity of the normal trace of ${\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)$ together with the single-valuedness of $\b v_F$ across each interface $F\in{{\mathcal{F}}_h^{\mathrm{i}}}$ and of the fact that $\b v_F=\b 0$ for every boundary face $F\in{{\mathcal{F}}_h^{\mathrm{b}}}$. Using an element by element integration by parts on the first term of $\mathcal T_1$ along with the definitions of ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}$ and of ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},T}}$, we can write $$\begin{aligned} \mathcal T_1 &= \cancel{\int_\Omega \left({\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)- {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\right) : {\b{\nabla}_{{\mathrm{s}},h}}\b v_h} \\ &\qquad + \sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T} \left(\int_F (\b v_F-\b v_T)\cdot({\b{\pi}_{T}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w))\b n_{TF}+\int_F\b v_T\cdot\left({\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\b n_{TF}\right) \right) \\ &= \sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T} \int_F (\b v_F-\b v_T)\cdot\left({\b{\pi}_{T}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)-{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)\right)\b n_{TF}, \end{aligned}$$ where we have used the definition of ${\b{\pi}_{h}^{k}}$ together with the fact that ${\b{\nabla}_{{\mathrm{s}},h}}\b v_h \in {\mathbb{P}}^{k-1}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}}) \subset {\mathbb{P}}^{k}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$ to cancel the term in the first line, and we have inserted and rearranged to conclude. Therefore, applying the Hölder inequality together with the bound $h_F \le h_T$, we infer $$\label{eq:consistency:ah:T1} \begin{aligned} \left|\mathcal T_1\right| &\le \left(\displaystyle\sum_{T \in {\mathcal{T}}_h}h_T \|{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)- {\b{\pi}_{T}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \|_{L^{{r}'}(\partial T,{{\mathbb{R}}^{d \times d}})}^{{r}'} \right)^\frac{1}{{r}'}\left(\sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T}h_F^{1-{r}}\| \b v_F-\b v_T\|_{L^{r}(F,{\mathbb{R}}^d)}^{r}\right)^\frac{1}{{r}} \\ &\lesssim h^{(k+1)({r^{\circ}}-1)} |{\b\sigma}(\cdot, {\b{\nabla}_{\mathrm{s}}}\b w)|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where the conclusion follows using the $((k+1)({r^{\circ}}-1),{r}')$-trace approximation properties of ${\b{\pi}_{T}^{k}}$ along with $h_T \le h$ for the first factor and the definition of the $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},h}$-norm for the second. For the second term, we use the Hölder inequality and the seminorm equivalence to write $$\label{eq:consistency:ah:T2} \begin{aligned} \left|\mathcal T_2\right| &= \| {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w)-{\b{\pi}_{h}^{k}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \|_{L^{{r}'}(\Omega,{{\mathbb{R}}^{d \times d}})} \| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}{\und{\b{v}}}_h\|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})} \\ &\lesssim h^{(k+1)({r^{\circ}}-1)}|{\b\sigma}(\cdot, {\b{\nabla}_{\mathrm{s}}}\b w)|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where the conclusion follows from the $((k+1)({r^{\circ}}-1),{r}',0)$-approximation properties of ${\b{\pi}_{T}^{k}}$ along with $h_T\le h$ for the first factor and the global norm equivalence for the second. For the third term, using the Hölder inequality and again , we get $$\label{eq:consistency:ah:T3:0} \left|\mathcal T_3\right| \le \|{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h)- {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \|_{L^{{r}'}(\Omega,{{\mathbb{R}}^{d \times d}})}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}.$$ We estimate the first factor as follows: $$\begin{aligned} &\|{\b\sigma}(\cdot,{\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h)- {\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b w) \|_{L^{{r}'}(\Omega,{{\mathbb{R}}^{d \times d}})} \\ &\quad \le \sigma_{\mathrm{hc}} \left\| \left(\sigma_{\mathrm{de}}^{r}+| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h |_{d \times d}^{r}+ | {\b{\nabla}_{\mathrm{s}}}\b w |_{d \times d}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h - {\b{\nabla}_{\mathrm{s}}}\b w |_{d \times d}^{{r^{\circ}}-1}\right\|_{L^{{r}'}(\Omega,{\mathbb{R}})} \\ &\quad \lesssim \sigma_{\mathrm{hc}} \left(\sigma_{\mathrm{de}}^{r}+\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}+\| {\b{\nabla}_{\mathrm{s}}}\b w \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}} {\und{\b{\hat w}}}_h - {\b{\nabla}_{\mathrm{s}}}\b w \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r^{\circ}}-1} \\ &\quad \lesssim \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+\|{\und{\b{\hat w}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+ |\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\b{\pi}_{h}^{k}}({\b{\nabla}_{\mathrm{s}}}\b w) - {\b{\nabla}_{\mathrm{s}}}\b w \|_{L^{r}(\Omega,{{\mathbb{R}}^{d \times d}})}^{{r^{\circ}}-1} \\ &\quad \lesssim h^{(k+1)({r^{\circ}}-1)}\sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+|\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1}, \end{aligned}$$ where we have used the Hölder continuity of ${\b\sigma}$ in the first bound, the $({r}';\frac{{r}}{{r}-{r^{\circ}}},\frac{{r}}{{r^{\circ}}-1})$-Hölder inequality in the second, the boundedness of $\Omega$ along with and the commutation property of ${\b{{\mathrm{G}}}^{k}_{{\mathrm{s}},h}}$ in the third, and we have concluded invoking the $(k+1,{r},0)$-approximation property of ${\b{\pi}_{T}^{k}}$. Plugging this estimate into , we get $$\label{eq:consistency:ah:T3} \left|\mathcal T_3\right| \lesssim h^{(k+1)({r^{\circ}}-1)}\sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+|\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}.$$ Finally, using the fact that $\gamma \le \sigma_{\mathrm{hc}}$ together with the consistency of ${\mathrm{s}}_h$ and the norm equivalence , we obtain for the fourth term $$\label{eq:consistency:ah:T4} \left|\mathcal T_4\right| \lesssim h^{(k+1)({r^{\circ}}-1)}\sigma_{\mathrm{hc}}|\b w|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{{r}-{r^{\circ}}}|\b w|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}.$$ Plug the bounds , , , and into and pass to the supremum to conclude. Consistency of the pressure-velocity coupling bilinear form ----------------------------------------------------------- Let $q \in W^{1,{r}'}(\Omega,{\mathbb{R}}) \cap W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})$. Let $\mathcal E_{{\mathrm{b}},h}(q;\cdot) : {{{\und{\b{U}}}}_{h}^{k}} \to {\mathbb{R}}$ be the pressure consistency error linear form such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h}^{k}}$, $$\label{eq:Ebh} \mathcal E_{{\mathrm{b}},h}(q;{\und{\b{v}}}_h) \coloneqq \int_\Omega {\b\nabla}q \cdot \b v_h - {\mathrm{b}}_h({\und{\b{v}}}_h,{\pi_{h}^{k}} q).$$ Then, we have that $$\label{eq:consistency:bh} \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \mathcal E_{{\mathrm{b}},h}(q;{\und{\b{v}}}_h) \lesssim h^{(k+1)({r^{\circ}}-1)} |q|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})}.$$ Let ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$. Integrating by parts element by element, we can reformulate the first term in the right-hand side of as follows: $$\label{eq:consistency:bh:1} \displaystyle\int_\Omega {\b\nabla}q\cdot \b v_h = - \sum_{T \in {\mathcal{T}}_h} \left( \int_T q({\b\nabla{\cdot}}\b v_T) +\sum_{F \in {\mathcal{F}}_T} \int_F q(\b v_F - \b v_T)\cdot \b n_{TF} \right),$$ where the introduction of $\b v_F$ in the boundary term is justified by the fact that the jumps of $q$ vanish across interfaces by the assumed regularity and that $\b v_F=\b 0$ on every boundary face $F \in {{\mathcal{F}}_h^{\mathrm{b}}}$. On the other hand, expanding, for each $T \in {\mathcal{T}}_h$, ${{\mathrm{D}}^{k}_{T}}$ according to its definition , we get $$\label{eq:consistency:bh:2} -{\mathrm{b}}_h({\und{\b{v}}}_h, {\pi_{h}^{k}} q) = \sum_{T \in {\mathcal{T}}_h} \left( \int_T {\pi_{T}^{k}} q~({\b\nabla{\cdot}}\b v_T) +\sum_{F \in {\mathcal{F}}_T} \int_F {\pi_{T}^{k}} q~(\b v_F - \b v_T)\cdot \b n_{TF} \right).$$ Summing and and observing that the first terms in parentheses cancel out by the definition of ${\pi_{T}^{k}}$ since ${\b\nabla{\cdot}}\b v_T \in {\mathbb{P}}^{k-1}(T,{\mathbb{R}}) \subset {\mathbb{P}}^k(T,{\mathbb{R}})$ for all $T \in{\mathcal{T}}_h$, we can write $$\begin{aligned} \mathcal E_{{\mathrm{b}},h}(q;{\und{\b{v}}}_h) &= \sum_{T \in {\mathcal{T}}_h} \left( \cancel{\int_T ({\pi_{T}^{k}} q-q) ({\b\nabla{\cdot}}\b v_T)} +\sum_{F \in {\mathcal{F}}_T} \int_F ({\pi_{T}^{k}} q-q) (\b v_F - \b v_T)\cdot \b n_{TF} \right) \\ &\le \left(\sum_{T \in {\mathcal{T}}_h} h_T\| {\pi_{T}^{k}} q-q \|_{L^{{r}'}(\partial T,{\mathbb{R}})}^{{r}'} \right)^\frac{1}{{r}'} \left(\sum_{T \in {\mathcal{T}}_h}\sum_{F \in {\mathcal{F}}_T} h_F^{1-{r}}\| \b v_F-\b v_T \|_{L^{{r}}(F,{\mathbb{R}}^d)}^{{r}} \right)^\frac{1}{{r}} \\ &\lesssim h^{(k+1)({r^{\circ}}-1)} |q|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})} \| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where we have used the Hölder inequality along with $h_F\ge h_T$ whenever $F \in {\mathcal{F}}_T$ in the second line and the $((k+1)({r^{\circ}}-1),{r}')$-trace approximation property of ${\pi_{T}^{k}}$ together with the bound $h_F \le h$ and the definition of the $\|{\cdot}\|_{{\boldsymbol{\varepsilon}},{r},h}$-norm to conclude. Passing to the supremum yields . Error estimate {#error-estimate} -------------- \[thm:error.estimate\] Let $(\b u,p) \in \b U \times P$ and $({\und{\b{u}}}_h,p_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ solve and , respectively. Assume the additional regularity $\b u \in W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)$, ${\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u) \in W^{1,{r}'}(\Omega,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}}) \cap W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$, and $p \in W^{1,{r}'}(\Omega,{\mathbb{R}}) \cap W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})$. Then, under Assumptions \[ass:stress\] and \[ass:sT\], \[eq:error.estimate\] $$\begin{aligned} \label{eq:error.estimate:velocity} \| {\und{\b{u}}}_h - {{\und{\b{I}}}_{h}^{k}} \b u \|_{{\boldsymbol{\varepsilon}},{r},h} &\lesssim h^\frac{(k+1)({r^{\circ}}-1)}{{r}+1-{r^{\circ}}} \left(\sigma_{\mathrm{sm}}^{-1}\mathcal N_{\b f}^{2-{r^{\circ}}}\mathcal N_{{\b\sigma},\b u,p}\right)^\frac{1}{{r}+1-{r^{\circ}}}, \\ \label{eq:error.estimate:pressure} \| p_h - {\pi_{h}^{k}} p \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim h^{(k+1)({r^{\circ}}-1)}\mathcal N_{{\b\sigma},\b u,p} + h^{\frac{(k+1)({r^{\circ}}-1)^2}{{r}+1-{r^{\circ}}}} \sigma_{\mathrm{hc}}\mathcal N_{\b f}^{|{r}-2|({r^{\circ}}-1)} \left(\sigma_{\mathrm{sm}}^{-1}\mathcal N_{{\b\sigma},\b u,p}\right)^\frac{{r^{\circ}}-1}{{r}+1-{r^{\circ}}}, \end{aligned}$$ where we have set, for the sake of brevity, $$\begin{aligned} \mathcal N_{{\b\sigma},\b u,p} &\coloneq \sigma_{\mathrm{hc}}\left(\sigma_{\mathrm{de}}^{r}+|\b u|_{W^{1,{r}}(\Omega,{\mathbb{R}}^d)}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}|\b u|_{W^{k+2,{r}}({\mathcal{T}}_h,{\mathbb{R}}^d)}^{{r^{\circ}}-1} \\ &\quad + |{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u)|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{{\mathbb{R}}^{d \times d}})} + |p|_{W^{(k+1)({r^{\circ}}-1),{r}'}({\mathcal{T}}_h,{\mathbb{R}})}, \\ \mathcal N_{\b f} &\coloneqq \sigma_{\mathrm{de}}+\left(\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}-1}+\left(\sigma_{\mathrm{de}}^{2-{r^{\circ}}}\sigma_{\mathrm{sm}}^{-1}\| \b f \|_{L^{{r}'}(\Omega,{\mathbb{R}}^d)}\right)^\frac{1}{{r}+1-{r^{\circ}}}. \end{aligned}$$ \[rem:ocv\] From , neglecting higher-order terms, we infer asymptotic convergence rates of $\mathcal O_{\mathrm{vel}}^k \coloneqq \frac{(k+1)({r^{\circ}}-1)}{{r}+1-{r^{\circ}}}$ for the velocity and $\mathcal O_{\mathrm{pre}}^k \coloneqq \frac{(k+1)({r^{\circ}}-1)^2}{{r}+1-{r^{\circ}}}$ for the pressure, that is, $$\label{eq:asymptotic.order} \mathcal O_{\mathrm{vel}}^k = \begin{cases} (k+1)(r-1) & \text{if $r<2$}, \\ \tfrac{k+1}{r-1} & \text{if $r\ge 2$,} \end{cases}\quad \text{and} \quad \mathcal O_{\mathrm{pre}}^k = \begin{cases} (k+1)(r-1)^2 & \text{if $r<2$}, \\ \tfrac{k+1}{r-1} & \text{if $r\ge 2$.} \end{cases}$$ Notice that, owing to the presence of higher-order terms in the right-hand sides of , higher convergence rates may be observed before attaining the asymptotic ones; see Section \[sec:num.res\]. Let $({\und{\b{e}}}_h, \epsilon_h) \coloneqq ({\und{\b{u}}}_h - {\und{\b{\hat u}}}_h,p_h - \hat p_h) \in {{{\und{\b{U}}}}_{h,0}^{k}} \times {P_{h}^{k}}$ where ${\und{\b{\hat u}}}_h \coloneqq{{\und{\b{I}}}_{h}^{k}} \b u \in {{{\und{\b{U}}}}_{h,0}^{k}}$ and $\hat p_h \coloneqq {\pi_{h}^{k}} p \in {P_{h}^{k}}$.\ **Step 1.** *Consistency error.* Let $\mathcal E_h : {{{\und{\b{U}}}}_{h,0}^{k}} \to {\mathbb{R}}$ be the consistency error linear form such that, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, $$\label{eq:Eh} \mathcal E_h({\und{\b{v}}}_h) \coloneqq \int_\Omega \b f \cdot \b v_h - {\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{v}}}_h)-{\mathrm{b}}_h({\und{\b{v}}}_h,{\hat p}_h).$$ Using in the above expression the fact that $\b f = -{\b\nabla{\cdot}}{\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u)+{\b\nabla}p$ almost everywhere in $\Omega$ to write $\mathcal E_h({\und{\b{v}}}_h)=\mathcal E_{{\mathrm{a}},h}(\b u;{\und{\b{v}}}_h) + \mathcal E_{{\mathrm{b}},h}(p;{\und{\b{v}}}_h)$, and invoking the consistency properties of ${\mathrm{a}}_h$ and of ${\mathrm{b}}_h$, we obtain $$\label{eq:error.estimate:step1:eh0} \$\coloneq \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1}\mathcal E_h({\und{\b{v}}}_h) \lesssim h^{(k+1)({r^{\circ}}-1)} \mathcal N_{{\b\sigma},\b u,p}.$$\ **Step 2.** *Error estimate for the velocity.* Using the strong monotonicity of ${\mathrm{a}}_h$, we get $$\label{eq:error.estimate:step2:eh0} \begin{aligned} \| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}} &\lesssim \sigma_{\mathrm{sm}}^{-1}\left( \sigma_{\mathrm{de}}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\|{\und{\b{\hat u}}}_h\|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{2-{r^{\circ}}}{{r}}\left( {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{e}}}_h) \right) \\ &\lesssim \sigma_{\mathrm{sm}}^{-1}\mathcal N_{\b f}^{2-{r^{\circ}}}\left( {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{e}}}_h) \right), \end{aligned}$$ where we have used the a priori bound on the discrete solution along with the boundedness of the global interpolator and the a priori bound on the continuous solution to conclude. Using then the discrete mass equation along with (written for $\b v=\b u$) and the continuous mass equation to write ${\mathrm{b}}_h({{\und{\b{I}}}_{h}^{k}} \b u,q_h) = b(\b u,q_h) = 0$, we get ${\mathrm{b}}_h({\und{\b{e}}}_h,q_h) = 0$ for all $q_h \in P^k_h$. Hence, combining this result with and the discrete momentum equation (with ${\und{\b{v}}}_h = {\und{\b{e}}}_h$), we obtain $$\label{eq:error.estimate:step1:eh1} {\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{e}}}_h)-{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{e}}}_h) = \int_\Omega \b f \cdot \b e_h - {\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{e}}}_h)-\cancel{{\mathrm{b}}_h({\und{\b{e}}}_h,p_h)} = \mathcal E_h({\und{\b{e}}}_h).$$ Plugging into , we get $$\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r}+2-{r^{\circ}}} \le \sigma_{\mathrm{sm}}^{-1}\mathcal N_{\b f}^{2-{r^{\circ}}}\$\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}.$$ Simplifying, using , and taking the $({r}+1-{r^{\circ}})$th root of the resulting inequality yields .\ **Step 3.** *Error estimate for the pressure.* Using the Hölder continuity of ${\mathrm{a}}_h$, we have, for all ${\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}}$, $$\label{eq:error.estimate:step3:ah} \begin{aligned} \left|{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{v}}}_h)-{\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)\right| &\lesssim \sigma_{\mathrm{hc}}\left( \sigma_{\mathrm{de}}^{r}+ \| {\und{\b{\hat u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}+\| {\und{\b{u}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{r}\right)^\frac{{r}-{r^{\circ}}}{{r}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} \\ &\lesssim \sigma_{\mathrm{hc}}\mathcal N_{\b f}^{{r}-{r^{\circ}}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1}\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}, \end{aligned}$$ where the first factor is estimated as in . Thus, using the inf-sup condition , we can write $$\label{eq:error.estimate:step3:epsilonh} \begin{aligned} \| \epsilon_h \|_{L^{{r}'}(\Omega,{\mathbb{R}})} &\lesssim \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} {\mathrm{b}}_h({\und{\b{v}}}_h,\epsilon_h) \\ &= \sup\limits_{{\und{\b{v}}}_h \in {{{\und{\b{U}}}}_{h,0}^{k}},\| {\und{\b{v}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h} = 1} \left(\mathcal E_h({\und{\b{v}}}_h)+{\mathrm{a}}_h({\und{\b{\hat u}}}_h,{\und{\b{v}}}_h)-{\mathrm{a}}_h({\und{\b{u}}}_h,{\und{\b{v}}}_h)\right)\\ &\lesssim \$+\sigma_{\mathrm{hc}}\mathcal N_{\b f}^{{r}-{r^{\circ}}}\| {\und{\b{e}}}_h \|_{{\boldsymbol{\varepsilon}},{r},h}^{{r^{\circ}}-1} \\ &\lesssim h^{(k+1)({r^{\circ}}-1)}\mathcal N_{{\b\sigma},\b u,p} +h^{(k+1)({r^{\circ}}-1)^2}\sigma_{\mathrm{hc}}\mathcal N_{\b f}^{|{r}-2|({r^{\circ}}-1)} \left(\sigma_{\mathrm{sm}}^{-1}\mathcal N_{{\b\sigma},\b u,p}\right)^\frac{{r^{\circ}}-1}{{r}+1-{r^{\circ}}}, \end{aligned}$$ where we have used the definition of the consistency error together with equation to pass to the second line, to pass to the third line (recall that $\$$ denotes here the supremum in the left-hand side of ), and the bounds and (proved in Step 2) to conclude. Numerical examples {#sec:num.res} ================== We consider a manufactured solution to problem in order to assess the convergence of the method, which was implemented within the SpaFEDTe library (cf. <https://spafedte.github.io>). Specifically, we take $\Omega=(0,1)^{2}$ and consider the $(1,0,1,{r})$-Carreau–Yasuda law (corresponding to the power-law model) with Sobolev exponent ${r}\in\{1.5, 1.75, 2, 2.25, 2.5, 2.75\}$. The exact velocity $\b u$ and pressure $p$ are given by, respectively, $$\b u(x,y) = \left(\sin\left(\tfrac{\pi}{2}x\right)\cos\left(\tfrac{\pi}{2}y\right),-\cos\left(\tfrac{\pi}{2}x\right)\sin\left(\tfrac{\pi}{2}y\right)\right),\quad p(x,y) = \sin\left(\tfrac{\pi}{2} x\right)\sin\left(\tfrac{\pi}{2} y\right)-\tfrac{4}{\pi^2}.$$ The volumetric load $\b f$ and the Dirichlet boundary conditions are inferred from the exact solution. This solution matches the assumptions required in Theorem \[thm:error.estimate\] for $k = 1$, except the case $r = 1.5$ for which ${\b\sigma}(\cdot,{\b{\nabla}_{\mathrm{s}}}\b u) \notin W^{1,{r}'}(\Omega,{{\mathbb{R}}^{d \times d}_{\mathrm{s}}})$. We consider the HHO scheme for $k=1$ on three mesh families, namely Cartesian orthogonal, distorted triangular, and distorted Cartesian; see Figure \[fig:meshes\]. Overall, the results are in agreement with the theoretical predictions, and in some cases the expected asymptotic orders of convergence are exceeded. Specifically, for ${r}\neq 2$, the convergence rates computed on the last refinement surpass in some cases the theoretical ones. As noticed in Remark \[rem:ocv\], this suggests that the asymptotic order is still not attained. A similar phenomenon has been observed on certain meshes for the $p$-Laplace problem; see [@Di-Pietro.Droniou:17*1 Section 3.5.2] and [@Di-Pietro.Droniou.ea:18 Section 3.7]. ![Coarsest Cartesian, distorted triangular, and distorted Cartesian mesh used in the numerical tests of Section \[sec:num.res\].\[fig:meshes\]](square.pdf "fig:") ![Coarsest Cartesian, distorted triangular, and distorted Cartesian mesh used in the numerical tests of Section \[sec:num.res\].\[fig:meshes\]](disttri.pdf "fig:") ![Coarsest Cartesian, distorted triangular, and distorted Cartesian mesh used in the numerical tests of Section \[sec:num.res\].\[fig:meshes\]](distsquare.pdf "fig:") ![Numerical results for the test case of Section \[sec:num.res\]. The slopes indicate the expected order of convergence expected from Theorem \[thm:error.estimate\], i.e. $\mathcal O_{\mathrm{vel}}^1 = 2(r-1)$ and $\mathcal O_{\mathrm{pre}}^1 = 2(r-1)^2$ for ${r}\in \{1.5,1.75,2\}$. \[tab:num.res.1\]](conv1.pdf) ![Numerical results for the test case of Section \[sec:num.res\]. The slopes indicate the expected order of convergence expected from Theorem \[thm:error.estimate\], i.e. $\mathcal O_{\mathrm{vel}}^1 = \mathcal O_{\mathrm{pre}}^1 = \frac{2}{r-1}$ for ${r}\in \{2.25,2.5,2.75\}$. \[tab:num.res.2\]](conv2.pdf) Power-framed functions {#sec:properties.stress} ====================== In the following theorem, we introduce the notion of power-framed function and discuss sufficient conditions for this property to hold. \[thm:1d.power-framed\] Let $U$ be a measurable subset of ${\mathbb{R}}^n$ with $n\ge1$, $(W,(\cdot,\cdot)_W)$ an inner product space, and $\b \sigma : U \times W \to W$. Assume that there exists a Carathéodory function $\varsigma : U \times \lbrack0,+\infty) \to {\mathbb{R}}$ such that, for all $\b\tau \in W$ and almost every $\b x \in U$, \[eq:1d.power-framed:stress\] $${\b\sigma}(\b x,\b\tau) = \varsigma(\b x,\|\b\tau\|_W)\b\tau,$$ where $\|{\cdot}\|_W$ is the norm induced by $(\cdot,\cdot)_W$. Additionally assume that, for almost every $\b x \in U$, $\varsigma(\b x,\cdot)$ is differentiable on $(0,+\infty)$ and there exist $\varsigma_{\mathrm{de}} \in \lbrack0,+\infty)$ and $\varsigma_{\mathrm{sm}},\varsigma_{\mathrm{hc}} \in (0,+\infty)$ independent of $\b x$ such that, for all $\alpha \in (0,+\infty)$, $$\begin{aligned} \varsigma_{\mathrm{sm}} (\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}} \leq \frac{\partial(\alpha\varsigma(\b x,\alpha))}{\partial \alpha} \leq \varsigma_{\mathrm{hc}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}}. \label{eq:1d.power-framed:eta} \end{aligned}$$ Then, ${\b\sigma}$ is an *${r}$-power-framed function*, i.e., for all $(\b\tau,\b\eta) \in W^2$ with $\b\tau \neq \b\eta$ and almost every $\b x \in U$, the function ${\b\sigma}$ verifies the Hölder continuity property \[eq:od:power-framed:holder.continuity.strong.monotonicity\] $$\|{\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\|_W \le \sigma_{\mathrm{hc}} \left(\sigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\| \b\tau-\b\eta \|_W,\label{eq:od:power-framed:holder.continuity}$$ and the strong monotonicity property $$\left({\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta),\b\tau-\b\eta\right)_W \ge \sigma_{\mathrm{sm}}\left(\sigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\|\b\tau-\b\eta\|_W^{2},\label{eq:od:power-framed:strong.monotonicity}$$ with $\sigma_{\mathrm{de}} \coloneqq \varsigma_{\mathrm{de}}$, $\sigma_{\mathrm{hc}} \coloneqq 2^{2-{r^{\circ}}+{r}^{-1}\left\lceil\hspace{0.02cm}2-{r^{\circ}}\right\rceil}({r^{\circ}}-1)^{-1} \varsigma_{\mathrm{hc}}$, and $\sigma_{\mathrm{sm}} \coloneqq 2^{{r^{\circ}}-{r}-\left\lceil{r}^{-1}({r}-{r^{\circ}})\right\rceil}({r}+1-{r^{\circ}})^{-1} \varsigma_{\mathrm{sm}}$, where ${r^{\circ}}$ is given by and $\lceil{\cdot}\rceil$ is the ceiling function. The boldface notation for the elements of $W$ is reminescent of the fact that Theorem \[thm:1d.power-framed\] is used with $W = {{\mathbb{R}}^{d \times d}_{\mathrm{s}}}$ in Corollary \[cor:Carreau–Yasuda\] to characterize the Carreau-Yasuda law as an ${r}$-power-framed function and in Lemma \[lem:sT\] with $W = {\mathbb{R}}^d$ to study the local stabilization function ${\mathrm{s}}_T$. Let $\b x \in U$ be such that holds, and $\b\tau,\b\eta \in W$. By symmetry of inequalities and the fact that ${\b\sigma}$ is continuous, we can assume, without loss of generality, that $\|\b\tau\|_W > \|\b\eta\|_W > 0$.\ (i) *Strong monotonicity.* Let $\beta \in (0,+\infty)$ and let $g : \lbrack\beta,+\infty) \to {\mathbb{R}}$ be such that, for all $\alpha \in \lbrack\beta,+\infty)$, $$g(\alpha) \coloneqq \alpha\varsigma(\b x,\alpha)-\beta\varsigma(\b x,\beta)-C_{\mathrm{sm}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}(\alpha-\beta), \;\text{ with }\ C_{\mathrm{sm}} \coloneqq \tfrac{2^{{r^{\circ}}-{r}}}{{r}+1-{r^{\circ}}}\varsigma_{\mathrm{sm}}.$$ Differentiating $g$ and using the first inequality in , we obtain, for all $\alpha \in \lbrack\beta,+\infty)$, $$\begin{aligned} \frac{\partial}{\partial \alpha}g(\alpha) &\geq \varsigma_{\mathrm{sm}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}}-C_{\mathrm{sm}}\left(({r}-2)(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^{-\frac{2}{{r}}}(\alpha-\beta)\alpha^{{r}-1}+(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}\right) \\ &\geq \varsigma_{\mathrm{sm}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}}-({r}+1-{r^{\circ}})C_{\mathrm{sm}} (\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}} \\ &\geq \varsigma_{\mathrm{sm}}2^{{r^{\circ}}-{r}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}-({r}+1-{r^{\circ}})C_{\mathrm{sm}} (\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}} = 0, \end{aligned}$$ where, to pass to the second line, we have removed negative contributions if ${r}< 2$ and used the fact that $(\alpha-\beta)\alpha^{{r}-1} \le \varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r}$ if ${r}\ge 2$, to pass to the third line we have used the fact that $t \mapsto t^{{r}-2}$ is non-increasing if ${r}< 2$, and the fact that $\beta \le \alpha$ otherwise, while the conclusion follows from the definition of $C_{\mathrm{sm}}$. This shows that $g$ is non-decreasing. Hence, for all $\alpha\in\lbrack\beta,+\infty)$, $g(\alpha)\geq g(\beta)=0$, i.e. $$\label{eq:1d.power-framed:bound:1} \alpha\varsigma(\b x,\alpha)-\beta\varsigma(\b x,\beta) \geq C_{\mathrm{sm}}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}(\alpha-\beta).$$ Moreover, for all $\alpha,\beta \in (0,+\infty)$, using (with $\beta = 0$) along with the fact that $t \mapsto t^{{r}-2}$ is decreasing if ${r}< 2$ and inequality if ${r}\ge 2$, we infer that $$\label{eq:1d.power-framed:bound:2} \begin{aligned} \varsigma(\b x,\alpha)+\varsigma(\b x,\beta) &\ge C_{\mathrm{sm}}\left((\varsigma_{\mathrm{de}}^{r}+\alpha^{r})^\frac{{r}-2}{{r}}+(\varsigma_{\mathrm{de}}^{r}+\beta^{r})^\frac{{r}-2}{{r}}\right) \ge C_{\mathrm{sm}}2^{1-\left\lceil\frac{{r}-{r^{\circ}}}{{r}}\right\rceil}(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r})^\frac{{r}-2}{{r}}. \end{aligned}$$ We conclude that ${\b\sigma}$ verifies by using and with $\alpha = \|\b\tau\|_W$ and $\beta = \|\b\eta\|_W$ as follows: $$\begin{aligned} &({\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta),\b\tau-\b\eta)_W \\ &\quad= (\b\tau\varsigma(\b x,\|\b\tau\|_W)-\b\eta\varsigma(\b x,\|\b\eta\|_W),\b\tau-\b\eta)_W \\ &\quad = \|\b\tau\|_W^2\varsigma(\b x,\|\b\tau\|_W)+\|\b\eta\|_W^2\varsigma(\b x,\|\b\eta\|_W) -(\b\tau,\b\eta)_W \left[\varsigma(\b x,\|\b\tau\|_W) + \varsigma(\b x,\|\b\eta\|_W)\right] \\ &\quad = \left[ \|\b\tau\|_W\varsigma(\b x,\|\b\tau\|_W)-\|\b\eta\|_W\varsigma(\b x,\|\b\eta\|_W) \right](\|\b\tau\|_W-\|\b\eta\|_W) \\ &\qquad + \left[ \varsigma(\b x,\|\b\tau\|_W)+\varsigma(\b x,\|\b\eta\|_W) \right](\|\b\tau\|_W\|\b\eta\|_W-(\b\tau,\b\eta)_W) \\ &\quad\geq C_{\mathrm{sm}}2^{-\left\lceil\frac{{r}-{r^{\circ}}}{{r}}\right\rceil}\left( \varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\left[ (\|\b\tau\|_W-\|\b\eta\|_W)^2+2(\|\b\tau\|_W\|\b\eta\|_W-(\b\tau,\b\eta)_W) \right] \\ &\quad= C_{\mathrm{sm}}2^{-\left\lceil\frac{{r}-{r^{\circ}}}{{r}}\right\rceil}\left( \varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\|\b\tau-\b\eta\|_W^2. \end{aligned}$$\ (ii) *Hölder continuity.* Now, setting $C_{\mathrm{hc}} \coloneqq \frac{\varsigma_{\mathrm{hc}}}{{r^{\circ}}-1}$ and reasoning in a similar way as for the proof of to leverage the second inequality in , we have, for all $\alpha \in \lbrack\beta,+\infty)$, $$\label{eq:1d.power-framed:bound:3} \alpha\varsigma(\b x,\alpha)-\beta\varsigma(\b x,\beta) \le C_{\mathrm{hc}}\left( \varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r}\right)^\frac{{r}-2}{{r}}(\alpha-\beta).$$ First, let ${r}\ge 2$. Using (with $\beta=0$) and the fact that $t \mapsto t^{{r}-2}$ is non-decreasing, we have, for all $\alpha,\beta \in (0,+\infty)$, $$\label{eq:1d.power-framed:bound:4} \varsigma(\b x,\alpha)\varsigma(\b x,\beta) \le C_{\mathrm{hc}}^2\left( \varsigma_{\mathrm{de}}^{r}+\alpha^{r}\right)^\frac{{r}-2}{{r}}\left( \varsigma_{\mathrm{de}}^{r}+\beta^{r}\right)^\frac{{r}-2}{{r}} \le \left[ C_{\mathrm{hc}}\left(\varsigma_{\mathrm{de}}^{r}+\alpha^{r}+\beta^{r}\right)^\frac{{r}-2}{{r}} \right]^2.$$ Thus, using inequalities and with $\alpha = \|\b\tau\|_W$ and $\beta = \|\b\eta\|_W$, we infer $$\label{eq:1d.power-framed:bound:5} \begin{aligned} &\|{\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\|_W^2 \\ &\quad = \left( \b\tau\varsigma(\b x,\|\b\tau\|_W)-\b\eta\varsigma(\b x,\|\b\eta\|_W), \b\tau\varsigma(\b x,\|\b\tau\|_W)-\b\eta\varsigma(\b x,\|\b\eta\|_W) \right)_W \\ &\quad = \left[\|\b\tau\|_W\varsigma(\b x,\|\b\tau\|_W)-\|\b\eta\|_W\varsigma(\b x,\|\b\eta\|_W)\right]^2 \\ &\quad \qquad + 2\varsigma(\b x,\|\b\tau\|_W)\varsigma(\b x,\|\b\eta\|_W)\left[ \|\b\tau\|_W\|\b\eta\|_W-(\b\tau,\b\eta)_W \right] \\ &\quad \le \left[ C_{\mathrm{hc}}\left(\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}} \right]^2\left[ (\|\b\tau\|_W-\|\b\eta\|_W)^2+2(\|\b\tau\|_W\|\b\eta\|_W-(\b\tau,\b\eta)_W) \right] \\ &\quad = \left[ C_{\mathrm{hc}}\left(\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{{r}-2}{{r}}\|\b\tau-\b\eta\|_W \right]^2, \end{aligned}$$ hence ${\b\sigma}$ verifies for ${r}\ge 2$. Assume now ${r}< 2$. Using a triangle inequality followed by and the left inequality in , it is inferred that $$\begin{aligned} \|{\b\sigma}(\b x,\b\tau)-{\b\sigma}(\b x,\b\eta)\|_W &\leq \varsigma(\b x,\|\b\tau\|_W)\|\b\tau\|_W+\varsigma(\b x,\|\b\eta\|_W)\|\b\eta\|_W \\ & \leq C_{\mathrm{hc}}\left((\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r})^\frac{{r}-1}{{r}}+(\varsigma_{\mathrm{de}}^{r}+\|\b\eta\|_W^{r})^\frac{{r}-1}{{r}}\right) \\ &\leq 2^\frac{1}{{r}}C_{\mathrm{hc}}(2\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r})^\frac{{r}-1}{{r}} \\ & = 2^\frac{1}{{r}} C_{\mathrm{hc}}(2\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r})^\frac{{r}-2}{{r}} (2\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r})^\frac1{r}, \\ & \leq 2^\frac{1}{{r}} C_{\mathrm{hc}}(\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r})^\frac{{r}-2}{{r}} (2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W), \end{aligned}$$ where the last line follows from the fact that $t \mapsto t^{{r}-2}$ is decreasing and again . If $2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W\le 2^{2-r}\|\b\tau-\b\eta\|_W$, from the previous bound we directly get the conclusion, i.e. with $\sigma_{\mathrm{hc}}=2^{2-r+\frac{1}{r}}C_{\mathrm{hc}}$. Otherwise, using and a triangle inequality yields $$\label{eq:est_else} \begin{aligned} (\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r})^\frac{1}{{r}}(\varsigma_{\mathrm{de}}^{r}+\|\b\eta\|_W^{r})^\frac{1}{{r}} &\ge 2^{-\frac{2}{r'}}(\varsigma_{\mathrm{de}}+\|\b\tau\|_W)(\varsigma_{\mathrm{de}}+\|\b\eta\|_W) \\ &= 2^{-2(\frac{1}{r'}+1)}\left[ \left(2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W\right)^2-\left(\|\b\tau\|_W-\|\b\eta\|_W\right)^2 \right] \\ &\ge 2^{-2(\frac{1}{r'}+1)}\left[ \left(2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W\right)^2-\|\b\tau-\b\eta\|_W^2 \right] \\ &\ge 2^{-2(\frac{1}{r'}+1)}(1-4^{r-2})\left(2\varsigma_{\mathrm{de}}+\|\b\tau\|_W+\|\b\eta\|_W\right)^2\\ &\ge 2^{\frac{2}{(r-2)r}-2}\left(\varsigma_{\mathrm{de}}^{r}+\|\b\tau\|_W^{r}+\|\b\eta\|_W^{r}\right)^\frac{2}{{r}}, \end{aligned}$$ where we concluded with together with the fact that $2^{-2(\frac{1}{r'}+1)}\left(1-4^{r-2}\right) \ge 2^{\frac{2}{(r-2)r}-2}$. Finally, raising both sides of to the power ${r}-2$, we get a relation analogous to . Hence, proceeding as in , we infer . \[cor:Carreau–Yasuda\] The strain rate-shear stress law of the $(\mu,\delta,a,{r})$-Carreau–Yasuda fluid defined in Example \[ex:Carreau–Yasuda\] is an ${r}$-power-framed function. Let $\b x \in \Omega$ and $g : (0,+\infty) \to {\mathbb{R}}$ be such that, for all $\alpha\in(0,+\infty)$, $$g(\alpha) \coloneqq \frac{\partial}{\partial \alpha}\left[ \alpha\mu(\b x)\left(\delta^{a(\b x)}+\alpha^{a(\b x)} \right)^\frac{{r}-2}{a(\b x)} \right] = \mu(\b x)\left( \delta^{a(\b x)}+\alpha^{a(\b x)} \right)^{\frac{{r}-2}{a(\b x)}-1}\left( \delta^{a(\b x)}+({r}-1)\alpha^{a(\b x)} \right).$$ We have for all $\alpha \in (0,+\infty)$, $$\mu_- ({r^{\circ}}-1) \left( \delta^{a(\b x)}+\alpha^{a(\b x)} \right)^{\frac{{r}-2}{a(\b x)}} \le g(\alpha) \leq \mu_+ ({r}+1-{r^{\circ}})\left( \delta^{a(\b x)}+\alpha^{a(\b x)} \right)^{\frac{{r}-2}{a(\b x)}},$$ and we conclude using together with Theorem \[thm:1d.power-framed\]. [^1]: [<[email protected]>]{} [^2]: [<[email protected]>]{} [^3]: [<[email protected]>]{} [^4]: [<[email protected]>]{}, corresponding author

ArXiv

--- abstract: 'We observed atmospheric gamma-rays around 10 GeV at balloon altitudes (15$\sim$25 km) and at a mountain (2770 m a.s.l). The observed results were compared with Monte Carlo calculations to find that an interaction model (Lund Fritiof1.6) used in an old neutrino flux calculation was not good enough for describing the observed values. In stead, we found that two other nuclear interaction models, Lund Fritiof7.02 and dpmjet3.03, gave much better agreement with the observations. Our data will serve for examining nuclear interaction models and for deriving a reliable absolute atmospheric neutrino flux in the GeV region.' author: - 'K. Kasahara' - 'E. Mochizuki' - 'S. Torii' - 'T. Tamura' - 'N. Tateyama' - 'K. Yoshida' - 'T. Yamagami' - 'Y. Saito' - 'J. Nishimura' - 'H. Murakami' - 'T. Kobayashi' - 'Y. Komori' - 'M.Honda' - 'T. Ohuchi' - 'S. Midorikawa' - 'T. Yuda' bibliography: - 'betsgamma.bib' title: 'Atmospheric gamma-ray observation with the BETS detector for calibrating atmospheric neutrino flux calculations' --- Introduction ============ The discovery of evidence for neutrino oscillation by the Super Kamiokande group[@skoscillation] is based on the comparison of the observed atmospheric neutrino flux with calculated values. Although the conclusion is so derived that it would not be upset by the uncertainty of the absolute flux value, it is desirable to obtain a reliable expected neutrino flux (under no oscillation assumption) for further detailed discussions. Two major sources of uncertainty in the atmospheric neutrino flux calculation are 1) the primary cosmic-ray spectrum and 2) the propagation of cosmic rays in the atmosphere, especially, modeling of the nuclear interaction. The absolute flux calculations so far made by various groups are expected to have uncertainty of $\sim$ 30 %[@GHreview]. The primary proton and He spectra recently measured with magnet spectrometers by the BESS [@bess1ry] and AMS[@ams1ry] groups agree very well and seem reliable. Therefore, we may take that the first problem mentioned above have now been almost settled at least up to 100 GeV/n. This means that if we have a reliable atmospheric cosmic-ray flux data, we may compare it with a calculation which uses such primaries and test the validity of nuclear interaction models. For such an atmospheric cosmic-ray component, one may first raise the muon and actually some new observations have been or being tried[@capricemuon1; @capricemuon2; @bessmuonnori]. As a secondary cosmic-ray component, we focused on gamma-rays which are easy to measure with our detector. A good model should be able to explain muons and gamma-rays simultaneously. Muons are important since they are directly coupled with neutrinos, but the flux is affected somehow by the structure of the atmosphere which is usually not well known. Compared to muons, the flux of gamma-rays is substantially lower but is almost insensitive to the atmospheric structure and depends only on the total thickness to the observation height. In 1998, we performed first gamma-ray observation with our detector at Mt. Norikura (2770m a.s.l) in Japan, and also made subsequent two successful observations at balloon altitudes (15 $\sim 25$ km) in 1999 and 2000. In the present paper, we report the final results of these observations and consequences. The Detector ============ For our observation, we upgraded the BETS (Balloon-born Electron Telescope with Scintillating fibers) detector which had been developed for the observation of cosmic primary electrons in the 10 GeV region. Its details before being upgraded for gamma-ray observation is in [@betsnim] and the electron observation result is in [@betselec]. The basic performance was tested at CERN using electron, proton and pion beams of 10 to 200 GeV[@betsnim; @betscern]. Although this was undertaken before the upgrading, we can essentially use that calibration for the current observeions partly with a help of Monte Carlo simulations. Figure \[det\] shows a schematic structure of the main body of BETS. The calorimeter has 7.1 r.l lead thickness and the cross-section is 28 cm $\times$ 28 cm. The whole detector system is contained in a pressure vessel made of thin aluminum. ![Schematic illustration of the main body of the detector. S1, S2 and S3 are 1 cm thick plastic scintillators used for trigger. Each fiber has 1mm diameter. Originally nuclear emulsion plates were placed on the upper scifi’s and also inserted between the upper thin lead plates for detailed investigation of tracking capability of scifi. They are kept in the present system to have the same structure at the calibration time. The inlaid cascade shows charged particle tracks by a simulation for a 30 GeV incident proton. \[det\]](detconfigwithshower.eps){width="92mm"} R.M.S energy resolution(%) 21, 18, 15 (for $\theta\sim 15^\circ$) ---------------------------------- ------------------------------------------- S$\Omega$(cm$^2$sr) 243, 240,218 (at $\sim$20 km) R.M.S angular resolution (deg) 2.3, 1.3, 1.0 (for $\theta\sim 15^\circ$) Total number of scifi’s 10080 Weght including electronics (kg) 230 Cross-section of the main body 28cm $\times $ 28cm Thickness (Pb radiation length) 7.1 : Basic characteristics of BETS\ (triple numbers in the table are for gamma-ray energy of 5, 10, and 30 GeV, respectively) \[basicchara\] The main feature of the BETS detector is that it is a tracking calorimeter; it contains a number of sheets consisting of 1 mm diameter scintillating fibers (scifi), many of which are sandwiched between lead plates. The total number of scifi’s are 10080. The sheets are grouped into two types; one is to serve for x and the other for y position measurement. Each of them is fed to an image intensifier which in turn is connected to a CCD. Thus, the two CCD output gives us an $x-y$ image of cascade shower development and enables us to discriminate gamma-rays, electrons from other (mainly hadronic) background showers. The proton rejection power against electron is $R\sim 2\times 10^3$ (i.e, one misidentification among $R$ protons) at 10 GeV[^1] The basic characteristics of the detector are summarized in Table \[basicchara\]. ![Image of cascade shower by a proton (120 GeV,left) and an electron(10 GeV, right) obtained at CERN. \[image\]](showerimage.eps){width="85mm"} In Fig.\[image\], we show examples of the CCD image of a cascade shower for a proton incident case and for an electron incident case. Figure \[anti\] illustrates the yearly change of anti-counters. In 1998 (Mt.Norikura observation), the main change was limited to the upgrading of trigger logic. In 1999, we added 4 side anti-counters (each 15 cm $\times$ 36 cm $\times$ 1.5 cm plastic scintillator. Nine optical fibers containing wave length shifter are embedded in each scintillator and connected to a Hamamatu H6780 PMT. ![Yearly change of the anti-counters. Left: 1998. No change from original BETS except for trigger logic. Middle: 1999. 1.5 cm thick plastic scintillator side anti-counters were added. Right: 2000. The whole top view was covered by a 1 cm thick plastic sintillator. \[anti\]](yearlychange.eps){width="85mm"} In 2000, we further added an anti-counter which covers the whole top view of the detector and also improved data acquisition speed. The top anti-counter is 38 cm $\times$ 38 cm $\times$ 1 cm plastic scintillator. We also embedded optical fibers; 8 in the $x$ and another 8 in the $y$ direction, all of which were fed to an H6780. Although we could remove background showers without the anti-counters, inclined particles (mainly protons) entering from the gap between top scintillator (S1) and the main body degrades the desired gamma-ray event rate. The addition of the top anti-counter greatly helped improve this rate. We emphasize that detection of gamma-rays is easier for us than that of electrons, since, for gamma-rays, we can utilize absence of incident charge. Observations ============ Table \[sumtab\] shows the summary of the observations. Observation Mt.Norikura(1998) --------------------- -------------------- ------ ------- ------- ------- ------- ------- ------- ------- ------- Period Aug.31$\sim$Sep.18 Altitude(km) 2.77 15.3 18.5 21.2 24.7 32.3 15.3 18.3 21.4 25.1 Depth(g/cm$^2$) 737 126 74.8 48.9 28.0 9.5 128 73 45.7 25.3 Obs. hour (s) $1.33\times 10^6$ 1260 1560 2100 4878 3120 1560 2160 4320 2320 Live time (s) $9.8\times 10^5$ 504 450 414 852 498 752 928 1805 789 Live time (%) 74.0 40.0 28.8 19.7 17.5 16.0 48.2 43.0 42.6 44.2 Triggered events $1.8\times 10^6 $ 9513 11288 13361 30439 16741 18808 25795 46675 17436 $\gamma$ events $4.7\times 10^4$ 700 650 611 848 345 1300 1485 2299 740 (%) 2.5 7.3 5.7 4.6 2.8 2.0 6.9 5.8 4.9 4.2 g-low trigger S1 $< 0.5$ condition (in mip). S2 $> 2.3$ S3 $> 1.7$ - Mt. Norikura observation. Our first gamma-ray observation was performed in 1998 at Mt.Norikura Observatory of Univ. of Tokyo, Japan (2770 m a.s.l, latitude 36.1$^\circ$N, longitude 137.55$^\circ$E, magnetic cutoff rigidity $\sim$ 11.5 GV). The atmospheric pressure during the observation is shown in Fig.\[noripress\]. The average atmospheric depth is 737 g/cm$^2$. ![Pressure change during Mt. Norikura observation. The last pressure drop is due to a typhoon. The average pressure is 723 hP (737 g/cm$^2$). \[noripress\]](norikurapressure.eps){width="85mm"} - Balloon flight We had two similar balloon filights in 1999 and 2000. Since the main outcome of the data is from the latter, we briefly describe it. A balloon of 43$\times 10^3$ m$^3$ was launched at 6:30 am, 5th June, 2000 from the Sanriku balloon center of the Institute of Space and Astronautical Science, Japan (latitude 39.2$^\circ$N, longitude 141.8$^\circ$E, magnetic cutoff rigidity $\sim$ 8.9 GV) and recovered with the help of the helicopter. at 17:59 on the sea not far from the center. The flight curve shown in Fig.\[flight\] confirms that we have good level flights at 4 different heights. As compared to the 1999 flight, this flight realized a smaller dead time and higher ratio of desired gamma-ray events. ![Flight curve of the 2000 observation. Pressure (upper) and altitude (lower) as a function of time. Each arrow shows the level flight region. The pressure change at around 15.3 km is rather rapid but the gamma-ray intensity is almost constant there and the change can be neglected. \[flight\]](flightcurve.eps){width="73mm"} Event trigger ------------- The basic event trigger condition is created by signals from the three plastic scintillators (S1, S2 and S3). We show the discrimination level in terms of the minimum ionizing particle number which is defined by the peak of the energy loss distribution of cosmic-ray muons passing both S1 and S3 with inclination less than 30 degrees. We prepare a multi-trigger system by which event trigger with different conditions is possible at the same time. The major two trigger modes are the g-low and g-high. The g-low is responsible for low energy gamma-rays and all anti-counters, when available, are used as veto counters. Its condition is listed in Table \[sumtab\]. High energy gamma-rays normally produce a lot of back splash particles which hit S1 and/or anti-counters, and thus the g-low trigger is suppressed. In such a case, i.e, if we have a large S3 signal, anti-counter veto is invalidated and the S1 threshold is relaxed (The g-high condition is S1$<3.0$, S2$>5.0$ and S3$>8.1$). The branch even point of the g-low and g-high mode efficiency is at $\sim $30 GeV. Since we deal with gamma-rays mostly below 30 GeV, and also to avoid complexity, we present results only by the g-low mode. Analysis ======== Event selection --------------- Among the triggered events, we selected gamma-ray candidates by imposing the following conditions: ![(left)Energy concentration distribution at 21.4 km. (right)the same by electrons at CERN []{data-label="conc"}](Econc.eps){width="8.5cm"} 1. The estimated shower axis passes S1 and S3. The axis position in S3 must be at least 2 cm apart from the edge of S3. 2. The estimated shower axis has a zenith angle less than 30 degrees. 3. The energy concentration (see below) must be greater than 0.7. According to a simulation, only neutrons could be a background against gamma-rays and the 3rd conditions above reduces the neutron contribution to a negligible level ($<1$%). The energy concentration is defined as the fraction of scintillating fiber light intensity within 5 mm from the shower axis. Figure \[conc\] shows the concentration of analysed events together with the result of CERN data. Hadrons make a distribution with a peak at around 0.5. We see that the contribution of hadrons in our observation is negligible. Energy Determination -------------------- The energy calibration was performed in 1996 at CERN using electrons with energy 10 $\sim $ 200 GeV[@betsnim; @betscern]. There is no direct calibration for gamma-rays, but, for the present detector thickness and energy range, a M.C simulation tells us that the calibration in 1996 can be used for gamma-rays, too[^2]. Therefore, for the 1998 and 1999 observations, energy is obtained as a function of the S3 output and zenith angle using the CERN calibration. In 2000, we made some change in the electronics so the CERN calibration could not be used directly. The effect by the change was absorbed by a M.C simulation of which the validity was verified by examining the 1998 and 1999 data. We used the sum of S2 and S3 outputs below 20 GeV since the energy resolution was found to be better than using S3 only. Figure \[eresol\] shows r.m.s energy resolution. ![R.m.s energy resolution. The resolution by S2+S3 or S3 only is shown. Different symbols indicate different incident angles. We used S2+S3 below 20 GeV for the year 2000 data. \[eresol\]](Eres.eps){width="8cm"} Correction of the gamma-ray intensity ------------------------------------- The gamma-ray vertical flux is obtained from the raw $dN/dE$ by dividing it by the live time of the detector and the effective $S\Omega$ (area $\times$ solid angle). The latter is obtained by a simulation[@someganu00]. It is dependent on the observation hight and energy. A typical value at 10 GeV is 240 cm$^2$sr (see Table\[basicchara\]). The energy spectrum is further corrected by the following factors which are not taken into account in the $S\Omega$ calculation. ![(upper)Multiple incidence rate. (lower) Correction factor for year 2000 due to spillover. The flux must be lowered. For Norikura, the factor below 20 GeV is larger by 1$\sim 3$ %. []{data-label="correc"}](turehuta.eps "fig:"){width="7cm"} ![(upper)Multiple incidence rate. (lower) Correction factor for year 2000 due to spillover. The flux must be lowered. For Norikura, the factor below 20 GeV is larger by 1$\sim 3$ %. []{data-label="correc"}](ER_hosei.eps "fig:"){width="7cm"} 1. Systematic bias in our estimation of the shower axis. We underestimate the zenith angle systematically and it leads to overestimation of the intensity about 4% for the balloon and 1.8 % for Mt.Norikura observations. 2. Multiple incidence of particles. A gamma-ray is sometimes accompanied by other charged particles and they enter the detector simultaneously (within 1 ns time difference in 99.9 % cases). They are a family of particles generated by one and the same primary particle[^3]. The charged particles fire the anti-counter and the g-low trigger is inhibited. In some case, multiple gamma-rays enter the detector simultaneously. The rate is smaller than the charged particle case. However, this is judged as a hadronic shower in most of cases. The multiple incidence leads to the underestimation of gamma-ray intensity. The portion of multiple incidence is shown in Fig.\[correc\] (upper). 3. Finite energy resolution. The rapidly falling energy spectrum leads to the spillover effect. This normally leads to the overestimation of flux (Fig.\[correc\], lower). Results and comparison with calculations ======================================== The flux values are summarized in Table \[flux\]. We put only the statistical errors in the flux values, since systematic errors coming from the uncertainty of the S$\Omega$ calculation, various cuts and flux corrections are expected to be order of a few percent and much smaller than the present statistical errors. [|l|l|l|l|l|l|l|l|l|l|]{}\ & & & &\ \ 5.48 & 2.42 $\pm$ 0.37 & 5.48 & 2.11 $\pm$ 0.39 & 5.47 & 2.11 $\pm$ 0.24 & 5.47 & 1.58 $\pm$ 0.25 & 5.47 & 0.49 $\pm$ 0.14\ 6.47 & 1.18 $\pm$ 0.27 & 6.47 & 1.10 $\pm$ 0.24 & 6.47 & 1.35 $\pm$ 0.21 & 6.47 & 0.82 $\pm$ 0.18 & 6.57 & 0.19 $\pm$ 0.09\ 7.47 & 0.89 $\pm$ 0.24 & 7.47 & 0.79 $\pm$ 0.21 & 7.47 & 0.82 $\pm$ 0.16 & 7.47 & 0.66 $\pm$ 0.16 & 7.47 & 0.24 $\pm$ 0.10\ 8.48 & 0.37 $\pm$ 0.15 & 8.48 & 0.92 $\pm$ 0.20 & 8.48 & 0.51 $\pm$ 0.13 & 8.48 & 0.49 $\pm$ 0.14 & 8.48 & 0.16 $\pm$ 0.08\ 9.48 & 0.54 $\pm$ 0.17 & 9.85 & 0.46 $\pm$ 0.11 & 9.48 & 0.50 $\pm$ 0.12 & 9.48 & 0.36 $\pm$ 0.12 & 9.48 & 0.16 $\pm$ 0.08\ 10.5 & 0.17 $\pm$ 0.10 & 11.5 & 0.35 $\pm$ 0.12 & 10.5 & 0.41 $\pm$ 0.09 & 10.5 & 0.34 $\pm$ 0.12 & 12.3 & 0.13 $\pm$ 0.037\ 12.1 & 0.28 $\pm$ 0.09 & 14.0 & 0.24 $\pm$ 0.06 & 11.8 & 0.23 $\pm$ 0.069 & 12.2 & 0.21 $\pm$ 0.054 & 17.0 & 0.032 $\pm$ 0.018\ 14.0 & 0.17 $\pm$ 0.05 & 18.3 & 0.072 $\pm$ 0.030 & 14.0 & 0.16 $\pm$ 0.030 & 14.0 & 0.076 $\pm$ 0.03 & 21.7 & 0.022$\pm$ 0.015\ 18.5 & 0.12 $\pm$ 0.04 & 26.8 & 0.040 $\pm$ 0.017 & 18.4 & 0.086 $\pm$ 0.023& 17.8 & 0.078 $\pm$ 0.029 & &\ 25.5 & 0.06 $\pm$ 0.02 & & & 27.1 & 0.026 $\pm$ 0.009& 21.7 & 0.064 $\pm$ 0.026 & &\ & & & & & & 26.8 & 0.024 $\pm$ 0.012 & &\ & & & & & & 36.0 & 0.012 $\pm$ 0.008 & &\ E(GeV) Flux ($10^{-4}/$m$^2\cdot$s$\cdot$sr$\cdot$GeV) -------- ------------------------------------------------- 5.48 274 $\pm$ 13 6.47 183 $\pm$ 11 7.47 133 $\pm$ 9 8.47 87.8 $\pm$ 7.5 9.47 86.5 $\pm$ 7.5 10.5 54.1 $\pm$ 5.9 11.5 46.6 $\pm$ 5.5 12.5 38.3 $\pm$ 5.0 13.5 32.6 $\pm$ 4.6 14.5 24.2 $\pm$ 4.0 15.5 25.7 $\pm$ 4.1 17.0 11.9 $\pm$ 2.0 19.0 15.3 $\pm$ 2.3 21.0 13.1 $\pm$ 2.1 23.0 5.80 $\pm$ 1.4 26.0 5.31 $\pm$ 0.95 30.0 3.00 $\pm$ 0.72 34.0 2.30 $\pm$ 0.64 38.0 1.07 $\pm$ 0.44 45.0 1.45 $\pm$ 0.32 55.0 0.52 $\pm$ 0.20 65.0 0.22 $\pm$ 0.13 75.0 0.30 $\pm$ 0.15 85.0 0.15 $\pm$ 0.10 : Flux values at Mt. Norikura\[noriflux\] The gamma-ray energy spectra thus obtained at balloon altitudes are shown in Fig.\[balspec\] together with the expected ones calculated by the Cosmos simulation code[@cosmos]. Except for 32.3 km altitude, we can disregard the small difference of the observation depths and we combine two flight data with statistical weight, although the main contribution is from the flight in 2000. In the simulation calculation, we employed 3 different nuclear interaction models: 1) fritiof1.6[@oldfri] used in the HKKM calculation[@hkkm95], which was widely used for comparison with the Kamioka data, 2)fritiof7.02[@newfri][^4] and 3) dpmjet3.03[@dpmjet]. As the primary cosmic ray, we used the BESS result on protons and He. The CNO component is also considered[@cno]. Besides these we included electron and positron data by AMS[@amselec]. Their data in the 10 GeV region is consistent with the HEAT[@heat] and BETS[@betselec] data. Bremstrahlung gamma-rays from the primary electrons could contribute order of $\sim 10$ % at very high altitudes. At balloon altitudes, the two models, fritiof7.02 and dpmjet3.03, give almost the same results which are close to the observed data, while fritof1.6 gives clearly smaller fluxes than the observation. Figure \[norispec\] shows the result from the observation at Mt.Norikura. It should be noted that the flux by fritiof1.6 becomes higher than the ones by the other models at this altitude. From these figures, we see fritiof7.02 and dpmjet3.03 give rapider increase and faster attenuation of intensity than fritiof1.6; the tendency is very consistent with the observed data. The transition curve of the flux integrated over 6 GeV shown in Fig.\[transition\] clearly demonstrates this feature. {width="6.5cm"} {width="6.5cm"} {width="6.5cm"} {width="6.5cm"} ![Gamma-ray spectra at 5 balloon heights are compared with 3 different models. The vertical axis is Flux$\times E^2$. Except for 1999 data at 32.3 km, 1999 and 2000 flights data are combined. From top to bottom, at 25.1, 21.4, 18.3, 15.3 and 32.3 km. The spectra expected from three interaction models are drawn by solid (dpmjet3.03), dash (fritiof7.02) and dotted (fritiof1.6) lines. []{data-label="balspec"}](spectrum5.eps){width="6.5cm"} ![Gamma-ray spectrum at Mt. Norikura (2.77 km a.s.l). The vertical axis is Flux$\times E^2$. Our data is at $<$ 100 GeV. Data above 300 GeV is from emulsion chamber experiments. For the latter, see Sec.\[discuss\] []{data-label="norispec"}](norikura.eps){width="7.5cm"} ![The altitude variation of the flux integrated over 6 GeV. The dpmjet3.03 and fritiof7.02 give almost the same feature consistent with the observation while the deviation of fritiof1.6 from the data is obvious. \[transition\]](transition.eps){width="7.5cm"} Discussions\[discuss\] ====================== Comparison with other data -------------------------- We found Fritiof7.02 and dpmjet3.03 give good agreement with the observed gamma-ray data at around 10 GeV. We briefly see whether these models can interpret other observations. More detailed inspection will be done elsewhere. - Muon data by the BESS group at Mt.Norikura[@bessmuonnori]. Recently, the BESS group reported detailed muon spectrum over several hundred MeV/c. In their paper, calculations by dpmjet3.03 and fritiof1.6 are compared with the data; agreement by dpmjet3.03 is quit good at least above GeV where Fritiof7.02 also gives more or less the same flux. On the other hand, fritiof1.6 shows too high flux. These features are consisten with our present analysis. - Higher energy gamma-ray data by emulsion chamber. In Fig. \[norispec\], we inlaid an emulsion chamber data[@ecc][^5] at Mt. Norikura. Our data seems to be smoothly connected to their data as the two interaction models (Fritiof7.02 and dpmjet3.03) predict. Since the emulsion chamber data extends to the TeV region and the primary particle energy responsible for such high energy gamma-rays is much higher than 100 GeV where we have no accurate information comparable to the AMS and BESS data, it would be premature to draw a definite conclusion on the primary and interaction model separately. However, the fact that smooth extrapolation of the primary spectra as shown in Table \[extendprim\] and the interaction model, dpmjet3.03 or fritiof7.02, give a consistent result with the data, seems to indicate that such combination would provide a good estimate on other components at $\gg$ 10 GeV. ------- ----------- ------- ----------- -------- ---------   E    flux    E    flux    E   flux 92.6 0.593E-01 79.4 0.549E-02 100. 9.0E-5 108 0.388E-01 100. 3.0E-3 400. 1.8E-6 126 0.276E-01 200. 5.0E-4 2.0E3 3.5E-8 147 0.179E-01 400. 7.0E-5 2.0E4 9.3E-11 171 0.124E-01 2.0E3 9.98E-7 2.0E5 2.3E-13 200 0.836E-02 2.0E4 2.5E-9 14.0E5 1.3E-15 1100 8.29E-5 2.0E5 3.97E-12 3.0E6 1.7E-16 1.1E4 1.47E-7 4.0E5 6.1E-13 3.0E7 2.0E-19 1.1E5 2.8E-10 8.0E5 7.0E-14 3.0E8 2.2E-22 2.2E5 3.7E-11 8.0E6 8.7E-17 4.4E5 5.0E-12 8.0E8 5.3E-23 4.4E8 2.8E-21 ------- ----------- ------- ----------- -------- --------- : Primary flux assumed in the simulation above 100 GeV/n\ (E in kinetic energy per nucleon (GeV), flux in /m$^2\cdot$s$\cdot$sr$\cdot$GeV) \[extendprim\] The $x$-distributions --------------------- The two models, fritiof7.02 and dpmjet3.03, give almost the same results in the present comparison. However, if we look into the $x$-distribution of the particle production, we note some difference, especially in the proton $x$-distribution. We define the $x$ as the kinetic energy ratio of the incoming proton and a secondary particle in the laboratory frame. The $x$ distribution for $p$Air collisions at incident proton energy of 40 GeV is presented for photons (from $\pi^0$ plus $\eta$ decay) and protons in Fig.\[xdist\]. Difference of the three models seen in the photon distribution is quite similar to the one for charged pions. The $x$ region most effective to atmospheric gamma-ray flux is around 0.2$\sim$0.3 where the difference is not so large but fritiof7.02 and dpmjet3.03 have higher gamma-ray yield than fritiof1.6. ![The $x$-distribution of photons from $\pi^0$ plus $\eta$ decay (upper) and protons (lower) for $p$Air collisions at 40 GeV. The three model results are shown. []{data-label="xdist"}](gammaxdist.eps "fig:"){width="7.5cm"} ![The $x$-distribution of photons from $\pi^0$ plus $\eta$ decay (upper) and protons (lower) for $p$Air collisions at 40 GeV. The three model results are shown. []{data-label="xdist"}](protonxdist.eps "fig:"){width="7.5cm"} On the other hand, the proton $x$ distribution has larger difference among the three models (we note, however, the difference may be exaggerated than the photon case due to the scale difference). It is interesting to see that, in spite of these large differences, the final flux is not so much different each other. Our gamma-ray data prefers to rather more inelastic feature of collisions than fritiof1.6, i.e rapider increase and faster attenuation of the flux. We should compare the distribution with accelerator data; however, there is meager stuff appropriate for our purpose. One such comparison has been done in a recent review paper[@GHreview] for $p$Air collisions at 24 GeV/c incident momentum. The charged pion distribution by fritiof1.6 and dpmjet3.03 well fit to some scattered data which prevents to tell the superiority of the two. As to the proton distribution, among the three models, fritiof1.6 is rather close to the data but deviation from the data is much larger than the pion case. The proton $x$-distribution would strongly affect the atmospheric proton spectrum. We calculated proton flux at Mt.Norikura to find a flux relation such that fritiof1.6 $>$ fritiof7.02 $>$ dpmjet3.03 as expected naturally from the $x$-distributions. The maximum difference is factor $\sim 2.5$ in the energy region of 0.3 to 3 GeV. The BESS group has measured the proton spectrum at Mt. Norikura in the same energy region. Their result expected to come soon[@sanukibess] will help select a better model for the proton $x$ distribution. summary ======= - We have made successful observation of atmospheric gamma-rays at around 10 GeV at Mt.Norikura (2.77 km a.s.l) and at balloon altitudes (15 $\sim$ 25 km). - The observed gamma-ray fluxes are compared with calculations by three interaction models; it is found that fritiof1.6 employed by the HKKM calculation [@hkkm95], which was used in comparison with the Kamioka data, is not a very good model. - Other two models (fritiof7.02 and dpmjet3.03) give better results consistent with the data, which shows rapider increase and faster attenuation of the flux than fritiof1.6 predicts. - Our data has complementary feature to muon data and will serve for checking nuclear interaction models used in atmospheric neutrino calculations. We sincerely thank the team of the Sanriku Balloon Center of the Institute of Astronautical Science for their excellent service and the support of the balloon flight. We also thank the staff of the Norikra Cosmic-Ray observatory, Univ. of Tokyo. for their help. We are also indebted to S.Suzuki, P.Picchi, and L. Periale for their spport at CERN in the beam test. For the management of X5 beam line of SPS at CERN, we would like to thank L. Gatignon and the tecnical staffs. One of the authors (K.K) thanks S. Roesler for his help in implementing dpmjet3.03. This work is partly supported by Grants-in Aid for Scientific Research B (09440110), Grants-in Aid for Scientific Research on Priority Area A (12047224) and Grant-in Aid for Project Research of Shibaura Institute of Technology. [^1]: We note electron showers of 10 GeV are normally simulated by $\sim$ 30 GeV protons when the latter start cascade at a shallow depth of the detector. [^2]: If we don’t impose the trigger condition, the gamma-ray case shows a small difference from the electron case. [^3]: The chance coincidence probability of uncorrelated particles is negligibly small. [^4]: It is used at energies greater than 10 GeV. At lower energies, model is the same as fritiof1.6 [^5]: Electrons included in the original data is subtracted statistically by use of cascade theory which is accurate at high energies.

ArXiv

--- author: - | \ Institut de Fisica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology (BIST)\ Campus UAB, 08193 Bellaterra (Barcelona), Spain\ E-mail: - | O. Blanch$^{a}$, E. de Oña Wilhelmi$^{b}$, D. Galindo$^{c}$, J. Herrera$^{d}$, M. Ribó$^{c}$, J. Rico$^{a}$, A. Stamerra$^{e}$ (for the MAGIC Collaboration), F. Aharonian$^{f,g}$, V. Bosch-Ramon$^{c}$ and R. Zanin$^{f}$\ $^{a}$ IFAE-BIST, Campus UAB, 08193 Bellaterra (Barcelona), Spain\ $^{b}$ CSIC/IEEC, E-08193 Barcelona, Spain\ $^{c}$ Departament de Fśica Quàntica i Astrofísica, Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, IEEC-UB, Barcelona, Spain\ $^{d}$ IAC and Universidad de La Laguna, E-38200/E-28206 La Laguna, Tenerife, Spain\ $^{e}$ INAF - National Institute for Astrophysics, I-00136 Rome, Italy\ $^{f}$ Max-Planck-Institut fur Kernphysik, 69029 Heidelberg, Germany\ $^{g}$ Dublin Institute for Advanced Studies, Dublin 2, Ireland\ title: 'Gamma rays from microquasars Cygnus X-1 and Cygnus X-3' --- Introduction ============ CygnusX-1 is an X-ray binary comprised by a (19.2$\pm$1.9) M$_{\odot}$ O9.7Iab supergiant star and a (14.8$\pm$1.0) M$_{\odot}$ BH [@Orosz2011], classified as a microquasar after the detection of a one-sided relativistic radio-jet [@Stirling2001]. The jet seems to create a 5 pc ring-like structure detected in the radio that extends up to $10^{19}$ cm from the BH [@Gallo2005]. The system follows an almost circular orbit of $\sim 5.6$ d period [@Brocksopp1999a]. Flux modulation with the orbital period is detected in X-ray and radio [@Wen1999; @Brocksopp1999b; @Szostek2007], produced by the absorption/scattering of the radiation by the stellar wind. CygnusX-1 displays the two principal X-ray states of BH transients, the soft state (SS) and the hard state (HS). Both are described by the sum of a blackbody-like emission from the accretion disk that peaks at $\sim 1$ keV (dominant in the SS) and a power-law tail with a cutoff at hundred keV, expected to be originated by inverse Compton (IC) scattering on disk photons by thermal electrons in the so-called *corona* (dominant in the HS). During HS the source displays persistent jets from which synchrotron radio emission is detected, whilst in the SS, these jets are disrupted. CygnusX-1 showed a $4\sigma$-hint above 100 MeV during HS reported by [@Malyshev], using 3.8 yr of *Fermi*-LAT data. Evidences of flaring activity were also reported by *AGILE* ($> 100$ MeV, [@AGILE2010Sabatini; @AGILE2010Bulgarelli; @AGILE2013]) and by MAGIC ($> 100$ GeV, [@Albert2007]). The microquasar CygnusX-3 hosts a Wolf-Rayet (WR) star, although it follows a short 4.8 hr-orbit. The compactness of the system produces an unusually high absorption, which complicates the identification of the compact object (1.4 M$_{\odot}$ neutron star (NS) [@Stark2003] or $< 10$ M$_{\odot}$ BH [@Hanson2000]). Despite this high absorption, its X-ray spectrum shows the two aforementioned states. CygnusX-3 is the strongest radio source among the X-ray binaries, whose flux can vary several orders of magnitude during its frequent radio outbursts. These major flares happen only during SS (see [@Szostek2008]). CygnusX-3 was detected above 100 MeV, during SS by AGILE [@Tavani2009] and *Fermi*-LAT [@Fermi2009]. Its spectrum was described as a power law with photon indices 1.8$\pm$0.2 and $2.70\pm0.25$, respectively. Here, we present the results for GeV and TeV searches on CygnusX-1 using 7.5yr of *Fermi*-LAT data and $\sim 100$ hr of MAGIC data. We also show the latest results of CygnusX-3 obtained with MAGIC during the August-September 2016 flare. Observations and Analysis ========================= *Fermi*-LAT is the principal scientific instrument on the Fermi Gamma-ray Space Telescope spacecraft that studies the gamma-ray sky within an energy range of $\sim 20$ MeV to $\sim 500$ GeV (see [@PerformanceFermi]). To study CygnusX-1 in the high-energy (HE; $>60$ MeV) regime, we used 7.5 years of `Pass8` *Fermi*-LAT data (from MJD 54682–57420). The analysis was performed using *Fermipy*[^1], a package of python tools to automatize the analysis with the FERMI SCIENCE TOOLS (v10r0p5 package). We selected photon-like events between 60 MeV and 500 GeV, within a 30$^{\circ}$ radius centered at the position of CygnusX-1. Find more details in [@Zanin2016]. MAGIC is a stereoscopic system of two 17 m diameter Cherenkov Telescopes located in La Palma (Spain). Until 2009, MAGIC consisted in just one telescope [@Aliu2009]. After autumn 2009, MAGICII started operation [@Alecksic2012] and between 2011-2012, both telescopes underwent a major upgrade [@Alecksic2016]. MAGIC observed CygnusX-1 for $\sim 100$ hours between 2007 and 2014 mostly during its HS (see [@FernandezBarral2017]). This analysis was carried out with standard MAGIC software (MARS, [@Zanin2013]). Upper limits (ULs) at 95% confidence level (CL) were computed with the full likelihood analysis developed by [@AleksicLikelihood], assuming 30% systematic uncertainty. Between August and September 2016, CygnusX-3 experienced strong flaring activity in radio and HE regimes during its SS [@RadioATel; @FermiATel]. MAGIC observed the source $\sim 70$ hours between MJD 57623 to 57653, under different moonlight conditions (moon analysis performed following [@MoonPerformance]). ULs at 95% CL were computed following Rolke method [@Rolke2005]. Results ======= CygnusX-1 --------- *Fermi*-LAT skymap, between 60 MeV and 500 GeV, showed a point-like source at the position of CygnusX-1 with a TS=53. Moreover, detection only happens during HS (Figure \[FermiSkypmaps\]) with TS=49 above 60 MeV (division between HS and SS done following [@Gringberg2013]). Therefore, CygnusX-1 is only detected while displaying persistent radio-jets, as claimed by [@Malyshev] and confirmed afterwards by [@Zdziarski2016]. Making use of the HS sample, we searched for orbital modulation (assuming ephemeris $T_{0}=52872.788$ HJD, [@Gies2008]). Orbital phases ($\phi$) were split into two bins, one centered at $\phi=0$, the superior conjunction of the compact object (0.75 $< \phi <$ 0.25) and other at the inferior conjunction (0.25 &lt; $\phi$ &lt; 0.75). Detection only occurred during superior conjunction (TS=31). CygnusX-1 spectrum, from 60 MeV up to $\sim 20$ GeV, is well defined by a power law with photon index $\Gamma=2.3\pm0.1$ and normalization factor of $N_{0}=(5.8\pm0.9)\times 10^{-13}$ MeV$^{-1}$ cm$^{-2}$ s$^{-1}$, at an energy pivot of 1.3 GeV. Daily basis analysis was also performed, but no short-term flux variability was observed. The results between 0.1-20 GeV can be found in Figure \[CygX1LC\]. ![TS maps above 1 GeV centered in CygnusX-1, using HS (*left*) and SS subsamples (*right*).[]{data-label="FermiSkypmaps"}](./FermiSkypmaps.pdf){width="0.9\linewidth"} ![Multi-wavelength light curve for CygnusX-1. *From top to bottom:* Daily MAGIC ULs ($> 200$ GeV), HE gamma rays from the *Fermi*-LAT analysis (flux points are computed when $TS>9$), hard X-rays from *Swift*-BAT (15-50 keV, [@Krimm2013]), soft X-rays from MAXI (2–20 keV, [@Matsuoka2009]) and *RXTE*-ASM (3–5 keV range), and radio from AMI (15 GHz) and RATAN-600 (4.6 GHz). In the HE pad, dashed lines correspond to *AGILE* transient events. The horizontal green line in *Swift*-BAT pad shows the limit at 0.09 cts cm$^{-2}$ s$^{-1}$ given by [@Gringberg2013] to differentiate between X-ray states. HS and SS periods are highlighted with grey and blue bands, respectively.[]{data-label="CygX1LC"}](./CygX1LC_pre.pdf){width="0.7\linewidth"} With MAGIC, we searched for steady emission at energies above 200 GeV, making use of the total data set of $\sim 100$ hr. No significant excess was found, which led to an integral UL of $2.6\times 10^{-12}$ photons cm$^{-2}$ s$^{-1}$, assuming a power-law function with photon index $\Gamma=3.2$ (following former MAGIC results, [@Albert2007]). We also looked for gamma-ray emission at each X-ray state separately. In the HS, the source was observed for $\sim83$ hours between 2007-2011, which yielded no significant excess. Differential ULs are included in the spectral energy distribution (SED) shown in Figure \[CygX1SED\]. Orbital phase-folded and daily analysis were also carried out, with no evidence of emission. Integral ULs in a night-by-night basis are depicted in Figure \[CygX1LC\]. During SS, this microquasar was observed for $\sim 14$ hours in 2014. We searched for steady, orbital and short-term variability modulation, resulting in no detection. ![SED of CygnusX-1. Soft X-rays from *BeppoSAX* are shown in green stars [@DiSalvo2001], while hard X-rays are taken from *INTEGRAL*-ISGRI (red diamonds,[@Rodriguez2015]) and *INTEGRAL*-PICsIT (brown diamonds, [@Zdziarski2012]). In the HE and VHE band, results presented in this proceeding obtained with *Fermi*-LAT (violet points) and MAGIC (black ULs) are depicted. Sensitivity curves for CTA-North for 50 hours (https://www.cta-observatory.org/science/cta- performance/) and scaled to 200 hours of observations are shown in light blue and dark blue, respectively. No statistical errors are drawn, apart from the *Fermi*-LAT butterfly.[]{data-label="CygX1SED"}](./CygX1SED_preliminary.pdf){width="0.7\linewidth"} CygnusX-3 --------- We searched for steady emission with the MAGIC telescopes, making use of the available $\sim 70$ hours. No excess was found at energies above 300 GeV (accounting for the energy threshold of the sample with the highest moonlight) nor 100 GeV (using $\sim 52$ hours of dark data, i.e. under absence of Moon). Differential ULs, assuming a power-law function with photon index $\Gamma=2.6$, are presented in Figure \[CygX3SED\]. In this figure, *Fermi*-LAT spectrum from [@Fermi2009] is taken, nevertheless *Fermi*-LAT data for the August-September 2016 flare is currently being studied. No orbital (assuming ephemeris $T_{0}=2440949.892\pm 0.001$ JD, [@Singh2002]) or daily modulation was detected either. ![SED of CygnusX-3. Blue butterfly corresponds to *Fermi*-LAT spectrum during 2009 flare [@Fermi2009]. MAGIC ULs for the August-September flare are represented in light orange ($\sim 52$ hours, dark data) and dark orange ($\sim 70$ hours, dark+moon data). Sensitivity curves for CTA-North for 50 hours (dot-dashed line) and 200 hours (dashed lines) observations are shown.[]{data-label="CygX3SED"}](./CygX3SED_ICRC.pdf){width="0.6\linewidth"} Discussion and conclusions ========================== HE and VHE gamma-ray emission were proposed in the literature from both leptonic and hadronic mechanisms (see e.g. [@BoschRamon2006; @Romero2003]). Among these mechanism, the most efficient process seems to be a leptonic one, the IC. The target photons depend on the distance of the production site with respect to the compact object: close to it, thermal photons from the disk or synchrotron photons would dominate [@Romero2002; @BoschRamon2006]; at a binary scales ($\sim R_{orb}$, the size of the system), IC would take place on stellar photons; and finally, gamma-ray emission could also be produced in the interaction between the jet and the medium (as seen in radio for CygnusX-1, [@Gallo2005]). In the first two scenarios, gamma rays may suffer high absorption due to pair creation. CygnusX-1 --------- At the base of the jet, GeV photons would be absorbed by $\sim 1$ keV X-rays. Given the detection achieved with *Fermi*-LAT, and following [@Aharonian] approach, we estimated the smallest region size for HE gamma-ray production at $2\times 10^{9}$ cm. The radius of the corona is $\sim 20-50~R_{g}\sim 5-10 \times 10^{7}$ cm [@Poutanen1998], which allows us to conclude that the observed GeV emission is not originated in the corona, but most likely inside the jets. This scenario is reinforced by the fact that *Fermi*-LAT detection only happens during HS. If the hint of orbital modulation here reported is finally confirmed, GeV emission must arrive from inside the jets and not from their interaction with the environment. Assuming so, we can set an UL on the largest distance of the production site at $< 10^{13}$ cm (few times $R_{orb}$ for this source). On the other hand, this flux variability is only expected if the radiative process that leads to GeV emission is anisotropic IC on stellar photons [@Khangulyan2014]. Given that the density of stellar photons is dominant over other photon fields at distances $>10^{11}$ cm, we place the GeV emitter at $10^{11}$–$10^{13}$ cm from the BH. On the other side, the MAGIC non-detection above 200 GeV allows us to discard jet-medium interaction as possible region for VHE emission above MAGIC sensitivity level, since these regions are not affected by photon-photon absorption. At binary scales this non-detection is less conclusive because of the pair production. Although VHE radiation is predicted in the models (see e.g. [@Pepe2015; @Khangulyan2008]), several factors can prevent detection: low flux below MAGIC sensitivity even under negligible absorption [@Zdziarski2016], no efficient acceleration on the jets or strong magnetic field. Nevertheless, transient events by relaxation of attenuation at some distance from the BH or extended pair cascade [@Zdziarski2009; @BoschRamon2008] cannot be discarded. Transient emission related to discrete radio-emitting-blobs between HS and SS could also happen, as observed in the HE regime for Cygnus X-3. Hint of transient event was indeed reported previously by MAGIC [@Albert2007]. More sensitive instruments, like the future CTA (see Figure \[CygX1SED\]), could provide interesting information on CygnusX-1. CygnusX-3 --------- Despite observing the source during strong radio and HE outbursts, no significant excess was found by MAGIC. One has to consider the extremely high absorption due to the WR, which may affect VHE gamma-ray emission. At energies above 300 GeV, the maximum absorption is produced by near-infrared (NIR) photons ($E_{target}\sim 1.7 $ eV). Following [@Aharonian2005], absorption can be estimated as $\tau\sim \sigma_{\gamma \gamma}\cdot n_{NIR} \cdot R$, where $\sigma_{\gamma \gamma}\sim 1\times 10^{-25}$ cm$^{2}$ is the cross-section of the process, $n_{NIR}\sim L_{NIR}/(4 \pi R^{2} c E_{target})$ is the density of NIR photons and $R$ the size of the emitting region. Assuming the $L_{NIR}$ to be the bolometric luminosity, $L_{NIR}=10^{38}$ erg s$^{-1}$, the absorption is not negligible until a radius $R\sim 10^{13}$ cm, i.e. outside the binary scale ($R_{orb,CygX3}\sim2.5\times 10^{11}$ cm). Given the MAGIC non-detection, acceleration up to VHE could still happen inside the jets at a distance $\lesssim 10^{13}$ cm, maybe related to the HE emission site (produced at $>10^{11}$ cm to avoid absorption by X-rays). On the other hand, MAGIC observed the source simultaneously with the strongest radio flare (at 9.5 Jy on MJD 57651), being the MAGIC significance for this day compatible with background. This could reinforce the idea that VHE gamma rays, if produced, are originated inside the binary scale and not at the radio-emitting regions of the jets far from the compact object. Note, however, that the amount of time observed during strong radio flares is very limited. Figure \[CygX3SED\] shows the CygnusX-3 SED with the results at VHE during the 2016 flare, along with *Fermi*-LAT spectrum taken from the 2009 flare [@Fermi2009]. As mentioned above, dedicated *Fermi*-LAT analysis for the August-Sept 2016 flare is currently being performed. Our constraining ULs are also put in context with the CTA-North sensitivity curve for 50 hours of observations[^2] and the scaled one for 200 hours. [99]{} Orosz J. A., et al., 2011, ApJ, 742, 84 Stirling A. 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ArXiv

--- abstract: 'We investigate the lazy states, entangled states and discordant states for 2-qubit systems. We show that many lazy states are discordant, many lazy states are entangled, and many mixed entangled states are not lazy. With these investigations, we provide a laziness-discord-entanglement hierarchy diagram about 2-qubit quantum correlations.' author: - Jianwei Xu title: 'Lazy states, discordant states and entangled states for 2-qubit systems' --- Introduction ============ Quantum correlation is one of the most striking features of quantum theory. Entanglement is the most famous kind of quantum correlation, and leads to powerful applications [@Horodecki2009]. Discord is another kind of quantum correlation, which captures more correlation than entanglement in the sense that a disentangled state may have no zero discord [@Modi2012]. Due to the theoretical and applicational interests, discord has been extensively studied [@Modi2012] and still in active research (for examples see [@Rulli2011; @Xu2013; @Chi2013; @Liu2013]). A bipartite state is called lazy, if the entropy rate of one subsystem is zero under any coupling to the other subsystem. Necessary and sufficient conditions have recently been established for a state to be lazy [@Rosario2011], and it was shown that almost all states are pretty lazy [@Hutter2012]. It is shown that a maximally entangled pure state is lazy[@Ferraro2010]. This indicates that the correlation described by lazy states is not the same by entanglement. So we are interested to clarify the question that, whether there are many lazy states which are entangled, and whether there are many entangled states which are lazy. This paper answers this question for the 2-qubit case. This paper is organized as follows. In Section 2, we briefly review the definitions of entangled states, discordant states and lazy states. In Section 3, we establish a necessary and sufficient condition for 2-qubit lazy states. In Section 4, we show that there are many 2-qubit lazy states which are discordant states. In Section 5, we show that there are many disentangled states which are not lazy. In Section 6, we show that there are many 2-qubit mixed lazy states which are entangled. In section 7, we briefly summary this paper by providing a laziness-discord-entanglement hierarchy diagram to characterize the bipartite quantum correlations. Entangled states, discordant states, lazy states ================================================ We briefly review the definitions about entangled states, discordant states and lazy states. Finite-dimensional quantum systems $A$ and $B$ are described by the Hilbert spaces $H^{A}$ and $H^{B}$ respectively, the composite system $AB$ is then described by the Hilbert space $H^{A}\otimes H^{B}$. Let $n_{A}=\dim H^{A}$, $n_{B}=\dim H^{B}$. A state $\rho ^{AB}$ is called a disentangled state (or separable state) if it can be written in the form$$\begin{gathered} \rho ^{AB}=\sum_{i}p_{i}\rho _{i}^{A}\otimes \rho _{i}^{B},\end{gathered}$$ where $p_{i}\geq 0,\sum_{i}p_{i}=1,\{\rho _{i}^{A}\}_{i}$ are density operators on $H^{A}$, $\{\rho _{i}^{B}\}_{i}$ are density operators on $% H^{B}. $If $\rho ^{AB}$ is disentangled we then say $E(\rho ^{AB})=0.$ A state $\rho ^{AB}$ is called a zero-discord state with respect to $A$ if it can be written in the form$$\begin{gathered} \rho ^{AB}=\sum_{i=1}^{n_{A}}p_{i}|\psi _{i}^{A}\rangle \langle \psi _{i}^{A}|\otimes \rho _{i}^{B},\end{gathered}$$ where $p_{i}\geq 0,\sum_{i}p_{i}=1,\{|\psi _{i}^{A}\rangle \}_{i}$ is an orthonormal basis for $H^{A}$, $\{\rho _{i}^{B}\}_{i}$ are density operators on $H^{B}. $If $\rho ^{AB}$ is in the form Eq.(2) we then say $D_{A}(\rho ^{AB})=0.$ Evidently, $$\begin{gathered} D_{A}(\rho ^{AB})=0 \ ^{\Rightarrow } _{\nLeftarrow } \ E(\rho ^{AB})=0.\end{gathered}$$ A state $\rho ^{AB}$ is called a lazy state with respect to $A$ if [@Rosario2011] $$\begin{gathered} C_{A}(\rho ^{AB})=[\rho ^{AB},\rho ^{A}\otimes I^{B}]=0,\end{gathered}$$ where $\rho ^{A}=tr_{B}\rho ^{AB}$, $I^{B}$ is the identity operator on $% H^{B}.$ An important physical interpretation of lazy states is that the entropy rate of $A$ is zero in the time evolution under any coupling to $B,$ $$\begin{gathered} C_{A}(\rho ^{AB}(t))=0\Leftrightarrow \frac{d}{dt}tr_{A}[\rho ^{A}(t)\log _{2}\rho ^{A}(t)]=0\text{.}\end{gathered}$$ $D_{A}(\rho ^{AB})=0$ and $C_{A}(\rho ^{AB})=0$ has the inclusion relation below [@Ferraro2010] $$\begin{gathered} D_{A}(\rho ^{AB})=0 \ ^{\Rightarrow} _{\nLeftarrow} \ C_{A}(\rho ^{AB})=0.\end{gathered}$$ Maximal pure entangled states are the examples of $C_{A}(\rho ^{AB})=0$ but $% D_{A}(\rho ^{AB})\neq 0$ [@Ferraro2010]. The direct product states have the form $$\begin{gathered} \rho ^{AB}=\rho ^{A}\otimes \rho ^{B},\end{gathered}$$ they are obviously zero-discord states. The form of 2-qubit lazy states =============================== Any 2-qubit state can be written in the form [@Fano1983] $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{i=1}^{3}x_{i}\sigma _{i}\otimes I+\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j} \notag \\ +\sum_{i,j=1}^{3}T_{ij}\sigma _{i}\otimes \sigma _{j}),\end{gathered}$$ where $I$ is the two-dimensional identity operator,$\{\sigma _{i}\}_{i=1}^{3} $ are Pauli operators, $\{x_{i}\}_{i=1}^{3},\{y_{j}% \}_{j=1}^{3},\{T_{ij}\}_{i,j=1}^{3},$ are all real numbers satisfying some conditions (we will explore these conditions when we need them) to ensure the positivity of $\rho ^{AB}$, $\rho ^{A}$ and $\rho ^{B}$. We often omit $% I $ for simplicity without any confusion. $ $ **Proposition 1.** The 2-qubit state $\rho ^{AB}$ in Eq.(8) is lazy if and only if $$\begin{gathered} \{x_{i}\}_{i=1}^{3} // \{T_{ij}\}_{i=1}^{3} \text{ for }j=1,2,3.\end{gathered}$$ $ $ **Proof.** For state in Eq.(8), $$\begin{gathered} \rho ^{A}=\frac{1}{2}(I+\sum_{k=1}^{3}x_{k}\sigma _{k}\otimes I), \\ [\rho ^{AB},\rho ^{A}]=\frac{1}{8}\sum_{ijk=1}^{3}T_{ij}x_{k}[\sigma _{i}\otimes \sigma _{j},\sigma _{k}\otimes I] \notag \\ =\frac{1}{8}\sum_{ijk=1}^{3}T_{ij}x_{k}[\sigma _{i},\sigma _{k}]\otimes \sigma _{j} \notag \\ =\frac{i}{4}\sum_{ijkl=1}^{3}T_{ij}x_{k}\varepsilon _{ikl}\sigma _{l}\otimes \sigma _{j}.\end{gathered}$$ In the last line, $\varepsilon _{ikl}$ is the permutation symbol. Let $[\rho ^{AB},\rho ^{A}]=0,$ then $$\begin{gathered} \sum_{ik=1}^{3}T_{ij}x_{k}\varepsilon _{ikl}=0,\end{gathered}$$ this evidently leads to Eq.(9). $\square $ Lazy but diacordant 2-qubit states ================================== It is easy to check that $C_{A}(\rho ^{AB})=0$ defined in Eq.(4) is invariant under locally unitary transformations for arbitrary $n_{A}$ and $n_{B}$. Under locally unitary transformations, any 2-qubit state in Eq.(8) can be written in the form [@Luo2008] $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{i=1}^{3}x_{i}\sigma _{i}\otimes I+\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j} \notag \\ +\sum_{i=1}^{3}\lambda _{i}\sigma _{i}\otimes \sigma _{i}),\end{gathered}$$ where $0\leq \lambda _{1}\leq \lambda _{2}\leq \lambda _{3}$ being the singular values of $\{T_{ij}\}_{ij}$ in Eq.(8). Note that $% \{x_{i}\}_{i=1}^{3},\{y_{j}\}_{j=1}^{3}$ in Eq.(9) are not the same with in Eq.(8). We now look for the conditions such that $D_{A}(\rho ^{AB})=0.$ Suppose $D_{A}(\rho ^{AB})=0$, then according to Eq.(2), there exists real vector $% \overrightarrow{n}=\{n_{1},n_{2},n_{3}\}$ with $% n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=1$ such that $$\begin{gathered} \rho ^{AB}=\Pi _{0}\otimes I\rho ^{AB}\Pi _{0}\otimes I+\Pi _{1}\otimes I\rho ^{AB}\Pi _{1}\otimes I,\end{gathered}$$ with $$\begin{gathered} \Pi _{0}=\frac{1}{2}(I+\overrightarrow{n}\cdot \overrightarrow{\sigma }), \\ \Pi _{1}=\frac{1}{2}(I-\overrightarrow{n}\cdot \overrightarrow{\sigma }).\end{gathered}$$ It can be check that $$\begin{gathered} \Pi _{0}\sigma _{i}\Pi _{0}+\Pi _{1}\sigma _{i}\Pi _{1}=n_{i} \overrightarrow{n}\cdot \overrightarrow{\sigma }.\end{gathered}$$ Then Eq.(14) becomes $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{i=1}^{3}x_{i}n_{i}\overrightarrow{n}% \cdot \overrightarrow{\sigma }\otimes I \notag \\ +\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j}+\sum_{i=1}^{3}\lambda _{i}n_{i}\overrightarrow{n}\cdot \overrightarrow{% \sigma }\otimes \sigma _{i}) \notag \\ =\frac{1}{4}(I\otimes I+\sum_{ij=1}^{3}x_{i}n_{i}n_{j}\sigma _{j}\otimes I \notag \\ +\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j}+\sum_{ij=1}^{3}\lambda _{i}n_{i}n_{j}\sigma _{j}\otimes \sigma _{i}).\end{gathered}$$ Comparing to Eq.(13), then for $j=1,2,3,$ $$\begin{gathered} \sum_{i=1}^{3}x_{i}n_{i}n_{j}=x_{j}\Rightarrow \overrightarrow{n}//% \overrightarrow{x}, \\ \lambda _{i}n_{i}n_{j}=\delta _{ij}\lambda _{j}=\delta _{ij}\lambda _{i} \Rightarrow \lambda _{i}=0 \ \text{or} \ n_{i}=\pm 1.\end{gathered}$$ (i).If $\lambda _{1}=\lambda _{2}=\lambda _{3}=0,$ let $\overrightarrow{n}//% \overrightarrow{x},$ then $D_{A}(\rho ^{AB})=0$. (ii).If $0=\lambda _{1}=\lambda _{2}<\lambda _{3}=0,$ then $\overrightarrow{n}% =(0,0,\pm 1),$ to satisfy $\overrightarrow{n}//\overrightarrow{x},$ we see that only when $% \overrightarrow{x}=(0,0,x_{3})$ we have $D_{A}(\rho ^{AB})=0$. (iii).If $0=\lambda _{1}<\lambda _{2}<\lambda _{3}=0,$ then Eq.(20) can not be satisfied, so $\rho ^{AB}$ is discordant. (iv).If $0<\lambda _{1}<\lambda _{2}<\lambda _{3}=0,$ then Eq.(20) can not be satisfied, so $\rho ^{AB}$ is discordant. Comparing with Proposition 1, we then get Proposition 2 below. $ $ **Proposition 2.** A 2-qubit state in Eq.(13) is lazy but discordant if and only if $\overrightarrow{x}=0$ and $0<\lambda _{2}<\lambda _{3}$. $ $ Since any locally unitary transformation keeps $\overrightarrow{x}=0$ invariant in Eq.(8), then we rewrite Proposition 2 as Proposition 2$'$ below. $ $ **Proposition 2$'$.** A 2-qubit state in Eq.(8) is lazy but discordant if and only if $\overrightarrow{x}=0$ and the matrix $\{T_{ij}\}_{ij}$ have at least two positive singular values. $ $ We make a note that some constraints about $\{y_{j}\}_{j=1}^{3},\lambda _{1},\lambda _{2},\lambda _{3}$ are required to guarantee the positivity of $% \rho ^{AB},\rho ^{A}$, $\rho ^{B}$ in Proposition 2$.$These constraints are rather complex since there are so many parameters. To show there indeed exist many states described in Proposition 2, we choose some special states. For the state $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j}+\sum_{i=1}^{3}\lambda _{i}\sigma _{i}\otimes \sigma _{i}),\end{gathered}$$ where $0\leq \lambda _{1}\leq \lambda _{2},0<\lambda _{2}<\lambda _{3},$ we have $\rho ^{A}=I$,and $$\begin{gathered} \rho ^{B}=\frac{1}{2}(I+\sum_{j=1}^{3}y_{j}\sigma _{j}).\end{gathered}$$ $\rho ^{B}$ is positive then $$\begin{gathered} \sum_{j=1}^{3}y_{j}^{2}\leq 1.\end{gathered}$$ Let $y_{2}=y_{3}=\lambda _{1}=0,$ then the four eigenvalues of $\rho ^{AB}$ in Eq.(21) are $$\begin{gathered} \frac{1}{4}(1\pm \sqrt{y_{1}^{2}+(\lambda _{3}\pm \lambda _{2})^{2}}).\end{gathered}$$ These eigenvalues are all nonnegtive then we need $$\begin{gathered} 0<\lambda _{2}<\lambda _{3}, \\ y_{1}^{2}+(\lambda _{3}+\lambda _{2})^{2}\leq 1.\end{gathered}$$ There are many triples $\{y_{1},\lambda _{3},\lambda _{2}\}$ satisfy Eqs.(25,26), then the corresponding states in Eq.(21) are lazy but discordant states. Some disentangled but not lazy 2-qubit states ============================================= To show there exist many 2-qubit states which are disentangled but not lazy, we consider the states of the form $$\begin{gathered} \rho ^{AB}=p|\psi _{1}^{A}\rangle \langle \psi _{1}^{A}|\otimes \rho _{1}^{B}+(1-p)|\psi _{2}^{A}\rangle \langle \psi _{2}^{A}|\otimes \rho _{2}^{B},\end{gathered}$$ where $p\in (0,1),\{|\psi _{i}^{A}\rangle \}_{i=1}^{2}$ are normalized states in $H^{A}$ but not necessarily orthogonal,$\{\rho _{i}^{B}\}_{i=1}^{2} $ are density operators on $H^{B}.$ Note that $p=0$ or $% p=1$ leads to direct product states, so we do not consider such cases. Under locally unitary transformations, we let $$\begin{gathered} |\psi _{1}^{A}\rangle \langle \psi _{1}^{A}|=\frac{I+(0,0,1)\cdot \overrightarrow{\sigma }}{2}, \\ |\psi _{2}^{A}\rangle \langle \psi _{2}^{A}|=% \frac{I+(\sin \alpha ,0,\cos \alpha )\cdot \overrightarrow{\sigma }}{2}, \\ \rho _{1}^{B}=\frac{I+a(0,0,1)\cdot \overrightarrow{\sigma }}{2}, \\ \rho _{2}^{B}=\frac{I+b(\sin \beta ,0,\cos \beta )\cdot \overrightarrow{\sigma }}{% 2},\end{gathered}$$ where $\alpha ,\beta \in \lbrack 0,\pi ],a,b\in \lbrack 0,1].$ Some special states can be apparently specified. (v).$\alpha =0,\rho ^{AB}$ in Eq.(27) are direct product states; (vi).$\alpha =\pi ,\rho ^{AB}$ in Eq.(27) are zero-discord states; (vii).$a=b=0,\rho ^{AB}$ in Eq.(27) are direct product states. Now we consider the cases excluding (v), (vi), (vii) above. Taking Eqs.(28-31) into Eq.(27), and using the notations in Eq.(8), we get $$\begin{gathered} \overrightarrow{x}=((1-p)\sin \alpha ,0,p+(1-p)\cos \alpha ), \\ \{T_{i1}\}_{i}=(b(1-p)\sin \alpha \sin \beta ,0,b(1-p)\cos \alpha \sin \beta ), \\ \{T_{i2}\}_{i}=(0,0,0), \\ \{T_{i3}\}_{i}=(b(1-p)\sin \alpha \cos \beta ,0,ap+b(1-p)\cos \alpha \cos \beta ).\end{gathered}$$ From Proposition 1, $\rho ^{AB}$ in Eq.(27) is lazy if and only if $% \overrightarrow{x}//\{T_{i1}\}_{i}$ and $\overrightarrow{x}//\{T_{i3}\}_{i}. $ Since $x_{1}=(1-p)\sin \alpha \neq 0,$ then $\overrightarrow{x}//\{T_{i1}\}_{i}$ and $\overrightarrow{x}//\{T_{i3}\}_{i}$ lead to $$\begin{gathered} b\sin \beta =0. \\ a=b\cos \beta .\end{gathered}$$ Eqs.(36,37) together correspond to direct product states since $\rho _{1}^{B}=\rho _{2}^{B} $. Otherwise, there are many states violate Eq.(36) or Eq.(37), so they are not lazy states. $ $ **Proposition 3**. 2-qubit disentangled state $\rho ^{AB}$ in Eq.(27), is a direct product state when $|\psi _{1}^{A}\rangle =|\psi _{2}^{A}\rangle $ or $\rho _{1}^{B}=\rho _{2}^{B}$, is a zero-discord state when $\langle \psi _{1}^{A}|\psi _{2}^{A}\rangle =0$. Otherwise, $\rho ^{AB}$ is not lazy. Some lazy but entangled states ============================== We know that a bipartite pure state is lazy only if under locally unitary transformations it can be written in the form [@Rosario2011] $|\psi^{AB}\rangle=\frac{1}{\sqrt{s}}\sum_{i=1}^{s}|\psi^{A}_{i}\rangle|\psi^{B}_{i}\rangle$, where $\{|\psi _{i}^{A}\rangle \}_{i}$ are orthonormal sets in $H^{A}$, $\{|\psi _{i}^{B}\rangle \}_{i}$ are orthonormal sets in $H^{B}$, $s\leq \min \{n_{A},n_{B}\}$. When $s= \min \{n_{A},n_{B}\}$ it is maximally entangled state. In this section we look for more 2-qubit mixed states which are lazy but entangled. From Proposition 1, we know the following 2-qubit Bell-diagonal states are lazy $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{i=1}^{3}\lambda _{i}\sigma _{i}\otimes \sigma _{i}),\end{gathered}$$ where $\{\lambda _{i}\}_{i=1}^{3}$ are real numbers satisfying some constraints to ensure the positivity of $\rho ^{AB}.$ In this section, for convenience, we do not assume $\{\lambda _{i}\}_{i=1}^{3}$ are all nonnegative. We represent the states in Eq.(38) in the $(\lambda _{1},\lambda _{2},\lambda _{3})$ space. The eigenvalues of $\rho ^{AB}$ in Eq.(38) are $$\begin{gathered} \frac{1}{4}\{1-\lambda _{1}+\lambda _{2}+\lambda _{3},1+\lambda _{1}-\lambda _{2}+\lambda _{3}, \notag \\ 1+\lambda _{1}+\lambda _{2}-\lambda _{3},1-\lambda _{1}-\lambda _{2}-\lambda _{3}\}.\end{gathered}$$ Then the positivity of $\rho ^{AB}$ requires that $\{\lambda _{i}\}_{i=1}^{3} $ are in the tetrahedron (with its boundary) with the vertices $(-1,-1,-1),(-1,1,1),(1,-1,1),(1,1,-1)$ in the $(\lambda _{1},\lambda _{2},\lambda _{3})$ space [@Horodecki1996]. Disentangled states in Eq.(38) are in the octahedron (with its boundary) with the vertices $(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1)$ [@Horodecki1996]. From Proposition 2, we know the zero-discord states in Eq.(38) are only three line segments $(\lambda _{1},0,0)$ with $\lambda _{1}\in \lbrack -1,1], (0,\lambda _{2},0)$ with $\lambda _{2}\in \lbrack -1,1]$, $(0,0,\lambda _{3})$ with $\lambda _{3}\in \lbrack -1,1].$ Then the states in the tetrahedron (with its boundary) but not in the octahedron (with its boundary) are lazy but entangled. Among these, only the states at the vertices of tetrahedron are (maximally entangled) pure states. Summary: a hierarchy diagram ============================ We explored some 2-qubit states, showed that many states are lazy but discordant, many states are lazy but entangled, and many states are disentangled but not lazy. With these investigations, we can surely give a hierarchy diagram (Figure 1) of 2-qubit states, including lazy states, disentangled states and zero-discord states. This hierarchy diagram enriches the entanglement-discord hierarchy, then provides more understandings about the structures of quantum correlations. This work was supported by the National Natural Science Foundation of China (Grant No.11347213) and the Research Start-up Foundation for Talents of Northwest A&F University of China (Grant No.2013BSJJ041). The author thanks Zi-Qing Wang and Chang-Yong Liu for helpful discussions. [99]{} R. Horodecki et al., Rev. Mod. Phys. **81**, 865 (2009), and references therein. K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, Rev. Mod. Phys. **84** 1655¨C1707 (2012) and references therein. C.C. Rulli, M.S. Sarandy, Phys. Rev. A **84** 042109 (2011). J. Xu, Phys. Lett. A **377** 238 (2013). D. P. Chi, J. S. Kim, and K. Lee, Phys. Rev. A **87** 062339 (2013). S.-Y. Liu, Y.-R. Zhang, L.-M. Zhao, W.-L. Yang, and H. Fan, arXiv:1307.4848. C. A. Rodriguez-Rosario, G. Kimura, H. Imai, and A. Aspuru-Guzik, Phys. Rev. Lett. **106** 050403 (2011). A. Hutter and S. Wehner, Phys. Rev. Lett. **108** 070501 (2012) A. Ferraro, L. Aolita, D. Cavalcanti, F. Cucchietti, and A. Acin, Phys. Rev. A **81** 052318 (2010). U. Fano, Rev. Mod. Phys. **55** 855 (1983). S. Luo, Phys. Rev. A **77** 042303 (2008). Horodecki, R. and M. Horodecki, Phys. Rev. A **54** 1838 (1996).

ArXiv

--- abstract: 'A pseudofinite group satisfying the uniform chain condition on centralizers up to finite index has a big finite-by-abelian subgroup.' address: 'Université Lyon 1; CNRS; Institut Camille Jordan UMR 5208, 21 avenue Claude Bernard, 69622 Villeurbanne-cedex, France' author: - 'Frank O. Wagner' date: '1/9/2015' title: 'Pseudofinite $\widetilde{\mathfrak M_c}$-groups' --- [^1] Introduction {#introduction .unnumbered} ============ We generalize the results of Elwes, Jaligot, MacPherson and Ryten [@ER08; @EJMR11] about pseudofinite superstable groups of small rank to the pseudofinite context, possibly of infinite rankl Rank ==== A [*dimension*]{} on a theory $T$ is a function $\dim$ from the collection of all interpretable subsets of a monster model to $\R^{\ge0}\cup\{\infty\}$ satisfying - [*Invariance:*]{} If $a\equiv a'$ then $\dim(\p(x,a))=\dim(\p(x,a'))$. - [*Algebraicity:*]{} If $X$ is finite, then $\dim(X)=0$. - [*Union:*]{} $\dim(X\cup Y)=\max\{\dim(X),\dim(Y)\}$. - [*Fibration:*]{} If $f:X\to Y$ is a interpretable surjection whose fibres have constant dimension $r$, then $\dim(X)=\dim(Y)+r$. For a partial type $\pi$ we put $\dim(\pi)=\inf\{\dim(\p):\pi\proves\p\}$. We write $\dim(a/B)$ for $\dim(\tp(a/B))$.It follows that $X\subseteq Y$ implies $\dim(X)\le\dim(Y)$, and $\dim(X\times Y)=\dim(X)+\dim(Y)$. Moreover, any partial type $\pi$ can be completed to a complete type $p$ with $\dim(\pi)=\dim(p)$. A dimension is [*additive*]{} if it satisfies - [*Additivity:*]{} $\dim(a,b/C)=\dim(a/b,C)+\dim(b/C)$. Additivity is clearly equivalent to [*fibration*]{} for type-definable maps.Examples for an additive dimension include: 1. coarse pseudofinite dimension (in the expansion by cardinality comparison quantifiers, in order to ensure invariance); 2. Lascar rank, $SU$-rank or -rank, possibly localised at some $\emptyset$-invariant family of types; 3. for any ordinal $\alpha$, the coefficient of $\omega^\alpha$ in one of the ordinal-valued ranks in (2) above. In any of the examples, we put $\dim(X)=\infty$ if $\dim(X)>n$ for all $n<\omega$.Note that additivity in the above examples follows from the Lascar inequalities for the corresponding rank in examples (2) and (3) and from [@Hr12] in example (1). Let $G$ be a type-definable group with an additive dimension, and suppose $0<\dim(G)<\infty$. A partial type $\pi(x)$ implying $x\in G$ is [*wide*]{} if $\dim(\pi)=\dim(X)$. It is [*broad*]{} if $\dim(\pi)>0$. An element $g\in G$ is [*wide*]{}/[*broad*]{} over some parameters $A$ if $\tp(g/A)$ is. We say that $\pi$ is [*negligible*]{} if $\dim(\pi)=0$.An additive dimension is invariant under definable bijections.Let $f$ be an $A$-definable bijection, and $a\in\text{dom}(f)$. Then $\dim(f(a)/A,a)=0$ and $\dim(a/A,f(a))=0$, whence $$\dim(a/A)=\dim(a,f(a)/A)=\dim(f(a)/A).\qquad\qed$$ If $\dim(a/A)<\infty$, we say that $a\indd_AB$ if $\dim(a/A)=\dim(AB)$.It follows that $\indd$ satisfies transitivity. Moreover, for any partial type $\pi$ there is a comple type $p\supseteq\pi$ (over any set of parameters containing dom$(\pi)$) with $\dim(p)=\dim(\pi)$, so $\indd$ satisfies extension. Let $\dim$ be an additive dimension. If $\dim(a/A)<\infty$ and $\dim(b/A)<\infty$, then $a\indd_Ab$ if and only if $b\indd_Aa$, if and only if $\dim(a,b/A)=\dim(a/A)+\dim(b/A)$.Obvious. \[generics\] Let $G$ be a type-definable group with an additive dimension, and suppose $0<\dim(G)<\infty$. If $g$ is wide over $A$ and $h$ is wide over $A,g$, then $gh$ and $hg$ are wide over $A,g$ and over $A,h$.As dimension is invariant under definable bijections $$\dim(gh/A,g)=\dim(h/A,g)=\dim(G).$$ The other statements follow similarly, possibly using symmetry. Big abelian subgroups ===================== \[centralizer\] Let $G$ be a pseudofinite group, and $\dim$ an additive dimension on $G$. Then there is an element $g\in G\setminus\{1\}$ such that $\dim(C_G(g))\ge\frac13\dim(G)$.Suppose first that $G$ has no involution. If $G\equiv\prod_I G_i/\U$ for some family $(G_i)_I$ of finite groups and some non-principal ultrafilter $\U$, then $G_i$ has no involution for almost all $i\in I$, and is soluble by the Feit-Thompson theorem. So there is $g_i\in G_i\setminus\{1\}$ such that $\langle g_i^{G_i}\rangle$ is commutative. Put $g=[g_i]_I\in G\setminus\{1\}$. Then $\langle g^G\rangle$ is commutative and $g^G\subseteq C_G(g)$. As $g^G$ is in definable bijection with $G/C_G(g)$, we have $$\dim(C_G(g))\ge\dim(g^G)=\dim(G/C_G(g))=\dim(G)-\dim(C_G(g)).$$ In particular $\dim(C_G(g))\ge\frac12\dim(G)$. Now let $i\in G$ be an involution and suppose $\dim(C_G(i))<\frac13\dim(G)$. Then $$\dim(i^G)=\dim(G/C_G(i))=\dim(G)-\dim(C_G(i))>\frac23\dim(G)$$ and there is $j=i^g\in i^G$ with $\dim(j)\ge\frac23\dim(G)$. Note that $$\dim(C_G(j))=\dim(C_G(i)^g)=\dim(C_G(i))<\frac13\dim(G).$$ Then $$\dim(j^Gj)=\dim(G/C_G(j))=\dim(G)-\dim(C_G(j))>\frac23\dim(G)$$ and there is $h=j^{g'}j\in j^Gj$ with $\dim(h/j)\ge\frac23\dim(G)$. Note that $h$ is inverted by $j$. By additivity, $$\dim(j/h)=\dim(j, h)-\dim(h)\ge\dim(h/j)+\dim(j)-\dim(G)>\frac13\dim(G).$$ If $H=\{x\in G:h^x=h^{\pm1}\}$, then $H$ is an $h$-definable subgroup of $G$, and $C_G(h)$ has index two in $H$. Since $j\in H$, we have $$\dim(C_G(h))=\dim(H)\ge\dim(j/h)>\frac13\dim(G).\qed$$ \[abelian\] Let $G$ be a pseudofinite -group, and $\dim$ an additive dimension on $G$. Then $G$ has a definable broad finite-by-abelian subgroup $Z$. More precisely, $Z=\Z(C)$ where $C$ is a minimal broad centralizer (up to finite index) of a finite tuple.By the  condition, there is a broad centralizer $C$ of some finite tuple, such that $C_C(g))$ is not broad for any $g\in C\setminus\Z(C)$. Put $Z=\Z(C)$, a definable finite-by-abelian normal subgroup of $C$. If $Z$ is broad, we are done. Otherwise $\dim(Z)=0$, and $$\dim(C/Z)=\dim(C)-\dim(Z)=\dim(C).$$ For $g\in C\setminus Z$ we have $\dim(C_C(g))=0$, whence for $\bar g=gZ$ we have $$\begin{aligned}\dim(\bar g^{C/Z})&=\dim(g^CZ/Z)\ge\dim(g^C)-\dim(Z)\\ &=\dim(C)-\dim(C_C(g))-\dim(Z)=\dim(C/Z).\end{aligned}$$ Hence for all $\bar g\in (C/Z)\setminus\{\bar1\}$ we have $$\dim(C_{C/Z}(\bar g))=\dim(C/Z)-\dim(\bar g^{C/Z})=0.$$ As $C$ and $Z$ are definable, $C/Z$ is again pseudofinite, contradicting Proposition \[centralizer\]. Theorem \[abelian\] holds in particular for any pseudofinite -group with the pseudofinite counting measure. Note that the broad finite-by-abelian subgroup is defined in the pure group, using centralizers and almost centres. Moreover, the -condition is just used in $G$, not in the section $C/Z$. \[corTh\] A superrosy pseudofinite group with $\Ut(G)\ge\omega^\alpha$ has a definable finite-by-abelian subgroup $A$ with $\Ut(A)\ge\omega^\alpha$.A superrosy group is . If $\alpha$ is minimal with $\Ut(G)<\omega^{\alpha+1}$ and we put $$\dim(X)\ge n\quad\text{if}\quad\Ut(X)\ge\omega^\alpha\cdot n,$$ then $\dim$ is an additive dimension with $0<\dim(G)<\infty$. The assertion now follows from Theorem \[abelian\]. For any $d,d'<\omega$ there is $n=n(d,d')$ such that if $G$ is a finite group without elements $(g_i:i\le d')$ such that $$|C_G(g_i:i<j):C_G(g_i:i\le j)|\ge d$$ for all $j\le d'$, then $G$ has a subgroup $A$ with $|A'|\le n$ and $n\,|A|^n\ge|G|$.If the assertion were false, then given $d,d'$, there were a sequence $(G_i:i<\omega)$ of finite groups satisfying the condition, such that $G_i$ has no subgroup $A_i$ with $|A_i'|\le i$ and $i\,|A_i|^i\ge|G_i|$. But any non-principal ultraproduct $G=\prod G_n/\U$ is a pseudofinite -group, and has a definable subgroup $A$ with $A'$ finite and $\dim(A)\ge\frac1n\dim(G)$ for some $n<\omega$. Unravelling the definition of the pseudofinite counting measure (and possibly increasing $n$) we get $|A_i'|\le n$ and $n\cdot|A_i|^n\ge|G_i|$ for almost all $i<\omega$, a contradiction for $i\ge n$. Pseudofinite -groups of dimension $2$ ===================================== \[dim2\] Let $G$ be a pseudofinite -group, and $\dim$ an additive integer-valued dimension on $G$. If $\dim(G)=2$, then $G$ has a broad definable finite-by-abelian subgroup $N$ whose normalizer is wide. Note that since $\dim$ is integer-valued, a broad subgroup has dimension at least $1$. By Corollay \[corTh\], if $C$ is a minimal broad centralizer (up to finite index), then $A=\Z(C)$ is broad and finite-by-abelian. If $A$ is commensurate with $A^g$ for all $g\in G$, then commensurativity is uniform by compactness. So by Schlichting’s Theorem there is a normal subgroup $N$ commensurate with $A$. But then $\Z(N)$ contains $A\cap N$, has finite index in $N$ and is broad; since it is characteristic in $N$, it is normal in $G$ and we are done. Suppose $g\in\N_G(H)$ is such that $A$ is not commensurable with $A^g$. Then clearly $C$ cannot be commensurable with $C^g$, and $$\dim(A\cap A^g)\le\dim(C\cap C^g)=0.$$ Hence $$\dim(AA^g)\ge\dim(AA^g/(A\cap A^g))=\dim(A/(A\cap A^g)+\dim(A^g/(A\cap A^g))=2\,\dim(A).$$ As $A$ is broad and $\dim(G)=2$, we have $\dim(A)=1$. Choose some $d$-independent wide $a,b_0,c_0$ in $A$ over $g$. Then $\dim(a^gb_0/g,c_0)=2$, and $$2\ge\dim(c_0a^gb_0/g)\ge\dim(c_0a^gb_0/g,c_0)=\dim(a^gb_0/g, c_0)=\dim(a^gb_0/g)=2$$ and $c_0\indd_gc_0a^gb_0$. Thus $c_0$ is wide in $A$ over $g,c_0a^gbc_0$. Similarly, $b_0$ is wide in $A$ over $g,c_0a^gbc_0$. Choose $d,b_1,c_1\equiv_{c_0a^gb_0,g}a,b_0,c_0$ with $d,b_1,c_1\indd_{c_0a^gb_0,g}a,b_0,c_0$. Then $c_0a^gb_0=c_1d^gb_1$, whence $$a^gb=cd^g,$$ where $c=c_0^{-1}c_1$ and $b=b_0b_1^{-1}$. Moreover, $$\dim(b/a,g)\ge\dim(b_0b_1^{-1}/a,b_0,c_0,g)\ge\dim(b_1/a,b_0,c_0, g)=\dim(b_1/c_0a^gb_0,g)=1,$$ so $b$ is wide in $A$ over $g,a$. Similarly, $c$ is wide in $A$ over $g,d$. Let $x$ and $y$ be two d-independent wide elements of $C_A(a,b,c,d)$ over $a,b,c,d,g$. Then they are d-independent wide in $A$, and for $z=xgy$ we have $$\dim(z/a,b,c,d,g)=2,$$ so $z$ is wide in $G$ over $g,a,b,c,d$. Moreover $$a^zb=a^{xgy}b=a^{gy}b^y=(a^gb)^y=(cd^g)^y=c^yd^{gy}=cd^{xgy}=cd^z.$$ Choose $z'\in G$ with $z'\equiv_{a,b,c,d}z$ and $z'\indd_{a,b,c,d}z$, and put $r=z'^{-1}z$. Then $r$ is wide in $G$ over $g,a,b,c,d,z$, and $$a^zb^r=a^{z'r}b^r=(a^{z'}b)^r=(cd^{z'})^r=c^rd^{z'r}=c^rd^z.$$ Hence $$c^{-1}a^zb=d^z=c^{-r}a^zb^r.$$ Putting $b'=bb^{-r}$ and $c'=cc^{-r}$, we obtain $$a^zb'=c'a^z,$$ where $a$ is wide in $A$. As $r\indd_{g,a,b,c,d}z$ we have $$\dim(z/a,b',c')\ge\dim(z/a,b,c,d,r)=\dim(z/a,b,c,d)=2,$$ and $z$ is wide in $G$ over $a,b',c'$. If $z''\equiv_{a,b',c'}z$ with $z''\indd_{a,b',c'}z$, then $a^{z''}b'=c'a^{z''}$, whence $$b'^{a^z}=c'=b'^{a^{z''}}$$ and $a^za^{-z''}=a'^z$ commutes with $b'$, where $a'=aa^{-z''z^{-1}}$. $\dim(b')\ge1$, $\dim(a'/b')\ge1$, $\dim(z/a',b')\ge2$ and $\dim(a'^z/b')\ge1$.If $\dim(b'/b)=0$, then $\dim(r/b,b')=\dim(r/b)=2$. Choose $r'\equiv_{b,b'}r$ with $r'\indd_{b,b'}r$. So $b^r=b'^{-1}b=b^{r'}$ and $r'r^{-1}\in C_G(b)$. Since $r'r^{-1}$ is wide in $G$ over $b$, so is $C_G(b)$, and it has finite index in $G$ by minimality. Thus $b\in\Z(G)$ and $\dim\Z(G)\ge1$, so we can take $N=\Z(G)$. Thus we may assume $\dim(b')\ge\dim(b'/b)\ge1$. Next, note that $z$ and $z''z^{-1}$ are wide and $d$-independent over $a,b',c'$ by Lemma \[generics\]. An argument similar to the first paragraph yields $\dim(a'/b')\ge\dim(a'/a,b',z''z^{-1})\ge1$ and $\dim(a'^z/b')\ge1$. To finish, note that $\dim(C_G(b'))\ge\dim(a'^z/b')\ge1$. If $\dim(C_G(b'))=2$, then $b'\in\Z(G)$ and $\dim(\Z(G))\ge\dim(b')\ge1$, so we are done again. Otherwise $\dim(C_G(b'))=1$. By the -condition there is a broad centralizer $D\le C_G(b')$ of some finite tuple, minimal up to finite index, and $\dim(D)=\dim(C_G(b'))=1$, whence $\dim(C_G(b')/D)=0$. Choose $z^*\in G$ with $z^*\equiv_{a',b'}z$ and $z^*\indd_{a',b'}z$, and put $h=z^{*-1}z$. Then $a'^{z^*}\in C_G(b')$, so $a'^z\in C_G(b',b'^h)$. Moreover, $h$ is wide in $G$ over $a',b'$ and $d$-independent of $z$, so $$\dim(C_G(b',b'^h))\ge\dim(a'^z/b',h)=\dim(a'^z/b')\ge1$$ and $\dim(C_G(b')/C_G(b',b'^h))=0$. It follows that $$\begin{aligned}\dim(D/D\cap D^h)&=\dim(D/(D\cap C_G(b'^h)))+\dim((D\cap C_G(b'^h))/(D\cap D^h))\\ &\le\dim(C_G(b')/C_G(b',b'^h))+\dim(C_G(b'^h)/D^h)=0,\end{aligned}$$ whence $\dim(D\cap D^h)=1$, and $D^h$ is commensurable with $D$ by minimality. Thus $$\dim(\N_G(D))\ge\dim(h/b')=2$$ and $\N_G(D)$ is wide in $G$. Since $D$ is finite-by-abelian by Corollary \[corTh\], we finish as before, using Schlichting’s Theorem. \[soluble\] Let $G$ be a pseudofinite group whose definable sections are , and $\dim$ an additive integer-valued dimension on $G$. If $\dim(G)=2$, then $G$ has a definable wide soluble subgroup.By Theorem \[dim2\], there is a definable finite-by-abelian group $N$ such that $N_G(N)$ is wide. Replacing $N$ by $C_N(N')$, we may assume that $N$ is (finite central)-by-abelian. If $\dim(N)=2$ we are done. Otherwise $\dim(N_G(N)/N)=1$; by Corollary \[corTh\] there is a definable finite-by-abelian subgroup $S/N$ with $\dim(S/N)=1$. As above we may assume that $S/N$ is (finite central)-by-abelian, so $S$ is soluble. Moreover, $$\dim(S)=\dim(N)+\dim(S/N)=1+1=2,$$ so $S$ is wide in $G$. A pseudofinite superrosy group $G$ with $\omega^\alpha\cdot2\le\Ut(G)<\omega^\alpha\cdot3$ has a definable soluble subgroup $S$ with $\Ut(S)\ge\omega^\alpha\cdot2$.Superrosiness implies that all definabel sections of $G$ are . We put $$\dim(X)=n\quad\Leftrightarrow\quad\omega^\alpha\cdot n\le\Ut(X)<\omega^\alpha\cdot(n+1).$$ This defines an additive dimension with $\dim(G)=2$. The result now follows from Corollary \[soluble\]. [99]{} Richard Elwes and Mark Ryten. , Math. Log. Quart. 54, No. 4, 374–386 (2008). R. Elwes, E. Jaligot, H.D. Macpherson and M. Ryten, , Proc. London Math. Soc. (3) 103 (2011), 1049-1082. E. Hrushovski. , J. Amer. Math. Soc. 25(1):189–243, 2012. [^1]: Partially supported by ValCoMo (ANR-13-BS01-0006)

ArXiv

--- author: - Nachi Gupta - Raphael Hauser bibliography: - 'ieee-tac3.bib' title: Kalman Filtering with Equality and Inequality State Constraints --- Introduction ============ Kalman Filtering [@Kalman1960] is a method to make real-time predictions for systems with some known dynamics. Traditionally, problems requiring Kalman Filtering have been complex and nonlinear. Many advances have been made in the direction of dealing with nonlinearities (e.g., Extended Kalman Filter [@BLK2001], Unscented Kalman Filter [@JU1997]). These problems also tend to have inherent state space [*equality*]{} constraints (e.g., a fixed speed for a robotic arm) and state space [*inequality*]{} constraints (e.g., maximum attainable speed of a motor). In the past, less interest has been generated towards constrained Kalman Filtering, partly because constraints can be difficult to model. As a result, constraints are often neglected in standard Kalman Filtering applications. The extension to Kalman Filtering with known equality constraints on the state space is discussed in [@SAP1988; @TS1988; @SC2002; @WCC2002; @Gupta2007]. In this paper, we discuss two distinct methods to incorporate constraints into a Kalman Filter. Initially, we discuss these in the framework of equality constraints. The first method, projecting the updated state estimate onto the constrained region, appears with some discussion in [@SC2002; @Gupta2007]. We propose another method, which is to restrict the optimal Kalman Gain so the updated state estimate will not violate the constraint. With some algebraic manipulation, the second method is shown to be a special case of the first method. We extend both of these concepts to Kalman Filtering with inequality constraints in the state space. This generalization for the first approach was discussed in [@SS2005].[^1] Constraining the optimal Kalman Gain was briefly discussed in [@Q1989]. Further, we will also make the extension to incorporating state space constraints in Kalman Filter predictions. Analogous to the way a Kalman Filter can be extended to solve problems containing non-linearities in the dynamics using an Extended Kalman Filter by linearizing locally (or by using an Unscented Kalman Filter), linear inequality constrained filtering can similarly be extended to problems with nonlinear constraints by linearizing locally (or by way of another scheme like an Unscented Kalman Filter). The accuracy achieved by methods dealing with nonlinear constraints will naturally depend on the structure and curvature of the nonlinear function itself. In the two experiments we provide, we look at incorporating inequality constraints to a tracking problem with nonlinear dynamics. Kalman Filter {#sec::kf} ============= A discrete-time Kalman Filter [@Kalman1960] attempts to find the best running estimate for a recursive system governed by the following model[^2]: $$\label{kfsm} x_{k} = F_{k,k-1} x_{k-1} + u_{k,k-1}, \qquad u_{k,k-1} \sim \mathcal{N}\left(0,Q_{k,k-1}\right)$$ $$\label{kfmm} z_{k} = H_{k} x_{k} + v_{k}, \qquad v_{k} \sim \mathcal{N}\left(0,R_{k}\right)$$ Here $x_{k}$ is an $n$-vector that represents the true state of the underlying system and $F_{k,k-1}$ is an $n \times n$ matrix that describes the transition dynamics of the system from $x_{k-1}$ to $x_{k}$. The measurement made by the observer is an $m$-vector $z_{k}$, and $H_{k}$ is an $m \times n$ matrix that transforms a vector from the state space into the appropriate vector in the measurement space. The noise terms $u_{k,k-1}$ (an $n$-vector) and $v_{k}$ (an $m$-vector) encompass known and unknown errors in $F_{k,k-1}$ and $H_{k}$ and are normally distributed with mean 0 and covariances given by $n \times n$ matrix $Q_{k,k-1}$ and $m \times m$ matrix $R_{k}$, respectively. At each iteration, the Kalman Filter makes a state prediction for $x_k$, denoted $\hat{x}_{k|k-1}$. We use the notation ${k|k-1}$ since we will only use measurements provided until time-step $k-1$ in order to make the prediction at time-step $k$. The state prediction error $\tilde{x}_{k|k-1}$ is defined as the difference between the true state and the state prediction, as below. $$\label{se1} \tilde{x}_{k|k-1} = x_{k} - \hat{x}_{k|k-1}$$ The covariance structure for the expected error on the state prediction is defined as the expectation of the outer product of the state prediction error. We call this covariance structure the error covariance prediction and denote it $P_{k|k-1}$.[^3] $$\label{P-outer1} P_{k|k-1} = \mathbb{E}\left[\left(\tilde{x}_{k|k-1}\right)\left(\tilde{x}_{k|k-1}\right)'\right]$$ The filter will also provide an updated state estimate for $x_{k}$, given all the measurements provided up to and including time step $k$. We denote these estimates by $\hat{x}_{k|k}$. We similarly define the state estimate error $\tilde{x}_{k|k}$ as below. $$\label{se2} \tilde{x}_{k|k} = x_{k} - \hat{x}_{k|k}$$ The expectation of the outer product of the state estimate error represents the covariance structure of the expected errors on the state estimate, which we call the updated error covariance and denote $P_{k|k}$. $$\label{P-outer2} P_{k|k} = \mathbb{E}\left[\left(\tilde{x}_{k|k}\right)\left(\tilde{x}_{k|k}\right)'\right]$$ At time-step $k$, we can make a prediction for the underlying state of the system by allowing the state to transition forward using our model for the dynamics and noting that $\mathbb{E}\left[u_{k,k-1}\right] = 0$. This serves as our state prediction. $$\label{kfsp} \hat{x}_{k|k-1} = F_{k,k-1} \hat{x}_{k-1|k-1}$$ If we expand the expectation in Equation , we have the following equation for the error covariance prediction. $$\label{kfcp} P_{k|k-1} = F_{k,k-1} P_{k-1|k-1} F_{k,k-1}' + Q_{k,k-1}$$ We can transform our state prediction into the measurement space, which is a prediction for the measurement we now expect to observe. $$\label{kfmp} \hat{z}_{k|k-1} = H_{k} \hat{x}_{k|k-1}$$ The difference between the observed measurement and our predicted measurement is the measurement residual, which we are hoping to minimize in this algorithm. $$\label{kfi} \nu_{k} = z_{k} - \hat{z}_{k|k-1}$$ We can also calculate the associated covariance for the measurement residual, which is the expectation of the outer product of the measurement residual with itself, $\mathbb{E}\left[\nu_k \nu_k'\right]$. We call this the measurement residual covariance. $$\label{kfic} S_{k} = H_{k} P_{k|k-1} H_{k}' + R_{k}$$ We can now define our updated state estimate as our prediction plus some perturbation, which is given by a weighting factor times the measurement residual. The weighting factor, called the Kalman Gain, will be discussed below. $$\label{kfsu} \hat{x}_{k|k} = \hat{x}_{k|k-1} + K_{k} \nu_{k}$$ Naturally, we can also calculate the updated error covariance by expanding the outer product in Equation .[^4] $$\label{kfcu} P_{k|k} = \left(\operatorname{I}- K_{k} H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K_{k} H_{k}\right)' + K_k R_k K_k'$$ Now we would like to find the Kalman Gain $K_k$, which minimizes the mean square state estimate error, $\mathbb{E}\left[\left|\tilde{x}_{k|k}\right|^2\right]$. This is the same as minimizing the trace of the updated error covariance matrix above.[^5] After some calculus, we find the optimal gain that achieves this, written below.[^6] $$\label{kfkg} K_{k} = P_{k|k-1} H_{k}' S_{k}^{-1}$$ The covariance matrices in the Kalman Filter provide us with a measure for uncertainty in our predictions and updated state estimate. This is a very important feature for the various applications of filtering since we then know how much to trust our predictions and estimates. Also, since the method is recursive, we need to provide an initial covariance that is large enough to contain the initial state to ensure comprehensible performance. For a more detailed discussion of Kalman Filtering, we refer the reader to the following book [@BLK2001]. Equality Constrained Kalman Filtering ===================================== A number of approaches have been proposed for solving the equality constrained Kalman Filtering problem [@TS1988; @SAP1988; @WCC2002; @SC2002; @Gupta2007]. In this paper, we show two different methods. The first method will restrict the state at each iteration to lie in the equality constrained space. The second method will start with a constrained prediction, and restrict the Kalman Gain so that the estimate will lie in the constrained space. Our equality constraints in this paper will be defined as below, where $A$ is a $q \times n$ matrix, $b$ a $q$-vector, and $x_k$, the state, is a $n$-vector.[^7] $$\label{constraints} A x_k = b$$ So we would like our updated state estimate to satisfy the constraint at each iteration, as below. $$\label{kfsu-con} A \hat{x}_{k|k} = b$$ Similarly, we may also like the state prediction to be constrained, which would allow a better forecast for the system. $$A \hat{x}_{k|k-1} = b$$ In the following subsections, we will discuss methods for constraining the updated state estimate. In Section \[sec::aic\], we will extend these concepts and formulations to the inequality constrained case, and in Section \[sec::csp\], we will address the problem of constraining the prediction, as well. Projecting the state to lie in the constrained space {#sec::pue} ---------------------------------------------------- We can solve the following minimization problem for a given time-step $k$, where $\hat{x}_{k|k}^{P}$ is the constrained estimate, $W_k$ is any positive definite symmetric weighting matrix, and $\hat{x}_{k|k}$ is the unconstrained Kalman Filter updated estimate. $$\label{eq-proj-problem} \hat{x}_{k|k}^{P} = \operatorname*{arg\,min}_{x \in \mathbb{R}^n} \ \left\{\left(x - \hat{x}_{k|k} \right)' W_k \left(x - \hat{x}_{k|k} \right) : A x = b\right\}$$ The best constrained estimate is then given by $$\label{bce-xP} \hat{x}_{k|k}^{P} = \hat{x}_{k|k} - W_k^{-1} A' \left( A W_k^{-1} A' \right)^{-1} \left(A \hat{x}_{k|k} - b \right)$$ To find the updated error covariance matrix of the equality constrained filter, we first define the matrix $\Upsilon$ below.[^8] $$\Upsilon = W_k^{-1} A' \left(A W_k^{-1} A' \right)^{-1}$$ Equation can then be re-written as following. $$\label{xeq} \hat{x}_{k|k}^P = \hat{x}_{k|k} - \Upsilon\left(A \hat{x}_{k|k} - b \right)$$ We can find a reduced form for $x_k - \hat{x}_{k|k}^P$ as below. $$\begin{aligned} x_k - \hat{x}_{k|k}^P &= x_k - \hat{x}_{k|k} +\Upsilon \left(A \hat{x}_{k|k} - b - \left(A x_k - b \right)\right) \\ &= x_k - \hat{x}_{k|k} +\Upsilon \left(A \hat{x}_{k|k} - A x_k\right) \\ &= -\left(\operatorname{I}- \Upsilon A \right) \left(\hat{x}_{k|k} - x_k\right)\end{aligned}$$ Using the definition of the error covariance matrix, we arrive at the following expression. \[bce-PP\] $$\begin{aligned} P_{k|k}^P &= \mathbb{E}\left[\left(x_k - \hat{x}_{k|k}^P\right)\left(x_k - \hat{x}_{k|k}^P\right)'\right] \\ &= \mathbb{E}\left[\left(\operatorname{I}- \Upsilon A \right) \left(\hat{x}_{k|k} - x_k\right) \left(\hat{x}_{k|k} - x_k\right)' \left(\operatorname{I}- \Upsilon A \right)'\right] \\ &= \left(\operatorname{I}- \Upsilon A \right) P_{k|k} \left(\operatorname{I}- \Upsilon A \right)' \\ &= P_{k|k} - \Upsilon A P_{k|k} - P_{k|k} A' \Upsilon' + \Upsilon A P_{k|k} A' \Upsilon' \\ &= P_{k|k} - \Upsilon A P_{k|k} \\ &= \label{Peq} \left(\operatorname{I}- \Upsilon A \right) P_{k|k} \end{aligned}$$ It can be shown that choosing $W_k = P_{k|k}^{-1}$ results in the smallest updated error covariance. This also provides a measure of the information in the state at $k$.[^9] Restricting the optimal Kalman Gain so the updated state estimate lies in the constrained space ----------------------------------------------------------------------------------------------- Alternatively, we can expand the updated state estimate term in Equation using Equation . $$A \left( \hat{x}_{k|k-1} + K_{k} \nu_{k} \right) = b$$ Then, we can choose a Kalman Gain $K_k^R$, that forces the updated state estimate to be in the constrained space. In the unconstrained case, we chose the optimal Kalman Gain $K_k$, by solving the minimization problem below which yields Equation . $$K_k = \operatorname*{arg\,min}_{K \in \mathbb{R}^{n \times m}} {\ensuremath{\textnormal{trace}}}\left[ \left(\operatorname{I}- K H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K H_{k}\right)' + K R_k K'\right]$$ Now we seek the optimal $K_k^R$ that satisfies the constrained optimization problem written below for a given time-step $k$. $$\label{min-con} \begin{split} K_k^R = \operatorname*{arg\,min}_{K \in \mathbb{R}^{n \times m}} & {\ensuremath{\textnormal{trace}}}\left[ \left(\operatorname{I}- K H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K H_{k}\right)' + K R_k K'\right] \\ \textnormal{s.t. } & A \left( \hat{x}_{k|k-1} + K \nu_{k} \right) = b \end{split}$$ We will solve this problem using the method of Lagrange Multipliers. First, we take the steps below, using the vec notation (column stacking matrices so they appear as long vectors, see Appendix \[app::kv\]) to convert all appearances of $K$ in Equation into long vectors. Let us begin by expanding the following term.[^10] $$\begin{gathered} \nonumber{\ensuremath{\textnormal{trace}}}\left[\left(\operatorname{I}- K H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K H_{k}\right)' + K R_k K' \right] \qquad \qquad \qquad \qquad \qquad \qquad \qquad\\ \begin{aligned} &\stackrel{\hphantom{\eqref{kfic}}}{=}{\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} - K H_{k} P_{k|k-1} - P_{k|k-1} H_{k}' K' + K H_{k} P_{k|k-1} H_{k}' K' + K R_k K' \right] \\ &\stackrel{\eqref{kfic}}{=} {\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} - K H_{k} P_{k|k-1} - P_{k|k-1} H_{k}' K' + K S_k K' \right] \\ &\stackrel{\hphantom{\eqref{kfic}}}{=}\label{trace-separated}{\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} \right] - {\ensuremath{\textnormal{trace}}}\left[ K H_{k} P_{k|k-1} \right] - {\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} H_{k}' K' \right] + {\ensuremath{\textnormal{trace}}}\left[ K S_k K' \right] \end{aligned}\end{gathered}$$ We now expand the last three terms in Equation one at a time.[^11] $$\label{KHP} \begin{aligned} {\ensuremath{\textnormal{trace}}}\left[ K H_{k} P_{k|k-1} \right] \stackrel{\eqref{tr-ab}}{=} {\ensuremath{\textnormal{vec}\left[{\left(H_k P_{k|k-1}\right)'}\right]}}' {\ensuremath{\textnormal{vec}\left[{K}\right]}} \\ = {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}' {\ensuremath{\textnormal{vec}\left[{K}\right]}} \end{aligned}$$ $${\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} H_{k}' K' \right] \stackrel{\eqref{tr-ab}}{=} {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}$$ $$\label{KSK} \begin{aligned} {\ensuremath{\textnormal{trace}}}\left[ K S_k K' \right] &\stackrel{\eqref{tr-ab}}{=} {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\textnormal{vec}\left[{K S_k}\right]}} \\ &\stackrel{\eqref{vec-ab}}{=} {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\left({S}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{K}\right]}} \end{aligned}$$ Remembering that ${\ensuremath{\textnormal{trace}}}\left[ P_{k|k-1} \right]$ is constant, our objective function can be written as below. $$\begin{aligned} {\ensuremath{\textnormal{vec}\left[{K}\right]}}' \left(\operatorname{I}\otimes S_k \right) {\ensuremath{\textnormal{vec}\left[{K'}\right]}} &- {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}' {\ensuremath{\textnormal{vec}\left[{K}\right]}}\\ &- {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}} \end{aligned}$$ Using Equation on the equality constraints, our minimization problem is the following. $$\begin{split} K_k^R = \operatorname*{arg\,min}_{K \in \mathbb{R}^{n \times m}}& \ {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{K}\right]}} \\ &- {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}' {\ensuremath{\textnormal{vec}\left[{K}\right]}} \\ & - {\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}} \\ \textnormal{s.t. } & \left( \nu_{k}' \otimes A \right) {\ensuremath{\textnormal{vec}\left[{K}\right]}} = b - A \hat{x}_{k|k-1} \end{split}$$ Further, we simplify this problem so the minimization problem has only one quadratic term. We complete the square as follows. We want to find the unknown variable $\mu$ which will cancel the linear term. Let the quadratic term appear as follows. Note that the non-“${\ensuremath{\textnormal{vec}\left[{K}\right]}}$" term is dropped as is is irrelevant for the minimization problem. $$\left({\ensuremath{\textnormal{vec}\left[{K}\right]}} + \mu \right)' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} \left( {\ensuremath{\textnormal{vec}\left[{K}\right]}} + \mu \right)$$ The linear term in the expansion above is the following. $${\ensuremath{\textnormal{vec}\left[{K}\right]}}' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} \mu + \mu' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{K}\right]}}$$ So we require that the two equations below hold. $$\begin{aligned} {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} \mu &= -{\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}} \\ \mu' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} &= -{\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}}' \end{aligned}$$ This leads to the following value for $\mu$. $$\begin{aligned} \mu &\stackrel{\eqref{kron-inv}}{=} - {\ensuremath{\left({S_k^{-1}}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k'}\right]}} \\ &\stackrel{\eqref{vec-abc}}{=} -{\ensuremath{\textnormal{vec}\left[{P_{k|k-1} H_k' S_k^{-1}}\right]}} \\ &\stackrel{\eqref{kfkg}}{=} -{\ensuremath{\textnormal{vec}\left[{K_k}\right]}} \end{aligned}$$ Using Equation , our quadratic term in the minimization problem becomes the following. $$\left({\ensuremath{\textnormal{vec}\left[{K - K_k}\right]}} \right)' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} \left( {\ensuremath{\textnormal{vec}\left[{K - K_k}\right]}} \right)$$ Let $l = {\ensuremath{\textnormal{vec}\left[{K - K_k}\right]}}$. Then our minimization problem becomes the following. $$\begin{aligned} K_k^R = \operatorname*{arg\,min}_{l \in \mathbb{R}^{mn}} & \ l' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} l \\ \textnormal{s.t. }& \left( \nu_{k}' \otimes A \right) \left(l + {\ensuremath{\textnormal{vec}\left[{K_{k}}\right]}}\right) = b - A \hat{x}_{k|k-1} \end{aligned}$$ We can then re-write the constraint taking the ${\ensuremath{\textnormal{vec}\left[{K_k}\right]}}$ term to the other side as below. $$\begin{aligned} \left( \nu_{k}' \otimes A \right) l & = b - A \hat{x}_{k|k-1} - \left( \nu_{k}' \otimes A \right) {\ensuremath{\textnormal{vec}\left[{K_{k}}\right]}} \\ & \stackrel{\eqref{vec-abc}}{=} b - A \hat{x}_{k|k-1} -{\ensuremath{\textnormal{vec}\left[{A K_{k} \nu_k}\right]}} \\ & = b - A \hat{x}_{k|k-1} - A K_{k} \nu_k \\ & \stackrel{\eqref{kfsu}}= b - A \hat{x}_{k|k} \end{aligned}$$ This results in the following simplified form. $$\label{first-SDPT3} \begin{aligned} K_k^R = \operatorname*{arg\,min}_{l \in \mathbb{R}^{mn}}&\ l' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} l \\ \textnormal{s.t. }& \left( \nu_{k}' \otimes A \right) l = b - A \hat{x}_{k|k} \end{aligned}$$ We form the Lagrangian $\mathcal{L}$, where we introduce $q$ Lagrange Multipliers in vector $ \lambda = \left( \lambda_1, \lambda_2, \ldots, \lambda_q \right)'$ $$\begin{aligned} \mathcal{L} = & l' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} l - \lambda' \left[\left( \nu_{k}' \otimes A \right) l - b + A \hat{x}_{k|k}\right] \end{aligned}$$ We take the partial derivative with respect to $l$.[^12] $$\label{partial1} \frac{\partial \mathcal{L}}{\partial l} = 2 l' {\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}} - \lambda' \left( \nu_{k}' \otimes A \right) \\$$ Similarly we can take the partial derivative with respect to the vector $\lambda$. $$\frac{\partial \mathcal{L}}{\partial \lambda} = \left( \nu_{k}' \otimes A \right) l - b + A \hat{x}_{k|k}$$ When both of these derivatives are set equal to the appropriate size zero vector, we have the solution to the system. Taking the transpose of Equation , we can write this system as $Mn = p$ with the following block definitions for $M,n$, and $p$. $$\label{M-matrix} M = \begin{bmatrix} 2 {\ensuremath{{S_k}\otimes{\operatorname{I}}}} & \nu_{k} \otimes A' \\ \nu_{k}' \otimes A & 0_{{\ensuremath{\left[{q}\times{q}\right]}}} \end{bmatrix}$$ $$\label{n-vector} n = \begin{bmatrix} l \\ \lambda \end{bmatrix}$$ $$\label{p-vector} p = \begin{bmatrix} 0_{{\ensuremath{\left[{mn}\times{1}\right]}}} \\ b - A \hat{x}_{k|k} \end{bmatrix}$$ We solve this system for vector $n$ in Appendix \[app::Mnp\]. The solution for $l$ is pasted below. $$\left(\left[S_k^{-1} \nu_k \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1}\right] \otimes \left[A' \left(A A' \right)^{-1} \right]\right) \left(b - A \hat{x}_{k|k}\right)$$ Bearing in mind that $b - A \hat{x}_{k|k} = {\ensuremath{\textnormal{vec}\left[{b - A \hat{x}_{k|k}}\right]}}$, we can use Equation to re-write $l$ as below.[^13] $${\ensuremath{\textnormal{vec}\left[{A' \left(A A' \right)^{-1}\left(b - A \hat{x}_{k|k} \right) \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \nu_k' S_k^{-1}}\right]}}$$ The resulting matrix inside the vec operation is then an $n$ by $m$ matrix. Remembering the definition for $l$, we notice that $K - K_k$ results in an $n$ by $m$ matrix also. Since both of the components inside the vec operation result in matrices of the same size, we can safely remove the vec operation from both sides. This results in the following optimal constrained Kalman Gain $K_k^R$. $$K_k - A' \left(A A' \right)^{-1}\left(A \hat{x}_{k|k} - b \right) \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \nu_k' S_k^{-1}$$ If we now substitute this Kalman Gain into Equation to find the constrained updated state estimate, we end up with the following. $$\hat{x}_{k|k}^R = \hat{x}_{k|k} - A' \left(A A' \right)^{-1}\left(A \hat{x}_{k|k} - b \right)$$ This is of course equivalent to the result of Equation with the weighting matrix $W_k$ chosen as the identity matrix. The error covariance for this estimate is given by Equation .[^14] Adding Inequality Constraints {#sec::aic} ============================= In the more general case of this problem, we may encounter equality and inequality constraints, as given below.[^15] $$\label{ineq-constraints} \begin{split} A x_{k} = b\\ C x_{k} \leq d \end{split}$$ So we would like our updated state estimate to satisfy the constraint at each iteration, as below. $$\begin{split} A \hat{x}_{k|k} = b \\ C \hat{x}_{k|k} \leq d \end{split}$$ Similarly, we may also like the state prediction to be constrained, which would allow a better forecast for the system. $$\begin{split} A \hat{x}_{k|k-1} = b \\ C \hat{x}_{k|k-1} \leq d \end{split}$$ We will present two analogous methods to those presented for the equality constrained case. In the first method, we will run the unconstrained filter, and at each iteration constrain the updated state estimate to lie in the constrained space. In the second method, we will find a Kalman Gain $\check{K}_k^R$ such that the the updated state estimate will be forced to lie in the constrained space. In both methods, we will no longer be able to find an analytic solution as before. Instead, we use numerical methods. By Projecting the Unconstrained Estimate {#sec::pue-ineq} ---------------------------------------- Given the best unconstrained estimate, we could solve the following minimization problem for a given time-step $k$, where $\check{x}_{k|k}^{P}$ is the inequality constrained estimate and $W_k$ is any positive definite symmetric weighting matrix. $$\begin{aligned} \check{x}_{k|k}^{P} = \operatorname*{arg\,min}_{x} &\ \left(x - \hat{x}_{k|k} \right)' W_k \left(x - \hat{x}_{k|k} \right) \\ \textnormal{s.t. } & A x = b \\ & C x \leq d \end{aligned}$$ For solving this inequality constrained optimization problem, we can use a variety of standard methods, or even an out-of-the-box solver, like `fmincon` in Matlab. Here we use an active set method [@Fletcher1981]. This is a common method for dealing with inequality constraints, where we treat a subset of the constraints (called the active set) as additional equality constraints. We ignore any inactive constraints when solving our optimization problem. After solving the problem, we check if our solution lies in the space given by the inequality constraints. If it doesn’t, we start from the solution in our previous iteration and move in the direction of the new solution until we hit a set of constraints. For each iteration, the active set is made up of those inequality constraints with non-zero Lagrange Multipliers. We first find the best estimate (using Equation for the equality constrained problem with the equality constraints given in Equation plus the active set of inequality constraints. Let us call the solution to this $\check{x}_{k|k,j}^{P*}$ since we have not yet checked if the solution lies in the inequality constrained space.[^16] In order to check this, we find the vector that we moved along to reach $\check{x}_{k|k,j}^{P*}$. This is given by the following. $$s = \check{x}_{k|k,j}^{P*} - \check{x}_{k|k,j-1}^P$$ We now iterate through each of our inequality constraints, to check if they are satisfied. If they are all satisfied, we choose $\tau_{\max}=1$. If they are not, we choose the largest value of $\tau_{\max}$ such that $\hat{x}_{k|k,j-1} + \tau_{\max} s$ lies in the inequality constrained space. We choose our estimate to be $$\check{x}_{k|k,j}^P = \check{x}_{k|k,j-1}^{P} + \tau_{\max} s$$ If we find the solution has converged within a pre-specified error, or we have reached a pre-specified maximum number of iterations, we choose this as the updated state estimate to our inequality constrained problem, denoted $\check{x}_{k|k}^P$. If we would like to take a further iteration on $j$, we check the Lagrange Multipliers at this new solution to determine the new active set.[^17] We then repeat by finding the best estimate for the equality constrained problem including the new active set as additional equality constraints. Since this is a Quadratic Programming problem, each step of $j$ guarantees the same estimate or a better estimate. When calculating the error covariance matrix for this estimate, we can also add on the safety term below. $$\left(\check{x}_{k|k,j}^P - \check{x}_{k|k,j-1}^{P}\right)\left(\check{x}_{k|k,j}^P - \check{x}_{k|k,j-1}^{P}\right)'$$ This is a measure of our convergence error and should typically be small relative to the unconstrained error covariance. We can then use Equation to project the covariance matrix onto the constrained subspace, but we only use the defined equality constraints. We do not incorporate any constraints in the active set when computing Equation since these still represent inequality constraints on the state. Ideally we would project the error covariance matrix into the inequality constrained subspace, but this projection is not trivial. By Restricting the Optimal Kalman Gain -------------------------------------- We could solve this problem by restricting the optimal Kalman gain also, as we did for equality constraints previously. We seek the optimal $K_k$ that satisfies the constrained optimization problem written below for a given time-step $k$. $$\label{min-con} \begin{aligned} \check{K}^R_k = \operatorname*{arg\,min}_{K \in \mathbb{R}^{n \times m}} & {\ensuremath{\textnormal{trace}}}\left[\left(\operatorname{I}- K H_{k}\right) P_{k|k-1} \left(\operatorname{I}- K H_{k}\right)' + K R_k K'\right] \\ \textnormal{s.t. } & A \left( \hat{x}_{k|k-1} + K_{k} \nu_{k} \right) = b \\ & C \left( \hat{x}_{k|k-1} + K_{k} \nu_{k} \right) \leq d \end{aligned}$$ Again, we can solve this problem using any inequality constrained optimization method (e.g., `fmincon` in Matlab or the active set method used previously). Here we solved the optimization problem using SDPT3, a Matlab package for solving semidefinite programming problems [@TTT1999]. When calculating the covariance matrix for the inequality constrained estimate, we use the restricted Kalman Gain. Again, we can add on the safety term for the convergence error, by taking the outer product of the difference between the updated state estimates calculated by the restricted Kalman Gain for the last two iterations of SDPT3. This covariance matrix is then projected onto the subspace as in Equation using the equality constraints only. Dealing with Nonlinearities {#sec::nl} =========================== Thus far, in the Kalman Filter we have dealt with linear models and constraints. A number of methods have been proposed to handle nonlinear models (e.g., Extended Kalman Filter [@BLK2001], Unscented Kalman Filter [@JU1997]). In this paper, we will focus on the most widely used of these, the Extended Kalman Filter. Let’s re-write the discrete unconstrained Kalman Filtering problem from Equations and below, incorporating nonlinear models. $$\label{kfsm-nl} x_{k} = f_{k,k-1} \left(x_{k-1}\right) + u_{k,k-1}, \qquad u_{k,k-1} \sim \mathcal{N}\left(0,Q_{k,k-1}\right)$$ $$\label{kfmm-nl} z_{k} = h_{k} \left(x_{k}\right) + v_{k}, \qquad v_{k} \sim \mathcal{N}\left(0,R_{k}\right)$$ In the above equations, we see that the transition matrix $F_{k,k-1}$ has been replaced by the nonlinear vector-valued function $f_{k,k-1}\left(\cdot\right)$, and similarly, the matrix $H_k$, which transforms a vector from the state space into the measurement space, has been replaced by the nonlinear vector-valued function $h_k\left(\cdot\right)$. The method proposed by the Extended Kalman Filter is to linearize the nonlinearities about the current state prediction (or estimate). That is, we choose $F_{k,k-1}$ as the Jacobian of $f_{k,k-1}$ evaluated at $\hat{x}_{k-1|k-1}$, and $H_k$ as the Jacobian of $h_k$ evaluated at $\hat{x}_{k|k-1}$ and proceed as in the linear Kalman Filter of Section \[sec::kf\].[^18] Numerical accuracy of these methods tends to depend heavily on the nonlinear functions. If we have linear constraints but a nonlinear $f_{k,k-1}\left(\cdot\right)$ and $h_k\left(\cdot\right)$, we can adapt the Extended Kalman Filter to fit into the framework of the methods described thus far. Nonlinear Equality and Inequality Constraints --------------------------------------------- Since equality and inequality constraints we model are often times nonlinear, it is important to make the extension to nonlinear equality and inequality constrained Kalman Filtering for the methods discussed thus far. Without loss of generality, our discussion here will pertain only to nonlinear inequality constraints. We can follow the same steps for equality constraints.[^19] We replace the linear inequality constraint on the state space by the following nonlinear inequality constraint $c\left(x_k\right) = d$, where $c\left(\cdot\right)$ is a vector-valued function. We can then linearize our constraint, $c\left(x_k\right) = d$, about the current state prediction $\hat{x}_{k|k-1}$, which gives us the following.[^20] $$c\left(\hat{x}_{k|k-1}\right) + C \left(x_k - \hat{x}_{k|k-1} \right) \lessapprox d$$ Here $C$ is defined as the Jacobian of $c$ evaluated at $\hat{x}_{k|k-1}$. This indicates then, that the nonlinear constraint we would like to model can be approximated by the following linear constraint $$\label{puenl} C x_k \lessapprox d + C \hat{x}_{k|k-1} - c\left(\hat{x}_{k|k-1}\right)$$ This constraint can be written as $\tilde{C} x_k \leq \tilde{d}$, which is an approximation to the nonlinear inequality constraint. It is now in a form that can be used by the methods described thus far. The nonlinearities in both the constraints and the models, $f_{k,k-1}\left(\cdot\right)$ and $h_k\left(\cdot\right)$, could have been linearized using a number of different methods (e.g., a derivative-free method, a higher order Taylor approximation). Also an iterative method could be used as in the Iterated Extended Kalman Filter [@BLK2001]. Constraining the State Prediction {#sec::csp} ================================= We haven’t yet discussed whether the state prediction (Equation ) also should be constrained. Forcing the constraints should provide a better prediction (which is used for forecasting in the Kalman Filter). Ideally, the transition matrix $F_{k,k-1}$ will take an updated state estimate satisfying the constraints at time $k-1$ and make a prediction that will satisfy the constraints at time $k$. Of course this may not be the case. In fact, the constraints may depend on the updated state estimate, which would be the case for nonlinear constraints. On the downside, constraining the state prediction increases computational cost per iteration. We propose three methods for dealing with the problem of constraining the state prediction. The first method is to project the matrix $F_{k,k-1}$ onto the constrained space. This is only possible for the equality constraints, as there is no trivial way to project $F_{k,k-1}$ to an inequality constrained space. We can use the same projector as in Equation so we have the following.[^21] $$F_{k,k-1}^P = \left(\operatorname{I}- \Upsilon A \right) F_{k,k-1}$$ Under the assumption that we have constrained our updated state estimate, this new transition matrix will make a prediction that will keep the estimate in the equality constrained space. Alternatively, if we weaken this assumption, i.e., we are not constraining the updated state estimate, we could solve the minimization problem below (analogous to Equation ). We can also incorporate inequality constraints now. $$\begin{aligned} \check{x}_{k|k-1}^{P} = \operatorname*{arg\,min}_{x} &\ \left(x - \hat{x}_{k|k-1} \right)' W_k \left(x - \hat{x}_{k|k-1} \right) \\ \textnormal{s.t. } & A x = b \\ & C x \leq d \end{aligned}$$ We can constrain the covariance matrix here also, in a similar fashion to the method described in Section \[sec::pue-ineq\]. The third method is to add to the constrained problem the additional constraints below, which ensure that the chosen estimate will produce a prediction at the next iteration that is also constrained. $$\begin{aligned} A_{k+1} F_{k+1,k} x_k &= b_{k+1} \\ C_{k+1} F_{k+1,k} x_k &\leq d_{k+1} \end{aligned}$$ If $A_{k+1}, b_{k+1}, C_{k+1}$ or $d_{k+1}$ depend on the estimate (e.g., if we are linearizing nonlinear functions $a\left(\cdot\right)$ or $b\left(\cdot\right)$, we can use an iterative method, which would resolve $A_{k+1}$ and $b_{k+1}$ using the current best updated state estimate (or prediction), re-calculate the best estimate using $A_{k+1}$ and $b_{k+1}$, and so forth until we are satisfied with the convergence. This method would be preferred since it looks ahead one time-step to choose a better estimate for the current iteration.[^22] However, it can be far more expensive computationally. Experiments =========== We provide two related experiments here. We have a car driving along a straight road with thickness 2 meters. The driver of the car traces a noisy sine curve (with the noise lying only in the frequency domain). The car is tagged with a device that transmits the location within some known error. We would like to track the position of the car. In the first experiment, we filter over the noisy data with the knowledge that the underlying function is a noisy sine curve. The inequality constrained methods will constrain the estimates to only take values in the interval $[-1,1]$. In the second experiment, we do not use the knowledge that the underlying curve is a sine curve. Instead we attempt to recover the true data using an autoregressive model of order 6 [@BJ1976]. We do, however, assume our unknown function only takes values in the interval $[-1,1]$, and we can again enforce these constraints when using the inequality constrained filter. The driver’s path is generated using the nonlinear stochastic process given by Equation . We start with the following initial point. $$\label{ickf1-x0} x_0 = \begin{bmatrix} 0 \text{\ m}\\ 0 \text{\ m} \end{bmatrix}$$ Our vector-valued transition function will depend on a discretization parameter $T$ and can be expressed as below. Here, we choose $T$ to be $\pi/10$, and we run the experiment from an initial time of 0 to a final time of $10 \pi$. $$f_{k,k-1} = \begin{bmatrix} \left(x_{k-1}\right)_1 + T \\ \sin \left(\left(x_{k-1}\right)_1 + T \right) \end{bmatrix}$$ And for the process noise we choose the following. $$Q_{k,k-1} = \begin{bmatrix} 0.1 \text{\ m}^2 & 0 \\ 0 & 0 \text{\ m}^2 \end{bmatrix}$$ The driver’s path is drawn out by the second element of the vector $x_k$ – the first element acts as an underlying state to generate the second element, which also allows a natural method to add noise in the frequency domain of the sine curve while keeping the process recursively generated. First Experiment ---------------- To create the measurements, we use the model from Equation , where $H_k$ is the square identity matrix of dimension 2. We choose $R_k$ as below to noise the data. This considerably masks the true underlying data as can be seen in Fig. \[fig-ickf1\].[^23] $$\label{ickf1-R} R_{k} = \begin{bmatrix} 10 \text{\ m}^2 & 0 \\ 0 & 10 \text{\ m}^2 \end{bmatrix}$$ ![We take our sine curve, which is already noisy in the frequency domain (due to process noise), and add measurement noise. The underlying sine curve is significantly masked.[]{data-label="fig-ickf1"}](ickf.ps){width="\columnwidth"} For the initial point of our filters, we choose the following point, which is different from the true initial point given in Equation . $$\hat{x}_{0|0} = \begin{bmatrix} 0 \text{\ m}\\ 1 \text{\ m} \end{bmatrix}$$ Our initial covariance is given as below.[^24]. $$P_{0|0} = \begin{bmatrix} 1 \text{\ m}^2 & 0.1\\ 0.1 & 1 \text{\ m}^2 \end{bmatrix}$$ In the filtering, we use the information that the underlying function is a sine curve, and our transition function $f_{k,k-1}$ changes to reflect a recursion in the second element of $x_k$ – now we will add on discretized pieces of a sine curve to our previous estimate. The function is given explicitly below. $$f_{k,k-1} = \begin{bmatrix} \left(x_{k-1}\right)_1 + T \\ \left(x_{k-1}\right)_1 + \sin \left(\left(x_{k-1}\right)_1 + T \right) - \sin \left(\left(x_{k-1}\right)_1\right) \end{bmatrix}$$ For the Extended Kalman Filter formulation, we will also require the Jacobian of this matrix denoted $F_{k,k-1}$, which is given below. $$F_{k,k-1} = \begin{bmatrix} 1 & 0 \\ \cos \left(\left(x_{k-1}\right)_1 + T \right) - \cos \left(\left(x_{k-1}\right)_1\right) & 1 \end{bmatrix}$$ The process noise $Q_{k,k-1}$, given below, is chosen similar to the noise used in generating the simulation, but is slightly larger to encompass both the noise in our above model and to prevent divergence due to numerical roundoff errors. The measurement noise $R_k$ is chosen the same as in Equation . $$Q_{k,k-1} = \begin{bmatrix} 0.1 \text{\ m}^2 & 0 \\ 0 & 0.1 \text{\ m}^2 \end{bmatrix}$$ The inequality constraints we enforce can be expressed using the notation throughout the chapter, with $C$ and $d$ as given below. $$C = \begin{bmatrix} 0 & 1 \\ 0 & -1 \end{bmatrix}$$ $$d = \begin{bmatrix} 1\\ 1 \end{bmatrix}$$ These constraints force the second element of the estimate $x_{k|k}$ (the sine portion) to lie in the interval $[-1,1]$. We do not have any equality constraints in this experiment. We run the unconstrained Kalman Filter and both of the constrained methods discussed previously. A plot of the true position and estimates is given in Fig. \[fig-ickf2\]. Notice that both constrained methods force the estimate to lie within the constrained space, while the unconstrained method can violate the constraints. ![We show our true underlying state, which is a sine curve noised in the frequency domain, along with the estimates from the unconstrained Kalman Filter, and both of our inequality constrained modifications. We also plotted dotted horizontal lines at the values -1 and 1. Both inequality constrained methods do not allow the estimate to leave the constrained space.[]{data-label="fig-ickf2"}](ickf2.ps){width="\columnwidth"} Second Experiment ----------------- In the previous experiment, we used the knowledge that the underlying function was a noisy sine curve. If this is not known, we face a significantly harder estimation problem. Let us assume nothing about the underlying function except that it must take values in the interval $[-1,1]$. A good model for estimating such an unknown function could be an autoregressive model. We can compare the unconstrained filter to the two constrained methods again using these assumption and an autoregressive model of order 6, or AR(6) as it is more commonly referred to. In the previous example, we used a large measurement noise $R_k$ to emphasize the gain achieved by using the constraint information. Such a large $R_k$ is probably not very realistic, and when using an autoregressive model, it will be hard to track such a noisy signal. To generate the measurements, we again use Equation , this time with $H_k$ and $R_k$ as given below. $$H_k = \begin{bmatrix} 0 & 1 \end{bmatrix}$$ $$R_k = \begin{bmatrix} 0.5 \text{\ m}^2 \end{bmatrix}$$ Our state will now be defined using the following 13-vector, in which the first element is the current estimate, the next five elements are lags, the six elements afterwards are coefficients on the current estimate and the lags, and the last element is a constant term. $$\hat{x}_{k|k} = \begin{bmatrix} y_k & y_{k-1} & \cdots & y_{k-5} & \alpha_1 & \alpha_2 & \cdots & \alpha_7 \end{bmatrix}'$$ Our matrix $H_k$ in the filter is a row vector with the first element 1, and all the rest as 0, so $y_{k|k-1}$ is actually our prediction $\hat{z}_{k|k-1}$ in the filter, describing where we believe the expected value of the next point in the time-series to lie. For the initial state, we choose a vector of all zeros, except the first and seventh element, which we choose as 1. This choice for the initial conditions leads to the first prediction on the time series being 1, which is incorrect as the true underlying state has expectation 0. For the initial covariance, we choose $\operatorname{I}_{{\ensuremath{\left[{13}\times{13}\right]}}}$ and add $0.1$ to all the off-diagonal elements.[^25] The transition function $f_{k,k-1}$ for the AR(6) model is given below. $$\begin{bmatrix} \min\left(1, \max\left(-1, \alpha_1 y_{k-1} + \cdots + \alpha_6 y_{k-6} + \alpha_7 \right) \right)\\ \min\left(1, \max\left(-1,y_{k-1} \right) \right)\\ \min\left(1, \max\left(-1,y_{k-2} \right) \right)\\ \min\left(1, \max\left(-1,y_{k-3} \right) \right)\\ \min\left(1, \max\left(-1,y_{k-4} \right) \right)\\ \min\left(1, \max\left(-1,y_{k-5} \right) \right)\\ \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_6 \\ \alpha_7 \end{bmatrix}$$ Putting this into recursive notation, we have the following. $$\begin{bmatrix} \min\left(1, \max\left(-1, \left(x_{k-1}\right)_7 \left(x_{k-1}\right)_1 + \cdots + \left(x_{k-1}\right)_{13} \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_1 \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_2 \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_3 \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_4 \right) \right)\\ \min\left(1, \max\left(-1, \left(x_{k-1}\right)_5 \right) \right)\\ \left(x_{k-1}\right)_7 \\ \left(x_{k-1}\right)_8 \\ \vdots \\ \left(x_{k-1}\right)_{12} \\ \left(x_{k-1}\right)_{13} \end{bmatrix}$$ The Jacobian of $f_{k,k-1}$ is given below. We ignore the $\min \left( \cdot \right)$ and $\max \left( \cdot \right)$ operators since the derivative is not continuous across them, and we can reach the bounds by numerical error. Further, when enforced, the derivative would be 0, so by ignoring them, we are allowing our covariance matrix to be larger than necessary as well as more numerically stable. $$\begin{bmatrix} \begin{BMAT}{c.c}{c.c} \begin{BMAT}{c.c}{c.c} \begin{BMAT}{cc}{c} \left(x_{k-1}\right)_7 & \cdots \end{BMAT} & \left(x_{k-1}\right)_{12} \\ \operatorname{I}_{{\ensuremath{\left[{5}\times{5}\right]}}} & 0_{{\ensuremath{\left[{5}\times{1}\right]}}} \end{BMAT} & \begin{BMAT}{c}{c.c} \begin{BMAT}{cccc}{c} \left(x_{k-1}\right)_{1} & \cdots & \left(x_{k-1}\right)_{6} & 1 \\ \end{BMAT} \\ 0_{{\ensuremath{\left[{5}\times{7}\right]}}} \end{BMAT} \\ 0_{{\ensuremath{\left[{7}\times{6}\right]}}} & \operatorname{I}_{{\ensuremath{\left[{7}\times{7}\right]}}} \end{BMAT} \end{bmatrix}$$ For the process noise, we choose $Q_{k,k-1}$ to be a diagonal matrix with the first entry as 0.1 and all remaining entries as $10^{-6}$ since we know the prediction phase of the autoregressive model very well. The inequality constraints we enforce can be expressed using the notation throughout the chapter, with $C$ as given below and $d$ as a 12-vector of ones. $$C = \begin{bmatrix} \begin{BMAT}{c.c}{c} \begin{BMAT}{c}{c.c} \operatorname{I}_{{\ensuremath{\left[{6}\times{6}\right]}}} \\ -\operatorname{I}_{{\ensuremath{\left[{6}\times{6}\right]}}} \end{BMAT} & 0_{{\ensuremath{\left[{12}\times{7}\right]}}} \end{BMAT} \end{bmatrix}$$ These constraints force the current estimate and all of the lags to take values in the range $[-1,1]$. As an added feature of this filter, we are also estimating the lags at each iteration using more information although we don’t use it – this is a fixed interval smoothing. In Fig. \[fig-ickfb\], we plot the noisy measurements, true underlying state, and the filter estimates. Notice again that the constrained methods keep the estimates in the constrained space. Visually, we can see the improvement particularly near the edges of the constrained space. ![We show our true underlying state, which is a sine curve noised in the frequency domain, the noised measurements, and the estimates from the unconstrained and both inequality constrained filters. We also plotted dotted horizontal lines at the values -1 and 1. Both inequality constrained methods do not allow the estimate to leave the constrained space.[]{data-label="fig-ickfb"}](ickfb.ps){width="\columnwidth"} Conclusions =========== We’ve provided two different formulations for including constraints into a Kalman Filter. In the equality constrained framework, these formulations have analytic formulas, one of which is a special case of the other. In the inequality constrained case, we’ve shown two numerical methods for constraining the estimate. We also discussed how to constrain the state prediction and how to handle nonlinearities. Our two examples show that these methods ensure the estimate lies in the constrained space, which provides a better estimate structure. Kron and Vec {#app::kv} ============ In this appendix, we provide some definitions used earlier in the chapter. Given matrix $A \in \mathbb{R}^{ m \times n}$ and $B \in \mathbb{R}^{p \times q}$, we can define the right Kronecker product as below.[^26] $$\left( A \otimes B \right) = \begin{bmatrix} a_{1,1} B & \cdots & a_{1,n} B \\ \vdots & \ddots & \vdots \\ a_{m,1} B & \cdots & a_{m,n} B \end{bmatrix}$$ Given appropriately sized matrices $A, B, C,$ and $D$ such that all operations below are well-defined, we have the following equalities. $$\label{kron-trans} \left( A \otimes B \right)' = \left( A' \otimes B' \right)$$ $$\label{kron-inv} \left( A \otimes B \right) ^{-1} = \left( A^{-1} \otimes B^{-1} \right)$$ $$\label{kron-dist} \left( A \otimes B \right) \left( C \otimes D \right) = \left( AC \otimes BD \right)$$ We can also define the vectorization of an ${\ensuremath{\left[{m}\times{n}\right]}}$ matrix $A$, which is a linear transformation on a matrix that stacks the columns iteratively to form a long vector of size ${\ensuremath{\left[{mn}\times{1}\right]}}$, as below. $${\ensuremath{\textnormal{vec}\left[{A}\right]}} = \begin{bmatrix} a_{1,1} \\ \vdots \\ a_{m,1} \\ a_{1,2} \\ \vdots \\ a_{m,2} \\ \vdots \\ a_{1,n} \\ \vdots \\ a_{m,n} \end{bmatrix}$$ Using the vec operator, we can state the trivial definition below. $$\label{vec-sum} {\ensuremath{\textnormal{vec}\left[{A+B}\right]}} = {\ensuremath{\textnormal{vec}\left[{A}\right]}} + {\ensuremath{\textnormal{vec}\left[{B}\right]}}$$ Combining the vec operator with the Kronecker product, we have the following. $$\label{vec-ab} {\ensuremath{\textnormal{vec}\left[{AB}\right]}} = {\ensuremath{\left({B'}\otimes{\operatorname{I}}\right)}} {\ensuremath{\textnormal{vec}\left[{A}\right]}}$$ $$\label{vec-abc} {\ensuremath{\textnormal{vec}\left[{ABC}\right]}} = \left(C' \otimes A \right) {\ensuremath{\textnormal{vec}\left[{B}\right]}}$$ We can express the trace of a product of matrices as below. $$\label{tr-ab} {\ensuremath{\textnormal{trace}\left[{AB}\right]}} = {\ensuremath{\textnormal{vec}\left[{B'}\right]}}'{\ensuremath{\textnormal{vec}\left[{A}\right]}}$$ $$\begin{aligned} {\ensuremath{\textnormal{trace}\left[{ABC}\right]}} &= \label{trace-1} {\ensuremath{\textnormal{vec}\left[{B}\right]}}' \left(\operatorname{I}\otimes C\right) {\ensuremath{\textnormal{vec}\left[{A}\right]}} \\ &= \label{trace-2} {\ensuremath{\textnormal{vec}\left[{A}\right]}}' \left(\operatorname{I}\otimes B \right) {\ensuremath{\textnormal{vec}\left[{C}\right]}} \\ &= \label{trace-3} {\ensuremath{\textnormal{vec}\left[{A}\right]}}' \left(C \otimes \operatorname{I}\right) {\ensuremath{\textnormal{vec}\left[{B}\right]}}\end{aligned}$$ For more information, please see [@LT1985]. Analytic Block Representation for the inverse of a Saddle Point Matrix {#app::spm} ====================================================================== $M_S$ is a saddle point matrix if it has the block form below.[^27] $$\label{spm} M_S = \begin{bmatrix} A_S & B_S' \\ B_S & -C_S \end{bmatrix}$$ In the case that $A_S$ is nonsingular and the Schur complement $J_S = -\left(C_S + B_S A_S^{-1} B_S'\right)$ is also nonsingular in the above equation, it is known that the inverse of this saddle point matrix can be expressed analytically by the following equation (see e.g., [@BGL2005]). $$M_S^{-1} = \begin{bmatrix} A_S^{-1} + A_S^{-1} B_S' J_S^{-1} B_S A_S^{-1} & -A_S^{-1} B_S' J_S^{-1} \\ -J_S^{-1} B_S A_S^{-1} & J_S^{-1} \end{bmatrix}$$ Solution to the system $Mn=p$ {#app::Mnp} ============================= Here we solve the system $Mn=p$ from Equations , , and , re-stated below, for vector $n$. $$\label{Mnp} \begin{bmatrix} 2 {\ensuremath{{S_k}\otimes{\operatorname{I}}}} & \nu_{k} \otimes A' \\ \nu_{k}' \otimes A & 0_{{\ensuremath{\left[{q}\times{q}\right]}}} \end{bmatrix} \begin{bmatrix} l \\ \lambda \end{bmatrix} = \begin{bmatrix} 0_{{\ensuremath{\left[{mn}\times{1}\right]}}} \\ b - A \hat{x}_{k|k} \end{bmatrix}$$ $M$ is a saddle point matrix with the following equations to fit the block structure of Equation .[^28] $$\begin{aligned} A_S & = 2 {\ensuremath{{S_k}\otimes{\operatorname{I}}}} \\ B_S & = \nu_{k}' \otimes A \\ C_S & = 0_{{\ensuremath{\left[{q}\times{q}\right]}}}\end{aligned}$$ We can calculate the term $A_S^{-1} B_S'$. $$\begin{aligned} A_S^{-1} B_S' & = \left[ 2{\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}}\right]^{-1} \left( \nu_{k}' \otimes A \right)' \\ &\stackrel{\eqref{kron-trans}\eqref{kron-inv}}{=} \frac{1}{2} {\ensuremath{\left({S_k^{-1}}\otimes{\operatorname{I}}\right)}} \left( \nu_{k} \otimes A' \right) \\ &\stackrel{\eqref{kron-dist}}{=} \frac{1}{2} \left( S_k^{-1} \nu_k \right) \otimes A'\end{aligned}$$ And as a result we have the following for $J_S$. $$\begin{aligned} J_S & = - \frac{1}{2} \left( \nu_{k}' \otimes A \right) \left[ \left( S_k^{-1} \nu_k \right) \otimes A' \right] \\ &\stackrel{\eqref{kron-dist}}{=} - \frac{1}{2} \left( \nu_{k}' S_k^{-1} \nu_k \right) \otimes \left(A A' \right) \end{aligned}$$ $J_S^{-1}$ is then, as below. $$\begin{aligned} J_S^{-1} & = -2 \left[ \left( \nu_{k}' S_k^{-1} \nu_k \right) \otimes \left( A A' \right)\right]^{-1} \\ &\stackrel{\eqref{kron-inv}}{=} -2 \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \otimes \left(A A' \right)^{-1}\end{aligned}$$ For the upper right block of $M^{-1}$, we then have the following expression. $$\begin{aligned} A_S^{-1} B_S' J_S^{-1} &= \left[\left( S_k^{-1} \nu_k \right) \otimes A' \right] \left[\left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \otimes \left(A A' \right)^{-1}\right] \\ &\stackrel{\eqref{kron-dist}}{=} \left[S_k^{-1} \nu_k \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1}\right] \otimes \left[A' \left(A A' \right)^{-1} \right]\end{aligned}$$ Since the first block element of $p$ is a vector of zeros, we can solve for $n$ to arrive at the following solution for $l$. $$\left(\left[S_k^{-1} \nu_k \left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1}\right] \otimes \left[A' \left(A A' \right)^{-1} \right]\right) \left(b - A \hat{x}_{k|k}\right) \\$$ The vector of Lagrange Multipliers $\lambda$ is given below. $$-2 \left[\left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1} \otimes \left(A A' \right)^{-1} \right] \left(b - A \hat{x}_{k|k}\right)$$ [^1]: The similar extension for the method of [@WCC2002] was made in [@GHJ2005]. [^2]: The subscript $k$ on a variable stands for the $k$-th time step, the mathematical notation $\mathcal{N}\left(\mu,\Sigma\right)$ denotes a normally distributed random vector with mean $\mu$ and covariance $\Sigma$, and all vectors in this paper are column vectors (unless we are explicitly taking the transpose of the vector). [^3]: We use the prime notation on a vector or a matrix to denote its transpose throughout this paper. [^4]: The $\operatorname{I}$ in Equation represents the $n \times n$ identity matrix. Throughout this paper, we use $\operatorname{I}$ to denote the same matrix, except in Appendix \[app::kv\], where $\operatorname{I}$ is the appropriately sized identity matrix. [^5]: Note that $v'v = {\ensuremath{\textnormal{trace}\left[{vv'}\right]}}$ for some vector $v$. [^6]: We could also minimize the mean square state estimate error in the $N$ norm, where $N$ is a positive definite and symmetric weighting matrix. In the $N$ norm, the optimal gain would be $K^N_k = N^{\frac{1}{2}}K_k$. [^7]: $A$ and $b$ can be different for different $k$. We don’t subscript each $A$ and $b$ to avoid confusion. [^8]: Note that $\Upsilon A$ is a projection matrix, as is $\left(\operatorname{I}- \Upsilon A\right)$, by definition. If $A$ is poorly conditioned, we can use a QR factorization to avoid squaring the condition number. [^9]: If $M$ and $N$ are covariance matrices, we say $N$ is smaller than $M$ if $M-N$ is positive semidefinite. Another formulation for incorporating equality constraints into a Kalman Filter is by observing the constraints as pseudo-measurements [@TS1988; @WCC2002]. When $W_k$ is chosen to be $P_{k|k}^{-1}$, both of these methods are mathematically equivalent [@Gupta2007]. Also, a more numerically stable form of Equation with discussion is provided in [@Gupta2007]. [^10]: Throughout this paper, a number in parentheses above an equals sign means we made use of this equation number. [^11]: We use the symmetry of $P_{k|k-1}$ in Equation and the symmetry of $S_k$ in Equation . [^12]: We used the symmetry of ${\ensuremath{\left({S_k}\otimes{\operatorname{I}}\right)}}$ here. [^13]: Here we used the symmetry of $S_k^{-1}$ and $\left(\nu_{k}' S_k^{-1} \nu_k \right)^{-1}$ (the latter of which is actually just a scalar). [^14]: We can use the unconstrained or constrained Kalman Gain to find this error covariance matrix. Since the constrained Kalman Gain is suboptimal for the unconstrained problem, before projecting onto the constrained space, the constrained covariance will be different from the unconstrained covariance. However, the difference lies exactly in the space orthogonal to which the covariance is projected onto by Equation . The proof is omitted for brevity. [^15]: $C$ and $d$ can be different for different $k$. We don’t subscript each $C$ and $d$ to simplify notation. [^16]: For the inequality constrained filter, we allow multiple iterations within each step. The $j$ subscript indexes these further iterations. [^17]: The previous active set is not relevant. [^18]: We can also do a midpoint approximation to find $F_{k,k-1}$ by evaluating the Jacobian at $\left(\hat{x}_{k-1|k-1} + \hat{x}_{k|k-1}\right)/2$. This should be a much closer approximation to the nonlinear function. We use this approximation for the Extended Kalman Filter experiments later. [^19]: We replace the ‘$\leq$’ sign with an ‘$=$’ sign and the ‘$\lessapprox$’ with an ‘$\approx$’ sign. [^20]: This method is how the Extended Kalman Filter linearizes nonlinear functions for $f_{k,k-1}\left(\cdot\right)$ and $h_k\left(\cdot\right)$. Here $\hat{x}_{k|k-1}$ can be the state prediction of any of the constrained filters presented thus far and does not necessarily relate to the unconstrained state prediction. [^21]: In these three methods, the symmetric weighting matrix $W_k$ can be different. The resulting $\Upsilon$ can consequently also be different. [^22]: Further, we can add constraints for some arbitrary $n$ time-steps ahead. [^23]: The figure only shows the noisy sine curve, which is the second element of the measurement vector. The first element, which is a noisy straight line, isn’t plotted. [^24]: Nonzero off-diagonal elements in the initial covariance matrix often help the filter converge more quickly [^25]: The bracket subscript notation is used through the remainder of this paper to indicate the size of zero matrices and identity matrices. [^26]: The indices $m,n,p$, and $q$ and all matrix definitions are independent of any used earlier. Also, the subscript notation $a_{1,n}$ denotes the element in the first row and $n$-th column of $A$, and so forth. [^27]: The subscript $S$ notation is used to differentiate these matrices from any matrices defined earlier. [^28]: We use Equation with $B_S'$ to arrive at the same term for $B_s$ in Equation .

ArXiv

--- abstract: 'Recent empirical studies have confirmed the key roles of complex contagion mechanisms such as memory, social reinforcement, and decay effects in information diffusion and behavior spreading. Inspired by this fact, we here propose a new agent–based model to capture the whole picture of the joint action of the three mechanisms in information spreading, by quantifying the complex contagion mechanisms as stickiness and persistence, and carry out extensive simulations of the model on various networks. By numerical simulations as well as theoretical analysis, we find that the stickiness of the message determines the critical dynamics of message diffusion on tree-like networks, whereas the persistence plays a decisive role on dense regular lattices. In either network, the greater persistence can effectively make the message more invasive. Of particular interest is that our research results renew our previous knowledge that messages can spread broader in networks with large clustering, which turns out to be only true when they can inform a non-zero fraction of the population in the limit of large system size.' author: - Pengbi Cui - Ming Tang - Zhixi Wu title: 'Message spreading in networks with stickiness and persistence: Large clustering does not always facilitate large-scale diffusion' --- Over the last few years, many empirical works [@contagion; @decay3; @science; @attention; @contagion2; @origin1; @origin2] or practical model [@zhou; @model2] have identified the strong relevance of complex contagion mechanisms such as memory effect, social reinforcement and decay effects to information diffusion or behavior spreading. On account of memory effect, the previous contact activities can affect the current spreading process [@memory; @attention]. Specifically, individual’s selection of message items can be naturally expedited by the increasing frequencies of the same choices of other people if they find the items interesting or crucial enough [@science; @zhou; @decay]. This is usually interpreted as the results of social reinforcement [@rein1; @model2; @theory2]. On the other hand, there are an increasing amount of new messages an individual is facing every day in modern real life, whereas the attention and processing abilities of people are finite and saturated [@attention; @decay; @attention1]. The novelty of a message usually trend to fade with time and hence the attention people pay to it, which is normally described as decay effects [@burst2; @attention; @contagion; @decay]. It is shown that the social reinforcement effect could be weakened or even counterbalanced by decay effects [@attention; @decay; @contagion2]. Although the competition between social reinforcement and decay effects has been emphasized and used as a guideline to measure the natural time scale that attention fades away [@decay], to our best knowledge few works have been attempted to model explicitly the competition and memory effect, and study deeply how it shapes the spreading of information on complex networks. Here we want to point out that the three mentioned effects in information spreading are quite different from those have been considered in the studies on Naming Game (NG) and Category Game (CG), since either NG or CG is a two-step multi-state negotiation process [@ng1; @ng2; @cg1; @cg2], whereas information spreading is not. First, herein memory effect performs as the storing of the times of contact of people with recipients of information [@science; @zhou], rather than the possible words (or names) for the object (or a category) in NG (or CG) [@ng1; @ng2; @cg1; @cg2]. Second, decay effect in information spreading reflects the decay of people’s interest or attention in a message owing to the competition with other news or stories [@attention; @decay], contrary to the NG (CG) in which it means the decrease of the number of different words used in the system (or average number of words per category) [@ng1; @ng2; @cg1; @cg2]. Third, unlike the phenomenon that an hearing would have more opportunities to add (or remain) one word only if more selected speakers try to transmit the same one to it [@ng1; @ng2; @cg1; @cg2], the reinforcement effect in information diffusion indicates the more simple situation that the more neighbors adopting the message, the higher likelihood an individual following them [@science; @contagion2]. Next, the big challenge we are confronted with is the possibility of modeling and studying the message spreading along with both social reinforcement and decay effects based on the memory effect. Recent researches [@origin1] have shown that the variation in the ways that different information spread is attributed to not only the stickiness – the probability of information adoption is mainly dominated by the first few exposures [@origin2; @origin1], but also the persistence – the relative extent to which more repeated exposures to the message continue to have durative effect. Similar results especially the exposure response behaviors were also confirmed by a lot of empirical studies [@contagion2; @origin2; @attention]. The two mechanisms, stickiness and persistence, thus enable us to quantitatively study the joint action of the three effects together. At the same time, the structures of complex social systems can be characterized by complex networks, on which many spreading activities may take place, ranging from the spreading of epidemics [@tang1; @tang2; @tang3; @tang4], the diffusion of behaviors and news [@science; @zhou], to the promotion of technique innovations [@innovation], etc. Consequently, motivated by the empirical studies [@science; @zhou; @attention; @contagion2; @origin1; @origin2; @decay] mentioned above, we propose a new agent-based model offering an opportunity to explore the impact of social reinforcement and decaying effects quantified by stickiness and persistence on the message (information) diffusion on various networks. In the presence of strong decay effects, we find that a message is more likely to outbreak (i.e., it can reach a non-zero fraction of the population in the thermodynamic limit) on the tree-like networks such as scale-free (SF) networks and Erd[ő]{}s-R[é]{}nyi (ER) random networks rather than on the regular lattices (RLs). Specifically, a message can spread broader in the RLs than that in the tree-like networks only if it can outbreak. The critical behaviors of the diffusion process can be reasonably estimated by the bond-percolation theory considering spatial correlations of the underlying networks through which message diffuses. In addition, we develop a verification approximation, whose solutions confirm well the non-negligible role of the dynamical correlations between transmission events in the RLs. Results {#results .unnumbered} ======= Here, we first carry out extensive simulations for the agent-based model of message diffusion on square lattice. We then compare the simulation results with the predictions from the analytical bond percolation theory and verification method involving time correlations of the spreading events. Finally we extend our model and analytical methods to other networks such as RLs, SF networks, and ER networks to validate the robustness of our findings. **Message diffusion on square lattice.** \[square\] We first consider the message diffusion on a square lattice of size $N=L\times L$ with periodic boundary conditions. The message starts spreading from the center node (selected as the seed), while all the others are in the susceptible state (i.e., they hear nothing about the message). ![**The time evolution of spatial patterns,** for two different values of stickiness (a1) $a=0.40$, (a2) $a=0.45$ (bottom panel) where $b=0$; and for two values of persistence (b1) $b=-1.00$, (b2) $b=1.00$ (bottom panel) where $a=0.45$. Red sites represent recovered or alerted nodes, bright green sites represent infected ones, and blue sites denote susceptible nodes. Other parameters are chosen as $n_{s}=2$ and $N=101\times 101$.[]{data-label="spatial"}](spatial.jpeg){width="\textwidth"} To intuitively grasp the roles of stickiness and persistence, we begin by presenting the time evolution of spatial patterns of message spreading in Fig. \[spatial\]. The message with stickiness $a=0.40$ (see the Methods for the precise definitions of $a$, $b$ and other parameters) spreads in an irregular manner (see Fig. \[spatial\](a1)). By comparison, the message with a slightly stronger stickiness $a=0.45$ diffuses outward to susceptible areas in a quasi-circular manner with a broader rim of informed (infected) individuals (see Fig. \[spatial\](a2)). This indicates that messages with different strength of stickiness could give rise quite different spreading patterns and behaviors. Figs. \[spatial\](b1) (b2) and Supplementary Fig. S1 show that the persistence $b$ also affects considerably the whole spreading size, by governing the number of the isolated susceptible islands (blue domains surrounded by red areas, the emergence of these islands arises from the fact that continually increasing number of infected neighbors fail to infect those individuals owing to small persistence). The above arguments suggest that the stickiness and persistence have great but different influences on the spreading of message. ![**The evolution of proportions of the transmission events.** The parameters are chosen at (a) subcritical point $a=0.20$, $b=0.20$; (b) critical point $a=0.35$, $b=0.20$; and (c) supercritical point $a=0.43$, $b=0.20$.[]{data-label="alround"}](alpharound.jpeg){width="\textwidth"} We next explore the critical behaviors of message diffusion for various stickiness and persistence, by providing a quantitative assessment of the message burstiness. By means of the theory of non-equilibrium phase transition in statistical physics which has been successfully generalized to study the epidemic dynamics, previous studies [@variance1; @variance2] on spreading dynamics have shown that the fluctuation of the order parameter is divergent at the critical point. We thus use the procedure proposed in [@variance2] to numerically determine the critical areas. To be more specific, a series of variabilities $v(a,b)$ are firstly obtained as $$\label{eq:variance} v(a,b)=\frac{\sqrt{<\rho^{2}_{R}(a,b)>-<\rho_{R}(a,b)>^{2}}}{<\rho_{R}(a,b)>},$$ where $\rho_{R}$ and $v(a,b)$ denote, respectively, the density of recovered individuals in the population and the relative standard deviation (RSD) at parameter point $(a,b)$. There exists a maximum variability $v_{max}(b)$ for each value of $b$ when varying $a$ from $0$ to $0.5$, and the values of $v_{max}(b)$ with $b\in[-1~1]$ can be used as the numerical estimation of the threshold position. Although a bond-percolation process can be mapped to the SIR model [@prl; @bond], its extension to the current model is not straightforward. First, in our model, the transmission probability that a susceptible individual approves the message varies with the times he has received it from his infected neighbors (i.e., the number of informed neighbor he has had). Second, the time correlations between different transmission events $E_{n}$ [@prl] (the transmission event $E_{n}(t)$ represents that an individual, who has received the message at least once before, successfully approves the message when he has received it another $n$ times until time $t$). To confirm the existence of the time correlation, in Fig. \[alround\] we compare the dynamic proportions of the four transmission events ($\alpha_{n}(a,b,t)=\frac{\omega_{n}(a,b,t)}{\sum^{\langle k\rangle -1}_{m=0}\omega_{m}(a,b,t)}$) obtained from numerical simulations with those predicted by the bond percolation considering spatial correlations ($\beta_{n}(a,b,t)=\frac{q_{n}T_{n}(a,b)}{\sum^{\langle k\rangle -1}_{m=0}q_{m}T_{m}(a,b)}$ and see the Methods section for definitions of $q_{n}$ and $T_{n}(a,b)$) for three parameter points $(a,b)$. Here, $\omega_{n}(a,b,t)$ represents the occurrence frequency of $E_{n}(t)$ obtained from numerical simulations, and $\langle k\rangle$ denotes the average degree of the networks. In Figs. \[alround\] (b) and (c), we see that $\alpha_{0}(t)$ ($\alpha_{2}(t)$) is greater (less) than $\beta_{0}(t)$ ($\beta_{2}(t)$) during the spreading process, whereas $\alpha_{1}(t)$ ($\alpha_{3}(t)$) is equal or close to $\beta_{1}(t)$ ($\beta_{3}(t)$). The reason is that the existence of time correlations of the transmission events $E_{n}(t)$ can determine whether the subsequent events $E_{m}(t)$ ($m>n$) happen or not in the spreading process. If $E_{n}(t)$ does not happen, the events $E_{m}(t)$ ($m>n$) will probably happen; otherwise $E_{m}(t)$ ($m>n$) will never happen since an informed individual transmits the message only one time and then becomes recovered (i.e., completely ignores the message) forever. Consequently, $E_{0}$ ($E_{2}$) contributes more (less) than predicted by percolation theory to the spreading course (also see Supplementary Fig. S2). To overcome the challenge, we develop a verification approximation involving the time correlations of the transmission events, besides the spatial correlations originating from the spatial structure of the lattice [@prl]. It is necessary to mention that the discrete-time synchronous transmission of the message enables us to avoid concerning about the additional synergistic effect [@synergy]. Moreover, Figs. \[alround\](b) and (c) show that $\alpha_{n}(a,b,t)$ is dynamically stable, which shows that this correlation always exists. It allows us to adopt average values of the four indices $\alpha_{n}(a,b,t)$ at the critical regions for the verification approximation. ![**The phase diagram of the message spreading.** The real numerical critical boundaries (crosses) are obtained by Eq.  for various $n_{s}$ on square lattice of size $N = 101\times 101$. For comparison, analytical boundary (black line) and verification boundary (dash black line) for $n_{s}=2$ are also shown (the cases for $n_{s}=3,~4,~5$ do not allow us to analytically identify the critical lines). Herein, we select a narrow parameter ranges $b\in[-1,~1]$ and $a\in[0.32,~0.36]$ containing the numerical critical boundary for the calculation of verification threshold (see more detailed method in the Methods section).[]{data-label="latthre"}](trelattice.jpg){width="\textwidth"} Based on the proposed methods in the Methods section and Eq. , we yield both the analytical prediction and the verification threshold for $n_{s}=2$, plus the numerical results for various $n_{s}$, as depicted in Fig. \[latthre\]. We note that the numerical thresholds stay at $a\approx 0.32$, which is mainly determined by the stickiness of message (i.e., the parameter $a$), regardless of the values of $b$ and $n_{s}$. This means that most informed individuals are actually infected by their first one or two infected neighbors (also see Supplementary Figs. S2–S6). Furthermore, the numerical estimations are fairly reproduced by the bond-percolation theory. Comparing the analytical boundary, the verification approximation involving both spatial and time correlations gives a higher accurate estimation than the bond-percolation method considering only spatial correlations (the dashed black line is clearly closer to the numeric markers than the black solid line is). **Message diffusion on regular lattice networks and regular random networks.** \[hm\] Centola’s work [@science] concludes that social behaviors can spread farther and faster across clustered-lattice networks than across corresponding regular random networks (RRNs), owing to the strong social reinforcement induced by clustered ties. RRNs are networks that all nodes have exactly the same degree while links are randomly distributed among nodes, avoiding self-connections and multiple connections. To check whether the findings by Centola are still fulfilled for information diffusion, we further investigate our model defined on the RLs (Hexagonal network and Moore network) and the RRNs. ![[]{data-label="lhgap"}](lhgap.jpg){width="\textwidth"} To get a comparison, we present in Fig. \[lhgap\] the differences of the final size of recovered population on the two networks with the same average degree. The blue areas characterize the parameter regions where the conclusion of Centola’s experiment (that the information spreads farther across the RLs than across the corresponding RRNs) does not hold. The violation is attributed to the presence of strong decay effects ($b<0$) in the vicinity of the critical regions. Specifically, the outbreaks of message can happen more easily in the RRNs than that in the RLs for negative persistence owing to that strong decay effects outcompetes the weak reinforcement effect (also see Supplementary Fig. S7, strong reinforcement effect (decay effects) is reflected by large stickiness and/or positive persistence (negative persistence) in our model). As $a$ and $b$ get larger, things turn out differently, the message is able to seize a larger population in the RLs, which is accordant with the anticipation of Centola’s experiment. This means that high level of clustering created by redundant ties that linked each node’s neighbors to one another in the RLs strengthens the reinforcement effect, and hence facilitates the diffusion of the message [@science]. Moreover, larger $n_{s}$ improves the performance of stickiness in facilitating the message diffusion, making social reinforcement effects be the most prominent for the RLs. Consequently, the blue areas shrink with increasing $n_{s}$. Thus, the above differences investigated indicate that network topology and $n_{s}$ (which can be regarded as one of the intrinsic characteristics of the message) simultaneously determine the effects of stickiness and persistence on the spreading dynamics. ![**The phase diagrams of the message spreading on RLs and RRNs.** The underlying networks are (a) RRN with $\left\langle k\right\rangle=6$, (b) RRN with $\left\langle k\right\rangle=8$, (c) Hexagonal lattice, and (d) Moore lattice. The predictions from the bond-percolation theory (solid lines) and verification approximations (dashed lines) for $n_{s}=2$ are illustrated to compare with the simulation data (markers) for various $n_{s}$. To get the verification trajectory on Hexagonal lattice (and Moore lattice), we select two parameter regions containing the numerical thresholds (see more detailed method in the Methods section). One region is $b\in[-0.3~1.0]$ and $a\in[0.06~0.24]$; the other one is $b\in[-0.3~1.0]$ and $a\in[0.01~0.19]$ in Hexagonal lattice. One region is $b\in[-1.0~-0.3]$ and $a\in[0.23~0.24]$; the other one is $b\in[-1.0~0.3]$ and $a\in[0.15~0.18]$ in Moore lattice.[]{data-label="rhthre"}](rhthre.jpg){width="\textwidth"} Using Eq. , we obtain the numerical thresholds for various values of $n_{s}$. According to the methods described in the Methods section, we can yield the analytical (theoretical) thresholds of message diffusion on both the RRNs and the RLs, in addition to the verification thresholds on the RLs for $n_{s}=2$ (Supplementary Fig. S10 and Fig. 11 show that the time correlations between the transmission events in Hexagonal lattice and Moore lattice are noticeable). In the case of RRNs (Figs. \[rhthre\](a) and (b)), we observe that the positions of the critical boundaries are mainly determined by both the stickiness and $\left\langle k\right\rangle$ instead of the persistence and $n_{s}$, on account of weaker social reinforcement [@science; @prl] resulting from the low clustering coefficient. Unlike the case of RRNs, the theoretical analysis by means of bond-percolation theory gives rather accurate predications of the position of the threshold with strong persistence, but not for negative persistence (i.e, with the presence of strong decay effects) on the RLs. When the underlying networks for message diffusion are Hexagonal lattice and Moore lattice, more available edges for message spreading will further strengthen the role of persistence in message diffusion, especially with stronger social reinforcement (positive $b$). That enables all transmission events involve in the diffusion (see Supplementary Figs. S10–S13), so that $\beta_{n}(a,b,t)$ gets close to $\alpha_{n}(a,b,t)$ (see Supplementary Figs. S10 and S11). On the other hand, the trajectories obtained by verification approximation are in good agreement with the simulations, from which one can conclude that the effect of time correlations of transmission events is indeed general on the RLs. Additionally, the message steps forward to arrive in half of the neighbors of the same host on the RLs by flowing through almost $\frac{\left\langle k\right\rangle}{2}$ edges connecting it. This makes the message with positive persistence ($b>0$) outbreak more easily in the presence of denser local connections, and the persistence thus imposes a greater influence on outbreaks of message on RLs with larger average degree for small $n_{s}$ ( $\frac{n_{s}}{\left\langle k\right\rangle}<\frac{1}{2}$). The message also has reached a saturation state when the subsequent events $E_{i}$ ($i>\frac{\left\langle k\right\rangle}{2}$) happen (see Supplementary Figs. S14–S21). Also in RLs, higher $n_{s}$ limits the effect of persistence, and the phase transitions are determined by not only the topologies of the networks but also the stickiness and persistence of the message. In addition, the reinforcement effect begins to work as the message is bursting and prevailing on both the RLs and RRNs. Therefore, the results for positive persistence near the thresholds (see Supplementary Figs. S12–S24) elucidate the actual phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"), where the majority do not believe the message as a truth until at least three neighbors have tried to transmit this message to them. **Message diffusion on SF networks and ER networks.** \[eb\] To further check the robustness of our above findings, we finally investigate our message diffusion model on SF and ER networks with $\langle k\rangle=6, 8, 10, 12$. Since two or more transmission events fail to last over the long time at the critical points (see Supplementary Fig. S24), we do not take the time correlations into consideration in theoretical analysis. Compared with the simulation data, the analytical results for SF networks and ER networks with different average degrees show that the theoretical approaches are already sufficient to give fairly precise expressions of the outbreaks of message completely determined by the stickiness $a$ (see Supplementary Fig. S25). Nevertheless, the persistence also partly boosts its impact on the size of message diffusion as $\left\langle k\right\rangle$ gets larger (see Supplementary Fig. S26). ![**The differences of spreading sizes between the RLs and SF (ER) networks ($\rho_{SF/ER}-\rho_{RL}$).** (a,b) the difference $\rho_{SF}-\rho_{RL}$, (c,d) the difference $\rho_{ER}-\rho_{RL}$. (a,c) The average degrees of the networks are $\left\langle k\right\rangle=6$ (a,c), and $\left\langle k\right\rangle=8$ (b,d), respectively. Here, $\rho_{SF}$ ($\rho_{ER}$, $\rho_{RL}$) represents the recovered density in SF networks (ER networks, RLs). Other parameter: $n_{s}=2$.[]{data-label="brhmgap"}](brhmgap.jpg){width="\textwidth"} Fig. \[brhmgap\] displays evident differences of the final spreading sizes on the RLs and SF (ER) networks with the same degree. The spreading sizes are larger in SF (ER) than those in the RLs at the parameter regions where the message has already outbroken on SF (ER) but not yet on the RLs, owing to the hubs and shorter shortest paths in SF (ER) [@shortest] (see Supplementary Fig. S25 for the critical boundaries of message on SF and ER). Just for strong persistence, denser local connectivity of the RLs can make an invasion take place more easily. As $a$ and $b$ increase beyond the critical boundaries displayed in Fig. \[rhthre\], the message can capture a larger population again on the RLs, despite of the existence of hubs the short characteristic path length in the SF (ER) networks. The reason is that very smaller clustering coefficient gives rise to weak social reinforcement effect [@science; @report], which again leads to weak performance of persistence in promoting the spread of message on the SF (ER) networks. Our results indicate that the role of reinforcement effect is more important than that of hubs or shortest paths in facilitating message spreading only when the message outbreaks on the RLs. Otherwise, especially for the presence of decay effects (negative persistence) the facilitation of hubs and shortest paths to the diffusion on SF (ER) cannot be neglected. In addition, the results for positive persistence on SF and ER networks can also be treated as evidence of the mechanism “Three men make a tiger” (see Supplementary Fig. S27 and Fig. S28). Discussion {#con .unnumbered} ========== In conclusion, taking into account social reinforcement and decay effects based on memory effect in reality, we have proposed a new agent-based message spreading model with stickiness and persistence, and carried out extensive computer simulations of our model on various types of networks. By means of the relative standard deviation method and the bond percolation theory involving the spatial correlations, we are able to determine numerically and analytically the positions of critical boundaries. Moreover, the remarkable accuracy of verification approximation involving the time correlations between different transmission events validates the wide existence of such correlations for the message diffusion on regular lattices. Our preliminary results show that in RLs, the persistence depends greatly on the position of inflection point $n_{s}$ and average degree $\left\langle k\right\rangle$ of the underlying networks, and begins to play a pivotal role in the spreading process with increasing $\left\langle k\right\rangle$ owning to the emergent large clustering coefficient [@science]. Stronger social reinforcement arising from larger clustering coefficient leads readily to stronger infectivity of the message, which can invade a great number of susceptible individuals in the RLs, confirming the conclusion of Centola’s work. By comparison, in tree-like networks such as RRNs, SF and ER, the critical thresholds of message diffusion are only dominated by the stickiness, and both the hubs and the short characteristic path length facilitate the outbreaks of message in the presence of decay effects. It worth emphasizing that the results presented in this paper has successfully substantiated the phenomenon “Three men make a tiger“ (or ”A lie, if repeated often enough, will be accepted as truth"). Placing our study in the context of social media, the hub nodes actually play a role of broadcasters, advertisements and so on, which are very important for the large scale spreading of information. However, the intrinsic contents of messages and their adaptation to hosts are extremely relevant in determining the message diffusion. In particular, the results for the message diffusion on the SF and ER networks constitute a proof that exposure to mass media can favour the outbreaks of behaviors, news or messages, despite the decay effects, only if the stickiness of the message is large enough. On the other hand, the RLs are efficient in taking advantage of social reinforcement effects to promote the global spreading of the message, owning to more cluster ties (or local pressures) that function as form of ’initial groups’ or ’small groups’ through interpersonal communication or ’machine-interpersonal communication’ [@fed]. It is also verified by our research that the local, personal communication is irreplaceable to lead to propagation of message despite of the developed media industry today, from the perspective of communication. Recent studies have started considering the memory effect [@science; @zhou], social reinforcement and decay effects [@attention] in information spreading. The mechanisms were yet investigated in isolation. Our work is the first attempt to account explicitly for the three key mechanisms together, and to evaluate the joint action of them by quantifying their effects as stickiness and persistence. It provides a quantitative guideline for future social experiments for message spreading. In reality, the ways in which information spread may be very complicated. In the present study, we do not capture the difference of individual-level preference [@leader] that might have also influenced their decisions to adopt one message. For example, in an online social network such as Twitter, individuals may prefer different hashtags, and significant variations in the ways that the hashtags on different topics spread were observed [@origin2]. In addition, one need to concern about the diverse cultural and societal backgrounds [@culture] which would lead to the different styles by which individuals contact with the medium, or even hamper communications among different groups of members [@fed]. Moreover, the volatilities of complex contagion of controversial topics, psychological status of individuals [@media] reveal that the status of the individuals in the communication systems are time dependent, which should be addressed in future research. Methods {#methods .unnumbered} ======= **Message spreading model with stickiness and persistence.** The message spreading model is implemented on a network consisting of $N$ nodes and $E$ edges, where the nodes represent the individuals in a population and the edges the social interactions among them, through which information propagates. Each individual is allowed to be in one of three states at each time step: (i) Susceptible (or uninformed) state—the individual has not yet heard the message or is aware of the news but not willing to transmit it. (ii) Infected state—the individual catches the message and forwards it to all his nearest neighbors. (iii) Recovered state—the individual will never transmit the message any more after having transmitted it once before. Specifically, the information propagation is modeled in a probabilistic framework at individual level [@zhou; @model2]. A susceptible individual $i$ will adopt the message with a probability $\lambda_{n^{o}}(a,b)$, given that he has heard it from his informed neighbors $n^{o}$ times (i.e., $n^{o}$ informed neighbors he has had), plus the first time. In detail, $n^{o}=n+1$ when individual $i$ has owned at least one informed neighbors, otherwise $n^{o}=0$. Here $\lambda_{n^{o}}(a,b)$ is a linear piecewise function of the times he has received message from his informed neighbors: $$\begin{aligned} \label{eq:lambda} &\lambda_{n^{o}}(a,b) = \begin{cases} \min\{an^{o}, 1\}, \quad 0\leq n\leq n_{s};\\ \min\{bn^{o}+n_{s}(a-b),1\}, \quad n_{s}< n\leq k_{i}; \end{cases}\\ \notag &\text{and} \quad \lambda_{n^{o}}(a,b) = 0, \qquad \text{if} \quad \lambda_{n^{o}}(a,b)<0; \end{aligned}$$ where $k_{i}$ is the degree of node $i$. $n_s$ is the inflection point beyond which persistence is the dominant factor for the infection probability of message, and the parameters $a$ (stickiness) and $b$ (persistence) characterize how $\lambda_{n^{o}}(a,b)$ change with $n^{o}$, as illustrated in Fig. \[model\]. Since empirical data [@science; @origin2; @origin1] have shown that social reinforcement sets in such that initial exposures generally increase infection probability, the parameter $a$ should be non-negative when $n^{o}\leq n_{s}$. For $n^{o}> n_{s}$, the competition between social reinforcement and decay effects, characterized by the parameter $b$, will be taken into account for the message adoption. If the reinforcement is strong enough the individual will be more likely to adopt the message with increasing $n^{o}$ ($b>0$). Otherwise, even if many infected neighbors try to transmit the information to the focal individual $i$, the multiple exposures will lead to a decreased probability for the information adoption ($b<0$). In the present study, we set $b\in[-1,1]$ and $a\in(0, 0.5]$ so that the spreading dynamics of the message can be comprehensively investigated. For simplicity, we do not consider the diversity of individuals’ response to the message, and all individuals behave identically with the same values of parameters $a$ and $b$. ![**The adopting probability of message as a function of $n^{o}$.** The degree of stickiness and persistence are quantified as $an_{s}$ and $b$, respectively. $n_{s}$ denotes the position of inflection point.[]{data-label="model"}](model.jpeg){width="\textwidth"} We perform Monte Carlo (MC) simulations with synchronous updating of the states of all the individuals. Each MC step consists of the following three procedures: (i) All susceptible individuals decide whether or not to adopt the message with probability $\lambda_{n^{o}}(a,b)$; (ii) If an individual adopts the message, he will try to transmit what he has approved to all his nearest susceptible neighbors in the next step, and then becomes recovered immediately; (iii) Otherwise, the susceptible individuals will wait to repeat the procedure (i) in the following MC steps. The above elementary spreading processes are repeated $T^{'}_{S}=500$ steps until there are no infected individuals anymore in the population. **Theoretical analysis of the model.** For the occurrence probability of transmission event $E_{n}$, $T_{n}(a,b)= 1-e^{-\lambda_{n+1}(a,b)\tau}\notag = 1-e^{-\lambda_{n+1}(a,b)}$ with $\tau =1$ [@newman]. We consider the spatial correlations that affect the process of diffusion, but do not yet influence the critical behaviour of the message spreading [@prl]. The transition point from susceptible to infected phase is determined by [@prl; @newman] $$\begin{aligned} T_{C}=\left\langle T\right\rangle, \label{eq:threshold}\end{aligned}$$ where $\left\langle T\right\rangle$ and $T_{C}$ are, respectively, the mean transmissibility and the critical topology-dependent bond-percolation threshold. On the other hand, the mean transmissibility can be gotten as $$\begin{aligned} \left\langle T\right\rangle & = & \sum^{\left\langle k\right\rangle -1}_{n=0}q_{n}T_{n}(a,b), \label{eq:meanthreshold}\end{aligned}$$ where $q_{n}=\binom{\left\langle k\right\rangle-1}{n}p^{n}(1-p)^{\left\langle k\right\rangle-n-1}$ is the probability that the recipient has other $n$ ($n=0, 1, 2, 3$) infected neighbors except for the one chosen beforehand when considering the spatial correlations ($p$ is the probability that one nearest neighbor of the focal individual is in infected state). Combine Eqs. , , , and $T_{n}(a,b) = 1-e^{-\lambda_{n+1}(a,b)}$, the mean transmissibility for the discrete case reads as $$\begin{aligned} \label{eq:percolation2} \left\langle T\right\rangle & = \sum^{\left\langle k\right\rangle-1}_{n=0}q_{n}(1-e^{-\lambda_{n+1}(a,b)}) \notag \\ & = \sum^{\left\langle k\right\rangle -1}_{n=0}q_{n}-\sum^{n_{s}-2}_{n=0}q_{n}e^{-\lambda_{n+1}(a,b)}-\sum^{\left\langle k\right\rangle -1}_{n=n_{s}-1}q_{n}e^{-\lambda_{n+1}(a,b)} \\ & = 1-\sum^{n_{s}-2}_{n=0}q_{n}e^{-(n+1)a}-e^{-n_{s}a}\sum^{\left\langle k\right\rangle -1}_{n=n_{s}-1}\binom{\left\langle k\right\rangle -1}{n}e^{-(n-1)b}.\notag\end{aligned}$$ Then we have $$\begin{aligned} \theta_{2}e^{-2a}+\theta_{1}e^{-a}+\theta_{0}=0, \label{eq:lattper}\end{aligned}$$ where $\theta_{2}=\sum^{\left\langle k\right\rangle-1}_{n=1} \frac{(\left\langle k\right\rangle-1)!}{n!(\left\langle k\right\rangle-n-1)!} p^{n}(1-p)^{\left\langle k\right\rangle-n-1} e^{(1-n)b}$, $\theta_{1}=(1-p)^{\left\langle k\right\rangle-1}$ and $\theta_{0}=T_{C}-1=-\frac{1}{2}$. The positive root is selected as the theoretical prediction $$a(b)=\ln(\frac{2\theta_{2}}{\sqrt{\theta^{2}_{1}-4\theta_{2}\theta_{0}}-\theta_{1}}). \label{eq:lattre}$$ In SF networks, the critical point $T_{C}$ beyond which the message can reach a finite faction of the population can be obtained [@newman] as $T_{c}=\frac{\left\langle k\right\rangle}{\left\langle k^{2}\right\rangle -\left\langle k\right\rangle}$. Since RRNs and ER networks are generated by connecting randomly selected pair of nodes, $T_{c}=\frac{1}{\left\langle k\right\rangle-1}$. Due to the randomness of connections in these networks, each edge of a neighbor of one host can probably connected to any other $N-2$ individuals. Therefore, the transmission of message from an informed neighbor to the susceptible host (i.e., message flows through the edge in the system) will happen only if there is at least one infected individual in the left $N-2$ ones to transmit the message to the recipient. In other words, the probability that the message flow can reach the susceptible host through the edges connecting to the recipient is $p=\frac{1}{N-2}$ for the three networks (i. e., the RRNs, SF networks, and ER networks). Also, two or more transmission events fail to last over the long time at the critical points (see Supplementary Fig. S8, Fig. S9, and Figs. S25), so the time correlations are ruled out in the theoretical analysis for the cases of RRNs, SF networks, and ER networks. In the regular lattices, $p=\frac{1}{\left\langle k\right\rangle-1}$ since each one has exact $\langle k\rangle$ specific neighbors. Specifically, $p=\frac{1}{3}\approx0.333$ for square lattice with von Neumann neighborhood, $p=\frac{1}{5}$ for Hexagonal lattice, and $p=\frac{1}{7}\approx0.143$ for the lattice with Moore neighborhood. The bond percolation threshold $T_{C}=\frac{1}{2}$ for a square lattice, $T_{C}=0.347$ for Hexagonal lattice, and $T_{C}=0.232$ for square lattice with Moore neighborhood [@prl; @tc]. **Verification approximation of information threshold.** In analogous to $q_{n}(a,b)$, we first set $Q^{'}_{n}(a,b) (n=0,~1,~2,~3)$. According to Eq. , $Q^{'}_{n}(a,b) (n=0,~1,~2,~3)$ can be calculated numerically by counting the relative number of successful attacks (i.e, transmission events) $\alpha_{n}(a,b)$ from infected neighbors to hosts for given values of $a$ and $b$ and equating this to $\alpha_{n}(a,b)=\frac{\sum_{t=0}^{T^{'}_{S}}\omega_{n}(a,b,t)}{\sum^{\langle k\rangle -1}_{m=0}\sum^{T^{'}_{S}}_{t=0}\omega_{m}(a,b,t)}=\frac{Q^{'}_{n}(a,b)T_{i}(a,b)}{<T(a,b)>}$. In other words, $\alpha_{n}(a,b)$ represents numerically the proportion of $E_{n}$ occurring at parameter point $(a,b)$. Furthermore, $Q^{'}_{n}(a,b)$ can be normalized as $$Q_{n}(a,b)= \frac{Q^{'}_{n}(a,b)}{\sum^{\langle k\rangle -1}_{m=0}Q^{'}_{m}(a,b)}= \frac{\frac{\alpha_{n}(a,b)<T(a,b)>}{T_{n}(a,b)}} {\sum^{\langle k\rangle -1}_{m=0}\frac{\alpha_{m}(a,b)<T(a,b)>}{T_{m}(a,b)}}= \frac{\frac{\alpha_{n}(a,b)}{T_{n}(a,b)}} {\sum^{\langle k\rangle -1}_{m=0}\frac{\alpha_{m}(a,b)}{T_{m}(a,b)}}.$$ $Q_{n}(a,b)$ actually represents the probability that the recipient owns $n$ ($n=0, 1, 2, 3$) infected neighbors expect for the preselected informed one, involving both the spatial and time correlations of the message diffusion. Herein, we select different parameter regions containing the numerical critical boundaries for corresponding lattices with $n_{s}=2$ for the calculation of $Q_{n}(a,b)$. We average all non-zero $Q_{n}(a,b)$ in the selected parameter ranges for the expected indices $Q_{n}$, and substitute them into the equation $$\begin{aligned} T_{C} = \left\langle T\right\rangle=\sum^{\langle k\rangle-1}_{n=0}Q_{n}T_{n}(a,b)\end{aligned}$$ to get the verification thresholds. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the National Natural Science Foundation of China under Grants No. 11135001 and No. 11105025, and by the Fundamental Research Funds for the Central Universities under Grant No. lzujbky-2014-28. [0]{} Centola, D. $\&$ Macy, M. Complex contagions and the weakness of long ties. [*Am. J. Sociol.*]{} **113**, 702–734 (2007). Portrat, S., Barrouillet, P. $\&$ Camos, V. Time-related decay or interference-based forgetting in working memory. [*J. Exp. Psychol. Learn Mem. Cogn.*]{} **34**, 1561 (2008). Centola, D. The spread of behavior in an online social network experiment. [*Science*]{} **329**, 1194 (2010). 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Spread of epidemic disease on networks. [*Phys. Rev. E*]{} **66**, 016128 (2002). Sykes, M. F. $\&$ Essam, J. W. Exact critical percolation probabilities for site and bond problems in two dimensions. [*J. Math. Phys.*]{} **5**, 1117–1127 (2004). Message diffusion on square lattice {#square} =================================== We first present additional results to support the arguments for message diffusion on square lattice. In Fig. S\[lattice1470\], we can observe that the size of message spreading depends more on the stickiness of the message (values of $a$ with fixed $n_{s}$) than on the persistence $b$. We observe that the information can reach the vast majority of population (more than $80\%$) for $a\gtrsim 0.45$. Fig. S\[laalpha\] shows the dependence of the size of recovered population on the parameters $a$ and $b$. For $a\lesssim0.3$, $\alpha_{0}\approx 0.7$, whose value is much larger than the other three indices $\alpha_{n}$ $(n=1,2,3)$, which indicates that the information has not yet outbreak. As $a$ increases, transmission events $E_{m}$ ($m>0$) contribute a lot to the message spreading, hinting the large scale outbreak of the message. Accordingly, $\alpha_{0}$ ($\alpha_{n}$) decreases (increases) sharply, as shown in Figs. S\[laalpha\](b)–(d). In addition, it is found that $E_{0}$ and $E_{1}$ (depending more closely on $a$) constitute most of the transmission events, whereas $E_{2}$ and $E_{3}$ rarely occurs during the spreading process. To provide support for the real phenomena “three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"), in what follows we investigate the changes of the accumulative indices $\eta_{i}(a,b)$ by evaluating verification indices $\alpha_{i}$ as $\eta_{i}=\sum_{j<i}\alpha_{j}$ ($i=1,2,3,4$). Here the accumulative indices $\eta_{i}$ represents the proportions of individuals that have adopted the message when they heard it from at most $i$ informed neighbors. Like the results in Fig. S\[laalpha\], high values of $\eta_{1}$ reflect that the occurrences of $E_{0}$ and $E_{1}$ account for most of the transmission events. In addition, the results in Figs. S\[lbar2\](b)(c) show that the vast majority will accept a message as truth if it is mentioned or reported by at least two or three neighbors, which supports the mechanisms “three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). Moreover, it also indicates that the reinforcement begins to work as the information is bursting and prevailing on the square lattice, where the growth of $\eta_{i}$ keep increasing with $n_{i}$ (Fig. S\[lbar2\](b)). To make the point clear, we plot the global graphs of the four accumulative indices $\eta_{i}(a,b)$ for different values of inflection point ($n_{s}$) in Fig. S\[lasatu2\], Fig. S\[lasatu3\], and Fig. S\[lasatu4\], respectively. Similar behaviors can be detected in the three figures. The results, especially for positive persistence, also reveal the more important roles of $E_{0}$ and $E_{1}$, and simultaneously provide the evidence for the real phenomenon “three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). Message diffusion on random regular networks (RRNs) and regular lattice networks (RLs) {#regular} ====================================================================================== In this section, we present additional results to support the arguments for message diffusion on RRNs and RLs. The sizes of message diffusion are presented in Fig. S\[rhinfor\] as a function of $a$ and $b$. In the presence of social reinforcement ($b>0$), we can observe that the message can more easily invade and reach the majority of the population in larger parameter regions beyond the thresholds on RLs, by comparing with that on RRNs with the same average degree. The denser regular lattices (Moore lattice in Fig. S\[rhinfor\](b) (d)) are much more efficient in promoting information spreading [@science] when the message outbreaks. The reason is that there are more local clustering links can be used for transmission with smaller $n_{s}$ [@science; @zhou; @report] and positive persistence. Instead, the message can more easily diffuse in the RRNs in the presence of strong decay effects ($b<0$). The results summarized in Fig. S\[r6alpharound\] and Fig. S\[r8alpharound\] show that the peaks of $E_{i}(t)$ ($i=0, 1, 2, 3$) brings about the peaks of subsequent transmission event $E_{i+1}(t)$, which indicates that transmission events $E_{i}(t)$ with $i>2$ fail to last stably simultaneously, even at the critical points throughout the spreading processes. There are thus no time correlations among different events. The time correlations of different transmission events can be completely ruled out in estimating the critical behaviors of the message spreading. By comparison, the occurrences of transmission events in Hexagonal lattices and Moore lattices can last stably for a long time at critical points (see Fig. S\[halpharound2\](b) and Fig. S\[malpharound2\](b)). This suggests the existence of the time correlations among different transmission events $E_{i}(t)$. Besides, the observed huge disparities between $\alpha_{i}(a,b,t)$ and the corresponding $\beta_{i}(a,b,t)$ illustrate that the time correlations among transmission events in the both lattices are considerable. As observed in Fig. S\[hbar2\](b)(c), Fig. S\[mbar2\](b)(c), and the figures ranging from Fig. S\[hsatu2\] to Fig. S\[msatu5\], the message captures the vast majority of the population until $E_{i}(t)$ ($i>2$) happens when message outbreaks and prevails. And the spreading reaches a saturation state for the case where $\frac{n_{s}}{\left\langle k\right\rangle}>\frac{1}{2}$, which means that the transmission events $E_{i}$ ($i>\frac{1}{2}\left\langle k\right\rangle$) rarely happen in the spreading process. Therefore, the results in Fig. S\[hbar2\]–Fig. S\[msatu5\] can be regarded as the evidence of the mechanism “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"), in addition to the results for positive persistence illustrated in Fig. S\[r6satu2\], and Fig. S\[r8satu2\]. Message diffusion on ER and SF networks ======================================= In this section, we provide additional results to support the arguments for message diffusion on ER networks and SF networks. We observe in Fig. S\[baalpharound\] that transmission events $E_{i}(t)$ ($i>2$) fail to last simultaneously and stably over the long time at the critical points throughout the spreading process. Therefore, the time correlations need not to be taken into consideration in theoretical analysis. We have found the same phenomena in the ER networks and the SF with other average degrees. As illustrated in Fig. S\[ae8thre\], the theoretical estimations are sufficient to give fairly precise value of the thresholds. It is apparent that the critical behaviors of the message spreading are completely determined by stickiness ($a$) of message. Moreover, in comparison with the case of ER networks, the analytical solutions are in better agreement with the simulation in SF networks, attributing to shorter shortest paths and hubs [@path1; @bapath; @shortest]. More interestingly, both the analytical boundaries and the numerical thresholds are shifting to left with average degree $\left\langle k\right\rangle$, instead of size of the populations as stressed in [@hub1]. This demonstrates that message can easily reach and infect a larger amount of susceptible individuals through paths from those high-degree vertices whose links increase rapidly with $\left\langle k\right\rangle$, although they can be infected only once. In Fig. S\[beinfor\], we find that the persistence also boosts its reasonable impact on the spreading as $\left\langle k\right\rangle$ gets larger. It implies that hub nodes of larger size and shorter shortest paths in the networks can transmit the information more efficiently [@path1; @hub1; @shortest]. It is in accordance with the conclusion in [@degree] that higher degrees and densities are relevant factors in improving the global spread of information. It should be noted that the both ER and SF networks with small average degrees (such as $\langle k \rangle =6, 8$) are tree-like, with few short loops, indicating that the critical transmissibility $T_{C}$ can be derived from equation $T_{c} = \frac{\left\langle k\right\rangle}{\left\langle k^{2}\right\rangle -\left\langle k\right\rangle}$ [@degree; @ther1]. More specifically, $T_{c} = \frac{1}{\left\langle k\right\rangle -1}$ for ER networks. Qualitatively, the contributions of the infection events $E_{n}~(n>\left\langle k\right\rangle-2)$ to the spread are insignificant (see Fig. S\[er6satu\] and Fig. S\[ba6satu\]). Therefore we neglect the contributions of the transmission events $E_{l}(t)$ ($l>\langle k\rangle$) to message spreading, and further assume that the following relationship $T_{C} = \left\langle T\right\rangle = \sum^{\left\langle k\right\rangle-1}_{n=0}q_{n}(1-e^{-\lambda_{n+1}(a,b)})$ is always satisfied in estimating the analytical thresholds. Similar to what has been demonstrated in Sec. \[regular\], both Fig. S\[er6satu\] (ER networks) and Fig. S\[ba6satu\] (SF networks) show that the results nearby the critical points for $b>0$ can also be considered as the evidence of the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). [0]{} Centola, D. The spread of behavior in an online social network experiment. [*Science*]{} **329**, 1194–1197 (2010). Lu, L., Chen, D. B. $\&$ Zhou, T. Small world yields the most effective information spreading. [*New Journal of Physics*]{} **13**, 123005 (2011). Gonzalez-Bailon, S., Borge-Holthoefer, J., Rivero, A. $\&$ Moreno, Y. The Dynamics of Protest Recruitment through an Online Network. [*Sci. Rep.*]{} **1**, 197 (2011). Valdez, L., Macri, P. $\&$ Braunstein, L. Intermittent social distancing strategy for epidemic control. [*Phys. Rev. E*]{} **85**, 036108 (2012). Callaway, D., Newman, M., Strogatz, S. $\&$ Watts, D. Network robustness and fragility: Percolation on random graphs. [*Phys. Rev. Lett.*]{} **85**, 5468–5471 (2000). Barrat, A. Barthlemy, M. $\&$ Vespignani, A. [*Dynamical processes on complex networks*]{} (Cambridge University Press, New York, 2008). Kalisky, T., Cohen, R., Mokryn, O., Dolev, D., Shavitt, Y. $\&$ Havlin, S. Tomography of scale-free networks and shortest path trees. [*Phys. Rev. E*]{} **74**, 066108 (2006). Friedkin, N. Theoretical foundations for centrality measures. [*American journal of Sociology*]{} 1478–1504 (1991). Castellano, C. $\&$ Pastor-Satorras, R. Thresholds for epidemic spreading in networks, [*Phys. Rev. Lett.*]{} **105**, 218701 (2010). {#figures .unnumbered} ![**The densities of recovered individuals as a function of $a$ and $b$.** Each data point is obtained by averaging $100$ independent realizations. The other parameters are $n_{s}=2$ and $L=101$.[]{data-label="lattice1470"}](latticeinfor.jpeg){width="\textwidth"} ![**The four indices (a) $\alpha_{0}(a,b)$, (b) $\alpha_{1}(a,b)$, (c) $\alpha_{2}(a,b)$, and (d) $\alpha_{3}(a,b)$ as a function of $a$ and $b$.** The other parameters are token as $n_{s}=2$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. As message outbreaks ($a\gtrsim 0.32$), it is clear in (a) and (b) that the transmission events $E_{0}$ and $E_{1}$ contribute the most to the whole spreading process. []{data-label="laalpha"}](laalpha2.jpeg){width="\textwidth"} ![**The four accumulative indices $\eta_{i}(a,b)$ in square latttice.** $n_{i}$ denotes the number of informed neighbors an individual has had at most when it approves the message. Analysis is performed at (a) subcritical point $a=0.20$, $b=0.20$; (b) critical point $a=0.33$, $b=0.20$; and (c) supercritical point $a=0.45$, $b=0.20$. The other parameters are token as $n_{s}=2$ and $L=101$. Results are obtained by averaging $100$ independent realizations. No matter which case, the diffusion of the message owes much to $E_{0}(t)$ and $E_{1}(t)$. In (b) and (c), the value of $\eta_{3}$ approaches to $1$ rather than $\eta_{i}$ ($i<4$) and $\eta_{i}$ ($i<3$), and the gaps between $\eta_{1}$ and $\eta_{2}$ are obvious. The results in (b) and (c) can be considered as an evidence of the emergence of “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth") – the vast majority of the population accept the message as truth only when it is repeated more than two times in their ears.[]{data-label="lbar2"}](labar2.jpeg){width="\textwidth"} ![**The four accumulative indices $\eta_{0}$ (a), $\eta_{1}$ (b), $\eta_{2}$ (c), and $\eta_{3}$ (d) as a function of $a$ and $b$.** The other parameters are token as $n_{s}=2$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>90\%$ ($i\geq 2$) rather than $\eta_{1}$, where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth").[]{data-label="lasatu2"}](lasatu2.jpeg){width="\textwidth"} ![**The four accumulative indices $\eta_{0}$ (a), $\eta_{1}$ (b), $\eta_{2}$ (c), and $\eta_{3}$ (d) as a function of $a$ and $b$.** The other parameters are token as $n_{s}=4$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>90\%$ ($i\geq 2$) rather than $\eta_{1}$, where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth").[]{data-label="lasatu4"}](lasatu3.jpeg){width="\textwidth"} ![**The four accumulative indices $\eta_{0}$ (a), $\eta_{1}$ (b), $\eta_{2}$ (c), and $\eta_{3}$ (d) as a function of $a$ and $b$.** The other parameters are token as $n_{s}=4$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>90\%$ ($i\geq 2$) rather than $\eta_{1}$, where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth").[]{data-label="lasatu4"}](lasatu4.jpeg){width="\textwidth"} ![**The densities of recovered individuals as a function of $a$ and $b$.** The other parameters are token as $n_{s}=2$, $N=1000$ for the RRNs, and $L=101$ for the RLs. The scales of spreading on RLs ((a) (b)) are compared with that in RRNs ((a) (b)). The degrees of the networks are $<k>=6$ (a, c) and $<k>=8$ (b, d), respectively. Each data is obtained by averaging $100$ independent realizations. By comparing (a) with (c), for $b>0$, it is clear in (c) that the message can more easily outbreak and capture a larger population, in contrast to what is observed in (a), owing to the function of social reinforcement effects. However, more individuals accept the message in RRNs for $b<0$, indicating the advantage of the RRNs in facilitating the diffusion of message in the presence of strong decay effects. As expected, the same conclusion can also be reached by comparing the plots in (b) with that in (d). []{data-label="rhinfor"}](rhinfor.jpeg){width="\textwidth"} ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from prediction of percolation theory (dashed lines) on the RRN with $\langle k\rangle=8$ are presented. Three different cases are considered here: (a) the information vanishes for $a=0.10$, $b=0.20$, (b) it outbreaks for $a=0.13$, $b=0.20$; and prevails for $a=0.30$, $b=0.20$ (c).The other parameters are token as $n_{s}=2$ and $N=1000$. It can be observed that the occurrences of all the transmission events fail to last stably throughout the whole spreading process. The time correlations among the transmission events can thus be neglected in estimating the critical behavior of the message spreading.[]{data-label="r8alpharound"}](r6alpharound.jpeg){width="\textwidth"} ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from prediction of percolation theory (dashed lines) on the RRN with $\langle k\rangle=8$ are presented. Three different cases are considered here: (a) the information vanishes for $a=0.10$, $b=0.20$, (b) it outbreaks for $a=0.13$, $b=0.20$; and prevails for $a=0.30$, $b=0.20$ (c).The other parameters are token as $n_{s}=2$ and $N=1000$. It can be observed that the occurrences of all the transmission events fail to last stably throughout the whole spreading process. The time correlations among the transmission events can thus be neglected in estimating the critical behavior of the message spreading.[]{data-label="r8alpharound"}](r8alpharound.jpeg){width="\textwidth"} ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from prediction of percolation theory (dashed lines) on the Moore lattice are presented. Three different cases are considered here: (a) the information vanishes for $a=0.05$, $b=0.20$; (b) it outbreaks for $a=0.10$, $b=0.20$ and prevails for $a=0.30$, $b=0.20$ (c). The other parameters are token as $n_{s}=2$ and $L=101$. In comparison with the dynamic behaviors of corresponding RRNs shown in Fig. S\[r8alpharound\], the occurrences of transmission events in Moore lattice can last stably for over $450$ MCs at the critical point, hence the time correlation among the events cannot be neglected in estimating the critical behavior of the message spreading. The inconsistencies between $\alpha_{i}(a,b,t)$ and the corresponding $\beta_{i}(a,b,t)$ have been chosen to demonstrate the existence of the time correlations between different transmisssion events $E_{i}(t)$. The phenomenon $\alpha_{i}(t)>0$ in (b) indicates that almost all transmission events involve in the spreading process at the critical point. In addition, $\beta_{i}(t)$ gets close to corresponding $\alpha_{i}(t)$ in (b) and (c).[]{data-label="malpharound2"}](halpharound2.jpeg){width="\textwidth"} ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from prediction of percolation theory (dashed lines) on the Moore lattice are presented. Three different cases are considered here: (a) the information vanishes for $a=0.05$, $b=0.20$; (b) it outbreaks for $a=0.10$, $b=0.20$ and prevails for $a=0.30$, $b=0.20$ (c). The other parameters are token as $n_{s}=2$ and $L=101$. In comparison with the dynamic behaviors of corresponding RRNs shown in Fig. S\[r8alpharound\], the occurrences of transmission events in Moore lattice can last stably for over $450$ MCs at the critical point, hence the time correlation among the events cannot be neglected in estimating the critical behavior of the message spreading. The inconsistencies between $\alpha_{i}(a,b,t)$ and the corresponding $\beta_{i}(a,b,t)$ have been chosen to demonstrate the existence of the time correlations between different transmisssion events $E_{i}(t)$. The phenomenon $\alpha_{i}(t)>0$ in (b) indicates that almost all transmission events involve in the spreading process at the critical point. In addition, $\beta_{i}(t)$ gets close to corresponding $\alpha_{i}(t)$ in (b) and (c).[]{data-label="malpharound2"}](malpharound2.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{i}(a,b)$ in Moore lattice.** $n_{i}$ denotes the number of informed neighbors an individual has had at most when it approves the message. The parameters are chosen at three selected parameter points: (a) subcritical point $a=0.10$, $b=0.20$; (b) critical point $a=0.18$, $b=0.20$ and (c) supercritical point $a=0.40$, $b=0.20$. The other parameters are token as $n_{s}=2$ and $L=101$. Results are obtained by averaging $100$ independent realizations. In (b), the gaps between $\eta_{i}$ and $\eta_{i+1}$ ($i<4$) are apparent, which indicates that almost all transmission events involve in the spreading process. In (b) and (c), the value of $\eta_{4}$ or $\eta_{3}$ approach to $1$, rather than $\eta_{i}$ ($i<6$) and $\eta_{i}$ ($i<4$). That can be considered as an evidence of the emergence of “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth") – the vast majority of the population accept the message as truth only when it is repeated more than two times in their ears.[]{data-label="mbar2"}](hbar2.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{i}(a,b)$ in Moore lattice.** $n_{i}$ denotes the number of informed neighbors an individual has had at most when it approves the message. The parameters are chosen at three selected parameter points: (a) subcritical point $a=0.10$, $b=0.20$; (b) critical point $a=0.18$, $b=0.20$ and (c) supercritical point $a=0.40$, $b=0.20$. The other parameters are token as $n_{s}=2$ and $L=101$. Results are obtained by averaging $100$ independent realizations. In (b), the gaps between $\eta_{i}$ and $\eta_{i+1}$ ($i<4$) are apparent, which indicates that almost all transmission events involve in the spreading process. In (b) and (c), the value of $\eta_{4}$ or $\eta_{3}$ approach to $1$, rather than $\eta_{i}$ ($i<6$) and $\eta_{i}$ ($i<4$). That can be considered as an evidence of the emergence of “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth") – the vast majority of the population accept the message as truth only when it is repeated more than two times in their ears.[]{data-label="mbar2"}](mbar2.jpeg){width="\textwidth"} ![ **The six accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), and $\eta_{6}$ (f) as a function of $a$ and $b$ for Hexagonal lattice.** The other parameters are token as $n_{s}=3$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions with positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $80\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="hsatu3"}](hsatu2.jpeg){width="\textwidth"} ![ **The six accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), and $\eta_{6}$ (f) as a function of $a$ and $b$ for Hexagonal lattice.** The other parameters are token as $n_{s}=3$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions with positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $80\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="hsatu3"}](hsatu3.jpeg){width="\textwidth"} ![**The six accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), and $\eta_{6}$ (f) as a function of $a$ and $b$ for Hexagonal lattice.** The other parameters are token as $n_{s}=5$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $80\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="hsatu5"}](hsatu4.jpeg){width="\textwidth"} ![**The six accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), and $\eta_{6}$ (f) as a function of $a$ and $b$ for Hexagonal lattice.** The other parameters are token as $n_{s}=5$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $80\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="hsatu5"}](hsatu5.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as a function of $a$ and $b$ for Moore lattice.** The other parameters are token as $n_{s}=3$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="msatu3"}](msatu2.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as a function of $a$ and $b$ for Moore lattice.** The other parameters are token as $n_{s}=3$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="msatu3"}](msatu3.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as a function of $a$ and $b$ for Moore lattice.** The other parameters are token as $n_{s}=5$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="msatu5"}](msatu4.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as a function of $a$ and $b$ for Moore lattice.** The other parameters are token as $n_{s}=5$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="msatu5"}](msatu5.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as function of $a$ and $b$ for RRs with $\langle k \rangle=8$.** The other parameters are token as $n_{s}=5$ and $N=10000$. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="r8satu2"}](r6satu2.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as function of $a$ and $b$ for RRs with $\langle k \rangle=8$.** The other parameters are token as $n_{s}=5$ and $N=10000$. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="r8satu2"}](r8satu2.jpeg){width="\textwidth"} \[be\] ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from predications of percolation theory (dashed lines) are presented for SF with $\left\langle k\right\rangle=8$. Three difference cases are considered here: (a) the information vanishes for $a=0.10$, $b=0.20$, (b) it outbreaks for $a=0.19$, $b=0.20$ and prevails for $a=0.30$, $b=0.20$ (c). The other parameters are token as $n_{s}=5$ and $N=10000$. It can be observed that the occurrences of all the transmission events fail to last simultaneously and stably in the whole spreading process, departing from what exhibited in Fig. S\[malpharound2\] for moore lattice. The time correlations among the transmission events can thus be neglected in estimating the critical behaviors of the message spreading. []{data-label="baalpharound"}](ba6alpharound2.jpeg){width="\textwidth"} ![**Locus of thresholds of message diffusion on SF networks and ER networks**. Respectively, analytical solutions (solid lines) for four SF networks (top panel) and four ER networks (bottom panel) with different average degrees are plotted to compare with the corresponding exact numerical data (markers). The other parameters are token as $n_{s}=2$ and $N=10000$. Each numerical data point is obtained by averaging $1000$ independent realizations. And (a)(e) $\left\langle k\right\rangle=6$, (b)(f) $\left\langle k\right\rangle=8$, (c)(h) $\left\langle k\right\rangle=10$ and (d)(i) $\left\langle k\right\rangle=12$. Both the simulations and the analytical predications show that the critical behaviors of the spreading are dominated by the stickiness of the message ($b$).[]{data-label="ae8thre"}](ae8thre.jpeg){width="\textwidth"} ![**The densities of recovered individuals as function of $a$ and $b$,** on SF networks (top panel) and Erdos-Renyi networks (bottom panel) with different average degrees. The other parameters are token as $n_{s}=2$ and $N=10000$. Each numerical data point is obtained by averaging $100$ independent realizations. Precisely, (a)(e) $\left\langle k\right\rangle =6$, (b)(f) $\left\langle k\right\rangle =8$, (c)(h) $\left\langle k\right\rangle=10$ and (d)(i) $\left\langle k\right\rangle =12$. Although the critical behaviors of the message spreading on SF and ER are dominated by the stickiness, the persistence of the message (i.e., $b$) has a reasonable impact on the sizes of message spreading, especially at the parameter regions nearby $b=0$. []{data-label="beinfor"}](beinfor.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g) $\eta_{8}$ (h) as function of $a$ and $b$ for ER networks with $\langle k \rangle=6$.** The other parameters are token as $n_{s}=2$ and $N=10000$. Each numerical data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results for $b>0$ nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). Similar results are obtained for SF networks with the same average degree. []{data-label="ba6satu"}](ba8satu.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g) $\eta_{8}$ (h) as function of $a$ and $b$ for ER networks with $\langle k \rangle=6$.** The other parameters are token as $n_{s}=2$ and $N=10000$. Each numerical data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results for $b>0$ nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). Similar results are obtained for SF networks with the same average degree. []{data-label="ba6satu"}](er8satu.jpeg){width="\textwidth"}

ArXiv

--- abstract: 'A local resolution of the Problem of Time has recently been given, alongside reformulation as a local theory of Background Independence. The classical part of this can be viewed as requiring just Lie’s Mathematics, albeit entrenched in subsequent Topology and Differential Geometry developments and extended to the setting of contemporary Physics’ state spaces. We now generalize this approach by mild recategorization to one based on Nijenhuis’ generalization of Lie’s Mathematics, as follows. 1) Relationalism is encoded using the Nijenhuis–Lie derivative. 2) Closure is assessed using the Schouten–Nijenhuis bracket, and a ‘Schouten–Nijenhuis Algorithm’ analogue of the Dirac and Lie Algorithms. This produces a class of Gerstenhaber algebraic structures of generators or of constraints. 3) Observables are defined by a Schouten–Nijenhuis brackets relation, reformulating the constrained canonical case as explicit PDEs to be solved using the Flow Method, and forming their own Gerstenhaber algebras of observables. Lattices of Schouten–Nijenhuis–Gerstenhaber constraint or generator algebraic substructures furthermore induce dual lattices of Gerstenhaber observables subalgebras. 4) Deformation of Gerstenhaber algebraic structures of generators or constraints encountering Rigidity gives a means of Constructing more structure from less. 5) Reallocation of Intermediary-Object Invariance gives the general Schouten–Nijenhuis–Gerstenhaber algebraic structure’s analogue of posing Refoliation Invariance for GR. We finally point to general Gerstenhaber bracket and Vinogradov bracket generalizations, with the former likely to play a significant role in Backgound-Independent Deformation Quantization and Quantum Operator Algebras.' --- [**Edward Anderson**]{}$^1$ $^1$ dr.e.anderson.maths.physics \*at\* protonmail.com Introduction {#Introduction} ============ It has been recently demonstrated that [@ABook; @ALRoPoT; @Higher-Lie; @XIV] Lie’s Mathematics suffices to construct A Local Resolution of the Problem of Time [@Battelle-DeWitt67; @Dirac; @K92-I93; @APoT-2; @ABook], which in turn can be reformulated as [@ABook; @A-CBI] A Local Theory of Background Independence. This locally-smooth approach is moreover sufficiently well-defined to extend to various other (at least locally) differential-geometric structures. The purpose of the current Article is to outline one of the more interesting cases: the ‘Nijenhuis Mathematics’ [@S40-53-N55; @FN56; @NR66] counterpart; see [@Nambu] for use of ‘Nambu Mathematics’ instead, while [@ABook; @XIV] already made mention of the simpler graded, alias supersymmetric, extension. Nijenhuis Mathematics’ distinctive primary structures are outlined in Sec 2, with further generalizations outlined in the concluding Sec 4: Vinogradov brackets [@V90; @KS] – a unification of Nijenhuis brackets – and general Gerstenhaber algebras. The latter motivates our current study, due to its Deformation Quantization [@L78; @Landsman; @Kontsevich; @Gengoux] and quantum operator algebra applications. The main part of the current Article is Sec 3, where the Abstract’s structures 1) to 5) – Nijenhuis parallels of the Lie structures used in A Local Resolution of the Problem of Time and its reformulation as A Local Theory of Background Independence – are outlined. These include in particular a ‘Nijenhuis Algorithm’ analogue of the Dirac Algorithm, and a more general theory of observables than those based on, sequentially, Dirac’s Mathematics [@DiracObs; @Dirac; @HTBook; @ABook] or Lie’s [@AObs; @XIV]. Nijenhuis Mathematics {#NM} ===================== The [*Schouten–Nijenhuis (SN) bracket*]{} [@S40-53-N55; @Gengoux][^1] on degree-r and thus shifted degree $\bar{r} := r - 1$ multivector fields $\FrX^r$ is given by $$\mbox{\bf [} \m \mbox{\bf ,} \, \m \mbox{\bf ]}_{\sS\sN} : \FrX^{\bar{p}} \times \FrX^{\bar{q}} \longrightarrow \FrX^{\bar{p} + \bar{q}} \label{NS-Bracket}$$ $$\mbox{\bf [} \, \biP \mbox{\bf ,} \, \biQ \, \mbox{\bf ]}_{\sS\sN} (F_1, \, ... \, , \, F_{\bar{p} + \bar{q} + 1} ) \:= \m \m \m \m \m \m \m \m \m \m \m \m \sum_{ \sigma \in S_{q, \bar{p}} } \, \mbox{sign}(\sigma) \biP ( \biQ ( F_{\sigma(1)}, \, ... \, F_{\sigma(q)}) , \, F_{\sigma(q + 1)} , \, ... \, F_{\sigma(q + \bar{p})} ) \m - \m$$ -(-1)\^[|[p]{}|[q]{}]{} \_[ S\_[p, |[q]{}]{} ]{} () ( ( F\_[(1)]{}, ... F\_[(p)]{}) , F\_[(p + 1)]{} , ... F\_[(p + |[q]{})]{} ) , where $\sigma$ denotes a shuffle and $S$ a permutation group formed by such. This obeys \_ -(-1)\^[|[p]{}|[q]{}]{} \_ , (-1)\^[|[p]{}|[r]{}]{} \_ \_ + 0 . The [*Nijenhuis–Lie derivative*]{} of $\biP \in \FrX^p$ with respect to $\biV \in \FrX^1$ is £\^\_ = \_ . \[NL-Deriv\] The algebras formed by equipping (graded) vector spaces $\bFrV$ with SN brackets are a subcase of [*Gerstenhaber algebras*]{} (defined in Sec 4), so we refer to them as [*SNG-algebras*]{}. Let us finally also extend consideration from algebras to algebroids [@CM], using the phrase ‘algebraic structures’ as a portmanteau of the two. Each of classical deformation theory [@CM; @Higher-Lie], kinematical quantization [@Landsman], and GR producing a constraint algebroid – the Dirac algebroid [@Dirac] – even before either of the previous are involved, justify this more generalized scope. [*SNG-algebroids*]{} and [*SNG-algebraic structures*]{} are thus in play. Nijenhuis Local Background Independence {#NLBI} ======================================= 1\) We here employ [*Nijenhuis–Lie derivatives*]{} (\[NL-Deriv\]) to encode Relationalism. A\) In the canonical case, we work with changes of configuration in place of velocities $\dot{\biQ} = \d \biQ/d t$ so as to stay free from time variables for the reasons given in Article I of [@ALRoPoT]. B\) We correct by Nijenhuis–Lie derivative along physically irrelevant group $\lFrg$’s changes $\d \ba$, - £\^\_ . C) We know what form these corrections take by solving the [*generalized Killing–Nijenhuis equation*]{} £\^ = 0 for geometrical level of structure $\bsigma$ to obtain the corresponding physically irrelevant automorphism group in question, $\lFrg$. D\) We complete this with a move using all of $\lFrg$ to obtain $\lFrg$-invariant objects. \[C.f. [*group averaging*]{} or Article II of [@ALRoPoT] for a detailed review of all of B) to D)\]. In the spacetime counterpart,[^2] we are free to use plain auxiliary corrections $\biA$ in place of change corrections $\d \bia$ on spacetime objects $\biS$: - £\^\_ , with steps B) and C) then applying unaltered. 2\) Closure is assessed using a) the SN bracket (\[NS-Bracket\]). b\) An ‘SN Algorithm’ analogue of the Dirac [@Dirac] and Lie [@Lie; @XIV] Algorithms. This permits six types of equation to arise from an initial set of generators $\sbcG$ or constraints $\sbcC$, as follows. i\) [*Inconsistencies*]{}: equations reducing to $0 = 1$ as envisaged by Dirac [@Dirac]. ii\) [*Identities*]{}: equations reducing to $0 = 0$. iii\) [*New secondary generators*]{} $\sbcG^{\prime}$ or secondary constraints $\sbcC^{\prime}$. iv\) ‘[*SN specifier equations*]{}’ are also possible if there is an appending process. I.e. a generalization of Dirac’s appending of constraints to Hamiltonians $H$ using Lagrange multipliers $\bLambda$, i.e.  H H + , by which equations relating these a priori for $\bLambda$ can also arise from one’s algorithm. v\) [*Rebracketing*]{} using [*SN–Dirac brackets*]{} in the event of encountering [*SN-secondary objects*]{}, i.e. ones that do not close under SN brackets. vi\) [*Topological obsruction terms*]{}: Nijenhuis Mathematics’ analogue of anomalies. if the SN Algorithm (using [@XIV]’s terminology) 0) [*hits an inconsistency*]{}, I) [*cascades to inconsistency*]{}, II) [*cascades to triviality*]{}, or III) [*arrives at an iteration producing no new objects*]{} while retaining some degrees of freedom. Successful candidates – terminating at iii) – produce an ‘SNG’ class of Gerstenhaber algebraic structures of generators \^ , or of SN-first-class constraints, $\bscF$ (those that do close under SN brackets) in the canonical case. 3\) We next consider [*observables*]{} $\sbiO$ [@AObs; @DiracObs; @K92-I93; @ABook] defined in the present context by [*zero-commutant SN brackets*]{} with the generators, $\sbcG$ (including with the $\bscF$ constraints as a subcase) , \_ 0 . \[NS-Commutants\] $\peq$ is here the portmanteau of Dirac’s notion of weak equality $\approx$ [@Dirac] – from equality up to a linear combination of constraints to [@XIV] equality up to a linear combination of generators – and of strong vanishing: the standard notion of equality, =. In the constrained canonical subcontext, these are zero-commutants of the SN-first-class constraints. In this context, these equations can moreover be recast as explicit PDEs to be solved using the Flow Method [@John; @Lee2]. This particular application gives an [*SN integral theory of invariants*]{}. I.e. the SN version of [*Lie’s Integral Approach to Invariants*]{} [@Lie; @XIV], albeit in both cases elevated to restricting functional dependence on a function space over the phase space geometry in question. The spacetime version is, rather, over the space of spacetimes. Such functions are, in each case, observables. These functions moreover constitute SNG-algebras: [*observables SNG-algebras*]{} \^ , whether canonical or spacetime. Each theory’s lattice of SNG generators or constraints subalgebraic structures additionally induces a dual lattice of SNG observables subalgebras. (If the reader is unsure what this means, they should consult the third Article of [@ALRoPoT] for the usual Lie subcase.) 4\) Deformation of SNG-algebraic structures encountering Rigidity [@G64; @NR66; @CM] gives a means of Constructing more structure from less. This generalizes, firstly, Spacetime Construction from space’s ‘passing families of theories through the Dirac Algorithm’ approach. Secondly, obtaining more structure from less for each of space and spacetime separately. (See e.g. Article IX in [@ALRoPoT] for Dirac and Lie Mathematics examples of each of these). On the one hand, we can [*pose*]{} rigidity for SNG-algebraic structures – that under deformations of generators \_ + . \[def\] for parameter $\alpha$ and functions $\phi$, the cohomology group condition \^2(\^, \^) = 0 diagnoses rigidity. On the other hand, we are currently rather lacking in theorems for this SNG-algebra case. The idea is moreover to take Rigidity to be [@Higher-Lie] a [*selection principle*]{} in the Comparative Theory of Background Independence. 5\) Reallocation of Intermediary-Object (RIO) Invariance gives the general SNG-algebraic structures’ commuting-pentagon analogue of posing Refoliation Invariance for GR. RIO refers to whether, in going from an initial object to a final object, proceeding via intermediary object 1 or intermediary object 2 causes one to be out by at most just an automorphism of the final object, O\_[21]{}\^ - O\_[12]{}\^ = Aut(O\^) . Refoliation Invariance is then the case in which our objects are spatial slices within a given spacetime. Consult Article III in [@ALRoPoT] for further details about Refoliation Invariance, or [@Higher-Lie] and Article XII in [@ALRoPoT] for RIO Invariance more generally. This is again a [*selection principle*]{} whose outcome remains unexplored in the SN case. 4\) and 5) are thus identifications of new research directions, whereas 1), 2) and 3) constitute new confirmed structures and results. Conclusion {#Conclusion} ========== We have considered Nijenhuis Mathematics, with concrete focus on Schouten–Nijenhuis brackets and subsequent algebraic structures and algorithms. This is a useful generalization and robustness check – a natural next port of call in developing the Comparative Theory of Background Independence – and also a stepping stone to further cases of interest that follow from introduction of each of the following two further structural elements. A\) The [*Vinogradov bracket*]{} [@V90; @KS] $\mbox{\bf [} \m \mbox{\bf ,} \, \m \mbox{\bf ]}_{\sV}$ unifies the Schouten–Nijenhuis and Frölicher–Nijenhuis brackets. B\) The general [*Gerstenhaber bracket*]{} [@G63; @Gengoux], is a bilinear product taking one between spaces of $k$-linear maps $\mH\mC^k(V)$ of a graded vector space $\bFrV$: \_ : \^[|[p]{}]{} \^[|[q]{}]{} \^[|[p]{} + |[q]{}]{} . Like the SN bracket, this is of graded Lie bracket type, thus obeying graded-antisymmetry and graded-Jacobi identities similar to (2, 3). Such $k$-linear maps are moreover the basic objects entering [*Hochschild cohomology*]{} [@Gengoux; @L98], which then plays a major role in Deformation Quantization [@L78; @Gengoux; @Kontsevich] and the theory of quantum-mechanically relevant operator algebras [@Landsman]. A\) and B) are known to support, firstly, [*Vinogradov algebraic structures*]{} and [*Gerstenhaber algebraic structures*]{}. Secondly – as new results – a [*Vinogradov Algorithm*]{} and a [*Gerstenhaber Algorithm*]{}: analogues of the Dirac and Lie Algorithms. Thirdly – as new notions (as far as the author is aware) – [*Vinogradov*]{} and [*Gerstenhaber notions of observables*]{}. I.e. the zero commutants with the corresponding type of algebraic structure’s generators under the corresponding defining type of bracket. Fourthly, one can moreover at least pose [*Vinogradov*]{} and [*Gerstenhaber deformations*]{} (not to be confused with Gerstenhaber having already worked out the deformations of simpler algebras). And then ask, as a selection principle, which subcases of each exhibit Rigidity. Finally, one can likewise pose [*RIO Invariance in*]{} both the [*Vinogradov*]{} and [*general Gerstenhaber contexts*]{}. I thank various friends for support and proofreading. [99]{} S. Lie and F. Engel, [*Theory of Transformation Groups*]{} Vols 1 to 3 (Teubner, Leipzig 1888-1893); for an English translation with modern commentary of Volume I, see J. Merker (Springer, Berlin 2015), arXiv:1003.3202. J.A. Schouten, “Über Differentialkonkomitanten zweier kontravarianten Grössen", Indag. Math. 2: 449-452; “On the Differential Operators of the First Order in Tensor Calculus", in Convegno Int. Geom. Diff. Italia. p1 (1953); A. Nijenhuis, “Jacobi-Type Identities for Bilinear Differential Concomitants of Certain Tensor Fields I". Indagationes Math. [**17**]{} 390 (1955). P.A.M. Dirac, “Forms of Relativistic Dynamics", Rev. Mod. Phys. [**21**]{} 392 (1949). K. Yano, “Theory of Lie Derivatives and its Applications (North-Holland, Amsterdam 1955). A. Frölicher and A. Nijenhuis, “Theory of Vector Valued Differential Forms. Part I.", Indagationes Math. [**18**]{} 338 (1956). P.A.M. Dirac, [*Lectures on Quantum Mechanics*]{} (Yeshiva University, New York 1964). M. Gerstenhaber, “The Cohomology Structure of an Associative Ring". Ann. Math. [**78**]{} 267 (1963). M. Gerstenhaber, “On the Deformation of Rings and Algebras", Ann. Math. [**79**]{} 59 (1964). A. Nijenhuis and R. Richardson, “Cohomology and Deformations in Graded Lie Algebras" Bull. Amer. Math. [**72**]{} 1 (1966); “Deformation of Lie Algebra Structures", J. Math. Mech [**67**]{} 89 (1967). J.A. Wheeler, in [*Battelle Rencontres: 1967 Lectures in Mathematics and Physics*]{} ed. C. DeWitt and J.A. Wheeler (Benjamin, New York 1968); B.S. DeWitt, “Quantum Theory of Gravity. I. The Canonical Theory." Phys. Rev. [**160**]{} 1113 (1967). F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, [*Deformation Theory and Quantization. I. Deformations of Symplectic Structures*]{} Ann. Phys. [**111**]{} 61 (1978). F. John, [*Partial Differential Equations*]{} (Springer, New York 1982). A.M. Vinogradov, “Unification of Schouten–Nijenhuis and Frölicher–Nijenhuis Brackets, Cohomology and Super Differential Operators". Sov. Math. Zametki. [**47**]{} (1990). M. Henneaux and C. Teitelboim, [*Quantization of Gauge Systems*]{} (Princeton University Press, Princeton 1992). K.V. Kuchař, “Time and Interpretations of Quantum Gravity", in [*Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics*]{} ed. G. Kunstatter, D. Vincent and J. Williams (World Scientific, Singapore 1992); C.J. Isham, “Canonical Quantum Gravity and the Problem of Time", in [*Integrable Systems, Quantum Groups and Quantum Field Theories*]{} ed. L.A. Ibort and M.A. Rodríguez (Kluwer, Dordrecht 1993), gr-qc/9210011. I. Kolář, P.W. Michor and J. Slovák, [*Natural Operations in Differential Geometry*]{} (Springer–Verlag, New York 1993). M. Kontsevich, “Deformation Quantization of Poisson Manifolds, I.", Lett. Math. Phys. [**66**]{} 157 (2003), q-alg/9709040. See e.g. Chapter 1 of J.-L. Loday, [*Cyclic Cohomology*]{} (Springer-Verlag, Berlin 1998). N.P. Landsman, [*Mathematical Topics between Classical and Quantum Mechanics*]{} (Springer–Verlag, New York 1998). Y. Kosmann-Schwarzbach, “Derived Brackets", Lett. Math. Phys. [**69**]{} 61 (2004) arXiv:math/0312524. M. Crainic and I. Moerdijk, “Deformations of Lie Brackets: Cohomological Aspects", J. European Math. Soc. [**10**]{} 4 (2008), arXiv:math/0403434. E. Anderson, “The Problem of Time", in [*Classical and Quantum Gravity: Theory, Analysis and Applications*]{} ed. V.R. Frignanni (Nova, New York 2011), arXiv:1009.2157. “Problem of Time", Annalen der Physik, [**524**]{} 757 (2012), arXiv:1206.2403. J.M. Lee, [*Introduction to Smooth Manifolds*]{} 2nd Ed. (Springer, New York 2013). E. Anderson, “Beables/Observables in Classical and Quantum Gravity", SIGMA [**10**]{} 092 (2014), arXiv:1312.6073. C. Laurent-Gengoux, A. Pichereau and P. Vanhaecke, [*Poisson Structures*]{} (Springer-Verlag, Berlin 2013). E. Anderson, [*Problem of Time. Quantum Mechanics versus General Relativity*]{}, (Springer International 2017) Found. Phys. [**190**]{}; free access to its extensive Appendices is at https://link.springer.com/content/pdf/bbm E. Anderson, “Shape Theories", arXiv:1811.06516; arXiv:1811.06528; arXiv:1812.08771. E. Anderson, “A Local Resolution of the Problem of Time", arXiv:1905.06200; arXiv:1905.06206; arXiv:1905.06212; arXiv:1905.06294; arXiv:1906.03630; arXiv:1906.03635; arXiv:1906.03641; arXiv:1906.03642; forthcoming. E. Anderson, “Problem of Time and Background Independence: Classical Version’s Higher Lie Theory", arXiv:1907.00912. E. Anderson, “A Local Resolution of the Problem of Time. XIV. Grounding on Lie’s Mathematics", arXiv:1907.13595. E. Anderson, “Nambu variant of Local Resolution of Problem of Time and Background Independence", arXiv:1908.\*\*\*\*\*. [^1]: Also note some related types of bracket are in use: Frölicher–Nijenhuis bracket[@FN56; @Michor] and Nijenhuis–Richardson bracket[@NR66], each on vector-valued differential forms rather than on multivector fields. [^2]: We consider both canonical and spacetime settings for the structures discussed in the current Article; see e.g. [@ABook; @ALRoPoT] for discussions of the merit of each of these positions, as well as for inter-relations between these two positions.

ArXiv

--- abstract: 'In a recent paper, a new parametrization for the dark matter (DM) speed distribution $f(v)$ was proposed for use in the analysis of data from direct detection experiments. This parametrization involves expressing the *logarithm* of the speed distribution as a polynomial in the speed $v$. We present here a more detailed analysis of the properties of this parametrization. We show that the method leads to statistically unbiased mass reconstructions and exact coverage of credible intervals. The method performs well over a wide range of DM masses, even when finite energy resolution and backgrounds are taken into account. We also show how to select the appropriate number of basis functions for the parametrization. Finally, we look at how the speed distribution itself can be reconstructed, and how the method can be used to determine if the data are consistent with some test distribution. In summary, we show that this parametrization performs consistently well over a wide range of input parameters and over large numbers of statistical ensembles and can therefore reliably be used to reconstruct both the DM mass and speed distribution from direct detection data.' author: - 'Bradley J. Kavanagh' bibliography: - 'Model\_Indep.bib' title: 'Parametrizing the local dark matter speed distribution: a detailed analysis' --- Introduction ============ The dark matter (DM) paradigm has enjoyed much success in explaining a wide range of astronomical observations (for a review, see e.g. Ref. [@Bertone:2005]). As yet no conclusive evidence has been provided for the identity of particle DM, though there are a variety of candidates, including the supersymmetric neutralino [@Jungman:1996], sterile neutrinos [@Dodelson:1994], axions [@Duffy:2009] and the lightest Kaluza-Klein particle [@Kolb:1984]. Here, we focus on the search for particles which belong to the generic class of Weakly Interacting Massive Particles (WIMPs). Direct detection experiments [@Goodman:1985; @Drukier:1986] aim to measure the energies of nuclear recoils induced by WIMP DM in the Galactic halo. Under standard assumptions about the DM halo, this data can be used to extract the WIMP mass and interaction cross section, allowing us to check for consistency with other search channels (such as indirect detection [@Lavalle:2012] and collider experiments [@Battaglia:2010]) and to probe underlying models of DM. Direct detection experiments are traditionally analyzed within the framework of the Standard Halo Model (SHM), in which WIMPs are assumed to have a Maxwell-Boltzmann speed distribution in the Galactic frame. The impact of uncertainties in the WIMP speed distribution has been much studied (see e.g. Refs. [@Green:2010; @Peter:2011; @Fairbairn:2012]), leading to the conclusion that such uncertainties may introduce a bias into any reconstruction of the WIMP mass from direct detection data. As yet, the speed distribution is unknown, while a number of proposals have been put forward for its form, including analytic parametrizations (e.g. Ref. [@Lisanti:2010]) and distributions reconstructed from the potential of the Milky Way [@Bhattacharjee:2012] or from N-body simulations [@Vogelsberger:2009; @Kuhlen:2010; @Kuhlen:2012; @Mao:2012]. Recent results from N-body simulations which attempt to include the effects of baryons on structure formation also report the possible presence of a dark disk in the Milky Way [@Read:2009; @Read:2010; @Kuhlen:2013]. With such a wide range of possibilities, we should take an agnostic approach to the speed distribution, not only to avoid introducing bias into the analysis of data, but also with the hope of measuring the speed distribution and thereby probing the formation history of the Milky Way. Several methods of evading these uncertainties have been proposed. These include simultaneously fitting the parameters of the SHM and dark matter properties [@Strigari:2009; @Peter:2009], fitting to empirical forms of the speed distribution (e.g. Ref. [@Pato:2011]) and fitting to a self-consistent distribution function [@Pato:2013]. However, these methods typically require that the speed distribution can be well fitted by a particular functional form. More model-independent methods, such as fitting the moments of the speed distribution [@Drees:2007; @Drees:2008] or using a step-function speed distribution [@Peter:2011], have also been presented. However, these methods can still introduce a bias into the measurement of the WIMP mass and perform less well with the inclusion of realistic experimental energy thresholds. In a recent paper [@Kavanagh:2013a] (hereafter referred to as Paper 1), a new parametrization of the speed distribution was presented, which allowed the WIMP mass to be extracted from hypothetical direct detection data without prior knowledge of the speed distribution itself. Paper 1 demonstrated this for a WIMP of mass 50 GeV, using several underlying distribution functions. In the present paper, we extend this analysis to a wider range of masses. We also aim to demonstrate the statistical properties of the method and show how realistic experimental parameters affect its performance. Finally, we will also elaborate on some of the technical details of the method and assess its ability to reconstruct the underlying WIMP speed distribution. Section \[sec:DDRate\] of this paper explains the direct detection event rate formalism and presents the parametrization of the speed distribution introduced in Paper 1. In Sec. \[sec:ParameterRecon\], the methodology for testing the parametrization is outlined. In Section \[sec:Parametrization\], we consider the choice and number of basis functions for the method. We then study the performance of the method as a function of input WIMP mass (Sec. \[sec:mass\]) and when Poisson fluctations in the data are taken into account (Sec. \[sec:stats\]). In Sec. \[sec:Recon\], we demonstrate how the speed distribution can be extracted from this parametrization and examine whether or not different distribution functions can be distinguished. Finally, we summarize in Sec. \[sec:Conclusions\] the main results of this paper. Direct detection event rate {#sec:DDRate} =========================== Dark matter direct detection experiments aim to measure the energies $E$ of nuclear recoils induced by interactions with WIMPs in the Galactic halo. Calculation of the event rate at such detectors has been much studied (e.g. Refs. [@Goodman:1985; @Drukier:1986; @Lewin:1996; @Jungman:1996]). For a target nucleus with nucleon number $A$, interacting with a WIMP of mass $m_\chi$, the event rate per unit detector mass is given by: $$\label{eq:Rate} \frac{\textrm{d}R}{\textrm{d}E} = \frac{\rho_0 \sigma_p}{2 m_\chi \mu_{\chi p}^2} A^2 F^2(E) \eta(v_\textrm{min})\,,$$ where $\rho_0$ is the local dark matter mass density, $\sigma_p$ is the WIMP-proton spin-independent cross section and the reduced mass is defined as $\mu_{A B} = m_A m_B/(m_A + m_B)$. The Helm form factor $F^2(E)$ [@Helm:1956] describes the loss of coherence of spin-independent scattering due to the finite size of the nucleus. A wide range of possible interactions have been considered in the literature, including inelastic [@Smith:2001], isospin-violating [@Kurylov:2003] and more general non-relativistic interactions [@Fan:2010; @Fitzpatrick:2012; @Fitzpatrick:2013]. We focus here on the impact of the WIMP speed distribution on the direct detection event rate. We therefore restrict ourselves to considering only spin-independent scattering, which is expected to dominate over the spin-dependent contribution for heavy nuclei, due to the $A^2$ enhancement in the rate. Information about the WIMP velocity distribution $f(\textbf{v})$ is encoded in the function $\eta$, sometimes referred to as the mean inverse speed, $$\label{eq:eta} \eta(v_\textrm{min}) = \int_{v > v_\textrm{min}} \frac{f(\textbf{v})}{v} \, \textrm{d}^3\textbf{v}\,,$$ where $\textbf{v}$ is the WIMP velocity in the reference frame of the detector. The integration is performed only over those WIMPs with sufficient speed to induce a nuclear recoil of energy $E$. The minimum required speed for a target nucleus of mass $m_N$ is $$\label{eq:v_min} v_\textrm{min}(E) = \sqrt{\frac{m_N E}{2\mu_{\chi N}^2}}\,.$$ We distinguish between the directionally averaged velocity distribution $$f(v) = \oint f(\textbf{v}) \, \textrm{d}\Omega_{\textbf{v}}\,,$$ and the 1-dimensional speed distribution $$f_1(v) = \oint f(\textbf{v}) v^2 \textrm{d}\Omega_{\textbf{v}}\,.$$ The distribution function should in principle be time-dependent, due to the motion of the Earth around the Sun. However, this is expected to be a percent-level effect (for a review, see e.g. Ref. [@Freese:2013]) and we therefore assume that $f_1(v)$ is time independent in the present work. We consider several benchmark speed distributions in this work, including the SHM and the SHM with the addition of a moderate dark disk which accounts for 23% of the total WIMP density [@Kuhlen:2013]. We model the speed distributions as combinations of Gaussian functions in the Earth frame $$\label{eq:gaussian} g(\textbf{v}) = N \exp\left(-\frac{(\textbf{v} - \textbf{v}_\textrm{lag})^2}{2\sigma_v^2}\right) \Theta(v_\textrm{esc} - |\textbf{v} - \textbf{v}_\textrm{lag}|)\,,$$ where $\textbf{v}_\textrm{lag}$ specifies the peak velocity of the distribution in the Earth frame and $\sigma_v$ the velocity dispersion. We truncate the distribution above the escape speed $v_\textrm{esc}$ in the Galactic frame and the factor $N$ is required to satisfy the normalization condition (Eq. \[eq:normalization\]). We use the value $v_\textrm{esc} = 544 {\,\textrm{km s}^{-1}}$, which lies within the 90% confidence limits obtained from the RAVE survey [@RAVE:2007; @RAVE:2013]. In addition, we also use the speed distribution of Lisanti et al. [@Lisanti:2010], which has the following form in the Earth’s frame: $$\label{eq:lisanti} f(\textbf{v}) = N \left[\exp\left(\frac{v_\textrm{esc}^2 - |\textbf{v} - \textbf{v}_0|^2}{k v_0^2}\right) -1\right]^k \Theta(v_\textrm{esc} - |\textbf{v} - \textbf{v}_0|)\,.$$ We use the parameter values $k = 2$ and $v_0 = 220 {\,\textrm{km s}^{-1}}$ in this work. We summarize in Tab. \[tab:distributions\] the different speed distributions considered. We also plot several of these in Fig. \[fig:Ensemble\_distributions\] for reference. [m[3cm]{}|ccc]{} Speed distribution benchmark & Fraction & $v_\textrm{lag} / {\,\textrm{km s}^{-1}}$ & $\sigma_v / {\,\textrm{km s}^{-1}}$\ SHM & 1 & 220 & 156\ & 0.77 & 220 & 156\ & 0.23 & 50 & 50\ Stream & 1 & 400 & 20\ & 0.97 & 220 & 156\ & 0.03 & 500 & 20\ & 0.5 & 200 & 20\ & 0.5 & 400 & 20\ Lisanti et al. & & $v_0 = 220 {\,\textrm{km s}^{-1}}$ & $k = 2$ ![Several of the benchmark speed distributions used in this work. They are defined in Eqs. \[eq:gaussian\] and \[eq:lisanti\] with parameters from Tab. \[tab:distributions\]. These distributions are the SHM (solid blue), SHM+DD (dashed green), Lisanti et al. (dot-dashed red) and the stream (dotted magenta).[]{data-label="fig:Ensemble_distributions"}](SpeedDistributions-Ensemble.pdf){width="49.00000%"} In Paper 1, a parametrization for the WIMP speed distribution was introduced, for use in the analysis of direct detection data. The parametrization of Paper 1 has the form: $$\label{eq:parametrization} f_1(v) = v^2 \exp\left\{ -\sum_{k =0}^{N-1} a_k P_k(v)\right\}\,$$ where $P_k(v)$ is some basis of polynomial functions of v. We fit the coefficients $\left\{a_1, ..., a_{N-1}\right\}$ using data, and fix $a_o$ by normalization $$\label{eq:normalization} a_0 = \ln\left(\int_{0}^\infty v^2 \exp\left\{ -\sum_{k = 1}^{N-1} a_k P_k(v)\right\} \, \textrm{d}v\right)\,.$$ This form of parametrization ensures that the distribution function $f_1(v)$ is everywhere positive and can be used to fit an arbitrary underlying directionally-averaged distribution function (given a sufficiently large number of polynomial basis functions). We explore in Sec. \[sec:Parametrization\] which basis functions should be used in the parametrization, as well as how many basis functions are required. Parameter Reconstruction {#sec:ParameterRecon} ======================== In order to assess the performance of the parametrization method, we attempt to reconstruct the WIMP mass $m_\chi$ and polynomial coefficients $\left\{a_1, ..., a_{N-1} \right\}$ using the nested sampling software <span style="font-variant:small-caps;">MultiNest</span> [@MultiNest1; @MultiNest2; @MultiNest3]. We also include the WIMP-proton spin-independent cross section $\sigma_p$ as a free parameter. However, we are forced to treat the cross section as a nuisance parameter. As has previously been noted [@Kavanagh:2012; @Kavanagh:2013a], taking an agnostic approach to the DM speed distribution means that we do not know what fraction of WIMPs lie above the energy thresholds of the experiments. While this does not adversely impact the reconstruction of the WIMP mass, it does result in a strong degeneracy, such that only lower limits can be placed on the cross section using such methods. In any case, the cross section appears in the event rate (Eq. \[eq:Rate\]) in the degenerate combination $\rho_0 \sigma_p$. Uncertainties on the local DM density $\rho_0$ are at present on the order of a factor of 2 (see e.g. [@Iocco:2011; @Bovy:2012; @Zhang:2013; @Nesti:2013]) and thus any reconstruction of the cross section would be subject to the same systematic uncertainty. In this work, we focus instead on reconstructing the WIMP mass and the shape of the speed distribution. For concreteness, we use the values $\sigma_p = 10^{-45} \textrm{ cm}^2$ and $\rho_0 = 0.3 \textrm{ GeV cm}^{-3}$ throughout this work. Experimental benchmarks {#sec:experiments} ----------------------- In order to generate mock data sets, we consider three idealized mock experiments, loosely based on detectors which are currently in development. As previous work has shown [@Kavanagh:2012; @Peter:2013a], the WIMP mass and speed distribution are degenerate when data from only a single experiment is considered. However, this degeneracy can be broken by including data from additional experiments with different nuclear target masses. The three target materials we consider here are Xenon, Argon and Germanium. We describe each experiment in terms of its nucleon number $A$, fiducial detector mass $m_\textrm{det}$, efficiency $\epsilon$ and energy sensitivity window $\left[E_\textrm{min}, E_\textrm{max}\right]$. We incorporate the effects of detector sensitivity, analysis cuts and detector down-time into the value of the efficiency $\epsilon$, which we take to be energy independent for simplicity. We consider a total exposure time for all experiments of $t_\textrm{exp} = \textrm{ 2 years}$. The experimental parameter values used in this work are summarized in Tab. \[tab:experiments\]. [c|m[1.2cm]{}m[1.7cm]{}m[1.5cm]{}m[1.7cm]{}]{} Experiment & Target Mass, $A$ & Detector Mass (fid.), $m_\textrm{det}$/kg & Efficiency, $\epsilon$ & Energy Range/keV\ Xenon & 131 & 1100 [@Aprile:2012a] & 0.7 [@Aprile:2012b] & 7-45 [@Aprile:2010]\ Argon & 40 & 1000 & 0.9 [@Benetti:2007] & 30-100 [@Grandi:2005]\ Germanium & 73 & 150 [@Bauer:2013b] & 0.6 [@Bauer:2013a] & 8-100 [@Bauer:2013a]\ The exact parameter values we used in this work do not strongly impact the results we present. However, it is important to note that the total mass and exposure of the experiments will affect the total number of events observed. This in turn will affect the precision of the reconstructions. For example, we have chosen a total Argon mass of 1000 kg. This is the stated target for Argon-based experiments which are in development (e.g. Ref. [@Badertscher:2013]), though at present typical fiducial masses for Argon prototypes are of the order of 100 kg [@Grandi:2005]. The data we have generated does not represent the ‘high-statistics’ regime: across all three experiments the total number of events observed is roughly 200-300 with as few as 10 events in the Germanium detector for some scenarios. Using a smaller exposure (or equivalently a smaller interaction cross section) will reduce the precision of the results, but should not introduce any additional bias. We also briefly consider the impact of a *larger* number of events in Sec. \[sec:Recon\]. Parameter sampling ------------------ We make parameter inferences using a combination of Bayesian and frequentist statistics. Bayes theorem for the probability of a particular set of theoretical parameters $\boldsymbol{\Theta}$ given the observed data $\textbf{D}$ is: $$P(\boldsymbol{\Theta}|\textbf{D}) = \frac{P(\boldsymbol{\Theta}) P(\textbf{D}|\boldsymbol{\Theta})}{P(\textbf{D})}\,,$$ where $P(\boldsymbol{\Theta})$ is the prior on the parameters and $P(\textbf{D})$ is the Bayesian evidence, which acts as a normalizing factor and has no impact on parameter inference. We summarize the priors used in this work in Tab. \[tab:priors\]. We also summarize in Tab. \[tab:MultiNest\] the MultiNest sampling parameters used. [m[1in]{}|cc]{} Parameter & Prior type & Prior range\ $m_\chi / \textrm{ GeV}$ & log-flat & $\left[10^{0}, 10^{3}\right]$\ $\sigma_p / \textrm{ cm}^2$ & log-flat & $\left[10^{-46}, 10^{-42}\right]$\ $\left\{a_k\right\}$ & linear-flat & $\left[-50, 50\right]$\ $R_{BG} / \textrm{dru}$ & log-flat & $\left[10^{-12}, 10^{-5}\right]$\ Parameter Value ------------------- ----------- $N_\textrm{live}$ 10000 efficiency 0.25 tolerance $10^{-4}$ : Summary of the MultiNest sampling parameters used in this work.[]{data-label="tab:MultiNest"} The factor $P(\textbf{D}|\boldsymbol{\Theta})$ is simply the likelihood of the data given the parameters $\boldsymbol{\Theta}$. In Sec. \[sec:Parametrization\] and Sec. \[sec:mass\], we consider the effects of varying the form of the parametrization and of varying the input WIMP mass. In order to eliminate the effects of Poisson noise, we use Asimov data [@Cowan:2013] for these sections. This means that we divide the energy window of each experiment into bins of width 1 keV. We then set the observed number of events $N_{o,i}$ in bin $i$ equal to the expected number of events $N_{e,i}$. In this case, we use the binned likelihood, calculated for $N_b$ energy bins: $$\label{eq:binnedL} \mathcal{L}_b = \prod_{i = 1}^{N_b} \frac{N_{e,i}^{N_{o,i}} \textrm{e}^{-N_{e,i}}}{N_{o,i}!}\,.$$ In Sec. \[sec:stats\] and Sec. \[sec:Recon\], we consider many realisations of data, including the effects of Poisson noise. We therefore use the extended likelihood which has previously been used by both the Xenon [@Aprile:2011] and CDMS [@Ahmed:2009] collaborations, which for a single experiment is given by: $$\label{eq:unbinnedL} \mathcal{L} = \frac{N_e^{N_o} \textrm{e}^{-N_e}}{N_o!} \prod_{i = 1}^{N_o} P(E_i)\,,$$ where the expected number of events is given by: $$\label{eq:N_expected} N_e = \epsilon m_\textrm{det}t_\textrm{exp}\int_{E_\textrm{min}}^{E_\textrm{max}} \frac{\textrm{d}R}{\textrm{d}E}\, \textrm{d}E\,,$$ and the normalised recoil spectrum is given by: $$\label{eq:eventdistribution} P(E) = \frac{ \epsilon m_\textrm{det} t_\textrm{exp}}{N_e} \frac{\textrm{d}R}{\textrm{d}E}\,.$$ The total likelihood is then the product over all experiments considered. Using nested sampling, we can extract the full posterior probability distribution of the parameters $P(\boldsymbol{\Theta}|\textbf{D})$, as well as the likelihood $\mathcal{L}(\Theta)$. However, we often want to make inferences not jointly for all parameters but for only a subset (treating the remaining as nuisance parameters). If we conceptually partition the parameter space into the parameters of interest $\boldsymbol{\psi}$ and the remaining nuisance parameters $\boldsymbol{\phi}$, we would like to make inferences about the values of $\boldsymbol{\psi}$, without reference to the values of $\boldsymbol{\phi}$. One option for doing this is to calculate the marginalized posterior distribution, obtained by integrating the posterior probability over the parameters we are not interested in: $$P_m(\boldsymbol{\psi}) = \int P(\boldsymbol{\psi}, \boldsymbol{\phi}) \, \textrm{d}\boldsymbol{\phi}\,.$$ This method performs well for small numbers of observations (compared to the number of free parameters in the fit). We take the mode of the distribution to be the reconstructed parameter value and construct p% *minimal* credible intervals, which include those parameter values with $P_m(\boldsymbol{\psi}) \geq h$, where $h$ is chosen such that p% of the probability distribution lies within the interval. The marginalized posterior method is used in Sec. \[sec:stats\] and Sec. \[sec:Recon\], where in some cases the number of events observed in an experiment is less than 10. An alternative method is to calculate the profile likelihood. This is obtained by maximizing the full likelihood function over the nuisance parameters: $$\label{eq:profilelikelihood} \mathcal{L}_p(\boldsymbol{\psi}) = \max_{\boldsymbol{\phi}} \mathcal{L}(\boldsymbol{\psi},\boldsymbol{\phi})\,.$$ For a large number of observations, we can take the value which maximizes $\mathcal{L}_p$ as the reconstructed value and construct confidence intervals using the asymptotic properties of the profile likelihood. We use the profile likelihood for parameter inferences in Sec. \[sec:Parametrization\] and Sec. \[sec:mass\], as the Asimov data sets provide a large number of measurements of $N_{e,i}$ over a large number of bins. The profile likelihood can also lead to less noisy reconstructions than the marginalized posterior, especially when the dimensionality of the parameter space becomes high, as in Sec. \[sec:Parametrization\] and Sec. \[sec:mass\]. Testing the parametrization {#sec:Parametrization} =========================== We now consider the two questions: how many basis functions are required and which polynomial basis should be used? In order to answer these questions, we use the two benchmark distribution functions illustrated in Fig. \[fig:VaryingN\_distributions\]. We have chosen these benchmarks not because they are necessarily realistic distribution functions but because they should be difficult to fit using standard techniques and fitting functions (e.g. [@Lisanti:2010]). The first distribution (referred to as ‘bump’) is a SHM distribution with the addition of a small bump, which contributes just 3% of the total WIMP population and could correspond to a small sub-halo or stream [@Vogelsberger:2009]. This should be difficult to fit because it represents only a very small deviation from the standard scenario. The second distribution (referred to as ‘double-peak’) has a sharp and rapidly varying structure, which we anticipate should be difficult to capture using a small number of basis functions. ![Benchmark speed distributions used in Sec. \[sec:Parametrization\] to test the performance of the parametrization as a function of the number and type of basis functions.[]{data-label="fig:VaryingN_distributions"}](SpeedDistributions-VaryingN.pdf){width="49.00000%"} Varying the number of basis functions ------------------------------------- We first investigate how the reconstructed WIMP mass $m_\textrm{rec}$ and uncertainty varies with the number of basis functions $N$. For now, we fix our choice of basis to shifted Legendre polynomials, as used in Paper 1: $$P_k(v) = L_k\left(2\frac{v}{v_\textrm{max}} - 1\right)\,,$$ where $L_k$ is the Legendre polynomial of order $k$, and $v_\textrm{max}$ is a cut off for the parametrization. We should choose $v_\textrm{max}$ to ensure that $f_1(v)$ is negligible above the cut off. However, too high a choice of $v_\textrm{max}$ will result in $f_1(v)$ being close to zero over a large range of the parametrization, making fitting more difficult. We use the value $v_\textrm{max} = 1000 {\,\textrm{km s}^{-1}}$, while lies significantly above the Galactic escape speed. The lower panel of Fig. \[fig:BUMP\_LEG\] shows the best fit mass and 68% confidence intervals as a function of $N$, using as input a WIMP of mass 50 GeV and the ‘bump’ distribution function. The reconstructed mass very rapidly settles close to the true value, using as few as three basis functions. This is because adding the bump near $v \sim 500 {\,\textrm{km s}^{-1}}$ still leaves the mean inverse speed relatively smooth, so a large number of basis function are not required. The correct mass is reconstructed and we emphasize in the lower panel of Fig. \[fig:BUMP\_LEG\] that the reconstruction is stable with the addition of more basis functions. We should also consider how the quality of the fit changes as a function of $N$. We would expect that adding fit parameters should always lead to a better fit. Eventually, the fit should be good enough that adding additional basis functions will no longer improve it significantly. We can then be confident that our reconstruction is accurate and not an artifact of using too few basis functions. In order to investigate this, we utilise the Bayesian Information Criterion (BIC) [@Schwarz:1978], which is given by: $$BIC = 2N_p\textrm{ln}(N_m) - \textrm{ln}(\mathcal{L}_\textrm{max}) \, ,$$ where $N_p$ is the number of free parameters, $N_m$ is the number of measurements or observations and $\mathcal{L}_\textrm{max}$ is the maximum likelihood value obtained in the reconstruction. For the case of binned data, $N_m$ corresponds simply to the total number of energy bins across all experiments. This criterion penalises the inclusion of additional free parameters and in comparing several models, we should prefer the one which minimises the BIC. ![Bayesian information criterion (BIC) as a function of the number of basis functions for an underlying ‘bump’ distribution function, 50 GeV WIMP and using Legendre polynomial basis functions (upper panel). Also shown (lower panel) are the reconstructed WIMP mass (dashed blue line), 68% confidence interval (shaded blue region) and underlying WIMP mass (solid horizontal black line).[]{data-label="fig:BUMP_LEG"}](VaryingN_BUMP_LEG.pdf){width="49.00000%"} The upper panel of Fig. \[fig:BUMP\_LEG\] shows the BIC (in arbitrary units) as a function of the number of basis functions for the ‘bump’ distribution function. The BIC is comparable for the cases of $N=2$ and $N=3$, indicating that the quality of the fit is improved slightly by the addition of another basis function. However, adding further basis functions does not have a significant impact on the maximum likelihood, leading to an increase in the BIC. This coincides with the stabilization of the reconstructed mass around the true value and we conclude that only two or three basis functions are required to provide a good fit to the data. Figure \[fig:DP\_LEG\] shows the corresponding results for the ‘double-peak’ distribution function. Here, we note that the bias induced by using too small a number of basis functions is larger than for the case of the ‘bump’ distribution, due to the more complicated structure in this case. The BIC is minimized for $N=7$, indicating that additional basis functions do not significantly improve the quality of the fit to data. This suggests that the shape of the speed distribution can be well fit by $N\geq7$ basis functions. As shown in the lower panel of Fig. \[fig:DP\_LEG\], the reconstruction of the WIMP mass is stable around the true mass for these values of $N$. ![As Fig. \[fig:BUMP\_LEG\] but for an underlying ‘double-peak’ distribution function.[]{data-label="fig:DP_LEG"}](VaryingN_DP_LEG.pdf){width="49.00000%"} We propose that such a procedure should be used in the case of real data should a dark matter signal be observed at multiple detectors. We have shown that by analyzing the reconstructed mass as a function of $N$ we can recover the true mass and that by using the BIC we can be confident that we have obtained an adequate fit to data. Choice of basis functions ------------------------- We now consider the second question posed at the start of Sec. \[sec:Parametrization\]: which polynomial basis should be used? We see immediately that a naive power series of the form $$\textrm{ln}f(v) \approx a_0 + a_1 v + a_2 v^2 + a_3 v^3 + ...\,,$$ is not practical for the purposes of parameter estimation. Higher powers of $v$ will have rapidly growing contributions to $\textrm{ln} f$, meaning that the associated coefficients must be rapidly decreasing in order to suppress these contributions. Fitting to the SHM using just 5 terms, the range of values for the $a_k$ in the case of a simple power series would span around 13 orders of magnitude. Ideally, we would like to specify an identical prior on each of the coefficients. However, in this scenario this would result in a highly inefficient exploration of the parameter space when some of the terms are so small. This problem can be significantly improved by rescaling $v$. We choose to rescale by a factor of $v_\textrm{max} = 1000 {\,\textrm{km s}^{-1}}$, and cut off the distribution function at $v_\textrm{max}$. The basis functions $(v/v_\textrm{max})^k$ are now less than unity by construction and the coefficients $a_k$ are now dimensionless: $$\textrm{ln}f(v) \approx a_0 + a_1 (v/v_\textrm{max}) + a_2 (v/v_\textrm{max})^2 + a_3 (v/v_\textrm{max})^3 + ...\,.$$ We now address the problem of *conditioning* of the polynomial basis (see e.g. Refs. [@Gautschi:1978; @Wilkinson:1984]). Conditioning is a measure of how much the value of a polynomial changes, given a small change in the coefficients. For a well-conditioned polynomial, small changes in the coefficient are expected to lead to small changes in the value of the polynomial. This is ideal for parameter estimation as it leads to a more efficient exploration of the parameter space. Orthogonal polynomial basis functions typically have improved conditioning [@Gautschi:1978] and we consider two specific choices: the Legendre polynomials which have already been considered and the Chebyshev polynomials. The Chebyshev polynomials are used extensively in polynomial approximation theory [@Mason:2002] and are expected to be well conditioned [@Gautschi:1978]. We have checked that the reconstruction results using Chebyshev polynomials are largely indistinguishable from the case of Legendre polynomials for both the ‘bump’ and ‘double-peak’ distributions and as a function of $N$. This leads us to conclude that the accuracy of the reconstruction is independent of the specific choice of basis. However, the reconstruction was much faster in the case of the Chebyshev basis. This is illustrated in Fig. \[fig:times\], which shows the time taken for reconstruction of the ‘bump’ benchmark as a function of $N$. The time taken grows much more slowly for the Chebyshev basis (roughly as $N^2$) than for the Legendre basis (roughly as $N^3$). We have also checked that this difference is not an artifact of how we calculate the basis functions. These results indicate that this choice of basis provides both reliable and efficient reconstruction for the WIMP mass and we therefore use the Chebyshev basis in the remainder of this work. ![Time taken (using 4 processors in parallel) for the reconstruction of the ‘bump’ benchmark, as a function of number of basis functions. The time taken using the Chebyshev basis (blue squares) grows more slowly with $N$ than for the Legendre basis (red triangles).[]{data-label="fig:times"}](RunTimes.pdf){width="49.00000%"} Varying $m_\chi$ {#sec:mass} ================ In previous work [@Kavanagh:2013a], this parametrization method was only tested for a single WIMP mass of $50 \textrm{ GeV}$. Here, we extend this analysis to a wider range of WIMP masses. We generate Asimov data for WIMP masses of 10, 20, 30, 40, 50, 75, 100, 200 and 500 GeV and reconstruct the best fit WIMP mass $m_\textrm{rec}$ and 68% and 95% confidence intervals from the profile likelihood. We use the SHM as a benchmark distribution function and use a fixed number of $N=5$ basis functions. The results are shown in Fig. \[fig:VaryingM\], along with the line $m_\textrm{rec} = m_\chi$ for reference. For large values of $m_\chi$, the shape of the event spectrum becomes independent of $m_\chi$ [@Green:2008], which results in a widening of the confidence intervals as the WIMP mass increases. For low mass WIMPs, fewer events are observed in each bin, again resulting in wider confidence intervals. It should be noted that for this analysis we have used Asimov data, in which the exact (non-integer) number of events is recorded in each bin. For low mass WIMPs, this means that the spectrum (and therefore the correct WIMP mass) is still well reconstructed using Asimov data, in spite of the small number of events. The tightest constraints are obtained when the input WIMP mass is close to the masses of several of the detector nuclei (in the range 30-80 GeV). There also appears to be no bias in the WIMP mass: the reconstruction matches the true mass across all values considered. ![Reconstructed WIMP mass $m_\textrm{rec}$ (central dashed blue line) as a function of input WIMP mass $m_\chi$ as well as 68% and 95% intervals (inner and outer blue dashed lines respectively). The line $m_\textrm{rec} = m_\chi$ (solid red line) is also plotted for reference.[]{data-label="fig:VaryingM"}](VaryingM.pdf){width="49.00000%"} An alternative parametrization method was proposed in Ref. [@Kavanagh:2012], in which the *momentum* distribution of halo WIMPs was parametrized. For a given speed distribution, the corresponding momentum distribution may be broad and easily reconstructed for high mass WIMPs. However, for low mass WIMPs the momentum distribution would be much narrower, owing to their lower momenta. The momentum parametrization method therefore performs poorly for low mass WIMPs. The parametrization presented in this paper does not suffer from similar problems. So far, we have only considered idealized direct detection experiments. We now apply the method to more realistic mock detectors, taking into account the effects of finite energy resolution, as well as unrejected background events. We assume here that each experiment has a gaussian energy resolution with fixed width $\sigma_E = 1 \textrm{ keV}$, such that the observed event rate for recoils of energy $E$ is given by: $$\frac{\textrm{d}R}{\textrm{d}E} = \int_{0}^{\infty} \frac{1}{\sqrt{2 \pi} \sigma_E}\exp\left\{-\frac{(E-E')^2}{2\sigma_E^2}\right\} \frac{\textrm{d}{R'}}{\textrm{d}E'} \, \textrm{d}E'\,,$$ where the primed event rate is the underlying (perfect resolution) rate. We also assume a constant flat background rate for each experiment $R_\textrm{BG} = 10^{-6}$ events/kg/keV/day (which has been suggested as a possible background rate for Xenon1T [@Aprile:2010] and WArP-100L [@Grandi:2005]) when generating mock data sets. However, we allow the flat background rate in each experiment to vary as free parameters during the fit. We have chosen relatively generic resolution and background parameters in this work, because the precise details of energy resolution and background shape and rate will depend on the specific experiment under consideration. Instead, we hope to show that the inclusion of more realistic experimental setups does not introduce an additional bias or otherwise spoil the good properties of the method presented here. Figure \[fig:VaryingM\_real\] shows the reconstructed mass as a function of input mass in this more realistic scenario. The 68% and 95% confidence intervals are now wider and the reconstructed mass does not appear to be as accurate. For input masses above $\sim$100 GeV, the uncertainties become very wide, with only a lower limit of $m_\textrm{rec} > 20 \textrm{ GeV}$ being placed on the WIMP mass. Due to the poorer energy resolution the shape of the energy spectrum is less well-determined. In addition, a flat background contribution can mimic a higher mass WIMP, as it leads to a flatter spectrum. This leads to a strong degeneracy, as a wide range of mass values can provide a good fit to the data. For high input masses, the profile likelihood is approximately constant above $m_\textrm{rec} \sim 20 \textrm{ GeV}$, indicating that there is no sensitivity to the underlying WIMP mass. In spite of this, the true mass values still lie within the 68% and 95% confidence intervals. In addition, the poor values for the reconstructed mass for heavy WIMPs are a side effect of the loss of sensitivity. Because the profile likelihood is approximately flat, the maximum likelihood point is equally likely to be anywhere within the 68% interval. These effects would be present even if we had considered a fixed form for the speed distribution. However, when we allow for a range of possible speed distributions, the effects become more pronounced. These results show that for more realistic experimental scenarios, the method presented in this paper remains reliable over a range of masses, though its precision may be significantly reduced. ![As fig. \[fig:VaryingM\] but including the effects of finite energy resolution and non-zero backgrounds, as described in the text.[]{data-label="fig:VaryingM_real"}](VaryingM_real.pdf){width="49.00000%"} Statistical properties {#sec:stats} ====================== We now consider the impact of statistical fluctuations on the reconstruction of the WIMP mass. In reality, the number of events observed $N_o$ at a given experiment will be Poisson distributed about the expected value $N_e$, while the observed distribution of recoil energies will not exactly match that expected from the calculated event rate. The fundamental statistical limitations of future direct detection experiments have been studied in detail in Ref. [@Strege:2012]. In this work, we generate 250 realisations of data from the mock experiments described in Tab. \[tab:experiments\]. Each realisation of the mock data is generated as follows: 1. Calculate the number of expected events $N_e$, given $\left\{m_\chi, \sigma_p, f(v)\right\}$, using Eq. \[eq:N\_expected\], 2. Pick the number of observed events $N_o$ from a Poisson distribution with mean $N_e$, 3. Pick recoil energies $\left\{E_1, E_2, ..., E_{N_o}\right\}$, from the distribution $P(E)$ in Eq. \[eq:eventdistribution\], 4. Repeat for all three experiments. For each realisation, we then use the method described in Sec. \[sec:ParameterRecon\] (using $N = 5$ basis functions) to reconstruct the WIMP mass and 68% and 95% credible intervals. Figure \[fig:Realisations\] shows the distribution of reconstructed masses for an input mass of 50 GeV for three benchmark speed distributions: SHM, SHM+DD and Lisanti et al. as described in Sec. \[sec:DDRate\]. In all three cases, the reconstructions are peaked close to the true value, regardless of the underlying distribution. For the SHM+DD distribution, the spread of reconstructions is slightly wider (with more reconstructions extending up to higher masses). This is due to the smaller number of events for this benchmark, making the data sets more susceptible to Poisson fluctuations. In order to assess the accuracy of the reconstructed value of the mass $m_\textrm{rec}$, we also calculate the bias $b$ for each realisation: $$\label{eq:bias} b = \textrm{ln}(m_\textrm{rec} / \textrm{GeV}) - \textrm{ln}(m_\textrm{true} / \textrm{GeV})\,.$$ We compare the logarithms of the mass values because we have used logarithmically-flat priors on the WIMP mass. In Tab. \[tab:bias\] we show the average bias across all 250 realisations for each of the three benchmark distributions. In all three cases, the average bias is consistent with zero. Even in the SHM+DD case, which shows larger fluctuations away from the true value, there is no statistical bias. ![Distribution of the reconstructed mass $m_\textrm{rec}$ for 250 mock data sets generated using several benchmark speed distributions, defined in Sec. \[sec:DDRate\]. These are the SHM (top), SHM+DD (middle) and Lisanti et al. (bottom) distributions. The input WIMP mass of $m_\chi = 50 \textrm{ GeV}$ is shown as a vertical dashed red line.[]{data-label="fig:Realisations"}](SHM_ensemble.pdf "fig:"){width="49.00000%"} ![Distribution of the reconstructed mass $m_\textrm{rec}$ for 250 mock data sets generated using several benchmark speed distributions, defined in Sec. \[sec:DDRate\]. These are the SHM (top), SHM+DD (middle) and Lisanti et al. (bottom) distributions. The input WIMP mass of $m_\chi = 50 \textrm{ GeV}$ is shown as a vertical dashed red line.[]{data-label="fig:Realisations"}](DD_ensemble.pdf "fig:"){width="49.00000%"} ![Distribution of the reconstructed mass $m_\textrm{rec}$ for 250 mock data sets generated using several benchmark speed distributions, defined in Sec. \[sec:DDRate\]. These are the SHM (top), SHM+DD (middle) and Lisanti et al. (bottom) distributions. The input WIMP mass of $m_\chi = 50 \textrm{ GeV}$ is shown as a vertical dashed red line.[]{data-label="fig:Realisations"}](LIS_ensemble.pdf "fig:"){width="49.00000%"} [m[1in]{}|c]{} Benchmark speed distribution & Mean bias $\langle b \rangle$\ SHM & 0.002 $\pm$ 0.008\ SHM+DD & 0.005 $\pm$ 0.007\ Lisanti et al. & 0.01 $\pm$ 0.01\ We also test the *coverage* of the credible intervals which have been constructed. For a $p\%$ credible interval, we expect that the true parameter value of the WIMP mass will lie within the interval in $p\%$ of realisations. In this case, we say that the method provides *exact coverage*. However, if the true parameter lies within the interval in fewer than $p\%$ of realisations, our reconstructed credible intervals are too narrow and provide *undercoverage*. Alternatively, we obtain *overcoverage* when the true parameter lies within the interval more often that $p\%$ of the time. Table \[tab:coverage\] shows the coverage values for the $68\%$ and $95\%$ intervals obtained in this section. In each case, there is very close to exact coverage. We have also checked that these intervals only provide exact coverage for the true WIMP mass of 50 GeV. Other values of $m_\textrm{rec}$ are contained within the intervals less frequently than the true value, again indicating that this parametrization allows for unbiased and statistically robust reconstructions of the WIMP mass. [m[1in]{}|cc]{} Benchmark speed distribution & 68% coverage & 95% coverage\ SHM & 71 $\pm$ 3 % & 94 $\pm$ 3 %\ SHM+DD & 68 $\pm$ 3 % & 91 $\pm$ 4 %\ Lisanti et al. & 70 $\pm$ 3 % & 95 $\pm$ 3 %\ Reconstructing $f_1(v)$ {#sec:Recon} ======================= Using the method described in this paper, we can obtain the posterior probability distribution for the coefficients $\left\{ a_1, ..., a_{N-1}\right\}$ given the data, which we refer to as $P(\textbf{a})$. We would like to be able to present this information in terms of the distribution function $f_1(v)$ in order to compare with some known distribution or look for particular features in the distribution. However, due to the fact that the distribution function is normalized, the values of $f_1$ at different speeds will be strongly correlated. We illustrate here how robust comparisons with benchmark distributions can be made. As a first step, we can attempt to sample from the $P(\textbf{a})$, in order to obtain $P(f_1(v))$. This is the probability distribution for the value of $f_1$ at a particular speed $v$, marginalizing over the values of $f_1$ at all other speeds. We can repeat for a range of speeds to obtain 68% and 95% credible intervals for the whole of $f_1(v)$. The result of this procedure is presented in Fig. \[fig:f\], for a randomly selected realisation from the SHM ensemble of Sec. \[sec:stats\]. The underlying SHM distribution is shown as a solid line, while the 68% and 95% marginalized intervals are shown as dark and light shaded regions respectively. In this naive approach, we see that there is little shape information which can be recovered from the reconstruction, with only upper limits being placed on the speed distribution. ![Reconstructed speed distribution for a single realisation of data, generated for a 50 GeV WIMP. 68% and 95% credible intervals are shown as dark and light shaded regions respectively, while the underlying SHM distribution function is shown as a solid blue line.[]{data-label="fig:f"}](f_SHM.pdf){width="49.00000%"} This method performs poorly because, as initially mentioned in Sec. \[sec:ParameterRecon\], we have no information about the fraction of dark matter particles below the energy threshold of our experiments. If this fraction is large, the event rate for a given cross-section is suppressed. However, increasing the cross-section will increase the total event rate. There is thus a degeneracy between the shape of the speed distribution and the cross-section, meaning that we can only probe the shape of $f_1(v)$, rather than its overall normalization. This degeneracy has not been accounted for in Fig. \[fig:f\]. We can attempt to correct for this by adjusting the normalization of $f_1(v)$. If we fix $f_1(v)$ to be normalized to unity above $v_a$ (where $v_a \approx 171 {\,\textrm{km s}^{-1}}$ is the lowest speed probed by the experiments for a WIMP of mass 50 GeV), we can compare the shapes of the underlying and reconstructed distribution functions. This is illustrated in Fig. \[fig:f\_scaled\], which shows that we now broadly reconstruct the correct shape of $f_1(v)$. Below $v_a$, the value of $f_1(v)$ is poorly constrained, because the experiments provide no information about the shape of the distribution below theshold. There remain several issues with this approach. In order to utilize this method, we must know the approximate value of the lowest speed probed by the experiments. However, this value is set by the WIMP mass. We could determine $v_a$ using the reconstructed WIMP mass, but this would be subject to significant uncertainty. In addition, direct reconstructions of the speed distribution are easily biased. The upper limit of the energy windows of the experiments corresponds to a particular WIMP speed (for a given WIMP mass). WIMPs above this speed still contribute to the total event rate, but contribute no spectral information. The reconstructed shape of the high speed tail of the distribution is therefore not constrained by the data, but may affect the reconstructed value of $f_1$ at lower speeds. ![Reconstructed speed distribution for the same realisation of data as Fig. \[fig:f\]. In this case, we have also normalized $f_1(v)$ to unity above $v_a \approx 171 {\,\textrm{km s}^{-1}}$ (vertical dashed line). This is the lowest speed accessible to the experiments for a WIMP of mass 50 GeV. 68% and 95% credible intervals are shown as dark and light shaded regions respectively, while the underlying SHM distribution function is shown as a solid blue line.[]{data-label="fig:f_scaled"}](f_SHM_scaled_line.pdf){width="49.00000%"} An alternative approach is to reconstruct the mean inverse speed $\eta(v)$ (defined in Eq. \[eq:eta\]) at some speed $v$. Because $\eta(v)$ is an integral function of $f_1$, it is less prone to bias as it takes into account the full shape of the distribution at speeds greater than $v$. However, we do not know the normalization of $f_1$ and so we must normalize $\eta$ appropriately. For each point sampled from $P(\textbf{a})$, we calculate $\eta$. We then divide by $\alpha(v)$, the fraction of WIMPs above speed $v$, calculated using the same parameter point: $$\label{eq:alpha} \alpha(v) = \int_{v}^{\infty} f_1(v') \, \textrm{d}v'\,.$$ We will write this rescaled mean inverse speed as $\eta^*(v) = \eta(v)/\alpha(v)$. The value of $\eta^*(v)$ is a measure of the shape of the distribution function above $v$. However, information about the normalization of the distribution has been factored out by dividing by $\alpha(v)$. We no longer need to know the value of $v_a$ in order to obtain information about the shape of the distribution at higher speeds. We may still need to decide the speed down to which we trust our reconstruction, but this no longer relies on an arbitrary choice of $v_a$ to normalize the reconstructions at all speeds. In Fig. \[fig:eta\_stats\], we plot the mean reconstructed value of $\eta^*$ at several values of $v$, using 250 realisations of the 50 GeV SHM benchmark. We also show the mean upper and lower limits of the 68% credible intervals as errorbars. The form of $\eta^*$ for the SHM is shown as a solid blue line. In all cases except for $v=100 {\,\textrm{km s}^{-1}}$, the mean reconstructed value is close to the true value, indicating that $\eta^*$ can be reconstructed without bias using this method. At low speeds, the reconstructed value deviates from the true value. In addition, the credible intervals lead to *under*coverage in the $v=100 {\,\textrm{km s}^{-1}}$ case. However, this point lies below the lowest speed to which the experiments are sensitive and therefore we cannot trust the reconstruction at this low speed. We have checked that for the remaining values of $v$ the method provides exact or overcoverage, indicating that at higher speeds we can use $\eta^*$ as a reliable and statistically robust measure of the shape of the distribution. ![Mean reconstructed values of the rescaled mean inverse speed $\eta(v)/\alpha(v)$ at several values of $v$, calculated over 250 realisations of data using a 50 GeV WIMP and underlying SHM distribution function. Errorbars indicate the mean upper and lower limits of the 68% credible intervals. The underlying form of $\eta(v)/\alpha(v)$ obtained from the SHM is shown as a solid blue line.[]{data-label="fig:eta_stats"}](Eta.pdf){width="49.00000%"} In the case of a single realisation of data, we would like to compare the probability distribution for $\eta^*(v)$ (obtained from $P(\textbf{a})$) to the value calculated from some test distribution. We note that several distributions may produce the same value of $\eta^*(v)$ at a given value of $v$. Thus, we may fail to reject a distribution function which is not the true distribution. However, if the calculated value of $\eta^*(v)$ does lie outside the $p\%$ interval, we can reject it at the $p\%$ level. We can increase the discriminating power of this method by repeating this reconstruction over all speeds and checking to see if the benchmark value of $\eta^*$ is rejected at any value of $v$. The result of this procedure is shown in Fig. \[fig:eta\] for a single realisation of data generated using an SHM distribution (the same data as in Figs. \[fig:f\] and \[fig:f\_scaled\]). We plot the 68%, 95% and 99% credible intervals as shaded regions, as well as the values of $\eta^*(v)$ calculated from several benchmark speed distribution. We will focus on the intermediate speed range ($v \gtrsim 200 {\,\textrm{km s}^{-1}}$), as we do not know *a priori* the lowest speed to which the experiments are sensitive. ![Rescaled mean inverse speed $\eta(v)/\alpha(v)$, reconstructed from a single realisation of data using a 50 GeV WIMP and underlying SHM distribution function. At each value of $v$ we calculate 68%, 95% and 99% credible intervals (shown as shaded intervals). We also show the calculated values of $\eta(v)/\alpha(v)$ for several possible benchmark speed distributions: SHM (solid blue), SHM+DD (dashed green), Lisanti et al. (dot-dashed red) and stream (dotted magenta). The benchmark curves are truncated when the underlying distribution function goes to zero.[]{data-label="fig:eta"}](SHM_lores.pdf){width="49.00000%"} The reconstructed intervals are consistent with a range of possible distribution functions. The SHM and SHM+DD distributions are identical over a wide range of speeds. This is because above $\sim 200 {\,\textrm{km s}^{-1}}$, the two distributions differ in normalization but not in shape. Differences appear between the two at low speeds where their shapes diverge. The Lisanti et al. distribution results in a larger deviation from the SHM, but not sufficiently large to differentiate between the two distributions given the size of the uncertainties. Finally, the stream distribution results in a significantly different form for $\eta^*(v)$. At approximately $400 {\,\textrm{km s}^{-1}}$, the curve for the stream distribution lies outside the reconstructed 99% credible interval. We can therefore use this method to reject the stream distribution at the 99% confidence level. Figure \[fig:eta\_hires\] shows the results of a reconstruction using a larger exposure. In this case, we generate data using the Lisanti et al. distribution and an exposure increased by a factor of $2.5$, resulting in approximately 1000 events across the three detectors. As expected, the resulting credible intervals are now substantially narrower. The stream distribution now lies significantly outside the 99% interval. In Fig. \[fig:eta\_hires\_zoom\], we show the same results, but focusing in on the region around $v \sim 400 {\,\textrm{km s}^{-1}}$. At certain points, the SHM and SHM+DD distributions now lie outside the 95% credible interval, suggesting that with a number of events of the order of 1000, we may be able to reject these benchmarks. ![As Fig. \[fig:eta\], but using as input a Lisanti et al. speed distribution and an exposure time which is 2.5 times longer.[]{data-label="fig:eta_hires"}](LIS_hires.pdf){width="49.00000%"} ![As Fig. \[fig:eta\_hires\], but focusing on the region around $v \sim 400 {\,\textrm{km s}^{-1}}$. Notice that in the range $400-550 {\,\textrm{km s}^{-1}}$, both the SHM and SHM+DD curves lie at or below the lower limit of the 95% credible interval.[]{data-label="fig:eta_hires_zoom"}](LIS_hires_zoom.pdf){width="49.00000%"} While the method displayed in Fig. \[fig:f\_scaled\] allows the approximate shape of the speed distribution to be reconstructed, reconstructions of $\eta^*(v)$ allow more statistically robust statements to be made about the underlying speed distribution. In particular, Fig. \[fig:eta\_hires\_zoom\] illustrates that with larger exposures deviations from Maxwellian speed distributions can be detected in a model-independent fashion. Conclusions {#sec:Conclusions} =========== We have studied in detail the parametrization for the local dark matter speed distribution introduced in Paper 1. This method involves writing the logarithm of the speed distribution as a polynomial in speed $v$ and fitting the polynomial coefficients (along with the WIMP mass and cross section) to the data. We have attempted to disentangle in this paper the influence of different benchmark speed distributions, different benchmark WIMP masses and different forms for the parametrization. We summarize our conclusions as follows: - We have shown that the reconstruction of the WIMP mass is robust under changes in the number of basis functions $N$. We have used the Bayesian Information Criterion (BIC) to compare models with different values of $N$ and have shown that minimizing the BIC allows us to determine how many basis functions are required for a reliable reconstruction. We have also demonstrated that the results of the method do not depend strongly on the choice of basis functions, but that the speed of reconstructions may improved by using the Chebyshev polynomial basis. - We have shown that the method leads to unbiased reconstructions of the WIMP mass for masses in the range 10-500 GeV. Including realistic experimental parameters, including non-zero backgrounds and finite energy resolution, reduces the precision of these reconstructions. In particular, for large values of the input mass, we can only place a lower limit of approximately 20 GeV on the reconstructed mass. This is significantly lower than in the idealized case, where we can typically constrain the WIMP mass to be heavier than around 50 GeV. - We have used several ensembles of data realisations to demonstrate the statistical properties of the method, including unbiased reconstructions and exact coverage of the WIMP mass. - We have presented several ways of displaying the reconstructed WIMP speed distribution using this method. In order to make robust statistical inferences about the speed distribution, we calculate the probability distribution of $\eta(v)/\alpha(v)$. This is the mean inverse speed $\eta(v)$, which appears in the direct detection event rate (eq. \[eq:Rate\]), rescaled by the fraction of WIMPs $\alpha(v)$ above speed $v$. This can be used as a measure of the *shape* of the distribution function, from which the unknown normalization has been factored out. We can then compare to the expected value of $\eta(v)/\alpha(v)$ from a given benchmark speed distribution, allowing us to distinguish between different underlying models. We have shown that this parametrization method is statistically robust and works well over a large range of input parameters, both in terms of particle physics and astrophysics. The inclusion of more realistic experimental parameters does not introduce any additional bias, but does reduce the precision of reconstructions. We obtain unbiased estimates of the WIMP mass over large numbers of data sets. Finally, we have shown that we can distinguish different forms of the speed distribution. With around 1000 events, it may be possible to detect minor deviations from the Standard Halo Model and begin to search for more interesting structure in the speed distribution of the Milky Way. The author thanks Anne M. Green and Mattia Fornasa for helpful comments. BJK is supported by STFC. Access to the University of Nottingham High Performance Computing Facility is also gratefully acknowledged.

ArXiv

--- abstract: 'We report in this paper the proofs that the pulse shape analysis can be used in some bolometers to identify the nature of the interacting particle. Indeed, while detailed analyses of the signal time development in purely thermal detectors have not produced so far interesting results, similar analyses on bolometers built with scintillating crystals seem to show that it is possible to distinguish between an electron or $\gamma$-ray and an $\alpha$ particle interaction. This information can be used to eliminate background events from the recorded data in many rare process studies, especially Neutrinoless Double Beta decay search. Results of pulse shape analysis of signals from a number of bolometers with absorbers of different composition (CaMoO$_4$, ZnMoO$_4$, MgMoO$_4$ and ZnSe) are presented and the pulse shape discrimination capability of such detectors is discussed.' address: - 'INFN - Milano Bicocca, Italy' - 'Dipartimento di Fisica - Università di Milano Bicocca, Italy' author: - 'C.Arnaboldi' - 'C.Brofferio' - 'O.Cremonesi' - 'L.Gironi' - 'M.Pavan' - 'G.Pessina' - 'S.Pirro' - 'E.Previtali' title: A novel technique of particle identification with bolometric detectors --- Bolometers ,Scintillators ,Pulse shape discrimination (PSD) 23.40B ,95.35.+d ,07.57.K ,29.40M ,66.70.-f Rare event searches {#RES} =================== Rare event studies, such as the search for Neutrinoless Double Beta decay () [@BBreview] or the identification of Weakly Interacting Massive Particle (WIMP) interactions with ordinary matter [@DMreview], are of extreme interest in astroparticle physics, since they would imply new physics beyond the Standard Model. In both cases, as in all the rare event studies, spurious events are a limiting factor to the reachable sensitivity of the experiment. Unfortunately natural radioactive background is often present in the detector itself or in the materials surrounding it, no matter how much one can try to reduce it with shieldings, selection of materials and complicated purification techniques. In order to handle the residual unavoidable background, all the envisaged approaches require both a good energy resolution (which always helps in the comprehension of the different structures of an energy spectrum) and the capability to identify the nature of the projectile that interacted with the detector. Indeed, the searched event has always well defined signatures helping to distinguish it from background, for instance two electrons with a fixed sum energy in the case of the . Bolometers [@bolometers] are based on the detection of phonons produced after an energy release by an interacting particle and can have both an excellent energy resolution and extremely low energy threshold with respect to conventional detectors. They can be fabricated from a wide variety of materials, provided they have a low enough heat capacity at low temperatures, which is the only requirement really unavoidable to build a working bolometer. The latter is a priceless feature for experiments that aim at detectors containing particular atomic or nuclear species to optimize the detection efficiency. If other excitations (such as ionization charge carriers or scintillation photons) are collected in addition to phonons, bolometers have already shown to be able to discriminate nuclear recoils from electron recoils, or $\alpha$ particles from $\beta$ particles and $\gamma$-rays. In this paper we will report on the possibility to obtain similar results just by pulse shape analysis, without the requirement of a double readout for phonons and ionization or scintillation light. Bolometric Technique and Scintillating Bolometers {#BolTec} ================================================= Bolometers can be essentially sketched as a two-component object: an energy absorber in which the energy deposited by a particle is converted into phonons, and a sensor that converts thermal excitations into a readable signal. The absorber must be coupled to a constant temperature bath by means of a weak thermal conductance. Denoting by C the heat capacity of the bolometer, the temperature variation induced by an energy release E in the absorber can be written as $$\label{eq:temperature} \Delta T = \frac{E}{C}$$ The accumulated heat flows then to the heat sink through the thermal link and the absorber returns to the base temperature with a time constant $\tau$ = C/G, where G is the thermal conductance of the link: $$\label{eq:signal} \Delta T(t) = \frac{E}{C} e^{ - \frac{t}{\tau}}$$ In order to obtain a measurable temperature rise the heat capacity of the absorber must be very small: this is the reason why bolometers need to be operated at cryogenic temperatures (of the order of 10-100 mK). A real bolometer is somewhat more complicated than the naive description presented above. It is made of different elements and it is therefore represented by more than one heat capacity and heat conductance. As such, the time development of the thermal pulse is characterized by various time constants. In principle, if the bolometer performs as an ideal calorimeter and if the conversion of the energy into heat deposited by the particle is instantaneous (as assumed in equation \[eq:signal\]), then the device is insensitive to the nature of the interacting particle. Although this situation is generally very far from reality, it is however true that the small differences are difficult to detect and the goal has been so far achieved only relying on complicated solutions. Among these are scintillating bolometers. The concept of a scintillating bolometer is very simple: a bolometer coupled to a light detector [@CaF2]. The first must consist of a scintillating absorber thermally linked to a phonon sensor while the latter can be any device able to measure the emitted photons. The driving idea of this hybrid detector is to combine the two available informations, the heat and the scintillation light, to distinguish the nature of the interacting particles, exploiting the different scintillation yield of $\beta$/$\gamma$, $\alpha$ and neutrons. Dark Matter as well as searches can benefit of this capability of tagging the different particles, and more generally this technique can be exploited in any research where background suppression or identification is important. Dark matter experiments look for a very rare signal generally hidden in a huge background. The signal is a nuclear recoil with an energy of few keV (or less) induced by the scattering of a WIMP off a target nucleus. Experiments like CDMS [@CDMS], Edelweiss [@Edelweiss] or CRESST [@CRESST] clearly show that in such energy region the background is dominated by $\beta$/$\gamma$ interactions. A second source of background are $\alpha$ decays, contributing through energy degraded $\alpha$’s and nuclear recoils. The capability to distinguish a nuclear recoil - candidate for a WIMP interaction - from $\alpha$ or $\beta$/$\gamma$ clearly allows to improve drastically the experimental sensitivity. A similar approach was proposed also for applications in searches [@CaF2]. More recently, such a possibility has been demonstrated to be viable for a number of candidate nuclei [@Pirr06]. In this case the major interest is the identification of $\alpha$ interactions. Indeed the other important source of background, namely $\gamma$-rays, is virtually indistinguishable from the signal. The suggested way to eliminate the problem of $\gamma$-rays contribution is to study isotopes with a transition energy above 2615 keV. This corresponds in fact to the highest energy $\gamma$-ray line from natural radioactivity and is due to $^{208}$Tl. Above this energy there are only extremely rare high energy $\gamma$’s from $^{214}$Bi (all the active isotopes with Q$_{\beta\beta}>$2615 keV are listed in Tab. \[tab:isotopes\]). Once $\gamma$-rays are no more a worrisome source of background, what is left - on the side of radioactivity- are $\alpha$ emissions. Indeed $\alpha$ surface contaminations not only can represent the dominant background source for searches based on high transition energy isotopes, but they are already recognized as the most relevant background source in the bolometric experiment CUORICINO [@CUORICINO; @CUOpotential; @ArtChambery] and as a limiting factor for the experiment CUORE [@CUOpotential; @CUORE]. Both the experiments search for the of $^{130}$Te whose transition energy is at 2527 keV, therefore in a region where $\gamma$ background (mainly due to Compton events produced by 2615 keV photons) can be still important. [ccc]{} Isotope & Q$_{\beta\beta}$ \[MeV\] & natural abundance\ $^{116}$Cd & 2.80 & 7.5 %\ $^{82}$Se & 3.00 & 9.2 %\ $^{100}$Mo & 3.03 & 9.6 %\ $^{96}$Zr & 3.35 & 2.8 %\ $^{150}$Nd & 3.37 & 5.6 %\ $^{48}$Ca & 4.27 & 0.19 %\ \[tab:isotopes\] The $\alpha$ contribution to the background in the region (i.e. at about 3 MeV) is the following. In the natural chains we have various nuclei that decay emitting an $\alpha$ particle with an energy between 4 and 8 MeV, their energy is quite higher than most Q-values. However, if the radioactive nucleus is located at a depth of a few $\mu$m inside a material facing the detector, the $\alpha$ particle looses a fraction of its energy before reaching the detector and its energy spectrum looks as a continuum between 0 and 4-8 MeV [@EPJA]. A similar mechanism holds in the case of surface contaminations on the bolometer. This radioactive source plays a role in almost all detectors but it turns out to be particularly dangerous for fully active detectors, as in the case of bolometers. It can be efficiently identified and removed with active background suppression technique such as that conceived with scintillating bolometers. The development of a hybrid detector, able to discriminate $\alpha$ particles and optimized for searches, was the main purpose of our studies on scintillating bolometers. We tested several devices, differing mainly in the scintillating crystal material and size, to study their thermal response, light yield and radio purity. The results obtained so far are reported in [@CDWO4; @CAMOO4; @ZNSE; @ZNMOO4]. On our way, we discovered an extremely interesting feature of some of the tested crystals: the different pulse shape of the thermal signals produced by $\alpha$’s and by $\beta$/$\gamma$’s. This feature opens the possibility of realizing a bolometric experiment that can discriminate among different particles, without the need of a light detector coupled to each bolometer. In the case of a huge, multi-detector array, such as CUORE [@CUORE] and EURECA [@EURECA], the benefits of employing this technique would be impressive: - more ease during assembly because the single element of the array would be a quite simpler device. - fewer readout channels, with not only an evident reduction of cost and work, but also a cryogenic benefit (in a cryogenic experiment particular care should be devoted to reduce any thermal link between room temperature and the bolometers working at few mK: the heat load of the readout channels must be taken into account and their reduction is always a good solution). - a significant cost reduction, saving money and work that would be necessary for the light detectors procurement and optimization. - no need of light collectors, this would simplify the structure of the assembly and it would allow the use of coincidences between facing crystals to further reduce the background. As a final remark, it is worth to be mentioned that these devices could be used also for the measurement of $\alpha$ emissions from surfaces, when extremely low counting rates are needed. Indeed, due to their lack of a dead layer and their high energy resolution, bolometers have an extraordinary sensitivity to low range particles like $\alpha$’s. However, a conventional bolometer cannot distinguish the nature of the interacting particle. It provides therefore only a limited diagnostic power (especially for $\alpha$ particles with energies lower than 2615 keV where the $\beta$/$\gamma$ induced background dominates the detector counting rate). On the other hand, a scintillating bolometer has to be surrounded by a reflector to properly collect the scintillation light (therefore cannot be faced to a sample whose radioactive emission has to be identified). The devices here discussed overcome these two difficulties. Traditionally the devices used in this field are Si surface barrier detectors. For low counting rates, large area low background detectors are needed. Today Si surface barriers detectors with an active area of about 10 cm$^2$, a typical energy resolution of about 25-30 keV FWHM, and counting rates of the order 0.05 count/h/cm$^2$ between 3 and 8 MeV are available [@CANBERRA]. A bolometer like those here discussed can easily reach a much larger active area, has a typical energy resolution of 10 keV and a background counting rate in the 3-8 MeV region that can be as low as 0.001 count/h/cm$^2$. Thanks to the particle identification technique discussed in this paper, it can distinguish an $\alpha$ emission from a $\beta/\gamma$ one[^1] and finally can reject the $\beta$/$\gamma$ background extending its measurement field to energies by far lower than 3 MeV. In the following section we report the results obtained with the pulse shape analysis on some of the tested crystals. Detectors, set-up and data analysis =================================== The results discussed in this paper have been obtained operating different scintillating bolometers in an Oxford 200 $^{3}$He/$^{4}$He dilution refrigerator located deep underground, in the National Laboratory of Gran Sasso (L’Aquila, Italy). The rock overburden (average depth $\sim$3650 m.w.e. [@Hime]) ensures a strong suppression of cosmic rays that in our case is mandatory to be able to operate the detectors without an overwhelming pile-up. A detailed description of the experimental setup can be found in [@SETUP]. In order to study the pulse shape characteristics of different materials (in particular those of interest for ), we operated a number of scintillating bolometers differing for size, geometry and, of course, the absorbing material. As light detector we have used a second bolometer able to absorb scintillation photons converting their energy into heat. This was realised using as absorber Ge wafers of about 5 g, covered - on the side facing the scintillating crystal - with a 600 Å thick layer of SiO$_2$ in order to increase the light absorption. In this way they provided measurable thermal signals over an extremely large band of scintillation wavelengths. Both the scintillating crystal and the Ge wafer were equipped with a Neutron Transmutation Doped Ge thermistor (NTD) [@NTD], glued on the crystal surface and used as a thermometer to measure the heat or light signal produced by particles traversing the scintillating crystal. A silicon resistance, glued on the crystals, was used to produce a calibrated heat pulse in order to monitor the thermal gain of the bolometer. This is indeed subject to variation upon temperature drifts of the cryostat that can spoil the energy resolution. In most cases this temperature drift could be re-corrected off-line on the basis of the measured thermal gain variation [@ALES98; @ARNA03]. Read-out and DAQ ---------------- The read-out [@ELE] of the thermistor was performed via a preamplifier stage, a second stage of amplification and an antialiasing filter (a 6 pole roll-off active Bessel filter 120 db/decade [@Bessel]) located in a small Faraday cage. The ADC was a NI USB-6225 device (16 bit 40 differential input channels). For each triggered signal the entire waveform (*raw-pulse*) is sampled, digitized and acquired for the off-line analysis. Since all the relevant parameters (including the amplitude) of the triggered signals are evaluated off-line, a particular care has to be dedicated to the optimization of the signal filtering and digitization. In the case of the scintillating bolometer the large heat capacity of the absorber, coupled to the finite conductance of the crystal-glue-thermistor interface results in quite slow signals, characterised by a rise-time of the order of few ms[^2] and a decay-time of hundreds of ms (determined by the crystal heat capacity and by its thermal conductance toward the heat sink). Consequently the sampling rate typically used for the signal is 1-4 kHz, over a time window of 200-2000 ms. The Bessel filter acts mainly as antialising, to avoid spurious contributions in the sampled signal. Generally it is preferred to fix its cut-off frequency at the lowest value that does not deteriorate the signal to noise ratio (i.e. to obtain the best results in terms of energy resolution). This results to be a frequency of the order of 10 Hz, which is by far lower than what needed for antialiasing purposes. In the studies here presented the Bessel cut-off frequency was fixed at 120 Hz in order to exploit the maximum available information in the signal bandwidth. Analysis techniques {#AnaTech} ------------------- Off-line analysis aims at determining the pulse amplitude and energy together with several pulse shape parameters associated with each raw-pulse waveform recorded by the data acquisition system. Starting from these quantities the physical informations that are relevant for the scientific goals can be extracted. The first step of the analysis consists in the correct evaluation of the pulse amplitude. Since thermal pulses are superimposed to stochastic noise, a simple maximum-minimum algorithm would not give the better achievable resolution. We therefore use the Optimum Filter (OF) approach [@GATTI86]. This algorithm has proven to provide the best estimate of the pulse amplitude and, as a consequence, the best energy resolution. The basic concept is to build a filter that, when applied to the raw-pulse, produces - as output - a pulse with the best signal to noise ratio. The filtered pulse is then used to evaluate the signal amplitude. It can be proven that in the frequency domain the OF transfer function H( $\omega$ ) is given by $$\label{eq:OF} H(\omega) = K ~ \frac{S^*(\omega)}{N(\omega)} ~ e^{-j\omega t_M}$$ where S($\omega$) is the Fourier transform of the ideal thermal signal (reference pulse in the absence of noise), N($\omega$) is the noise power spectrum, $t_M$ is the delay of the current pulse with respect to the reference pulse and K is a proper normalising factor usually chosen in order to obtain the correct event energy. The role of the optimum filter is to weight the frequency components of the signal in order to suppress those frequencies that are more affected by noise. It can be seen from eq. \[eq:OF\] that, in order to build the filter, the shape of the reference pulse S($\omega$) and the noise power spectrum N($\omega$) must be known. S($\omega$) is usually estimated by averaging a large number of recorded raw-pulses, so that the noise associated with each of them averages to zero. N($\omega$) is obtained according to the Wiener-Khintchine theorem by acquiring many detector baselines in absence of thermal pulses and averaging the corresponding noise power spectra. Once the pulse amplitude has been evaluated, gain instability corrections are applied to data. Due to the dependence of the detector response on the working temperature, the same amount of released energy can produce thermal pulses of different amplitudes. Gain instabilities are corrected monitoring the time behaviour of thermal pulses of fixed energy, generated every few minutes across a Si heater resistor attached on the crystal absorber [@ALES98; @ARNA03]. Finally the amplitude to energy conversion (calibration) is determined by measuring the pulse amplitudes corresponding to fixed calibration lines. In the measurements here reported the signal of the scintillating bolometer (we will refer to this signal as to the heat or thermal signal) has been calibrated on the basis of the full energy peaks visible in the spectrum collected when the detector was exposed to an (external to the cryostat) source. These peaks have a nominal energy of: 511, 583, 911, 968 and 2615 keV. Below 511 keV and above 2615 keV the energy calibration is extrapolated. However, the heat response for $\alpha$ particles is slightly different from the $\beta$/$\gamma$ response in scintillating bolometers [@CDWO4]. For the molybdates that are here reported this heat quenching factor is lower than few percent while for ZnSe it is a little bit higher, about 10 percent for $\sim$6 MeV alpha particles [@ZNSE]. This miscalibration of the $\alpha$ band however does not imply an appreciable change in the discrimination confidence level described below. In the case of the light signal the energy calibration is not needed and we therefore present its value in arbitrary units. Besides the amplitude, few other characteristic parameters of the pulse are computed by the off-line analysis. Some of them are: $\tau_{rise}$ and $\tau_{decay}$, TVL and TVR. The rise-time ($\tau_{rise}$) and the decay-time ($\tau_{decay}$) are determined on the recorded raw-pulse as (t$_{90\%}$-t$_{10\%}$) and (t$_{30\%}$-t$_{90\%}$) respectively. TVR (Test Value Right) and TVL (Test Value Left) are computed on the optimally filtered pulse A(t). They are the root mean square differences between the current signal A(t) and the reference pulse after OF filtering A$_{0}$(t)= H(t) $\otimes$ S(t). In more detail, the filtered response function A$_{0}$(t) is synchronized with the filtered signal A(t), making their maxima to coincide, then the least square differences of the two functions are evaluated on the right (TVR) and left (TVL) side of the maximum on a proper time interval which is usually chosen depending on the shape of the OF signals. Although these two parameters do not have a direct physical meaning, however they are very sensitive (even in noisy conditions) to any difference between the shape of the analyzed pulse and the response function. Consequently, they are used either to reject fake triggered signals (e.g. spikes) or to identify variations in the pulse shape with respect to the reference response function (and this will be our case). {width="1.\linewidth"} Pulse shape signature in the heat pulse {#PSA} ======================================= A series of measurements was carried out, in which different scintillating bolometers, each coupled to a light detector (but in one case), were exposed to $\gamma$ and $\alpha$ sources. This allowed us to study the response of our devices to different radiations. The light signal was used to identify - on the basis of the heat to light ratio - the particle producing the event under study. As mentioned in section \[AnaTech\], for each triggered signal different pulse shape parameters are computed by the off-line analysis, generally to isolate spurious and pile-up events. In the case of scintillating bolometers, looking at the distribution of the pulse shape parameters for the heat signals, we realized that it was possible to distinguish $\beta$/$\gamma$ from $\alpha$ events. This is clearly evident in Fig. \[fig:camoo4\] where the scatter plot of the amplitudes measured for the light and heat signals (Light vs. Heat) is compared with the scatter plot (obtained for the same events) of the linearized rise-time (see later in the text) vs. amplitude for the heat signal ($\tau_{rise}^{lin}$ vs. Heat). In this detector, $\alpha$ and $\beta$/$\gamma$ interactions draw different distributions in both the scatter plots, definitely proving that the shape of the thermal pulse induced by an $\alpha$ particle is different from that of a $\beta$/$\gamma$ interaction. This behavior can be explained by the dependence of light yield on the nature of the interacting particle. The high ionization density of $\alpha$ particles implies that all the scintillation states along their path are occupied. This saturation effect does not occur or at least is much less for $\beta$/$\gamma$ particles. Therefore, in $\alpha$ interactions a larger fraction of energy flow in the heat channel with respect to $\beta$/$\gamma$ events. This leads not only to a different light and heat yield but also to a different time evolution of both signals. The pulse shape of the thermal signal then can be explained by the partition of energy in the two channels with different decay constants. In particular, as shown by [@LightDep1; @LightDep2] the scintillation produced in molybdates by $\alpha$ and $\beta$/$\gamma$ particles presents few decay-time constants with different relative intensities. Also a strong temperature dependence of the averaged decay-time of the light pulses in these same crystals was reported [@TimeDep1; @TimeDep2]. For this reason - while at room temperature the averaged decay-time of molybdate or tungstate scintillators is of the order of tens of $\mu$s (thus almost instantaneous in the heat pulse timescale) - at low temperatures it increases to hundreds of $\mu$s and is therefore comparable to the typical rise-time of the heat signal of our scintillating bolometers. In the following sections we analyse the results obtained with different crystals, each being a possible candidate for a experiment. We quote for each crystal the discrimination power achieved - in the region - between $\alpha$ and $\beta$/$\gamma$ particles on the basis of the light/heat ratio or simply on the basis of the pulse shape of the heat signal. We discuss in more detail the case of CaMoO$_4$, briefly summarizing the results obtained for other crystals. CaMoO$_{4}$ {#CaMoO4} =========== Recently CaMoO$_{4}$ has been intensively studied, for its possible application as a scintillating bolometer for and Dark Matter experiments [@Pirr06; @Seny06; @Mikh06-JPDAP]. This crystal contains two isotopes that could undergo : $^{48}$Ca (Q$_{\beta\beta}$=4.27 MeV) and $^{100}$Mo (Q$_{\beta\beta}$=3.03 MeV). Actually, while the large content of $^{100}$Mo makes this crystal very attractive, the presence of $^{48}$Ca is a problem. Indeed, the natural isotopic abundance of $^{48}$Ca (a.i.=0.19%) is too low to study the without enrichment, which is extremely difficult, and at the same time, it is too high to study the of $^{100}$Mo, since the background due to the of $^{48}$Ca in the region of $^{100}$Mo will limit the reachable sensitivity for the latter isotope. A possible solution to this problem was suggested by Annenkov et al. [@CAMOO4_DEP] who proposed an experiment with CaMoO$_{4}$ depleted in $^{48}$Ca. Despite these possible problems, we discovered that CaMoO$_{4}$ is an extremely interesting crystal because of its capability to discriminate $\beta$/$\gamma$ from $\alpha$, thanks to the different shape of the thermal pulses. The sample we used was a cylindric CaMoO$_{4}$ crystal with a mass of 158 g (h = 40mm, $\varnothing$ = 35mm). The crystal was faced to two $\alpha$ sources. Source A was obtained by implantation of $^{224}$Ra in an Al reflecting stripe. The shallow implantation depth allows to reduce to a minimum the energy released by the $\alpha$’s in the Al substrate so that monochromatic $\alpha$ lines (those produced in the decay chain of $^{224}$Ra to the stable $^{208}$Pb isotope) can be observed in the scintillating bolometer. Besides these $\alpha$ particles - all with energies above 5 MeV - the source emits a $\beta$ with a maximum energy of 5 MeV, due to the decay of . Since our main goal was to study the efficiency of $\alpha$ particle rejection in the region (i.e. at about 3 MeV), a second source (B) was also used. This was obtained contaminating an Al stripe with a liquid solution and later covering the stripe with an alluminated Mylar film (6 $\mu$m thick). Thus the source produced a continuous spectrum of $\alpha$ particles, extending from about 3 MeV down to 0. In Fig. \[fig:camoo4\] we show the Light vs. Heat scatter plot, collected while the crystal was exposed to an external source. The total live time of this measurement was $\sim$43 h. The FWHM energy resolution, on the heat signal, ranges from 2.7 keV at 243 keV to 8.7 keV at 2615 keV on $\beta$/$\gamma$ events and is about 10 keV on 5 MeV $\alpha$’s. The light yield of the crystal, evaluated calibrating the light detector with a $^{55}$Fe source, is 1.87 keV/MeV (for more details on the procedure used for the evaluation of the light yield see references [@CDWO4; @ZNSE]). The two separate bands - clearly visible in the Light vs. Heat scatter plot - are ascribed to $\beta$/$\gamma$’s (upper band) and $\alpha$’s (lower band). The upper band is dominated by the source $\gamma$’s (plus the environmental $\gamma$’s). The lower band is due to $\alpha$’s from source A and from Uranium and Thorium internal contamination of the crystal (these are the monochromatic lines above 4 MeV) and source B (the continuum counting rate below 4 MeV). Above 8 MeV we observe a group of events ascribed to $\alpha$+$\beta$ summed signals due to internal contamination in Bismuth and Polonium. Indeed, due to the long rise-time of this device (5 ms), the beta decay of $^{214}$Bi or $^{212}$Bi (respectively of $^{238}$U and $^{232}$Th chains) followed immediately by $\alpha$ decay of $^{214}$Po ($\tau$=164 $\mu$s) and $^{212}$Po ($\tau$=298 $\mu$s ), may lead to a pile up on the rise-time of the thermal pulses that can hardly be recognized as a double signal. The two decays produce therefore a single pulse, with an energy that is the sum of the two. The discrimination between the $\alpha$ and the $\beta$/$\gamma$ populations, provided by the scintillation signal, can be evaluated by measuring the difference between the average amplitude of the light signal produced by the two kinds of particles (Light$_{\beta/\gamma}$ and Light$_{\alpha}$), considering a group of events releasing the same energy in the scintillating crystal. This difference is then compared with the width of the two distributions ($\sigma_{\beta/\gamma}$ and $\sigma_{\alpha}$). The discrimination confidence level D$_{Light}$ can be then defined as: $$D_{Light} = \frac{Light_{\beta/\gamma}-Light_{\alpha}}{\sqrt{\sigma_{\beta/\gamma}^2+\sigma_{\alpha}^2}}$$ To evaluate this discrimination power in the region we selected events belonging to the 2615 keV $\gamma$-line full energy peak and compared their light pulse distribution with that of events of similar energy in the $\alpha$ band (Fig. \[fig:camoo4\_light\] left panel). D$_{light}$ results to be 12.6 sigma. The reason for using events with the same energy to evaluate D$_{light}$ is in the energy dependence of the distance (and width) of the $\alpha$ and $\beta$/$\gamma$ bands, that induce an energy dependence of the discrimination confidence level. In the energy range where both the bands are populated (i.e. between 1 and 3 MeV), D$_{Light}$(E) appears to be linearly decreasing with energy. Its extrapolated value at E=0 being 3 sigma. As already anticipated, we discovered that the heat signal shape is enough to discriminate $\beta$/$\gamma$’s from $\alpha$’s. This is shown in the right panel of Fig. \[fig:camoo4\] where we report the (thermal pulse) $\tau_{rise}^{lin}$ vs. Heat scatter plot for the same events shown in the left panel. Two separate bands, ascribed to $\beta$/$\gamma$ and $\alpha$ events can be identified. The former with an average rise-time of 5.8 ms, the latter with 5.6 ms. In this plot the weak energy (Heat) dependence of $\tau_{rise}$ observed for the two populations was corrected by fitting their rise-time distributions with two lines having the same slope. To do this we used $\beta$/$\gamma$ events in the 0.5-2.6 MeV range and $\alpha$ events in the 1.5-7 MeV range. The $\tau_{rise}$ is then linearized re-defining it as $\tau_{rise}^{lin}=\tau_{rise} - slope \times E$ with $slope$=0.0075 ms/MeV. In order to emphasize the correspondence in the identification of $\beta$/$\gamma$ and $\alpha$ events, signals selected in the Light vs. Heat scatter plot as having an energy between 2 and 4 MeV and belonging to the $\beta$/$\gamma$ or $\alpha$ bands, are reported in different colors. We will refer in the following to these two groups of events as to 2-4 MeV $\gamma$’s and 2-4 MeV $\alpha$’s, although - above the 2615 keV line - the $\gamma$ group is empty. These two samples will be used to evaluate the difference in shape among $\beta$/$\gamma$ and $\alpha$ signals. In Fig. \[fig:Impulso\] we compare two pulses obtained by averaging - separately - signals belonging to the 2615 keV $\gamma$-line and signals due to $\alpha$ particles that release a similar energy. The average is here needed to get rid of the noise that can mask the small differences among the pulses. The difference in shape can be appreciated in the bottom panel of Fig. \[fig:Impulso\]. The discriminating power D$_{RiseTime}$ of the heat pulse shape method can be defined exactly as done for D$_{Light}$: $$D_{RiseTime}(E) = \frac{RiseTime_{\beta/\gamma}-RiseTime_{\alpha}}{\sqrt{\sigma_{\beta/\gamma}^2+\sigma_{\alpha}^2}}$$ where, however, the dependence on energy is here due only to the variation of the width of the rise-time distributions since the difference in rise-time appears to be independent on energy. The distribution obtained projecting the rise-time of 2-4 MeV $\gamma$’s and 2-4 MeV $\alpha$’s is shown in Fig. \[fig:camoo4\_light\] (right panel). A gaussian fit of the two peaks yields a rise-time of (5.788$\pm$0.017) ms for $\gamma$’s and of (5.649$\pm$0.013) ms for $\alpha$’s. D$_{RiseTime}$ results equal to 6.5 sigma. As for the D$_{Light}$ case, we can extrapolate the D$_{RiseTime}$ value also in the energy region where the $\alpha$ band is poorly populated, simply assuming that the distance between the two bands remains constant (as we observe) and D$_{RiseTime}$ changes because the width of the two distributions becomes larger, at low energies, due to noise. The result is that D$_{RiseTime}$ becomes lower than 2 sigma at 500 keV. In other words the 150 $\mu$s difference in rise-time of $\alpha$ and $\beta$/$\gamma$ events cannot be appreciated when the pulse amplitude is too low and, therefore, the noise modifies appreciably the signal shape. Finally, although in this measurement the rise-time is the most efficient shape parameter for $\alpha$ event discrimination, good discrimination levels were observed also in other parameters such as the decay-time and the TVR. The possibility to increase the discrimination power by combining the information from different shape parameters or the fit on the rise-time is under study. We note that either in the Light vs. Heat scatter plot as in the $\tau_{rise}^{lin}$ vs. Heat one we can see some outliers that could indicate a possible failure of the particle identification technique. However such events can be accounted for if one considers the following two effects. If a $\gamma$-ray interacts both in the light detector and in the scintillator the light signal that is read-out has a wrong amplitude since it is not only ascribed to scintillating photons but also to a direct $\gamma$ interaction in the Ge wafer. The measurement is performed in a high rate condition, therefore we expect to have a number of pile-up events in each of the two detectors, this leads to an erroneous evaluation of the pulse amplitude and pulse rise-time. To conclude, in the case of decay the discrimination power provided by the use of the scintillation signal is comparable with that provided by the pulse shape analysis. For what concerns the use of this device for the measurements of $\alpha$ particle emissions from an external sample, a D$_{RiseTime}$ better than 2 sigma above 500 keV means that this detector has a good sensitivity even in the region where Si surface barrier detectors start to be dominated by $\gamma$ background. On the contrary the applicability of this technique to Dark Matter searches cannot be proved directly. Other crystals -------------- Other scintillating crystals have shown the possibility to discriminate interacting particles through the thermal pulse shape differences. Among the tested crystals, other molybdates (ZnMoO$_{4}$, MgMoO$_{4}$) and also other crystals such as ZnSe showed a good discrimination power. ZnMoO$_{4}$ ----------- We tested a 19.8 g ZnMoO$_{4}$ crystal, having the shape of a prism with height of 11mm and a regular hexagonal base, with a diagonal of 25 mm [@ZNMOO4]. The FWHM energy resolution, on the heat signal, is 4.2 keV FWHM on $\beta$/$\gamma$ events at 2615 keV, and about 6 keV on the 5.4 MeV $\alpha$ line. The light yield of the crystal is 1.1 keV/MeV. The total live time of this measurement was $\sim$195 h. Also for this analysis we have used the Light vs. Heat scatter plot to select $\beta$/$\gamma$ and $\alpha$ between 2 and 4 MeV, in order to evaluate the discrimination power provided by pulse shape studies. Unlike for the CaMoO$_4$ case where the most efficient parameter for the discrimination is the rise-time, this measurement showed a higher discrimination power on the decay-time of the thermal pulse. In order to emphasize the discrimination power, all signals have been fitted with a function obtained as the sum of two exponentials: $$\label{eq:vs. energy} \Delta V(t) = (e^{ - t/ \tau_1 + A_1} + e^{ - t/ \tau_2 + A_2})$$ where the $\tau_1$ and $\tau_2$ parameters are obtained by fitting the decay of the thermal signals in the raw-pulse acquired for each event. It was observed that the best discrimination power is obtained by the ratio of the two decay constants (RDC): $$\label{RDC_def} RDC = \frac{\tau_1}{\tau_2}$$ The scatter plot of RDC vs. Heat is reported in the right panel of Fig. \[fig:znmoo4\] (in the left panel the corresponding Light vs. Heat scatter plot). The discrimination between $\alpha$ and $\beta$/$\gamma$ populations provided by RDC is evaluated linearizing the RDC vs. Heat relationship (fitting the $\alpha$ band in scatter plot of Fig. \[fig:znmoo4\] with a polynomial as done for the $\tau_{rise}$ of CaMoO$_4$) and then projecting the distribution of RDC$^{lin}$ for events in the 2-4 MeV range. The two peaks (${\beta/\gamma}$ and ${\alpha}$) are then fitted with a gaussian (Fig. \[fig:ZnMoO4\_RDC\_Proj\]). The discrimination power D$_{RDC}$ is 6.4 sigma. MgMoO$_{4}$ ----------- This compound contains, as in the two previous cases, the active isotope $^{100}$Mo which is here present in a larger concentration (52% in mass). The crystal tested is 32x31x24 mm$^3$ with a weight of 89.1 g. The total live time of this measurements was $\sim$22 h. In this run it was not possible to face the crystal to a light detector because of the assembly structure so we can’t use the Light vs. Heat scatter plot in order to tag $\alpha$ events. The performances of the bolometer were quite poor, most probably this was due to a problem with the gluing of the NTD thermistor: at the end of the measurement, when the crystal was back to room temperature, we discovered that under the thermistor the crystal showed a crack. This could explain why the signal to noise ratio was so bad (the energy resolution measured on $\alpha$ lines was 150 keV FWHM) and consequently also the resolution in the evaluation of the signal shape parameters was limited. Despite these problems, we were able to observe in the scatter plot of RDC$^{lin}$ vs. Heat (fig. \[fig:MgMoO4\_RT\]) clear difference between events due to the interaction of $\alpha$ particles and events due to $\beta$/$\gamma$ events. The discrimination between the two populations provided by the RDC parameter can be evaluated by means of a gaussian fit of the two peaks obtained by projecting selected events after linearization. The resulting discrimination power D$_{RDC}$ is 1.8 sigma. These preliminary results, even if limited because of the problems reported above, lead us to program new more detailed measurements to better study this promising crystal. ZnSe ---- $^{82}Se$ is a emitter with an isotopic abundance of 9.2% and a Q-value of (2995.5 $\pm$ 2.7) keV. It has always been considered a good candidate for studies because of its high transition energy and the favorable nuclear factor of merit. For these reasons in the last years an R&D work was carried out in which we have extensively studied the performances of ZnSe detectors in different conditions. For our studies we have used different ZnSe crystals. Characteristics of measurements done and obtained results are reported in detail in [@ZNSE]. Here we report the discrimination power on the RDC obtained with the Huge ZnSe crystal (h = 50mm, $\varnothing$ = 40mm, 337g). In Fig. \[fig:znse\] the Light vs. Heat scatter plot of a measurement of $\sim$70 h of live time is shown. In order to have a high number of $\alpha$ counts in the 2-3 MeV region also in these measurements a degraded $^{238}$U source was placed in front of the crystal. An explanation of the Quenching Factor larger than one (i.e. the $\alpha$ band lies above the $\beta$/$\gamma$ band) and other anomalies observed in this crystal can be found in [@ZNSE]. Also in this case, the RDC$^{lin}$ vs. Heat (fig. \[fig:znse\_RDC\]) showed a difference between events due to the interaction of $\alpha$ particles and events due to $\beta$/$\gamma$ events. The discrimination power D$_{RDC}$ is 2.2 sigma. Conclusion ========== The possibility to discriminate the nature of the particle interacting in a bolometric detector, simply on the base of the shape of the thermal pulse, is now definitely proved and opens new possibilities for the application of these devices in the field of rare events searches. In particular, the high rejection capability that could allow to completely rule out the $\alpha$ background in experiments was demonstrated. Unfortunately, based on the results obtained so far, the applicability of this technique to Dark Matter searches cannot yet be proved directly. This feature was observed in different scintillating crystals (CaMoO$_4$, ZnMoO$_4$, MgMoO$_4$ and ZnSe) and new tests are under preparation in order to investigate if a similar behavior can be observed also in other compounds. A discrimination confidence level of $\sim$6.5 sigma was obtained both for CaMoO$_4$ and ZnMoO$_4$ crystals in the 2-4 MeV energy region. Discrimination confidence levels reached with MgMoO$_4$ (D$_{RDC}$=1.8$\sigma$) and ZnSe (D$_{RDC}$=2.2$\sigma$) are not so high but they indicate that even with these crystals the discrimination based on pulse shape analysis is possible. New techniques aiming at improving the discrimination power of the pulse shape analysis are being studied. Acknowledgments =============== This work was funded and developed under the Bolux experiment of INFN. Thanks are due to E. Tatananni, A. Rotilio, A. Corsi and B. Romualdi for continuous and constructive help in the overall setup construction. Finally, we are especially grateful to Maurizio Perego for his invaluable help in the development and improvement of the Data Acquisition software. [10]{} V. I. Tretyak and Yu. G. Zdesenko, Atomic Data and Nuclear Data Tables, **80 (2002) 83 S. Elliott and P. Vogel, Ann. Rev. Nucl. Part. Sci. **52 (2002) 115 A. Morales and J. Morales, Nucl. Phys. B (Proc. Suppl.) **114 (2003) 141 F. T. Avignone III, S. R. Elliott and J. Engel (2006), Rev. Mod. Phys. **80 (2008) 481 G. Jungman, M. Kamionkowski, K. Griest, Physics Reports **267 (1996) 195 J.R. Primack, B. Sadoulet and D. Seckel, Annu. Rev. Nucl. Part. Sci. B **38 (1988) 751 P.F. Smith and J.D. Lewin, Phys. Rep. **187 (1990) 203 Proc. of the 5th Int. 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Sci **57 (2010) 1475************ [^1]: This feature is extremely important when the sample which is investigated produces a continuous counting rate which could be due either to $\alpha$ emissions from a thick contamination or to a $\beta/\gamma$ continuum. [^2]: The minimum observable signal rise-time is limited by the integration on the parasitic capacitance of the signal wires that connect the NTD thermistor to the front-end electronics.

ArXiv

--- abstract: 'In this paper, I address the oscillation probability of $O$(GeV) neutrinos of all active flavours produced inside the Sun and detected at the Earth. Flavours other than electron-type neutrinos may be produced, for example, by the annihilation of WIMPs which may be trapped inside the Sun. In the GeV energy regime, matter effects are important both for the “1–3” system and the “1–2” system, and for different neutrino mass hierarchies. A numerical scan of the multidimensional three-flavour parameter space is performed, “inspired” by the current experimental situation. One important result is that, in the three-flavour oscillation case, $P_{\alpha\beta}\neq P_{\beta\alpha}$ for a significant portion of the parameter space, even if there is no $CP$-violating phase in the MNS matrix. Furthermore, $P_{\mu\mu}$ has a significantly different behaviour from $P_{\tau\tau}$, which may affect expectations for the number of events detected at large neutrino telescopes.' --- CERN-TH-2000-168\   hep-ph/0006157\ .3in [**The Oscillation Probability of GeV Solar Neutrinos of All Active Species**]{} 0.5in André de Gouvêa 0.1in [*CERN - Theory Division\ CH-1211 Geneva 23, Switzerland*]{} .2in Introduction ============ In the Standard Model of particle physics, neutrinos are strictly massless. Any evidence for neutrino masses would, therefore, imply physics beyond the Standard Model. Even though the direct experimental determination of a neutrino mass is (probably) far beyond the current experimental reach, experiments have been able to obtain indirect, and recently very strong, evidence for neutrino masses, via neutrino oscillations. The key evidence for neutrino oscillations comes from the angular dependent flux of atmospheric muon-type neutrinos measured at SuperKamiokande [@atmospheric], combined with a large deviation of the muon-type to electron-type neutrino flux ratio from theoretical predictions. This “atmospheric neutrino puzzle” is best solved by assuming that $\nu_{\mu}$ oscillates into $\nu_{\tau}$ and that the $\nu_e$ does not oscillate. For a recent analysis of all the atmospheric neutrino data see [@atmos_analysis]. On the other hand, measurements of the solar neutrino flux [@Cl; @Kamiokande; @GALLEX; @SAGE; @Super-K] have always been plagued by a large suppression of the measured solar $\nu_e$ flux with respect to theoretical predictions [@SSM]. Again, this “solar neutrino puzzle” is best resolved by assuming that $\nu_e$ oscillates into a linear combination of the other flavour eigenstates [@bksreview; @rate_analysis] (for a more conservative analysis of the event rates and the inclusion of the “dark side” of the parameter space, see [@dark_side]). The most recent analysis of the solar neutrino data which includes the mixing of three active neutrino species can be found in [@solar_3]. Neutrino oscillations were first hypothesised by Bruno Pontecorvo in the 1950’s [@Pontecorvo]. The hypothesis of three flavour mixing was first raised by Maki, Nakagawa and Sakata [@MNS]. In light of the solar neutrino puzzle, Wolfenstein [@W] and Mikheyev and Smirnov [@MS] realized that neutrino–matter interactions could affect in very radical ways the survival probability of electron-type neutrinos which are produced in the solar core and detected at the Earth (MSW effect). Since then, significant effort has been devoted to understanding the oscillation probabilities of electron-type neutrinos produced in the Sun. For example, in [@KP_3] the survival probability of solar electron-type neutrinos was discussed in the context of three-neutrino mixing including matter effects, and solutions to the solar neutrino puzzle in this context were studied (for example, in [@KP_3; @MS_3; @solar_3]). In this paper, the understanding of solar neutrino oscillations is extend to the case of other active neutrino species ($\nu_{\mu}$, $\nu_{\tau}$, and antineutrinos) produced in the solar core. Even though only electron-type neutrinos are produced by the nuclear reactions which take place in the Sun’s innards, it is well know that, in a number of dark matter models, dark matter particles can be trapped gravitationally inside the Sun, and that the annihilation of these should yield a flux of high energy neutrinos ($E_{\nu}\gtrsim 1$ GeV) of all species which may be detectable at the Earth [@DM_review]. Indeed, this is one of the goals of very large “neutrino telescopes,” such as AMANDA [@Amanda] or BAIKAL [@Baikal]. It is important to understand how neutrino oscillations will affect the expected event rates at these experiments.[^1] The oscillation probability of all neutrino species has, of course, been studied in different contexts, such as in the case of neutrinos produced in the core of supernovae [@supernova] or in the case of neutrinos propagating in constant electron number densities [@barger_etal]. The latter case has been receiving a considerable amount of attention from neutrino factory studies [@nufact]. The case at hand (GeV solar neutrinos) differs significantly from these mentioned above, in at least a few of the following: source-detector distance, electron number density average value and position dependency, energy average value and spectrum. Neutrino factory studies, for example, are interested in $O$(1000) km base-lines, $O$(10) GeV electron-type and muon-type neutrinos produced via muon decay propagating in roughly constant, Earth-like (matter densities around 3 g/cm$^3$) electron number densities. The paper is organised as follows. In Sec. 2, the well known case of two-flavour oscillations is reviewed in some detail, while special attention will be paid to neutrinos produced inside the Sun. In Sec. 3 the same discussion is extended to the less familiar case of three-flavour oscillations. Again, special attention is paid to neutrinos produced in the Sun’s core. In Sec. 4 the results presented in Sec. 3 will be analysed numerically, and the three-neutrino multi-dimensional parameter space will be explored. Sec. 5 contains a summary of the results and the conclusions. It is important to comment at this point that one of the big challenges of studying three-flavour oscillations is the multi-dimensional parameter space, composed of three mixing angles, two mass-squared differences, and one complex phase, plus the neutrino energy. For this reason, the discussions presented here will take advantage of the current experimental situation to constrain the parameter space, and of the possibility of producing neutrinos of all species via dark matter annihilations to constrain the neutrino energies to the range from a few to tens of GeV. Two-Flavour Oscillations ======================== In this section, the well studied case of two-flavour oscillations will be reviewed [@general_review]. This is done in order to present the formalism which will be later extended to the case of three-flavour oscillations and describe general properties of neutrino oscillations and of neutrinos produced in the Sun’s core. Generalities ------------ Neutrino oscillations take place because, similar to what happens in the quark sector, neutrino weak eigenstates are different from neutrino mass eigenstates. The two sets are related by a unitary matrix, which is, in the case of two-flavour mixing, parametrised by one mixing angle $\vartheta$.[^2] $$\left(\matrix{\nu_{e} \cr \nu_{x} }\right)= \left(\matrix{U_{e1}&U_{e2}\cr U_{x1}&U_{x2}}\right) \left(\matrix{\nu_{1} \cr \nu_{2} }\right)= \left(\matrix{\cos\vartheta&\sin\vartheta\cr -\sin\vartheta&\cos\vartheta}\right) \left(\matrix{\nu_{1} \cr \nu_{2} }\right),$$ where $\nu_1$ and $\nu_2$ are neutrino mass eigenstates with masses $m_1$ and $m_2$, respectively, and $\nu_x$ is the flavour eigenstate orthogonal to $\nu_e$. All physically distinguishable situations can be obtained if $0\leq\vartheta\leq\pi/2$ and $m_1^2\leq m_2^2$ or $0\leq\vartheta\leq\pi/4$ and no constraint is imposed on the masses-squared. In the case of oscillations in vacuum, it is trivial to compute the probability that a neutrino produced in a flavour state $\alpha$ is detected as a neutrino of flavour $\beta$, assuming that the neutrinos are ultrarelativistic and propagate with energy $E_{\nu}$: $$P_{\alpha\beta}=|U_{\beta1}|^2|U_{\alpha1}|^2+|U_{\beta2}|^2|U_{\alpha2}|^2+2Re\left( U_{\beta1}^*U_{\beta2}U_{\alpha1}U_{\alpha2}^* e^{i\frac{\Delta m^2x}{2E_{\nu}}}\right).$$ Here $\Delta m^2\equiv m^2_{2}-m_1^2$ is the mass-squared difference between the two mass eigenstates and $x$ is the distance from the detector to the source. It is trivial to note that $P_{\alpha\beta}=P_{\beta\alpha}$ since all $U_{\alpha i}$ are real and the theory is $T$-conserving. Furthermore, note that $\vartheta$ is indistinguishable from $\pi/2-\vartheta$ (or, equivalently, the sign of $\Delta m^2$ is not physical), and all physically distinguishable situations are obtained by allowing $0\leq\vartheta\leq\pi/4$ and choosing a fixed sign for $\Delta m^2$. In the case of nontrivial neutrino–medium interactions, the computation of $P_{\alpha\beta}$ can be rather involved. Assuming that the neutrino–medium interactions can be expressed in terms of an effective potential for the neutrino propagation, one has to solve $$\frac{\rm d}{{\rm d}t}\left(\matrix{\nu_{1}(t) \cr \nu_{2}(t) }\right)= -i\left[\left(\matrix{E_1 & 0 \cr 0 & E_2}\right)+\left(\matrix{V_{11}(t) & V_{12}(t) \cr V_{12}(t)^* & V_{22}(t)}\right)\right]\left(\matrix{\nu_{1}(t) \cr \nu_{2}(t) }\right),$$ with the appropriate boundary conditions (either a $\nu_e$ or a $\nu_x$ as the initial state, for example). In the ultrarelativistic limit one may approximate $E_2-E_1\simeq \Delta m^2/2E_{\nu}$, ${\rm d}/{\rm d}t=\simeq{\rm d}/{\rm d}x$, and $V_{ij}(t)\simeq V_{ij}(x)$. A very crucial assumption is that there is no kind of neutrino absorption due to the neutrino–medium interaction, [*i.e.,*]{} the $2\times 2$ Hamiltonian for the neutrino system is Hermitian. It is interesting to argue what can be said about $P_{\alpha\beta}$ in very general terms. First, the conservation of probability requires that $$\begin{aligned} P_{ee}+P_{ex}&=&1, \\ P_{xe}+P_{xx}&=&1.\end{aligned}$$ Second, given that the Hamiltonian evolution is unitary, $$P_{ee}+P_{xe}=1. \label{extra_constraint}$$ It is easy to show that the extra constraint $P_{ex}+P_{xx}=1$ is redundant. Eq. (\[extra\_constraint\]) can be understood by the following “intuitive” argument: if the same amount of $\nu_e$ and $\nu_x$ is produced, independent of what happens to $\nu_e$ and $\nu_x$ during flight, the number of $\nu_e$ and $\nu_x$ detected in the end has to be the same. In light of the constraints above, one can show that there is only one independent $P_{\alpha\beta}$, which is normally chosen to be $P_{ee}$. The others are given by $P_{ex}=P_{xe}=1-P_{ee}$ and $P_{xx}=P_{ee}$. Note that the equality $P_{ex}=P_{xe}$ is [*not*]{} a consequence of $T$-invariance, but a consequence of the unitarity of the Hamiltonian evolution and particular only to the two-flavour oscillation case, as will be shown later. Oscillation of Neutrinos Produced in the Sun’s Core --------------------------------------------------- It is well known [@W; @MS] that neutrino–Sun interactions affect the oscillation probabilities of neutrinos produced in the Sun’s core in very nontrivial ways. Indeed, all but one solution to the solar neutrino puzzle rely heavily on neutrino–Sun interactions [@bksreview; @rate_analysis; @dark_side]. The survival probability of electron-type solar neutrinos has been computed in many different approximations by a number of people over the years, and can be understood in very simple terms [@general_review]. In the presence of electrons, the differential equation satisfied by the two neutrino system is, in the flavour basis, $$\frac{\rm d}{{\rm d}x}\left(\matrix{\nu_{e}(x) \cr \nu_{x}(x) }\right)=-i\left[ \frac{\Delta m^2}{2E_{\nu}}\left(\matrix{|U_{e2}|^2 & U_{e2}^*U_{\mu2} \cr U_{e2}U_{\mu2}^* & |U_{\mu2}|^2 }\right) +\left(\matrix{A(x) & 0 \cr 0 & 0}\right)\right] \left(\matrix{\nu_{e}(x) \cr \nu_{x}(x) }\right), \label{eq_2ns}$$ where terms proportional to the $2\times 2$ identity matrix were neglected, since they play no role in the physics of neutrino oscillations. $$A(x)=\sqrt{2}G_FN_e(x) \label{A(x)}$$ is the charged current contribution to the $\nu_e$-$e$ forward scattering amplitude, $G_F$ is Fermi’s constant, and $N_e(x)$ is the position dependent electron number density. In the case of the Sun [@SSM] (see also [@bahcall_www]), $A\equiv A(0)\simeq 6\times 10^{-3}$ eV$^2$/GeV, assuming an average core density of 79 g/cm$^3$, and $A(x)$ falls roughly exponentially until close to the Sun’s edge. It is safe to say that significantly far away from the Sun’s edge $A(x)$ is zero. A particularly simple way of understanding the propagation of electron-type neutrinos produced in the Sun’s core to the Earth is to start with a $\nu_e$ state in the basis of the eigenstates of the Hamiltonian evaluated at the production point, $|\nu_e\rangle=cos\vartheta_M(0)|\nu_L\rangle+\sin\vartheta_M(0)|\nu_H\rangle$, where $|\nu_H\rangle$ ($|\nu_L\rangle$) correspond to the highest (lowest) instantaneous Hamiltonian eigenstate. The matter mixing angle $\vartheta_M\equiv\vartheta_M(0)$ is given by $$\cos 2\vartheta_M=\frac{\Delta m^2\cos 2\vartheta-2E_{\nu}A} {\sqrt{(\Delta m^2)^2+A^2-4E_{\nu}A\Delta m^2\cos 2\vartheta}}. \label{cos2tm}$$ The evolution of this initial state from the Sun’s core is described by an arbitrary unitary matrix until the neutrino reaches the Sun’s edge. From this point on, one can rotate the state to the mass basis and follow the vacuum evolution of the state. Therefore, $P_{ee}(x)$, where $x$ is is the distance from the Sun’s edge to some point far away from the Sun (for example, the Earth), is $$P_{ee}(x)=\left|\left(\matrix{U_{e1}^* & U_{e2}^*}\right)\left(\matrix{1 & 0 \cr 0 & e^{-i\frac{\Delta m^2x}{2E_{\nu}}}}\right)\left(\matrix{A & B \cr -B^* & A^*}\right) \left(\matrix{\cos\vartheta_M \cr \sin\vartheta_M}\right)\right|^2, \label{Peex}$$ where overall phases in the amplitude have been neglected. The matrix parametrised by $A,B$ represents the evolution of the system from the Sun’s core to vacuum, and also rotates the state into the mass basis.[^3] Expanding Eq. (\[Peex\]), and assuming that there is no coherence in the Sun’s core between $\nu_L$ and $\nu_H$,[^4] one arrives at the well known expression (these have been first derived using a different language in [@Petcov_eq] and [@PP]) $$P_{ee}(x)=P_1\cos^2\vartheta + P_2\sin^2\vartheta -\cos 2\vartheta_M \sqrt{P_c(1-P_c)}\sin 2\vartheta\cos\left(\frac{\Delta m^2x}{2E_{\nu}}+\delta \right), \label{pee}$$ where $\delta$ is the phase of $AB^*$, $P_c\equiv |B|^2=1-|A|^2$ is the “level crossing probability”, and $P_1=1-P_2=\frac{1}{2}+\frac{1}{2}\left(1-2P_c\right)\cos 2\vartheta_M$ is interpreted as the probability that the neutrino exits the Sun as a $\nu_1$. Eq. (\[pee\]) should be valid in all cases of interest, and contains a large amount of features. In the case of the solar neutrino puzzle, the neutrino energies of interest range between hundreds of keV to ten MeV, and matter effects start to play a role for values of $\Delta m^2$ as high as $10^{-4}$ eV$^2$. In the adiabatic limit ($P_c\rightarrow 0$) very small values of $P_{ee}$ are attainable when $\cos 2\vartheta_M\rightarrow -1$ and $\sin^2\vartheta$ is small. More generally, in this limit $P_{ee}=\sin^2\vartheta$. This is what happens for all solar neutrino energies in the case of the LOW solution,[^5] for solar neutrino energies above a few MeV in the case of the LMA solution, and for 400 keV $\lesssim E_{\nu}\lesssim 1$ MeV energies in the case of the SMA solution. In the extremely nonadiabatic limit, which is reached when $\Delta m^2/2E_{\nu}\ll A$, $P_c\rightarrow \cos^2\vartheta$ and $\cos 2\vartheta_M\rightarrow -1$, the original vacuum oscillation expression is obtained, up to the “matter phase” $\delta$. This is generically what happens in the VAC solution to the solar neutrino puzzle. If the electron number density is in fact exponential, one can solve Eq. (\[eq\_2ns\]) exactly [@exponential; @PC]. For $N_e(x)=N_e(0)~e^{-x/r_0}$, where $x=0$ is the centre of the Sun, $$P_c=\frac{e^{-\gamma\sin^2\vartheta}-e^{-\gamma}}{1-e^{-\gamma}}, \label{pc}$$ [@PC; @check] where $$\gamma=2\pi r_0\frac{\Delta m^2}{2E_{\nu}}=1.05\left(\frac{\Delta m^2} {10^{-6}~{\rm eV}^2}\right)\left(\frac{1~{\rm GeV}}{E_{\nu}}\right), \label{gamma}$$ for $r_0=R_{\odot}/10.54=6.60\times 10^4$ km [@bahcall_www]. In the case of the Sun, the exponential profile approximation has been examined [@check], and was shown to be very accurate, especially if one allows $r_0$ to vary as a function of $\Delta m^2/2E_{\nu}$. The exact expression for $\delta$ has also been obtained [@Petcov_eq], and the readers are referred to [@P_phase] for details concerning physical implications of the matter phase. Its effects will not be discussed here any further. The Case of Antineutrinos ------------------------- Antineutrinos that are produced in the Sun’s core obey a differential equation similar to Eq. (\[eq\_2ns\]), except that the sign of the matter potential changes, [*i.e.*]{} $A(x)\leftrightarrow -A(x)$, and $U_{\alpha i}\leftrightarrow U^*_{\alpha i}$ (this is immaterial since, in the two-flavour mixing case, all $U_{\alpha i}$ are real). Instead of working out the probability of an electron-type antineutrino being detected as an electron-type antineutrino $P_{\bar{e}\bar{e}}$ from scratch, there is a very simple way of relating it to $P_{ee}$. One only has to note that, if the following transformation is applied to Eq. (\[eq\_2ns\]): $\vartheta\rightarrow \pi/2-\vartheta$, subtract the matrix $A(1_{2\times 2})$, where $1_{2\times 2}$ is the $2\times 2$ identity matrix and relabel $\nu_e(x)\leftrightarrow \nu_x(x)$, the equation of motion for antineutrinos is obtained.[^6] Therefore, $P_{\bar{e}\bar{e}}(\vartheta)= P_{xx}(\pi/2-\vartheta)=P_{ee}(\pi/2-\vartheta)$ (this was pointed out in [@Chizhov]). Remember that, in the case of vacuum oscillations, $\vartheta$ is physically equivalent to $\pi/2-\vartheta$, so $P_{\bar{e}\bar{e}}=P_{ee}$. In the more general case of nontrivial matter effects, this is clearly not the case, since the presence of matter (or antimatter) explicitly breaks $CP$-invariance. It is curious to note that, in the case of two-flavour oscillations, there is no $T$-noninvariance, [*i.e.,*]{} $P_{\alpha\beta}=P_{\beta\alpha}$, while there is potentially large $CP$ violation, [*i.e.,*]{} $P_{\alpha\beta}\neq P_{\bar{\alpha}\bar{\beta}}$, even if the Hamiltonian for the system is explicitly $T$-noninvariant and $CP$-noninvariant, as is the case of the propagation of neutrinos produced in the Sun (namely $A(t)$ is a generic function of time and $A(t)$ for neutrinos is $-A(t)$ for antineutrinos). Three Flavour Oscillations ========================== Currently, aside from the solar neutrino puzzle, there is an even more convincing evidence for neutrino oscillations, namely the suppression of the muon-type neutrino flux in atmospheric neutrino experiments [@atmospheric]. This atmospheric neutrino puzzle is best solved by $\nu_{\mu}\leftrightarrow\nu_{\tau}$ oscillations with a large mixing angle [@atmos_analysis]. Furthermore, the values of $\Delta m^2$ required to solve the atmospheric neutrino puzzle are at least one order of magnitude higher than the values required to solve the solar neutrino puzzle. For this reason, in order to solve both neutrino puzzles in terms of neutrino oscillations, three neutrino families are required. In this section, the oscillations of three neutrino flavours will considered. In order to simplify the discussion, I will concentrate on neutrinos with energies ranging from a few to tens of GeV, which is the energy range expected for neutrinos produced by the annihilation of dark matter particles which are possibly trapped inside the Sun. Furthermore, a number of experimentally inspired constraints on the neutrino oscillation parameter space will be imposed, as will become clear later. Generalities ------------ Similar to the two-flavour case, the “mapping” between the flavour eigenstates, $\nu_e$, $\nu_{\mu}$ and $\nu_{\tau}$ and the mass eigenstates $\nu_i$, $i=1,2,3$ with masses $m_i$ can be performed with a general $3\times 3$ unitary matrix, which is parametrised by three mixing angles ($\theta$, $\omega$, and $\xi$) and a complex phase $\phi$. In short hand notation $\nu_{\alpha}=U_{\alpha i}\nu_{i}$ where $\alpha=e,\mu,\tau$ and $i=1,2,3$. The MNS mixing matrix [@MNS] will be written, similar to the standard CKM quark mixing matrix [@PDG], as [$$\left(\matrix{U_{e1} & U_{e2} & U_{e3} \cr U_{\mu1} & U_{\mu2} & U_{\mu3} \cr U_{\tau1} & U_{\tau2} & U_{\tau3}}\right) =\left(\matrix{c\omega~c\xi & s\omega~c\xi & s\xi e^{i\phi} \cr -s\omega~c\theta-c\omega~s\theta~s\xi e^{-i\phi} & c\omega~c\theta- s\omega~s\theta~s\xi e^{-i\phi} & s\theta~c\xi \cr s\omega~s\theta-c\omega~c\theta~s\xi e^{-i\phi} & -c\omega~s\theta-s\omega~c\theta~s\xi e^{-i\phi} & c\theta~c\xi}\right), \label{MNSmatrix}$$ ]{} where $c\zeta\equiv\cos\zeta$ and $s\zeta\equiv\sin\zeta$ for $\zeta=\omega,\theta,\xi$. If the neutrinos are Majorana particles, two extra phases should be added to the MNS matrix, but, since they play no role in the physics of neutrino oscillations, they can be safely ignored. All physically distinguishable situations can be obtained if one allows $0\leq\phi\leq\pi$, all angles to vary between $0$ and $\pi/2$ and no restriction is imposed on the sign of the mass-squared differences, $\Delta m^2_{ij}\equiv m^2_i-m^2_j$. Note that there are only two independent mass-squared differences, which are chosen here to be $\Delta m^2_{21}$ and $\Delta m^2_{31}$. All experimental evidence from solar, atmospheric, and reactor neutrino experiments [@atmospheric; @Cl; @Kamiokande; @GALLEX; @SAGE; @Super-K; @reactor] can be satisfied,[^7] somewhat conservatively, by assuming [@general_review]: $10^{-4}$ eV$^2\lesssim |\Delta m^2_{31}|\simeq|\Delta m^2_{32}|\lesssim 10^{-2}$ eV$^2$, $0.3\lesssim\sin^2\theta\lesssim 0.7$, $10^{-11}$ eV$^2\lesssim |\Delta m^2_{21}|\lesssim 10^{-4}$ eV$^2$, $\sin^2\xi\lesssim 0.1$, while $\omega$ is mostly unconstrained. There is presently no information on $\phi$. In determining these bounds, it was explicitly assumed that only three active neutrinos exist. A few comments about the constraints imposed above are in order. First, one may complain that $\omega$ is more constrained than mentioned above by the solar neutrino data. The situation is far from definitive, however. As pointed out recently in [@dark_side] if the uncertainty on the $^8$B neutrino flux is inflated or if some of the experimental data is not considered (especially the Homestake data [@Cl]) in the fit, a much larger range of $\Delta m^2_{21}$ and $\omega$ is allowed. Furthermore, if three-flavour mixing is considered [@solar_3], different regions in the parameter space $\Delta m^2_{21}$-$\sin^2\omega$ are allowed for different values of $\sin^2\xi$, even if $\sin^2\xi$ is constrained to be small. Second, the limit from the Chooz and Palo Verde reactor experiments [@reactor] do not constrain $\sin^2\xi$ for $|\Delta m^2_{31}|\lesssim 10^{-3}$ eV$^2$. Furthermore, their constraints are to $\sin^2 2\xi$, so values of $\sin^2\xi$ close to one should also be allowed. However, the constraints from the atmospheric neutrino data require $\cos^2\xi$ to be close to one. This is easy to understand. Assuming that $L_{21}^{\rm osc}$ is much larger than the Earth’s diameter and that $\Delta m^2_{31}=\Delta m^2_{32}$, $$P_{\mu\mu}^{\rm atm}=1-4\cos^2\xi\sin^2\theta(1-\cos^2\xi\sin^2\theta) \sin^2\left(\frac{\Delta m^2_{31}x}{4E_{\nu}}\right),$$ according to upcoming Eq. (\[p3vac\]). Almost maximal mixing implies that $\cos^2\xi\sin^2\theta\simeq 1/2$. With the further constraint from $P_{ee}^{\rm atm}$, namely $\sin^2 2\xi\simeq 0$, one concludes that $\cos^2\xi\simeq 1$ and $\sin^2\theta\simeq 1/2$. In the case of oscillations in vacuum, it is straight forward to compute the oscillation probabilities $P_{\alpha\beta}$ of detecting a flavour $\beta$ given that a flavour $\alpha$ was produced. $$\label{p3vac} P_{\alpha\beta}=\sum_{i,j}U_{\alpha i}^*U_{\alpha j}U_{\beta i} U_{\beta j}^*e^{i\frac{\Delta m^2_{ij}x}{2E_{\nu}}}$$ The three different oscillation lengths, $L_{\rm osc}^{ij}$, are numerically given by $$L_{\rm osc}^{ij}= \frac{4\pi E_{\nu}}{\Delta m^2_{ij}}=2.47\times10^{8}{\rm km}\left(\frac{E} {1~\rm GeV}\right) \left(\frac{10^{-8}~\rm eV^2}{\Delta m^2_{ij}}\right),$$ which are to be compared to the Earth-Sun distance (1 a.u.$=1.496\times 10^{8}$ km). In the energy range of interest, 1 Gev$\lesssim E_{\nu}\lesssim 100$ GeV and given the experimental constraints on the parameter space described above, it is easy to see that $L_{\rm osc}^{31}$ and $L_{\rm osc}^{32}$ are much smaller than 1 a.u., and that its effects should “wash out” due to any realistic neutrino energy spectrum, detector energy resolution, or other “physical” effects. Such terms will therefore be neglected henceforth. In contrast, $L_{\rm osc}^{21}$ maybe as large as (and maybe even much larger than!) the Earth-Sun distance. Note that a nonzero phase $\phi$ implies $T$-violation, [*i.e.,*]{} $P_{\alpha\beta}\neq P_{\beta\alpha}$, unless $L_{\rm osc}^{21}\gg 1$ a.u.. This will be discussed in more detail later. In the presence of neutrino–medium interactions, the situation is, in general, more complicated (indeed, much more!). Similar to the two-neutrino case, it is important to discuss what is known about the oscillation probabilities. From the conservation of probability one has $$\begin{aligned} P_{ee}+P_{e\mu}+P_{e\tau}&=&1, \nonumber \\ P_{\mu e}+P_{\mu\mu}+P_{\mu\tau}&=&1, \\ P_{\tau e}+P_{\tau\mu}+P_{\tau\tau}&=&1, \nonumber \end{aligned}$$ and, similar to the two-neutrino case, unitarity of the Hamiltonian evolution implies $$\begin{aligned} P_{ee}+P_{\mu e}+P_{\tau e}&=&1, \nonumber \\ P_{e\mu}+P_{\mu\mu}+P_{\tau\mu}&=&1, \label{const3}\end{aligned}$$ A third equation of this kind, $P_{e\tau}+P_{\mu\tau}+P_{\tau\tau}=1$, is redundant. As before, Eqs. (\[const3\]) can be understood by arguing that, if equal numbers of all neutrino species are produced, the number of $\nu_{\beta}$’s to be detected should be the same, regardless of $\beta$, simply because the neutrino propagation is governed by a unitary operator. One may therefore express all $P_{\alpha\beta}$ in terms of only four quantities. Here, these are chosen to be $P_{ee}$, $P_{e\mu}$, $P_{\mu\mu}$, and $P_{\tau\tau}$. The others are given by $$\begin{aligned} P_{e\tau} & = & 1-P_{ee}-P_{e\mu}, \nonumber \\ P_{\mu e} & = & 1+P_{\tau\tau}-P_{ee}-P_{\mu\mu}-P_{e\mu}, \nonumber \\ P_{\mu\tau} & = & P_{ee}+P_{e\mu}-P_{\tau\tau}, \\ P_{\tau e} &= & P_{\mu\mu}+P_{e\mu}-P_{\tau\tau}, \nonumber \\ P_{\tau\mu} & = & 1-P_{\mu\mu}-P_{e\mu}. \nonumber \end{aligned}$$ Note that, in general, $P_{\alpha\beta}\neq P_{\beta\alpha}$. Oscillation of Neutrinos Produced in the Sun’s Core --------------------------------------------------- The propagation of neutrinos in the Sun’s core can, similar to the two-neutrino case, be described by the differential equation $$\frac{\rm d}{{\rm d}x}\nu_{\alpha}(r)=-i\left( \sum_{i=2}^{3}\left(\frac{\Delta m^2_{i1}}{2E_{\nu}}\right)U_{\alpha i}^*U_{\beta i} +A(x) \delta_{\alpha e}\delta_{\beta e}\right) \nu_{\beta}(r), \label{eq_3nus}$$ where $\delta_{\eta\zeta}$ is the Kronecker delta symbol. Terms proportional to the identity $\delta_{\alpha\beta}$ are neglected because they play no role in the physics of neutrino oscillations. The matter induced potential $A(x)$ is given by Eq.(\[A(x)\]). As in the two-neutrino case, it is useful to first discuss the initial states $\nu_{\alpha}$ in the Sun’s core, and to express them in the basis of instantaneous Hamiltonian eigenstates, which will be referred to as $|\nu_H\rangle$, $|\nu_M\rangle$, and $|\nu_L\rangle$ ($H=$ high, $M$= medium, and $L$= low). Therefore $$|\nu_{\alpha}\rangle=H_{\alpha}|\nu_H\rangle+M_{\alpha}|\nu_M\rangle+L_{\alpha} |\nu_L\rangle,$$ where $\langle\nu_{\alpha}|\nu_{\alpha '}\rangle=\delta_{\alpha\alpha '}$. As before (see Eq. [\[Peex\]]{}), the probability of detecting this initial state as a $\beta$-type neutrino far away from the Sun ([*e.g.,*]{} at the Earth) is given by $$P_{\alpha\beta}=\left|\left(\matrix{U_{\beta1}^* & U_{\beta2}^* & U_{\beta3}^*}\right) \left(\matrix{1 & 0 & 0 \cr 0 & e^{-i\frac{\Delta m^2_{21}x}{2E_{\nu}}} & 0 \cr 0 & 0 & e^{-i\frac{\Delta m^2_{31}x}{2E_{\nu}}}}\right)\left(V_{3\times 3}\right) \left(\matrix{L_{\alpha} \cr M_{\alpha} \cr H_{\alpha}}\right)\right|^2, \label{palphabetax}$$ where $V_{3\times 3}$ is an arbitrary $3\times 3$ unitary matrix which takes care of propagating the initial state until the edge of the Sun and rotating the state into the mass basis. In order to proceed, it is useful take advantage of the constraints on the neutrino parameter space and the energy range of interest. Note that $A\gtrsim\frac{|\Delta m^2_{31}|}{2E_{\nu}}\gg \frac{|\Delta m^2_{21}|}{2E_{\nu}}$ (remember that the energy range of interest is 1 GeV$\lesssim E_{\nu}\lesssim 100$ GeV and that $A\simeq 6\times 10^{-3}$ eV$^2$/GeV). It has been shown explicitly [@KP_3], assuming the neutrino mass-squared hierarchy to be $m_3^2>m_2^2>m_1^2$, [^8] that, if the mass-squared differences are very hierarchical ($|\Delta m^2_{31}|\gg |\Delta m^2_{21}|$), the three-level system “decouples” into two two-level systems, [*i.e.,*]{} one can first deal with matter effects in the “$H-M$” system and then with the matter effects in the “$M-L$” system. One way of understanding why this is the case is to realize that the “resonance point” corresponding to the $\Delta m^2_{31}$ is very far away from the resonance point corresponding to $\Delta m^2_{21}$. With this in mind, it is fair to approximate (this is similar to what is done, for example, in [@P_3]) $$V_{3\times 3}=\left(\matrix{A^L & B^L & 0 \cr -B^{L*} & A^{L*} & 0 \cr 0 & 0 & 1} \right)\left(\matrix{1 & 0 & 0 \cr 0 & A^{H} & B^{H} \cr 0 & -B^{H*} & A^{H*}} \right),$$ where $|B^H|^2=1-|A^H|^2\equiv P_c^H$, $|B^L|^2=1-|A^L|^2\equiv P_c^L$. The superscripts $H$, $L$ correspond to the “high” and the “low” resonances, respectively. It also possible to obtain an approximate expression for the initial states in the Sun’s core. Following the result outline above, this state should be described by two matter angles, $\xi_M$ and $\omega_M$, corresponding to each of the two-level systems. Both should be given by Eq. (\[cos2tm\]), where, in the case of $\cos 2\xi_M$, $\vartheta$ is to be replaced by $\xi$ and $\Delta m^2$ by $\Delta m^2_{31}$, while in the case of $\cos 2\omega_M$, $\vartheta$ is to be replaced by $\omega$, $\Delta m^2$ by $\Delta m^2_{21}$ and $A$ is to be replaced by $A\cos\xi$ [@P_3; @solar_3]. Furthermore, because $A\cos\xi\gg\frac{|\Delta m^2_{21}|}{2E_{\nu}}$, $\cos 2\omega_M$ can be safely replaced by -1 (remember that $\cos^2\xi\gtrsim0.9$). Within these approximations, in the Sun’s core, $$\begin{aligned} \label{inistate3} |\nu_{e}\rangle&=&\sin\xi_M|\nu_H\rangle+\cos\xi_M|\nu_M\rangle, \nonumber \\ |\nu_{\mu}\rangle&=&\sin\theta\cos\xi_M|\nu_H\rangle- \sin\theta\sin\xi_M|\nu_M\rangle -\cos\theta|\nu_L\rangle, \\ |\nu_{\tau}\rangle&=&\cos\theta\cos\xi_M|\nu_H\rangle- \cos\theta\sin\xi_M|\nu_M\rangle+\sin\theta|\nu_L\rangle. \nonumber\end{aligned}$$ The accuracy of this approximation has been tested numerically in the range of parameters of interest, and the difference between the “exact” result and the approximate result presented in Eq. (\[inistate3\]) is negligible. Keeping all this in mind, it is straight forward to compute all oscillation probabilities, starting from Eq. (\[palphabetax\]). From here on, $\phi=0$ (no $T$-violating phase in the mixing matrix, such that all $U_{\alpha i}$ are real) will be assumed, in order to simplify expressions and render the results cleaner. In the end of the day one obtains $$\begin{aligned} \label{pall3} P_{\alpha\beta}&=&a_{\alpha}^2 (U_{\beta 1})^2 + b_{\alpha}^2 (U_{\beta 2})^2 + c_{\alpha}^2 (U_{\beta 3})^2 + 2a_{\alpha}b_{\alpha} (U_{\beta1}U_{\beta2}) \cos\left(\frac{\Delta m^2_{21}x}{2E_{\nu}}+\delta^L\right) \nonumber \\ &&{\rm or} \\ P_{\alpha\beta}&=&\left(a_{\alpha}U_{\beta 1} + b_{\alpha}U_{\beta 2}\right)^2 + c_{\alpha}^2 (U_{\beta 3})^2 - 4a_{\alpha}b_{\alpha} (U_{\beta1}U_{\beta2}) \sin^2\left(\frac{\Delta m^2_{21}x}{4E_{\nu}}+\delta^L\right), \nonumber\end{aligned}$$ where $\delta^L$ is the matter phase, induced in the low resonance, and $$\begin{aligned} a_e&=&\sqrt{P_2^HP_c^L}, \nonumber \\ b_e&=&\sqrt{P_2^H(1-P_c^L)}, \nonumber \\ c_e&=&\sqrt{P_3^H}, \nonumber \\ a_{\mu}&=&-\sqrt{(1-P_c^L)}\cos\theta-\sqrt{P_3^HP_c^L}\sin\theta, \nonumber \\ b_{\mu}&=&\sqrt{P_c^L}\cos\theta-\sqrt{P_3^H(1-P_c^L)}\sin\theta, \\ c_{\mu}&=&\sqrt{P_2^H\sin^2\theta}, \nonumber \\ a_{\tau}&=&\sqrt{(1-P_c^L)}\sin\theta-\sqrt{P_3^HP_c^L}\cos\theta, \nonumber \\ b_{\tau}&=&-\sqrt{P_c^L}\sin\theta-\sqrt{P_3^H(1-P_c^L)}\cos\theta, \nonumber \\ c_{\tau}&=&\sqrt{P_2^H\cos^2\theta}, \nonumber $$ and $P_2^H=1-P_3^H=(|A^H|^2\cos^2\xi_M+|B^H|^2\sin^2\xi_M)$, which can also be written as $P_2^H=\frac{1}{2}+\frac{1}{2}\left(1-2P_c^H\right)\cos 2\xi_M$. This is to be compared with the expression for $P_1$ obtained in the two-flavour case. Note that $a^2_{\alpha}+b^2_{\alpha}+c^2_{\alpha}=1$. The effect of $\delta^L$ will not be discussed here and from here on $\delta^L$ will be set to zero. For details about the significance of $\delta^L$ for solar neutrinos in the two-flavour case, readers are referred to [@Petcov_eq; @P_phase]. Many comments are in order. First, in the nonadiabatic limit which can be obtained for very large energies, $P_c^H\rightarrow \cos^2\xi$, $P_c^L\rightarrow \cos^2\omega$ and $\cos 2\xi_M\rightarrow -1$. It is trivial to check that in this limit $a_{\alpha}\rightarrow U_{\alpha 1}$, $b_{\alpha}\rightarrow U_{\alpha 2}$, $c_{\alpha}\rightarrow U_{\alpha 3}$, and the vacuum oscillation result is reproduced, up to the matter induced phase $\delta^L$. Second, $P_{ee}$ can be written as $$P_{ee}=P_2^H\cos^2\xi(P_{ee}^{2\nu})+P_3^H\sin^2\xi, \label{pee3}$$ where $P_{ee}^{2\nu}$ is the two-neutrino result obtained in the previous section (see Eq. (\[pee\])) in the limit $\cos 2\vartheta_M\rightarrow -1$. It is easy to check that Eq. (\[pee\]) would be exactly reproduced (with $\vartheta_M$ replaced by $\omega_M$, of course) if the $\cos2\omega_M= -1$ approximation were dropped. For solar neutrino energies (100 keV$\lesssim E_{\nu}\lesssim 10$ MeV), $\xi_M\rightarrow\xi$, $P_c^H\rightarrow 0$ and therefore $P_2^H~(P_3^H)\rightarrow\cos^2\xi~(\sin^2\xi)$, reproducing correctly the result of the survival probability of electron-type solar neutrinos in a three-flavour oscillation scenario (see [@general_review] and references therein). In this scenario there is no “$H-L$” resonance inside the Sun, because $\frac{|\Delta m^2_{31}|}{2E_{\nu}}\gg A$ for solar neutrino energies. On the other hand, in the case $P_c^H\rightarrow 0$ and $\cos 2\xi_M\rightarrow -1$, $P_3^H\rightarrow 1$ and electron-type neutrinos exit the Sun as a pure $\nu_3$ mass eigenstate, and do not undergo vacuum oscillations even if $\Delta m^2_{21}$ is very small. In contrast, $\nu_{\mu}$ and $\nu_{\tau}$ always undergo vacuum oscillations if $\Delta m^2_{21}$ is small enough. The reason for this is simple. The generic feature of matter effects is to “push” $\nu_e$ into the heavy mass eigenstate, while $\nu_{\mu}$ and $\nu_{\tau}$ are “pushed” into the light mass eigenstates. This situation is changed by nonadiabatic effects, as argued above. Finally, it is important to note that all equations obtained are also valid in the case of inverted hierarchies ($m_3^2<m_{1,2}^2$ or $m_{2}^2<m_1^2$). This has been discussed in detail in the two-neutrino oscillation case [@earth_matter], and is also applicable here. It is worthwhile to point out that, in the approximation $\Delta m^2_{31} \simeq\Delta m^2_{32}$ the transformation $\Delta m^2_{21} \rightarrow-\Delta m^2_{21}$ can be reproduced by transforming $\omega\rightarrow\pi/2-\omega$, $\theta\rightarrow \pi-\theta$ and redefining the sign of $\nu_{\tau}$. Therefore, one is in principle allowed to fix the sign of $\Delta m^2_{21}$ as long as $\theta$ is allowed to vary between $0$ and $\pi$. In the case of inverted hierarchies (especially when $\Delta m^2_{31}<0$) one expects to see no “level crossing” (indeed, matter effects tend to increase the distance between the “energy” levels in this case), but matter effects are still present, because the initial state in the Sun’s core can be nontrivial ( [*i.e.*]{}, $\vartheta_M\neq\vartheta$). Note that $\nu_e$ is still “pushed” towards $\nu_H$, even in the case of inverted hierarchies, and the expressions for the matter mixing angles Eq. (\[cos2tm\]), and the initial states inside the Sun Eq. (\[inistate3\]) are still valid. The consequence of no “level crossing” is that the adiabatic limit does not connect, for example, $\nu_H\rightarrow\nu_3$ but $\nu_H\rightarrow\nu_2$ (or $\nu_1$, depending on the sign of $\Delta m^2_{21}$). This information is in fact contained in the equations above. The crucial feature is that, for example, when $\Delta m^2_{31}<0$, $P_c^H\rightarrow 1$ in the “adiabatic limit,” and the matrix $V_{3\times 3}$ correctly “connects” $\nu_H\rightarrow\nu_2$ (or $\nu_1$)! Another curious feature is that, in the limit $|\Delta m^2_{31}|/2E_{\nu}\gg A$, $\cos 2\xi_M\rightarrow-\cos2\xi$, $P_c^H\rightarrow 1$ and Eq. (\[pee3\]) correctly reproduces the survival probability of electron-type solar neutrinos in the three-flavour oscillation case. Note that on this case the sign of $\Delta m^2_{31}$ does not play any role, as expected. On the other hand, it is still true that $P_c^{(H,L)}\rightarrow\cos^2(\xi,\omega)$ in the extreme nonadiabatic limit, and vacuum oscillation results are reproduced, as expected. Again, in this limit, one is not sensitive to the sign of $\Delta m^2_{31}$, as expected. The Case of Antineutrinos ------------------------- As in the two-neutrino case, the difference between neutrinos and antineutrinos is that the equivalent of Eq. (\[eq\_3nus\]) for antineutrinos can be obtained by changing $A(x)\rightarrow -A(x)$ and $U_{\alpha i}\leftrightarrow U_{\alpha i}^*$. Unlike the two-flavour case, however, there is no set of variable transformations that allows one to exactly relate the differential equation for the neutrino and antineutrino systems. One should, however, note that if the signs of both $\Delta m^2$ are changed and $U_{\alpha i}\leftrightarrow U_{\alpha i}^*$, the neutrino equation turns into the antineutrino equation, up to an overall sign. This means, for example, that the instantaneous eigenvalues of the antineutrino Hamiltonian can be read from the eigenvalues of the neutrino Hamiltonian with $\Delta m^2_{ij}\leftrightarrow-\Delta m^2_{ij}$, $U_{\alpha i}\leftrightarrow U_{\alpha i}^*$ plus an overall sign. When it comes to computing $P_{\bar{\alpha}\bar{\beta}}$ this global sign difference is not relevant, and therefore $P_{\bar{\alpha}\bar{\beta}}(\Delta m^2_{ij},U_{\alpha i})= P_{\alpha\beta}(-\Delta m^2_{ij},U_{\alpha i}^*)$. Results and Discussions ======================= This section contains the compilation and discussion of a number of results concerning the oscillation of GeV neutrinos of all species produced in the Sun’s core. The goal here is to explore the multidimensional parameter space spanned by $\Delta m^2_{21}$, $\Delta m^2_{31}$, $\sin^2\omega$, $\sin^2\theta$, and $\sin^2\xi$ (and $E_{\nu}$). It will be assumed throughout that the electron number density profile of the Sun is exponential, so that Eq. (\[pc\]) can be used. As mentioned before, the numerical accuracy of this approximation is quite good, and certainly good enough for the purposes of this paper. Therefore, both $P_c^H$ and $P_c^L$ which appear in Eq. (\[pall3\]) will be given by Eqs. (\[pc\], \[gamma\]), with $\vartheta\rightarrow\xi$, $\Delta m^2\rightarrow\Delta m^2_{31}$ in the former, and $\vartheta\rightarrow\omega$, $\Delta m^2\rightarrow\Delta m^2_{21}$ in the latter. When computing $P_{\alpha\beta}$, an averaging over “seasons” is performed, which “washes out” the effect of very small oscillation wavelengths. Furthermore, integration over neutrino energy distributions is performed. Finally, all $P_{\alpha\beta}$ to be computed should be understood as the value of $P_{\alpha\beta}$ in the Earth’s surface, [*i.e.,*]{} Earth matter effects are not included. This is done in order to make the Sun matter effects in the evaluation of $P_{\alpha\beta}$ more clear. It should be stressed that Earth matter effects may play a significant role for particular regions of the parameter space, but the discussion of such effects will be left for another opportunity. Because the parameter space to be explored is multidimensional, it is necessary to make two-dimensional projections of it, such that “illustrative” points are required. The following points in the parameter space are chosen, all inspired by the current experimental situation: [$\bullet$]{} [ATM: $\Delta m^2_{31}=3\times 10^{-3}$ eV$^2$, $\sin^2\theta=0.5$, and $\sin^2\xi=0.01$,]{} [LMA: $\Delta m^2_{21}=2\times 10^{-5}$ eV$^2$, $\sin^2\omega=0.2$,]{} [SMA: $\Delta m^2_{21}=6\times 10^{-6}$ eV$^2$, $\sin^2\omega=0.001$,]{} [LOW: $\Delta m^2_{21}=1\times 10^{-7}$ eV$^2$, $\sin^2\omega=0.4$,]{} [VAC: $\Delta m^2_{21}=1\times 10^{-10}$ eV$^2$, $\sin^2\omega=0.55$.]{} ATM corresponds to the best fit point of the solution to the atmospheric neutrino puzzle [@atmos_analysis], and a value of $\sin^2\xi=0.01$ which is consistent with all the experimental bounds. Note that some “subset” of ATM will always be assumed (for example, $\sin^2\theta$ is fixed while exploring the ($\Delta m^2_{31}\times\sin^2\xi$)- plane). For each analysis, it will be clear what are the “variables” and what quantities are held fixed at their “preferred point” values. All other points refer to sample points in the regions which best solve the solar neutrino puzzle [@bksreview; @rate_analysis; @dark_side], and the notation should be obvious. Initially a flat neutrino energy distribution with $E_{\nu}^{min}=1$ GeV and $E_{\nu}^{max}=5$ GeV is considered (for concreteness), and the case of higher average energies is briefly discussed later. The Case of Vacuum Oscillations ------------------------------- If neutrinos were produced and propagated exclusively in vacuum, the oscillation probabilities would be given by Eq. (\[p3vac\]). This would be the case of neutrinos produced in the Sun’s core if either the electron number density were much smaller than its real value or if very low energy neutrinos were being considered. Nonetheless, it is still useful to digress some on the “would be” vacuum oscillation probabilities in order to understand better the matter effects. In the case of pure vacuum oscillations, it is trivial to check that $P_{\alpha\beta}=P_{\beta\alpha}$ (remember that the MNS matrix phase $\phi$ has been set to zero), and therefore all $P_{\alpha\beta}$ can be parametrised by three quantities, namely $P_{\alpha\alpha}$, $\alpha=e,\mu,\tau$. It is easy to show that $$P_{\alpha\beta}=P_{\beta\alpha}\Leftrightarrow P_{e\mu}=\frac{1}{2}(1+P_{\tau\tau}-P_{\mu\mu}-P_{ee}). \label{pab=pba}$$ From Eq. (\[p3vac\]) $$\begin{aligned} \label{p3vac_aa} P_{\alpha\alpha}&=&U_{\alpha1}^4+U_{\alpha2}^4+U_{\alpha3}^4+ 2U_{\alpha1}^2U_{\alpha2}^2 \cos\left(\frac{\Delta m^2_{21}x}{2E_{\nu}}\right)\nonumber \\ &&{\rm or} \\ P_{\alpha\alpha}&=&(1-U_{\alpha3}^2)^2+U_{\alpha3}^4-4U_{\alpha1}^2U_{\alpha2}^2 \sin^2\left(\frac{\Delta m^2_{21}x}{4E_{\nu}}\right). \nonumber \end{aligned}$$ Note that there is no dependency on $\Delta m^2_{31}$. Particularly simple limits can be reached when $L_{\rm osc}^{21}$ is either very small or very large compared with the Earth-Sun distance. In both limits $P_{\alpha\alpha}$ is independent of $\Delta m^2_{21}$ and, in the latter case, $P_{\alpha\alpha}$ depends only on $U_{\alpha3}^2$. Fig. \[dm21\_vacuum\] depicts constant $P_{\alpha\beta}$ contours in the ($\Delta m^2_{21}\times\sin^2\omega$)-plane, at ATM. Remember that, here, $P_{e\mu}$ is not an independent quantity but is a linear combination of all $P_{\alpha\alpha}$. Note that $P_{ee}$ is symmetric for $\omega\rightarrow\pi/2-\omega$, and that $P_{\mu\mu}\leftrightarrow P_{\tau\tau}$ when $\omega\rightarrow\pi/2-\omega$. The latter property is a consequence of $\theta=\pi/4$. Also, in the case of $P_{ee}$, the $L_{\rm osc}^{21}\rightarrow\infty$ coincides with the $\omega\rightarrow 0, \pi/2$ limit for any $L_{\rm osc}^{21}$ (this is because either $U_{e1}$ or $U_{e2}$ go to zero). This is not true of $P_{\mu\mu}$ or $P_{\tau\tau}$ unless $\sin^2\xi=0$. Another important consequence of $L_{\rm osc}^{21}\gg 1$ a.u. is that $T$-violating effects are absent, even if $\phi$ is nonzero. This can be seen by looking at the second expression in Eq. (\[p3vac\_aa\]), which is a function only of $|U_{\alpha 3}|^2$ in the limit $L_{\rm osc}^{21}\rightarrow\infty$. \[t\] Finally, one should note that oscillatory effects are maximal for $\Delta m^2_{21}\simeq 2\times 10^{-8}$ eV$^2$. In this region $ \cos\left(\frac{\Delta m^2_{21}x}{2E_{\nu}}\right)\simeq -1$, and the largest suppression to all $P_{\alpha\alpha}$ is obtained when $U_{\alpha1}^2U_{\alpha2}^2$ is maximum. For example, $P_{ee}$ is smallest when $\omega=\pi/4$, since $U_{e1}^2U_{e2}^2\propto\sin^2 2\omega$. There are no “localised” maxima for $P_{\alpha\alpha}$ because $U_{\alpha1}^2U_{\alpha2}^2$ is positive definite. “Normal” Neutrino Hierarchy --------------------------- When matter effects are “turned on,” the situation can be dramatically different. This is especially true in the case of normal neutrino mass hierarchies ($m_1^2<m_2^2<m_3^2$), which will be discussed first. The first effect one should observe is that, even though $L_{\rm osc}^{31}\ll 1$ a.u., $P_{\alpha\beta}$ depend rather nontrivially on $\Delta m^2_{31}$. This dependency comes from the terms $P_3^H$ and $P_2^H=1-P_3^H$ in Eq. (\[pall3\]). Remember that $P_3^H$ is interpreted as the probability that a $\nu_e$ produced in the Sun’s core exits the Sun as a $\nu_3$ mass eigenstate. When matter effects are negligible (such as in the limit of small neutrino energies) $P_3^H\rightarrow\sin^2\xi$, its “vacuum limit.” Fig. \[p3h\] depicts constant $P_3^H$ contours in the ($\Delta m^2_{31}/E_{\nu}\times\sin^2\xi$)-plane. \[t\] Note that, for $\Delta m^2_{31}/E_{\nu}\sim 10^{-2}$ eV$^2$/GeV, $P_3^H\rightarrow 1$, even for small values of $\sin^2\xi$. In this region, $\nu_e$’s produced in the Sun’s core exit the Sun as pure $\nu_3$’s. Therefore, $P_{e\alpha}\simeq U_{\alpha3}^2$. Because of unitarity in the propagation, $\nu_{\mu}$’s and $\nu_{\tau}$’s exit the Sun as linear combinations of the light mass eigenstates, and may not only undergo vacuum oscillations but are also susceptible to further matter effects (dictated by the “$M-L$” system, as described in Sec. 3). For future reference, at ATM, $P_3^H\simeq0.87$ when averaged over the energy range mentioned in the beginning of this section. As $\Delta m^2_{31}/E_{\nu}$ decreases (as is the case for higher energy neutrinos) the nonadiabaticity of the “$H-M$” system starts to become relevant, and $P_3^H\rightarrow\sin^2\xi$, as argued in Sec. 3.2. A hint of this behaviour can already be seen in Fig. \[p3h\], for small values of $\Delta m^2_{31}/E_{\nu}$. The information due to the “$M-L$” matter effect is encoded in $P_c^L$, present in Eq. (\[pall3\]). Fig. \[1-pcl\] depicts contours of constant $1-P_c^L$ in the ($\Delta m^2_{21}/E_{\nu}\times\sin^2\omega$)-plane. One should note that $1-P_c^L$ reaches its extreme nonadiabatic limit, $\sin^2\omega$, when $\Delta m^2_{21}/E_{\nu}\lesssim 10^{-7}$ eV$^2$/GeV. For $\Delta m^2_{21}/E_{\nu}\gtrsim 10^{-7}$ eV$^2$/GeV, matter effects increase the value of $1-P_c^L$. \[t\] One can use the intuition from the two-flavour solution to the solar neutrino puzzle to better appreciate the results presented here. In the case of the solutions to the solar neutrino puzzle, the energies of interest range from 100 keV to 10 MeV, and large matter effects happen around $\Delta m^2\sim 10^{-5}$ eV$^2$. Furthermore, at $\Delta m^2\sim 10^{-10}$ eV$^2$ one encounters the “just-so” solution, which is characterised by very long wave-length vacuum oscillations. Rescaling to $O$(GeV) energies, the equivalent of the “just-so” solution happens for $\Delta m^2_{21}\sim (10^{-8}-10^{-7})$ eV$^2$, while large matter effects would be present at $\Delta m^2\sim (10^{-3}-10^{-2})$ eV$^2$. Indeed, one observes large matter effects for $\Delta m^2_{31}\sim (10^{-3}-10^{-2})$ eV$^2$. $\Delta m^2_{21}\sim (10^{-5}-10^{-6})$ eV$^2$ corresponds to the region between the LOW and VAC solutions, where matter effects distort $P_{\alpha\beta}$ from its pure vacuum value, but no dramatic suppression or enhancement takes place. Incidently, this behaviour has physical consequences in the solution to the solar neutrino problem, as was first pointed out in [@alex]. Figs. \[dm31\_ssxi\_lma\] and \[dm31\_ssxi\_low\] depict contours of constant $P_{\alpha\alpha}$ and $P_{e\mu}$ in the ($\Delta m^2_{31}\times\sin^2\xi$)-plane. As expected, in the region where $P_3^H\sim 1$, $P_{ee}$ and $P_{e\mu}$ do not depend on $\Delta m^2_{21}$ or $\sin^2\omega$, namely $P_{ee}\sim\sin^2\xi$ and $P_{e\mu}\sim 0.5\cos^2\xi$. Remember that the results depicted in Figs. \[dm31\_ssxi\_lma\] and \[dm31\_ssxi\_low\] (and all other plots from here on) are for an energy band from 1 to 5 GeV. On the other hand, $P_{\mu\mu}$ and $P_{\tau\tau}$ do depend on the point (LMA, SMA, etc), even for $P_3^H\sim 1$, as foreseen. This dependence will be discussed in what follows. \[pt\] \[pt\] In the limit $P_3^H=1$, $\sin^2\theta=0.5$ $$\begin{aligned} c_{\mu}^2=&c_{\tau}^2=&0, \\ a_{\mu}^2=&b_{\tau}^2=&0.5(1+2\sqrt{P_c^L(1-P_c^L))}, \\ b_{\mu}^2=&a_{\tau}^2=&0.5(1-2\sqrt{P_c^L(1-P_c^L))}, \\ \label{amubmu} a_{\mu}b_{\mu}=&-a_{\tau}b_{\tau}=&0.5(1-2P_c^L),\end{aligned}$$ and $$\begin{aligned} \label{pmmtt} P_{(\mu\mu,\tau\tau)}&=&\frac{1}{2}(1-U_{(\mu,\tau)3}^2)\pm\sqrt{P_c^L(1-P_c^L)} (U_{(\mu,\tau)1}^2-U_{(\mu,\tau)2}^2) \\ &\pm& (1-2P_c^L)U_{(\mu,\tau)1}U_{(\mu,\tau)2} \cos\left(\frac{\Delta m^2_{21}x}{2E_{\nu}}\right). \nonumber\end{aligned}$$ At both LMA and SMA, the oscillatory term averages out to zero, while at VAC $\cos\left(\frac{\Delta m^2_{21}x}{2E_{\nu}}\right)=1$. It is only at LOW that the oscillatory term is nontrivial, as was mentioned in the analogy between the situation at hand and the solutions to the solar neutrino puzzle. Furthermore, at SMA, $1-P_c^L$ is tiny (see Fig. \[1-pcl\]), so it is fair to approximate $P_{\mu\mu}\simeq P_{\tau\tau}\simeq 0.5(1-0.5\times0.99) \simeq 0.25$, in agreement with Fig. \[dm31\_ssxi\_lma\](bottom). At LMA, it is fair to approximate $\sin^2\xi=0$. In this limit, $U_{\mu1}^2-U_{\mu2}^2\simeq -0.5\cos 2\omega+\sin 2\omega\sin\xi$, while $U_{\tau1}^2-U_{\tau2}^2\simeq -0.5\cos 2\omega-\sin 2\omega\sin\xi$. Therefore, because $\cos 2\omega=0.6>0$, $P_{\mu\mu}$ is significantly less than $P_{\tau\tau}$, since $\sqrt{P_c^L(1-P_c^L)}$ is nonnegligible. Roughly, $P_{\mu\mu}\simeq 0.15$ and $P_{\mu\mu}\simeq 0.4$, using the approximations above. Again, there is agreement with Fig. \[dm31\_ssxi\_lma\](top). In order to understand the behaviour at LOW and VAC, one should take advantage of the fact that $1-P_c^L\rightarrow\sin^2\omega$. In this case, it proves more advantageous to use the second form of Eq. (\[pall3\]) in order to express all $P_{\alpha\beta}$ $$\begin{aligned} \label{1-pcl=sso} P_{ee}&=&P_2^H\cos^2\xi+P_3^H\sin^2\xi-{\rm (Osc)}_{ee}, \nonumber \\ P_{e\mu}&=&\left(P_2^H\sin^2\xi+P_3^H\cos^2\xi\right)\sin^2\theta- {\rm (Osc)}_{e\mu}, \\ P_{\mu\mu}&=&\left(\cos^2\theta+\sqrt{P_3^H}\sin\xi\sin^2\theta\right)^2 +P_2^H\cos^2\xi\sin^4\theta-{\rm (Osc)}_{\mu\mu}, \nonumber \\ P_{\tau\tau}&=&\left(\sin^2\theta+\sqrt{P_3^H}\sin\xi\cos^2\theta\right)^2 +P_2^H\cos^2\xi\cos^4\theta-{\rm (Osc)}_{\tau\tau}, \nonumber\end{aligned}$$ where $${\rm (Osc)}_{\alpha\beta}=4a_{\alpha}b_{\alpha}U_{\beta1}U_{\beta2} \sin^2\left(\frac{\Delta m^2_{12}x}{4E_{\nu}}\right)$$ are the oscillatory terms. When $L_{\rm osc}^{21}\gg 1$ a.u., the oscillatory terms are zero, and $P_{\alpha\beta}$ are particularly simple. Note that on this limit many simplifications happen: $P_{\alpha\beta}$ is independent of $\omega$ and $\Delta m^2_{21}$, and $P_{\mu\mu}=P_{\tau\tau}$ if $\sin^2\theta=\cos^2\theta$, as can be observed in Fig. \[dm31\_ssxi\_low\](bottom). A very important fact is that, when the oscillatory terms are neglected, $2P_{e\mu}=1+P_{\tau\tau}-P_{\mu\mu}-P_{ee}$, as one may easily verify directly. As argued before, when this condition is satisfied, $P_{\alpha\beta}=P_{\beta\alpha}$. This is not the case in the presence of nonnegligible oscillation effects or when $P_c\neq\cos^2\omega$. Both statements are trivial to verify directly. For example, $$\label{abmumu} 4a_e b_e U_{\mu1} U_{\mu2}=P_2^H\sin 2\omega\left[\sin 2\omega\left(\sin^2\xi \sin^2\theta-\cos^2\theta\right)-\sin\xi\sin 2\theta\cos 2\omega\right]$$ while $$\label{abee} 4a_{\mu} b_{\mu} U_{e1} U_{e2}=\cos^2\xi\sin 2\omega\left[\sin 2\omega \left(P_3^H\sin^2\theta-\cos^2\theta\right)- \sqrt{P_3^H}\sin 2\theta\cos 2\omega\right],$$ so ${\rm Osc}_{e\mu}\neq{\rm Osc}_{\mu e}$. Figs. \[sst\_lma\] and \[sst\_low\] depict $P_{\alpha\beta}$ as a function of $\sin^2\theta$ at LMA and SMA, and at LOW and VAC, respectively. In these figures, all $P_{\alpha\beta}$ are plotted, in order to illustrate that $P_{\alpha\beta}\neq P_{\beta\alpha}$ at LMA, SMA and VAC. Note that at LMA and SMA, the difference comes from the fact that $P_3^H\neq \sin^2\xi$ [*and*]{} $P_c^L\neq \cos^2\omega$. At LOW, $P_c^L\simeq\cos^2\omega$, but $P_3^H\neq \sin^2\xi$ [*and*]{} nontrivial oscillatory terms render $P_{\alpha\beta}\neq P_{\beta\alpha}$. At VAC, even though $P_3^H\neq \sin^2\xi$, $P_{\alpha\beta}=P_{\beta\alpha}$ because $P_c^L=\cos^2\omega$ [*and*]{} because “1-2” oscillations don’t have “time” to happen. \[pt\] \[pt\] From Eqs. (\[1-pcl=sso\]) one can roughly understand the dependency of $P_{\alpha\beta}$ on $\sin^2\theta$. Obviously $P_{ee}$ does not depend on $\theta$ (by the very form of the MNS matrix, Eq. (\[MNSmatrix\])), while $P_{e\mu}$ ($P_{e\tau}$) depends almost exclusively on $\sin^2\theta$ ($\cos^2\theta$). This is guaranteed by the fact that $P_3^H\gg P_2^H$ even at LMA and LOW, when one expects the interference terms to play a significant role. It is also worthwhile to note that, as expected, at VAC and SMA the curves are very similar, a behaviour that can be understood from earlier discussions. Finally, Fig. \[dm21\_std\] depicts constant $P_{\alpha\beta}$ contours in the ($\Delta m^2_{21}\times\sin^2\omega$)-plane, at ATM. In light of the previous discussions, the shapes and forms can be readily understood. First note that the shapes of the constant $P_{ee}$ and $P_{e\mu}$ regions resemble those of the pure vacuum oscillations depicted in Fig. \[dm21\_vacuum\], with two important differences. First, the constant values of the contours are quite different. For example, $P_{ee}$ varies from a few percent to less then 15%, while in the case of pure vacuum oscillations, $P_{ee}$ varies from 30% to 100%. This can be roughly understood numerically by noting that $P_{(ee,e\mu)}\simeq P_2^H P_{(ee,e\mu)}^{\rm vac}$ (remember that $P_2^H=1-P^H_3\simeq 0.13$ when averaged over the energy range of interest). Second, at high $\Delta m^2_{21}$, the regions are distorted. This is due to nontrivial matter effects in the “$M-L$” system. Note that the contours follow the constant $1-P_c^L$ curves depicted in Fig. \[1-pcl\]. \[t\] The $P_{\mu\mu}$ and $P_{\tau\tau}$ contours are a lot less familiar, and require some more discussion. Many features are rather prominent. For example, the plane is roughly divided into a $\sin^2\omega>0.5$ and $\sin^2\omega<0.5$ structure, and large (small) values of $P_{\mu\mu}$ ($P_{\tau\tau}$) are constrained to the $\sin^2\omega>0.5$ half, and vice-versa. Also, there is a rough $P_{\mu\mu}(\omega) \leftrightarrow P_{\tau\tau}(\pi/2-\omega)$ symmetry in the picture, which was present in the pure vacuum case (see Fig. \[dm21\_vacuum\]). This symmetry is absent for large values of $\Delta m^2_{21}$, similar to what happens in the case of $P_{ee}$, and is due, as mentioned in the previous paragraph, to the fact that $P_c^L$ is significantly different from $\cos^2\omega$ in this region. The other features are also fairly simple to understand, and are all due to fact that $P_3^H\gg\sin^2\xi$. It is convenient to start the discussion in the limit when $L_{21}^{\rm osc}\gg 1$ a.u. (the very small $\Delta m^2_{21}$ region). As was noted before, $P_{\alpha\beta}$ are given by Eq. (\[1-pcl=sso\]) where the Osc$_{\alpha\beta}$ terms vanish. It is therefore easy to see that $P_{\alpha\beta}$ do not depend on $\omega$ or $\Delta m^2_{21}$ (as mentioned before), and furthermore it is trivial to compute the value of $P_{\alpha\beta}$ given that we are at ATM and that $P_3^H\simeq 0.87$. The next curious feature is that there is a “band” around $\sin^2\omega=1/2$ where $P_{\mu\mu}\simeq P_{\mu\mu}(L_{21}^{\rm osc}\rightarrow\infty)$. The same is true of $P_{\tau\tau}$. This is due to the fact that, around $\sin^2\omega\simeq 1/2$, $a_{\mu}b_{\mu}$ and $a_{\tau}b_{\tau}$ vanish when $P_3^H$ is large. In the limit $P_3^H=1$ one can use Eq.(\[amubmu\]) and note that indeed both $a_{\mu}b_{\mu}$ and $a_{\tau}b_{\tau}$ vanish at $P_c^L=1/2$. However, for values of $\Delta m^2_{21}\lesssim 10^{-7}$ eV$^2$ $P_c^L\simeq\cos^2\omega$, which explains the band around $\omega\simeq \pi/4$. Slight distortions are due to the fact that $P_3^H\neq1$, and are easily computed from the exact expressions. Again in the limit $P_3^H=1$, $P_c^L=\cos^2\omega$, the coefficient of the $\cos\left(\frac{\Delta m^2_{21}x}{2E_{\nu}}\right)$ term Eq. (\[pmmtt\]) is $$\pm\frac{1}{2}(1-2\cos^2\omega)\left(-\frac{1}{2}\sin 2\omega\mp 0.1\cos 2\omega\right),$$ if $\sin^2\xi$ terms are neglected. The $+,-$ signs are for $P_{\mu\mu}$ while the $-,+$ signs for $P_{\tau\tau}$. It is trivial to verify numerically (if a little tedious) that the $P_{\mu\mu}$ term has a maximum at $\sin^2\omega\simeq 0.1$ and a minimum at $\sin^2\omega\simeq 0.8$. For $P_{\tau\tau}$ the maximum (minimum) is at $\sin^2\omega\simeq 0.9 (0.2)$. It is important to comment that the minima are negative numbers. On the other hand, from Fig. \[dm21\_vacuum\] (as mentioned before) it is easy to see that $\cos\left(\frac{\Delta m^2_{21}x}{2E_{\nu}}\right)$ is minimum for $\Delta m^2_{21}\simeq 2\times 10^{-8}$ eV$^2$ (this is where all $P_{\alpha\alpha}$ are maximally suppressed in Fig. \[dm21\_vacuum\]). Combining both informations, it is simple to understand the maxima/minima of $P_{\mu\mu}$ and $P_{\tau\tau}$ at $\Delta m^2_{21}\simeq 2\times 10^{-8}$ eV$^2$: Minima occur when the coefficient is maximum ([*e.g.,*]{} at $\sin^2\omega\simeq 0.1$ for $P_{\mu\mu}$) while maxima occur when the coefficient is minimum ([*e.g.,*]{} at $\sin^2\omega\simeq 0.8$ for $P_{\mu\mu}$). A description of what has happened is the following: The matter effects “compress” the constant $P_{\mu\mu}$ ($P_{\tau\tau}$) contours from the pure vacuum oscillation case (presented in Fig. \[dm21\_vacuum\]) to the $\sin^2\omega<1/2$ ($>1/2$) half of the plane, and a new region “appears” on the other half. This other region is characterised by negative values to the coefficients of the oscillatory terms, which are not attainable in the case of pure vacuum oscillations (see Eq. (\[p3vac\_aa\])). At last, the contours in the region where the oscillatory effects average out, $P_{\mu\mu}$ and $P_{\tau\tau}$ are also best understood from Eq. (\[pmmtt\]) and the paragraphs which follow it, in the limit that $\cos\left(\frac{\Delta m^2_{21}x}{2E_{\nu}}\right)\rightarrow 0$. It is simple to see, for example, that $P_{\mu\mu}<P_{\tau\tau}$ if $\cos 2\omega>0$ ($\sin^2\omega>1/2$), while the situation is reversed if $\cos 2\omega<0$. This is indeed what one observes in Fig. \[dm21\_std\]. “Inverted” Neutrino Hierarchy ----------------------------- Here I turn to the case of an “inverted” neutrino hierarchy, namely $\Delta m^2_{31}<0$. Currently, there is no experimental hint as to what the sign of $\Delta m^2_{31}$ should be, so there is no reason to believe that the “normal” hierarchy is to be preferred over the “inverted” hierarchy. Indeed, even from a theoretical/ model building point of view, there are no strong reasons for or against a particular neutrino mass hierarchy [@theory_review]. The discussion will be restricted to $\Delta m^2_{21}>0$ for two reasons. First, the $\Delta m^2_{21}<0$ can be approximately read off from the $\Delta m^2_{21}>0$ case by changing $\omega\rightarrow\pi/2-\omega$, as mentioned before. Second, and most important, there is some experimental hints as to what is the sign of $\Delta m^2_{21}$ [@solar_3; @dark_side]. For example, the SMA solution only exists for one sign of $\Delta m^2_{21}$, while the LMA and LOW solutions prefer one particular sign. Even in the case of VAC there is the possibility of obtaining information concerning the sign of $\Delta m^2_{21}$ from solar neutrino data [@alex]. Therefore, the notation introduced in the beginning of this section (ATM, SMA, LMA, LOW, VAC) still applies, and one should simply remember that here $\Delta m^2_{31}<0$. As advertised, the largest effect of $\Delta m^2_{31}<0$ is the typical values of $P_{c}^H$. From Eq. (\[pc\]), keeping in mind that here $\gamma$ is negative, $$P_c^H=\frac{1-e^{-|\gamma|\cos^2\xi}}{1-e^{-|\gamma|}},$$ where $\gamma$ is given by Eq. (\[gamma\]) with $\Delta m^2\rightarrow\Delta m^2_{31}$. Since $|\gamma|\gg 1$ (see Eq. (\[gamma\])), $P_c^H=1$ for all values of $\Delta m^2_{31}$ and $\sin^2\xi$ of interest. Indeed, this is true for any value of $\sin^2\xi$ as long as $\Delta m^2_{31}\gg 10^{-6}$ eV$^2$. This is to be contrasted to the normal hierarchy case, when there is always some value of $\sin^2\xi$ (which is a function of $\Delta m^2_{31}$) below which $P_c$ deviates significantly from its adiabatic limit. All of the $\Delta m^2_{31}$ dependency of $P_{\alpha\beta}$ is therefore encoded in $\xi_M$. However, in the case $\Delta m^2_{31}<0$ it is trivial to show that $-1<\cos 2\xi_M<-\cos 2\xi$, where the upper bound is reached in the limit $|\Delta m^2_{31}/2E_{\nu}|\gg A$ (this has been mentioned before. The minus sign takes care of the “unorthodox” $P_c\rightarrow 1$ adiabatic limit). Since one is interested in $\sin^2\xi<0.1$ ($-\cos 2\xi<-0.8$), the range for $\xi_M$ is rather limited, and therefore any $\Delta m^2_{31}$ effects are bound to be very small. Larger $\Delta m^2_{31}$ effects are expected for larger $\sin^2\xi$. In light of this, Fig. \[ssxi\_min\] depicts $P_{\alpha\beta}$ and $P_{e\mu}$ as a function of $\sin^2\xi$ at the various points (LMA, SMA, LOW, VAC), for $\Delta m^2_{31}=-3\times 10^{-3}$ eV$^2$ and $\sin^2\theta=0.5$. \[t\] It is interesting to compare the results presented here with the pure vacuum case. In the limit $P_2^H=1$ $$P_{ee}=\cos^2\xi\left[P_c^L\cos^2\omega+(1-P_c^L)\sin^2\omega+\sqrt{P_c(1-P_c^l)} \sin 2\omega\cos\left(\frac{\Delta m^2_{21}L}{2E_{\nu}}\right)\right],$$ while the pure vacuum result in the same region of the parameter space is $$P_{ee}^{\rm vac}=\cos^4\xi\left[\cos^4\omega+\sin^4\omega+2\sin^2\omega\cos^2\omega \cos\left(\frac{\Delta m^2_{21}L}{2E_{\nu}}\right)\right]+\sin^4\xi.$$ In the limit $P_c^L=\cos^2\omega$, the difference $$P_{ee}-P_{ee}^{\rm vac}=\left(P_{ee}^{2\nu, \rm vac}\right) \cos^2\xi\sin^2\xi-\sin^4\xi,$$ where $P_{ee}^{2\nu, \rm vac}$ is the electron neutrino survival probability in the two-flavour case with $\Delta m^2=\Delta m^2_{21}$ and vacuum mixing angle $\omega$. This difference vanishes at $\sin^2\xi=0$, and $\sin^2\xi=\frac{P_{ee}^{2\nu, \rm vac}}{1+ P_{ee}^{2\nu, \rm vac}}$, (which is between 0 and 0.5). Furthermore, it is a convex function of $\sin^2\xi$, which means that $P_{ee}$ is [*larger*]{} than the pure vacuum case for values of $\sin^2\xi< \frac{P_{ee}^{2\nu, \rm vac}}{1+ P_{ee}^{2\nu, \rm vac}}$. Away from the limit $P_c^L=\cos^2\omega$, keeping in mind that the oscillatory terms average out, $P_{ee}$ is still larger than the pure vacuum case if $\cos^2\omega>\sin^2\omega$ since $P_c^L\leq\cos^2\omega$, as one can easily verify. Also, in the limit $P_2^H=1$, $\sin^2\theta=1/2$, $$P_{\mu\mu}=\frac{1}{2}\left[(1-P_c^L)U_{\mu1}^2+P_c^LU_{\mu2}^2+ U_{\mu3}^2+2\sqrt{P_c^L (1-P_c^L)}U_{\mu1}U_{\mu2}\cos\left(\frac{\Delta m^2{21}L}{2E_{\nu}}\right)\right].$$ The same expression applies for $P_{\tau\tau}$ with $U_{\mu i}\rightarrow U_{\tau i}$. This is a consequence of $\sin^2\theta=\cos^2\theta$. Furthermore, in the limit $\sin^2\xi\rightarrow 0$ (and for $\sin^2\theta=\cos^2\theta$), $U_{\mu i}=U_{\tau i}$, which explains why $P_{\mu\mu}=P_{\tau\tau}$ for $\sin^2\xi\lesssim 10^{-2}$. At VAC this equality remains for all values of $\sin^2\xi$. The reason for this is that, at VAC, the expression simplifies tremendously and $P_{\mu\mu}=P_{\tau\tau}=\frac{1}{4}\left(1+\cos^2\xi\right)$. In the same region of the parameter space, the pure vacuum oscillation case yields $P_{\mu\mu}^{\rm vac}=P_{\tau\tau}^{\rm vac}=\frac{1}{2}\cos^4\xi-\cos^2\xi+1$. Note that, in this region of the parameter space $P_{\mu\mu}^{\rm vac}\geq P_{\mu\mu}$, the inequality being saturated at $\cos^2\xi=1$. The same result also applies (approximately) at SMA, since the oscillatory terms are proportional to $\sqrt{P_c^L(1-P_c^L)}$ and $1-P_c^L$ is very small at SMA (see Fig \[1-pcl\]). The equality $P_{\mu\mu}=P_{\tau\tau}$ is broken at larger values of $\sin^2\xi$ because $P_c^L\neq \cos^2\omega$ at SMA. It remains to discuss how $P_{\mu\mu}$ and $P_{\tau\tau}$ diverge from the pure vacuum case at LMA and LOW. In the limit $P_c^L=\cos^2\omega$, and averaging out the oscillatory terms, $$\label{eq_diff} P_{\mu\mu}-P_{\mu\mu}^{\rm vac}=\frac{\sin\xi}{2}\left[\sin\xi\left(U_{\mu3}^2- (\cos^2\omega U_{\mu1}^2+\sin^2\omega U_{\mu2}^2)\right)- \sin 2\omega(U_{\mu1}^2-U_{\mu2}^2) \right].$$ This difference goes to zero as $\sin^2\xi\rightarrow 0$. This is to be expected, since in this limit the difference of $P_2^H$ and $\cos^2\xi$ disappears. For small values of $\sin^2\xi$, the last term in Eq. \[eq\_diff\] dominates, and, as discussed before, $U_{\mu1}^2-U_{\mu2}^2=-0.5\cos 2\omega+O(\sin\xi)$. Therefore, $P_{\mu\mu}-P_{\mu\mu}^{\rm vac}>0$ ($<0$) for $\cos 2\omega>0$ ($<0$). The expression for $P_{\tau\tau}$ can be obtained from Eq.(\[eq\_diff\]) by replacing $U_{\mu i}\rightarrow U_{\tau i}$ and changing the sign of the last term. Therefore, since $U_{\tau1}^2-U_{\tau2}^2=-0.5\cos 2\omega+O(\sin\xi)$, $P_{\tau\tau}-P_{\tau\tau}^{\rm vac}>0$ ($<0$) for $\cos 2\omega<0$ ($>0$). When the oscillatory terms do not average out, it is easy to verify explicitly that the behaviour of the oscillatory terms follows the behaviour of the average terms, discussed above, and the inequalities obtained above still apply. \[t\] The situation, however, changes, when $P_c^L\neq \cos^2\omega$, [*i.e.,*]{} when matter effects due to the “M-L” system are relevant. In this region, a behaviour similar to the one observed in the “normal” hierarchy case is expected, since $\Delta m^2_{21}>0$. Fig. \[sso\_min\] depicts constant $P_{\alpha\beta}$ contours in the ($\Delta m^2_{21}\times\sin^2\omega$)-plane. One should be able to see upon close inspection that the region $P_{ee}<30\%$ is smaller in Fig. \[sso\_min\] than the same region in the pure vacuum oscillation case, Fig \[dm21\_vacuum\]. Also, the constant $P_{\mu\mu}$ ($P_{\tau\tau}$) contours are shifted to larger (smaller) values of $\sin^2\omega$. The other prominent (and expected, as mentioned above) feature is the distortion of the contours at large values of $\Delta m^2_{21}$. This behaviour is similar to the one observed in Fig. \[dm21\_std\]. I conclude this subsection with a comment on antineutrinos. As discussed previously, $P_{\bar{\alpha}\bar{\beta}}(\Delta m^2_{21},\Delta m^2_{31}) =P_{\alpha\beta}(-\Delta m^2_{21},-\Delta m^2_{31})$, such that the “normal” hierarchies yield “inverted” hierarchy results for antineutrinos, and vice-verse. One cannot, however, apply Fig. \[dm21\_std\] and Fig. \[sso\_min\] for the antineutrinos because both $\Delta m^2_{ij}$ have to change sign, not just $\Delta m^2_{31}$. Qualitatively, however, it is possible to understand the constant $P_{\bar{\alpha}\bar{\beta}}$ contours by examining figures Fig. \[dm21\_std\] and Fig. \[sso\_min\] reflected in a mirror positioned at $\sin^2\omega=0.5$, meaning that $P_{\bar{\alpha}\bar{\beta}}(\sin^\omega,\Delta m^2_{31})\simeq P_{\alpha\beta}(\cos^2\omega,-\Delta m^2_{31})$. The equality is not complete because one is also required to exchange $\theta\rightarrow \pi-\theta$, as mentioned earlier. Higher Neutrino Energies ------------------------ As the average neutrino energy increases, the values of $P_{\alpha\beta}$ start to resemble more the pure vacuum case. This is easy to see from Figs. \[p3h\] and \[1-pcl\]. Any deviation of $1-P_c^L$ from $\sin^2\omega$ goes away even at LMA for $E_{\nu}\simeq 50$ GeV, while “H-M” effects remain important up to $E_{\nu}\simeq 1$ TeV, even though quantitatively the effect decreases noticeably. This can be illustrated by the value of $P_3^H$ at ATM, for example, which drops from 0.87 for energies which range from 1 to 5 GeV (see the previous subsections) to 0.058, for energies which range from 100 to 110 GeV. Furthermore, all $L^{\rm osc}_{ij}$ increase as the energy increases, for fixed values of $\Delta m^2_{ij}$. Therefore, LOW becomes indistinguishable from VAC at $E_{\nu}\simeq 100$ GeV. For $O$(TeV) neutrinos the sensitivity to $\Delta m^2_{21}$ remains only for its highest allowed values, while one should start worrying about nontrivial oscillatory effects due to $L_{31}^{\rm osc}$. The case of higher energy neutrinos contains a more serious complications: neutrino absorption inside the Sun. As the neutrino energy increases, one has to start worrying about the fact that absorptive neutrino interactions can take place. According to [@absorption], for neutrinos produced in the Sun’s core, absorption becomes important for $E_{\nu}\gtrsim 200$ GeV. In this case, $\nu_e$ and $\nu_\mu$ interact with nuclear matter and produce electrons and muons, respectively. The former are capture and “lost” inside the Sun, while the latter stop before decaying into low energy neutrinos. The case of $\nu_{\tau}$-Sun interactions is more interesting, because the $\tau$-leptons produced via charged current interactions decay before “stopping”, yielding $\nu_{\tau}$’s with slightly reduced energies. Therefore, it is possible to get a flux of very high energy initial state $\tau$-neutrinos but not muon or electron-type neutrinos. Such effects have been studied for high energy galactic neutrinos traversing the Earth [@absorption_earth]. The effect of neutrino oscillations inside the Sun in the presence of nonnegligible neutrino absorption is certainly of great interest but is beyond the scope of this paper. Conclusions =========== The oscillation probability of $O$(GeV) neutrinos of all flavours produced in the Sun’s core has been computed, including matter effects, which are, in general, nontrivial. In particular, it was shown that, unlike the two-flavour oscillation case, in the three-flavour case the probability of a neutrino produced in the flavour eigenstate $\alpha$ to be detected as a flavour eigenstate $\beta$ ($P_{\alpha\beta}$) is (in general) different from $P_{\beta\alpha}$, even if the $CP$-violating phase of the MNS matrix vanishes. This is, of course, expected since Sun–neutrino interactions explicitly break $T$-invariance. Indeed, it is the case of two-flavour oscillations which is special, in the sense that the number of independent oscillation probabilities is too small because of unitarity. The results of a particular scan of the parameter space are presented in Sec. 4. In this case, special attention was paid to the regions of the parameter space which are preferred by the current experimental situation. It turns out that, in the case of a “normal” neutrino mass hierarchy, it is possible to suppress $P_{ee}$ tremendously with respect to its pure vacuum oscillation values, by a mechanism that is similar to the well known MSW effect in the case of two-flavour oscillations: the parameters are such that electron-type neutrinos produced in the Sun’s core exit the Sun (almost) as pure mass eigenstates, and the $\nu_e$ component of this eigenstate is small. Both $P_{\mu\mu}$ and $P_{\tau\tau}$ can be significantly suppressed, and the constant $P_{\mu\mu}$ and $P_{\tau\tau}$ contours as a function of the “solar” angle and the smaller mass-squared differences are nontrivial. One important feature is that when $P_{\mu\mu}$ is significantly suppressed, $P_{\tau\tau}$ is not, and vice-versa. One consequence of this is that, for some regions of the parameter space, it is possible to have an enhancement of $\nu_{\tau}$’s detected in the Earth with respect to the number of $\nu_{\mu}$’s (or vice-versa). This may have important implications for solar WIMP annihilation searches at neutrino telescopes, and will be studied in another oportunity. It is important to note that the effect of neutrino oscillations on the expected event rate at neutrino telescopes will depend on the expected production rate of individual neutrino species inside the Sun, which is, of course, model dependent. In the case of an “inverted” mass hierarchy, the situation is very similar to the pure vacuum case, and no particular suppression of any $P_{\alpha\alpha}$ is possible. Indeed, for a large region of the parameter space $P_{ee}$ is in fact enhanced, a feature which is also observed in the two-flavour case [@earth_matter]. The case of higher energy neutrinos was very briefly discussed, and the crucial point is to note that, for neutrino energies above a few hundred GeV, the absorption of neutrinos by the Sun becomes important. The study of absorption effects is beyond the scope of this paper. Finally, it is important to reemphasise that the values of $P_{\alpha\beta}$ computed here are to be understood as if they were evaluated at the Earth’s surface. No Earth-matter effects have been included. It is possible that Earth-matter effects are important, especially the ones related to $\Delta m^2_{31}$, in the advent that $U_{e3}^2\equiv\sin^2\xi$ turns out to be “large.” Acknowledgements {#acknowledgements .unnumbered} ================ I would like to thank John Ellis for suggesting the study of GeV solar neutrinos, and for many useful discussions and comments on the manuscript. 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[^3]: The most general form of a $2\times 2$ unitary matrix is $\left(\matrix{A & B \cr -B^* & A^*}\right)\left(\matrix{1 & 0 \cr 0 & e^{i\zeta}} \right)$, where $|A|^2+|B|^2=1$ and $0\leq\zeta\leq 2\pi$. In the case of neutrino oscillations, however, the physical quantities are $|A|^2$ and the phase of $AB^*$, and therefore $\zeta$ can be ignored. [^4]: This is in general the case, because one has to consider that neutrinos are produced at different points in space and time. [^5]: See [@bksreview; @rate_analysis; @dark_side] for the labelling of the regions of the parameter space that solve the solar neutrino puzzle [^6]: If one decides to limit $0\leq\vartheta\leq\pi/4$, a similar result can be obtained if $\Delta m^2\rightarrow-\Delta m^2$, explicitly $P_{\bar{e}\bar{e}}(\Delta m^2)=P_{ee}(-\Delta m^2)$. [^7]: There is evidence for neutrino oscillations coming from the LSND experiment [@LSND]. Such evidence has not yet been confirmed by another experiment, and will not be considered in this paper. If, however, it is indeed confirmed, it is quite likely that a fourth, sterile, neutrino will have to be introduced into the picture. [^8]: I will work under this assumption for the time being.

ArXiv

--- abstract: 'In this paper we study the ratio between the number of $p$-elements and the order of a Sylow $p$-subgroup of a finite group $G$. As well known, this ratio is a positive integer and we conjecture that, for every group $G$, it is at least the $(1-\frac{1}{p})$-th power of the number of Sylow $p$-subgroups of $G$. We prove this conjecture if $G$ is $p$-solvable. Moreover, we prove that the conjecture is true in its generality if a somewhat similar condition holds for every almost simple group.' address: 'Dipartimento di Matematica e Informatica “U. Dini”,Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy.' author: - Pietro Gheri title: 'On the number of $p$-elements in a finite group' --- Introduction ============ Let $G$ be a finite group and $p$ be a prime dividing the order of $G$. Moreover, let $$\mathfrak{U}_p(G) = \bigcup_{P \in Syl_p(G)} P,$$ be the set of $p$-elements of $G$. A celebrated theorem of F.G. Frobenius ([@frobenius:sylow]) states that if $P$ is a Sylow $p$-subgroup of $G$, then $|P|$ divides $|\mathfrak{U}_p(G)|$. We will call the positive integer $|\mathfrak{U}_p(G)|/|P|$ the *$p$-Frobenius ratio of $G$*. The number of $p$-elements of a finite group is a fundamental invariant in finite group theory. Several different proofs of Frobenius’ theorem have been given (see, for example, [@isaacs:frobenius] and [@speyer:frobenius]). Moreover in [@steinberg:endom Theorem 15.2] it is proven that the $p$-Frobenius ratio in a finite group of Lie type is equal to the size of a Sylow $p$-subgroup. Nevertheless, it is still unknown if the Frobenius ratio has a combinatorial meaning. It is clear that the $p$-Frobenius ratio is $1$ if and only if $G$ contains a normal Sylow $p$-subgroup. In [@miller:theoryapplications], it is proven with a nice and easy argument that if the $p$-Frobenius ratio is not $1$, then it must be greater or equal than $p$. In this paper, we focus on the search for “good” bounds for the $p$-Frobenius ratio in terms of the number $n_p(G)$ of Sylow $p$-subgroups of $G$. Of course, a trivial upper bound is obtained when every pair of Sylow p-subgroups of G has trivial intersection, so that, given a Sylow $p$-subgroup $P$ of $G$, $$\frac{|\mathfrak{U}_p(G)|}{|P|} \leq n_p(G)- \frac{n_p(G)-1}{|P|} \leq n_p(G).$$ It is not hard to find examples of sequences of groups that show that a lower bound on the $p$-Frobenius ratio cannot be linear in $n_p(G)$. We state the following conjecture. Let $G$ be a finite group, $p$ be a prime dividing $|G|$ and $P$ a Sylow $p$-subgroup of $G$. Then $$\label{conj pfrobratio} \frac{|\mathfrak{U}_p(G)|}{|P|} \geq n_p(G)^{1-\frac{1}{p}}.$$ We will show in Example \[tight frob ratio\] that this bound is “asymptotically tight”. We show that Conjecture \[conj pfrobratio\] is true for $p$-solvable groups. Namely, we prove the following. \[bound on Omega psolv\] Let $G$ be a finite $p$-solvable group and $P$ be a Sylow $p$-subgroup of $G$. If $n_p(G)$ denotes the number of Sylow $p$-subgroups in $G$, then $$\label{bound p el} \frac{|\mathfrak{U}_p(G)|}{|P|} \geq n_p(G)^{\frac{p-1}{p}}.$$ Inspired by the proof of Theorem \[bound on Omega psolv\], we show that a sufficient condition for Conjecture \[conj pfrobratio\] to be true in general is that $$\label{conj lambda} \left( \prod_{x \in P} \lambda_G(x) \right)^{1/|P|} \leq n_p(G)^{\frac{1}{p}},$$ where for every $p$-element $x$ of $P$, $\lambda_G(x)$ denotes the number of Sylow $p$-subgroups of $G$ containing $x$. For this condition we give a reduction to almost simple groups. \[red alm simp\] Inequality (\[conj lambda\]) holds for every finite group if and only if it holds for every finite almost simple group. One of the most important tools used here is the so-called Wielandt’s subnormalizer (see Definition \[subnormalizer\]), which is related to the number of $p$-elements (see Lemma \[form Omega subnor\]). This connection is mainly due to the works of C. Casolo on subnormalizers ([@casolo:subnor], [@casolo:subnorsolv]). Another fundamental tool for the proof of our result is a theorem by G. Navarro and N. Rizo, concerning the number of fixed points in a coprime action of a $p$-group. Throughout the paper $G$ will be a finite group and $p$ a prime dividing $|G|$. Also, for all $x \in G$ we denote with $x^G$ the conjugacy class of $x$ in $G$. The p-solvable case =================== In this section we prove Theorem \[bound on Omega psolv\]. First of all we introduce the we introduce the concept of subnormalizer, whose definition (see [@lennox:subnormal pag. 238]) is inspired by the celebrated Wielandt’s subnormality criterion, which says that a subgroup $H$ of $G$ is subnormal in $G$ if and only if $H$ is subnormal in $\langle H, g \rangle$ for every $g \in G$. \[subnormalizer\] Let $H$ be a subgroup of $G$. The *subnormalizer of $H$ in $G$* is the set $$S_G(H) = \lbrace g \in G \ | \ H {\unlhd \unlhd \ }\langle H,g \rangle \rbrace.$$ where ${\unlhd \unlhd \ }$ means “is subnormal in”. A useful link between subnormalizers and the number of p-elements in a finite group (see Lemma \[form Omega subnor\]) is established by using a beautiful theorem by C. Casolo In order to state this theorem we introduce some notation. Let $H$ be a $p$-subgroup of $G$ and $P$ be a Sylow $p$-subgroup of $G$. We write $\lambda_G(H)$ for the number of Sylow $p$-subgroups of $G$ containing $H$ and $\alpha_G(H)$ for the number of $G$-conjugates of $H$ contained in $P$ (note that this number does not depend on the Sylow subgroup $P$ we are considering). When $H = \langle x \rangle $ is a cyclic subgroup, we simply write $S_G(x)$ and $\lambda_G(x)$, in place of $S_G(\langle x \rangle)$ and $\lambda_G(\langle x \rangle)$. In a similar fashion, we write $\alpha_G(x)$ for the number of $G$-conjugates of the element $x$ contained in $P$. We thus have that $$\label{alpha x} \alpha_G(x)=\alpha_G \left( \langle x \rangle \right) |N_G(\langle x \rangle)|/|C_G(x)|.$$ We can now state the aforementioned theorem by C. Casolo. \[form subnor theorem\] Let $H$ be a $p$-subgroup of $G$. Then the following holds and $P$ be a Sylow $p$-subgroup of $G$. - $$|S_G(H)|=\lambda_G(H) |N_G(P)| = \alpha_G(H) |N_G(H)|.$$ - If $G$ is $p$-solvable and $\mathcal{M}$ is the set of all $p'$-factors in a given normal $\lbrace p,p' \rbrace$-series of $G$, then $$|S_G(H)|= |P| \prod_{U/V \in \mathcal{M}} \left| C_{U/V} \left( HV/V \right) \right|.$$ A first easy application of this result is a formula that expresses the number of $p$-elements in $G$ in terms of the orders of the subnormalizers of the cyclic subgroups of a Sylow $p$-subgroup of $G$. \[form Omega subnor\] Let $P$ be a Sylow $p$-subgroup of $G$. We have $$|\mathfrak{U}_p(G)|=\sum_{x \in P} \frac{|G|}{|S_G(x)|}.$$ In the sum $$\sum_{x \in P} |x^G|$$ every class of $p$-elements is involved and its contribution is repeated as many times as the cardinality $|x^G \cap P|=\alpha_G(x)$. Hence $$|\mathfrak{U}_p(G)| = \sum_{x \in P} \frac{|x^G|}{\alpha_G(x)}=\sum_{x \in P} \frac{|G|}{\alpha_G(x)|C_G(x)|}=\sum_{x \in P} \frac{|G|}{|S_G(x)|},$$ by part *a)* of Theorem \[form subnor theorem\] and formula (\[alpha x\]). We now turn to the proof of Theorem \[bound on Omega psolv\]. Another fundamental tool that we are going to use in the proof is the following formula proved by Navarro and Rizo. \[nav riz\] Suppose that $P$ is a $p$-group acting on a $p'$-group $G$. Then $$|C_G(P)| = \left( \prod_{x \in P} \frac{|C_G(x)|}{|C_G(x^p)|^{1/p}} \right)^{\frac{p}{(p-1)|P|}}.$$ We can now prove Theorem \[bound on Omega psolv\]. By Lemma \[form Omega subnor\] we have that the $p$-Frobenius ratio of $G$ is the arithmetic mean of the ratios $$\frac{|G|}{|S_G(x)|}$$ when $x$ runs across $P$. By the Arithmetic-Geometric Mean Inequality, we get $$\label{FrobRatio arit geom} \frac{|\mathfrak{U}_p(G)|}{|P|} = \frac{1}{|P|} \left( \sum_{x \in P} \frac{|G|}{|S_G(x)|} \right) \geq \left( \prod_{x \in P} \frac{|G|}{|S_G(x)|} \right)^{1/|P|}.$$ Since $G$ is $p$-solvable we can take a normal $\lbrace p,p' \rbrace$-series, whose set of $p'$-factors we call $\mathcal{M}$. Then, by part *b)* of Theorem \[form subnor theorem\], we have for all $x \in P$ $$\frac{|G|}{|S_G(x)|} = \frac{|G|/|P|}{\prod_{U/V \in \mathcal{M}}|C_{U/V}(Vx)|} = \prod_{U/V \in \mathcal{M}} \frac{|U/V|}{|C_{U/V}(Vx)|}.$$ We insert this last term in (\[FrobRatio arit geom\]) and swap the products to get $$\begin{aligned} \frac{|\mathfrak{U}_p(G)|}{|P|} & \geq & \left( \prod_{U/V \in \mathcal{M}} \left( \prod_{x \in P} \frac{|U/V|}{|C_{U/V}(xV)|} \right) \right)^{1/|P|} \\ & = & \left( \prod_{U/V \in \mathcal{M}} \left| \frac{U}{V} \right|^{|P|} \left( \prod_{x \in P} \frac{1}{|C_{U/V}(xV)|} \right) \right)^{1/|P|}.\end{aligned}$$ Now for all $U/V \in \mathcal{M}$, $P$ is a $p$-group that acts on the $p'$-group $U/V$. We can then apply Theorem \[nav riz\] and use the trivial inequality $|C_{U/V}((xV)^p)| \leq |U/V|$, so that we have $$\begin{aligned} \prod_{x \in P} \frac{1}{|C_{U/V}(xV)|} & = & \left( \prod_{x \in P} \frac{1}{|C_{U/V}((xV)^p)|^{1/p}} \right) \frac{1}{|C_{U/V}(P)|^{|P|(p-1)/p}} \\ & \geq & \left( \frac{1}{|U/V|^{|P|/p}}\right)\frac{1}{|C_{U/V}(P)|^{|P|(p-1)/p}}\end{aligned}$$ and so $$\frac{|\mathfrak{U}_p(G)|}{|P|} \geq \left( \prod_{U/V \in \mathcal{M}} \frac{|U/V|}{|C_{U/V}(P)|} \right)^{\frac{p-1}{p}} = \left( \frac{|G|}{|S_G(P)|} \right)^{1-\frac{1}{p}},$$ again by part *b)* of Theorem \[form subnor theorem\]. Finally, we observe that for a Sylow $p$-subgroup $S_G(P)=N_G(P)$, so that $$\frac{|\mathfrak{U}_p(G)|}{|P|} \geq \left( \frac{|G|}{|S_G(P)|} \right)^{1-\frac{1}{p}} = \left( \frac{|G|}{|N_G(P)|} \right)^{1-\frac{1}{p}} = n_p(G)^{1-\frac{1}{p}}.$$ It is worth mentioning that the bound in Theorem \[bound on Omega psolv\] is asymptotically tight in the sense specified by the following example. \[tight frob ratio\] Let $p$ be a prime and, for $n$ a positive integer, let $P$ be an elementary abelian group of order $p^n$. Moreover set $\mathcal{M}$ to be the set of the maximal subgroups of $P$. Choose a prime $q$ such that $p$ divides $q-1$. Then for any $M \in \mathcal{M}$ we have that $P/M \simeq C_p$ acts fixed point freely as a group of automorphisms on a cyclic group $\langle a_M \rangle \simeq C_q$. We denote the image of the generator $a_M$ under this action by $a_M^{xM}$, for every $xM \in P/M$. Since $\bigcap_{M \in \mathcal{M}} M =1$, it follows that $P$ acts faithfully on the direct product $N$ of the groups $\langle a_M \rangle$. To be more explicit the following map $$\begin{aligned} P & \rightarrow Aut(N) \\ x & \mapsto \phi_x,\end{aligned}$$ where $\phi_x(a_M)=a_M^{xM}$, for all $M \in \mathcal{M}$ is an injective homomorphism. We consider the semidirect product $G_n = N \rtimes P$. The normalizer of $P$ in $G_n$ is $C_N(P)P=P$, hence the number of Sylow $p$-subgroups of $G_n$ is $$\label{npgn} n_p(G_n)= |N| = q^{|\mathcal{M}|}=q^{\frac{p^n-1}{p-1}}.$$ In order to count the number of $p$-elements in $G_n$ we use the equality $$| \mathfrak{U}_p(G_n) | = \sum_{x \in P} \frac{n_p(G_n)}{\lambda_{G_n}(x)},$$ which follows from Lemma \[form Omega subnor\] and part *a)* of Theorem \[form subnor theorem\]. We thus have to compute $\lambda_{G_n}(x)$, for $x \in P \setminus \lbrace 1 \rbrace$. Using again part *a)* of Theorem \[form subnor theorem\], we have $$\lambda_{G_n}(x)= \frac{\alpha_{G_n}(x)n_p(G_n)}{|x^{G_n}|}.$$ Now since $P$ is abelian and $G_n$ has a normal $p$-complement, we have $\alpha_{G_n}(x)=1$ and $|x^{G_n}|=|N|/|C_N(x)|$, so that $\lambda_{G_n}(x)=|C_N(x)|$. Given $x \in P \setminus \lbrace 1 \rbrace$, the centralizer of $x$ in $N$ is generated by those $a_M$ such that $a_M^{xM}=a_M$. Since $P/M$ acts fixed point freely on $\langle a_M \rangle$, this holds if and only if $x \in M$, hence $$C_N(x)= \langle a_M \ | \ x \in M \rangle.$$ The number of maximal subgroups in $P$ containing a fixed nontrivial element is $\frac{p^{n-1}-1}{p-1}$, and so $$|C_N(x)|=q^{\frac{p^{n-1}-1}{p-1}}.$$ We can then calculate the $p$-Frobenius ratio of $G_n$ $$\begin{split} \frac{|\mathfrak{U}_p(G_n)|}{|P|} &= \frac{1}{|P|} \sum_{x \in P} \frac{n_p(G_n)}{\lambda_{G_n}(x)} \\ & = \frac{1}{p^n} \left( 1 + (p^n-1) \frac{q^\frac{p^{n}-1}{p-1}}{q^\frac{p^{n-1}-1}{p-1}} \right) \\ & = \frac{1}{p^n} + \frac{p^n-1}{p^n} q^{p^{n-1}}. \end{split}$$ By (\[npgn\]) we have $$n_p(G_n)^{\frac{p-1}{p}}= q^{\frac{p^n-1}{p}}.$$ We can now compare the two members of the inequality stated by Theorem \[bound on Omega psolv\]. By considering the limit $$\lim_{n \rightarrow \infty} \frac{|\mathfrak{U}_p(G_n)|/|P|}{n_p(G_n)^{\frac{p-1}{p}}} = \lim_{n \rightarrow \infty} \left( \frac{1}{p^n q^{\frac{p^n-1}{p}}} + \frac{p^n-1}{p^n} \frac{q^{p^{n-1}}}{q^{\frac{p^n-1}{p}}} \right) = q^{1/p},$$ we see that the $p$-Frobenius ratio of $G_n$ and the $\left( 1-\frac{1}{p} \right)$-th power of the number of Sylow $p$-subgroups have the same asymptotic behaviour. The general case ================ In this section we explain why inequality (\[conj lambda\]) is sufficient for establishing Conjecture \[conj pfrobratio\] and we prove Theorem \[red alm simp\]. Let $P$ a Sylow $p$-subgroup of $G$. Since Lemma \[form Omega subnor\] is true for any group, by applying the Arithmetic-Geometric Mean Inequality as in \[FrobRatio arit geom\], we get $$\frac{|\mathfrak{U}_p(G)|}{|P|} \geq \left( \prod_{x \in P} \frac{|G|}{|S_G(x)|} \right)^{1/|P|}$$ and, recalling Theorem \[form subnor theorem\], we can write $$\frac{|\mathfrak{U}_p(G)|}{|P|} \geq \left( \prod_{x \in P} \frac{n_p(G)}{\lambda_G(x)} \right)^{1/|P|}.$$ A sufficient condition for (\[bound p el\]) is then $$\left( \prod_{x \in P} \frac{n_p(G)}{\lambda_G(x)} \right)^{1/|P|} \geq n_p(G)^{\frac{p-1}{p}},$$ that is $$\left( \prod_{x \in P} \lambda_G(x) \right)^{1/|P|} \leq n_p(G)^{\frac{1}{p}},$$ which is inequality (\[conj lambda\]). In [@gheri:degnil] it is proven that if $x$ is a $p$-element of $G$ which is not contained in the $O_p(G)$, then $\lambda_G(x)$ is at most $n_p(G)/(p+1)$. Focusing on a single element, this is the best one can get. Inequality (\[conj lambda\]), if true, would give a better bound on average, as it states that the geometric mean of the number of Sylow $p$-subgroups containing an element of a Sylow $p$-subgroup is at most the $p$-th root of the total number of Sylow $p$-subgroups. The bound (\[conj lambda\]), if true, is best possible in a strict sense. If we compute the terms of inequality (\[conj lambda\]) for the groups $G_n$ defined in Example \[tight frob ratio\], an equality occurs. \[Op 1\] For the proof of Theorem \[red alm simp\], we can assume $O_p(G)=1$. This is because if $N$ is a normal $p$-subgroup of $G$, then, for all $x \in \mathfrak{U}_p(G)$, we have that $\lambda_G(x)=\lambda_{G/N}(xN)$, and so (\[conj lambda\]) holds for $G$ if and only if it holds for $G/N$. First of all, we can reduce (\[conj lambda\]) to nonsolvable groups all of whose proper quotients are solvable (see Proposition \[reduction conj mns\]). We need some technical lemmas, the first of whom is proved in [@gheri:unsesto Lemma 3.3]. \[sp x cresce sui sottogruppi\] Let $H$ be a subgroup of $G$ and $x \in H$ be a $p$-element. Then $$\frac{\lambda_G(x)}{n_p(G)} \leq \frac{\lambda_H(x)}{n_p(H)}.$$ Moreover, if $H \unlhd G$, then the equality holds. \[SNx\] Let $N$ be a normal subgroup of $G$, $P$ be a Sylow $p$-subgroup of $G$ and $x$ an element of $P$. Assume that $G=NP$. Then $$|S_G(x)|=|S_N(x)| |NP/N|$$ If $x \in N$, the thesis follows from the fact that $|S_G(x)|/|G|=|S_N(x)|/|N|$, which can be easily derived from part *a)* of Theorem \[form subnor theorem\]. We work by induction on $m$, where $|NP/N|=p^m$. If $m=0$, then $G=N$ and there is nothing to prove. Suppose that $m > 0$ and $G \neq N \langle x \rangle$. Let $M$ be a maximal subgroup of $G$ containing $N \langle x \rangle$. Then, as $M \unlhd G$, by inductive hypothesis $$|S_G(x)|=|S_M(x)|p=|S_N(x)|p^{m-1}p.$$ Finally, if $G=N \langle x \rangle$, we observe that, given $a \in N$ and $t$ a positive integer, $ax^t \in S_G(x)$ if and only if $\langle x \rangle$ is subnormal in $\langle x, ax^t \rangle = \langle x, a \rangle $, that is if and only if $a \in S_G(x)$. It follows that $$|S_G(x)|=|S_N(x)| |NP/N|.$$ \[lambda np\] Let $N$ be a normal subgroup of $G$, $P$ be a Sylow $p$-subgroup of $G$ and $x \in P$. Then $$\lambda_G(x)= \lambda_{\frac{G}{N}}(Nx) \lambda_{NP}(x).$$ First of all we show that given a $p$-element $x$, the value $\lambda_{NP}(x)$ is independent of the particular Sylow $p$-subgroup $P$ containing $x$. By Theorem \[form subnor theorem\] and Lemma \[SNx\], we have $$\lambda_{NP}(x)= \frac{|S_{NP}(x)|}{|N_{NP}(P)|} =\frac{|S_{N}(x)|}{|N_{NP}(P)|} \left| \frac{NP}{N} \right| = \frac{|S_{N}(x)|}{|N|} |n_p(NP)| .$$ If $Q$ is another Sylow $p$-subgroup such that $x \in Q$, then of course $n_p(NP)=n_p(NQ)$, since $NP$ and $NQ$ are conjugated in $G$. Moreover $|S_N(x)|$ depends only on $N$ and $x$. Now let $\Delta_G^x$ be the set of the Sylow $p$-subgroups of $G$ containing $x$. We define the map $$\begin{split} \Delta_G^x & \rightarrow \Delta_{\frac{G}{N}}^{xN} \\ Q & \mapsto NQ/N. \end{split}$$ For all $\tilde{Q} \in \Delta_G^x$, the fiber of $N\tilde{Q}/N \in \Delta_{\frac{G}{N}}^{xN}$ is the set of Sylow $p$-subgroups $Q$ of $G$ containing $x$ and such that $NQ=N\tilde{Q}$, that is, $\Delta_{N \tilde{Q}}^{x}$. Since we proved that $\lambda_{N\tilde{Q}}(x)$ is independent of $\tilde{Q}$, we have $$\lambda_G(x)= | \Delta_{G}^{x}| = \left| \Delta_{\frac{G}{N}}^{xN}\right| \left| \Delta_{NP}^{x} \right| = \lambda_{\frac{G}{N}}(xN) \lambda_{NP}(x).$$ \[reduction conj mns\] A counterexample of minimal order to inequality (\[conj lambda\]) is a nonsolvable group having a unique minimal normal subgroup $M$, which is nonsolvable, and such that $G=MP$, where $P$ is a Sylow $p$-subgroup of $G$. Let $G$ be a counterexample of minimal order to inequality \[conj lambda\] and let $P$ be a Sylow $p$-subgroup of $G$. By Remark \[Op 1\] we have that $O_p(G)=1$. By Theorem \[bound on Omega psolv\], $G$ is nonsolvable. We show that every proper quotient of $G$ is solvable. Let $M$ be a minimal normal subgroup of $G$. By Lemma \[lambda np\], we have $$\begin{aligned} \prod_{x \in P} \lambda_G(x) & = & \prod_{x \in P} \lambda_{\frac{G}{M}}(Mx) \lambda_{MP}(x) \\ &=& \left( \prod_{xM \in \frac{PM}{M}} \lambda_{\frac{G}{M}}(Mx) \right)^{|P \cap M|} \left( \prod_{x \in P} \lambda_{MP}(x) \right). \end{aligned}$$ By the minimality of $G$, we have $$\prod_{xM \in \frac{PM}{M}} \lambda_{\frac{G}{M}}(Mx) \leq n_p \left( G/M \right)^{\frac{|PM/M|}{p}}.$$ If $MP<G$, we can again assume that the inequality is true for $MP$ and so $$\prod_{x \in P} \lambda_G(x) \leq n_p\left( \frac{G}{M} \right)^{\frac{|PM/M|}{p}|P \cap M|} n_p(MP)^\frac{|P|}{p}=n_p(G)^\frac{|P|}{p}.$$ Since this is true for every minimal normal subgroup of $G$, the usual subdirect product argument gives that $G$ has a unique minimal normal subgroup $M$, which is nonsolvable and such that $G=MP$. Referring to the notation of Lemma \[reduction conj mns\], $M$ is the direct product of simple groups permuted by $P$. The next easy lemma loosely bounds the number of $\langle x \rangle$-invariant Sylow $p$-subgroups of $M$, where $x \in P$, in terms of the action of $\langle x \rangle$ on the direct factors of $M$. \[num orb\] Let $M \unlhd G$ be a direct product of $k$ copies of a group $L$, $M=L_1 \times \dots \times L_k$, let $x \in G$ be an element that permutes the factors $L_i$ of $M$ and let $p$ be a prime number. Then the number of $\langle x \rangle$-invariant Sylow $p$-subgroups of $M$ is at most $n_p(L)^s$, where $s$ is the number of orbits of $\langle x \rangle$ on $\Delta= \lbrace L_1, \dots, L_k \rbrace$. A Sylow $p$-subgroup $Q$ of $M$ is the direct product of $k$ Sylow $p$-subgroups of $L$, $Q = Q_1 \times \dots \times Q_k$. Suppose that $Q$ is normalized by $x$. If $L_i=L_j^{x^r}$, for $r \in \mathbb{Z}$, then $Q_i=Q_j^{x^r}$ and so one has at most $n_p(L)$ choices for each $\langle x \rangle$-orbit in $\Delta$. We can now prove Theorem \[red alm simp\] Suppose that inequality (\[conj lambda\]) is true for all finite almost simple groups and let $G$ be a counterexemple of minimal order. By Proposition \[reduction conj mns\], $G=MP$ where $P$ is a Sylow $p$-subgroup of $G$ and $M$ is the unique minimal normal subgroup $$M = L_1 \times \dots \times L_k, \ L_i \simeq L, \ \forall i \in \lbrace 1, \dots , k \rbrace.$$ for some nonabelian simple group $L$. Moreover $P$ acts transitively on the set $\Delta = \lbrace L_1, \dots , L_k \rbrace$. Since we are assuming the result for almost simple groups, we have $k>1$. Let $Q=P \cap M$. For any subgroup $Q \leq X \leq P$ we set $m_X$ to be the ratio $$\label{mX} m_X=\frac{|N_M(X)|}{|N_M(P)|}= \frac{n_p(G)}{n_p(MX)}.$$ The last equality holds since $n_p(G)=[MP:N_{MP}(P)]=[M:N_M(P)]$ and $n_p(MX)=[MX:N_{MX}(X)]=[M:N_M(X)]$. Moreover observe that if $g \in N_M(P)$ and $x\in X$ then $$x^g=x[x,g] \in X(M \cap P)=XQ=X,$$ and so $N_M(P) \leq N_M(X)$. With a slight abuse of notation, we denote with $\lambda_M(x)$ the number of Sylow $p$-subgroups in $M$ normalized by $x$ even for $x \notin M$. It is easy to check that $\lambda_M(x)=\lambda_{M\langle x \rangle}(x)$. Let $H_0=N_P(L_1)$ be the stabilizer of $L_1$ in the action of $P$ on $\Delta$. Since $P$ is transitive on $\Delta$, $H_0 \neq P$. Choose now a maximal subgroup $H$ of $P$ containing $H_0$. Since $H$ is normal in $P$ and the stabilizers of the subgroups $L_i$ are all conjugated in $P$, we have that $H$ contains all of them. It follows that every element $x \in P \setminus H$ has at most $k/p$ orbits on $\Delta$ and so by Lemma \[num orb\] $$\label{orbits} \lambda_M(x) \leq n_p(L)^{\frac{k}{p}}.$$ We now consider separately the elements inside and outside $H$. As for the elements inside $H$, since $MH$ is normal in $G$, using Lemma \[sp x cresce sui sottogruppi\], we get $$\begin{split} \prod_{x \in H} \lambda_G(x) &= \prod_{x \in H} \frac{n_p(G)}{n_p(MH)}\lambda_{MH}(x) \\ &= \prod_{x \in H} m_H \lambda_{MH}(x)=(m_H)^{|H|} \prod_{x \in H} \lambda_{MH}(x). \end{split}$$ Since $H$ is a Sylow $p$-subgroup of $MH$ and inequality (\[conj lambda\]) holds for $MH<G$ we get $$\label{in H} \begin{split} \prod_{x \in H} \lambda_G(x) & = (m_H)^{|H|} \prod_{x \in H} \lambda_{MH}(x) \\ & \leq (m_H)^{|H|} n_p(MH)^\frac{|H|}{p}=m_H^{\frac{p-1}{p}|H|} n_p(G)^\frac{|H|}{p}, \end{split}$$ where we applied (\[mX\]). Now we turn our attention on elements in $P \setminus H$. Let $\mathcal{T}$ be a set of representatives for the right cosets of $Q$ in $P$ that are not contained in $H$. The cardinality of $\mathcal{T}$ is then $$|\mathcal{T}|=[P:Q]-[H:Q]=\frac{|P|-|H|}{|Q|}=(p-1) \frac{|H|}{|Q|}.$$ We have, by Lemma \[sp x cresce sui sottogruppi\] $$\begin{aligned} \prod_{x \in P \setminus H} \lambda_G(x) & = & \prod_{x \in \mathcal{T}} \prod_{g \in Q} \lambda_G(gx) \leq \prod_{x \in \mathcal{T}} \left( \prod_{g \in Q} m_{Q\langle gx \rangle} \lambda_{M\langle gx \rangle}(gx) \right) \\ &=& \prod_{x \in \mathcal{T}} \left( m_{Q\langle x \rangle}^{|Q|} \prod_{g \in Q} \lambda_{M}(gx) \right).\end{aligned}$$ Now the elements $gx$ in the previous product are not in $H$ and so by (\[orbits\]) $$\label{prod lambda outside H} \begin{split} \prod_{x \in P \setminus H} \lambda_G(x) & \leq \left( \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} \right)^{|Q|} n_p(L)^{\frac{k}{p}|Q||\mathcal{T}|}\\ & = \left( \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} \right)^{|Q|} n_p(M)^{\frac{p-1}{p}|H|}. \end{split}$$ We now want to evaluate the product $\prod_{x \in \mathcal{T}} m_{Q\langle x \rangle}$. In the following we use the bar notation for the quotients modulo $Q$. If $R=N_M(Q)$ we have a coprime action of $\bar{P}$ on the $p'$-group $\bar{R}$. We apply Theorem \[nav riz\] to this action and get $$|C_{\bar{R}}(\bar{P})|^{|\bar{P}|\frac{p-1}{p}} = \prod_{x \in \bar{P}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(x^p)|^{1/p}}.$$ Separating the elements inside $\bar{H}$ and those outside $\bar{H}$ and applying twice Theorem \[nav riz\], we get $$\begin{split} |C_{\bar{R}}(\bar{P})|^{|\bar{P}|\frac{p-1}{p}} & = \left( \prod_{x \in \bar{H}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(x^p)|^{1/p}} \right) \left( \prod_{x \in \bar{P} \setminus \bar{H}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(x^p)|^{1/p}} \right) \\ & = |C_{\bar{R}}(\bar{H})|^{|\bar{H}|\frac{p-1}{p}} \left( \prod_{x \in \bar{P} \setminus \bar{H}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(x^p)|^{1/p}} \right). \end{split}$$ Using the bound $|C_{\bar{R}}(x^p)| \leq |\bar{R}|$ and the fact that $|\bar{P} \setminus \bar{H}|=(p-1) \left| \bar{H}\right|$, we have $$\label{prodCRx} \begin{split} \prod_{x \in \bar{P} \setminus \bar{H}} |C_{\bar{R}}(x)| & = |C_{\bar{R}}(\bar{P})^{|\bar{P}|\frac{p-1}{p}} |C_{\bar{R}}(\bar{H})|^{-|\bar{H}|\frac{p-1}{p}} \left( \prod_{x \in \bar{P} \setminus \bar{H}} |C_{\bar{R}}(x^p)|^{1/p} \right) \\ &\leq |C_{\bar{R}}(\bar{P})|^{|\bar{P}|\frac{p-1}{p}} |C_{\bar{R}}(\bar{H})|^{-|\bar{H}|\frac{p-1}{p}} |\bar{R}|^{|\bar{H}|\frac{p-1}{p}}. \end{split}$$ We now observe that if $X$ is a subgroup of $P$ containing $Q$, then $Q=X \cap M$, so that $N_M(X) \leq N_M(Q)=R$ and we have $$\label{NNX CRX} \overline{N_M(X)}=C_{\bar{R}}(\bar{X}).$$ and since $Q \leq N_M(X)$ $$m_X=\frac{|N_M(X)|}{|N_M(P)|} = \frac{\left| \overline{N_M(X)} \right|}{\left| \overline{N_M(P)} \right|} = \frac{|C_{\bar{R}}(\bar{X})|}{|C_{\bar{R}}(\bar{P})|}$$ Hence, for all $x \in \mathcal{T}$, $$m_{Q\langle x \rangle} = \frac{\left| C_{\bar{R}}(xQ) \right|}{\left| C_{\bar{R}}(\bar{P}) \right|}.$$ Going back to our product and using (\[prodCRx\]), we then get $$\begin{split} \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} & = \prod_{x \in \mathcal{T}} \frac{|C_{\bar{R}}(xQ)|}{|C_{\bar{R}}(\bar{P})|} =\prod_{x \in \bar{P} \setminus \bar{H}} \frac{|C_{\bar{R}}(x)|}{|C_{\bar{R}}(\bar{P})|} \\ & \leq |C_{\bar{R}}(\bar{P})|^{-|\bar{P}|\frac{p-1}{p}} |C_{\bar{R}}(\bar{P})|^{|\bar{P}|\frac{p-1}{p}} |C_{\bar{R}}(\bar{H})|^{-|\bar{H}|\frac{p-1}{p}} |\bar{R}|^{|\bar{H}|\frac{p-1}{p}} \\ & = |C_{\bar{R}}(\bar{H})|^{-|\bar{H}|\frac{p-1}{p}} |\bar{R}|^{|\bar{H}|\frac{p-1}{p}}. \end{split}$$ We now remove the bar notation using the definition of $R$ and (\[NNX CRX\]), $$\begin{split} \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} & = \left( \frac{|N_M(H)|}{|Q|} \right)^{-\frac{|H|}{|Q|}\frac{p-1}{p}} \left( \frac{|N_M(Q)|}{|Q|} \right)^{\frac{|H|}{|Q|}\frac{p-1}{p}} \\ & = \left( \frac{|N_M(Q)|}{|N_M(H)|} \right)^{\frac{|H|}{|Q|}\frac{p-1}{p}}. \end{split}$$ Using this bound in inequality (\[prod lambda outside H\]) we get $$\begin{split} \prod_{x \in P \setminus H} \lambda_G(x) & \leq \left( \prod_{x \in \mathcal{T}} m_{Q\langle x \rangle} \right)^{|Q|} n_p(M)^{\frac{p-1}{p}|H|} \\ & \leq \left( \frac{|N_M(Q)|}{|N_M(H)|} \right)^{\frac{p-1}{p}|H|} n_p(M)^{\frac{p-1}{p}|H|} \\ & \leq \left( \frac{|N_M(Q)|}{|N_M(H)|} n_p(M)\right)^{\frac{p-1}{p}|H|}, \end{split}$$ and since $Q$ is a Sylow $p$-subgroup of $M$, $$\label{not in H} \begin{split} \prod_{x \in P \setminus H} \lambda_G(x) \leq \left( \frac{|N_M(Q)|}{|N_M(H)|} \frac{|M|}{|N_M(Q)|}\right)^{\frac{p-1}{p}|H|} = \left( \frac{|M|}{|N_M(H)|}\right)^{\frac{p-1}{p}|H|} \end{split}$$ By combining (\[in H\]) and (\[not in H\]) we obtain $$\begin{split} \prod_{x \in P} \lambda_G(x) & = \left( \prod_{x \in H} \lambda_G(x) \right)\left( \prod_{x \in P \setminus H} \lambda_G(x) \right) \\ & \leq \left( m_H^{\frac{p-1}{p}|H|} n_p(G)^\frac{|H|}{p} \right) \left( \frac{|M|}{|N_M(H)|}\right)^{\frac{p-1}{p}|H|} \\ & = n_p(G)^\frac{|H|}{p} \left( m_H \frac{|M|}{|N_M(H)|}\right)^{\frac{p-1}{p}|H|}. \end{split}$$ Finally, recalling (\[mX\]) and the fact that $|H|=|P|/p$ we get $$\begin{split} \prod_{x \in P} \lambda_G(x) & \leq n_p(G)^\frac{|H|}{p} \left( \frac{|N_M(H)|}{|N_M(P)|} \frac{|M|}{|N_M(H)|}\right)^{\frac{p-1}{p}|H|} \\ &= n_p(G)^\frac{|H|}{p} n_p(G)^{\frac{p-1}{p}|H|} = n_p(G)^\frac{|P|}{p} \end{split}$$ which is against the fact that $G$ is a counterexample. Acknowledgements {#acknowledgements .unnumbered} ---------------- This article is part of the author’s PhD thesis, which was written under the great supervision of Carlo Casolo, whose contribution to this work was essential. Thanks are also due to Francesco Fumagalli and Silvio Dolfi for his valuable comments and suggestions. This work was partially funded by the Istituto Nazionale di Alta Matematica “Francesco Severi" (Indam). [99]{} C. Casolo, *On the subnormalizer of a $p$-subgroup*, J. Pure Appl. Algebra 77 (1992), no.3. C. Casolo, *Subnormalizers in finite groups*, Comm. Algebra 18 (1990), no. 11. G. Frobenius, *Verallgemeinerung des Sylowschen Satzes*, Sitzungsberichte der Preussischen Akademie, Berlin (1895). P. Gheri, *Subnormalizers and the degree of nilpotence in finite groups*, Proc. Amer. Math. Soc., DOI: https://doi.org/10.1090/proc/15080 (to appear in print). P. Gheri, *Subnormalizers and solvability in finite groups*, preprint, (2020). I.M. Isaacs and G.R. Robinson, *On a theorem of Frobenius: solutions of $x^n=1$ in finite groups*, Amer. Math. Monthly 99 (1992), no. 4. J.C. Lennox and S.E. Stonehewer, *Subnormal subgroups of groups*, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1987. G.A. Miller, H.F. Blichfeldt and L.E. Dickson, *Theory and applications of finite groups*, Dover Publications, Inc., New York 1961. G. Navarro and N. Rizo, *A Brauer-Wielandt formula (with an application to character tables)*, Proc. Amer. Math. Soc. 144 (2016), no. 10. Speyer, David E.(1-MI) A counting proof of a theorem of Frobenius. (English summary) Amer. Math. Monthly 124 (2017), no. 4. R. Steinberg, *Endomorphisms of linear algebraic groups*, Memoirs of the American Mathematical Society, No. 80 American Mathematical Society, Providence, R.I. 1968.

ArXiv

--- abstract: 'We conclude from an analysis of high resolution NYSE data that the distribution of the traded value $f_i$ (or volume) has a finite variance $\sigma_i$ for the very large majority of stocks $i$, and the distribution itself is non-universal across stocks. The Hurst exponent of the same time series displays a crossover from weakly to strongly correlated behavior around the time scale of $1$ day. The persistence in the strongly correlated regime increases with the average trading activity $\ev{f_i}$ as $H_i=H_0+\gamma\log\ev{f_i}$, which is another sign of non-universal behavior. The existence of such liquidity dependent correlations is consistent with the empirical observation that $\sigma_i\propto\ev{f_i}^\alpha$, where $\alpha$ is a non-trivial, time scale dependent exponent.' author: - Zoltán Eisler - János Kertész bibliography: - 'Eisler.bib' title: The dynamics of traded value revisited --- The recent years have seen a number of important contributions of physics to various areas, among them to finance [@bouchaud.book; @stanley.book]. The application of physical concepts often seems well suited to the analysis of financial time series, however, it is not without caveats. Often, the theoretical background of these methods is deeply rooted in physical laws that – naturally – do not apply to stock markets. In particular, observations regarding power laws [@futurepl], universality [@culturecrash], and other empirical regularities [@gallegati.etal] are often criticized. We carried out a thorough study of the traded value per unit time [@eisler.non-universality; @eisler.sizematters; @eisler.unified] and have arrived at the result that some earlier conclusions have to be modified. Here we present an analysis of some new data, which supports our earlier findings. The paper is organized as follows. Section \[sec:intro\] introduces notations. Section \[sec:value\] shows that the distribution of traded volume/value is not universal, and it is not in the Levy stable regime as suggested by Ref. [@gopi.volume]. Section \[sec:correl\] shows, that traded value displays only weak correlations for time scales shorter than one day. On longer horizons there is stronger persistence whose degree depends logarithmically on the liquidity of the stock. Finally, Section \[sec:alpha\] surveys the concept of fluctuation scaling, shows how it complements the observed liquidity dependence of correlations, and how those two form a consistent scaling theory. Notations and data {#sec:intro} ================== For a fixed time window size $\Delta t$, let us denote the total traded value of the $i$th stock at time $t$ by $$f_i^{\Delta t}(t) = \sum_{n, t_i(n)\in [t, t+\Delta t]} V_i(n), \label{eq:flow}$$ where $t_i(n)$ is the time when the $n$th transaction of the $i$th stock takes place. Tick-by-tick data are denoted by $V_i(n)$, this is the value traded in transaction $n$, calculated as the product of the price and the traded volume. Since price changes very little from trade to trade while variations of trading volume are much faster, the fluctuations of the traded value $f_i(t)$ are basically determined by those of traded volume. Price merely acts as a weighting factor that enables one to compare different stocks, while this also automatically corrects the data for stock splits and dividends. The correlation properties and the normalized distribution are nearly indistinguishable between traded volume and traded value. This study is based on the complete Trades and Quotes database of New York Stock Exchange for the period $1994-1995$. Note that throughout the paper we use $10$-base logarithms. Traded value distributions revisited {#sec:value} ==================================== In this section, we first revisit the analysis done in Ref. [@gopi.volume]. That work finds that the cumulative distribution function of traded volume for time windows of $\Delta t = 15$ minutes decays as a power-law with a tail exponent $\lambda = 1.7 \pm 0.1$ for a wide range of stocks. This is the so called *inverse half cube law*, and it can be written as $${\mathbb P}_{\Delta t}(f) \propto f^{-(\lambda + 1)}, \label{eq:pl}$$ where $\mathbb P_{\Delta t}$ is the probability density function of the same quantity. The estimation of tail exponents is often difficult due to poor statistics of rare events, large stock-to-stock variations and the presence of correlations. For the same $1994-1995$ period of data and the same $15$ minute time window certain stocks have $\lambda$ values significantly higher than $1.7$ \[see Fig. \[fig:distrib\](left)\]. The tails of these distributions can be fitted by a power law over an order of magnitude, for the top $3-10\%$ of the events. The exponent $\lambda$ is around $2.8$ for these examples. The question arises: Which value (if any) is correct? In order to address this question we carried out a systematic investigation comprising the $1000$ stocks with the highest total traded value in the TAQ database. We used variants of Hill’s method [@hill; @alves] to estimate the typical tail exponent, see Ref. [@eisler.sizematters] for details. The results of this Section are summarized in Table \[tab:DETRlambda94-95\]. Note that in all cases the $U$-shaped intraday pattern of trading activity was removed. Most descendants of Hill’s method, including the ones applied here, contain a free parameter, namely the fraction $p$ of top events to be considered to belong to the tail of the distribution (see Ref. [@alves] and refs. therein). According to Fig. \[fig:distrib\](left) this should be set around $p\approx 3-10\%$. First, let us follow the methodology of Ref. [@gopi.volume]. In that paper, the authors first they deduct the mean from the time series by taking $f_i(t)-\ev{f_i}$, where $\ev{\cdot}$ denotes time averaging. Then this series is used to estimate value of $\lambda$ by applying Hill’s method [@gopi.personal]. The choice $p=0.03$ provides results in line with Ref. [@gopi.volume], for $\Delta t = 15$ min time windows one finds $\lambda = 1.67 \pm 0.20$. There are several issues with this approach: 1. $p$ is a parameter that can be chosen arbitrarily. With the variation of $p$ the same procedure can produce estimates ranging from $\lambda = 1.1\pm 0.2$ ($p=0.10$) to $\lambda = 2.15 \pm 0.2$ ($p=0.005$). 2. The transformation significantly decreases the estimates of $\lambda$, down to the range of Levy stable distributions ($\lambda < 2$). Estimates for the untransformed data are given in Table \[tab:DETRlambda94-95\] for comparison. It is simple to show, that the first issue emerges, i.e. the estimates systematically depend on $p$, when one applies Hill’s method to a finite sample from a distribution of the form $${\mathbb P}_{\Delta t}(f) \propto (f+f_0)^{-(\lambda + 1)}, \label{eq:pl2}$$ where $f_0$ is a non-zero constant. The transformation to $f_i(t)-\ev{f_i}$ does not resolve the problem, but biases the estimates further. Instead, to correct for these biases one can (i) either find the proper constant $f_0$, remove it from the data, and apply Hill’s estimator afterwards (ii) or apply the estimator of Fraga-Alves [@alves], which is insensitive to such shifts. Both of these estimates were found to be significantly higher [^1]: $\lambda > 2$, see Table \[tab:DETRlambda94-95\]. The methods are described in detail in Ref. [@eisler.sizematters]. The two corrected estimators show a strong tendency of increasing $\lambda$ with increasing $\Delta t$. Monte Carlo simulations on surrogate datasets show that this is beyond what could be explained by decreasing sample size. For distributions with $\lambda < 2$ increasing window size should result in a convergence to the corresponding Levy distribution, and the measured $\lambda$’s should be independent of $\Delta t$. Only when $\lambda > 2$ can the measured effective value of $\lambda$ systematically increase with $\Delta t$. For the $95\%$ of the stocks the increasing tendency is observed and for a window size $\Delta t = 1$ day the respective $\lambda$’s are greater than $2$. These are strong indications that the distributions are not in the Levy stable regime, and thus the second moment exists. Note that our calculations *assume* that the variable is asymptotically distributed as and do not *prove* it. Still, the existence of the second moment is guaranteed by the absence of convergence to a Levy distribution. Consequently, it is possible to define the Hurst exponent for $f_i(t)$. $\Delta t$ Hill’s $\lambda$ ($p=0.06$) Ref. [@gopi.volume], $p=0.03$ Shifted Hill’s $\lambda$ $f_0/\ev{f}$ Fraga Alves ($p=0.1$) ------------ ----------------------------- ------------------------------- -------------------------- -------------- ----------------------- $1$ min $1.43 \pm 0.09$ $1.45\pm0.10$ $2.15\pm0.15$ $3.0$ $1.98 \pm 0.25$ $5$ min $1.56 \pm 0.13$ $1.55\pm0.15$ $2.29 \pm 0.25$ $2.8$ $2.04 \pm 0.25$ $15$ min $1.71 \pm 0.20$ $1.67\pm0.20$ $2.55 \pm 0.35$ $2.8$ $2.1 \pm 0.3$ $60$ min $2.06 \pm 0.30$ $1.90\pm0.25$ $2.85 \pm 0.45$ $1.8$ $2.1 \pm 0.4$ $120$ min $2.3 \pm 0.4$ $2.0\pm0.3$ $3.15 \pm 0.70$ $1.6$ $2.1 \pm 0.4$ $390$ min $2.7 \pm 0.6$ $2.1\pm0.5$ $3.7 \pm 0.9$ $1.2$ no estimate Regardless of the absence of the convergence to Levy stability there are qualitative similarities in the shape of the traded value distributions of various stocks \[cf. Fig. \[fig:distrib\](left)\]. Nevertheless, the existence of a universal distribution can be rejected by a simple test [^2]. If the form of the normalized distribution was universal, then the ratio of the standard deviation and the mean would have to obey $\sigma_i/\ev{f_i}=h$, where $h$ is a constant independent of the stock. Equivalently, a relationship $$\sigma_i \propto \ev{f_i}^\alpha \label{eq:alpha_first}$$ would have to hold with an exponent $\alpha = 1$, at least on average. Even though one finds a monotonic dependence between the two quantities \[as shown in Fig. \[fig:distrib\](right)\], the exponent is significantly less than $1$. This means, that the ratio $\sigma/\ev{f}$ decreases with growing $\ev{f}$, i.e., the normalized distribution of $f$ is narrower for larger stocks, so their trading exhibits smaller relative fluctuations. We will return to this observation in Section \[sec:alpha\]. {height="205pt"}{height="205pt"} Non-universality of correlations in traded value time series {#sec:correl} ============================================================ One of the classical tools of both financial analysis and physics is the measurement of the correlation properties of time series [@bouchaud.book; @stanley.book; @tumminello]. In particular, scaling methods [@dfa] have a long tradition in the study of physical systems, where the Hurst exponent $H_i$ is often calculated. For the traded value time series $f_i^{\Delta t}(t)$ of stock $i$ this is defined as $$\label{eq:hurst} \sigma_i^2(\Delta t) = \ev{\left [f_i^{\Delta t}(t)-\ev{f_i^{\Delta t}(t)} \right ]^2}\propto\Delta t^{2H_i},$$ Note that it follows from the results of Section \[sec:value\] that the variance on the left hand side exists regardless of stock and for any window size $\Delta t$. The measurements were carried out for all $2474$ stocks that were continuously available on the market during $1994-1995$ [^3]. Then we sorted the stocks into $6$ groups according to the order of magnitude of their average traded value: $0\leq \ev{f}\leq 10^4$, $10^4\leq \ev{f}\leq 10^5$, …, $10^8\leq\ev{f}$, all values in USD/min. Finally we averaged $\sigma_i^2(\Delta t)$ within each group. The obtained scaling plots are shown in Fig. \[fig:scavg\]. ![The normalized variance $\frac{1}{2}\log \sigma_i^2(\Delta t)-\frac{1}{2}\log \Delta t$ for the six groups of companies, with average traded values $\ev{f}\in[0,10^4)$, $\ev{f}\in[10^4,10^5)$, …, $\ev{f}\in[10^8,\dots)$ USD/min, increasing from bottom to top. A horizontal line would mean the absence of autocorrelations in the data. Instead, one observes a crossover phenomenon in the regime $\Delta t = 60-390$ mins, indicated by darker background. Below the crossover all stocks show very weakly correlated behavior, $H^-\approx 0.5$. Above the crossover, the strength of correlations, and thus the slope corresponding to $H^+-\frac{1}{2}$, increases with the liquidity of the stock. The asymptotic values of $H^{\pm}$ are indicated in the plot.[]{data-label="fig:scavg"}](EislerFig2){height="200pt"} ![Value of the Hurst exponents of traded value for the time period $1994-1995$. For short time windows (O, $\Delta t < 60$ min) all signals are nearly uncorrelated, $H^-\approx 0.51 - 0.53$. The fitted slope is $\gamma^-=0.00\pm 0.01$. For larger time windows ($\blacksquare$, $\Delta t > 390$ min) the strength of correlations depends logarithmically on the mean trading activity of the stock, $\gamma^+=0.053\pm 0.01$ for $1994-1995$. Shuffled data ($\bigstar$) display no correlations, thus $H_{\mathrm{shuff}} = 0.5$, which also implies $\gamma_\mathrm{shuff} = 0$. [*Note*]{}: Groups of stocks were binned, and their Hurst exponents were averaged. The error bars correspond to the standard deviations in the bins.[]{data-label="fig:hurst"}](EislerFig3){height="205pt"} All stocks display a crossover around window sizes of $\Delta t = 60-390$ min, and there are two sets of Hurst exponents: $H^-_i$ valid below, and $H^+_i$ above the crossover. These characterize the strength of intraday and long time correlations, respectively. The behavior on these two time scales is very different. 1. For intraday fluctuations, regardless of stock $H^-\approx 0.51-0.52$. This means that intraday fluctuations of traded value are nearly uncorrelated. 2. For long time fluctuations the data are correlated, but the strength of correlations depends strongly on the liquidity of the stock. As one moves to groups of larger $\ev{f}$, the strength of correlations ($H^+$) grows, up to $H^+\approx 0.8$. 3. If one shuffles the time series, correlations are destroyed, and $H_{\rm shuff} = 0.5$. The same phenomenon can be characterized by directly plotting the dependence of $H^\pm$ on $\ev{f}$, as done in Fig. \[fig:hurst\]. Such a dependence is well described by a logarithmic law: $$\begin{aligned} H_i^{\pm} = H_0^\pm + \gamma^\pm \log \ev{f_i}, \label{eq:hurst_scaling}\end{aligned}$$ where $\gamma^-=0.00\pm0.01$, and $\gamma^+=0.053\pm0.01$. For the shuffled time series $\gamma_{\rm shuff}=0$. These results indicate, at least in the case of traded value, the absence of universal behavior. Liquidity (or, analogously, company size) is a relevant quantity, which acts as a *continuous* parameter of empirical observables, in particular the strength of correlations and the distribution of $f$. Related results can be found in Refs. [@eisler.liquidity; @bonanno.dynsec; @ivanov.itt; @eisler.sizematters]. ![The dependence of the scaling exponent $\alpha$ on the window size $\Delta t$ for the years $1994-1995$. The lighter shaded intervals have well-defined Hurst exponents and values of $\gamma$, the crossover is indicated with a darker background. Without shuffling ($\blacksquare$) there are two linear regimes: For shorter windows $\alpha = 0.74 \pm 0.02$, the slope is $\gamma^-=\gamma(\Delta t<60$ min$)=0.00\pm 0.01$ (solid line), while for longer windows $\alpha$ grows logaritmically, with a slope $\gamma^+=\gamma(\Delta t>390$ min$)=0.052\pm 0.01$ (dashed line). For shuffled data (O) the exponent is independent of window size, $\alpha (\Delta t)=0.74\pm0.02$.[]{data-label="fig:alpha"}](EislerFig4){width="255pt"} Fluctuation scaling {#sec:alpha} =================== Fluctuation scaling is a general phenomenon, observed in a wide range of complex systems [@barabasi.fluct; @eisler.non-universality; @eisler.unified]: A scaling law connects the standard deviation $\sigma_i$ and the average $\ev{f_i}$ of the same quantity. In the case of the trading activity of stocks we have already presented this result in Section \[sec:value\] \[cf. Eq. \]. Now we give a more detailed discussion. Let us start from our observation that $$\sigma_i(\Delta t) \propto \ev{f_i}^{\alpha (\Delta t)}, \label{eq:alpha}$$ where the scaling variable is $\ev{f_i}$, or more appropriately the stock $i$, and $\Delta t$ is kept constant \[see Fig. \[fig:distrib\](right)\]. Notice that $\sigma_i(\Delta t)$ is the same as in the definition of the Hurst exponent in Eq. , where $i$ was constant and $\Delta t$ was varied. In Eq. the window size $\Delta t$ is a free parameter. This scaling law persists for any $\Delta t$, but $\alpha$ strongly depends on its value, as shown in Fig. \[fig:alpha\]. For small time windows (up to $60$ min), $\alpha(\Delta t)\approx 0.74$, then, after a crossover regime, when $\Delta t > 390$ min, there is a logarithmic trend. This can be summarized as $$\alpha(\Delta t) = \alpha_0^\pm + \gamma^\pm\log \Delta t, \label{eq:alpha_scaling}$$ where $\cdot^\pm$ refers to the regimes $\Delta t < 60$ min and $\Delta t > 390$ min. The constants are $\alpha_0^- = 0.74$, $\gamma^- =0$, and $\gamma^+ = 0.052\pm 0.01$. For shuffled time series, $\alpha(\Delta t) = 0.74$ regardless of $\Delta t$, i.e., $\gamma_{\rm shuff}=0$. A visual comparison of Figs. \[fig:scavg\] and \[fig:alpha\] reveals that the crossover in the behavior of $\alpha(\Delta t)$ and $H$ falls into the same interval. Moreover, when $\Delta t < 60$ min, both $\alpha(\Delta t)$ and $H^-(\ev{f})$ are constant. For $\Delta t > 390$ min, both $\alpha(\Delta t)$ and $H^+(\ev{f})$ vary logarithmically with their arguments (see Figs. \[fig:hurst\] and \[fig:alpha\]). In order to better understand the connection between temporal correlations and fluctuation scaling, let us repeat here Eqs. and : $$\begin{aligned} \alpha(\Delta t)=\alpha_0^\pm + \gamma^\pm \log \Delta t, \nonumber \\ H_i=H_0^\pm + \gamma^\pm \log \ev{f_i}. \nonumber\end{aligned}$$ Beyond the obvious symmetry of these two logarithmic laws, notice that the prefactors are equal: in both equations $\gamma^- \approx 0$ and $\gamma^+\approx 0.05$. It is easy to show [@eisler.unified] that none of this is a simple coincidence. If both fluctuation scaling and long range autocorrelations are present in data, there are only two possible ways for their coexistence: 1. Correlations are homogeneous throughout the system, $H_i=H_0$, $\gamma = 0$, and $\alpha$ is independent of $\Delta t$. This is realized for $\Delta < 60$ min. For shuffled time series correlations are absent, thus such data also fall into this category. 2. Both the $H(\ev{f_i})$ and $\alpha(\Delta t)$ are logarithmic functions of their arguments with the same coefficient $\gamma^+$. This is realized for $\Delta > 390$ min. In other words the coexistence of the two scaling laws is so restrictive, that if the strength of correlations depends on $\ev{f}$ at all, then the realized logarithmic dependence is the only possible scenario. Conclusions =========== In this paper, we analyzed the empirical properties of trading activity on the New York Stock Exchange. We showed that, in contrast to earlier findings, the distribution of traded value is not in the Levy stable regime, and is not universal. Traded value is nearly uncorrelated on an intraday time scale, while on daily or longer scales fluctuations show strong persistence, whose strength grows logarithmically with the liquidity of the stock. This effect is in harmony with findings on fluctuation scaling, a general scaling framework for complex systems. All our results imply, that the notion of universality must be used with extreme care in the context of financial markets, where the concepts and the theoretical background are radically different from those in physics. The liquidity of a stock strongly affects the distribution and the correlation structure of its trading activity. This dependence is continuous, which means the absence of universality classes in trading dynamics. The dynamical process responsible for such a dependence is yet to be identified. Finally, we would like to make two remarks. Firstly, in Refs. [@gopi.volume] and [@gabaix.volatility] it is stated that the so called inverse half cubic law is observable not only on the $15$ minute level but also in the tick-by-tick data. Our analysis dealt with data aggregated for $1$ minute and more, and we showed that the assumption of a power law decay is not consistent with the inverse half cubic law for these cases. Secondly, in Ref. [@gabaix.volatility] a footnote mentions the possibility of an exponential cutoff in the distribution. This assumption would influence the estimators strongly and we did not consider this case. We thank the anonymous referee for calling our attention to these points. The authors thank György Andor and Ádám Zawadowski for their help with the data. ZE is grateful for the hospitality of l’Ecole de Physique des Houches. JK is member of the Center for Applied Mathematics and Computational Physics, BME. Support by OTKA T049238 is acknowledged. [^1]: The Fraga-Alves estimator converges very slowly, and it underestimates the actual values of $\lambda$ from small samples. Its estimates can be interpreted as lower bounds for $\lambda$. [^2]: Similar techniques were used in Refs. [@lillo.variety; @eisler.sizematters] to show non-universality in the distribution of returns and intertrade times. [^3]: For a similar analysis of the years $2000-2002$, see Ref. [@eisler.sizematters].

ArXiv

--- abstract: | We propose a solution to the ‘cuspy-core’ problem by extending the geodesic equations of motion using the Dark Energy length scale $\lambda_{DE}=c/(\Lambda_{DE} G)^{1/2}$. This extension does not affect the motion of photons; gravitational lensing is unchanged. A cosmological check of the theory is made, and $\sigma_8$ is calculated to be $0.68_{\pm0.11}$, compared to $0.761_{-0.048}^{+0.049}$ for WMAP. We estimate the fractional density of matter that cannot be determined through gravity at $0.197_{\pm 0.017}$, compared to $0.196^{+0.025}_{-0.026}$, the fractional density of nonbaryonic matter. The fractional density of matter that can be determined through gravity is estimated at $0.041_{-0.031}^{+0.030}$, compared to $0.0416_{-0.0039}^{+0.0038}$ for $\Omega_B$. author: - 'A. D. Speliotopoulos' date: 'November 30, 2007' title: 'Connecting the Galactic and Cosmological Scales: Dark Energy and the Cuspy-Core Problem' --- Introduction ============ The recent discovery of Dark Energy [@Ries1998; @Perl1999] has not only broadened our knowledge of the universe, it has brought into sharp relief the degree of our understanding of it. Only a small fraction of the mass-energy density of the universe is made up of matter that we have characterized; the rest consists of Dark Matter and Dark Energy, both of which have not been experimentally detected, and both of whose precise properties are not known. Both are needed to explain what is seen on an extremely wide range of length scales. On the galactic ($\sim 100$ kpc parsec), galactic cluster ($\sim$  10 Mpc), and supercluster ($\sim$ 100 Mpc) scales, Dark Matter is used to explain phenomena ranging from the formation of galaxies and rotation curves, to the dynamics of galaxies and the formation of galactic clusters and superclusters. On the cosmological scale, both Dark Matter and Dark Energy are needed to explain the evolution of the universe. While the need for Dark Matter is ubiquitous on a wide range of length scales, our understanding of how matter determines dynamics on the galactic scale is lacking. Recent measurements by WMAP [@WMAP] have validated the $\Lambda$CDM model to an unprecedented precision; such is not the case on the galactic scale, however. Current understanding of structure formation is based on [@Peebles1984], and both analytical solutions [@Gunn] and numerical simulations [@JNav; @Krav; @Moore; @PeeblesRev; @Silk] of galaxy formation have been done since then. These simulations have consistently found a density profile that has a cusp-like profile [@Moore; @JNav; @Silk], instead of the pseudoisothermal profile commonly observed. Indeed, De Blok and coworkers [@Blok-1] has explicitly shown that the density profile from [@JNav] attained through simulation does not fit the density profile observed for Low Surface Brightness galaxies; the pseudoisothermal profile is the better fit. This is the cuspy-core problem. There have been a number of attempts to solve it within $\Lambda$CDM [@PeeblesRev; @Silk], with varying degrees of success. While the problem does not exist for MOND [@Mil], there are other hurdles MOND must overcome. Our approach to this problem, and to structure formation in general, is more radical; therefore, its consequences are correspondingly broader. It is based on the observation that with the discovery of Dark Energy, $\Lambda_{DE}$, there is a length scale, $\lambda_{DE} = c/(\Lambda_{DE}G)^{1/2}$, associated with the universe. Extensions of the geodesic equations of motion (GEOM) can now be made that will satisfy the equivalence principal, while not introducing an observable fifth force. While affecting the motion of massive test particles, photons will still travel along null geodesics, and gravitational lensing is not changed. For a model galaxy, the extend GEOM results in a nonlinear evolution equation for the density of the galaxy. This equation is the minimum of a functional of the density, which is interpreted as an effective free energy for the system. We conjecture that like Landau-Ginzberg theories in condensed matter physics, the system prefers to be in a state that minimizes this free energy. Showing that the pseudoisothermal profile is preferred over cusp-like profiles reduces to showing that it has a lower free energy. Here, phenomena on the galactic scale are inexorably connected to phenomena on the cosmological scale, and a cosmological check of our theory is made. The Hubble length scale $\lambda_H = c/hH_0$ naturally appears in our approach, *even though a cosmological model is not mentioned either in its construction, or in its analysis*. Using the average rotational velocity and core sizes of 1393 galaxies obtained through four different sets of observations [@Blok-1; @Rubin1980; @Cour; @Math] spanning 25 years, we calculate $\sigma_8$ to be $0.68_{\pm 0.11}$, in excellent agreement with $0.761^{+0.049}_{-0.048}$ from [@WMAP]. We also calculate $\Omega_{\hbox{\scriptsize asymp}}$, the fractional density of matter that *cannot* be determined through gravity, to be $0.197_{\pm 0.017}$, which is nearly equal to the fractional density of nonbaryonic matter $\Omega_m-\Omega_{B} = 0.196^{+0.025}_{-0.026}$ [@WMAP]. We then find the fractional density of matter in the universe that can be determined through gravity, $\Omega_{\hbox{\scriptsize Dyn}}$, to be $0.041^{+0.030}_{- 0.031}$, which is nearly equal to $\Omega_B=0.0416^{+0.0038}_{-0.0039}$. Details of our calculations and theory is in [@ADS]. Extending the GEOM and Galactic Structure ========================================= Any extension of the geodesic action requires a dimensionless, scalar function of some property of the spacetime folded in with some physical property of matter. While before no such properties existed, with the discovery of Dark Energy there is now $\lambda_{DE}$ and these extensions can be made. As we work in the nonrelativistic, linearized gravity limit, we consider the simplest extension: $$\mathcal{L}_{\hbox{\scriptsize{Ext}}} = mc\Big(1+\mathfrak{D}\left[Rc^2/ \Lambda_{DE}G\right]\Big)^{\frac{1}{2}} \left(g_{\mu\nu}\frac{d x^\mu}{dt}\frac{d x^\nu}{dt}\right)^{\frac{1}{2}} \equiv mc\mathfrak{R}[Rc^2/\Lambda_{DE}G] \left(g_{\mu\nu}\frac{d x^\mu}{dt}\frac{d x^\nu}{dt}\right)^{\frac{1}{2}} \label{extendL}$$ with the constraint $v^2=c^2$ for massive test particles. Here, $\mathfrak{D}(x)$ is a function function given below, and $R$ is the Ricci scalar. For massive test particles, the extended GEOM is $v^\nu\nabla_\nu v^\mu = c^2\left(g^{\mu\nu} - v^\mu v^\nu/c^2\right)\nabla_\nu \log\mathfrak{R}[4+8\pi T/\Lambda_{DE}c^2]$, where $v^\mu$ is the four-velocity of a test particle, $T_{\mu\nu}$ is the energy-momentum tensor, $T=T_\mu^\mu$, and we take $\Lambda_{DE}$ to be the cosmological constant. As the action for gravity+matter is a linear combination of the Hilbert action and the action for matter, any changes to the equation of motion for test particles can be accounted for in $T_{\mu\nu}$, and we still have $R=4\Lambda_{DE}G/c^2+8\pi GT/c^4$ in Eq. $(\ref{extendL})$. For massless particles, $v^\nu\nabla_\nu \left(\mathfrak{R}[4+8\pi T/\Lambda_{DE}c^2]v^\mu\right)=0$ instead. With the reparametization $dt \to \mathfrak{R} dt$, the extended GEOM for massless test particles reduces to the GEOM. Our extended GEOM does not affect the motion of photons. Because the geodesic Lagrangian is extended covariantly, Eq. $(\ref{extendL})$ explicitly satisfies the strong equivalence principal. For $T_{\mu\nu}$, we may still take $T_{\mu\nu} = (\rho+p/c^2)v_\mu v_\nu - p g_{\mu\nu}$ for an inviscid fluid with density $\rho$ and pressure $p$ [@ADS]. While for the GEOM $T^{\hbox{\scriptsize{geo-Dust}}}_{\mu\nu}=\rho v_\mu v_\nu$ for dust, for the extended GEOM the pressure does not vanish [@ADS]; it is a functional of $\rho$ and $\mathfrak{R}$. Nevertheless, in the nonrelativistic limit $p<<\rho c^2$, and $T_{\mu\nu}^{\hbox{\scriptsize{Ext-Dust}}}\approx\rho v_\mu v_\nu$ still [@ADS]. Moreover, because $v^\mu v_\mu =c^2$ for the extended GEOM, the first law of thermodynamics still holds for the fluid, and *the standard thermodynamical analysis of the evolution of the universe under the extended GEOM follows much in the same way as before.* All dynamical effects of extension can be interpreted as the rest energy gained or lost by the test particle due to variations in the local curvature. For these effects not to have already been seen, $\mathfrak{D}(4+8\pi T/\Lambda_{DE}c^2)$ must change very slowly at current experimental limits. As such, we take $ \mathfrak{D}(x) = \chi(\alpha_\Lambda)\int_x^\infty(1+s^{1+\alpha_\Lambda})^{-1}ds$, where $\alpha_\Lambda \ge 1$ and $\chi(\alpha_\Lambda)$ is set by $\mathfrak{D}(0)=1$. This $\mathfrak{D}(x)$ was chosen for three reasons. First, there is only one free parameter, $\alpha_\Lambda$, to determine. Second, it ensures that the effects of the additional terms in the extended GEOM will not already have been observed; $\Lambda_{DE} = (7.21^{ +0.82}_{-0.84}) \times 10^{-30}$ g/cm$^3$, and $\rho \gg \Lambda_{DE}/2\pi$ in all current experimental environments so that $\mathfrak{D}\approx 0$. A lower experimental bound of 1.35 for $\alpha_\Lambda$ can be found [@ADS]. Third, $\mathfrak{D}'(x)$ is negative, and will contribute an effective repulsive potential to the extended GEOM that mitigates the Newtonian $1/r$ potential. While definitive, a first principles calculation of the galactic rotation curves using the extended GEOM would be analytically intractable. Instead, we show that *given* a model, stationary galaxy with a specific rotation velocity curve $v(r)$, we can *derive* the mass density profile of the galaxy. We use a spherical model for the galaxy that has three regions. Region I $=\{r \>\>\vert \>\> r\le r_H \hbox{, and } \rho \gg \Lambda_{DE}/2\pi\}$, where $r_H$ is the galactic core radius. Region II $=\{r \>\>\vert \>\> r> r_H, \>r \le r_{II}, \hbox{ and } \rho \gg \Lambda_{DE}/2\pi\}$ is the region outside the core containing stars undergoing rotations with constant rotational velocity; it extends out to $r_{II}$, which is determined by the theory. A Region III $=\{r \>\>\vert \>\> r > r_{II} \hbox{, and } \rho \ll \Lambda_{DE}/2\pi\}$ also appears in the theory. As all the stars in the model galaxy undergo circular motion, the acceleration of a star, $\mathbf{a} \equiv \ddot{\mathbf{x}}$, is a function of is location, $\mathbf{x}$, only. Taking the divergence of the extended GEOM, $$f(\mathbf{x})= \rho - \frac{1}{\kappa^2(\rho)} \left\{\mathbf{\nabla}^2\rho - \frac{1+{\alpha_\Lambda}}{4+8\pi\rho/\Lambda_{DE}} \left(\frac{8\pi}{\Lambda_{DE}}\right) \vert\mathbf{\nabla}\rho\vert^2\right\}, \label{rhoGEOM}$$ where $\kappa^2(\rho) \equiv \left\{1+\left(4+ 8\pi\rho/\Lambda_{DE}\right)^{1+{\alpha_\Lambda}}\right\}/\chi\lambda_{DE}^2$, and $f(\mathbf{x}) \equiv -\mathbf{\nabla}\cdot\mathbf{a}/4\pi G$. We do not differentiate between baryonic matter and Dark Matter in $\rho$. Near the galactic core $1/\kappa(\rho)\sim \lambda_{DE}[\Lambda_{DE}/8\pi\rho_H]^{(1+\alpha_\Lambda)/2}$, where $\rho_H$ is the core density. Even though $\lambda_{DE}= 14010^{+800}_{-810}$ Mpc, because $\rho_H\gg\Lambda_{DE}/2\pi$, $\alpha_\Lambda$ can be chosen so that $1/\kappa(r)$ is comparable to typical $r_H$. Doing so sets $\alpha_\Lambda\approx 3/2$. Given a $v(r)$, $\mathbf{a}(r)$ can be found and $f(r)$ determined. We idealize the observed velocity curves as $v^{\hbox{\scriptsize ideal}}(r) =v_H r/r_H$ for $r \le r_H$, while $v^{\hbox{\scriptsize ideal}}(r)=v_H$ for $r>r_H$, where $v_H$ is the observed asymptotic velocity. This $v^{\hbox{\scriptsize{ideal}}}(r)$ is more tractable than the pseudoisothermal velocity curve, $v^{\hbox{\scriptsize{p-iso}}}(r)$, used in [@Blok-1]. As it has the same limiting forms in both the $r\ll r_H$ and $r\gg r_H$ limits, $v^{\hbox{\scriptsize ideal}}(r)$ is also an idealization of $v^{\hbox{\scriptsize p-iso}}(r)$. For cusp-like density profiles [@Silk], it is the density profile that is given. While it is possible to integrate the general density profile to find the corresponding curves $v_{\hbox{\scriptsize cusp}}(r)$, both the maximum value of $v_{\hbox{\scriptsize cusp}}(r)$ and the size of the core are different depending on the profile. These core sizes would thus have to be scaled appropriately to compare one profile with another. Doing so is possible in principle, but would be analytically intractable in practice. We instead take $f(r) = \rho_H \left(r_H/r\right)^\gamma $ if $r \le r_H$, and $f(r) = \rho_H \left(r_H/r\right)^\beta/3 $ if $r>r_H$ for the density profiles. Here, $\gamma < 2$ and $\beta\ge 2$ agrees with the parameters for the generic cusp-like density profile [@Krav], with the core size set to $r_H$. The $\gamma=0, \beta=2$ case corresponds to the idealized psuodoisothermal profile. Since $\rho\gg\Lambda_{DE}/2\pi$ in Regions I and II, Eq. $(\ref{rhoGEOM})$ minimizes $$\begin{aligned} \mathcal{F}[\rho] = \frac{\Lambda_{DE}c^2}{8\pi}\left(\chi^{1/2}\lambda_{DE}\right)^3 \int d^3\mathbf{{u}} &{}& \Bigg\{ \frac{1}{2\alpha_\Lambda} \Bigg\vert \mathbf{\nabla} \left(\frac{\Lambda_{DE}}{8\pi\rho}\right)^{\alpha_\Lambda} \Bigg\vert^2 - \frac{\alpha_\Lambda}{\alpha_\Lambda-1} \left(\frac{\Lambda_{DE}}{8\pi\rho}\right)^{\alpha_\Lambda-1} + \nonumber \\ &{}& \left(\frac{\Lambda_{DE}}{8\pi\rho}\right)^{\alpha_\Lambda} \frac{8\pi f({u})}{\Lambda_{DE}}\Bigg\}, \label{free-energy}\end{aligned}$$ which we identify as a free energy functional; here, $u=r/\chi^{1/2}\lambda_{DE}$. For $\gamma=0$, Eq. $(\ref{rhoGEOM})$ gives $\rho(r) = \rho_H$ in Region I; the free energy for this solution is ${}^I\mathcal{F}_{\gamma=0} = -\Lambda_{DE}r_H^3 \left( \Lambda_{DE}/8\pi\rho_H \right)^{\alpha_\Lambda-1}/6(\alpha_\Lambda-1)$. While for $\gamma>0$ perturbative solutions can be found, all such solutions have a $^{I}\mathcal{F}_\gamma$ greater than ${}^I\mathcal{F}_{\gamma=0}$ [@ADS]. This results because $\sim \vert\nabla \rho\vert^2 \ge0$ in Eq. $(\ref{free-energy})$; just as in a Landau-Ginzberg theory, $\vert\nabla\rho\vert^2$ only vanishes for the constant density solution. For Region II, the density, $\rho_{II}$, is first found asymptotically in the large $r$ limit. With the anzatz $f(r)\ll\rho(r)$ for large $r$, Eq. $(\ref{rhoGEOM})$ reduces to a homogeneous equation [@ADS] with the solution $\rho_{\hbox{\scriptsize{asymp}}} (u)= \Lambda_{DE} \Sigma({\alpha_\Lambda})/8\pi u^{\frac{2}{1+\alpha_\Lambda}}$, where $\Sigma({\alpha_\Lambda}) = \left[2(1+3\alpha_\Lambda)/ (1+{\alpha_\Lambda})^2\right]^{\frac{1}{1+\alpha_\Lambda}}$. To include the galaxy’s structural details, we take $\rho_{II}(r) = \rho_{\hbox{\scriptsize{asymp}}}(r) + \rho^1_{II}(r)$ and to first order in $\rho_{II}^{1}$, $$\rho_{II}(r) = \rho_{\hbox{\scriptsize{asymp}}}(r) + \frac{1}{3} A_\beta \rho_H \left(\frac{r_H}{r}\right)^\beta+ \left(\frac{r_H}{r}\right)^{5/2} \left(C_{\cos}\cos\left[\nu_0\log r/r_H\right] + C_{sin}\sin\left[\nu_0\log r/r_H\right]\right). \label{rho-beta}$$ where $\nu_0 = \left[2(1+3\alpha_\Lambda)/(1+\alpha_\Lambda)^2 - 1/4\right]^{1/2}$, $C_{\cos}$ and $C_{\sin}$ are determined by boundary conditions, and $A_\beta=1$ for $\beta = 2,3$. The first part, $\rho_{\hbox{\scriptsize{asymp}}}(r)$, of $\rho_{II}(r)$ corresponds to a background density. *It is universal, and has the same form irrespective of the detailed structure of the galaxy.* The second part, $\rho_{II}^1(r)$, gives the structural details. The free energy, ${}^{II}\mathcal{F}$, for Region II separates into the sum of three parts. The first part depends only on $\rho_{\hbox{\scriptsize{asymp}}}$; it is positive, and is independent of $\beta$. The second part is $$\frac{{}^{II}\mathcal{F}_{\hbox{\scriptsize{asymp}}-\beta}} {(\chi^{1/2}\lambda_{DE})^3} = c^2 \int_{D_{II}} d^3\mathbf{u} f(u)\left(\frac{\Lambda_{DE}}{8\pi\rho_{\hbox{\scriptsize{asymp}}}} \right)^{\alpha_\Lambda} +\frac{8\pi \alpha_\Lambda c^2}{\Lambda_{DE}\Sigma^{2(1+\alpha_\Lambda)}} \int_{\partial D_{II}} u^4\rho_{II}^1(u)\mathbf{\nabla}\rho_{\hbox{\scriptsize{asymp}}}\cdot d\mathbf{S},$$ where $D_{II}$ is Region II. It is negative because the minimum $\rho$ must be positive. Indeed, we find that ${}^{II}\mathcal{F}_{\hbox{\scriptsize{asymp}}-\beta} \sim - (r_H/r_{II})^{\beta}$ for $\beta < 5/2$; ${}^{II}\mathcal{F}_{\hbox{\scriptsize{asymp}}-\beta} \sim - (r_H/r_{II})^{5/2}$ for $5/2 \le \beta < 5-2/(1+\alpha_\Lambda)$; and ${}^{II}\mathcal{F}_{\hbox{\scriptsize{asymp}}-\beta} \sim \pm (r_H/\chi^{1/2}\lambda_{DE})^{5-2/(1+\alpha_\Lambda)}$ for $5-2/(1+\alpha_\Lambda)<\beta$. Clearly, free energy is lowest for $\beta=2$. The third part depends on $(\rho_{II}^1(r))^2$, and is negligibly small. The total free energy in this region is thus smaller for $\beta = 2 $ than for $\beta >2$. Combined with the calculation for ${}^{I}\mathcal{F}$, we conclude that the pseudoisothermal density profile has the lowest free energy, and is the preferred state of the system. We thus take $\gamma=0$ and $\beta=2$ in the following. In Region III, $\rho\ll\Lambda_{DE}/2\pi$, and $\kappa^2(\rho)\approx (1+4^{1+\alpha_\lambda})/\chi\lambda_{DE}^2$; Eq. $(\ref{rhoGEOM})$ reduces to the undriven, modified Bessel equation. As such, the density vanishes exponentially fast in this region on the scale $1/\kappa(r)$. This sets $r_{II} = [\chi/(1+4^{1+\alpha_\lambda})]^{1/2}\lambda_{DE}$. The extended GEOM can be written as $\ddot{\mathbf{x}} = - \mathbf{\nabla}\mathfrak{V}$. The dynamics of test particles is governed by an effective potential $\mathfrak{V}(\mathbf{x}) = \Phi(\mathbf{x}) + c^2\log\left(\mathfrak{R}[4+8\pi\rho\Lambda_{DE}]\right)$, and not by the gravitational potential $\Phi(\mathbf{x})$. For $\Phi(r)$ in Region I, we obtain the Newtonian gravity result $\Phi(r) = v^2_H r^2/2r_H^2 + \hbox{constant}$. In Region II, $\Phi(r)$ is dominated by four terms. The first is the usual $1/r$ term. The second is a $\log(r/r_H)$ term due to $f(r)$. This term is long ranged, and in addition to galactic rotation curves, could explain the interaction observed between galaxies and galactic clusters. The third is a $\rho_{II}^1(r) r^2$ term, and contains terms $\sim 1/r^{1/2}$. The fourth term is a $r^{2\alpha_\Lambda/(1+\alpha_\Lambda)}$ term due to $\rho_{\hbox{\scriptsize{asymp}}}$, and is proportional to $c^2$. This last term grows as $r^{6/5}$ for $\alpha_\Lambda=3/2$, and would dominate the motion of test particles in the galaxy if the extended GEOM depended on $\Phi(\mathbf{x})$ instead of $\mathfrak{V}(\mathbf{x})$. We instead find that $ \mathfrak{V}(\mathbf{x}) \approx \Phi(\mathbf{x}) - \left[u^2\chi c^2(1+\alpha_\Lambda)^2/4\alpha_\Lambda(1+3\alpha_\Lambda)\right] (\Lambda_{DE}/8\pi) (\rho_{\hbox{\scriptsize{asymp}}} -\alpha_\Lambda\rho_{II}^{1})$. The last two terms in this expression cancel both the $\rho_{II}^1(r) r^2$ and the $r^{2\alpha_\Lambda/(1+\alpha_\Lambda)}$ terms in $\Phi(r)$; the resultant $\mathfrak{V}(r)$ increases as $\log{r/r_H}$, agreeing with observation. The $r^{2\alpha_\Lambda/(1+\alpha_\Lambda)}$ term in $\Phi(\mathbf{x})$ comes from the background density $\rho_{\hbox{\scriptsize{asymp}}}$. Thus, a good fraction of the mass in the observable galaxy *does not contribute to the motion of test particles in the galaxy*. It is rather the near-core density $\rho_{II}^1(r)$ that contributes to $\mathfrak{V}(\mathbf{x})$. As inferring the mass of structures through observations of the dynamics under gravity of their constituents is one of the main ways of estimating mass, the motion of stars in galaxies can only be used to estimate $\rho_{II}^1$; the matter in $\rho_{\hbox{\scriptsize{asymp}}}(r)$ is present, but cannot be “seen” in this way. Moreover, as $\rho_{\hbox{\scriptsize{asymp}}}(r)\gg \rho_{II}^1(r)$ when $r\gg r_H$, *the majority of the mass in the universe cannot be seen using these methods*. A Cosmological Check ==================== We have extrapolated our results for a single galaxy to the cosmological scale. This is possible because recent measurements from WMAP, the Supernova Legacy Survey, and the HST key project show that the universe is essentially flat; $h=0.732_{-0.032}^{+0.031}$ and of the age of the universe $t_0=13.73_{-0.15}^{+0.16}$ Gyr were determined using this assumption. The largest distance between galaxies is thus $ct_0\equiv \mathfrak{K}(\Omega) \lambda_{H}$, where $\mathfrak{K}(\Omega) =1.03_{\pm0.05}$. Next, the density of matter of our model galaxy dies off exponentially fast at $r_{II}$; the extent of matter in the galaxy is fundamentally limited to $2r_{II}$. This size does not depend on the detailed structure of the galaxy; it is inherent to the theory. Given a $\Omega_\Lambda = 0.716_{\pm 0.055}$, we can express $r_{II} =[8\pi\chi/3\Omega_\Lambda(1+4^{1+\alpha_\Lambda})]^{1/2}\lambda_H$ [@ADS] as well [@WMAP], and numerically $r_{II}=0.52\lambda_H$ for $\alpha_\Lambda = 3/2$. Although $\alpha_\Lambda$ was set to $3/2$ based on analysis at the galactic scale, $\rho(r)$ naturally cuts off at $\lambda_H/2$. To accomplish the extrapolation, we consider our model galaxy to be the representative galaxy for the observed universe. This representative galaxy could, in principal, be found by sectioning the observed universe into three-dimensional, non-overlapping cells of different sizes centered on each galaxy. By surveying these cells, a representative galaxy, with an average $v_H^{*}$ and $r^{*}_H$, can be found, and used as inputs for the model galaxy. Even though such a survey has not yet been done, a large repository of galactic rotation curves and core radii [@Blok-1; @Cour; @Math] is present in the literature. Taken as a whole, these 1393 galaxies are reasonably random, and are likely representative of the observed universe at large. While we were able to estimate of $\alpha_\Lambda=3/2$ by looking at the galactic structure, the accuracy of this estimate is unknown; comparison with experiment is not possible. We instead *require* that $r_{II} = \mathfrak{K}(\Omega)\lambda_H/2$, which in turn gives $\alpha_\Lambda$ as the solution of $\mathfrak{K}(\Omega)^2(1+4^{1+\alpha_\Lambda}) = 32\pi \chi(\alpha_\Lambda)/3\Omega_\Lambda$; this sets $\alpha_\Lambda = 1.51_{\pm 0.11}$. A calculation of $\sigma_8^2$ has been done [@ADS] using Eq. $(\ref{rho-beta})$. The resultant $\sigma_8^2$ is dominated by two terms. The first is due to the background density $\rho_{\hbox{\scriptsize{asymp}}}$. It depends only on $\alpha_\Lambda$, and contributes a set amount of 0.141 to $\sigma_8^2$. The second is the larger one, and is due primarily to the $1/r^2$ term in Eq. $(\ref{rho-beta})$. It depends explicitly on the rotation curves through the term $(v_H^{*}/c)^4(8h^{-1}\hbox{Mpc}/r_H^{*})$. Although there have been a many studies of galactic rotation curves in the literature, both $v_H$ and $r_H$ are needed here. This requires fitting the observed velocity curve to some model. To our knowledge, both values are available from four places in the literature: The de Blok et. al. data set [@Blok-1]; the CF data set [@Cour]; the Mathewson et. al. data set [@Math; @Pers-1995] analysed in [@Cour]; and the Rubin et. al. data set [@Rubin1980]. Except the last set, the observed velocity curves is fitted to either $v^{\hbox{\scriptsize{p-iso}}}(r)$, or to a functionally similar velocity curve [@Cour]. The last set gives only the galactic rotation curves, and they have been fitted to $v^{\hbox{\scriptsize{p-iso}}}(r)$ in [@ADS]. While the URC of [@Pers-1996] has a constant asymptotic velocity, it has a $r^{0.66}$ behavior for $r$ small. This behavior is different from $v^{\hbox{\scriptsize{ideal}}}$, and was not considered here [@ADS]. While $v_H$ is easily identified for all four data sets, determining $r_H$ is more complicated; this is determination is done in [@ADS]. The resultant values are used to obtain $v^{*}_H$ and $r^{*}_H$ for each set, which are then used to calculate the $\sigma_8$ and $\Delta\sigma_8$ for it. Results of these calculations are in Table \[summary\]. Four of the five data sets give a $\sigma_8$ that agrees with the WMAP value at the 95% CL. The Rubin et. al. set does not, but it is known that these galaxies were not randomly selected [@Rubin1980]. *Data Set* 0.1in $v^{*}_H$ 0.1in $\Delta v^{*}_H$ 0.1in $r^{*}_H$ 0.1in $\Delta r^{*}_H$ $\qquad\sigma_8$ $\qquad\Delta \sigma_8$ 0.1in *t-test* ------------------------- ----------------- ------------------------ ----------------- ------------------------ ------------------ ------------------------- ---------------- -- deBlok et. al. (53) 119.0 6.8 3.62 0.33 0.613 0.097 1.36 CF (348) 179.1 2.9 7.43 0.35 0.84 0.18 0.43 Mathewson et. al. (935) 169.5 1.9 15.19 0.42 0.625 0.089 1.34 Rubin et. al. (57) 223.3 7.6 1.24 0.14 2.79 0.82 2.46 Combined (1393) 172.1 1.6 11.82 0.30 0.68 0.11 0.70 : The $v_H^{*}$ (km/s), $r_H^{*}$ (kps), and resultant $\sigma_8$, $\Delta\sigma_8$, and t-test comparison with the WMAP value of $\sigma_8$.[]{data-label="summary"} We have estimated $\Omega_{\hbox{\scriptsize asymp}}$ by averaging $\rho_{\hbox{\scriptsize asymp}}(r)$ over a sphere of radius $r_{II}$, and found $\Omega_{\hbox{\scriptsize{asymp}}} = 0.197_{\pm0.017}$. In calculating this average, we assumed that there is only a single galaxy within the sphere, however. While this is a gross under counting of the number of galaxies in the universe, $\rho_{\hbox{\scriptsize{asymp}}}$ is an asymptotic solution, and $\rho_{II}^1 \to0$ rapidly with $r$. Additional galaxies may change the form of $\rho_{\hbox{\scriptsize asymp}}$, but these changes are expected to be equally short ranged; we expect that our calculation is an adequate estimate of $\Omega_{\hbox{\scriptsize asymp}}$. Such is not the case for $\Omega_{\hbox{\scriptsize Dyn}}$, however. Direct calculation of $\Omega_{\hbox{\scriptsize Dyn}}$ would require knowing both the detailed structure of galaxies, and the distribution of galaxies in the universe. Instead, we note that $\Omega_m = \Omega_{\hbox{\scriptsize{asymp}}}+ \Omega_{\hbox{\scriptsize{Dyn}}}$, and using $\Omega_m=0.238_{-0.026}^{+0.025}$ from WMAP, find $\Omega_{\hbox{\scriptsize{Dyn}}}=0.041^{+0.030}_{-0.031}$. Concluding Remarks ================== Given how sensitive $\sigma_8$ is to $v_H^{*}$, $r_H^{*}$, and $\alpha_\Lambda$, that our predicted values of $\sigma_8$ is within experimental error of the WMAP value is surprising. Even in the absence of a direct experimental search for $\alpha_\Lambda$, this agreement provides a compelling argument for the validity of our extension of the GEOM. It also supports our free energy conjecture; our calculation of $\sigma_8$ would be very different if $\beta=3$, say, was used instead of $\beta=2$. With $\alpha_\Lambda=1.51$ so close to the experimental lower bound for $\alpha_\Lambda$ of $1.35$, direct measurement of $\alpha_\Lambda$ may also be possible in the near future. Interestingly, $\Omega_m-\Omega_{B} = 0.196_{-0.026}^{+0.025}$ is nearly equal to $\Omega_{\hbox{\scriptsize asymp}}$ in value. Correspondingly, $\Omega_B$ [@WMAP] is nearly equal to $\Omega_{\hbox{\scriptsize Dyn}}$. It would be tempting to identify $\Omega_{\hbox{\scriptsize asymp}}$ with $\Omega_m-\Omega_B$, especially since matter in $\rho_{\hbox{\scriptsize{asymp}}}(r)$ is not “visible” to inferred-mass measurements. That $\Omega_{\hbox{\scriptsize Dyn}}$ would then be identified with $\Omega_B$ is consistent with the fact that most of the mass inferred through gravitational dynamics are indeed made up of baryons. We did not differentiate between normal and dark matter in our theory, however. Without a specific mechanism funneling nonbaryonic matter into $\rho_{\hbox{\scriptsize{asymp}}}$ and baryonic matter into $\rho-\rho_{\hbox{\scriptsize{asymp}}}$, we cannot at this point rule out the possibility that $\Omega_m-\Omega_B=\Omega_{\hbox{\scriptsize{asymp}}}$ and $\Omega_B\approx\Omega_{\hbox{\scriptsize Dyn}}$ is a numerical accident. The author would like to thank John Garrison for his numerous suggestions, comments, and generous support during this research. He would also like to thank K.-W. Ng H. T. Cho and Clifford Richardson for their helpful criticisms. 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ArXiv

--- abstract: 'We study the phase diagram of director structures in cholesteric liquid crystals of negative dielectric anisotropy in homeotropic cells of thickness $d$ which is smaller than the cholesteric pitch $p$. The basic control parameters are the frustration ratio $d/p$ and the applied voltage $U$. Upon increasing $U$, the direct transition from completely unwound homeotropic structure to the translationally invariant configuration ($TIC$) with uniform in-plane twist is observed at small $d/p\lessapprox 0.5$. Cholesteric fingers that can be either isolated or arranged periodically occur at $0.5\lessapprox d/p<1$ and at the intermediate $U$ between the homeotropic unwound and $TIC$ structures. The phase boundaries are also shifted by (1) rubbing of homeotropic substrates that produces small deviations from the vertical alignment; (2) particles that become nucleation centers for cholesteric fingers; (3) voltage driving schemes. A novel re-entrant behavior of $TIC$ is observed in the rubbed cells with frustration ratios $0.6\lessapprox d/p\lessapprox 0.75,$ which disappears with adding nucleation sites or using modulated voltages. In addition, Fluorescence Confocal Polarising Microscopy (FCPM) allows us to directly and unambiguously determine the 3-D director structures. For the cells with strictly vertical alignment, FCPM confirms the director models of the vertical cross-sections of four types of fingers previously either obtained by computer simulations or proposed using symmetry considerations. For rubbed homeotropic substrates, only two types of fingers are observed, which tend to align along the rubbing direction. Finally, the new means of control are of importance for potential applications of the cholesteric structures, such as switchable gratings based on periodically arranged fingers and eyewear with tunable transparency based on $TIC$.' address: - | Liquid Crystal Institute and Chemical Physics\ Interdisciplinary Program, Kent State University, Kent, Ohio 44242 - | Department of Mathematical Sciences, Kent State University, Kent,\ Ohio 44242 - 'AlphaMicron Inc., Kent, Ohio 44240.' author: - | I. I. Smalyukh ($\thanks{% Author for correspondence (e-mail: [email protected])}$), B. I. Senyuk, P. Palffy-Muhoray, and O. D. Lavrentovich - 'H. Huang and E. C. Gartland, Jr.' - 'V. H. Bodnar, T. Kosa, and B. Taheri' title: 'Electric-field-induced nematic-cholesteric transition and 3-D director structures in homeotropic cells' --- Introduction ============ The unique electro-optic and photonic properties of cholesteric liquid crystals (CLCs) make them attractive for applications in displays, switchable diffraction gratings, eyeglasses with voltage-controlled transparency, for temperature visualization, for mirrorless lasing, in beam steering and beam shaping devices, and many others [Blinov-Ch-book,McManamon,NatureChColor,DKYang,EyeLC1,Patel,ChGrating1,Bistable-chiral,ChLaser1,ChLaserAPL,ChLaser3,de Gennes-book,KlemanLavrentovichBook,ChiralityInLC]{}. In nearly all of these applications, CLCs are confined between flat glass substrates treated to set the orientation of molecules at the liquid crystal (LC)-glass interface along some well-defined direction (called easy axis) and an electric field is often used to switch between different textures. In the confined CLCs, the magnitude of the free energy terms associated with elasticity, surface anchoring, and coupling to the applied field are frequently comparable; their competition results in a rich variety of director structures that can be obtained by appropriate surface treatment, material properties of CLCs, and applied voltage. Understanding these structures and transitions between them is of great practical interest and of fundamental importance[@de; @Gennes-book; @KlemanLavrentovichBook; @ChiralityInLC]. CLCs have a twisted helicoidal director field in the ground state. The axis of molecular twist is called the helical axis and the spatial period over which the liquid crystal molecules twist through $2\pi $ is called the cholesteric pitch $p$. CLCs can be composed of a single compound or of mixtures of a nematic host and one or more chiral additives. Cholesterics usually have the equilibrium pitch $p$ in the range $100nm-100\mu m$; the pitch $p$ can be easily modified by additives. When CLCs are confined in the cells with different boundary conditions or subjected to electric or magnetic fields, one often observes complex three-dimensional (3-D) structures. The cholesteric helix can be distorted or even completely unwound by confining CLCs between two substrates treated to produce homeotropic boundary conditions [@Oswald-review00]. Interest in this subject was initiated by Cladis and Kleman [@CladisKleman], subsequently a rich variety of spatially periodic and uniform structures have been reported [Oswald-review00,LGil,ourreview,PressArrott,MCLC-PressArrott,Flow-PressArrott,Gil-PRL98,Baudry-PRE99,CHnegativeDeltaE,Tarasov,Oswald2004,InvWalls,T-Junktions,FLequeux,Ishikawa-fingerprint]{}.  These structures can be controlled by varying the cell gap thickness $d$, pitch $p$, applied voltage $U$, and the dielectric and elastic properties of the used CLC. The complexity of many LC structures usually does not allow simple analytic descriptions of the director configuration. Since the pioneering work of Press and Arrott[@PressArrott; @MCLC-PressArrott], a great progress has been made in computer simulations of static director patterns in CLCs confined into homeotropic cells (see, for example [Oswald-review00,LGil,ourreview,PressArrott,Flow-PressArrott,Gil-PRL98,Baudry-PRE99]{}), which brought much of the current understanding of these structures. The first goal of this work is to study phase diagram and director structures that appear because of geometrical frustration of CLCs in the cells with either strictly vertical or slightly tilted ($<2^{\circ }$) easy axis at the confining substrates. We start with the phase diagram in the plane of $\rho =d/p$ and $U$ similar to the one reported in Refs.[Oswald-review00,CHnegativeDeltaE]{} and then proceed by studying influence of such extra parameters as rubbing, introducing nucleation sites, and voltage driving schemes. We use CLCs with negative dielectric anisotropy; the applied voltages are sufficiently low and the frequencies are sufficiently high to avoid hydrodynamic instabilities [@de; @Gennes-book]. Cell gap thicknesses $d$ are smaller than $p$ and the phase diagrams are explored for frustration ratios $\rho =0-1$ and $U=(0-4)Vrms$. For small $\rho $ and $U$, the boundary conditions force the LC molecules throughout the sample to orient perpendicular to the glass plates. Above the critical values of $\rho $ and/or $U$, cholesteric twisting of the director takes place [Oswald-review00]{}. Depending on $\rho ,$ $U,$ and other conditions, the twisted director structures can be either uniform or spatially periodic, with wave vector in the plane of the cell. Upon increasing $U$ for $\rho <0.5 $, the direct transition from completely unwound homeotropic structure to the translationally invariant configuration ($TIC$) [Oswald-review00,PressArrott,MCLC-PressArrott]{} with uniform in-plane twist is observed. Cholesteric fingers ($CF$s) of different types that can be either isolated or arranged periodically are observed for $0.5\lessapprox \rho <1$ and intermediate $U$ between the homeotropic unwound and $TIC$ structures. The phase diagrams change if the homeotropic alignment layers are rubbed, if particles that become the nucleation centers for $CF$s are present, and if different driving voltage schemes are used. Upon increasing U in rubbed homeotropic cells with $0.6<\rho <0.75$, we observe a re-entrant behavior of $TIC$ and the following transition sequence: (1) homeotropic untwisted state, (2) translationally invariant twisted state, (3) periodic fingers structure (4) translationally invariant twisted state with larger in-plane twist. This sequence has not been observed in our own and in the previously reported[@Oswald-review00] studies of unrubbed homeotropic cells. The second goal of this work is to unambiguously reconstruct director field of $CF$s and other observed structures in the phase diagram. For this we use the Fluorescence Confocal Polarizing Microscopy (FCPM) [@FCPMCPL]. Although the fingers of different types look very similar under the polarizing microscope (which may explain some confusion in the literature [@Baudry-PRE99]), FCPM allows clear differentiation of $CF$s, as well as other structures. We directly visualize the $TIC$ with the total director twist ranging from $0$ to $2\pi $, depending on $\rho $ and $U,$ and rubbing. We reconstruct director structure in the vertical cross-sections of four different types of $CF$s that are observed for cells with strictly vertical alignment. We unambiguously prove the models described recently by Oswald et al. [@Oswald-review00] while disproving some of the other models that were proposed in the early literature (see, for example[Baudry-PRE99]{}). Only two types of $CF$ structures are observed in CLCs confined to cells with slightly tilted easy axes at the substrates. The third goal of our work stems from the importance of the studied structures for practical applications. The spatially-uniform $TIC$ and homeotropic-to-$TIC$ transition are used in the electrically driven light shutters, intensity modulators, eyewear with tunable transparency, and displays [@Blinov-Ch-book; @EyeLC1; @Patel; @Bistable-chiral]. In these applications, it is often advantageous to work in the regime of high $\rho $ but fingers are not desirable since they scatter light. In our study, we therefore focus on obtaining maximum effect of different factors on the phase diagrams. We demonstrate that the combination of rubbing and low-frequency voltage modulation can stabilize the uniformly twisted structures up to $\rho \approx 0.75$, much larger than $\rho \approx 0.5$ reported previously [@Oswald-review00]. The presence of nucleation centers, such as particles used to set the cell thickness, tends to destroy homogeneously twisted cholesteric structure even at relatively low $\rho \approx 0.5$ confinement ratios; this information is important for the optimal design of the finger-free devices. On the other hand, periodic finger patterns with well controlled periodicity and orientation may be used as voltage-switchable diffraction gratings. Our finding, which enables the very possibility of such application, is that rubbing can set the unidirectional orientation of periodically arranged fingers. The article is organized as follows. We describe materials, cell preparation, and experimental techniques in section II. The phase diagrams are described in section III.A and the reconstructed director structures in section III.B. Section IV gives an analytical description of the transition from homeotropic to a twisted state as well as a brief discussion of other structures and transitions along with their potential applications. The conclusions are drawn in section V. Experiment ========== Materials and cell preparation ------------------------------ The cells with homeotropic boundary conditions were assembled using glass plates coated with transparent ITO electrodes and the polyimide JALS-204 (purchased from JSR, Japan) as an alignment layer. JALS-204 provides strong homeotropic anchoring; anchoring extrapolation length, defined as the ratio of the elastic constant to the anchoring strength, is estimated to be in the submicron range. Some of the substrates with thin layers of JALS-204 were unidirectionally buffed (5 times using a piece of velvet cloth) in order to produce an easy axis at a small angle $\gamma $ to the normal to the cell substrates. $\gamma $ was measured by conoscopy and magnetic null methods [@MagneticNull].  The value of $\gamma $ weakly depends on the rubbing strength, but in all cases it was small, $\gamma <2^{\circ }$. The cell gap thickness was set using either the glass micro-sphere spacers uniformly distributed within the area of a cell (one spacer per approximately $100\mu m\times 100\mu m$ area) or strips of mylar film placed along the cell edges. The cell gap thickness $d$ was measured after cell assembly using the interference method [@BornWolf] with a LAMBDA18 (Perkin Elmer) spectrophotometer. In order to study textures as a function of the confinement ratio $\rho =d/p$, we constructed a series of cells, with identical thickness, but filled with CLCs of different pitch $p$. To minimize spherical aberrations in the FCPM, observations were made with immersion oil objectives, using glass substrates of thickness $0.15mm$ with refractive index $1.52$ [@FCPMCPL].  Regular ($1mm$) and thick ($3mm$) substrates were used to construct cells for polarizing microscopy (PM) observations. Cholesteric mixtures were prepared using the nematic host AMLC-0010 (obtained from AlphaMicron Inc., Kent, OH) and the chiral additive ZLI-811 (purchased from EM Industries). The helical twisting power $HTP=10.47\mu m^{-1}$ of the additive ZLI-811 in the AMLC-0010 nematic host was determined using the method of Grandjean-Cano wedge [@SmalyukPRE; @TKosaMCLC].  The obtained mixtures had pitch $2<p<500\mu m$ as calculated from $% p=1/(c_{chiral}\cdot HTP)$ where $c_{chiral}$ is the weight concentration of the chiral agent, and verified by the Grandjean-Cano wedge method [SmalyukPRE,TKosaMCLC,SmalyukhPRL]{}. The low frequency dielectric anisotropy of the AMLC-0010 host$\ $is $\Delta \varepsilon =-3.7$ ($\varepsilon _{\parallel }=3.4,$ $\varepsilon _{\perp }=7.1$) as determined from capacitance measurements for homeotropic and planar cells using an SI-1260 Impedance/Gain-phase analyzer (Schlumberger) [@Blinov-Ch-book; @deJeu]. The birefringence of AMLC-0010 is $\Delta n=0.078$ as measured with an Abbe refractometer.  The elastic constants describing the splay, twist, and bend deformations of the director in AMLC-0010 are $K_{11}=17.2pN$, $% K_{22}=7.51pN $, $K_{33}=17.9pN$ as determined from the thresholds of electric and magnetic Freedericksz transition in different geometries [Blinov-Ch-book,deJeu]{}.  The cholesteric mixtures were doped with a small amount of fluorescent dye n,n’-bis(2,5-di-tert-butylphenyl)-3,4,9,10-perylenedicarboximide (BTBP) [FCPMCPL]{} for the FCPM studies. Small quantities ( $0.01$wt. % ) of BTBP dye were added to the samples; at these concentrations, the dye is not expected to affect properties of the CLCs used in our studies. Constant amplitude and modulated amplitude signals were applied to the cells using a DS345 generator (Stanford Research Systems) and a Model 7602 Wide-band Amplifier (Krohn-Hite) which made possible the use of a wide range of carrier and modulation frequencies ($10-100000$) $Hz$. The transitions from the homeotropic untwisted to a variety of twisted structures were monitored via capacitance measurements and by measuring the light transmittance of the cell between crossed polarizers. The transitions between different director structures and textures were characterized with PM and FCPM [@FCPMCPL] as described below. Polarizing Microscopy and Fluorescence Confocal Polarizing Microscopy --------------------------------------------------------------------- Polarizing Microscopy (PM) observations were performed using the Nikon Eclipse E600 POL microscope with the Hitachi HV-C20 CCD camera. The PMstudies were also performed using a BX-50 Olympus microscope in the PM mode. In order to directly reconstruct the vertical cross-sections of the cholesteric structures, we performed further studies in the FCPM mode of the very same modified BX-50 microscope[@FCPMCPL] as described below. The PM and FCPM techniques are used in parallel and provide complementary information. The FCPM set-up was based on a modified BX-50 fluorescence confocal microscope [@FCPMCPL]. The excitation beam ($\lambda =488nm$, from an Ar laser) is focused by an objective onto a small submicron volume within the CLC cell. The fluorescent light from this volume is detected by a photomultiplier in the spectral region $510-550nm$. A pinhole is used to discriminate against emission from the regions above and below the selected volume. The pinhole diameter $D$ is adjusted according to magnification and numerical aperture ${\rm NA}$ of the objective; $D=$ $100\,\mu m$ for an immersion oil $60\times $ objective with ${\rm NA}=1.4$. A very same polarizer is used to determine the polarization of both the excitation beam and the detected fluorescent light collected in the epifluorescence mode. The relatively [low birefringence (]{}${\Delta n\approx 0.0}78$) [of the AMLC-0010 nematic host ]{}mitigates two problems that one encounters in FCPM imaging of CLCs: (1) defocussing of the extraordinary modes relative to the ordinary modes [@FCPMCPL] and (2) the Mauguin effect, where polarization follows the twisting director field [@ourreview; @Yehbook]. The used BTBP dye has both absorption and emission transition dipoles parallel to the long axis of the molecule [SmalyukPRE,FordKamat,SmalyukhPRL]{}. The FCPM signal, resulting from a sequence of absorption and emission events, strongly depends on the angle $% \beta $ between the transition dipole moment of the dye (assumed to be parallel to the local director ${\bf \hat{n}}$) and the polarization ${\bf \hat{P}}$.$\ $The intensity scales as $I_{FCPM}\sim $ $\cos ^{4}\beta $, [@FCPMCPL] as both the absorption and emission are proportional to $\cos ^{2}\beta $[**.**]{} The strongest FCPM signal corresponds to ${\bf \hat{n}% \parallel \hat{P}}$, where $\beta =0$, and sharply decreases when $\beta $ becomes non-zero [@ourreview; @FCPMCPL; @SmalyukPRE; @SmalyukhPRL]. By obtaining the FCPM images for different ${\bf \hat{P}}$, we reconstruct director structures in both in-plane and vertical cross-sections of the cell from which then the entire 3-D director pattern is reconstructed. We note that in the FCPM images the registered fluorescence signal from the bottom of the cell can be somewhat weaker than from the top, as a result of light absorption, light scattering caused by director fluctuations, depolarization, and defocussing. To mitigate these experimental artefacts and to maintain both axial and radial resolution within $1\mu m$, we used relatively shallow ($\leq $ $20$ $\mu m$) scanning depths [ourreview,FCPMCPL]{}. The other artefacts, such as light depolarization by a high NA objective, are neglected as they are of minor importance [ourreview,FCPMCPL,SmalyukPRE,SmalyukhPRL]{}. Results ======= Phase diagrams of textures and structures ----------------------------------------- We start with an experimental phase diagram of cholesteric structures in the homeotropic cells similar to the one reported in [Oswald-review00,CHnegativeDeltaE]{} and then explore how this diagram is affected by rubbing of homeotropic substrates, using different voltage driving schemes, and introducing nucleation sites. We note that for pitch $% p\gtrapprox 5\mu m$ and the cell gap $d\gtrapprox 5\mu m$ much larger than the anchoring extrapolation length ($<1\mu m,$ describing polar anchoring at the interface of CLC and JALS-204 layer), the observed structures depend on $% \rho =d/p$ but not explicitly on $d$ and $p$. We therefore construct the diagrams of structures in the plane of the applied voltage $U$ and the frustration ratio $\rho $; to describe the phase diagram we adopt the terminology introduced in Ref. [@Oswald-review00]. The diagrams display director structures (phases) of homeotropic untwisted state, isolated $CF$s and periodically arranged $CF$s, the $TIC$ and the modulated (undulating) $% TIC$. The phase boundary lines are denoted as $V0-V3$, $V01$, $V02$, Fig.1, similarly to Refs. [@Oswald-review00; @CHnegativeDeltaE] (for comparison with the phase diagrams reported for other LCs). As we show below, the phase diagram can be modified to satisfy requirements for several electro-optic applications of the CLC structures. ### Cells with unrubbed homeotropic substrates The diagram for unrubbed homeotropic cells is shown in Fig.1. The completely unwound homeotropic texture is observed at small $U$ and $\rho $, Fig.1. At high $U$ above $V0$, $V01$, and $V02$, the $TIC$ with some amount of director twist (up to $2\pi $, helical axis along the cell normal) is observed; the twist in $TIC$ is accompanied by splay and bend deformations. The $TIC$ texture is homogeneous within the plane of a cell except that it often contains the so-called umbilics, defects in direction of the tilt [de Gennes-book]{}, Fig.2f. Periodically arranged $CF$s are observed for voltages $U\approx 1.5-3.5Vrms$ and for $0.5<\rho <1$, Fig.1 and Fig.2b-e. If the values of $U$ and $\rho $ are between the $V0$ and $V01$ boundary lines, Fig.1, a transient $TIC$ appears first but then it is replaced (within $0.1-10s$ after voltage pulse, depending on $\rho $ and $U$) by a periodic pattern of $CF$s, which also undergoes slow relaxation; equilibrium is reached only in $3-50s$, Fig.2d. The isolated $CF$s coexisting with the homeotropic state are observed at $U\lessapprox 1.8$ and for $0.75<\rho <1$, Fig.1 and Fig.2a,b. For $\rho $ and $U$ between $V0$ and $V1$, the isolated fingers start growing from nucleation sites such as spacers, Fig.2b, or from already existing fingers. In both cases the $CF$s separated by homeotropic regions split in order to fill in the entire space with a periodic texture of period $\sim p$, similar to the one shown in Fig.2c. In the region between $V1$ and $V2$, isolated $CF$s nucleate and grow but they do not split and do not fill in the whole sample; fingers in this part of diagram coexist with homeotropic untwisted structure, Fig.2a. Hysteresis is observed between $V2$ and $V3$ lines: a homeotropic texture is observed if the voltage is increased, but isolated fingers coexisting with untwisted homeotropic structure can be found if $U$ is decreased from the initial high values. Even though the neighboring $CF$s in the fingers pattern are locally parallel to each other, Fig.2c,d, there is no preferential orientation of the fingers in the plane of the cell on the scales $\gtrapprox 10mm$. Finally, the periodic structure observed between $V01$ and $V02$ does not contain interspersed homeotropic regions, Fig.2e. The director field of $CF$s as well as other structures of the diagram will be revealed by FCPM below, see Sec. III.B. The behavior of the voltage-driven transitions between untwisted homeotropic and different types of twisted structures is reminiscent of conventional temperature-driven phase transitions with voltage playing a role similar to temperature. The phase diagram of structures has a Landau tricritical point $% \rho =\rho _{tricritical}$ at which $V2$ and $V0$ meet. The order of the transition changes from the second order (continuous) at $\rho <\rho _{tricritical}$ to the first order (discontinuous, proceeding via nucleation) at $\rho >\rho _{tricritical}$, Fig.1. The phase diagram also has a triple point at $\rho =\rho _{triple},$ where $V0$ and $V01$ meet. At the triple point, the untwisted homeotropic texture coexists with two different twisted structures, spatially uniform $TIC$ and periodic fingers pattern. The phase diagram and transitions in homeotropic cells with perpendicular easy axes at the substrates are qualitatively similar to those reported by Oswald et al. [@Oswald-review00; @CHnegativeDeltaE] for other materials; both qualitative and quantitative differences are observed when the homeotropic substrates are rubbed to produce slightly tilted easy axes at the confining substrates as discussed below. ### Effects of rubbing and nucleation centers Rubbing of the homeotropic alignment layers induces small pretilt angle from the vertical axis, $\gamma <2^{\circ }$. The azimuthal degeneracy is therefore broken, and the projection of easy axis defines a unique direction in the plane of a cell. Therefore, even rubbing of only one of the cell substrates has a strong effect on the CLC structures: (1) no umbilics are observed in the $TIC$, Fig.3a; (2) $CF$s preferentially align along the rubbing direction, Fig.3b. In addition, the homeotropic-$TIC$ transition, which is sharp in cells with vertical alignment, becomes somewhat blurred for rubbed homeotropic substrates with small $\gamma $, Fig.3c,d. In principle, one can set opposite rubbing directions on the substrates; we report a phase diagram of structures for such anti-parallel rubbing in Fig.4. The cells used to obtain the diagram were constructed from thick $3mm$ glass plates and only the mylar spacers at the cell edges were used to set the cell gap thickness. Compared to phase diagrams of structures with unrubbed substrates, dramatic changes are observed at $\rho \gtrapprox 0.5$. The direct homeotropic to $TIC$ transition is observed up to $\rho \approx 0.75$. The experimental triple and tricritical points are closer to each other than for unrubbed cells (compare  Fig.1 and Fig.4). Interestingly, within the range $0.6<\rho <0.75$ and upon increasing $U$, one first observes a homogeneous $TIC$, Fig.5b, which is then replaced by a periodic fingers pattern at higher $U$, Fig.5c, and again a uniform $TIC$ at even higher $U>3-3.5V$, Fig.5d. The same sequence, $TIC$-fingers-$TIC$-homeotropic, is also observed upon decreasing $U$ from initial high values. Pursuing the analogy with temperature-driven phase transitions, the $TIC$ texture between the fingers pattern and homeotropic texture can be considered as a re-entrant $TIC$ phase. As compared to unrubbed cells, the antiparallel rubbing has little effect on $V0$, but shifts the other boundary lines towards increasing $\rho $. The effects of anti-parallel rubbing on the phase diagram can be explained as follows. At $\rho \sim 0.5$ anti-parallel rubbing matches the director twist of $TIC,$ which at high $U$ is $\approx \pi $. Therefore, $TIC$ is stabilized by anti-parallel rubbing and $CF$s do not appear until higher $\rho $, Fig.4. The transient $TIC$ disappears if large quantities of spacers ($>100/mm^{2}$) or other nucleation sites for fingers are present in the cells with anti-parallel rubbing; in this case the phase diagram is closer to the one shown in Fig.1. The spacers with perpendicular surface anchoring produce director distortions in their close vicinity even in the part of diagram corresponding to homeotropic unwound state, Fig.6a. In the vicinity of the homeotropic-$TIC$ transition, Fig.6b, the director realignment starts in the vicinity of spacers. Similar to the observations in Refs. [InvWalls,PECladis]{}, particles with perpendicular surface anchoring cause inversion walls ($IW$s) and disclinations. The $TIC$ with $0.5<\rho <0.75$, Fig.6c, is eventually replaced by fingers, which are facilitated by the particles, Fig.6d. Moreover, even at high $U$, $TIC$ remains spatially non-uniform and contains different types of $IW$s and disclination lines [@InvWalls; @PECladis], which are caused by the boundary conditions at the surfaces of the particles. ### Phase diagrams for different voltage driving schemes The phase diagrams of structures shown in Figs.1 and 4 were obtained with constant amplitude sinusoidal voltages applied to the cells. The diagram changes dramatically if the applied voltage is modulated. The effect is especially strong in the cells with rubbed homeotropic substrates, for which we present results in Fig.7a-c; somewhat weaker effect is also observed for unrubbed substrates. We explored modulation with rectangular-type, triangular, and sinusoidal signals of different duration and modulation depth. The strongest effect is observed with $100\%$ modulation depth and sinusoidal modulation signal at frequencies (10-200)$Hz$. The fingers patterns are shifted towards increasing $\rho $, Fig.7. At the same time, the $rms$ voltage values of homeotropic-$TIC$ transition are practically the same for different voltage driving schemes, Figs.1,4,7. We assume that the effect of amplitude modulation is related to the very slow dynamics of some of the structures (see Secs. III.A.1, III.A.2), such as $CF$s; the corresponding parts of the diagram are the most sensitive to voltage driving schemes. The substantial combined effect of rubbing and voltage driving schemes is important for practical applications of the homeotropic-$TIC$ transition when it is important to have strongly twisted but finger-free field-on state [@EyeLC1; @Bistable-chiral]. We therefore present only the diagrams corresponding to the largest $\rho $ values at which fingers do not appear for given surface rubbing conditions, Fig.7. On the other hand, voltage modulation could be a way to study the stability of different parts of the diagram in the $\rho ,U$ plane and deserves to be explored in more details; we leave this for forthcoming publications. Finally, to understand the diagrams and transitions explored in this section it is important to know the director fields that are behind different textures; this will be explored in the following section. Director structures ------------------- ### Spatially-homogeneous twisted structures, umbilics, and inversion walls In this section, we take advantage of the FCPM and study the director field $% {\bf \hat{n}}\left( x,y,z\right) $ in the vertical cross-sections (i.e., along the $z$, normal to the cell substrates) of the cholesteric structures. This is important as, for example, in the $TIC$ ${\bf \hat{n}}$ varies only along $z$ and not in the plane of a cell. The $TIC$, observed above the $V0$ and $V02$ lines in the phase diagrams, Figs.1,4,7, can be visualized as having ${\bf \hat{n}}$ rotating with distance from the cell wall on a cone whose axis is along $z$; the half angle of this cone varies from $\theta =0$ at the substrates to $\theta _{\max }$ in the middle plane of a cell ($% \theta _{\max }<\pi /2$), Fig.8. FCPM reveals that the in-plane twist of the director in the $TIC$ depends on $\rho $. For small $\rho \approx 0,$ the $% TIC$ contains practically no in-plane twist. When $\rho \approx 1/2$, the in-plane twist at high $U$ reaches $\pi $, Fig.8a,c. Finally, when $\rho \approx 1$, the twist of the $TIC$ structure at high $U$ can reach $2\pi $, Fig.8b,d. The maximum in-plane twist at high $U$ is $\approx 2\pi \rho $; we stress that the twist of $TIC$ depends not only on $\rho $, but also on $U.$ In addition, for the cells with rubbed homeotropic plates, the in-plane twist is affected by the rubbing direction. For example, the re-entrant $TIC$ in the rubbed cells of $0.5<\rho <0.75$ has twist $\approx \pi $ at small $U$ just above the $V0$ and the twist close to $2\pi $ at high voltages $U>4Vrms$. If both of the homeotropic substrates are rubbed, the natural twist of the $TIC$ structure may or may not be compatible with the tilted easy axes at the confined substrates. Since the amount of twist in the $TIC$ depends both on $\rho $ and $U$, it is impossible to match tilted homeotropic boundary conditions to a broad range of $\rho $ and $U$. However, since the in-plane anchoring is weak, the effect of rubbing on the twist in $TIC$ is not as strong as in the case of planar cells. The FCPM also allows us to probe the defects that appear in $TIC$. We confirm that the defects with four brushes, Fig.2f, are umbilics of strength $\pm 1$[@de; @Gennes-book] rather than disclinations with singular cores. We also verify that the umbilics are caused by degeneracy of director tilt when $U$ is applied; such degeneracy is eliminated by rubbing, Fig.3a. Within $TIC$, we also observe $IW$s[@InvWalls; @PECladis]. The appearance of these walls was previously attributed to a variety of factors, such as flow of liquid crystal, hydrodynamic instabilities, alignment induced at the edges of the sample, and others [@InvWalls; @PECladis]. FCPM observations indicate that in the presence of spacers with perpendicular anchoring, the $% IW$s appear at the particles when $U$ above the threshold for homeotropic-$% TIC$ transition is applied. This is believed to be caused by director distortions in the vicinity of the particles [@InvWalls; @PECladis]. When the confinement ratio is $0.5<\rho <0.75$, the distorted $TIC$ with umbilics, disclinations, and $IW$s is replaced by $CF$s with the spacers serving as nucleation sites for the fingers, Fig.6. ### Fingers structures; nonsingular fingers of $CF1$-type Fingers structures have translational invariance along their axes ($y$-direction) and can be observed as isolated between $V3$ and $V0$ or periodically arranged between $V0$ and $V01$ boundary lines, Figs.1,4,7 (see Sec.III.A for details). We again take advantage of FCPM by visualizing the vertical cross-sections and then reconstruct ${\bf \hat{n}}\left( x,y,z\right) $ of the fingers directly from the experimental data. To describe the results, we use the $CFs$ classification of Oswald et al. [Oswald-review00]{}. The finger of $CF1$ type is the most frequently observed in cells with vertical as well as slightly tilted alignment, Fig.9. $CF1$ is isolated and co-existing with the homeotropic untwisted structure between $% V3 $ and $V0$ and is a part of the spatially periodic pattern between $V0$ and $V01$ lines, Figs.1,4,7. The reason for abundance of $CF1$, is that it can form from $TIC$ by continuous transformation of director field above the $V0$ line and it also can easily nucleate from homeotropic untwisted structure below the $V0$ boundary line, Figs.1,4,7[@Oswald-review00]. The director structure of $CF1$ reconstructed from the FCPM vertical cross-section, Fig.9, is in a good agreement with the results of computer simulations [Oswald-review00,LGil,ourreview,PressArrott,MCLC-PressArrott]{}. In the $CF1$, the axis of cholesteric twist is tilted away from the cell normal $z$, Fig.9; the in-plane twist in direction perpendicular to the finger in the middle of a cell is $2\pi $. The isolated $CF1s$ that are separated from each other by large regions of homeotropic texture, Fig.2a, assume random tilt directions. The width of an isolated $CF1$ is somewhat larger than $d$; this is in a good agreement with computer simulations of L. Gil [@LGil]. $CF1$ is nonsingular in ${\bf \hat{n}}$ (i.e., the spatial changes of ${\bf \hat{n}}$ are continuous and ${\bf \hat{n}}$ can be defined everywhere within the structure) as the twist is accompanied with escape of ${\bf \hat{n% }}$ into the third dimension along its center line. An isolated $CF1$ can be represented as a quadrupole of the non-singular $\lambda $-disclinations, two of strength $+1/2$ and two of strength $-1/2$, as shown in Fig.9b. The $% \lambda $-disclinations, with core size of the order of $p$, cost much less energy than the disclinations with singular cores [KlemanLavrentovichBook]{}. The pair of disclinations $\lambda ^{+1/2}\lambda ^{-1/2}$ introduces $2\pi $-twist at one homeotropic substrate; this $2\pi $-twist is then terminated by introducing another $\lambda ^{+1/2}\lambda ^{-1/2}$ pair in order to satisfy the homeotropic boundary conditions at another substrate, Fig.9b. A segment of an isolated $CF1$ has different ends; one is rounded while the another is pointed. Behavior of these ends is different during growth; the pointed end remains stable, while the rounded end continuously splits, as also discussed in Ref. [@Oswald2004]. The FCPM vertical cross-section, Fig.10, reveals details of $CF1$ tiling into periodically arranged structures that are observed above $V0$ line, Figs.1,4,7. When $U$ or $\rho $ are relatively large, the $CF1$ fingers are close to each other so that the homeotropic regions in between cannot be clearly distinguished. The tilt of the helical axis in the periodic $CF$ structures is usually in the same direction, Fig.10. A possible explanation is that the elastic free energy is minimized since the structure of unidirectionally tilted $CFs$ is essentially space-filling. On rare occasions, the tilt direction of neighboring $CF1s$ is opposite. Upon increasing $U$, the width of fingers originally separated by homeotropic regions, Fig.11a, gradually increases, Fig.11b-e; the fingers then merge to form a periodically modulated $TIC$, Fig.11f. Finally, at high applied voltages, the transition to uniform $TIC$ is observed, Fig.11g. The details of transformation of periodically arranged fingers into the in-plane homogeneous $TIC$ via the modulated (undulating) twisted structure were not known before and would be difficult to grasp without FCPM. $TIC$ can also be formed by expanding one of the $CF1$s; structure of coexisting fingers and $% TIC$ contains only $\lambda $-disclinations nonsingular in ${\bf \hat{n}}$ again demonstrating the natural tendency to avoid singularities, Fig.12. Periodically arranged $CF1$s slowly (depending on rubbing, $U$ and $\rho $; usually up to $1s$) appear from $TIC$ if $U$ is between $V0$ and $V01,$ and quickly disappear (in less than $50ms$) if $U$ is increased above $V02$. This allowed us to use the amplitude-modulated voltage driving schemes in combination with rubbing and obtain finger-free $TIC$ up to $\rho \approx 0.8 $ (Sec. III.A.3), as needed for applications of $TIC$ in the electrically driven light shutters, intensity modulators, eyewear with tunable transparency, displays, etc. [Blinov-Ch-book,EyeLC1,Patel,Bistable-chiral]{} ### Fingers of $CF2$, $CF3$, and $CF4$ types containing defects; T-junctions of fingers Another type of fingers is $CF2$, Fig.13, which is observed for vertical as well as slightly tilted alignment in the same parts of the diagram as $CF1$, Figs.1,4,7. However, in contrast to the case of nonsingular $CF1$, a segment of $CF2$ has point defects at its ends. Because of this, $CF2$ does not nucleate from the homeotropic or $TIC$ structures as easily as $CF1$ and normally dust particles, spacers, or irregularities are responsible for its appearance. Therefore, fingers of $CF2$-type, Fig.13, are found less frequently than $CF1$. Using FCPM, we reconstruct the director structure in the vertical cross-section of $CF2$, Fig.13; the experimental result resembles the one obtained in computer simulations by Gil and Gilli [Gil-PRL98]{}, proving the latest model of $CF2$ [Oswald-review00,Gil-PRL98]{} and disproving the earlier ones [Baudry-PRE99]{}. Within the $CF2$ structure, one can distinguish the non-singular $\lambda ^{+1}$ disclination in the central part of the cell and two half-integer $\lambda ^{-1/2}$ disclinations in the vicinity of the opposite homeotropic substrates. The total topological charge of the $CF2$ is conserved, similarly to the case of $CF1$. Polarizing microscopic observations show that unlike in $CF1$-type fingers, the ends of $CF2$ segments have similar appearance. FCPM reveals that the point defects (of strength $1$) at the two ends have different locations being closer to the opposite substrates of a cell. Unlike the $CF1$ structure, $CF2$ is not invariant by $\pi $-rotation around the $y$-axis along the finger, as also can be seen from the FCPM cross-section, Fig.13. Different symmetries of $% CF1 $ and $CF2$ are responsible for their different dynamics under electric field [@Oswald-review00; @Gil-PRL98]. This, along with computer simulations, allowed Gil and Gilli [@Gil-PRL98] to propose the model of $% CF2$; our direct imaging using FCPM unambiguously proves that this model is correct, Fig.13b. The isolated $CF2$ fingers (coexisting with homeotropic state) expand, when $U\gtrapprox 2.1Vrms$ is applied, Fig.13c,d. The structures with non-singular $\lambda $-disclinations often separate the parts of a cell with different twist, Fig.14; they resemble the structures of thick lines that are observed in Grandjean-Cano wedges with planar surface anchoring [@SmalyukPRE; @SmalyukhPRL]. The appearance of these lines in homeotropic cells is facilitated by sample thickness variations and spacers. The width of $CF2$ coexisting with the homeotropic state is usually the same or somewhat (up to 30%) smaller than $CF1$; this can be seen in Fig.15 showing a $T$-junction of the $CF1$ and $CF2$ fingers. Even though $CF1$ and $CF2$ have similar appearance under a polarizing microscope [@Oswald-review00], FCPM allows one to clearly distinguish these structures. Note also the tendency to avoid singularities in ${\bf \hat{n}}$ evidenced by the reconstructed structure of the $T$-junction, Fig.15b. The metastable cholesteric fingers of $CF3$-type, Fig.16, occur even less frequently than $CF2s$. The director structure of this finger was originally proposed by Cladis and Kleman [@CladisKleman]. In polarizing microscopy observations, the width of $CF3$ fingers is about half of that in $CF1$ and $% CF2$. The reconstructed FCPM structure of $CF3$ indicates that the director ${\bf \hat{n}}$ rotates through only $\pi $ along the axis perpendicular to the finger ($x$-axis). This differs from both $CF1$ and $CF2$, which both show a rotation of ${\bf \hat{n}}$ through $2\pi $.  Two twist disclinations of opposite signs near the substrates allow the cholesteric $% \pi $-twist in the bulk to match the homeotropic boundary conditions, Fig.16. The structure of $CF3$ is singular in ${\bf \hat{n}}$; the disclinations are energetically costly and this explains why $CF3$ is observed rarely even in the cells with vertical alignment. In cells with rubbed homeotropic substrates $CF3$ was never observed. This is likely due to the easy axis having the same tilt on both sides of the finger on a rubbed substrate, whereas the $\pi $-twisted configuration of  $CF3$ requires director tilt in opposite directions. The $CF4$-type metastable finger shown in Fig.17 is also singular, and is usually somewhat wider than the other $CFs$. It can be found in all regions of existence of $CF1$, Fig.1, but is very rare and usually is formed after cooling the sample from isotropic phase. $CF4$ contains two singular disclinations at the same substrate. In the plane of a cell, the director $% {\bf \hat{n}}$ rotates by $2\pi $ with the twist axis being along ${\bf x}$ and perpendicular to the finger. Using the direct FCPM imaging, we reconstruct the director structure of $CF4$, Fig.17b, which is in a good agreement with the model of Baudry et al. [@T-Junktions]. The bottom part of this finger, Fig.17, is nonsingular, and is similar in this respect to $CF1$ and $CF2$; its top part, however, contains two singular twist disclinations. The $CF4$ structure is observed only in cells with no rubbing. Similar to the case of $CF3$, the structure of $CF4$ is not compatible with uniform tilt produced by rubbing of homeotropic substrates. Of the four different fingers structures, $CF4$ might be the least favorable energetically, since it usually rapidly transforms into $CF1$ or $CF2$; less frequently, it also splits into two $CF3$ fingers. Transformation of other fingers into $CF4$ was never observed. Discussion ========== The system that we study is fairly rich and complicated; some of the structures and transitions can be described analytically while the other require numerical modeling. Below we first restrict ourselves to translationally uniform structures (i.e., homeotropic and $TIC$) which can be described analytically. We then discuss the other experimentally observed structures and transitions comparing them to the analytical as well as numerical results available in literature [Oswald-review00,CHnegativeDeltaE]{}, as well as our own numerical study of the phase diagram that will be published elsewhere[@Garlandetal]. Finally, we discuss the practical importance of the obtained results on the phase diagrams of director structures. Translationally uniform homeotropic and $TIC$ structures -------------------------------------------------------- We represent the director ${\bf \hat{n}}$ in terms of the polar angle $% \theta $ (between the director and the $z$-axis) and the azimuthal angle $% \phi $ (the twist angle); $\psi $ is electric potential. For the $TIC$ configurations, these fields are functions of $z$ only, and the Oseen-Frank free-energy density takes the form $$\begin{aligned} 2f &=&(K_{11}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta )\theta _{z}^{2}+(K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta )\sin ^{2}\!\theta \,\phi _{z}^{2} \nonumber \\ &&{}-K_{22}\frac{4\pi }{p}\sin ^{2}\!\theta \,\phi _{z}+K_{22}\frac{4\pi ^{2}% }{p^{2}}-(\varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta )\psi _{z}^{2}, \label{eqn:fe}\end{aligned}$$where, $K_{11}$, $K_{22}$, and $K_{33}$ are the splay, twist, and bend elastic constants, respectively; $\varepsilon _{\parallel }$,$\varepsilon _{\perp }$ are the dielectric constants parallel and perpendicular to ${\bf \hat{n},}$ respectively; $\theta _{z}=d\theta /dz,$ $\phi _{z}=d\phi /dz,$ and $\psi _{z}=d\psi /dz$. The associated coupled Euler-Lagrange equations are $$\begin{aligned} \lefteqn{\frac{d}{dz}\left[ \left( K_{11}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta \right) \theta _{z}\right] =\sin \theta \cos \theta \Bigl\{(K_{11}-K_{33})\theta _{z}^{2}} \nonumber \\ &&{}+\left[ (2K_{22}-K_{33})\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta % \right] \phi _{z}^{2}-K_{22}\frac{4\pi }{p}\phi _{z}-\Delta \varepsilon \psi _{z}^{2}\Bigr\}, \label{eqn:ELa}\end{aligned}$$ $$\frac{d}{dz}\left\{ \sin ^{2}\!\theta \left[ (K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta )\phi _{z}-K_{22}\frac{2\pi }{p}\right] \right\} =0, \label{eqn:ELb}$$ $$\frac{d}{dz}\left[ \left( \varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta \right) \psi _{z}\right] =0, \label{eqn:ELc}$$ with associated boundary conditions $\theta (0)=\theta (d)=0$; $\phi (0),\phi (d)$ undefined; and $\psi (0)=0$, $\psi (d)=U$. Dielectric anisotropy is negative for the studied material, $\Delta \varepsilon =\varepsilon _{\parallel }-\varepsilon _{\perp }<0$. Representative solutions of these equations are plotted in Fig.18 and describe how $\theta (z),$ $\phi (z),$ and $\psi (z)$ vary across the cell. Equations (\[eqn:ELb\]) and (\[eqn:ELc\]) above admit first integrals, $$\phi _{z}=\frac{K_{22}}{K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta }% \frac{2\pi }{p} \label{eqn:phiz}$$and $$\left( \varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta \right) \psi _{z}=\frac{U}{\displaystyle\int_{0}^{d}\frac{1}{% \varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta }dz},$$which allow to express the free energy in terms of the tilt angle $\theta $ only: $$\begin{aligned} {\cal F}[\theta ] &=&\frac{1}{2}\int_{0}^{d}\left[ (K_{11}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta )\theta _{z}^{2}+\frac{K_{2}K_{3}\cos ^{2}\!\theta }{K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta }\frac{4\pi ^{2}}{p^{2}}% \right] dz \nonumber \\ &&{}-\frac{1}{2}U^{2}\left[ \int_{0}^{d}\frac{1}{\varepsilon _{\perp }\sin ^{2}\!\theta +\varepsilon _{\parallel }\cos ^{2}\!\theta }dz\right] ^{-1}.\end{aligned}$$This is similar to Eq. (3.221) of Ref. [@stewart:04] (see p. 91), where the splay Freedericksz transition with a coupled electric field is discussed. We expand the free energy in terms of $\theta (z)$ about the undistorted $\theta =0\,$homeotropic configuration to obtain$${\cal F}[\theta ]=\frac{1}{2}\left[ \frac{4\pi ^{2}dK_{22}}{p^{2}}-\frac{% \varepsilon _{\parallel }U^{2}}{d}\right] +\frac{1}{2}\int_{0}^{d}\left[ K_{33}\theta _{z}^{2}-\left( \frac{\Delta \varepsilon U^{2}}{d^{2}}+\frac{% 4\pi ^{2}K_{22}^{2}}{p^{2}K_{33}}\right) \theta ^{2}\right] dz+O(\theta ^{4}).$$The first term is the free energy of the uniform homeoptropic configuration. The second-order term is positive definite if $U$ and $\rho $ are sufficiently small. Ignoring higher order terms, we find that the loss of stability occurs when $$\frac{4K_{22}^{2}}{K_{33}^{2}}\rho ^{2}+\frac{\Delta \varepsilon }{K_{33}\pi ^{2}}U^{2}=1. \label{eqn:ellipse}$$Eq. (\[eqn:ellipse\]) is the spinodal ellipse. The homeotropic configuration is metastable with respect to translationally homogeneous perturbations for the $\rho $ and $U$ parameters inside the ellipse ([eqn:ellipse]{}), which corresponds to the boundary line $V0$, Figs.1,4,7. Eq. (\[eqn:ellipse\]) gives the threshold voltage for transition between homeotropic and $TIC$ structures: $$U_{th}=\pi \sqrt{K_{33}/\Delta \varepsilon }\cdot \sqrt{1-4\rho ^{2}K_{22}^{2}/K_{33}^{2}}. \label{U0-threshold}$$ Eq. (\[U0-threshold\]) is in a good agreement with our experimental results described in Sec. III above and with Ref. [@Rosenblatt]. The experimental data for boundary line $V0$ are well described by Eq. ([eqn:ellipse]{}) for rubbed and unrubbed homeotropic cells, Figs.1,4,7. According to the linear stability analysis above, the ellipse in the $\rho $-$U$ plane (\[U0-threshold\]) defines the limit of metastability of the homeotropic phase: for $\rho $ and $U$ inside this ellipse, the uniform homeotropic configuration is metastable, while outside the ellipse, it is a locally unstable equilibrium. In an idealized cell with infinitely strong homeotropic anchoring and no pretilt, the transition from homeotropic to $% TIC $ is a forward pitchfork bifurcation, that is, a second-order transition. For voltages $U$ below the $V0$ line (inside the ellipse), the $% TIC$ configuration does not exist. On the other hand, when there is a slight tilt of the easy axis, the reflection symmetry is broken. The pitchfork is unfolded into a smoother transition, i.e., the transition becomes supercritical and the precise transition threshold is not well defined. The experimentally observable artefact of this is the somewhat blurred transition, which is described in Sec. III.A.2, for the cells with rubbing and resembles a similar effect in planar cells with small pretilt[Yehbook]{}. The above analysis allows one to understand the dependence of the total in-plane twist of $TIC$ on $\rho $ and $U$ that was described in Sec. III.B.1. The first integral (\[eqn:phiz\]) gives the tilt-dependence of the local twist rate. The total twist across a cell of thickness $d$ is $$\Delta \phi =\frac{2\pi }{p}\int_{0}^{d}\frac{K_{22}}{K_{22}\sin ^{2}\!\theta +K_{33}\cos ^{2}\!\theta }\,dz. \label{eqn:Dphi}$$For the AMLC-0010 material with $K_{22}/K_{33}\approx 0.42$ for given $\rho $ the total twist $\Delta \phi $ can be varied $$0.42\ast 2\pi \rho <\Delta \phi <2\pi \rho , \label{eqn:DphiLimits}$$by changing $U$. $\Delta \phi $ approaches the lower limit for relatively small $U$ that are just above $U_{th}$ and $\theta \approx 0$ and the upper limit for $U\gg U_{th}$ and $\theta \approx \pi /2$ . This analysis is in a good agreement with the FCPM images of the vertical cross-sections of $TIC$ for different $\rho $, as described in Sec. III. Finally, knowledge of $% \Delta \phi $ variation with changing $U$ is important for the practical applications of $TIC$ as it will be discussed below. Other structures and transitions of the phase diagram ----------------------------------------------------- Modeling of transitions associated with $CFs,$ in which ${\bf \hat{n}}$ is a function of two coordinates, is more complicated than in the case of $TIC$. Ribière, Pirkl, and Oswald[@CHnegativeDeltaE] obtained complete phase diagram in calculations assuming a simplified model of a cholesteric finger. This theoretical diagram qualitatively resembles our experimental result for the cells with vertical alignment, Fig.1. We explored the phase diagram using 2-D numerical modeling in which the equilibrium structure of the $CFs$ and equilibrium period of periodically arranged fingers are determined from energy minimization in a self-consistent way and the nonlocal field effects are taken into account[@Garlandetal]. The numerical phase diagrams show a good quantitative agreement with the experimental results presented here, predicting even the re-entrant behavior of $TIC$ that we experimentally obtain for cells with rubbed substrates (Sec.III.A.2). Presentation of these results requires detailed description of numerical modeling and will be published elsewhere[@Garlandetal]. Therefore, we only briefly discuss the qualitative features of the phase diagrams shown in Figs.1,4,7 in the light of the previous theoretical studies[@Oswald-review00; @CHnegativeDeltaE] and also summarize the new experimental results below. The important feature of the studied diagram is that the nematic-cholesteric transition changes order: it is second order for $0<\rho <\rho _{tricritical} $ and first order for $\rho >\rho _{tricritical}$, Fig.1, in agreement with Refs.[@Oswald-review00; @CHnegativeDeltaE] The phase diagram has a triple point at $\rho =\rho _{triple}$, where $V0$ and $V01$ meet and the untwisted homeotropic texture coexists with two twisted structures, $TIC$ and periodically arranged $CFs$, Figs1,4,7. For vertical alignment, the direct voltage-driven homeotropic-$TIC$ transition is observed at small $\rho \lessapprox 0.5$. Structures of isolated $CFs$ and periodically arranged $CFs $ occur for $0.5\lessapprox \rho <1$ and intermediate $U$ between the homeotropic state and $TIC$. The theoretical analysis of Ref. [@CHnegativeDeltaE] allows one to determine $\rho $ corresponding to the tricritical and triple points in the phase diagrams. Solving the equations given in Ref.[@CHnegativeDeltaE] numerically[error]{}  and using the material parameters of the AMLC-0010 host doped with the chiral agent ZLI-811, we find $\rho $ corresponding to the triple and tricritical points: $\rho _{triple}=0.816$ and $\rho _{tricritical}=0.861.$ These values are somewhat larger than $\rho _{triple}$ and $\rho _{tricritical}$ determined experimentally for the cells with vertical alignment, Figs.1,4,7, as also observed in [@CHnegativeDeltaE] for other CLCs. The calculated $\rho _{triple}$ and $\rho _{tricritical}$ are closer to the experimental ones in the case of rubbed substrates; this may indicate the possible role of umbilics and $IWs$ in the $TIC,$ which were not taken into account in the model [@CHnegativeDeltaE] (umbilics and $IWs$ are nucleation sites for fingers and may also increase elastic energy of $TIC$). Agreement is improved when phase diagrams are obtained using 2-D numerical modeling [@Garlandetal]. An interesting new finding revealed by the FCPM is that upon increasing $U$ the periodically arranged fingers merge with each other forming modulated (undulating) $TIC$ that is observed in a narrow voltage range between the structures of $TIC$ and periodically arranged $CFs$, Figs.1,4,7. We also find that the phase boundaries can be shifted in a controlled way by rubbing-induced tilt ($<2^{\circ }$) of easy axis from the vertical direction, by introducing particles that become nucleation sites for $CFs,$ as well as by using different amplitude-modulated voltage schemes. A novel and unexpected result is the re-entrant behavior of $TIC$ in the rubbed cells with $0.6\lessapprox \rho \lessapprox 0.75,$ which, however, disappears if nucleation sites are present. FCPM allows us to directly and unambiguously determine the 3-D director structures corresponding to different parts of the phase diagram. In particular, we unambiguously reconstruct the structures of four types of $CFs$. In all parts of diagrams corresponding to stability or metastability of $CFs$, the fingers of $CF1$-type are the most frequently observed. $CF2$ fingers are less frequent; the metastable $CF3$ and $CF4$ are very rare. Such findings indicate that fingers of $CF1$-type have the lowest free energy out of four fingers; this is consistent with the reconstruction of the structure of $CF1,$ which is nonsingular in ${\bf \hat{n}}$. It is also natural that $CF2$ with singular point defects and especially metastable $CF3$ and $CF4$ with singular line defects are less frequently observed. In the case of rubbed homeotropic substrates, only $CF1$ and $CF2$ are observed whereas $CF3$ and $CF4$ newer appear because the rubbing-induced tilting of the easy axis at one or both substrates contradicts with their symmetry. Control of phase diagrams to enable practical applications ---------------------------------------------------------- The combination of rubbing and amplitude-modulated voltage driving allows one to suppress appearance of fingers up to high $\rho \approx 0.75$, compare Fig.1 and Fig.7. This is a valuable finding for many practical applications such as eyeglasses with voltage-tunable transparency and light shutters,[@EyeLC1] bistable[@Bistable-chiral] and inverse twisted nematic displays,[@Patel] etc. In these applications of the homeotropic-$% TIC$ transition, it is important to have a broad range of well controlled total twist $\Delta \phi $ in the finger-free $TIC$. The broad range of voltage-tuned $\Delta \phi $ allows one to control optical phase retardation in the displays and electro-optic devices[@Patel; @Bistable-chiral] as well as light absorption when the dye-doped CLC is used in the tunable eyeglasses and light shutters[@EyeLC1]. A very subtle tilt of easy axis from the vertical direction not only makes the director twist in $TIC$ vary in a controlled way but also suppresses the appearance of fingers, $IW$s, and umbilics, Figs.3-7. Slow appearance of fingers from $TIC$ and untwisted homeotropic states allowed us to magnify the effect of rubbing via using amplitude-modulated voltage schemes and suppress appearance of fingers up to even higher $\rho $, Fig.7. For example, we can control $\Delta \phi =55^{\circ }-270^{\circ }$ in the finger-free $TIC$. Even stronger effects of rubbing and voltage modulation can be expected if the tilt of the easy axis is increased. This might be implemented by using the approach recently developed by Huang and Rosenblatt [@Homeotrop-tilt] in which case a tilt of easy axis up to $30^{\circ }$ could be achieved. On the other hand, when constructing cells for all of the above applications of tightly-twisted $TIC$, it is important to remember about the effect of particles, which become nucleation sites for fingers and can cause their appearance at lower $\rho $. Such particles are often used as spacers to set cell thickness and it is, therefore, important to either avoid their usage or limit (optimize) their concentration in order to obtain finger-free $TIC$. The finding that fingers align along the rubbing direction, Sec. III.A, may enable the use of periodically-arranged $CF$s in switchable diffraction gratings with the diffraction pattern corresponding to the field-on state. The spatial periodicity and the diffraction properties of such gratings can be easily controlled by selecting proper pitch $p$ and cell gap $d$, which can be varied from sub-micron to tens of microns. Our preliminary study shows that the grating periodicity can be changed in range $1-50\mu m$. More detailed studies of the cholesteric diffraction gratings based on voltage-induced well-oriented pattern of $CF$s will be published elsewhere. Conclusions =========== The major findings of this work are threefold: (1) we obtained phase diagram of CLC structures as a function of confinement ratio $\rho =d/p$ and voltage $U$ for different extra parameters such as rubbing, voltage driving, presence of nucleation sites; (2) we enabled new applications of finger-free tightly twisted $TIC$ and well-oriented fingers; (3) we unambiguously deciphered 3-D director fields associated with different structures and transitions in CLCs using FCPM. In the phase diagram, the direct homeotropic-$TIC$ transition upon increasing $U$ was observed for $\rho \lessapprox 0.5$; the analytical model of this transition is in a very good agreement with the experiment. Structures of isolated and periodically arranged fingers were found at $0.5\lessapprox \rho <1$ and intermediate $U$ between the homeotropic and $TIC$ phases. We observed the re-entrant behavior of $TIC$ in the rubbed cells of $0.6\lessapprox \rho \lessapprox 0.75$ for which the following sequence of transitions has been observed upon increasing $U$: (1) homeotropic untwisted - (2) $TIC$ - (3) periodically arranged fingers - (4) $% TIC$ with larger in-plane twist. The re-entrant behavior of $TIC$ is also observed in our numerical study of the phase diagram that will be published elsewhere[@Garlandetal]. The re-entrant $TIC$ disappears if nucleation sites are present or amplitude-modulated driving schemes are used. Rubbing also eliminates non-uniform in-plane structures such as umbilics and inversion walls that otherwise are often observed in $TIC$ and also influence the phase boundary lines. The lowest $\rho $ for which periodically arranged fingers start to be observed upon increasing $U$ can be shifted for up to $0.3$ towards higher $\rho $-values by rubbing and/or voltage driving schemes. The FCPM allowed us to unambiguously determine and confirm the latest director models [@Oswald-review00] for the vertical cross-sections of four types of $CFs$ ($CF1-CF4$) while disproving some of the earlier models[@Baudry-PRE99]. The $CF1$-type fingers are observed in all regions of the phase diagrams where the fingers are either stable or metastable; other fingers appear in the same parts of the diagram but less frequently. For the rubbed cells, only two types of $CFs$ ($CF1$ and $CF2$) are observed, which align along the rubbing direction. The new means to control structures in CLCs are of importance for potential applications, such as switchable gratings based on periodically arranged $CF$s and eyewear with tunable transparency based on $TIC$. Acknowledgements ================ The work is part of the AlphaMicron/TAF collaborative project “Liquid Crystal Eyewear”, supported by the State of Ohio and AlphaMicron, Inc. I.I.S. and O.D.L. acknowledge support of the NSF, Grant DMR-0315523. I.I.S. acknowledges Fellowship of the Institute for Complex and Adaptive Matter. E.C.G. acknowledges support under NSF Grant DMS-0107761, as well as the hospitality and support of the Department of Mathematics at the University of Pavia (Italy) and the Institute for Mathematics and its Applications at the University of Minnesota, where part of work was carried out. We are grateful to M. Kleman, L. Longa, Yu. Nastishin, and S. Shiyanovskii for discussions. Figure Captions: FIG.1. Phase diagram of structures in the $U-\rho $ parameter space for CLCs in cells with homeotropic surface anchoring. The boundary lines $V0-V3$, $% V01 $, $V02$ separate different phases (cholesteric structures). The two dashed vertical lines mark $\rho _{triple}=0.816$ and $\rho _{tricritical}=0.861$ as estimated according to Ref. [@CHnegativeDeltaE] for the material parameters of AMLC-0010 - ZLI-811 LC mixture. The solid line $V0$ was calculated using Eq.(1) and parameters of the used CLC; the solid lines $V1-V3$, $V01$, $V02$ connect the experimental points to guide the eye. FIG.2. Polarizing microscope textures observed in different regions of the phase diagram of structures shown in Fig.1: (a) isolated $CFs$ coexisting with the homeotropic untwisted state between the boundary lines $V1$ and $V2$ of Fig.1; (b) dendritic-like growth of $CFs$ (observed between the boundary lines $V0$ and $V1$); (c) branching of $CFs$ with increasing voltage, between the boundary lines $V0$ and $V1$; (d) periodically arranged $CFs$ where the individual $CFs$ are separated by homeotropic narrow stripes, observed between $V0$ and $V01$; (e) $CFs$ merge producing undulating $TIC$, observed between $V01$ and $V02$; (f) $TIC$ with umbilics, observed above the lines $V0$ and $V02$. Picture shown in part (b) was taken about 2 seconds after voltage was applied; it shows an intermediate state in which the circular domains grow from nucleation sites and will eventually fill in the whole area of the cell by the fingers. FIG.3. (a,b) Polarizing microscope textures of (a) the $TIC$ with no umbilics and (b) periodic fingers pattern in a homeotropic cell with one of the substrates rubbed along the black bar. (c,d) Light transmission through the cell with rubbed homeotropic substrates placed between crossed polarizers for (c) $\rho =0$ and (d) $\rho =0.5$. The insets in (c,d) show details of intensity changes in the vicinity of homeotropic-$TIC$ transition; note that the rubbing-induced pretilt makes these dependencies not as sharp as normally observed in non-rubbed homeotropic cells (see, for example, Ref. [@Rosenblatt]). FIG.4. Phase diagram of structures in the $U$ vs. $\rho $ parameter space for CLCs in the cells with homeotropic boundary conditions and substrates rubbed in anti-parallel directions. The cell has mylar spacers at the edges; no spacer particles are present in the bulk. The lines $V0-V3$, $V01$, $V02$ separate different phases and the two dashed vertical lines mark $\rho _{triple}=0.816$ and $\rho _{tricritical}=0.861$ corresponding to the triple and tricritical points, similar to Fig.1. The solid line $V0$ was calculated using Eq.(1) and is the same as in Fig.1; the solid lines $V1-V3$, $V01$, $% V02$ connect the experimental points to guide the eye. FIG.5. Polarizing microscope textures illustrating the transition from (a) homeotropic untwisted state to (b) $TIC$ with no umbilics and total twist $% \Delta \phi \approx \pi $ between the substrates, and then to (c) fingers pattern that slowly ($\sim 1s$) appears from $TIC$, and then to (d) uniform $% TIC$ with $\ \lessapprox 2\pi $ twist. The applied voltages are indicated. The homeotropic cell has substrates rubbed in anti-parallel directions; $% \rho =0.65$. The cell was assembled by using mylar spacers at the cell edges; no particles or other nucleation sites are present in the working area of the cell. FIG.6. Influence of spherical particles with perpendicular surface anchoring on the CLC structures in homeotropic cells: (a) particle-induced director distortions in the homeotropic state; (b) director distortions in the $TIC$ at $U\approx U_{th}$; (c) $TIC$ $10ms$ after $U>U_{th}$ is applied and (d) relaxation of the distortions in $TIC$ via formation of fingers as observed $% \approx 1s$ after $U$ is applied (the particles become nucleation sites for the fingers). FIG.7. Phase diagram of structures in the $U$ vs. $\rho $ parameter space for CLCs in homeotropic cells with rubbed substrates : (a) anti-parallel (i.e., at 180$^{\circ }$); (b) at 90$^{\circ }$; (c) at 270$^{\circ }$. The frequency of the applied voltage is $1kHz$, which is amplitude-modulated with a $50Hz$ sinusoidal signal. The boundary lines $V0-V3$, $V01$, $V02$ separate different phases and the two dashed vertical lines mark $\rho _{triple}=0.816$ and $\rho _{tricritical}=0.861$, similar to Figs.1 and 4. The solid line $V0$ was calculated using Eq.(1) and is the same as in Figs.1,4; the solid lines $V1-V3$, $V01$, $V02$ connect the experimental points to guide the eye. FIG.8. FCPM cross-sections (a,b) and schematic of director structures (c,d) of $TIC$ with twist: (a,c) $\approx \pi $ at $U=5Vrms$ and $\rho =1/2$; (b,d) $\approx 2\pi $ at $U=5Vrms$ and $\rho =1$. The polarization of the probe light in FCPM marked by ”P” is in the $y$-direction, along the normal to the pictures in (a,b). FIG.9. FCPM vertical cross-sections (a) and schematic of director structures (b) of a $CF1$-type isolated finger. The polarization of the probe light in FCPM marked by ”P” is in the $y$-direction, along the normal to the picture in (a). The non-singular $\lambda $-disclinations are marked by circles in (b); the open circles correspond to the $\lambda ^{-1/2} $ and the solid circles correspond to the $\lambda ^{+1/2}$ disclinations. FIG.10. FCPM cross-sections (a) and schematic of director structures (b) of a periodic finger pattern composed of $CF1$s separated by homeotropic stripes. The polarization of the FCPM probe light marked by ”P” is normal to the picture in (a). FIG.11. FCPM vertical cross-section illustrating the voltage-induced transition from (a) isolated fingers coexisting with homeotropic state to (f) periodically-modulated $TIC$ and then to (g) a uniform $TIC$. The fingers gradually widen (b-e) and then merge in order to form the modulated $% TIC$ (f). The polarization of the FCPM probe light marked by ”P” is normal to the pictures in (a). The applied voltages are indicated, the confinement ratio is $\rho =0.9$. FIG.12. FCPM cross-section (a) and reconstructed director structure (b) illustrating the $CF$ expanding into $TIC$. The polarization of the FCPM probe light marked by ”P” is along $y$, normal to the picture in (a). FIG.13. FCPM cross-section (a) and reconstructed director structure (b) of $% CF2$ finger; the $CF2$ can expand in one (c) or two in-plane directions (d) forming $TIC$. The non-singular $\lambda $-disclinations are marked by circles in (b); the open circles correspond to the $\lambda ^{-1/2}$ disclinations and the solid circle corresponds to the $\lambda ^{+1}$ disclination. The FCPM polarization is normal to the picture in (a). FIG.14. FCPM cross-section (a) and reconstructed director structures (b,c)formed between the parts of a cell with different in-plane twist and helical axis along the $z$: (b) $TIC$ with $\approx \pi $ twist, coexisting with homeotropic untwisted state and separated by $CF2$-like structure with two nonsingular $\lambda ^{-1/2}$ disclinations; (c) $TIC$ with $\approx 2\pi $ twist coexisting with $TIC$ with $\approx \pi \,\ $twist. The FCPM polarization marked by ”P” is normal to the picture in (a). FIG.15. FCPM images and director structure of a $T$-junction of $CF1$ and $% CF2$: (a) in-plane FCPM cross-section; (b) perspective view of the 3-D director field of the $T$-junction; (c,d,e) FCPM cross-sections along the lines $c-c$, $d-d$, and $e-e$ in part (a). The FCPM cross-sections in (c,e) correspond to $CF2$ and cross-section (d) corresponds to $CF1$. The FCPM polarization marked by ”P” is normal to the pictures in (c,d,e). FIG.16. FCPM cross-section (a) and reconstructed director structures (b) of $% CF3$ finger. The FCPM polarization marked by ”P” is normal to the picture in (a). Two $CF3$s can be seen in (a); the director structure of only one $% CF3$ is shown in (b). The singular disclinations at the two substrates are marked by circles in (b). FIG.17. FCPM cross-sections (a,c) and reconstructed director structure (b) of the $CF4$ fingers. The FCPM polarization is along $y$ in (a,c). The singular disclinations at one of the substrates are marked by circles in (b). The $CF4$s in (c) have singular disclinations at the same substrate. FIG.18. A representative $TIC$ equilibrium configuration obtained as numerical solution of Eqs. (\[eqn:ELa\]–\[eqn:ELc\])): (a) tilt angle $% \theta $, (b) twist angle $\phi $, (c) electric potential $\psi $. The material parameters used in the calculations were taken for the AMLC-0010host doped with ZLI-811, $d=5\,\mu m$, $\rho =0.5$, $U=3.5Vrms$. L. M. Blinov and V. G. 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ArXiv

--- abstract: 'A fast algorithm for Antoine and Vandergheynst’s (1998) directional Continuous Spherical Wavelet Transform () is presented. Computational requirements are reduced by a factor of $\complexity(\sqrt{\num_{\rm pix}})$, when $\num_{\rm pix}$ is the number of pixels on the sphere. The spherical  wavelet Gaussianity analysis of the  1-year data performed by Vielva  (2003) is reproduced and confirmed using the fast . The proposed extension to directional analysis is inherently afforded by the fast  algorithm.' address: 'Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, U.K.' author: - 'J.D. MCEWEN, M.P. HOBSON, A.N. LASENBY and D.J. MORTLOCK' title: | A FAST DIRECTIONAL CONTINUOUS SPHERICAL\ WAVELET TRANSFORM --- Introduction ============ A range of primordial processes imprint signatures on the temperature fluctuations of the  (). For instance, various cosmic defect and non-standard inflationary models predict non-Gaussian anisotropies. By studying the Gaussianity of the  anisotropies evidence may be provided for competing scenarios of the early Universe. In addition, a number of astrophysical processes introduce secondary sources of non-Gaussianity. Measurement systematics or contamination may also be highlighted by Gaussianity analysis. Wavelets are a powerful tool for probing the Gaussianity of anisotropies. Previous wavelet analysis of the , however, has been restricted to simple spherical Haar and isotropic Mexican Hat wavelets. A directional analysis on the full sky has previously been prohibited by the computational infeasibility of any implementation. We rectify this problem by providing a fast algorithm for Antoine and Vandergheynst’s[@antoine:1998] Continuous Spherical Wavelet Transform (), based on the fast spherical convolution proposed by Wandelt and Górski[@wandelt:2001]. The remainder of this paper is organised as follows. The  is presented in and the fast implementation in . In the fast  is applied to reproduce the non-Gaussianity  analysis performed by Vielva [@vielva:2003]. Concluding remarks are made in . A directional Continuous Spherical Wavelet Transform {#sec:cswt} ==================================================== Antoine and Vandergheynst[@antoine:1998] extend Euclidean wavelet analysis to spherical geometry by constructing a wavelet basis on the sphere. The natural extension of Euclidean motions on the sphere are rotations, defined by where we parameterise $\rho$ by the Euler angles $(\eulers)$. Dilations on the sphere, denoted $(\dil_\scale f)(\sa) = f_\scale(\sa)$, are constructed by first lifting the sphere  to the plane by a norm preserving stereographic projection from the South pole, performing the usual Euclidean dilation in the plane, before re-projecting back onto . Mother spherical wavelets are constructed simply by projecting Euclidean planar wavelets onto the sphere by a norm preserving inverse stereographic projection. A wavelet basis on  may be constructed from rotations and dilations of an admissible mother spherical wavelet. The corresponding wavelet family $\{ \wav_{\scale,\rho} \equiv \rot_\rho \dil_\scale \wav, \, \rho \in SO(3), \, \scale \in \real_{\ast}^{+} \}$ provides an over-complete set of functions in $L^2(\sphere)$. The  is given by the projection onto each wavelet basis function $$\skywav(\scale, \eulers) = \int_{\sphere} (\rot_{\eulers} \wav_\scale)^\conj(\sa) \: \sky(\sa) \: d\mu(\sa) \spcend , \label{eqn:cswt}$$ where the  denotes complex conjugation and $d\mu(\sa)=\sin(\saa) \, d\saa \, d\sab$ is the usual rotation invariant measure on the sphere. Fast algorithm {#sec:fast} ============== A direct implementation of the  is simply not computationally feasible for a data set of any practical size; a fast algorithm is essential. At a particular scale the  is essentially a spherical convolution, hence it is possible to apply Wandelt and Górski’s[@wandelt:2001] fast spherical convolution algorithm to rapidly evaluate the transform. Fast implementation {#sec:harmonic} ------------------- There does not exist any finite point set on the sphere that is invariant under rotations, hence it is more natural, and in fact more accurate for numerical purposes, to recast the  in spherical harmonic space. The Wigner rotation matrices (defined by Brink and Satchler[@brink:1993], for example) introduced to characterise the rotation of a spherical harmonic may be decomposed as $ \dmatbig_{mm\p}^{l}(\eulers) = e^{-\img m\eulera} \: \dmatsmall_{mm\p}^l(\eulerb) \: e^{-\img m\p\eulerc} $. This decomposition is exploited by factoring the rotation into two separate rotations, both of which contain a constant $\pm \pi/2$ polar rotation: $ \rot_{\eulers} = \rot_{\eulera-\pi/2, \; -\pi/2, \; \eulerb} \:\: \rot_{0, \; \pi/2, \; \eulerc+\pi/2} $. By uniformly sampling and applying some algebra the  may be recast as $$\skywav [\ind_\eulera, \ind_\eulerb, \ind_\eulerc] = e^{-\img 2\pi [ \frac{\ind_\eulera \lmax}{\num_\eulera} + \frac{\ind_\eulerb \lmax}{\num_\eulerb} + \frac{\ind_\eulerc \mmax}{\num_\eulerc}]} \sum_{j=0}^{\num_\eulera-1} \: \sum_{j\p=0}^{\num_\eulerb-1} \: \sum_{j\pp=0}^{\num_\eulerc-1} \cswtfftterm_{j, j\p, j\pp} \: e^{ \img 2\pi [ \frac{j\ind_\eulera}{\num_\eulera} + \frac{j\p \ind_\eulerb}{\num_\eulerb} + \frac{j\pp \ind_\eulerc}{\num_\eulerc}]} \spcend , \label{eqn:cswt_fast}$$ where the summation is simply the unnormalised  inverse discrete Fourier transform of $$\cswtfftterm_{m+\lmax, m\p+\lmax, m\pp+\mmax} = e^{i(m\pp-m)\pi/2} \sum_{l=\max(\mid m \mid, \mid m\p \mid, \mid m\pp \mid)}^{\lmax} \dmatsmall_{m\p m}^l(\pi/2) \: \dmatsmall_{m\p m\pp}^l(\pi/2) \: \shcoeff{\wav}_{lm\pp}{}^\conj \: \shcoeff{\sky}_{lm} \spcend , \label{eqn:cswt_fast_term}$$ where $\shcoeff{\cdot}_{lm}$ denote spherical harmonic coefficients, $\lmax$ and $\mmax$ define the general and azimuthal band limits of the wavelet respectively and the shifted indices show the conversion between the harmonic and Fourier conventions. The  may be performed very rapidly in spherical harmonic space by using fast Fourier techniques to rapidly and simultaneously evaluate , once is precomputed.[^1] Comparison with other algorithms -------------------------------- Direct and semi-fast (where only one rotation is performed in Fourier space) implementations of the  are also possible. A comparison of the theoretical complexity and typical execution times of each algorithm is presented in . The fast  algorithm provides a saving of $\complexity(\sqrt{\num_{\rm pix}})$ for $\num_{\rm pix}$ pixels on the sphere.  non-Gaussianity analysis {#sec:cmb} ========================= We reproduce the Gaussianity analysis of Vielva [@vielva:2003], preprocessing the  data in the same manner. The resolution of the co-added map defined by Komatsu [@komatsu:2003] is down-sampled by a factor or 4, before the *[Kp0]{}* exclusion mask is applied to remove emissions due to the Galactic plane and known point sources. Spherical wavelet analysis -------------------------- The  is a linear operation, hence the wavelet coefficients of a Gaussian map will also obey a Gaussian distribution. To test for deviations from Gaussianity, skewness and kurtosis statistics are calculated for each wavelet coefficient map at each scale. Monte Carlo simulations are performed to construct confidence bounds on the test statistics. The application of the *Kp0* mask distorts coefficients corresponding to wavelets that overlap with the mask exclusion region. These wavelet coefficients must be removed from any subsequent analysis. Our construction of an extended coefficient exclusion mask differs to that of Vielva  and inherently accounts for the dominant distortion (either point-source or Galactic plane) at each scale. The only non-zero coefficients in a  of the original mask are those that are distorted (due to the zero-mean property of spherical wavelets). These may be easily detected and the coefficient exclusion mask extended accordingly. Results ------- We reproduce the results of Vielva [@vielva:2003] for the spherical  wavelet analysis of the co-added  data. The  wavelet scales $\{ \scale_i \}_{i=1}^{11} = \{ 14, 25, 50, 75, 100, 150, 200, \linebreak 250, 300, 400, 500\}_{i=1}^{11}$ acrmin are considered, corresponding to an effective size of the sky of $\effsize_i=4\tan^{-1}( a_i/\sqrt{2} ) \approx 2\sqrt{2} \, \scale_i $ (defined as the angular separation between opposite zero-crossings). shows the skewness and kurtosis of the coefficients at each scale. The wavelet analysis inherently allows one to localise signal components on the sky, as illustrated in . We make similar observations to Vielva [@vielva:2003], although the different coefficient exclusion masks [produce]{} slight discrepancies. These discrepancies do not alter the general findings of the analysis. Conclusions and future work {#sec:conc} =========================== A fast algorithm is presented and evaluated for performing a directional  on the sphere. The fast implementation reduces the complexity of the  by $\complexity(\sqrt{\num_{\rm pix}})$, where $\num_{\rm pix}$ is the number of pixels on the sphere. Furthermore, the numerical accuracy of the  is improved by elegantly representing rotations in harmonic space. The Gaussianity analysis of the  1-year data performed by Vielva [@vielva:2003] has been reproduced and confirmed using the fast . We consider the extension to a full directional analysis in an upcoming publication by McEwen [@mcewen:2004]; preliminary findings indicate deviations from Gaussianity outside of the 99% confidence level. References {#references .unnumbered} ========== [99]{} , [*J. Math. Phys.*, 39, 8, 3987–4008]{}. [(1998)]{}. , [*Angular Momentum* (3rd Ed.)]{}, [Clarendon Press, Oxford]{}, [(1993)]{}. , [*ApJ*, 148, 119]{}, [(2003)]{}. , [*Preprint* (/0406604)]{}, submitted to MNRAS, [(2004)]{}. , [*Preprint* (/0310273)]{}, submitted to ApJ, [(2003)]{}. , [*Phys. Rev.*, 63, 123002, 1–6]{}, [(2001)]{}. [^1]: Memory and computational requirements may be reduced by a further factor of two for real signals by exploiting the conjugate symmetry relationship $\cswtfftterm_{-m,-m\p,-m\pp}=\cswtfftterm_{m,m\p,m\pp}^\conj$.

ArXiv

--- abstract: 'We investigate whether the final state interaction (FSI) effect plays a significant role in the large hidden charm decay width of X(3872) and Y(3940) using a model. Our numerical result suggests (1) the FSI contribution to $X(3872)\to J/\psi\rho $ is tiny; (2) $\Gamma[ Y(3940)\to D\bar{D}^{*}+\text{h.c.}\to J/\psi\omega ]$ from FSI is around several keV, far less than Belle’s experimental value 7 MeV.' author: - Xiang Liu - Bo Zhang - 'Shi-Lin Zhu' title: ' The Hidden Charm Decay of $X(3872), Y(3940)$ and Final State Interaction Effects' --- Introduction {#sec1} ============ The underlying structure of $X(3872)$ is still very controversial. It was discovered by Belle collaboration [@belle-3872] and confirmed by Babar [@babar-3872], CDF [@CDF-3872] and D0 [@D0-3872] collaborations. Recently, Belle collaboration reported a new decay mode $X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ [@DDpi-3872]. The mass of $X(3872)$ from various experiments reads [@belle-3872; @babar-3872; @CDF-3872; @D0-3872; @DDpi-3872] $$\begin{aligned} M_{_{X(3872)}}= \left \{ \begin{array}{lc} 3875\pm0.7^{+1.2}_{-2.0}\; {\text MeV}/{\text c}^2&{\text Belle} \\ 3871.8 \pm 3.1 \pm 3.0\; {\text MeV}/{\text c}^2 & {\text D0} \\ 3871.3 \pm 0.7\pm 0.4 \; {\text MeV}/{\text c}^2& {\text CDF} \\ 3873.4 \pm 1.4 \; {\text MeV}/{\text c}^2& {\text BaBar} \\3872.0 \pm 0.6 \pm 0.5 \; {\text MeV}/{\text c}^2& {\text Belle}. \end{array}\right.\end{aligned}$$ The available experimental information indicates $J^{PC}=1^{++}$ for $X(3872)$ [@angular-3872]. Theoretical interpretations of $X(3872)$ include a charmonium state [@charmonium], a molecular state [@mole], or the mixture of charmonium with molecular state [@mix]. At present the observed decay modes of $X(3872)$ include $J/\psi\pi^{+}\pi^{-}$ [@belle-3872], $\gamma J/\psi$ [@jpsi-gamma], $J/\psi\pi^{+}\pi^{-}\pi^{0}$ [@jpsi-gamma] and $D^{0}\bar{D}^{0}\pi^{0}$ [@DDpi-3872]. The dipion in $J/\psi\pi^{+}\pi^{-}$ seems to originate from $\rho\to \pi^{+}\pi^{-}$ because the peak of the dipion invariant mass spectrum locates around 775 MeV. $J/\psi\pi^{+}\pi^{-}\pi^{0}$ comes from the sub-threshold decay $X(3872)\to J/\psi\omega$ [@jpsi-gamma]. Meanwhile the ratio of $B(X(3872)\to J/\psi \pi^{+}\pi^{-}\pi^{0})$ to $B(X(3872)\to J/\psi\pi^{+}\pi^{-})$ given by experiment is $1.0\pm 0.4({\rm stat})\pm 0.3({\rm syst})$ [@jpsi-gamma]. Recently Belle experiment indicated $B(X(3872)\to D^{0}\bar{D}^{0}\pi^{0}K^{+})=9.4^{+3.6}_{-4.3} B(X(3872)\to J/\psi\pi^{+}\pi^{-}K^{+})$ [@DDpi-3872]. Based on the above experimental data, one concludes (1) the $D^{0}\bar{D}^{*0}$ is the dominant decay of $X(3872)$; (2) the isospin violating mode $X(3872)\to J/\psi\rho\to J/\psi\pi^{+}\pi^{-}$ is not suppressed, compared with the isospin conserving mode $X(3872)\to J/\psi\omega\to J/\psi \pi^{+}\pi^{-}\pi^{0}$. The Final State Interaction (FSI) effect sometimes plays a crucial role in many processes [@FSI]. In this work, we study if the hidden charm decay $J/\psi\rho(\pi^{+}\pi^{-})$ mainly arises from the FSI effect of $X(3872)\to \bar{D}^{*0}D^{0}+{\rm{h.c.}}$. $X(3872)$ decays to $D^{0}\bar{D}^{*0}+\text{h.c.}$ but not $D^{+}D^{*-} +\text{h.c.}$ because $D^{0}+\bar{D}^{*0}= 3871.3 \;\text{MeV}<M_{_{X(3872)}}$ and $D^{+}+D^{*-}=3879.3\;\text{MeV} >M_{_{X(3872)}}$. Thus the isospin violating process $X(3872)\to J/\psi\rho$ can occur via the $\bar{D}^{*0}D^{0}+{\rm{h.c.}}$ re-scattering effect. This paper is organized as follows. We present the formulation about $X(3872)\to D^{0}\bar{D}^{*0}+{\rm{h.c.}}\to J/\psi\rho$ in Section \[sec2\]. Then we present our numerical results. The last section is a short discussion. Formalism {#sec2} ========= The Feynman diagrams for the $X(3872)\to J/\psi\rho$ through $\bar{D}^{*0}D^{0}+{\rm{h.c.}}$ re-scattering are depicted in Fig. \[FSI\]. In Refs. [@lagrangian-jpsi; @lagrangian-hl; @Casalbuoni], the effective Lagrangians, which are relevant to the present calculation, are constructed based on the chiral symmetry and heavy quark symmetry: $$\begin{aligned} \mathcal{L}&=&g_{X}X^{\mu}[D^{0}\bar{D}^{*0}_{\mu}-\bar{D}^{0}{D}^{*0}_{\mu} ]\nonumber\\&&+i g_{_{J/\psi \mathcal{D}\mathcal{D}}}^{} \psi_\mu \left( \partial^\mu \mathcal{D} {\mathcal{D}}^{\dagger} - \mathcal{D} \partial^\mu {\mathcal{D}}^{\dagger} \right)\nonumber\\&& -g_{_{J/\psi \mathcal{D}^* \mathcal{D}}}^{} \varepsilon^{\mu\nu\alpha\beta} \partial_\mu \psi_\nu \left( \partial_\alpha \mathcal{D}^*_\beta {\mathcal{D}}^{\dagger} + \mathcal{D} \partial_\alpha {\mathcal{D}}^{*\dagger}_\beta \right)\nonumber\\ &&-i g_{_{J/\psi \mathcal{D}^* \mathcal{D}^*}}^{} \Bigl\{ \psi^\mu \left( \partial_\mu \mathcal{D}^{*\nu} {\mathcal{D}}_\nu^{*\dagger} - \mathcal{D}^{*\nu} \partial_\mu {\mathcal{D}}_\nu^{*\dagger} \right) \nonumber\\&&+ \left( \partial_\mu \psi_\nu \mathcal{D}^{*\nu} - \psi_\nu \partial_\mu \mathcal{D}^{*\nu} \right) {\mathcal{D}}^{*\mu\dagger} \mbox{} + \mathcal{D}^{*\mu} \big( \psi^\nu \partial_\mu {\mathcal{D}}^{*\dagger}_{\nu} \nonumber\\&&- \partial_\mu \psi_\nu {\mathcal{D}}^{*\nu\dagger} \big) \Bigr\} +\Big\{-ig_{_{\mathcal{D}\mathcal{D}\mathbb{V}}}\mathcal{D}_{i}^{\dagger}{\stackrel{\leftrightarrow}{\partial}} _{\mu}\mathcal{D}^{j}(\mathbb{V}^{\mu})^{i}_{j}\nonumber\\&& -2f_{_{\mathcal{D^{*}}\mathcal{D}\mathbb{V}}}\varepsilon_{\mu\nu\alpha\beta}(\partial^{\mu}\mathbb{V}^{\nu}) ^{i}_{j}(\mathcal{D}_{i}^{\dagger} {\stackrel{\leftrightarrow}{\partial}}^{\alpha}\mathcal{D^{*}}^{\beta j}-\mathcal{D^{*}}_{i}^{\beta \dagger}{\stackrel{\leftrightarrow}{\partial}}^{\alpha}\mathcal{D}^{j})\nonumber\\ &&+ig_{_{\mathcal{D^{*}}\mathcal{D^{*}}\mathbb{V}}}\mathcal{D^{*}}_{i}^{\nu \dagger}{\stackrel{\leftrightarrow}{\partial}}_{\mu}\mathcal{D^{*}}_{\nu}^{j}(\mathbb{V}^{\mu})^{i}_{j}\nonumber\\&&+ 4if_{_{\mathcal{D^{*}}\mathcal{D^{*}}\mathbb{V}}}\mathcal{D^{*}}_{i\mu}^{\dagger}(\partial^{\mu}\mathbb{V}^{\nu}-\partial^{\nu} \mathbb{V}^{\mu})^{i}_{j} \mathcal{D^{*}}_{\nu}^{j}\Big\},\label{lagrangian}\end{aligned}$$ where $\mathcal{D}$ and $\mathcal{D^*}$ are the pseudoscalar and vector heavy mesons respectively, i.e. $\mathcal{D^{(*)}}$=(($\bar{D}^{0})^{(*)}$, $(D^{-})^{(*)}$, $(D_{s}^{-})^{(*)}$). The values of the coupling constants will be given in the following section. $\mathbb{V}$ denotes the nonet vector meson matrices $$\begin{aligned} \mathbb{V}&=&\left(\begin{array}{ccc} \frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\ \rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}& K^{*0}\\ K^{*-} &\bar{K}^{*0}&\phi \end{array}\right).\end{aligned}$$ By the Cutkosky rule, the absorptive part of Fig. 1 (a) which comes from the re-scattering process of $X(3872)\to D^{0}(p_{1})+\bar{D}^{*0}(p_{2})\to J/\psi(p_{3})+\rho(p_{4})$ is written as $$\begin{aligned} \textbf{Abs}(a)&=&\frac{1}{2}\int\frac{d^{3}p_{1}}{(2\pi)^{3}2E_{1}}\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}} \nonumber\\&&\times(2\pi)^{4}\delta^{4}(m_{_X}-p_{1}-p_{2}) [ig_{X}\varepsilon_{\xi}]\nonumber\\&&\times\Big[-g_{_{J/\psi DD}}i(p_{1}-q)\cdot \varepsilon_{J/\psi}\Big]\nonumber\\ &&\times\Big[-\frac{2}{\sqrt{2}}i\;f_{_{D^{*}DV}}\epsilon_{\mu\nu\alpha\beta}ip_{4}^{\mu}\varepsilon_{\rho}^{\nu}(iq^{\alpha} +ip_{2}^{\alpha})\Big]\nonumber\\&&\times\bigg[-g^{\xi\beta}+\frac{p_{2}^{\xi}p_{2}^{\beta}}{m_{2}^{2}}\bigg]\bigg[\frac{i}{q^2 -m_{D}^{2}}\bigg]\mathcal{F}^{2}(m_{D},q^2)\nonumber\\ &=&\int d\Omega\frac{|\mathbf{p}_{1}|}{32\pi^{2}m_{_X}}[\sqrt{2}g_{X}g_{_{J/\psi DD}}f_{_{D^{*}DV}}]\nonumber\\&&\times[(2p_{1}-p_{3})\cdot \varepsilon_{J/\psi}] \epsilon_{\mu\nu\alpha\beta}p_{4}^{\mu}\varepsilon_{\rho}^{\nu}(p_{3}^{\alpha}+p_{2}^{\alpha}-p_{1}^{\alpha}) \nonumber\\&&\times\bigg[-\varepsilon^{\beta}+p_{2}^{\beta}\frac{p_{2}\cdot \varepsilon}{m_{2}^{2}}\bigg] \frac{\mathcal{F}^{2}(m_{D},q^2)}{q^{2}-m_{D}^{2}}\end{aligned}$$ with $q^{2}=m_{1}^{2}+m_{3}^{2}-2E_{1}E_{3}+2|\mathbf{p}_{1}||\mathbf{p}_{3}|\cos\theta$, where $\mathcal{F}^{2}(m_{i},q^2)$ etc denotes the form factors which compensate the off-shell effects of mesons at the vertices and are written as $$\begin{aligned} \mathcal{F}^{2}(m_{i},q^2)=\bigg(\frac{\Lambda^{2}-m_{i}^2 }{\Lambda^{2}-q^{2}}\bigg)^2,\end{aligned}$$ where $\Lambda$ is a phenomenological parameter. As $q^2\to 0$ the form factor becomes a number. If $\Lambda\gg m_{i}$, it becomes unity. As $q^2\rightarrow\infty$, the form factor approaches to zero. As the distance becomes very small, the inner structure would manifest itself and the whole picture of hadron interaction is no longer valid. Hence the form factor vanishes and plays a role to cut off the end effect. The expression of $\Lambda$ is [@HY-Chen] $$\begin{aligned} \Lambda(m_{i})=m_{i}+\alpha \Lambda_{QCD},\end{aligned}$$ where $m_{i}$ denotes the mass of exchanged meson. $\Lambda_{QCD}=220$ MeV. $\alpha$ is a phenomenological parameter. Similarly we obtain the absorptive contributions from Fig. \[FSI\] (b)-(d) respectively. $$\begin{aligned} \textbf{Abs}(b)&=&\frac{1}{2}\int\frac{d^{3}p_{1}}{(2\pi)^{3}2E_{1}}\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\nonumber\\&& \times(2\pi)^{4}\delta^{4}(m_{_X}-p_{1}-p_{2}) [ig_{X}\varepsilon_{\xi}]\nonumber\\&& \times\Big[i\;g_{_{J/\psi DD^{*}}}\epsilon_{\mu\nu\kappa\sigma}\varepsilon_{J/\psi}^{\mu}(-i)p_{1}^{\nu}(-i)q^{\sigma}\Big]\nonumber\\&&\times \bigg\{-\frac{g_{_{D^{*}D^{*}V}}}{\sqrt{2}} i(q+p_{2})\cdot\epsilon_{\rho}g_{\alpha\beta}\nonumber\\&& -\frac{4f_{_{D^{*}D^{*}V}}}{\sqrt{2}}\Big[ip_{4\beta}{\epsilon_{\rho}}_{\alpha}-i {\epsilon_{\rho}}_{\beta}p_{4\alpha}\bigg]\bigg\}\nonumber\\&& \times\bigg[-g^{\kappa\beta}+\frac{p_{2}^{\kappa}p_{2}^{\beta}}{m_{2}^{2}}\bigg] \bigg[-g^{\xi\alpha}+\frac{q^{\xi}q^{\alpha}}{m_{D^{*}}^{2}}\bigg]\nonumber\\&&\times\bigg[\frac{i}{q^2 -m_{D^{*}}^{2}}\bigg]\bigg(\frac{\Lambda^{2}-m_{D^{*}}^2 }{\Lambda^{2}-q^{2}}\bigg)^2,\end{aligned}$$ $$\begin{aligned} \textbf{Abs}(c)&=&\frac{1}{2}\int\frac{d^{3}p_{1}}{(2\pi)^{3}2E_{1}}\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}} \nonumber\\&& \times(2\pi)^{4}\delta^{4}(m_{_X}-p_{1}-p_{2}) [ig_{X}\varepsilon_{\xi}]\nonumber\\&& \times\bigg[\frac{g_{_{DDV}}}{\sqrt{2}}i(q-p_{1})\cdot\varepsilon_{\rho}\bigg]\nonumber\\&& \times \Big[ig_{_{J/\psi DD^{*}}} \epsilon_{\mu\nu\alpha\beta}\varepsilon_{J/\psi}^{\mu}iq^{\nu}(-i)p_{2}^{\beta}\Big]\nonumber\\&& \times\bigg[-g^{\xi\alpha}+\frac{p_{2}^{\xi}p_{2}^{\alpha}} {m_{2}^{2}}\bigg]\bigg[\frac{i}{q^2 -m_{D}^{2}}\bigg]\bigg(\frac{\Lambda^{2}-m_{D}^2 }{\Lambda^{2}-q^{2}}\bigg)^2,\nonumber\\\end{aligned}$$ $$\begin{aligned} \textbf{Abs}(d)&=&\frac{1}{2}\int\frac{d^{3}p_{1}}{(2\pi)^{3}2E_{1}}\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\nonumber\\&& \times(2\pi)^{4}\delta^{4}(m_{_X}-p_{1}-p_{2}) [ig_{X}\varepsilon_{\xi}]\nonumber\\&& \times\bigg[-\frac{2}{\sqrt{2}}if_{_{D^{*}DV}}\epsilon_{\mu\nu\alpha\beta}ip_{3}^{\mu}\varepsilon_{\rho}^{\nu} i(q^{\alpha}-p_{1}^{\alpha})\bigg]\nonumber\\&&\times\Big\{-g_{_{J/\psi D^{*}D^{*}}}\Big[iq^{\kappa}\varepsilon_{J/\psi}^{\sigma}+ip_{2}^{\sigma}\varepsilon_{J/\psi}^{\kappa}\nonumber\\&& \times+i(p_{2}+q)\cdot\varepsilon_{J/\psi}g^{\kappa\sigma} \Big] \Big\}\bigg[-g^{\xi}_{\kappa}+\frac{{p_{2}}_{\kappa}p_{2}^{\xi}}{m_{2}^{2}}\bigg] \nonumber\\&&\times\bigg[-g^{\beta}_{\sigma}+\frac{q_{\sigma}q^{\beta}}{m_{D^{*}}^{2}}\bigg]\bigg[\frac{i}{q^2 -m_{D^{*}}^{2}}\bigg]\bigg(\frac{\Lambda^{2}-m_{D^{*}}^2 }{\Lambda^{2}-q^{2}}\bigg)^2.\nonumber\\\end{aligned}$$ The contributions from Fig. \[FSI\] (e), (f), (g), (h) is the same as that corresponding to Fig. \[FSI\] (a), (b), (c), (d) respectively. The total amplitude of $X(3872)\to D^{0}\bar{D}^{*0}+\bar{D}^{0}D^{*0}\to J/\psi\rho$ can be written as $$\begin{aligned} \mathcal{M}&=&2[\textbf{Abs}(a)+\textbf{Abs}(b)+\textbf{Abs}(c)+\textbf{Abs}(d)],\end{aligned}$$ where the pre-factor “2” comes from the consideration that the contribution from $D^{0}\bar{D}^{*0}$ re-scattering is the same as that from $\bar{D}^{0}D^{*0}$ re-scattering. Because the $\rho$ meson is a broad resonance with $\Gamma_{\rho}\sim 150$ MeV, the decay width of $X(3872)\to D^{0}\bar{D}^{*0}+\bar{D}^{0}D^{*0}\to J/\psi\rho$ is written as $$\begin{aligned} \Gamma=\int^{(M_{_{X(3872)}}-m_{J/\psi})^{2}}_{0}{\rm{d}}s f(s,m_{\rho},\Gamma_{\rho})\frac{|\mathbf{k}||\mathcal{M}(m_{\rho}\to \sqrt{s})|^{2}}{24\pi M^{2}_{_{X(3872)}}},\nonumber\end{aligned}$$ where the Breit-Wigner distribution function $f(s,m_{\rho},\Gamma_{\rho})$ and the decay momentum $|\mathbf{k}|$ are $$\begin{aligned} &f(s,m_{\rho},\Gamma_{\rho})=\frac{1}{\pi} \frac{m_{\rho}\Gamma_{\rho}}{(s-m_{\rho}^{2})^{2}+m_{\rho}^{2}\Gamma_{\rho}^{2}},\\ &|\mathbf{k}|=\frac{\sqrt{[M_{_{X(3872)}}^{2}-(\sqrt{s}+m_{J/\psi})^{2}][M_{_{X(3872)}}^{2}-(\sqrt{s}-m_{J/\psi})^{2}]}}{2M_{_{X(3872)}}}\;.\end{aligned}$$ Numerical Results {#sec3} ================= The coupling constants related to our calculation include [@HY-Chen]: $$\begin{aligned} g_{_{DDV}}&=&g_{_{D^{*}D^{*}V}}=\frac{\beta g_{_{V}}}{\sqrt{2}},\;\;\;f_{_{D^{*}DV}}=\frac{f_{_{D^{*}D^{*}V}}}{m_{_{D^*}}}=\frac{\lambda g_{_{V}}}{\sqrt{2}},\nonumber\\ g_{_{V}}&=&\frac{m_{_{\rho}}}{f_{\pi}},\end{aligned}$$ where $f_{\pi}=132$ MeV, $g_{_{V}},\;\beta$ and $\lambda$ are parameters in the effective chiral Lagrangian that describes the interaction of heavy mesons with the low-momentum light vector mesons [@Casalbuoni]. Following Ref. [@Isola], we take $g=0.59$, $\beta=0.9$ and $\lambda=0.56$. Based on the vector meson dominance model and using $J/\psi$’s leptonic width, the authors of Ref. [@Achasov] determined ${g_{_{J/\psi \mathcal{D} \mathcal{D}}}^2}/{(4\pi)}=5$. As a consequence of the spin symmetry in the heavy quark effective field theory, $g_{_{J/\psi \mathcal{D}{\mathcal{D}}^{*}}}$ and $g_{_{J/\psi \mathcal{D}^{*}{\mathcal{D}}^{*}}}$ satisfy the relations: $g_{_{J/\psi \mathcal{D}{\mathcal{D}}^{*}}}={g_{_{J/\psi \mathcal{D}{\mathcal{D}}}}}/{m_{_{D}}}$ and $g_{_{J/\psi \mathcal{D}^{*}{\mathcal{D}}^{*}}}=g_{_{J/\psi \mathcal{D}{\mathcal{D}}}}$ [@JPsi-relation]. By fitting the upper limit of the total width of $X(3872)$ (2.3 MeV), one obtains the coupling constant $g_{X}$ in Eq. (\[lagrangian\]) $$\begin{aligned} g_{X}=\left\{\begin{array}{cc} 2.2 \;\mathrm{GeV}, &\mathrm{for}\quad M_{_{X(3872)}}=3875.0\;\mathrm{MeV} ,\\ 2.5\;\mathrm{GeV},&\mathrm{for}\quad M_{_{X(3872)}}=3873.4\;\mathrm{MeV},\\ 3.1\;\mathrm{GeV},&\mathrm{for}\quad M_{_{X(3872)}}=3872.0\;\mathrm{MeV}, \end{array}\right.\end{aligned}$$ where we approximately take $\bar{D}^{*0}D^{0}$ as the dominant decay mode of $X(3872)$ considering the experimental result [@DDpi-3872]: $B(X(3872)\to D^{0}\bar{D}^{0}\pi^{0}K^{+})=9.4^{+3.6}_{-4.3} B(X(3872)\to J/\psi\pi^{+}\pi^{-}K^{+})$. The value of $\alpha$ in the form factor usually is of order unity [@HY-Chen]. In this work we take the range of $\alpha=0.5\sim 3$. The dependence of the branching ratio of $X(3872)\to D^{0}\bar{D}^{*0}\to J/\psi\rho $ on $\alpha$ is presented in Fig. 2. In Table \[numerical\], we list the typical values of the branching ratio of $X(3872)\to D^{0}\bar{D}^{*0}+{\rm{h.c.}} \to J/\psi\rho$ when we take several masses of $X(3872)$ from various experiments and different $\alpha$. ---------------------------------------------------------------------------------------------------------------------------------------------------------------- 0.5 1.0 1.5 2.0 2.5 3.0 ---------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -- -- -- -- 3872.0 [@belle-3872] $2.1\times $2.0\times10^{-6}$ $6.6\times10^{-6}$ $1.4\times10^{-5}$ $2.4\times10^{-5}$ $3.6\times10^{-5}$ 10^{-7}$ 3873.4 [@babar-3872] $3.2\times10^{-7}$ $3.2\times10^{-6}$ $1.0\times10^{-5}$ $2.2\times10^{-5}$ $3.7\times10^{-5}$ $5.6\times10^{-5}$ 3875.0 [@DDpi-3872] $4.2\times10^{-7}$ $4.1\times10^{-6}$ $1.3\times10^{-5}$ $2.9\times10^{-5}$ $4.9\times10^{-5}$ $7.2\times10^{-5}$ ---------------------------------------------------------------------------------------------------------------------------------------------------------------- : The typical values of branching ratio of $X(3872)\to D^{0}\bar{D}^{*0}+{\rm{h.c.}} \to J/\psi\rho$ for different values of $M_{X(3872)}$ and $\alpha$.[]{data-label="numerical"} Discussion {#sec4} ========== Understanding the large $J/\psi\rho$ decay width of $X(3872)$ may help reveal the nature of $X(3872)$. In this work, we study if the large branching ratio of $X(3872)\to J/\psi\rho$ can be explained by the $X(3872)\to D^{0}\bar{D}^{0*}+{\rm{h.c.}}$ re-scattering effect. The numerical results indicate that $B(X(3872)\to D^{0}\bar{D}^{0*}+{\rm{h.c.}}\to J/\psi \rho)$ is about $10^{-5}\sim 10^{-7}$. Thus the large isospin violating $X(3872)\to J/\psi\rho$ decay width can hardly be attributed to the FSI effect of $X(3872)\to D^{0}\bar{D}^{0*}+{\rm{h.c.}}$. The suppression from the phase space of $X(3872)\to D^{0}\bar{D}^{0*}+{\rm{h.c.}}$ is huge because the experimental values of $X(3872)$ mass is only barely above the $J/\psi+\rho$ or $D^{*0}+D^{0}$ threshold, although the re-scattering effect is obvious. The reliable dynamical calculation of the hidden charm decay width has been a challenging theoretical issue for decades. In Ref. [@swanson], the explicit $J/\psi\rho$ component is introduced into the $X(3872)$ wave function in order to explain the large $J/\psi\pi\pi$ decay width of $X(3872)$. In Ref. [@Yan-Kuang], QCD multipole expansion technique was used to calculate hadronic transitions such as $\psi(2S)\to J/\psi\pi\pi$ and $\Upsilon(ns)\to \Upsilon(1s)\pi\pi$. If the main component of $X(3872)$ is $c\bar{c}$ [@mix; @suzuki], the large $X(3872)\to J/\psi\pi\pi$ decay might also be understood with this approach. Last year, Belle reported a new charmonium state $Y(3940)$ in the channel $B\to J/\psi\omega K$ and obtained $B(B\to Y(3940)+K)\cdot B(Y(3940)\to \omega J/\psi)=(7.1\pm1.3\pm 3.1)\times 10^{-5}$. Its mass and width are $3946\pm11(stat)\pm13(syst)$ MeV and $87\pm22(stat)\pm 26(syst)$ MeV [@3940] respectively. In particular, $\Gamma[Y(3940)\to J/\psi \omega] > 7$ MeV. Godfrey suggested $Y(3940)$ as the $\chi_{c1}'$ state with quantum number $2^{3}P_{1}$ [@Godfrey] and indicated that $Y(3940)\to J/\psi\omega$ might come from the FSI effect of $Y(3940)\to D\bar{D}^{*}+{\rm{h.c.}}$. In Fig. \[result-3940\], we present the dependence of the width of $Y(3940)\to D\bar{D}^{*}+{\rm{h.c.}}\to J/\psi\omega$ on $\alpha$[^1], where one takes $D\bar{D}^{*}$ as the dominant decay mode of $Y(3940)$ as suggested in Ref. [@Godfrey]. The order of magnitude of $\Gamma[Y(3940)\to D\bar{D}^{*}+{\rm{h.c.}}\to J/\psi\omega]$ is keV, which is far less than Belle’s data. Clearly more experimental information on $Y(3940)$ will be very helpful. Acknowledgments {#acknowledgments .unnumbered} =============== S.L.Z thanks Prof. K.T. Chao for helpful discussions. This project was supported by the National Natural Science Foundation of China under Grants 10375003, 10421503 and 10625521, Key Grant Project of Chinese Ministry of Education (No. 305001). [99]{} S.K. 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Godfrey, eConf C060409: 015, 2006. arXiv: hep-ph/0605152. [^1]: For the calculation of $Y(3940)\to D\bar{D}^{*}\to J/\psi\omega$, we only replace relevant masses and coupling constant in the formulas of $X(3872)\to D^{0}\bar{D}^{*0}+{\rm{h.c.}} \to J/\psi\rho$ to make a rough estimate. Other intermediate mesons for $Y(3940)\to J/\psi\omega$ are also allowed.

ArXiv

--- abstract: 'We investigate interfacial properties between two highly incompatible polymers of different stiffness. The extensive Monte Carlo simulations of the binary polymer melt yield detailed interfacial profiles and the interfacial tension via an analysis of capillary fluctuations. We extract an effective Flory-Huggins parameter from the simulations, which is used in self-consistent field calculations. These take due account of the chain architecture via a partial enumeration of the single chain partition function, using chain conformations obtained by Monte Carlo simulations of the pure phases. The agreement between the simulations and self-consistent field calculations is almost quantitative, however we find deviations from the predictions of the Gaussian chain model for high incompatibilities or large stiffness. The interfacial width at very high incompatibilities is smaller than the prediction of the Gaussian chain model, and decreases upon increasing the statistical segment length of the semi-flexible component.' author: - | M. Müller${}^{1,2}$ and A. Werner${}^{1}$\ [${}^1$ Institut f[ü]{}r Physik, Johannes Gutenberg Universit[ä]{}t]{}\ [D-55099 Mainz, Germany]{}\ [${}^2$Department of Physics, Box 351560, University of Washington,]{}\ [Seattle, Washington 98195-1560]{} title: | Interfaces between highly incompatible polymers of different stiffness:\ Monte Carlo simulations and self-consistent field calculations --- epsf Introduction ============== Melt blending of polymers has proven useful in designing new composite materials with improved application properties. In many practical situations the constituents of the blend are characterized by some degree of structural asymmetry. For example, a flexible component might contribute to a higher resistance to fracture, while blending it with a stiffer polymer can increase the tensile strength of the material. Since the entropy of mixing in polymeric systems decreases with increasing degree of polymerization, a small unfavorable mismatch in enthalpic interactions, entropic packing effects or the combination of both, generally leads to materials which are not homogeneous on mesoscopic scales, but rather fine dispersions of one component in another. Therefore properties of interfaces between unmixed phases are crucial in controlling the application properties of composites[@GENERAL] and have found abiding experimental interest[@FRIEND1; @FRIEND2; @FRIEND3; @FRIEND4]. Recently, the bulk phase behavior and surface properties[@WU] of polyolefins[@BATES; @OLEFINS] with varying microstructure has attracted considerable experimental and theoretical interest. These mixtures are often modeled[@BATES; @LIU; @SCHWEIZER] as blends of polymers with different bending rigidities, the less branched polymer corresponding to the more flexible component. For pure hard core interactions, field theoretical calculations by Fredrickson, Liu and Bates[@LIU], polymer reference interaction site model (P-RISM) computations by Singh and Schweizer[@SCHWEIZER], lattice cluster theories by Freed and Dudowicz[@FREED] and Monte Carlo simulations[@M1] find a small positive contribution to the Flory-Huggins parameter $\chi$. Monte Carlo simulations which include a repulsion between unlike species reveal an additional increase of the effective Flory-Huggins parameter with chain stiffness, because a back folding of chains becomes less probable with increasing stiffness and the number of intermolecular contacts increases[@M1] respectively. Qualitatively similar effects were found analytically in P-RISM[@SCHWEIZER] and lattice cluster[@FREED] theories. In spite of their ubiquitous occurrence, interfacial properties in asymmetric blends have attracted comparably little interest. When entropic packing contributions to the Flory-Huggins parameter $\chi$ are small and composition fluctuations are negligible, the self-consistent field theory is expected to yield an adequate quantitative description. Helfand and Sapse[@HS] extended the self-consistent field theory to Gaussian chains with different statistical segment lengths. In the limit of infinite long Gaussian chains and strong segregation, they obtained analytical expressions for the interfacial width $w$ and the interfacial tension $\sigma$. Both increase upon increasing the statistical segment length of one component, leaving $\chi$ and the architecture of the other component unaltered. However, there are other models, that incorporate structural disparities on the monomer level. Freed and coworkers model monomers as clusters of various shape on a lattice[@FREED2] and have explored corrections to the energy of mixing and entropic contributions to the Flory-Huggins parameter. Stiffness disparities have also been investigated using the worm-like chain model[@WORM], which captures the crossover between rod-like behavior on small length scales and Gaussian statistics on length scales much larger than the persistence length. Morse and Fredrickson[@MORSE] extended the self-consistent field calculation to a symmetric blend of worm-like chains. For vanishing bending rigidity $\kappa$ they reproduced the Gaussian chain result. In the limit of high bending rigidities and strong segregation ($\kappa\chi \gg 1$), however, they found that the width $w$ of the monomer density profile can be considerably smaller than for a Gaussian chain with the same long distance behavior. At large $\kappa \chi$, increasing the statistical segment length even leads to a decrease of the interfacial width in qualitative contrast to the Gaussian chain result. They also observed that the width of the bond orientation profile is of the order of the persistence length, which is much larger than $w$ in that limit. Thus the interfacial width $w$ and the persistence length constitute two independent length scales of the interfacial profiles. A reduction of the interfacial width in the case of small bending rigidities was obtained numerically by Schmid and Müller[@SCHMID1]. They noted that the local structure might become important if its length scale is comparable to the interfacial width; a situation which occurs at rather large incompatibilities. In the present study we extend our Monte Carlo studies[@M1] of structural asymmetric blends to the investigation of interfacial properties between well segregated phases of flexible and semi-flexible polymers. We consider rather small bending rigidities of the semi-flexible component, so that the long distance behavior of both species is Gaussian. However, we chose the incompatibility $\chi$ high enough, such that the interfacial width and the persistence length are comparable for the higher bending rigidities. The Monte Carlo simulations highlight the architectural influences and give a detailed picture of interfaces between structural asymmetric polymers. They yield density and orientation profiles for bonds and chains as a whole. Extracting an effective Flory-Huggins parameter $\chi$ from the simulation data, we compare our Monte Carlo results to self-consistent field calculations which take due account of the chain architecture via a partial enumeration procedure[@SZLEIFER; @M2; @M2A], and to Gaussian chain results. Therefore we can assess the importance of the level of coarse graining on the interfacial properties. Our paper is organized as follows: In the next section we describe our polymer model, especially the dependence of single chain properties on the stiffness. We comment on some computational aspects of the Monte Carlo simulations and describe the measurement of the interfacial tension. We also introduce the salient features of our self-consistent field calculations for arbitrary molecular architecture. In the following, we present our simulational results and compare them to the self-consistent field calculations. We close with a brief discussion of our findings and an outlook on future work. Model and technical details ============================= Bond fluctuation model and single chain properties -------------------------------------------------- In the framework of our coarse grained lattice model, a small number of chemical repeat units, say 3-5, is mapped onto a lattice monomer, such that the relevant features - chain connectivity and excluded volume interaction between monomeric units - are retained. We use the three dimensional bond fluctuation model (BFM)[@BFM], which has found widely spread application in computer simulations, because it combines the computational efficiency of lattice models with a rather faithful approximation of continuous space properties. Each effective monomer blocks a cube of 8 neighboring sites from further occupancy on a simple cubic lattice. Due to the extended monomer size, the model captures some nontrivial packing effects. We consider a blend of $n_A$ flexible polymers of length $N_A$ and $n_B$ semi-flexible B-polymers comprising $N_B$ monomers in a volume $V$. At a total monomer density $\Phi_0 = (N_An_A+N_Bn_B)/V= 0.5/8$, the model reproduces many properties of a dense polymeric melt. We use chain lengths $N=N_A=N_B=32$ and $64$, which correspond to a degree of polymerization of the order 120 and 240 in more chemically realistic polymer models. Monomers are connected via one of 108 bond vectors with lengths $2,\sqrt{5},\sqrt{6},3$ or $\sqrt{10}$, where here, and henceforth, all lengths are measured in units of the lattice spacing. The large number of bond vectors permits 87 different bond angles. The persistence length of the semi-flexible B-polymers is tuned by imposing an intermolecular potential, which favors straight bond angles. We use a particular simple choice[@M1]: $E(\theta) = f k_BT \cos(\theta)$ where $\theta$ denotes the complementary angle to two successive bonds. Previous Monte Carlo simulations[@M1] of the bulk thermodynamics for $N=32$ and $f=1.0$ revealed a purely entropic Flory-Huggins parameter $\Delta \chi = 0.0018(2)$ for the athermal blend. This small value is in good quantitative agreement with theories[@LIU; @SCHWEIZER]. These packing effects result in a slight increase of the osmotic pressure with the bending energy, which gives rise to a monomer density difference of about $1\%$ between the coexisting phases. Since $\Delta \chi \ll 2/N = 0.0625$ for this combination of chain length and stiffness disparity, we introduce an additional enthalpic repulsion to induce phase separation. For simplicity, these thermal interactions are modeled as a square well potential comprising all 54 neighbor sites up to a distance $\sqrt{6}$. The contact of monomers of the same species lowers the energy by $\epsilon k_BT$, whereas the contact of different monomers increases the energy by the same amount. A finite size scaling analysis yields accurate estimates for the critical point of the binary blends ($N=32$): $\epsilon_c=0.01442(6), \phi_{Ac}=0.5$ and $\epsilon_c=0.0127(1), \phi_{Ac}=0.516(10)$ for $f=0$ and $1$[@M1], respectively. In the present study we chose $\epsilon=0.05$ which corresponds roughly to $\chi \approx 0.27$. This value is much higher than typical values for polyolefin blends[@BATES]. Our results correspond to rather strongly immiscible blends (e.g. interfaces between polystyrene (PS) and polyvinylpropylene PVP [@FRIEND4]). The conformational data for $N=32$ and $\epsilon=0.05$ as a function of the bending energy $f$ are presented in Fig. \[fig:conf\] and Table \[tab:konf\]. The inset shows the growth of the chain extension upon increasing $f$. The ratio between the square end-to-end distance $R^2$ and the square radius of gyration $R_g^2$ remains very close to the Gaussian value $6$ (within $5\%$ even for $f=2$). Also the small wave vector regime of the single chain structure function $$S(q)=\frac{1}{N} \left \langle \left| \sum_{i=1}^{N} \exp(i\vec{q}\vec{r}_i)\right|^2 \right \rangle$$ is well describable by a Debye function $S_D(q)/N = 2[\exp(-q^2R_g^2)-1+q^2R_g^2]/(q^2R_g^2)^2$[@FLORY] for $q<0.3$. Thus the [*long*]{} range behavior of our chains is characterized by Gaussian statistics for all values of the bending energy $f$ studied and we define the statistical segment length $b$ according $b^2 = R^2/(N-1)$. Note that the statistical segment length grows from 3.06 for $f=0$ to $4.63$ for $f=2$. This asymmetry in the statistical segment length is of similar magnitude as in polyolefin blends[@BATES]. However, for length scales of the order of the statistical segment length, we find deviations from the Gaussian behavior. The plateau $q^2S(q)=12/b^2$ for large $q$ in the Kratky Porod plot is only observed for flexible chains ($f=0$) and yields a slightly higher estimate for the statistical segment length $b=3.4$. For the semi-flexible chains the slope of $q^2S(q)$ in the range $0.3<q<1$ increases upon increasing the bending energy. Defining an effective bending rigidity of an equivalent worm-like chain $\kappa = R^2/2\langle b^2\rangle (N-1)$ ($\langle b^2\rangle$: mean squared bond length), $\kappa$ grows from 0.68 to 1.57 ($\kappa\chi = 0.18 \cdots 0.47$) upon increasing the bending energy $f$. For wave vectors $q R_g\approx 2\pi R_g/2 w$ (denoted by the arrows in the Fig. \[fig:conf\]), where $w=3.4$ corresponds roughly to the width of the monomer density profile in the self-consistent field (SCF) calculations, we find deviations from the Gaussian behavior for higher bending energies and anticipate corrections to the predictions of the Gaussian model. Local fluid structure and effective Flory-Huggins parameter $\chi$ ------------------------------------------------------------------ In order to compare our simulational results to self-consistent field (SCF) calculations, which cannot account for the local fluid structure of our model, we have to identify an effective Flory-Huggins parameter $\chi$. For the bulk behavior in the one phase region this has been discussed in ref. [@M1; @M0]: We define a dimensionless monomer density $\phi_{A(B)}$ as the ratio between the local number density of A(B)-monomers and the total monomer density $\Phi_0$. Then, the density of intermolecular contacts $n_{AB}$ takes the form: $$\frac{2n_{AA}}{\Phi_0\phi_A^2} = \Phi_0 \int_{r \leq \sqrt{6}}d^3r\; g_{AA}(r) \equiv z_{AA} \qquad \mbox{and} \qquad \frac{ n_{AB}}{\Phi_0\phi_A\phi_B} = \Phi_0 \int_{r \leq \sqrt{6}}d^3r\; g_{AB}(r) \equiv z_{AB}$$ where $g_{IJ}$ denotes the $IJ$ interchain correlation function, which is normalized such that $g_{IJ}(r \to \infty)=1$. The integration is extended over the spatial extension of the square well potential and $z_{IJ}$ corresponds to the effective coordination number of the Flory-Huggins lattice. If the coupling between chain conformations and effective monomer repulsion is negligible, only the [*inter*]{}molecular energy drives the phase separation. In this case (as we shall see in the next subsection), the $\chi$ parameter takes the form: $\chi=\epsilon(z_{AA}+2z_{AB}+z_{BB})/2k_BT$, where $z_{IJ}$ denote the coordination numbers obtained from the [*inter*]{}molecular pair-correlation functions. At the critical temperatures the coordination numbers have been measured in the simulations at composition $\phi_A=\phi_B=1/2$[@M1]. From its very definition the Flory-Huggins parameter $\chi$ accurately describes the intermolecular interaction energy, and it agrees nicely with values obtained from the semi-grandcanonical equation of state and the estimate from the long wavelength behavior of the collective structure factor. It also yields estimates of the critical temperature, which agree with the Monte Carlo results up to $1/\sqrt{N}$ corrections due to composition fluctuations[@M0]. In the pure system, the intermolecular coordination number of the flexible component is lower than the corresponding value for the semi-flexible chains[@M1]. The number of intramolecular contacts[@C2] is higher for the flexible chains. Therefore, the $\chi$-parameter grows upon increasing $f$[@M1]. Due to the larger chain extension for the semi-flexible component, the correlation hole has a larger spatial extent, but is more shallow. The intermolecular pair correlation function is presented in the inset of Fig. \[fig:ginter\]. Due to the extended monomer size $g(r)$ vanishes for distances $r<2$. At short distances, the presence of single site vacancies introduces local packing effects, which gives rise to several neighbor shells in the fluid. The extended structure of the polymer manifests itself in a reduction of contacts with [*other*]{} chains on the length scale of the end-to-end distance. On short distances, the intermolecular pair-correlation function for the stiffer chains is larger than for the flexible ones. For flexible chains it is possible to separate the monomeric packing effect from the polymeric correlation hole by dividing $g(r)$ by its monomeric equivalent[@M0], which exhibits only packing effects. The ratio $g(r)/g_{N=1}(r)$ presents the conditional probability of finding a monomer of a different chain at a distance $r$, if there would be one in the monomer system. This ratio, presented in Fig. \[fig:ginter\], is a rather smooth function, indicating, that the chain connectivity hardly affects the monomeric packing. If the correlation hole would be characterized by a single length scale, i.e. the end-to-end distance $R$ in the Gaussian chain model, one expects a scaling behavior of the form: $$1-\frac{g(r)}{g_{N=1}(r)} = \frac{N}{R^3} f\left(\frac{r}{R}\right)$$ Such a scaling plot is shown in Fig. \[fig:ginter\]. The data collapse well for the different bending rigidities at large distances, whereas there are deviations for small distances. This is a further indication, that the chain structure is characterized by two independent length scales, the end-to-end distance and the persistence length. In the well segregated regime (far below the critical temperature), it is very difficult to measure the $AB$ intermolecular correlation function in the bulk. Therefore, unlike ref. [@M1], we make an [*additional*]{} ad-hoc assumption: $z_{AB}=(z_{AA}+z_{BB})/2$. For symmetric blends near the critical point P-RISM calculations[@YETH] predict that deviations from this behavior die out with growing chain length like $1/\sqrt{N}$. However, the validity of this random-packing like assumption for highly incompatible structural asymmetric blends is not obvious. We explore the interfacial structure by simulating a system in a $L\times L\times 2L$ geometry with $L=64$ and periodic boundary conditions in the canonical ensemble. The system contains two interfaces parallel to the $xy$ plane. The chain conformations are generated via local monomer displacements and slithering snake moves, which are applied at a ratio 1:3 (except for $f=0$, where only local monomer displacements were employed). The systems were equilibrated over 125,000 attempted local moves per monomer (AMM) and 375,000 slithering snake tries per chain (SS). Every 12,500 AMM and 37,500 SS movements a configuration was stored for detailed analysis, at least 898 configuration were generated. We use a trivial parallelization strategy on a CRAY T3E, running typically 8 or 32 configurations in parallel. Profiles across the interface are measured according to the following procedures: “Apparent” profiles are obtained by locating the instantaneous position of the interface across the whole lateral system extension in each snapshot and averaging over profiles with respect to the instantaneous, but laterally averaged midpoint. These profiles exhibit a system size dependent broadening due to capillary fluctuations, which is not accounted for in the SCF calculations. To avoid this broadening, we define “reduced” profiles by laterally dividing the system into subsystems of size $B \times B$. We choose $B=16$. One could reduce the effect of capillary fluctuations further by chosing a smaller block size $B$, however, one should take care not to cut off “intrinsic” fluctuations[@SEM_CAP]. Since on the scale $B$ fluctuations are still reasonably described by a Helfrich Hamiltonian[@HELFRICH](see below), our block size $B$ is larger than the length scale of “intrinsic” fluctuations. This is consistent with Semenov’s[@SEM_CAP] estimate for the corresponding length scale $L_{\rm cutoff}=\pi w \approx 10<B$. Thus this averaging procedure reduces the influence of capillary fluctuations, but does not eliminate it completely[@AW]. The presence of an interface gives rise to a spatial dependence of the local monomer densities and chain conformations, which in turn is reflected in the intermolecular pair correlation functions. In Fig. \[fig:zcoord\] we present the reduced profiles of the intermolecular and intramolecular coordination numbers as a function of the distance from the center of the interface for the bending energies $f=0$ and $2$. The individual coordination numbers exhibit a considerable spatial dependence; this is however partially due to the spatial range of interactions and the remaining capillary fluctuations[@C1]. To illustrate the effect we plot the apparent and reduced profile of the AB intermolecular coordination number. The value at the center of the interface increases upon reducing $B$; the intrinsic value can not be estimated from these data with high precision. However, the average value of the “reduced” profile is close to $z_{AB}=(z_{AA}+z_{BB})/2$, the value used in the SCF calculations. The total number of intermolecular contacts $z^{inter}=(n_{AA}^{inter}+n_{BB}^{inter}+n_{AB}^{inter})/ \Phi_0(\phi_A+\phi_B)^2$ is much less sensitive to the intrinsic (local) profiles and shows a gradual transition between the corresponding bulk values, with a reduction at the center of the interface[@M4] of about $8\%$. This spatial dependence of the effective Flory Huggins parameter is neglected. Interestingly, the sum of all contacts $z^{all}$ (both intermolecular and intramolecular) is largely independent of the stiffness or the distance from the interface, i.e. the bending energy or the unfavorable interactions at the interface causes the chains to rearrange (e.g. exchange unfavorable interchain contacts by energetic favorable intrachain contacts) but hardly affect the structure of the underlying monomer fluid. Therefore, the local fluid structure is dominated by the packing constraints and the excluded volume interactions. The chain connectivity, bending energies, and the thermal interactions are of minor importance for the monomer fluid. The chain conformations are strongly influenced by the bending energies but depend only slightly on the thermal interactions. The Flory-Huggins parameter is determined by the thermal interactions and also depends on the bending energies via the correlation hole effect. The disparity in the packing behavior of the flexible and the stiff polymers is of minor importance for $\chi$ for the chain lengths studied. Measuring the surface tension via the capillary fluctuation spectrum -------------------------------------------------------------------- Due to the stiffness disparity between the species, straightforward application of semi-grandcanonical identity changes between different polymer types are rather inefficient (note that the efficiency drops by about 3 orders of magnitude[@M1] upon increasing $f$ from 0 to 1 for $N=32$) and therefore limited to small chain length and stiffness. Measurement of the interfacial tension via the reweighting of the composition distribution, which has been successfully applied to structural symmetric blends ($f=0$), is therefore difficult. In principle, the interfacial tension can be determined via the anisotropy of the pressure tensor. This method has been successfully applied in off-lattice simulations[@PRESSURE], but the generalization to lattice models is difficult[@CIFRA]. However, the spectrum of capillary fluctuations offers an alternative[@M3] for measuring the interfacial tension; a method which does not rely on identity switches. Let $u(x,y)$ be the local interfacial position. Then the free energy cost for deviations from a flat planar interface is given by the Helfrich expression[@HELFRICH]: $${\cal H} = \int dxdy\; \frac{\sigma}{2} (\nabla u)^2+ \cdots$$ where higher order gradient terms are neglected. In our simulation, we define local $x$- and $y$-averaged interface positions by minimizing the quantity $$\left| \sum_{z=u(y)-6}^{u(y)+6}\sum_{x=0}^{x=L-1} {\Large (}\phi_A(x,y,z)-\phi_B(x,y,z) {\Large )}\right|$$ for the $x$-averaged position $u(y)$ and a similar expression for the $y$-averaged one. This averaged interfacial position is Fourier decomposed according to: $ u(y)=\frac{a_0}{2} + \sum_{l=0}^{L/2} a(q_l)\cos(q_ly) + b(q_l)\sin(q_ly) $ with $q_l=2\pi l/L$. The Helfrich Hamiltonian predicts that the Fourier components $a(q_l)$ and $b(q_l)$ are Gaussian distributed with a width $$\frac{2}{L^2\left\langle a^2(q)\right\rangle} = \frac{2}{L^2\left\langle b^2(q)\right\rangle} = \frac{\sigma}{k_BT} q^2$$ In Fig. \[fig:sigma\_all\] we present the distribution of the Fourier components for two different bending energies $f=0,2$ and the 4 smallest wave vectors $q$. This long wavelength part of the fluctuation spectrum is well described by the quadratic Helfrich expression. The straight line marks the expected Gaussian distribution for the Fourier amplitudes, to which the simulation data comply. The inverse width of the distribution determines the interfacial tension. The extracted value for the symmetric blend agrees with the independent measurement obtained via the reweighting scheme[@M4]. (The latter scheme measures the interfacial free energy via the ratio of the probability for finding the system in a homogeneous bulk state or a configuration comprising two interfaces. ) To estimate the errors of measuring the interfacial tension via the capillary fluctuation spectrum, it would be desirable to increase the lateral system size. However, the error in extrapolating the simulation data to $q \to 0$ is smaller than $7\%$. Thus the analysis of the capillary fluctuation spectrum is an efficient alternative for measuring interfacial tensions in structurally asymmetric systems; the results are compared to the predictions of the SCF calculations in Sec. IIIa. Self-consistent field calculations ---------------------------------- The mean field approach is similar to Helfand[@HT; @HS], Noolandi[@NOOLANDI], and Shull[@SHULL], except for the treatment of the chain architecture[@M2]. The partition function of a binary polymer blend has the general form[@H75]: $${\cal Z} \sim \frac{1}{n_A!n_B!} \int \Pi_{\alpha=1}^{n_A} {\cal D}[r_\alpha] {\cal P}_A[r_\alpha] \Pi_{\beta=1}^{n_B} {\cal D}[r_\beta] {\cal P}_B[r_\beta] \exp \left( -\frac{\Phi_0}{k_BT} \int d^3r\; {\cal E}(\hat{\phi}_A,\hat{\phi}_B)\right)$$ where the functional integrals ${\cal D}[r]$ sum over all polymer conformations and ${\cal P}[r]$ denotes the probability distribution characterizing the noninteracting, single chain conformations. ${\cal E}$ represents a segmental interaction free energy, and the dimensionless monomer density takes the form[@H75]: $$\hat{\phi}_A(r) = \frac{1}{\Phi_0} \sum_{\alpha=1}^{n_A}\sum_{i_A=1}^{N_A} \delta(r-r_{\alpha,i_A})$$ where the sum runs over all monomers in the A-polymer $\alpha$. A similar expression holds for $\hat{\phi}_B(r)$. The segment free energy ${\cal E}$ comprises two contributions: a free volume part arising from hard core interactions and an energetic term from the thermal interactions. Since the melt is nearly incompressible, we approximate the free volume part by a simple quadratic expression introduced by Helfand[@HT], which reproduces the relative reduction of the total monomer density by about $4\%$[@SCHMID1]. However the difference of the bulk densities of the coexisting phases has a different sign in the simulations and than in the SCF calculations. In the simulations the higher osmotic pressure of the semi-flexible component results in a slightly lower bulk density of the semi-flexible component in the simulations, an effect neglected in the SCF calculations. Moreover, in the SCF calculations the more negative intermolecular energy density (see below) of the B component results in a slightly higher density of semi-flexible polymers. The total density differences between the coexisting phases is however only about $1\%$. The pairwise intermolecular interactions $V_{IJ}(r)$ ($I,J$=A,B) are treated as point interactions of strength $\epsilon z_{IJ} \delta(r)/\Phi_0$. $z_{IJ}$ parameterizes the local fluid structure of the underlying microscopic model, as discussed above. The coupling between individual chain conformations and the coordination numbers, which results in the spatial dependence of the $\chi$-parameter observed in the simulations, is neglected. Furthermore, we ignore purely entropic contributions (which have been determined to be small by Monte Carlo simulations) and do not include orientation dependent segmental interactions, which will eventually lead to a nematic phase at much higher bending energies $f$. Thus we take the interactions to be $$\frac{{\cal E}(\phi_A,\phi_B)}{k_BT} = \frac{\zeta}{2} \left( \phi_A + \phi_B - 1\right)^2 - \frac{\epsilon z_{AA}}{2} \phi_A^2 - \frac{\epsilon z_{BB}}{2} \phi_B^2 + \epsilon z_{AB} \phi_A \phi_B$$ The inverse compressibility $\zeta$ has been measured in simulations of the athermal model; $\zeta=4.1$[@WM1]. A Hubbard-Stratonovich transformation rewrites the many chain problem in terms of independent chains in external, fluctuating fields $W_A$ and $W_B$. $${\cal Z} \sim \int {\cal D}[W_A,W_B,\Phi_A,\Phi_B] \exp \left(-{\cal F}[W_A,W_B,\Phi_A,\Phi_B]/k_BT\right)$$ where the free energy functional is defined by $$\begin{aligned} \frac{{\cal F}[W_A,W_B,\Phi_A,\Phi_B]}{\Phi_0 k_BT V} &=& \frac{\bar{\phi}_A}{N_A} \ln \bar{\phi}_A + \frac{\bar{\phi}_B}{N_B} \ln \bar{\phi}_B + \frac{1}{V} \int d^3r \; {\cal E}(\Phi_A,\Phi_B) \nonumber \\ && - \frac{1}{V} \int d^3r \left\{ W_A\Phi_A + W_B\Phi_B \right\} - \frac{\bar{\phi}_A}{N_A} \ln q_A[W_A] - \frac{\bar{\phi}_B}{N_B} \ln q_B[W_B]\end{aligned}$$ $\bar{\phi}_A = \frac{n_A N_A}{\Phi_0 V} = 1 -\bar{\phi}_B$ denotes the average A-monomer density and $q_A[W_A]$ the single chain partition function in the external field $W_A$ $$q_A[W_A] = \frac{1}{V} \int {\cal D}_1[r] {\cal P}_A[r] \exp \left(- \Phi_0 \int d^3r\; \hat{\phi}_A W_A \right)$$ respectively. The leading contributions to the partition function stem from those values $\phi_A,\phi_B,w_A,w_B$ of the collective variables which extremize the free energy functional, and the mean field approximation amounts to retaining only these contributions. The values are determined by: $$\begin{aligned} \frac{\delta {\cal F}}{\delta \phi_A} = 0 &\Rightarrow & w_a = \frac{\delta}{\delta \phi_A} \int d^3r\; {\cal E}(\phi_A,\phi_B) = \zeta (\phi_A + \phi_B -1) - \epsilon z_{AA} \phi_A + \epsilon z_{AB} \phi_B \\ \frac{\delta {\cal F}}{\delta w_A} = 0 &\Rightarrow & \phi_A = \frac{\bar{\phi}_A V}{N_A q_A} \frac{\delta q_A}{\delta w_A} \label{eq:d}\end{aligned}$$ and similar expressions for $w_B$ and $\phi_B$. The saddle point integration approximates the original problem of mutually interacting chains by one of a single chain in an external field, which is determined, in turn, by the monomer density. Composition fluctuations are ignored, but the coupling between chain conformations (e.g. orientations) and the monomer density is retained. The free energy of a homogeneous system takes the Flory-Huggins form: $$\frac{{\cal F}}{\Phi_0k_BTV} = \frac{\bar{\phi}_A}{N_A}\ln\left(\bar{\phi}_A\right) +\frac{1-\bar{\phi}_A}{N_B}\ln \left(1-\bar{\phi}_A\right) -\frac{1}{2}\epsilon\left\{ \left( z_{AA}+2z_{AB}+z_{BB}\right)\bar{\phi}_A^2 -2\left(z_{AB}+z_{BB}\right)\bar{\phi}_A +z_{BB} \right\}$$ where we identify the Flory-Huggins parameter $\chi=( z_{AA}+2z_{AB}+z_{BB} )\epsilon/2$. In the strongly segregated regime, the free energy of a system containing one interface is given by: ${\cal F}=-\epsilon(\bar{\phi}_Az_{AA}+(1-\bar{\phi}_A)z_{BB})\Phi_0k_BTV/2+\sigma k_BTL^2$. The definition of the interfacial tension as the difference of the free energy of a system containing an interface and the homogeneous bulk system corresponds literally to the measurement of the interfacial tension via the reweighting scheme[@M4] in the Monte Carlo simulations. As shown in Sec. IIc for the symmetric blend, these values agree with the measurement of the interfacial tension via the capillary fluctuation spectrum, so that we can compare the results of the SCF calculations and the values extracted from the capillary wave spectrum quantitatively. For the special cases of Gaussian chains[@HT; @EDWARDS] and worm-like polymers[@WORM; @MORSE] one can treat the single chain problem in an arbitrary external field in limiting cases (e.g $N \to \infty$) analytically. For general parameters, however, one has to resort to numerical procedures even for these simple models. The BFM chains used in the simulations are characterized by structure on different length scales. The conformations are rod-like for length smaller than the persistence length, which depends on the bending energy $f$. On intermediate length scales, they obey self-avoiding walk statistics, while on the largest scale, the excluded volume interactions are screened in the melt, and the chains exhibit Gaussian statistics. Since we want to explore dependence on the explicit chain structure, we evaluate the single chain partition function via a partial enumeration scheme, introduced by Szleifer and coworkers[@SZLEIFER]. The method is conceptually straightforward and applicable to [*arbitrary*]{} architecture[@M2; @M2A]. It can use experimental or simulational data as input. Note that no adjustable parameters are involved in the chain structure (such as the statistical segment length in the Gaussian model or the bond length and the bending rigidity in the worm-like polymer model) and the chain structure is correctly represented on [*all*]{} length scales. Using Monte Carlo simulations of the pure melt, we generated 40,960 independent polymer conformations for each bending energy. Rotating and translating those original conformations, we obtain a sample of 7,864,320 polymer conformations for chain length $N=32$. (Note only the $z$ coordinates of the chains are employed for a flat interface parallel to the $xy$ plane.) For $N=64$ we use twice as many conformations. Within this framework, the A-monomer density (c.f. eq. \[eq:d\]) is simply the statistical average of independent A-polymers in the external field $w_A$: $$\phi_A = \bar{\phi}_A \frac{ \sum_{\alpha=1}^{C} \frac{1}{N_A} \sum_{i=1}^{N_A} V \delta(r-r_{\alpha,i}) \exp \left( -\sum_{i=1}^{N_A} w_A(r_{\alpha,i}) \right) } { \sum_{\alpha=1}^{C} \exp \left( -\sum_{i=1}^{N_A} w_A(r_{\alpha,i}) \right) }$$ Other single chain quantities are given by corresponding averages over independent chains in the fields $w_A$ and $w_B$. The set of nonlinear equations is expanded in a Fourier series[@MATSEN] and solved by a Newton-Raphson like method. Convergence is usually reached within 3-6 steps. The evaluation of the partition function [@M2A] in the external fields poses rather high memory demands (several Gbytes). Therefore we employ a CRAY T3E, assigning a subset of conformations to each processing element. Typically we use 64 or 128 processors in parallel, and the program scales very well with the number of processors employed[@M2A]. One needs about 1200 seconds for each set of parameters, which is roughly 2 orders of magnitude less than for the detailed Monte Carlo simulations. Comparison between Monte Carlo simulations and self-consistent field (SCF) calculations ========================================================================================= In the following we compare our Monte Carlo simulations to the results of the SCF calculations. Both, large length scale thermodynamic properties (e.g. interfacial tension) as well as the local interfacial structure (e.g. orientation of individual bonds) are investigated. The temperature dependence of most quantities for symmetric blends ($f=0$) has been studied previously[@M4] and compared to predictions of the Gaussian and worm-like chain model[@SCHMID1]. The results for vanishing bending energy compare well to our calculations. Interfacial tension ------------------- The interfacial tension $\sigma$ between the coexisting phases has an important impact on the morphology of the compound material[@MORPHOLOGY]. The control of domain size and shape is a key to tailoring the application properties of the blend. The size of minority droplets often is the smaller, the smaller the interfacial tension between the coexisting phases[@MORPHOLOGY; @MILNER]. In the strong segregation limit, Helfand and Sapse[@HS] obtained for infinite long Gaussian chains in an incompressible blend the analytic expression: $$\sigma = \Phi_0 \sqrt{ \chi /6 } \left( \frac{2}{3} \frac{b_A^2+b_A b_B+b_B^2}{b_A+b_B} \right)$$ The interfacial tension $\sigma$ grows upon increasing the bending energy (i.e. the statistical segment length) of the semi-flexible component. This behavior is presented in Fig. \[fig:Ssigma\], as well as our simulation results and the SCF calculations, which take account of the detailed chain architecture. All data exhibit an increase of the interfacial tension of about $30\%$. The simulation data and the SCF calculations agree nicely on the growth of the interfacial tension upon increasing the bending energy. The almost quantitative agreement indicates that our identification of the $\chi$-parameter yields reasonable results for structural asymmetric mixtures. However, the Gaussian chain result is about a factor $1.3$ higher than the simulation data. Recently, Ermoshkin and Semenov[@ER] calculated corrections to the interfacial tension due to effects of finite chain length $N$. For symmetric blends, they found that chain end effects reduce the interfacial tension by a factor $(1-4\ln2/\chi N) \approx 0.67$, which accounts well for the discrepancies between the Helfand-Sapse result and the Monte Carlo data. Similar reductions are found in numerical SCF calculations[@SCHMID1; @SHULL]. Note that purely entropic, packing contributions to the Flory-Huggins parameter $\Delta\chi$ are less than $1\%$ of the total $\chi$ value for $f=1$. That is somewhat smaller than the uncertainties in identifying the enthalpic contributions of $\chi$ and the accuracy of our interfacial tension measurement in the simulations. Therefore, purely entropic effects derived from packing are irrelevant to the interfacial behavior for the chain lengths, stiffness asymmetries, and temperatures investigated in the present study. Monomer density profiles ------------------------ Another important characterization of the interface are the density profiles of the individual components. Experiments[@ENTANGLE] indicate, that entanglements in the interfacial zone are of major importance for the mechanical properties of the blend. Of course, our chain lengths are too small to observe entanglements, however static properties can be extracted from our simulation data. The density profiles obtained from the SCF calculation are presented in Fig. \[fig:profile\], as well as the “apparent” profiles in the Monte Carlo simulation. The width of the apparent profile in the Monte Carlo simulations is about a factor 1.5 larger than the SCF result; this is not unexpected, because capillary fluctuations increase the squared width by a term proportional to $\ln(L)/\sigma$. However, the profiles are qualitative similar: both data show a reduction of the total monomer density at the center of the interface (the relative reduction is roughly $\chi/2\zeta$[@M4]) and almost no dependence on the bending energy $f$. The dependence of the interfacial width on the bending energy is shown in Fig. \[fig:width\][@C3]. The width $w_r$ of the reduced profile is smaller than the apparent width $w_a$, and agrees better with the SCF results. Due to remaining capillary wave effects it is an upper bound on the intrinsic width. All profiles presented below are obtained by the reduced averaging procedure. The excess energy density of the interface can also be used to estimate the intrinsic width. Since the relative increase in interfacial area due to capillary fluctuations is of the order $\sigma\ln L/L^2$, this quantity (as well as the interfacial tension) is not strongly affected by fluctuations of the local interfacial position. A tanh-shaped profile $\phi_A = \frac{1}{2}(1+\tanh\frac{z}{w})$ yields in the SCF framework: $$\frac{e_s}{k_BT} = \Phi_0 \int dz\; \left\{ -\frac{\epsilon z_{AA}}{2}\phi_A^2 -\frac{\epsilon z_{BB}}{2}\phi_B^2 + \epsilon z_{AB}\phi_A\phi_B \right\} -\frac{\Phi_0 L (z_{AA}+z_{BB})}{4} \approx \frac{1}{2} w_e\Phi_0\chi$$ where we have assumed incompressibility and neglected the finite range of interactions and any contribution of the intramolecular interactions to the excess energy density. Of course, this measure relies crucially on the identification of the Flory-Huggins parameter. However, the method is computational very convenient and can be combined[@M3] with the reweighting methods of measuring interfacial tensions. It results in values which are between the reduced width and the SCF results, which shows again the consistent parameterization of the local fluid structure. The Gaussian chain model for $N \to \infty$ predicts for symmetric blends ($f=0$) a width which is about $20\%$ smaller than the SCF result. SCF calculations[@SCHMID1] of Gaussian chains with the same long distance behavior and which include chain end effects and the finite compressibility agree within $2\%$ with our results for symmetric, flexible mixtures. An increase of the chain length from $N=32$ to $64$ reduces the effective $\chi$-parameter by $4\%$ and reduces the broadening due to finite chain length effects. The latter effect is stronger, such that the width decreases slightly. Most notably, the apparent width of the Monte Carlo data, the energetic width $w_e$ and the results of the SCF calculations show almost no dependence on the bending energy $f$, whereas the analytic expression obtained by Helfand and Sapse $$w = \sqrt{ \frac{b_A^2+b_B^2}{12 \chi} }$$ predicts an increase of about $28\%$ due to the variation of $b_B$. Taking account of the stiffness dependence of the effective Flory-Huggins parameter $\chi$, the formula above predicts an increase of $21\%$. Qualitatively, calculations for worm-like chains[@MORSE; @SCHMID1] indicate that increasing the bending rigidity results in a reduction of the interfacial width compared to the Gaussian chain result. For the present combination of parameters, both effects seem to cancel, resulting in an interfacial width, which is nearly independent of the bending rigidity. For lower incompatibilities, the interfacial width is larger, the Gaussian description on the length scale of the interfacial width becomes more appropriate. Therefore the difference between the width in the flexible/semi-flexible blend and the width in the symmetric flexible mixture increases in accord with the Helfand Sapse description. This is confirmed by SCF calculations (c.f. Fig. \[fig:dw\]), where we have assumed that the effective coordination numbers are temperature independent. However, upon increasing the incompatibility further ($\epsilon>0.082$), one finds that an increase of the statistical segment length results in a [*smaller*]{} interfacial width of the asymmetric blend in qualitative contrast to the predictions of the Gaussian model. Distribution of chain ends and orientations ------------------------------------------- The enrichment of chain ends at the center of the interface[@M4] and at hard walls[@SKKUMAR; @BITSANIS; @SCHMID; @JOERG] has attracted abiding interest. Chain ends are important for the interdiffusion and healing properties at interfaces between long polymers[@WU]. They also play an important role for reactions at interfaces. In many experimental systems, chain ends have slightly different interactions than inner chain segments,which might result in a modification of the interfacial properties. On the theoretical side, the behavior of chain ends is related to corrections to the ground state approximation. Therefore it is a sensitive test for a quantitative theoretical description. Chain end effects give rise to large corrections to the interfacial width and tension, and they also play an important role for long range interactions between interfaces[@ER]. The distribution of chain ends for symmetric blends has been investigated by Monte Carlo simulations[@M4], and in the framework of SCFT for Gaussian chains[@WU; @SCHMID1]. In Fig. \[fig:end\] the simulational results and the SCF calculations are presented; both agree almost quantitatively. As in symmetric blends, chain ends are enriched at the center of the interface, and this effect goes along with a depletion away from the interface. The fact that the depletion zone in the wings shifts outwards with increasing chain length, indicates that the length scale of the rearrangement of chain ends is the radius of gyration. A-polymers stick their ends into the B-rich phase and vice versa. The effect on the semi-flexible chains becomes more pronounced with growing stiffness, while the A-polymers are hardly influenced by the stiffness of the B-polymers. The instantaneous shape of a polymer coil is a prolate ellipsoid[@M4]. Polymers orient themselves by putting their ends preferentially at the center of the interface. This is quantified by the orientational parameter[@M4] for the end-to-end vector (cf. Fig. \[fig:qe\]): $$q_e(z) = \frac{3\langle R^2_z\rangle_z-\langle \vec{R}^2\rangle_z}{2\langle \vec{R}^2\rangle_z}$$ where the outer index $z$ at the brackets denotes the z coordinate of the midpoint of the end-to-end vector $\vec{R}$, and the inner indices its Cartesian components. The chains align their two long axis parallel to the interface in their majority phases, similar to the behavior at a hard wall. The chain orientation of semi-flexible polymers increases for growing stiffness, while the flexible A-polymers are not affected. The agreement between Monte Carlo simulation and SCF calculations is again almost quantitative. In the SCF framework, the orientations of the chains in the minority phase is accessible. The polymers align perpendicular to the interface, as to reach with one end their corresponding bulk phase. The length scale of the ordering increases with the bending energy $f$ and with chain length $N$. The orientation of individual bond vectors $q_b$ shows a similar behavior. Bonds align parallel to the interface; the effect for the semi-flexible component grows with increasing bending energy and its range is largely independent of the chain length. The agreement between simulations and SCF calculations is very good. The Gaussian chain model cannot predict any nonzero orientation of the bonds. The orientation of bonds in our model is, in fact, much smaller than for the end-to-end distance[@M4]. In contrast to the width of the density profile, the spatial range over which the orientation of bonds extends grows upon increasing the bending energy. Therefore, the orientational width and the width of the composition profile are two independent microscopic length scales. Conclusions and outlook ======================== In summary, we have presented extensive simulations of highly incompatible polymers with different stiffness. The local structure of the interface has been characterized by density profiles of different monomer species and chain ends and orientational profiles of whole chains and individual bonds. The interfacial tension has been measured via analyzing the spectrum of capillary fluctuations. Using the pair correlation functions of the pure components and a random-packing like assumption for the intermolecular contacts between different species, we have extracted an effective Flory-Huggins parameter, which takes account of the stiffness dependence of the structure of the polymeric fluid. The effective Flory-Huggins parameter grows upon increasing the stiffness, because back folding is less probable and the number of intermolecular contacts increases respectively. This effective Flory Huggins parameter was then employed in SCF calculations, as well as the chain conformations in the pure melt. These calculation incorporate the chain structure on [*all*]{} length scales via a partial enumeration scheme; there is no free parameter in describing the chain architecture. Using the detailed local structure of the bulk (as obtained by simulations) in the SCF calculations, we predict the interfacial properties. Monte Carlo results and SCF calculations for the interfacial tension, the excess interfacial energy, the redistribution of chain ends and orientations of whole chains and individual bonds agree very well provided that the analysis accounts for capillary fluctuations. However, comparing our results to the analytical predictions of the Gaussian chain model for infinite chain length, we find qualitative deviations, especially for the dependence of the interfacial width on the chain stiffness. This finding might be important for extracting the Flory-Huggins parameter from interfacial profiles in highly incompatible polymer blends. Therefore, our results emphasize that the local structure, both of the underlying monomer fluid and of the chain architecture, is important for a quantitative description. The radius of gyration determines the range of orientation of whole chains and the distribution of chain ends. Furthermore, we identify two independent microscopic length scales of the interfacial profile; one controls the width of the monomer density profile, the other corresponds to the range of orientations. This behavior resembles the findings in symmetric blends of worm-like chains in the limit $\kappa\chi \gg 1$[@MORSE] and the behavior of a homopolymer melt at a hard wall which is the limiting case for infinite incompatibility. However, in the present study this behavior is found for a different model which can be described neither by Gaussian nor by worm-like statistics on small length scales. Deviations from the Gaussian model occur under rather mild conditions which correspond roughly to $\kappa\chi=0.18 \cdots 0.47$ in the equivalent worm-like chain model. Furthermore our self consistent field approach as well as the simulation techniques are applicable to arbitrary chain architecture[@M2]. Assuming that the chain conformations and the local fluid structure are approximately independent of temperature, we have extended the self consistent field calculations to other incompatibilities. The results indicate that chain architecture becomes important when its length scale is comparable with the interfacial width. At very high incompatibility, increasing the stiffness of the semi-flexible component results in a decrease of the interfacial width. 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Binder, Macromolecules [**28**]{}, 6808 (1995). $N$ $f$ $\langle b^2\rangle$ $R^2$ $R_g^2$ $z_{BB}$ $\langle e \rangle/k_BT$ $e_s/k_BT$ $w_a$ $w_r$ $w_e$ $w_{\rm scf}$ ----- ----- ---------------------- ------- --------- ---------- -------------------------- ------------ ------- ------- ------- --------------- 32 0.0 6.92 290.4 48.8 2.65 -0.00732 0.0290 4.77 3.88 3.51 3.11 0.5 6.88 350.1 58.1 2.84 -0.00731 0.0293 4.63 3.83 3.40 3.13 1.0 6.86 431.8 70.8 3.02 -0.00730 0.0300 4.57 3.80 3.38 3.16 1.5 6.84 536.9 86.5 3.17 -0.00730 0.0304 4.49 3.77 3.34 3.19 2.0 6.84 665.2 105.1 3.29 -0.00730 0.0310 4.53 3.75 3.34 3.23 64 0.0 6.92 609.3 101.7 2.53 -0.00732 1.0 6.86 987.3 148.3 2.92 -0.00730 0.0265 4.13 3.43 3.11 2.99 : Single chain conformations and interfacial properties as a function of the bending energy $f$. Interfacial data refer to blends of flexible ($f=0$) and semi-flexible ($f$ as indicated) chains. $\langle b^2\rangle$: mean squared bond length, $R^2$: mean squared end-to-end distance, $R_g^2$ mean squared radius of gyration, $z_{BB}$ effective intermolecular coordination number as measured in simulations of the bulk system, $\langle e \rangle/k_BT$: bulk energy density, $e_s/k_BT$: interfacial energy excess per unit area, $w_a$ apparent interfacial with for $L=64$, $w_r$ reduced interfacial width for block size $B=16$, $w_e$ estimated interfacial with from the excess energy, $w_{\rm scf}$ interfacial width in the SCF calculations.[]{data-label="tab:konf"}

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