we study the total valuation adjustment ( xva ) of a european style claim under differential borrowing and lending rates .we adopt the framework introduced in the companion paper of , where the authors define buyer s and seller s xva in terms of solutions to non - linear bsdes .these characterize the portfolio process replicating long and short positions in the claim , and take into account early termination of the contract due to counterparty credit risk , funding spreads , as well as collateral servicing costs .when funding spreads are zero , security lending and borrowing rates coincide , and collateral rates paid by the taker equal the ones received by the provider , give explicit expressions for the xva as well as for the corresponding replicating strategies .however , in most realistic market scenarios , rates are asymmetric and funding spreads constitute a significant driver of xva . despite the unavailability of closed form expressions , in this paper we show that we can exploit the connection between the bsdes and the corresponding nonlinear pdes to study how funding costs , collateral and counterparty risk affect the total valuation adjustment .our study extends previous literature along two directions .first , we develop a rigorous study of the semilinear pdes associated with the nonlinear bsdes introduced in and yielding the xva prices .previous studies in this direction include , who gives the pde representations for the lower and upper bound prices of options in an extended s model where borrowing and lending rates differ .however , he does not take into account counterparty risk .this is accounted for in and , who consider an extended black - scholes framework where two corporate bonds are introduced in order to hedge the default risk of the trader and of his counterparty .they generalize their framework in to include collateral mitigation and evaluate the impact of different funding strategies .we also mention who study a bsde with random terminal time of the type appearing in a funding valuation context , but do not develop connections to the resulting pde .we prove the existence and uniqueness of a viscosity solution to the pde .we extend standard results of by using the locality of the viscosity solution , and also by accounting for the possibility of default risk .we then show that the pde also admits a classical solution implying existence and uniqueness of the classical solution .our result contrasts with earlier studies in the funding literature , e.g. , where smoothness of the price of the derivative contract with respect to the underlying spot asset price is usually assumed to hold , and the pde is derived after assuming that the market does not admit arbitrage . using the classical solution , we compute the replicating strategies of the traded claim in a portfolio consisting of a stock , and risky bonds of the trader and of his counterparty . to the best of our knowledge ,this is the first study to establish this result . develops a connection between the funding bsde and the solution of a classical pde , but in a different framework where the resulting pde is linear and the market model complete , see section 5.2 therein .we also mention the recent work of who establish classical solutions to quasi - linear pdes corresponding with bsdes yielding the fair bilateral prices of a european contingent claim .their work , however , does not consider the possibility that the trader or his counterparty can default .  a second noticeable contribution of our study is a comprehensive numerical analysis made possible by the previously established existence and uniqueness result .earlier studies on funding , such as , , and lack the computational component .we find strong sensitivity of xva to funding costs , counterparty risk and collateralization levels .viewing both buyer s and seller s xva as functions of collateral levels defines a no - arbitrage band whose width increases with the funding spread . as the position becomes more collateralized , the value of the closeout payment increases and leads the trader to implement a riskier strategy .this yields increased buyer s and seller s xva . interestingly , both of these quantities may decrease in the counterparty s default intensity as the default premium compensation demanded by the investor for holding counterparty bonds becomes higher than the funding costs incurred for replicating the closeout position . moreover , increased values of counterparty s default intensity reduce both seller s and buyer s xva by a similar amount .our findings serve as a useful guide to risk - management and trading desks , who can decide on the terms of the trades ( collateralization levels , borrowing rates charged by treasury , etc ... ) based on the incurred costs as measured by xva .the rest of the paper is organized as follows .section [ sec : model ] reviews the framework for computing xva introduced in .section [ sec : fk ] analyzes the semilinear pdes .section [ sec : numanalysis ] develops a comprehensive numerical analysis .section [ sec : conclusions ] concludes the paper .we review the setup introduced in the companion paper by and refer to sections 2 and 3 of their paper for additional details .we give here a self - contained exposition needed to understand the pde analysis presented in the next section .we fix a probability space , where denotes the physical measure .the market filtration is , , where the filtrations and will be specified later in the section .we take the viewpoint of an investor `` '' ( also referred to as trader or hedger ) who wants to compute the total valuation adjustment , abbreviated as xva , of a european claim traded with a counterparty `` '' .the claim is written on a stock whose time price is denoted by .the claim matures at and has terminal payoff , where is a piecewise continuously differentiable real valued function of polynomial growth .the value of the claim as determined by a third party evaluator is given by  ,\ ] ] where the valuation measure associated with the publicly available discount rate chosen by the valuation agent is equivalent to and given by the radon - nikodm density    this section describes the securities at disposal of the investor to construct his replicating portfolio .they include the default - free stock on which the financial claim is written , and two risky bonds underwritten by the trader and by his counterparty . under the physical measure , the dynamics of the stock price processis given by where and are constants .moreover , is a standard brownian motion under the probability measure .we set to be the -augmentation of the filtration generated by .let and be the default times of the trader and of his counterparty , respectively .these are independent exponentially distributed random variables with constant intensities , .we use , , to denote the default indicator process of the -th name .the default event filtration is set to .the two risky bonds underwritten by the trader and by his counterparty mature have the same maturity of the claim . for , ,their price processes admit dynamics given by where is constant . throughout the paper , we set .we also recall the following relation , given in , between the physical measure and the valuation measure :    this section describes the various trading accounts used by the hedger to finance the strategy replicating the price process of the claim .financing and lending activities related to the stock security happen via the repo market .we denote by the rate received by the hedger when he lends money to the repo market .symmetrically , we denote by the rate he has to pay when he borrows money from the repo market .let be the number of shares of the security account at time .the value of this account at is given by where    the hedger finances the portfolio strategy to replicate the claim from his treasury .we denote by the rate at which the hedger lends cash to the treasury , and by the rate at which he borrows cash from it .let denote the number of shares of the funding account at time , whose value is given by where    each party posts collateral to mitigate potential losses incurred by the other party in case of default .collateral is in the form of variation margins , i.e. cash transfers are made by the out - of - the money party to cover unfavourable moves in the value of the underlying transaction .on , the collateral process is given by where the collateral level determines the amount of covered exposure .notice that in our setup , if the investor sold the claim to his counterparty , then he is always the collateral taker ( ) , while if he purchased the claim from this counterparty he is always the collateral provider ( ) .we denote by the rate on the collateral amount received by the hedger if he has posted collateral , and by the rate paid by the hedger if he has received collateral .the collateral account is defined by where we use to denote the number of shares of collateral account held by the hedger at time .it then holds that    since both the hedger and his counterparty can default , the transaction can terminate prematurely . in the event of a default ,the mark - to - market value of the transaction equals the residual value , after mitigating on - default related losses with the available collateral .let , and , be respectively the positive and negative parts of a real number . denote by the value of the transaction which must be replicated by the hedger at the earlier of the two : the hedger s , or the counterparty s default time ( if positive the hedger owes this amount to the counterparty ) .this is given by here and are constant loss rates .we refer to section 3.3 of for a detailed explanation .we review the definition of xva given in .denote by the wealth process of the hedger associated with the strategy , where .we recall from above that is the number of shares of the funding account .we use to denote the number of stock shares .moreover , and denote the number of shares of the bonds underwritten by the trader , and by his counterparty , respectively .the wealth process is given by the following expression   show that , under the valuation measure , the dynamics of the wealth process is given by    setting the dynamics  may be rewritten as next , we distinguish between and .we use to describe the wealth process of the hedger when he replicates the claim ( hence hedging the position after selling the claim with terminal payoff ) . on the other hand describes the wealth process when replicating the claim ( hence hedging the position after buying the claim with terminal payoff ) . to this purpose , define where the driver depends on the market evaluation process ( via the collateral ) .in particular \times { \mathbb{r}}^4 ] are free of hedger s arbitrage . in particular, we have that and .  the _ total valuation adjustment _ is defined as the amount that needs to be added to the black - scholes price ( exclusive of funding costs ) to get the actual price ( inclusive of funding costs ) .this is asymmetric for sell- and buy - prices . define the as follows .the seller s is the -adapted stochastic process defined by while the buyer s is defined as   corresponds to the total costs that the hedger incurs when replicating the payoff of a claim he sold , whereas corresponds to the total costs that he incurs when replicating the payoff of a claim he purchased .we note that the difference of the also describes the width of the no - arbitrage interval , as section derives the pde representations corresponding to the master bsdes and . for brevity , we will state everything with regards to the upper no - arbitrage price , while noting that the treatment for is identical .for notational convenience , we will drop the plus superscript , and refer to , and simply as , and .[ rembsde ] in the sequel , we write the solution of the bsde on as , where .the existence of such a measurable function , i.e. the fact that is markovian , is shown in proposition 4.1.1 in .  to simplify notation , we will not specify the argument of the latter functions .then we can rewrite the bsde in as where , are the compensated poisson random measures such that on  ] be a smooth function , such that and for some fixed . instead of working with directly , we will work with a local approximation .let be small enough , such that .as the default intensities are constant , it follows that  ] are both strictly positive .additionally , w.l.o.g .we may assume that is bounded .specifically , let be a subsequence such that \times { \mathbb r_{>0}}\times{\mathbb{r}}\times{\mathbb{r}}\bigr) ] next , choose this allows us to make the further assumption that and by possibly extending the open neighborhood , because we have that and .we now have that  ]  next , define ,  \end{aligned}\ ] ] where since is lipschitz , we can show in a similar way as in that for it holds that \le c h^\frac12,\label{eq : estim1}\\   & { { \mathbb{e}}}^{{\mathbb{q}}}\biggl[\int_{t_0\wedge\tau}^{(t_0+h)\wedge\tau } { \left \vert { \bar{\bar{{z}}}}_r \right \vert}^2dr  + \sum_{j\in\{i , c\}}\int_{t_0\wedge\tau}^{(t_0+h)\wedge\tau}\int_{{\mathbb{r } } } { \left \vert { \bar{\bar{{z}}}}_r^j \right \vert}^2 \nu^j(d\rho)dr\biggr ] \le c h^\frac32,\label{eq : estim2}\\   & { \mathbb{q}}[\tau_1 \le ( t_0+h)\wedge\tau ] \le ch,\label{eq : estim3 }  \end{aligned}\ ] ] where we have introduced another stopping time   from  > 0 ] , it also follows that \\   & \ge 1- { \mathbb{q } } [ \tau \le t_0 ] - { \mathbb{q}}[\tau_1\le ( t_0+h)\wedge\tau ] - { \mathbb{q } } [ \tau \in ( t_0,t_0+h)]\ge q_1 - ch ,  \end{aligned}\ ] ] and similarly that \le ch .\end{aligned}\ ] ] here and throughout this proof , we will abuse notation and use to denote a generic constant , which may be different in each inequality below .it follows that \\   & \quad+ \frac1{h}{{\mathbb{e}}}^{{\mathbb{q } } } \biggl [ \int_{t_0\wedge\tau}^{(t_0+h)\wedge\tau } \psi(t , s_t,\varpi_t^{i,{\mathbb{q } } } , \varpi_t^{c,{\mathbb{q}}})dt \ind_{\{\tau_1 \le ( t_0+h)\wedge\tau\ } \cup \ {  \tau \in(t_0,t_0+h)\ } } \biggr]\\   & \le -\epsilon { \mathbb{q}}\bigl[\{\tau_1>(t_0+h)\wedge\tau\ } \cap \ {  \tau \ge t_0+h \}\bigr ] \\   & \quad+ \frac1{h}\sqrt{{{\mathbb{e}}}^{{\mathbb{q } } } \biggl [ \int_{t_0}^{t_0+h } 1dt \ind_{\{\tau_1 \le ( t_0+h)\wedge\tau\ } \cup \ {  \tau \in(t_0,t_0+h)\ } } \biggr ] { { \mathbb{e}}}^{{\mathbb{q } } } \biggl [  \int_{t_0\wedge\tau}^{(t_0+h)\wedge\tau } \psi^2(t , s_t,\varpi_t^{i,{\mathbb{q } } } , \varpi_t^{c,{\mathbb{q}}})dt  \biggr]}\\   & \le -\epsilon { \mathbb{q}}\bigl[\{\tau_1>(t_0+h)\wedge\tau\ } \cap \ {  \tau \ge t_0+\}\bigr ] \\   & \quad+ \frac { \sqrt{h{\mathbb{q}}\bigl[\{\tau_1 \le ( t_0+h)\wedge\tau \ } \cup \ {  \tau \in(t_0,t_0+h ) \}\bigr ] } } { h } \sqrt{{{\mathbb{e}}}^{{\mathbb{q } } } \biggl [ \int_{t_0\wedge\tau}^{(t_0+h)\wedge\tau } \psi^2(t , s_t,\varpi_t^{i,{\mathbb{q } } } , \varpi_t^{c,{\mathbb{q}}})dt  \biggr]}\\   & \le -\epsilon ( q_1-ch)+ c \sqrt{{h } } \sqrt{1+{{\mathbb{e}}}^{{\mathbb{q } } } \biggl [ \sup_{t \in [ t_0\wedge\tau , ( t_0+h)\wedge\tau ] } s_t^4\biggr]}.  \end{aligned}\ ] ] this shows that for sufficiently small , we have that   from the comparison theorem a.3 in we get that . using this along with, we conclude that \bigg\vert  \\   & \le c \sup_{r\in[t_0,t_0+h ] } { { \mathbb{e}}}^{{\mathbb{q}}}\biggl [ { \left \vert { \bar{\bar{{v}}}}_{r\wedge\tau } \right \vert}\ind_{\{\tau > r\}}\biggr ] \\   & \quad + \frac{c}{\sqrt{h}}\biggl ( \sqrt { { { \mathbb{e}}}^{{\mathbb{q}}}\biggl [ \int_{t_0\wedge\tau}^{(t_0+h)\wedge\tau } { \left \vert  { \bar{\bar{{z}}}}_r \right \vert}^2dr\biggr ] } +  \sum_{j\in\{i , c\ } }  \sqrt { { { \mathbb{e}}}^{{\mathbb{q}}}\biggl[\int_{t_0\wedge\tau}^{(t_0+h)\wedge\tau } \int_{{\mathbb{r } } } { \left \vert { \bar{\bar{{z}}}}_r^j \right \vert}^2 \nu^j(d\rho)dr\biggr ] } \biggr)\\   & \le c ( h^\frac12 + h^\frac14 ) ,  \end{aligned}\ ] ] where the second inequality follows from the fact that the driver of our bsde is lipschitz along with the hlder s inequality ( setting the exponent to ) , while the last inequality follows from the estimates given by the inequalities  and .this yields a contradiction , as we let .hence does not violate the subsolution property .  in order to incorporate the jump condition at the jump time , note that we can rewrite the subsolution property for as where we set under the no arbitrage conditions given by, it follows that is an increasing function in both arguments and by continuity of ( see e.g. lemma 4.1.1 of ) , it follows that and similarly together with it follows that this shows that is a subsolution of the pde  as claimed .  finally ,the uniqueness result , follows from the uniqueness of the solution to the bsde , which in turn follows from the comparison theorem a.3 in .[ remark : pde1 ] since we are only concerned with before any default occurs , there is no need to keep track of the martingale terms s .these are only needed to realize that a default has happened .otherwise , it then follows that reduces to a function of only two variables , so that we can simply define in other words , in this case , the pde - becomes   [ remark : pde2 ] additionally , we will also employ the standard change of variables so that and .note that this change of variables allows us to get rid of boundary conditions at then , the pde together with the boundary condition becomes hence , we can express the whole pricing problem as a cauchy problem for a two - dimensional system of semilinear pdes : where the differential operators are defined by   it turns out that the pde - ( resp .- ) not only has a unique viscosity solution , but this solution is also a classical solution if we assume that and ( where defined ) have at most polynomial growth , i.e. , the classical argument of theorem 20.2.1 in assumes a bounded terminal condition , but we can employ a change of variables and divide by to utilize their framework : in this case , our pde - becomes :   we can then prove the following  [ prop : sol - exists ] assume that is piecewise continuously differentiable and as well as ( where defined ) have at most polynomial growth , i.e. , then the pde with terminal condition has a classical solution .assume first that is continuosly differentiable . using the transformation given above , it is then suffices to prove that - has a classical solution .the above transformation guarantees that both and are bounded .then the existence of a smooth ( and bounded ) solution to - follows from theorem 20.2.1 in . in the casethat is only piecewise smooth , the original proof can be modified following a similar procedure to . hence , using the change of variables , we conclude that there exists a classical solution to the pde with terminal condition .combining proposition [ prop : sol - exists ] with the uniqueness result from theorem [ thm : pde ] allows us to conclude that there exists a unique classical solution to the pde with polynomial growth .moreover , the hedging strategies can be easily found by an application of theorem [ thm : strat ] . using the relations in  we find that on the set  where we have used that the bond price on , by virtue of and the relations between the measures and given in .we perform a comparative statics analysis to analyze the dependence of xva and portfolio replicating strategies on funding rates , default intensities , and collateral levels .we consider the relative xva , i.e. express the adjustment as a percentage of the price of the claim , given by , where depending on whether we are considering buyer s or seller s xva .the claim is chosen to be a european - style call option on the stock security , i.e. .we consider one at - the - money option , with maturing at . in order to focus on the impact of funding costs ( which in practice is the most relevant ) and separate it from additional contributions to the xvacoming from asymmetries in collateral and repo rates , we set , and .as the derivatives contract does not only specify the price of the option but also the levels of collateralization of the deal , the no - arbitrage region appears as a ( two - dimensional ) band in and rather than as a ( one - dimensional ) interval in only .we also use the following other benchmark parameters : , , , , , , , , , and .we compute the numerical solution of the pde using a finite difference crank - nicholson scheme .the main finding of our analysis are discussed in the sequel :  [ [ higher - funding - rates - increase - the - width - of - the - no - arbitrage - band . ] ] higher funding rates increase the width of the no - arbitrage band .+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  figure [ fig : alpharfm ] displays the no - arbitrage band whose width is increasing in the funding rate . as gets higher , the band noticeably shrinks reaching its minimum around before widening again .notice that buyer s and seller s xva do not have a symmetric behavior .this can be better understood by analyzing the dependence of the band on the collateral level in figure [ fig : alpharfm ] .if is not too high ( ) , the widening of the no - arbitrage band with respect to the funding rate is due to decreasing buyer s xva . on the other hand ,if is high the buyer s xva is insensitive to changes in whereas the seller s xva increases with , contributing to widen the no - arbitrage band .this is further supported by the numerical values reported in table [ tab : tablealrfm ] . when , the position in the funding account for the seller s xva is long and the same regardless of the funding rate . on the other hand , the size of the long position for the buyer s xva increases in . in presence of full collateralization ,i.e. , the situation reverses . the position in the funding account for the buyers xva is short and stays constant with respect to .vice versa , for the seller s xva the size of the short position increases in size with respect to .if is high , the trader will have to post more collateral and consequently reduce the cash resources for his replicating strategy .he will then have to borrow more from the funding desk , resulting in higher funding costs .this drives up both the seller s xva and the number of shares of stocks and bonds needed for the replication strategy .   for different .the seller s lies above the buyer s xva and the same line style is used for both .top right : number of stock shares in the replication strategy .bottom left : number of trader bond shares in the replication strategy .bottom right : number of counterparty bond shares in the replication strategy .we plot the strategies for the portfolio replicating the seller s xva.,title="fig:",width=249 ]  for different .the seller s lies above the buyer s xva and the same line style is used for both .top right : number of stock shares in the replication strategy .bottom left : number of trader bond shares in the replication strategy .bottom right : number of counterparty bond shares in the replication strategy .we plot the strategies for the portfolio replicating the seller s xva.,title="fig:",width=249 ]  for different .the seller s lies above the buyer s xva and the same line style is used for both .top right : number of stock shares in the replication strategy .bottom left : number of trader bond shares in the replication strategy .bottom right : number of counterparty bond shares in the replication strategy .we plot the strategies for the portfolio replicating the seller s xva.,title="fig:",width=249 ]  for different .the seller s lies above the buyer s xva and the same line style is used for both .top right : number of stock shares in the replication strategy .bottom left : number of trader bond shares in the replication strategy .bottom right : number of counterparty bond shares in the replication strategy .we plot the strategies for the portfolio replicating the seller s xva.,title="fig:",width=249 ]  .the columns give the dollar position in the funding account corresponding to the replicating strategies of seller s xva and buyer s xva . [ cols="^,^,^,^",options="header " , ]   consistently with figure [ fig : alphahc ] , figure [ fig : hcalpha ] shows that the seller s xva decreases when the default intensity of the counterparty increases .this can be understood as follows . using the relations between the default intensities under the probability measures and given in eq ., we can see that increasing is equivalent to increasing the bond rate while keeping the default probability under constant .hence , the trader would earn higher premium from his long position in counterparty bonds ( see also bottom panels of figure [ fig : hcalpha ] ) .such a gain dominates over the funding costs incurred when replicating a larger closeout position ( eq .indicates that the closeout payment increases to the risk - free payoff as increases ) .altogether , this means that the funding costs of the investor would be reduced as increases .hence , the dependence of xva on counterparty s default intensity contrasts with its sensitivity to collateral levels numerically illustrated in figure [ fig : alpharfm ] .this is because higher default risk of the counterparty also means higher return on the bond underwritten by the counterparty , whereas higher only means that a larger value of the closeout payment needs to be replicated .   for different .top right : number of stock shares in the replication strategy .bottom left : number of trader bond shares in the replication strategy .bottom right : number of counterparty bond shares in the replication strategy .the replicating portfolio refers to the seller s xva.,title="fig:",width=249 ]  for different .top right : number of stock shares in the replication strategy .bottom left : number of trader bond shares in the replication strategy .bottom right : number of counterparty bond shares in the replication strategy .the replicating portfolio refers to the seller s xva.,title="fig:",width=249 ]  for different .top right : number of stock shares in the replication strategy .bottom left : number of trader bond shares in the replication strategy .bottom right : number of counterparty bond shares in the replication strategy .the replicating portfolio refers to the seller s xva.,title="fig:",width=249 ]  for different .top right : number of stock shares in the replication strategy .bottom left : number of trader bond shares in the replication strategy .bottom right : number of counterparty bond shares in the replication strategy .the replicating portfolio refers to the seller s xva.,title="fig:",width=249 ]we have developed a rigorous analysis of the semilinear pde associated with the bsde characterizing the price process of a portfolio replicating a european option , when funding , collateral and closeout costs are taken into account .we have shown the existence and uniqueness of a classical solution to the pde under mild assumptions on the coefficients . using this result , we have conducted a thorough numerical study analyzing the sensitivity of xva and of the claim s replication strategy to collateral levels , default risk and rates asymmetries .our findings support the introduction of centralized xva desks to manage and hedge all costs related to over - the - counter transactions .it shows that funding costs originating from the different trading components can not be easily separated and hence attributed to different business units ( cva , dva and fva desks ) because they are highly interdependent .recall the measurable function from remark [ remark : pde1 ] that was defined by the following theorem shows how the function can be used to compute the hedging strategies . noting that the law of is absolutely continuous , the proof of the theorem becomes analogous to the proof of theorem 4.1.4 in .hence , we omit it here and only give the statement of the theorem .a. pallavicini , d. perini , and d. brigo .funding , collateral and hedging : uncovering the mechanics and the subtleties of funding valuation adjustments .preprint available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2161528 .f. mercurio .bergman , piterbarg and beyond : pricing derivatives under collateralization and differential rates ._ actuarial sciences and quantitative finance _ springer proceedings in mathematics & statistics * 135 * 6595 2015 .

Abstract: we study the semilinear partial differential equation ( pde ) associated with the non - linear bsde characterizing buyer s and seller s xva in a framework that allows for asymmetries in funding , repo and collateral rates , as well as for early contract termination due to counterparty credit risk . we show the existence of a unique classical solution to the pde by first proving the existence and uniqueness of a viscosity solution and then its regularity . we use the uniqueness result to conduct a thorough numerical study illustrating how funding costs , repo rates , and counterparty risk contribute to determine the total valuation adjustment .


the numerical renormalization group ( nrg) is a powerful method that provides a highly systematic non - perturbative approach to the wide realm of so - called quantum impurity systems . these consist of an arbitrary small quantum system ( the _ impurity _ ) in contact with a macroscopic non - interacting usually fermionic bath .each part is simple to solve exactly on its own . in presence of interaction at the location of the impurity, however , the combination of both gives rise to strongly - correlated quantum - many - body phenomena . wilson s logarithmic coarse - graining of the bath leads to a semi - infinite chain with exponentially decaying couplings , which justifies the concept of energy scale separation .that is the wilson chain can be diagonalized iteratively by adding one site at a time and retaining the lowest only .the obvious question , however , is how many states should one keep on average for convergence in this procedure ? at a given iteration there is no quantitative _ a priori _ measure that indicates how many low - energy states are required for a proper description of the remaining low - energy physics .usually , the only way to check convergence within the nrg is by repeating the entire calculation and showing that the results no longer change when further increasing .therefore an nrg calculation is typically run somewhat blindly for some pre - determined .   by including and analyzing nrg iterations .the impurity ( dot ) is entirely contained in the first site , while the bath is coarse - grained and mapped onto the remaining semi - infinite tight - binding chain of sites . ]this somewhat uncontrolled truncation in the nrg is in stark contrast to the situation in the density matrix renormalization group ( dmrg ) . dmrg is based on a ( strictly ) variational principle , and as such has a clean well - defined truncation of the state space for part of the system through the discarded weight in its reduced density matrix . in contrast to the less suggestive plain number states kept , the discarded weight represents a reliable _ quantitative _ measure for the accuracy of a calculation . within the dmrg , be easily adjusted according to some predefined threshold in the discarded weight , instead .motivated by dmrg then , an approximate similar criterion can be established within the nrg as will be shown in the following .the analysis requires a slightly longer chain , as shown schematically in fig .[ fig : wilson - chain ] . withthe extra traced out again from the ground state space of the enlarged system , this allows to estimate the discarded weight .the latter offers a quantitative convergence measure that is specifically of interest for numerically expensive models such as multi - channel models , or models where the energy scale separation along the wilson chain might be in question due to modifications in the discretized hamiltonian . in either case, a small discarded weight provides a strong indication for converged nrg data .furthermore , the reduced density matrices generated for the evaluation of the discarded weight also allow a quite different analysis in terms of their _ dominant _ correlations .in particular , combining their entanglement spectra into _ entanglement flow diagrams _ , this offers a complementary view to the usual nrg energy flow diagram , which is entirely based on the analysis of the low - energy state space of a prior nrg run .the paper is thus organized as follows . in sec .i the essentials of the numerical renormalization group are revisited , including the construction of reduced density matrices . sec .ii then uses a specific set of reduced density matrices in the definition and analysis of the discarded weight within the nrg . sec .iii offers a complementary view on these reduced density matrices by analyzing their entanglement content in terms of entanglement spectra .iv , finally , summarizes and includes an outlook .   within the nrg ,the continuum of the non - interacting bath of half - bandwidth is logarithmically coarse - grained in energy space , followed by an exact mapping onto a semi - infinite so - called wilson - chain . the impurity space is coupled to the first site of this chain only , as depicted schematically in fig .[ fig : wilson - chain ] .the logarithmic coarse - graining is defined through the dimensionless discretization parameter . with the chemical potential at energy zero , the continuum of states in the energy intervals ] ( _ i.e._without the hat ) represents the space of matrix elements } ] and }_n ] in eq .( [ eq : backprop ] ) . through this operation ,the schmidt rank will rise , in general , by a factor of , with the state space dimension of a wilson site . repeating this process iteratively ,this allows to trace out the smallest energy shells in .thus with , this leads to the reduced density matrix , }_{0}}\xspace } &  \equiv   \bigl(\prod_{l = n+1 }  ^{n+{\ensuremath{n_{0}}\xspace } } \mathcal{\hat{p}}_l\bigr ) { \ensuremath{\hat{\rho}_{0,n+{\ensuremath{n_{0}}\xspace}}}\xspace } \nonumber\\ &  \equiv\sum_{ss^{\prime } } ^{{\ensuremath{m_{{\mathrm{k}}}}\xspace}}\rho_{ss^{\prime}}^{[n;{\ensuremath{n_{0}}\xspace } ] }   \vert s_{n}^{{\mathrm{k } } } \rangle \langle s^{\prime{\mathrm{k}}}_{n } \vert \text { , } \label{eq : rho_nn0}\end{aligned}\ ] ] which , by construction , is defined in the _ kept _ space of iteration .the schmidt rank will grow quickly , _i.e._exponentially , in this process , until after iterations , with \qquad ( { \ensuremath{n_{0}}\xspace}\ll n ) \text { , } \label{eq : def_n0}\ ] ] it reaches the full dimension the kept space .typically , is much smaller compared to the full length of the wilson chain considered , and conversely also specifies the initial number of nrg iterations in a forward direction that can be typically performed without truncation . for the definition of the discarded weight below , it is sufficient to stop the back - propagation of at this point .  the reduced density matrix }_{0} ] within the nrg eigenstates kept at iteration , } |s_n^{\prime{\mathrm{k}}}\rangle , \nonumber\\  \text{with}\quad & { \ensuremath{\hat{\rho}^{[n;{\ensuremath{n_{0}}\xspace}]}_{0}}\xspace}|r_{n;{\ensuremath{n_{0}}\xspace}}\rangle  = \rho_r^{[n;{\ensuremath{n_{0}}\xspace } ] } |r_{n;{\ensuremath{n_{0}}\xspace } } \rangle \text{,}\label{eq : rho - eig}\end{aligned}\ ] ] where the index shall refer to the eigenstates of the reduced density matrix , in contrast to the index for the energy eigenstates . here, the eigenvalue }_{r} ] .( i ) adhering to the energy eigenbasis of the nrg , the importance of the kept state eigenenergy the latter low - energy physics is given by the expectation value }_{s}}\xspace}\equiv { \ensuremath{\langle s_{n}^{{\mathrm{k}}}\vert}\xspace}{\ensuremath{\hat{\rho}^{[n;{\ensuremath{n_{0}}\xspace}]}_{0}}\xspace}{\ensuremath{\vert s_{n}^{{\mathrm{k}}}\rangle}\xspace}\text{,}\label{eq : rdiag}\end{aligned}\ ] ] _ i.e._the diagonal matrix elements } ] , while now their energies are given by the expectation values }_{r}}\xspace}\equiv \langle r_{n;{\ensuremath{n_{0}}\xspace}}| \hat{h}_n |r_{n;{\ensuremath{n_{0}}\xspace}}\rangle \text{.}\label{eq : reig}\end{aligned}\ ] ] both routes will be analyzed and compared in the following .however , the actual eigendecomposition of the reduced density matrices will be preferred for the remainder of the paper as explained .  in either case, a set of states with ( average ) energy is given together with their respective ( average ) weight that represents the states importance for latter iterations .for the first [ second ] route above this data is given by }_{s}}\xspace}) ] ] , respectively .given that the reduced density matrix }_{0}]as in eq .( [ eq : rdiag ] ) .data is shown only for those iterations where truncation occurred , with data from the same iteration shown in the same color .the two iterations with smallest ( largest ) energy range , ( ) , are highlighted in strong colors ( black diamonds ( red crosses ) ) , respectively , while light colors are used for all other iterations .the top [ right ] panel shows the energy [ weight ] distribution [ , eq .( [ nu : e ] ) [ eq .( [ nu : rho ] ) ] , respectively , for the data in the main panel , with matching energy [ weight axis .the binning referred to in the text to eqs .( [ nu : e ] ) and ( [ nu : rho ] ) is indicated by the intervals between the data points in the top and right panel . ]the resulting data then is clearly correlated .it is analyzed threefold , ( i ) in terms of the average distribution of the rescaled energies  [ nu : rho+e](ii ) the average distribution of the weights , and ( iii ) their average dependence on each other    [ eq : rhoe ]  here some appropriate linear ( logarithmic ) binning of the data is assumed with energy ( weight ) intervals ( ) , respectively . in particular , the densities in eqs .( [ nu : e ] ) and ( [ nu : rho ] ) are clearly dependent on these binning intervals , which therefore will properly indicated in the subsequent plots .the prime in the summation and the normalization indicates that only those iterations are included where state space truncation occurred , _i.e._typically .the total number of these iterations is given by .with chosen normalization then , the sum over the binned and data both yield the average number of kept states , while the integrated weight distribution in eq .( [ eq : def - rhoe ] ) is normalized to 1 , since .as will be seen later , the weight distribution typically shows a clear exponential decay with a characteristic exponent , as indicated already in eq .( [ eq : rhoe - decay ] ) , with the prefactor chosen such that it also preserves normalization .the correlation between the eigenenergies their corresponding weights }_{s}]clearly diminish exponentially with energy , which is intuitively expected as a consequence of energy scale separation within the nrg .the integrated weight distribution ( dashed black line , _ cf._eq .[ eq : def - rhoe ] ) , shows a clear exponential decay with an exponent . as seen in fig .[ fig : nrgtruncdd ] , this distribution clearly also serves as an upper bound of the weights }_{s} ] plotted in the main panel ( matching vertical axis ) .this distribution is peaked around the largest weights }_{s} ] in the main panel of fig .[ fig : nrgtruncdd ] show significant vertical spread , which translates into a pronounced tail towards exponentially smaller in the distribution in the right panel . for a given energy therefore , many of the states have _ order of magnitudes _ lower weight than the top - most weights close to in the main panel .this indicates that the energy representation with its corresponding diagonal weights }_{s}](fig .[ fig : nrgtruncdd ] ) , the eigenvalues }_{r}]do represent a clearly better choice for the analysis of accuracy or entanglement in the system ( fig . [ fig : nrgtruncee ] ) , and thus will be used henceforth .this prescription also shows a more systematic exponential decay all the way down to numerical double precision noise ( ) , with the decay rate of roughly independent of the discretization parameter .  for the siam [ eq .( [ eq : h - siam ] ) : ( same parameters as in fig .[ fig : nrgtruncdd ] ) , with .panel ( a ) shows the discarded weight defined in eq .( [ eq : nrg - epsd ] ) vs.  .the data is shown in solid lines , while the data based on the minimum eigenvalue of }_{0}]in eq . ( [ eq : rho_nn0 ] ) , using the combined data of eq .( [ eq : rho - eig ] ) and eq .( [ eq : reig ] ) . in terms of fig .[ fig : nrgtruncee ] , adding more states to the calculation essentially extends the data to larger energies and smaller weights , while the large - weight low - energy sector already remains widely intact .therefore the largest discarded weight , _ i.e._the weight missing by states not included and hence not available , can be estimated , to a good approximation , up to an overall prefactor by the smallest weights in the kept state space , which are easily accessible . given the exponential decay of the weights together with the residual spread in the data as seen in fig .[ fig : nrgtruncee ] , the discarded weight at given iteration can thus be defined through the _ average _ weights }_{r}]in the kept space ,  }_{r}}\xspace}\bigr\rangle _ { { \ensuremath{e^{[n;{\ensuremath{n_{0}}\xspace}]}_{r}}\xspace}\geq ( 1-\chi ) \max({\ensuremath{e^{[n;{\ensuremath{n_{0}}\xspace}]}_{r}}\xspace } ) } \text{.}\label{eq : epsknn0}\end{aligned}\ ] ]  the parameter is considered small , yet is chosen large enough ( typically ) to average over the residual spread of weights . alternatively and for comparison , an even simpler measure in terms of the minimum eigenvalue of }_{0} n ] , _ cf ._ eq . ( [ eq : epsmnn0 ] ) , will in general lie a ( constant ) few orders of magnitude lower , as it happens , for example , for the data in fig .[ fig : nrgtruncee ] . nevertheless , as will be shown in the following , up to an overall global prefactor the discarded weight based on either,  , both behave in an essentially similar fashion .the discarded weight in eq .( [ eq : nrg - epsd ] ) sensitively depends on the number states kept or the energy threshold . from fig .[ fig : nrgtruncee ] one expects a strongly diminishing discarded weight with increasing  , a quantitative analysis of which is presented in figs .[ fig : nrgtrunc_n ] and [ fig : nrgtrunc_e ] for the siam .figure [ fig : nrgtrunc_n ] analyzes the dependence of the discarded weight the number states kept . as seen in panel ( a ), the discarded weight decays with , with minor variations when a new wilson shell is fully included without truncation , _e.g._at . with panel (a ) being a log - log plot , the decay of the discarded weight with resembles a polynomial convergence , yet with very large power ( on the order of 10 ) .the reason for the slower than exponential decay is due to the strong increase in the density of states of the full many - body eigenspectrum with increasing as discussed with figs .[ fig : nrgtruncdd ] and [ fig : nrgtruncee ] .  together with the analysis of the discarded weight in fig .[ fig : nrgtrunc_n ] , an independent physical check for convergence is provided by the numerically computed conductance in units of shown in fig .[ fig : nrgtrunc_n](b ) .the conductance was calculated via the ( spin - resolved ) spectral function of the impurity level , with . herethe fermi function and the spectral function are evaluated at small but finite temperature , which is much smaller than the kondo temperature of for given parameter set and corresponds to the energy scale close to the end of the wilson chain , having and .expecting for the symmetric siam , the data in fig .[ fig : nrgtrunc_n ] indicates convergence for .the data for smaller not yet converged , and therefore ( strongly ) depends on numerical details , such as non - averaged z - shifts .  with constant , the energy of the topmost kept states can vary significantly with wilson shell , which directly also leads to a clear dependence of the discarded weight  .this is shown in the inset to panel ( a ) for the set of different values of in the main panels by the vertical dashed lines .the discarded weight clearly varies over more than three orders of magnitude within a single nrg run , irrespective of the actual .in particular , one can see that earlier iterations dominate the discarded weight for physical reasons . in the strong - coupling regime for ( with iteration corresponding to the energy scale of ) , the discarded weight is smallest , while for the intermediate free orbital or local moment regime for , these regimes require a _ larger _ number of states for comparable numerical accuracy from a physical point of view , indeed .  , yet for truncation with respect to fixed energy . for several values of by the vertical dashed lines in the main panels ,the distribution of the discarded weight along the wilson shell is shown in the inset to panel ( a ) . with to vary over a wider range , panel ( c ) shows the correlation of  , plotting average , minimum , and maximum of the wilson chain .for the average , data for different z - shifts is shown ( several lines on top of each other , with same color coding as in panel b ) . ]  given the underlying energy scale separation of the nrg , a straightforward way to obtain a more equally distributed is achieved using an energy cutoff , as demonstrated in fig .[ fig : nrgtrunc_e ] for exactly the same system as in fig .[ fig : nrgtrunc_n ] otherwise . for the values of by the vertical dashed lines in the main panels , the inset to fig .[ fig : nrgtrunc_e](a ) shows the distribution of . by construction , the discarded weight is , up to even - odd oscillations , clearly more uniformly distributed over the wilson shells as compared to the case of fixed in fig .[ fig : nrgtrunc_n](a ) .the discarded weight in panel ( a ) clearly diminishes exponentially with , yet with pronounced intermediate plateaus since the discrete eigenenergies within an nrg run are usually bunched around certain energies .the corresponding average function of , nevertheless , follows a rather smooth monotonic behavior , as shown in panel ( c ) .given fixed , however , clear variations of are seen within a given nrg run , hence also smallest and largest shown in panel ( c ) . ignoring iterations without truncation , in given example , typically the largest required at early iterations , while the smallest encountered in the strong coupling regime at late iterations .the calculated conductance shown in panel ( b ) converges clearly more uniformly with increasing compared to fig .[ fig : nrgtrunc_n](b ) .in particular , it indicates converged nrg data for , which corresponds to . therefore in both settings , for constant in fig .[ fig : nrgtrunc_n ] as well as for constant in fig .[ fig : nrgtrunc_e ] , convergence of the physical data is found for a similar discarded weight of with a negligible dependence on .this value therefore is considered a sufficient bound in accuracy to capture the main physics , with other quantities such as the nrg energy flow diagram already also well converged .  alternatively , using the plain minimum of the eigenvalues of the reduced density matrices in eq .( [ eq : epsmnn0 ] ) , this leads to convergence for .given that refers to the minimum eigenvalue in the kept space , consistently lies about three orders of magnitudes lower than and is considered a lower bound to the actual discarded weight . while fluctuates slightly more strongly compared to owing to the fact that it is not an averaged quantity such as , it nevertheless follows a similar consistent picture in terms of convergence with the number states kept or the energy for truncation . in this sense , either discarded weight , as well as , can be used quite generally as a quantitative measure , indeed , to demonstrate accuracy within the nrg . in order to avoid confusion ,however , it shall be made clear which one is used .the reduced density matrices }_{0}]are of interest , which do vary with a wider range depending on the underlying physics .hence , in the following , the actual entanglement spectra will be calculated with respect to the reduced density matrices }_{0}]for finite small , with in context .motivated by the discussion following eq .( [ eq : fdm - rho ] ) , the latter analysis can be linked to finite temperature settings .   in the sc regime ( ) .panel ( a)[b ] show the energy [ entanglement ] fixed point spectrum , respectively , _vs._spin symmetry quantum number .for all low - energy multiplets the underlying ( approximate ) degeneracy is indicated .the entanglement spectrum is shifted _w.r.t._to its lowest energy and scaled to match the energy fixed point spectrum in panel ( a ) . ]   in the sc regime ( ) .panel ( a)[b ] show the energy [ entanglement ] fixed point spectrum , respectively , _vs._spin symmetry quantum number .for all low - energy multiplets the underlying ( approximate ) degeneracy is indicated .the entanglement spectrum is shifted _w.r.t._to its lowest energy and scaled to match the energy fixed point spectrum in panel ( a ) . ]   the partitioning of the wilson chain into two parts , the chain up to and including site ( part a ) , and the traced out remainder of the system ( part b ) is generic . in particular , this allows to make use of the recently introduced entanglement spectra ( es ) for the physical characterization of a given wave function . herethese entanglement spectra provide a powerful tool for the systematic analysis of the physical correlations in the reduced density matrices }_{0}]in eq .( [ eq : rhon0 ] ) is computed and diagonalized , with its eigenspectrum mapped onto the entanglement spectrum in eq .( [ eq : es - def - logrho ] ) . collecting these spectra and plotting them _ vs. _ iteration index for even and odd iterations separately, the result will be referred to as _ entanglement flow diagram _ , in complete analogy to the standard energy flow diagrams of the nrg . for comparison ,also the truncated entanglement spectra for finite small be analyzed , which in their combination will be referred to as _ truncated _ entanglement flow diagram . in either case , the entanglement spectra are obtained in a _ backward _ sweep , purely based on the iterative low - energy hilbert - space decomposition of a prior nrg run in terms of the a - tensors in eq .( [ eq : atensor ] ) .this is in contrast to the energy flow diagram , which is calculated with increasing shell index in a _ forward _ sweep making explicit reference to the hamiltonian .   at the even iteration ( similar analysis as in fig .[ fig : espec : scs ] otherwise ) . ]   at the even iteration ( similar analysis as in fig .[ fig : espec : scs ] otherwise ) . ]the entanglement spectra were calculated for the symmetric siam in the absence of magnetic field .the resulting entanglement flow diagram is presented in fig .[ fig : entangle - spec ] together with a direct comparison to the standard nrg energy flow diagram .the data is plotted for even ( odd ) wilson shells in the upper ( lower ) panels , respectively .the energy flow diagram , shown in the left panels , clearly distinguish the well - known physical regimes of the siam , namely the free orbital regime ( fo ; ) , the local moment regime ( lm ; ) , and the strong coupling regime ( sc ; ) , where corresponds to the energy scale of the kondo temperature , having .all degeneracies for are explicitly specified in fig .[ fig : entangle - spec ] . in particular , for even iterations ,the ground state is unique throughout , _e.g._the kondo singlet for ( panel a ) , while for odd - iterations the ground state space at small energies is four - fold degenerate due to the particle - hole symmetric parameter set ( panel c ) .  interestingly , a very similar picture emerges from the entanglement flow diagram in right panels , fig .[ fig : entangle - spec](b+d ) ( black lines ) . for comparison ,also the truncated entanglement spectra are shown using ( orange ( gray ) lines ) , which in given case converge rapidly , in fact exponentially , with increasing towards the actual entanglement flow diagram .the latter then mimic the energy flow diagram in the left panels over a wide range .for example , the convergence towards the kondo fixed point occurs around similar iterations , and even the degeneracies of the lowest states of the energy flow diagram are exactly recovered by the entanglement spectra .the latter also holds on the symmetry - resolved level , as demonstrated in fig .[ fig : espec : scs ] for the even iteration ( see later discussion ) . nevertheless , looking more closely , a few notable qualitative differences of the entanglement flow diagrams in right panels of fig .[ fig : entangle - spec ] are seen compared to the energy flow diagrams in the left panels .overall , the entanglement flow diagrams appear shifted by about 5 iterations to larger energies .this can be understood , considering that the entanglement spectra are calculated for enlarged systems together with the rapid convergence with increasing in given case .however , there are further pronounced differences with the energy flow diagram for the earliest iterations in the fo regime , .these latter differences in the entanglement flow diagram can be significantly enhanced by turning on a magnetic field on the order of the kondo temperature , as shown in fig .[ fig : entangle - specb ] for .this corresponds to the energy scale at iteration , given .the magnetic field has been chosen such , that for late iterations the fixed point spectrum for even and odd iterations become essentially the same ( compare the low - energy fixed point spectra in panels a(b ) to c(d ) , respectively ) . due to the magnetic field , the kondo singlet ( previously the unique state at even iterations )is largely destroyed for with a sizeable magnetization at the impurity .clearly , the nrg eigenbasis at early iterations does not yet know about the small energy physics to come ( _ e.g._the small applied in given case ) .therefore the energy flow diagram essentially remains unaltered there , when compared to the case without magnetic field in fig .[ fig : entangle - spec](a+c ) .the flow changes strongly only starting from the energy scale of the magnetic field value , _i.e._for where it moves into a different fixed - point spectrum .in particular , there also emerges a unique state now in the energy flow diagram for odd iterations for , _ i.e._the symmetry broken spinful state favored by the magnetic field . by including magnetic field , the entanglement flow diagram shows pronounced differences from the energy flow diagram for , which includes large portions of the lm regime . while the energy spectrum up to and including site is ignorant of the low - energy physics to come , this very low - energy physics is captured by the reduced density matrices and thus reflected in the entanglement flow diagram . at the impurity at small temperatures in the presence of a magnetic field redistributes significant spectral weight at _ large _ energies , which accounts for the breakup of the kondo singlet . within the nrg ,this thus translates into a _ feedback _ from small to large energies , which is captured correctly only after including the reduced density matrices for the remainder of the system . yet contributions from all nrg shells are required to cover the full spectral range of dynamical correlation functions . a clean combined prescription for this was finally provided by the fdm - nrg approach based on complete basis sets . ]  consider the entanglement spectra derived from the overall ground state ( black lines ) in fig .[ fig : entangle - specb](b+d ) . in panel( d ) the ground state remains unique throughout , _ i.e._remembers the symmetry broken magnetic state , determined at much lower energy scales , all the way up to the largest energies . within the split - up lowest energy space with subsequent degeneracies [ 1 - 2 - 1 ] in panel ( d ) for ( to be called [ 1 - 2 - 1 ] configuration ) ,the first and second excited states cross each other with decreasing leading to a [ 1 - 1 - 2 ] configuration for small , _ i.e._large energies . nevertheless , the singly degenerate excited state clearly remains split - off , and does not merge with the ground state , which is in strong contrast to the energy flow diagram in panel ( c ) with a [ 2 - 2 ] configuration for .this degeneracy in the ground state space that is ignorant of the small magnetic field is partly reflected only in the _ truncated _ entanglement flow diagram . using small (orange ( gray ) lines in panelsd ) , this eventually also misses the low energy physics .therefore these spectra in panel ( d ) eventually are also in a [ 2 - 2 ] configuration for the smallest , with a more irregular transient behavior with increasing .a similar trend is also observed for even iterations in panels ( a+b ) .while the ground state remains unique for all iterations in both panels , the entanglement flow in panel ( b ) tends to split off the excited levels right above the lowest [ 1 - 2 - 1 ] state space configuration for small . for the truncated entanglement flow , on the other hand ,the lines of these excited levels remain entangled with higher excitations , which is similar to the situation in the energy flow diagram in panel ( a ) .nevertheless , the low - energy fixed - point spectra for again agree well for both the energy and entanglement flow diagram in fig .[ fig : entangle - specb ] , which again also holds for the symmetry - resolved spectra , as demonstrated for the even iteration in fig .[ fig : espec : sca ] .this agreement in the spectra of the stable low - energy fixed point , present in both the non - magnetic as well as the magnetic case , is understood as a generic feature . there both , the energy eigenstates as well as the reduced density matrices are deeply rooted in the low - energy physics , _i.e._of the overall ground state of the system at , and hence present a consistent description of the system .the detailed structure of the energy fixed point spectra provides clear physical information . this includes , for example , phase shifts if a fermi - liquid point of view is supported as is the case for the siam .this then directly explains all of the splittings and degeneracies in the low energy sector of the energy fixed point spectra .for example , consider the energy spectrum in fig .[ fig : espec : scs](a ) for the fully symmetric siam in the non - magnetic case . note that while spin - resolved spectra are shown in fig .[ fig : espec : scs ] , in given case the charge - resolved spectra would look exactly the same due to particle - hole symmetry . with the spectra shown for an even iteration , the ground state is unique , _ i.e._represents the kondo singlet with .the first excited states for , correspond to an extra particle with spin - up or a hole with spin - down . given particle - hole symmetry ,both processes have the same energy ( in rescaled energy units ) , and hence are two - fold degenerate , indicated by the number on top of the level in fig . [ fig : espec : scs ] . by symmetry ,the same excitations exist for , leading to the [ 2 - 2 ] degeneracy ( 4 states ) in the lowest excitations in fig . [fig : espec : scs](a ) .the next higher excitation combines two of above processes .this leads to a total of 6 excitations , all with energy and distributed over . heretwo of the excitations at correspond to the extraction or annihilation of two particles with opposite spin .this fully explains the [ 1 - 4 - 1 ] degeneracy of the excited states at energy in fig .[ fig : espec : scs](a ) , and also the combined 6-fold degeneracy seen in the energy flow diagram seen at this energy in fig .[ fig : entangle - spec](a ) .the argument can be continued along similar lines to explain the [ 4 - 4 ] ( 8 states ) and [ 4 - 9 - 4 ] ( 17 states ) degenerate subspaces of the next higher excitations . excitations with even higher energy eventually have missing levels due to nrg truncation .the same analysis as for the energy spectra , however , can not be applied with equal rigor to the entanglement spectra . while the ground state [ 1 ] and the lowest [ 2 - 2 ] and [ 1 - 4 - 1 ] excitations in fig .[ fig : espec : scs](b ) fully agree in symmetries , degeneracy and also in the precise relative level spacing , the next higher [ 4 - 4 ] excitation in panel ( a ) is broken up in fig .[ fig : espec : scs](b ) , with some of the levels shifting to higher entanglement energy. nevertheless , the degenerate set [ 2 - 10 - 10 - 2 ] further up in energy still again equally appears for both , energy and entanglement spectra .  the same analysis as in fig .[ fig : espec : scs ] , is repeated for the magnetic case in fig .[ fig : espec : sca ] for the same even iteration . despite the rather different level spectrum for large in the flow diagram in fig .[ fig : entangle - specb ] , the actual spin - resolved fixed point spectrum is qualitatively very similar to the non - magnetic case in fig .[ fig : espec : scs ] . aside an overall tilt of the level structure , all degeneracies and level positions of the lower part of the energy spectrum in panel ( a ) are again fully described by elementary single - particle excitations .the underlying reason for this similarity of the fixed points spectra in the magnetic and non - magnetic case is that , apart from the ( screened ) impurity spin , the system is well described by an effective fermi - liquid picture . with the low energy fixed point spectra well reflected in the entanglement spectra , a similar tilt in the level structureis also observed in fig .[ fig : espec : sca](b ) when compared to fig .[ fig : espec : scs](b ) . note , for example , that to the lower left of the spectrum the same [ 1 - 2 - 1 ] , as well as the [ 2 - 4 - 2 , 2 - 4 - 2 ] state sequence with increasing energy is seen .the reduced density matrices of the nrg by tracing out the low - energy sector have been analyzed in detail .the low end of their eigenspectra was used to estimate the discarded weight in eqs .( [ eq : epsxnn0]-[eq : nrg - epsd ] ) as a quantitative and site - resolved measure of the accuracy within the nrg . while , in principle , the same reduced density matrices could also be utilized as the basis for an altered truncation criteria similar to the dmrg , this, however , requires sufficiently large start with . in practice , this is sufficiently close to a truncation with respect to an energy cutoff . either way , all of this can be easily and quickly checked using the proposed analysis in terms of the discarded weight which provides a useful quantitative tool .furthermore , the dominant correlations of the reduced density matrices were analyzed in terms of their entanglement spectra . due to the nrg flow towards small energy scales ,these spectra can be combined into entanglement flow diagrams . theredifferent physical regimes can be identified similar to the standard nrg energy flow diagrams .considering that the entanglement spectra are obtained solely based on the wave function , the agreement of the low - energy fixed point spectra are stunning . a possible larger disagreement at higher energies , _ i.e._for earlier wilson shells , on the other hand , depends on the specific physical situation . given the nrg background , as an outlook this appears to suggest the following .for all energy shells ( iterations ) where the entanglement spectrum is quantitatively comparable to the nrg energy spectrum for the lowest set of states , the reduced density matrices themselves are not crucially important in the description of the system . instead , they may be replaced by thermal density matrices in the nrg eigenbasis . in a sense , by tracing out the low - energy sector , the resulting reduced density matrices maintain an approximate thermal character , with implications to thermalization at a given energy shell . for energy shells with a qualitative difference between the energy and entanglement spectra , however , the reduced density matrices are crucially important to capture the correct physics in the nrg calculation that explicitly uses data from such energy shells .a detailed analysis of the deeper connection and the explicit differences between the energy and the entanglement spectra appears interesting , yet is out of the scope of this paper . in particular, it also appears instructive to analyze the entanglement spectra for non - fermi liquid systems such as the symmetric two - channel kondo model , as the analysis presented in this paper suggests a strong physical connection of the entanglement spectra to the underlying physics .i want to thank jan von delft for a critical review of the script , and also ronny thomale for helpful comments on entanglement spectra .this work has received support from the german science foundation ( dfg : tr-12 , sfb631 , nim , and we4819/1 - 1 ) .james demmel , jack dongarra , axel ruhe , and henk van der vorst . _ templates for the solution of algebraic eigenvalue problems : a practical guide_. society for industrial and applied mathematics , philadelphia , pa , usa , 2000 .isbn 0 - 89871 - 471 - 0 .

Abstract: a quantitative criterion to prove and analyze convergence within the numerical renormalization group ( nrg ) is introduced . by tracing out a few further nrg shells , the resulting reduced density matrices carry relevant information on numerical accuracy as well as entanglement . their spectra can be analyzed twofold . the smallest eigenvalues provide a sensitive estimate of how much weight is discarded in the low energy description of latter iterations . as such , the discarded weight indicates in a site - specific manner whether sufficiently many states have been kept in a single nrg run . the largest eigenvalues of the reduced density matrices , on the other hand , lend themselves to a straightforward analysis in terms of entanglement spectra , which can be combined into entanglement flow diagrams . the latter show strong similarities with the well - known standard energy flow diagram of the nrg , supporting the prevalent usage of entanglement spectra to characterize different physical regimes .