Quantum Quenches in Extended Systems
arXiv:0704.1880v2  [cond-mat.stat-mech]  19 Apr 2007
Quantum Quenches in Extended Systems
Pasquale Calabrese1 and John Cardy2
Dipartimento di Fisica dell’Università di Pisa and INFN, Pisa, Italy
Institute for Theoretical Physics, University of Amsterdam, 1018 XE Amsterdam, The Netherlands. and
Oxford University, Rudolf Peierls Centre for Theoretical Physics,
Oxford, United Kingdom
All Souls College, Oxford, United Kingdom
(Dated: October 23, 2018)
We study in general the time-evolution of correlation functions in a extended quantum system after
the quench of a parameter in the hamiltonian. We show that correlation functions in d dimensions
can be extracted using methods of boundary critical phenomena in d + 1 dimensions. For d = 1
this allows to use the powerful tools of conformal field theory in the case of critical evolution.
Several results are obtained in generic dimension in the gaussian (mean-field) approximation. These
predictions are checked against the real-time evolution of some solvable models that allows also to
understand which features are valid beyond the critical evolution.
All our findings may be explained in terms of a picture generally valid, whereby quasiparticles,
entangled over regions of the order of the correlation length in the initial state, then propagate
with a finite speed through the system. Furthermore we show that the long-time results can be
interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible
future developments.
I. INTRODUCTION
Suppose that an extended quantum system in d dimensions (for example a quantum spin system), is prepared at
time t = 0 in a pure state |ψ0〉 which is the ground state of some hamiltonian H0 (or, more generally, in a thermal
state at a temperature less than the mass gap to the first excited state). For times t > 0 the system evolves unitarily
according to the dynamics given by a different hamiltonian H , which may be related to H0 by varying a parameter
such as an external field. This variation, or quench, is supposed to be carried out over a time scale much less than
the inverse mass gap. How does the state |ψ(t)〉 = e−iHt|ψ0〉 evolve? For a finite number of degrees of freedom,
the system generically shows a periodic (or quasiperiodic) behavior with a period that typically increases when the
number of degrees of freedom grows. This is the well-known phenomenon of quantum recurrence. However, in the
thermodynamic limit this is no longer necessarily the case, and the natural question arises as to whether the system
(or, rather, a macroscopically large subsystem) reaches a stationary state for very large times. To attack this question
we consider the simpler question of how the correlation functions, expectation values of products of local observables,
evolve and whether they reach constant values for large times.
This problem has its own theoretical interest, being a first step towards the understanding of equilibration in
quantum systems. On the same fundamental level we can also ask whether the approach can provide a new tool for
the characterization of the collective excitations in strongly correlated systems. However, until recently it has been
considered a largely academic question, because the time scales over which most condensed matter systems can evolve
coherently without coupling to the local environment are far too short, and the effects of dissipation and noise are
inescapable. Recent developments of experimental tools for studying the behavior of ultra-cold atoms have revised
completely this negative attitude. In fact, thanks to the phenomenon of Feshbach resonance it is possible to tune the
coupling of an interacting system to essentially any value and in extremely short times. In addition, the coupling to
dissipative degrees of freedom can be much weaker than in ordinary condensed matter systems. This has allowed the
observation, among other things, of the collapse and revival of a Bose-Einstein condensate [1] and of the quenching
of a spinor condensate [2]. Furthermore, using highly anisotropic optical lattices, it has been possible to build and
study essentially one-dimensional systems [3], and, of direct interest for this paper, the coherent non-equilibrium
dynamics of these integrable models has been measured [4]. (Other interesting experiments are considered in Refs.
[5].) These striking experimental results have largely motivated the development of new numerical methods to study
non-equilibrium dynamics, the most successful one being the time-dependent density matrix renormalization group
(DMRG) [6].
On the purely theoretical side these kind of questions were first considered (as far as we are aware) in the seventies
in the context of the quantum Ising-XY model in Refs. [7, 8] (see also [9–14] for recent developments). However, in the
very last few years, after the previously mentioned experimental progress, the study of quantum quenches has been
pursued in a systematic manner and a large number of results is nowadays available. To quote only a few examples, the
models considered include several realizations of one-dimensional (1D) Bose gases [15–25], Luttinger liquids [26, 27],
coupled 1D condensates [28], strongly correlated 1D fermions [29], and mean field fermion condensates [30, 31]. A
http://arxiv.org/abs/0704.1880v2
closely related topic, that will not be considered here, concerns the formation of defects when crossing a critical point
with changing the external parameter at a fixed rate, i.e. the so-called quantum Kibble-Zurek mechanism [32].
Most of these papers concern the exact or approximate solutions of very specific models. These are often strongly
relevant, since in many cases they can be directly compared with existing experimental results or give very accurate
predictions for future investigations. However, from the study of specific models it is difficult to draw general con-
clusions on the physics of quantum quenches. There have been at least two notable exceptions. In Ref. [15] it was
conjectured that the asymptotic state at very large times can be described by a generalized Gibbs ensemble (for more
details we refer the reader to section VI where this conjecture will be explicitly discussed in relation to some of our
findings). In Ref. [33] we studied the time evolution after a quench in general, exploiting the path integral approach
and the well-known mapping of the quantum problem to a classical one in d + 1 dimensions. The translational in-
variance in the (imaginary) time direction is explicitly broken and thus the initial state plays the role of a boundary
condition. This limits the range of applicability of field-theoretical methods, but when the hamiltonian H is at or
close to a quantum critical point we could use the renormalization group (RG) theory of boundary critical behavior
(see, e.g., [34]). From this point of view, particularly powerful analytic results are available for d = 1 because then
the 1 + 1-dimensional problem is described asymptotically by a boundary conformal field theory (BCFT)[35, 36].
The aim of this paper is twofold: on the one hand we give a detailed description of the methods and the results
reported briefly in Ref. [33], and on the other we generalize these results and give some new physical insights. Several
results, in fact, appear here for the first time. The paper is organized as follows. In Sec. II we introduce the path-
integral formalism that is applied in the following section III to one-dimensional critical systems by means of CFT. In
Sec. IV we consider two simple 1D models whose non-equilibrium dynamics is exactly solvable to check the correctness
of the previous results and to extend our findings to gapped and lattice systems. In Sec. V we generalize the method
to higher dimensions. In the last section VI we analyze all our results and explain most of the general findings in
quantum quenches by means of simple physical arguments. We broadly discuss some open questions and possible
future developments.
II. PATH INTEGRAL FORMULATION AND SURFACE CRITICALITY
Let us consider a lattice quantum theory in d space dimensions. The lattice spacing is a, and the lattice variables
are labelled by a discrete vector variable r. Time is considered to be continuous. The dynamics of the theory is
described by the hamiltonian H . Suppose we prepare this system in a state |ψ0〉 that is not an eigenstate of H and
unitarily evolve it according to H . The expectation value of a local operator O({ri}) at time t is
〈O(t, {ri})〉 = 〈ψ0|eiHtO({ri})e−iHt|ψ0〉 . (1)
We modify this time-dependent expectation value as
〈O(t, {ri})〉 = Z−1〈ψ0|eiHt−ǫHO({ri})e−iHt−ǫH |ψ0〉 , (2)
where we have included damping factors e−ǫH in such a way as to make the path integral representation of the
expectation value absolutely convergent. The normalization factor Z = 〈ψ0|e−2ǫH |ψ0〉 ensures that the expectation
value of the identity is one. At the end of the calculation we shall set ǫ to zero.
Eq. (2) may be represented by an analytically continued path integral in imaginary time over the field variables
φ(τ, r), with initial and final values weighted by the matrix elements with |ψ0〉:
[dφ(τ, r)]〈ψ0|φ(τ2, r)〉〈φ(τ1 , r)|ψ0〉 e−
L[φ]dτ
where
L[φ]dτ is the (euclidean) action. The operator O is inserted at τ = 0, and the width of the slab is 2ǫ. τ1
and τ2 should be considered as real numbers during the calculation, and only at the end should they be continued to
their effective values ±ǫ− it. In this way we have reduced the real-time non-equilibrium evolution of a d dimensional
systems to the thermodynamics of a d + 1 field theory in a slab geometry with the initial state |ψ0〉 playing the role
of boundary condition at both the borders of the slab. The validity of our results relies on the technical assumption
that the leading asymptotic behavior given by field theory, which applies to the Euclidean region (large imaginary
times), may simply be analytically continued to find the behavior at large real time.
Eq. (2) can be in principle used to describe the time evolution in any theory and for all possible initial conditions.
Unfortunately, in confined geometries only a few field theories with very specific boundary conditions can be solved
analytically in such a way to have results that can be continued from real to complex values. Some examples of these
will be given in Sec. V. (An interesting case concerns integrable massive boundary field theories as for example the
recent application to the dynamics of coupled 1D condensates in Ref. [28].) However, the treatment greatly simplifies
if a system is at or close to a (quantum) phase transition. In fact, in this case, we can use the powerful tools of
Renormalization Group (RG) theory of boundary critical phenomena (see, e.g. [34]). For example, in the case of a
scalar order parameter (corresponding to the Ising universality class), the slab geometry of above is usually described
by the action [34]
S(φ) =
(∂φ)2 +
ddr c[φ(τ = τ1, r) + φ(τ = τ2, r)] , (4)
where m2 measures the distance from criticality, g > 0 ensures the stability in the broken-symmetry phase, and c
denotes the surface enhancement, i.e. the difference of the interactions on the surface with respect to the bulk. The
value of c depends on the boundary conditions. For example, the choice c = +∞ forces the field to vanish on the
boundary, thus corresponding to Dirichlet boundary conditions. On the other hand c = −∞ forces the field to diverge
at the boundary. Finite values of c correspond to intermediate boundary conditions. According to the RG theory,
the different boundary universality classes are characterized by the fixed point of c. A complete RG analysis (see,
e.g.,[34]) shows that the possible fixed points of c are c∗ = ∞, 0,−∞, with c∗ = ±∞ stable and c∗ = 0 unstable. Any
other value of c flows under RG transformations to ±∞. These universality classes are called ordinary (c∗ = ∞),
special (c∗ = 0), and extraordinary (c∗ = −∞) [34].
Thus, for the purpose of extracting the asymptotic behavior, as long as |ψ0〉 is translationally invariant, we may
replace it by the appropriate RG-invariant boundary state |ψ∗0〉 to which it flows. The difference may be taken into
account, to leading order, by assuming that the RG-invariant boundary conditions are not imposed at τ = τ1 and
τ2 but at τ = τ1 − τ0 and τ = τ2 + τ0. In the language of boundary critical behavior, τ0 is called the extrapolation
length [34]. It characterizes the RG distance of the actual boundary state from the RG-invariant one. It is always
necessary because scale-invariant boundary states are not in fact normalizable[36]. It is expected to be of the order of
the typical time-scale of the dynamics near the ground state of H0, that is the inverse gap m
0 . (This can be checked
explicitly for a free field theory, see later). The effect of introducing τ0 is simply to replace ǫ by ǫ + τ0. The limit
ǫ→ 0+ can now safely be taken, so the width of the slab is then taken to be 2τ0. For simplicity in the calculations, in
the following we will consider the equivalent slab geometry between τ = 0 and τ = 2τ0 with the operator O inserted
at τ = τ0 + it. This is illustrated in the left part of Fig. 1. In fact, in this geometry we can consider products of
operators at different times tj , by analytically continuing their labels to τ0 + itj .
Let us discuss to what initial conditions the fixed points correspond. In the case of free boundary conditions, when
the order parameter is unconstrained, the continuum limit forces it to vanish there, so it corresponds to the ordinary
transition (c = ∞). For a lattice model, this corresponds to a completely disordered initial state (e.g. for the Ising
chain in a transverse field corresponds to infinite field). In contrast for non-vanishing fixed boundary conditions (e.g.
+ or − for Ising-like systems), the continuum limit makes the order parameter to diverge at the boundary, thus
corresponding to the extraordinary transition (c = −∞).
III. ONE SPACE DIMENSION AND CONFORMAL FIELD THEORY
In this section we specialize the methods just introduced to the case when H is at a quantum critical point whose
long-distance behavior is given by a 1+1-dimensional CFT, with dispersion relation Ωk = v|k|. We set v = 1 in the
following. RG-invariant boundary conditions then correspond to conformally invariant boundary states. In this case
the correlation functions are accessible through the powerful tools of boundary CFT.
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w w z
FIG. 1: Left: Space-imaginary time region in (2). Imwi = τ , that will be analytically continued to τ → τ0 + it. Right:
Conformal mapping of the left geometry to the upper half-plane (c.f. Eq. (5)). Note that argzi = θ = πτ/2τ0.
The main property we will repeatedly use in the following is the relation of correlation functions of (primary)
operators among two geometries connected by a conformal transformation. For example, the slab geometry of above
is just a two-dimensional strip whose points are labelled by a complex number w = r + iτ with 0 < Imw < 2τ0. The
strip can be obtained from the upper half-plane (UHP) Imz > 0 by the conformal mapping
w(z) =
log z , (5)
with the images of points at the same imaginary time on the strip lying along argzi = θ = πτ/2τ0. In the case where
O is a product of local primary scalar operators Φi(wi), the expectation value in the strip is related to the one in the
UHP by the standard transformation
Φi(wi)〉strip =
|w′(zi)|−xi〈
Φi(zi)〉UHP , (6)
where xi is the bulk scaling dimension of Φi. Note that the (eventual) expectation values of the Φi in the ground state
of H are supposed to have been subtracted off. The asymptotic real time dependence is obtained via the analytic
continuation τ → τ0 + it, and taking the limit t, rij ≫ τ0.
In the following subsections we apply these methods to some specific cases.
A. The one-point function
In the UHP, the one-point function of a scalar primary field with bulk scaling dimension x is 〈Φ(z)〉UHP =
AΦb [2Im(z)]
−x, as a simple consequence of scaling invariance. The normalization factor AΦb is a non-universal ampli-
tude. In CFT the normalizations are chosen in such a way that the bulk two-point functions have unit amplitude
(i.e. 〈Φ(z1)Φ(z2)〉bulk = |z2 − z1|−2x). This choice fixes unambiguously the amplitude AΦb that turns out to depend
both on the considered field Φ and on the boundary condition on the real axis b. It vanishes if Φ corresponds to an
operator whose expectation value in |ψ0〉 vanishes, and thus 〈Φ(t)〉 = 0, for all times.
When the primary field is not vanishing on the boundary, performing the conformal mapping (5) we obtain
〈Φ(w)〉strip = |w′(z)|−x〈Φ(z(w))〉UHP = AΦb
sin(πτ/(2τ0))
that continued to real time τ = τ0 + it gives
〈Φ(t)〉 = AΦb
cosh(πt/(2τ0))
≃ AΦb
e−xπt/2τ0 . (8)
Thus the order parameter (and any other observables described by a primary field) decays exponentially in time to
the ground-state value, with a non-universal relaxation time tOrel = 2τ0/xOπ. The ratio of the relaxation times of two
different observables equals the inverse of the ratio of their scaling dimensions and it is then universal.
The normalization factor AΦb is known for the simplest boundary universality classes [37]. In the case of Φ being
the order parameter and the boundary condition is fixed (ψ0(x) = ∞) AΦb is 1 for the free boson and 21/4 for the
Ising model.
An important exception to this law is the local energy density (or any piece thereof). This corresponds to the tt
component of the energy-momentum tensor Tµν . In CFT this is not a primary operator. Indeed, if it is normalized
so that 〈Tµν〉UHP = 0, in the strip [38] 〈Ttt(r, τ)〉 = πc/24(2τ0)2 (where c is the central charge of the CFT) so that
it does not decay in time. Of course this is to be expected since the dynamics conserves energy. A similar feature is
expected to hold for other local densities corresponding to globally conserved quantities which commute with H , for
example the total spin in isotropic models.
B. The two-point function
In the case of the one-point function, scaling invariance was enough to fix the functional dependence on the position
in the UHP. However, the form of the two-point function depends explicitly on the boundary universality class and
on the operator considered. In the following subsections we will consider the equal-time correlation function for the
order parameter in the gaussian and in the Ising universality classes that are easily treated in full generality. At the
quantum level they describe (among the other things) a chain of harmonic oscillators (explicitly considered in Sec.
IVA) and the Ising model in a transverse field (whose real time evolution has been considered in Refs. [7–9, 11] and
is briefly reviewed in Sec. IVB). Finally we will discuss the general form of the two-point function for asymptotically
large time and distance, that can be obtained from general CFT arguments.
1. The gaussian model
The content of this subsection has been already reported in Ref. [39], during the study of the time evolution of the
entanglement entropy, that transforms like the two-point function of a primary field in a boundary gaussian theory
[40]. We report it here for sake of completeness.
For a free boson the two-point function in the UHP is [35]
〈Φ(z1)Φ(z2)〉UHP =
z12̄z21̄
z12z1̄2̄z11̄z22̄
, (9)
with zij = |zi − zj | and zk̄ = zk. Note that Φ is not the gaussian field θ(z), but its exponential Φ(z) = eiθ(z) (see Sec.
IVA).
Under the conformal mapping (5) we obtain the two-point function on the strip at imaginary time τ , at distance r
apart
〈Φ(r, τ)Φ(0, τ)〉strip = |w′(z1)|−x|w′(z2)|−x〈Φ(z1(w))〈Φ(z2(w))〉UHP =
cosh(πr/2τ0)− cos(πτ/τ0)
8 sinh2(πr/4τ0) sin
2(πτ/2τ0)
, (10)
that continued to real time τ = τ0 + it gives
〈Φ(r, t)Φ(0, t)〉 =
cosh(πr/2τ0) + cosh(πt/τ0)
8 sinh2(πr/4τ0) cosh
2(πt/2τ0)
. (11)
In the case where r/τ0 and t/τ0 are large this simplifies to
〈Φ(r, t)Φ(0, t)〉 = (π/2τ0)2x
eπr/2τ0 + eπt/τ0
eπr/2τ0 · eπt/τ0
e−xπt/τ0 for t < r/2
e−xπr/2τ0 for t > r/2
. (12)
i.e. the two point function at fixed r decays exponentially in time up to t∗ = r/2 and then saturates to a value that
depends exponentially on the separation.
In the case of fixed initial condition, with one-point function given by Eq. (8), the connected correlation function is
〈Φ(r, t)Φ(0, t)〉conn = 〈Φ(r, t)Φ(0, t)〉 − 〈Φ(0, t)〉2 ∝
0 for t < r/2 ,
e−xπr/2τ0 − e−xπt/τ0 for t > r/2 ,
i.e. correlations start developing at t∗ = r/2 and, being t ≫ τ0, at t∗ the connected two-point function almost
immediately jumps to its asymptotic value. In the case of disordered initial conditions (ψ0(r) = 0), connected and
full correlation functions are equal.
2. The Ising universality class
For the Ising model the two-point function in the UHP is [35]
〈Φ(z1)Φ(z2)〉UHP =
z12̄z21̄
z12z1̄2̄z11̄z22̄
F (η) , (14)
where F (η) is given by
F (η) =
1 + η1/2 ±
1− η1/2√
, (15)
and η is the four-point ratio
z11̄z22̄
z12̄z21̄
. (16)
The sign ± depends on the boundary conditions. + corresponds to fixed boundary conditions and and − to disordered
ones.
The only difference with respect to the gaussian case is that we have also to map F (η) according to the conformal
transformation (5). After simple algebra we have
2 sin2(πτ/2τ0)
cosh(πr/2τ0)− cos(πτ/τ0)
, (17)
and so
〈Φ(r, τ)Φ(0, τ)〉strip =
cosh(πr/2τ0)− cos(πτ/τ0)
8 sinh2(πr/4τ0) sin
2(πτ/2τ0)
2 sin(πτ/2τ0)
cosh(πr/2τ0)− cos(πτ/τ0)
2 sin(πτ/2τ0)
cosh(πr/2τ0)− cos(πτ/τ0)
. (18)
Analytically continuing to real time τ = τ0 + it we obtain
〈Φ(r, t)Φ(0, t)〉 =
cosh(πr/2τ0) + cosh(πt/τ0)
8 sinh2(πr/4τ0) cosh
2(πt/2τ0)
2 cosh(πt/2τ0)
cosh(πr/2τ0) + cosh(πt/τ0)
2 cosh(πt/2τ0)
cosh(πr/2τ0) + cosh(πt/τ0)
, (19)
that for r/τ0 and t/τ0 much larger than 1 simplifies to
(π/2τ0)
1/4 1√
eπr/2τ0 + eπt/τ0
eπr/2τ0 · eπt/τ0
eπt/2τ0√
eπr/2τ0 + eπt/τ0
πt/2τ0
eπr/2τ0 + eπt/τ0
. (20)
Note that the exponential terms in the square root are always ≪ 1. Thus for fixed boundary condition we get the
free boson result Eq. (12) with x = 1/8. For the connected part we need to subtract 〈Φ(0, t)〉2 given by Eq. (8) with
AΦ+ = 2
1/4. We finally obtain
〈Φ(r, t)Φ(0, t)〉conn = 〈Φ(r, t)Φ(0, t)〉 − 〈Φ(0, t)〉2 ∝
0 for t < r/2 ,
e−πr/16τ0 − e−πt/8τ0 for t > r/2 .
Thus, also for the Ising model with fixed boundary conditions, connected correlations start developing at t = t∗ = r/2.
In the case of disordered initial condition, we have
〈Φ(r, t)Φ(0, t)〉 ∝
eπr/2τ0 + eπt/τ0
eπr/2τ0 · eπt/τ0
eπt/2τ0√
eπr/2τ0 + eπt/τ0
e−π(r−3/2t)/4τ0 for t < r/2 ,
e−πr/16τ0 for t > r/2 ,
resulting in an exponential space dependence even for t < r/2 (clearly in this case the connected correlation function
equals the full one).
3. The general two point-function
From the results reported for the gaussian and Ising models, it is now relatively simple to understand the general
properties of the time dependence of the two-point function in the very general case. The two-point function in the
half-plane has the general form [35]
〈Φ(z1)Φ(z2)〉UHP =
z12̄z21̄
z12z1̄2̄z11̄z22̄
F (η) , (23)
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FIG. 2: Left: Space-time region for the correlation functions at different times. Right: Conformal mapping to the upper
half-plane.
where the function F (η) depends explicitly on the considered model. Under the conformal map to the strip we know
that the first part of Eq. (23) transforms according to Eq. (10). Thus we need only to map F (η) that, in the
general case, is an unknown function. However, during the study of the Ising model we showed that the analytical
continuation of η for t, r ≫ τ0 is
η ∼ e
πt/τ0
eπr/2τ0 + eπt/τ0
. (24)
Thus for t < r/2 we have η ∼ eπ(t−r/2)/τ0 ≪ 1 and in the opposite case t > r/2 we have η ∼ 1. As a consequence
to have the asymptotic behavior of the two-point function we only need to know the behavior close to η ∼ 0 (i.e.
the behavior close to the surface) and for η ∼ 1 (i.e. deep in the bulk). Fortunately they are both exactly known.
Indeed when η ∼ 1 the two points are deep in the bulk, meaning F (1) = 1. Instead for η ≪ 1, from the short-distance
expansion, we have
F (η) ≃ (AΦb )2ηxb , (25)
where xb is the boundary scaling dimension of the leading boundary operator to which Φ couples and A
b is the
bulk-boundary operator product expansion coefficient that equals the one introduced in Eq. (8) [see e.g. Ref. [37]].
All the previous observations and the explicit calculations of the previous sections lead for t > r/2 to
〈Φ(r, t)Φ(0, t)〉 ∝ e−xπr/2τ0 , (26)
while for t < r/2 we get
〈Φ(r, t)Φ(0, t)〉 ∝ (AΦb )2e−xπt/τ0 × eπxb(t−r/2)/τ0 . (27)
Note that if 〈Φ〉 6= 0, xb = 0 and the last factor is absent. The leading term is then just 〈Φ〉2. Thus the leading
term in the connected two-point function vanishes for t < r/2, and its first non-vanishing contributions is given by
subleading terms either in F or in the bulk-boundary short-distance expansion.
It is very interesting that we only have to know the behavior as η → 0 and 1 to get the results we need for large r
and t. However, we stress that only a complete calculation (as those performed in the preceding sections) gives the
full analytic structure of the CFT result needed to justify the analytical continuation from imaginary to real time.
Moreover, the behavior within a distance O(τ0) of the horizon r = 2t depends on the detailed form of F .
C. Correlations functions at different times
Let us consider the case of the two-point function calculated at different real times 〈Φ(r, t)Φ(0, s)〉. This is again
obtained by mapping the imaginary time strip to the UHP, but in this case the two points are w1 = r + iτ1 and
w2 = 0+ iτ2, that, at the end of the calculation, must be analytically continued to τ1 = τ0 + it and τ2 = τ0 + is. See
Fig. 2 for a pictorial representation of the space-time domain in the strip and the resulting mapping to the UHP.
Let us start the discussion with the free boson. The distances appearing in Eq. (9) are
z212 = 1 + e
πr/τ0 − 2eπr/2τ0 cos(θ1 − θ2) , z11̄ = 2 sin θ1 ,
z212̄ = 1 + e
πr/τ0 − 2eπr/2τ0 cos(θ1 + θ2) , z22̄ = 2eπr/2τ0 sin θ2 , (28)
where θi = πτi/2τ0. Thus the correlation function on the strip is
〈Φ(r, τ1)Φ(0, τ2)〉strip = |w′(z1)|−x|w′(z2)|−x〈Φ(z1(w))〈Φ(z2(w))〉UHP = (29)
cosh(πr/2τ0)− cos(π(τ1 + τ2)/2τ0)
4 sin(πτ1/2τ0) sin(πτ2/2τ0)(cosh(πr/2τ0)− cos(π(τ1 − τ2)/2τ0))
that for τ1 = τ2 reduces to Eq. (10) as it should. Continuing to real times and considering r, t, s, |t − s| ≫ τ0 we
obtain
τ0 + e
(t+s)
(t+s)
τ0 + e
|t−s|
e−xπ(t+s)/4τ0 for r > t+ s ,
e−xπr/4τ0 for t− s < r < t+ s ,
e−xπ|t−s|/4τ0 for r < |t− s| .
Following the line sketched in the previous subsection, it is easy to generalize this result to the most general CFT.
In the case of a theory with fixed initial conditions (i.e., 〈Φ〉 6= 0) the asymptotic result is the same as before, with
only the crossover points being affected by the precise expression for F (η). Instead, in the case where 〈Φ〉 = 0, the
first case gains an additional factor e−πxb(t+s−r)/4τ0 .
Note that the autocorrelation function (i.e. r = 0) has only an exponential dependence on the time separation t− s
and does not exhibit aging in this regime.
D. Evolution with boundaries
We now consider the case of time evolution of a half-chain with some boundary condition at r = 0. For simplicity
we assume that the (conformal) boundary condition is of the same kind of the initial boundary condition (for example
we fix all the spins at t = 0 and at the boundary r = 0 to point in the same direction). The space-time region we
have to consider is depicted in Fig. 3. If different initial and boundary were considered, one needs to insert boundary
conditions changing operators at the corners of the figure.
The w plane is mapped into the UHP by
z(w) = sin
, (31)
with the corners at ±τ0 mapped to ±1. The mapping of w1 is
z1 ≡ z(w1) = z(−τ0 + τ1 + ir) = − cos(πτ1/2τ0) cosh(πr/2τ0) + i sin(πτ1/2τ0) sinh(πr/2τ0) . (32)
In the z plane the 1-point function is
〈Φ(z1)〉UHP ∝ |Imz1|−x → [sin(πτ1/2τ0) sinh(πr/2τ0)]−x , (33)
|w′(z1)|2 =
|1− z2|
cosh(πr/τ0)− cos(πτ1/τ0)
. (34)
Thus on the strip we have
〈Φ(w1)〉strip = |w′(z1)|−x〈Φ(w(z1))〉UHP ∝
sin2(πτ1/2τ0) sinh
2(πr/2τ0)
cosh(πr/τ0)− cos(πτ1/τ0)
]−x/2
, (35)
that continued to real time τ1 = it is
〈Φ(t, r)〉 ∝
cosh(πt/τ0) + cosh(πr/τ0)
cosh(πt/2τ0)2 sinh
2(πr/2τ0)
, (36)
and for t, r ≫ τ0 simplifies to
〈Φ(t, r) ∝
eπr/τ0 + eπt/τ0
eπr/τ0 · eπt/τ0
e−πxt/2τ0 for t < r ,
e−πxr/2τ0 for t > r .
Note that in this case the characteristic time is t∗ = r and not r/2. This explains also why the entanglement entropy
of a semi-infinite chain with free boundary condition at x = 0, has characteristic time t∗ = r as firstly noted in Ref.
[41].
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FIG. 3: Left: Space-time region for the one-point function in a boundary (at r = 0) geometry. Note w = τ + ir. Right:
Conformal mapping to the upper-half plane, c.f. Eq. (31).
E. Discussion and interpretation of the CFT results
All the correlation functions calculated so far display two very general features: first there is a sharp horizon (or
light-cone) effect at t = t∗ = r/2 (or r) resulting in a behavior before and after t∗ completely different; second the
asymptotic long-time correlation functions are the same as those at finite temperature βeff = 4τ0. The light-cone
effect is a very general phenomenon and and will be discussed in section VI. In the following sections we will point
out that also the “effective temperature” is a general phenomenon, but it has some specific CFT features that are
worthy of comment.
In fact, it is very easy to understand the technical reason why we find an effective temperature despite the fact that
we are studying a pure state at T = 0. The finite temperature correlations can be calculated by studying the field
theory on a cylinder of circumference β = 1/T . In CFT a cylinder can be obtained by mapping the complex plane
with the logarithmic transformation β/(2π) log z. The form for the two-point function in the slab depends in general
on the function F (η) –cf. Eq. (15)– but when we analytically continue and take the limit of large real time, we find
that effectively the points are far from the boundary, i.e. at η = 1. Thus we get the same result as we would get if we
conformally transformed from the full plane to a cylinder, and from Eq. (5) the effective temperature is βeff = 4τ0.
A similar argument can be worked out for the multi-point functions as well.
An addtional comment concerns what we expect for correlation functions of general operators, not only primary.
We have clearly seen that the local energy density does not relax, of course, and this is consistent with its not being
primary. But, we also know that at finite temperature 〈Ttt〉β = πc/6β2, that is perfectly compatible with the previous
result with βeff = 4τ0. Furthermore, for large real times, the two-point function of Ttt − 〈T 〉tt does behave as though
it was at finite temperature. This means, in particular, that the energy fluctuations in a large but finite volume (the
specific heat) are the same as those at finite temperature. Thus one is tempted to extend this finite temperature
interpretation to non-primary operators, and indeed it is the case. In fact, in the argument of above for the equivalence
of the long-time correlations and finite T , there are essentially three steps. First, we need to write down the form of
the correlation function in the half-plane, but this only depends on special conformal transformation and so is valid
for any quasi-primary operator like Tµν . Second we have to transform from the slab to the half-plane: for non-primary
operators this can have some anomalous term. Finally we need to compare the large limit of this to what one would
get transforming directly from the cylinder to the full plane. However, the two conformal transformations we are
comparing are both logarithmic (in one case 2τ0/π log z, in the other case β/2π log z), so the anomalous terms should
be the same. Thus the two correlations functions are the same. This argument works for all quasi-primary operators.
Then, since we can get all the non-quasiprimaries by considering successive operator product expansions with the
stress tensor, it also works for all operators.
IV. EXACT REAL-TIME DYNAMICS IN SIMPLE INTEGRABLE CHAINS
The results of the previous section rely on the technical assumption that the leading asymptotic behavior given
by CFT, which applies to the euclidean region (large imaginary times), may simply be analytically continued to
find the behavior at large real time. While such procedures have been shown to give the correct behavior for the
time-dependent correlations in equilibrium, it is important to check them in specific solvable cases for non-equilibrium
evolution.
Thus, as a complement to the CFT calculations, in this section we consider the real-time evolution of two simple
analytically tractable models. We solve the dynamics of a chain of coupled harmonic oscillators and we review and
re-analyze some known results for the Ising-XY chain in a transverse magnetic field. Beyond providing examples of
the CFT results (with central charge c = 1 and 1/2 respectively) in the critical case, these models allow us to take
into account the effects of a finite mass-gap and of the lattice.
A. The chain of harmonic oscillators
The simplest model with an exactly solvable non-equilibrium dynamics is surely a chain of coupled harmonic
oscillators with hamiltonian
π2r +m
2φ2r +
ω2j (φr+j − φr)2
. (38)
We introduce a coupling more general than simple nearest-neighbor hopping so as to allow for a general dispersion
relation below. For simplicity we also assume periodic boundary conditions. ϕn and πn are the position and the
momentum operators of the n-th oscillator, with equal time commuting relations
[ϕm, πn] = iδnm [ϕn, ϕm] = [πn, πm] = 0 . (39)
The hamiltonian can be written in diagonal form H(m) =
k ΩkA
kAk with modes
(Ωkϕk + iπk) , (40)
(Ωkϕ−k − iπ−k) , (41)
Ω2k = m
2 + 2
ωj (1− cos(2πkj/N)) . (42)
Note that we use the same symbols for the operators and their Fourier transforms (ϕk = 1/
n=0 e
2πikn/Nϕn
and analogously for πk).
We consider the scenario in which the system is prepared in a state |ψ0〉, that is ground-state of H(m0), and at the
time t = 0 the mass is quenched to a different value m 6= m0. We use the notation Ω0k for the dispersion relation for
t < 0 and the Ωk for the one for t > 0.
Since 〈ψ0|ϕn|ψ0〉 = 0, the expectation value of the field ϕn vanishes at any time. This example in fact corresponds
to the quench from the disordered phase in the language of the previous section. Thus we concentrate our attention
on the two-point function
〈ψ(t)|ϕnϕ0|ψ(t)〉 = 〈ψ0|ϕHn (t)ϕH0 (t)|ψ0〉 , (43)
where we introduced the operator in the Heisenberg picture ϕHn (t), whose time evolution is given by
ϕHn (t) =
ei(pkn−Ωkt)Ak + e
−i(pkn−Ωkt)A
, (44)
where pk = 2πk/N . Accordingly, the product of the two fields is
ϕHn (t)ϕ
0 (t) =
ΩkΩk′
ei(pkn−Ωkt)Ak + e
−i(pkn−Ωkt)A
e−iΩk′ tAk′ + e
. (45)
In order to have the time dependent two-point function we need the expectation values of the bilinear combinations
of A’s on the initial state, that is annihilated by the A0k. Thus it is enough to write A’s as functions of A0’s, i.e.
≡ ckA0k + dkA†0−k , (46)
+A0−k
≡ ckA†0k + dkA0−k , (47)
leading to (we understand 〈·〉 = 〈ψ0| · |ψ0〉)
〈AkAk′ 〉 = 〈(ckA0k + dkA†0−k)(ck′A0k′ + dk′A
0−k′ )〉 = ckdk′〈A0kA
0−k′〉 = ckdkδk,−k′ , (48)
〈AkA†k′ 〉 = 〈(ckA0k + dkA
0−k)(ck′A
0k′ + dk′A0−k′ )〉 = ckck′〈A0kA
0k′ 〉 = c
kδk,k′ , (49)
〈A†kAk′ 〉 = 〈(ckA
0k + dkA0−k)(ck′A0k′ + dk′A
0−k′ )〉 = dkdk′〈A0−kA
0−k′〉 = d
kδk,k′ , (50)
〈A†kA
k′ 〉 = 〈(ckA
0k + dkA0−k)(ck′A
0k′ + dk′A0−k′ )〉 = dkck′〈A0−kA
0k′ 〉 = ckdkδk,−k′ . (51)
Finally we arrive at
〈ϕHn (t)ϕH0 (t)〉 =
ckdke
i(pkn−2Ωkt) + c2ke
ipkn + d2ke
−ipkn + ckdke
−i(pkn−2Ωkt)
, (52)
which, in the thermodynamic limit (N → ∞), may be written as
〈ϕHr (t)ϕH0 (t)〉 − 〈ϕHr (0)ϕH0 (0)〉 =
(Ω20k − Ω2k)(1 − cos(2Ωkt))
Ω2kΩ0k
, (53)
where the integral is on th first Brillouin zone |k| < π/a. Note that for t = 0 and for m = m0 this two-point function
reduces to the static one, as it should. This result can also be found by integrating the Heisenberg equations of
motion for each mode. For future reference it is also useful to write down explicitly the Fourier transform known as
momentum distribution function
ρ(k) =
(Ω20k +Ω
k)− (Ω20k − Ω2k) cos(2Ωkt)
Ω2kΩ0k
. (54)
Note that when considering correlation functions in momentum space, a long-time limit does not exist and we need
to take the time average, in contrast to what happens in real space.
1. The continuum limit
In Eq. (52) everything is completely general and applies to any chain with finite lattice spacing. Let us know discuss
the continuum limit that is achieved by sending N → ∞ in such a way that (1/N)
dp/(2π), pn → p, and
Ω2k → Ω2p = m2 + p2.
In this limit the correlation function becomes (∆m2 = m20 −m2)
G(r, t) ≡ 〈ϕHr (t)ϕH0 (t)〉 =
−∆m2 cos(2
p2 +m2t) +m2 +m20 + 2p
(m2 + p2)
m20 + p
. (55)
Since a closed form for such integral is quite difficult to write down in the most general case, we will only consider
some particular cases.
Let us first consider the conformal evolution (m = 0) from an initial state with a very large mass m0 → ∞. This
should reproduce the CFT result previous section for all the time t, since the correlations of the initial state shrink
to a point. We have
G(r, t) = m0
1− cos(2pt)
0 for t < r/2 ,
(2t− r) for t > r/2 .
To compare such result with the conformal result given by Eq. (12), we have to keep in mind that the primary field is
not the gaussian one ϕH(r, t), but its imaginary exponential. Thus we need 〈eiqϕH (r,t)e−iqϕH (0,t)〉, with an arbitrary
q. Despite of the apparent complexity of such correlator, it is very simple to obtain it using the standard property of
gaussian integrals
〈eiqϕ
H (r,t)e−iqϕ
H(0,t)〉 = e−q
2〈(ϕH(r,t)−ϕH(0,t))2〉/2 = eq
2(G(r,t)−G(0,t)) , (57)
leading to
〈eiqϕ
H (r,t)e−iqϕ
H (0,t)〉 =
2m0t for t < r/2 ,
2m0r/2 for t > r/2 ,
0 0.2 0.4 0.6 0.8 1
0 2 4 6 8
0 1 2
FIG. 4: Left: G(r, t)/m0 given by Eq. (60) as function of t, at fixed r = 1. Three different values of m0 = 10, 3, 1 (from the
bottom to the top) are shown. Inset: Lattice effects showing the cos 4t oscillations on top of the continuum result. Right:
G(r, t)/m0 given by the numerical integral of Eq. (62) as function of t, at fixed m, r = 1. It is compared with the asymptotic
behavior for 0 < 2t− r ≪ m−1 and for t ≫ r.
that is exactly the same of Eq. (12) with xΦ ∝ q2 and τ0 ∝ m−10 , confirming that τ0 is just proportional to the
correlation in the initial state, as its interpretation in terms of the extrapolation length suggests.
The case m = 0 and m0 finite corresponds to a conformal evolution from a generic state with correlations propor-
tional to m−10 . Thus we expect the CFT result Eq. (12) to be true for asymptotic large times and separations.
From Eq. (55), the two-point function of the gaussian field is
G(r, t) =
−m20 cos(2pt) +m20 + 2p2
m20 + p
, (59)
that can be written as
2πG(r, t) = 2K0(m0r) + f(m0r) −
f(m0(r − 2t)) + f(m0(r + 2t))
, (60)
where K0(y) is the modified Bessel function and
f(y) = 1 +
G2113
0 1 1/2
, (61)
with G2113 the Meijer G-function (see e.g. [42]). Note that f(x) is characterized by f
′′(x) = −K0(x) and f(0) = f ′(0) =
0. In the limit m0 → ∞, it is easy to show that G(r, t) reduces to the previous result. In the left panel of Fig. 4 , we
report G(r, t) as function of t at fixed r = 1 for m0 = 10, 3, 1. It is evident that a finite m0 results in smoothing the
curve close to t = r/2 and giving an offset in zero. Both the effects are more pronounced as m0 decreases. For large
t, independently on m0 6= 0, we have G(r, t) = t + O(t0), confirming that the CFT result is correct for asymptotic
large times.
The case with arbitrary m and m0 is quite cumbersome to be worked out analytically and not really illuminating.
For this reason we concentrate here on the massive evolution from a state with m0 → ∞. In this case the correlation
function reads
G(r, t) = m0
1− cos(2
p2 +m2t)
(m2 + p2)
. (62)
In the limit t→ ∞ the cosine term averages to zero, giving
G(r, t = ∞) = m0
(m2 + p2)
. (63)
To understand the time dependence, let us first note that, despite the presence of a square-root, the integrand of
Eq. (62) is analytic in p, since the square-root is the argument of the even cosine function. Thus we can make the
integral in the complex plane and use the Cauchy theorem. As long as r > 2t, the behavior for p→ i∞ is dominated
by eipr and we can safely close the contour path in the upper half-plane where the residue at p = im is zero. As a
consequence G(r, t < r/2) = 0.
For t > r/2, the integral is more difficult, since we can not close the contour in the upper half-plane, because the
cosine is “larger” than eipr for p→ i∞. However, the approach to the asymptotic value is easily worked out. In fact,
for t≫ r, the term eipr in the integral (62) can be approximated with 1, since it is slowly oscillating. Thus we have
G(r, t ≫ r)−G(r, t = ∞) ≃ −m0
cos(2
p2 +m2t)
(m2 + p2)
= −m0fm(t) , (64)
where fm(t) = 1/(2m) − t1F2(1/2; 1, 3/2;−m2t2), that satisfies f ′m(t) = −J0(2mt) and fm(0) = 1/(2m). Note that
fm(t) is just the result for r = 0. In the complementary region 0 < 2t−r ≪ m−1, the integral in Eq. (62) is dominated
by the modes with p≫ m, and so it can be described by the conformal result.
In the right panel of Fig. 4, we plot the time-dependence of G(r, t) at fixed r as obtained by numerically integrating
Eq. (62) for m = r = 1. The plot shows that the conformal result t − r/2 describes the behavior close to t = r/2,
while for larger times the asymptotic expression gives an excellent approximation.
We finally note that G(r, t) at fixed t displays spatial oscillations (for r < 2t, else it vanishes), as it can be simply
realized by a stationary phase argument.
2. Lattice effects
Another advantage of this simple model is that the effects of the lattice can be easily understood. In fact, by
a stationary phase argument, the dominant contribution to (53) in the limits of large t and r comes from where,
the group velocity vk ≡ Ω′k = r/2t, independently of the explicit form of Ωk. Consequently, the two-point function
of a gapless model with dispersion relation Ωk = 2 sink/2 differs from the continuum limit previously derived for
t > r/2vm, where vm is the maximum group velocity. In particular G(r, t) receives a contribution from the slowest
mode (k = π with vπ = 0) whose effect is to add to Eq. (56) fast oscillations going as cos(2Ωπt) = cos 4t. The
resulting G(r, t) is plotted in the inset of Fig. 4 in the gapless case.
Thus, lattice effects play a more important role in the cases where the asymptotic result vanishes. A typical example
is the energy density that in this model is proportional to (ϕr+1 − ϕr)2. The continuum calculation would just give
zero (the mean-energy is conserved), but the approach to this value is governed by lattice effects and is dominated by
the smallest group velocity that comes from the zone boundary at |k| = π. In fact, just with a trivial calculation we
〈(ϕr+1 − ϕr)2〉 = 2(〈ϕ20〉 − 〈ϕ0ϕ1〉) =
(1− cos k) (1− cos 2Ωkt)
2 sin2 k/2
(1− cos 2Ωkt) ∝ J0(4t) ∼ t−1/2 cos 4t , (65)
that displays the typical cos 4t from the zone boundary at |k| = π.
The lattice dispersion relations are sensitive to the microscopical details of the model and consequently quantities
like the energy-density show a dependence on this. For example, a lattice massless fermion has dispersion relation
Ωk = 2 sink/2 and energy density given by sin
2 k. In this case the time evolution of the energy density is
sin2 k
(1− cos 2Ωkt)
2 sin2 k/2
, (66)
that for large times goes like t−3/2 cos 4t, resulting in a different power law compared to before.
If the system is quenched to a gapped H with Ω2k = m
2 + 2(1 − cos k) the maximum group velocity corresponds
to a non-zero wave number. This gives rise to spatial oscillations in the correlation function (this is true also in the
continuum limit as already discussed in the previous subsection).
B. The Ising-XY chain in a transverse magnetic field
The most studied one dimensional quantum spin model is the so-called XY chain in a transverse field, defined by
the hamiltonian
(1 + γ)
σxj σ
j+1 +
(1− γ)
j+1 − hσz
, (67)
where σ
x,y,z
i are the Pauli matrices, γ is called anisotropy parameter, and h is the applied external transverse field.
It is well known that for any γ 6= 0 the model undergoes a phase transition at h = 1, that is in the universality class
of the Ising model (defined by γ = 1). For simplicity we will just consider the Ising case, other values of γ 6= 0 being
equivalent.
We consider the non-equilibrium unitary dynamics that follows from a quench of the magnetic field at t = 0 from
h0 to h1 6= h0. Earlier works on this subject by Barouch et al. date back to seventies [7, 8]. Exploiting the mapping
of this model onto a free fermion, the time evolution of the transverse magnetization (that is not the order parameter)
was obtained exactly [7]:
mz(t) =
(h0 − h1) sin2 k cos(2ǫ1t)− (cos k − h1)[(cos k − h0)(cos k − h1) + sin2 k]
, (68)
where ǫi = ǫ(hi) with ǫ(h) =
sin2 k + (h− cos k)2. In Ref. [7] it was shown that (for non-exceptional parameters)
the approach to the asymptotic value for t→ ∞ is of the form t−3/2 cos 4t. This is simply shown in the case of h0 = ∞
and h1 = 1, when the integral simplifies and we obtain (see also [9])
mz(t) =
sin2 k cos(2ǫ1t)
4 sin2 k/2
J1(4t)
, (69)
where J1 is the Bessel function of the first kind whose asymptotic expansion for large argument is J1(x) ∼
2/πx cos(x+ π/4).
To understand this result we should keep in mind that the transverse magnetization is not the order parameter:
it corresponds to the product of two “disorder parameters” at neighbor sites on the dual lattice. Thus it must have
the symmetry of an energy operator, for which CFT just predicts a constant asymptotic result. The power law term
t−3/2 cos(4t) is just the lattice correction to the asymptotic. This correction clearly shows the fermionic nature of the
model on the basis of what discussed in the previous section.
The asymptotic result of the two-point function for t → ∞ has been studied in Ref. [11]. Calling Gn =
〈Φ(n,∞)Φ(0,∞)〉, in the case h0 = ∞ it has been found
2nhn1
, forh1 ≥ 1 ,
cos[n arccos(h1)] , forh1 ≤ 1 ,
instead for h0 = 0
, forh1 ≥ 1 ,
(n+ 1) ln
, forh1 ≤ 1 ,
that for large n decay exponentially with n. It has been shown that Gn is decaying exponentially with n for general
h0 and h1 [11], although closed forms are not available. The exponential decay is the prediction of CFT that (maybe
surprisingly) applies even far from the critical point h1 = 1.
The time dependence of two-point function has been studied in Ref. [9] by means of exact diagonalization of the
model with open boundary condition at the two ends r = 0, L. The results of interest for this paper are
• The connected two-point function of σz in the thermodynamic limit and at the critical point is
〈σzr (t)σz0(t)〉c =
J2r(4t)
2 − 1
J2r+1(4t)J2r−1(4t) , (72)
which is valid both for h0 = 0,∞. Neglecting fast oscillations, 〈σzr (t)σz0(t)〉c increases as r2 for r < 2t and then
it drops almost immediately to 0.
• The autocorrelation function (not connected) of σz at different times is
〈σz0(t1)σz0(t2)〉 = J20 (2t2 − 2t1)−
[f(t2 + t1)± g(t2 − t1)] , (73)
where f(x) = J2(2x) + J0(2x), g(x) = J2(2x)− J0(2x) and the sign + (−) refers to h0 = 0 (h0 = ∞).
Let us comment on these results in view of the general understanding we found so far. For t < t∗ all the connected
correlation functions are zero, in agreement with CFT. All the oscillation terms of the asymptotic form cos(4t) are,
as we discussed in the previous section, a lattice effect. Concerning the σz correlator, the r
2 dependence for t > r/2
is a consequence of the fact that σz is not primary. The same is true for the two-time correlations function of σz that
also decays as a power law of t− s for large times, instead of the exponential prediction by CFT for primary field.
V. HIGHER DIMENSIONS
Until now we just considered one-dimensional systems. Despite the fact that in low dimensions the effect of fluctu-
ations is more pronounced making the physics highly non-trivial it is desirable to have results in higher dimensions
as well. The method presented in Sec. II to obtain the non-equilibrium dynamics of a quantum model close to
a critical point from the critical behavior of a system confined in a slab geometry applies to generic dimension d
through the study of the hamiltonian (4). Its analysis proceeds via field-theoretical RG that may provide the all
scaling quantities of the model in an expansion close to the upper critical dimensions (u.c.d.), that is D = d+ 1 = 4.
Above the u.c.d. mean-field (or gaussian) results are exact, with logarithmic correction at the u.c.d.. For dimensions
lower than the u.c.d., the scaling quantities are obtained as series in ǫ = 4 − (d + 1). Thus for the time-evolution
problem the simple mean-field solution represents an exact scaling result (a part log corrections) for the physically
relevant three-dimensional case. An alternative method to attack analytically the hamiltonian (4) is to consider an
N component field φ and taking the limit N → ∞, but this will not be employed here.
A. Dirichlet boundary conditions: the two-point function
The D-dimensional slab geometry with Dirichlet boundary conditions has been the subject of several investigations.
The two-point function has been calculated at the first order in ǫ expansion in Ref. [43]. The gaussian two-point
function, with partial Fourier transform in the parallel directions reads [43]
G(p, z1, z2) =
e−b|z1−z2| − e−b(z1+z2) + e
−b(z1−z2) + e−b(z2−z1) − e−b(z1+z2) − eb(z1+z2)
e2bL − 1
, (74)
with b =
p2 +m2. We are interested in the case where L = 2τ0, z1 = z2 = τ that we will analytically continue to
τ → τ0 + it, and for computational simplicity we will restrict to the massless case m = 0. Thus in real space and
imaginary time, we have (p = |p|)
G(r, τ) =
(2π)3
e−ip·r
1− e−2pτ + 2(1− cosh(2pτ))
e4pτ0 − 1
(2π)2
1− e−2pτ + 2(1− cosh(2pτ))
e4pτ0 − 1
d(cos θ)eipr cos θ =
(2π)2r
dp sin pr
1− e−2pτ + 2(1− cosh(2pτ))
e4pτ0 − 1
. (75)
This integral can be performed by making the sum over all the residues coming from the denominator e4pτ0 − 1. The
calculation is rather involved, but the final result is very simple:
G(r, τ) =
coth(πr/4τ0) sin
2(πτ/2τ0)
cosh(πr/2τ0)− cos(πτ/τ0)
. (76)
Continuing to real time τ = τ0 + it we obtain
G(r, t) =
coth(πr/4τ0) cosh
2(πt/2τ0)
cosh(πr/2τ0) + cosh(πt/τ0)
, (77)
that for t, r ≫ τ0 simplifies to
G(r, t) ≃ 1
eπt/τ0
eπt/τ0 + eπr/2τ0
eπ(t−r/2)/2τ0/r for t < r/2 ,
1/r for t > r/2 .
We recall that this result in d = 3 is exact a part log corrections. Thus the basic structure of the two-point function
in 3D is the same as in 1D, with a characteristic time t∗ = r/2. Using the result in Ref. [43] it is in principle possible
to calculate the correlation functions for d < 3 in the ǫ expansion framework. However, this requires the analytical
continuation of complicated functions, resulting in a quite cumbersome algebra, as we shall see in the following for a
simpler observable.
Eq. (74) can be used in principle to determine the gaussian behavior in any d < 3. Unfortunately the integral
one gets is not analytically tractable for d 6= 1, 3. A possible strategy would be to perform the analytic continuation
before of the integral and then evaluate it through a saddle-point approximation. It is straightforward to show that
the result obtained in this way is equivalent to what we discuss in section VD where we remand for the analysis of
1 < d < 3.
B. Dirichlet boundary conditions: a non-trivial one-point function
In the case of Dirichlet boundary conditions, the order parameter profile in the slab geometry is trivially vanishing.
However not all the one-point expectation values are zero. Let us consider as a typical example the operator O = φ2
that has been calculated at the first order in ǫ expansion in Ref. [44]. It has the scaling form
〈O(z, L)〉 ≃ L−d+1/νH(z/L) , (79)
where ν is the correlation length exponent. In D = 4, the function H(x) is [44]
H(x) =
sin2(πx)
, (80)
that, using L = 2τ0 and continuing to z = τ = τ0 + it, leads to the real time evolution
〈O(t)〉 − 〈O(t = ∞)〉 ≃ τ−20
(πt/2τ0)
∼ τ−20 e−πt/2τ0 , (81)
i.e. 〈O(t)〉 approaches its asymptotic value exponentially, as it as been found for all primary operators in d = 1.
The one-loop result for the O(N) model in D = 4− ǫ dimensions is [44]
H(x) =
sin2(πx)
1 + ǫ
N + 2
N + 8
sinπx
− ǫ[ζ′(2, x) + ζ′(2, 1− x)] + const , (82)
where const stands for terms do not depend on x and ζ′(α, x) = ∂αζ(α, x) and ζ is the generalized Riemann function.
This example clearly shows that within the ǫ expansion, the scaling functions contain logarithmic contributions that
originate from the expansion of power law as e.g. sinǫ πx = 1 + ǫ ln sinπx. To have a function with good analytical
structure to perform the real time continuation, it is desirable to “exponentiate” such logarithms. In Ref. [44] the
exponentiation procedure leaded to
H(x) =
sin(πx)
)2−1/ν
[ζ(D − 2, x) + ζ(D − 2, 1− x)− 2ζ(D − 2)]
(2 − 1/ν)
sin(πx)
)D−2−1/ν
, (83)
with ν = 1/2 + (N + 2)/(N + 8) ǫ/4. Performing the analytical continuation we have the sum of two exponentials,
and in any dimension d < 3 the second one has a largest “relaxation time” that hence is dominating for large t.
Considering only the second term we have
〈O(t ≫ τ0)〉 − 〈O(t = ∞)〉 ∝ e−(d−1−1/ν)πt/2τ0 . (84)
Note that the subleading term e−(2−1/ν)πt/2τ0 is multiplied by a log t term arising from the ζ function.
This example put forward the idea that (at least for Dirichlet boundary conditions) the exponential relaxation of
the one-point functions is not only a property of one-dimensional systems, but holds in any dimension (with eventually
log corrections) with relaxation times related to the scaling dimensions of the operator.
C. Fixed boundary conditions: the one-point function
The case of the extraordinary transition, that corresponds to fixed boundary conditions, has been considered in
Ref. [45]. The magnetization profile can be written as [45]
φ(z) =
dn(2K4z/L)
sn(2K4z/L)
, (85)
where K4 = K(k4), where K(k) is the elliptic integral, k4 the elliptic modulus that in terms of the parameter of the
model is φ2L2 = (2K4)
2(2k24 − 1), and sn(x) and dn(x) and the Jacobi functions. Continuing to real time, and using
the properties of the Jacobi functions we obtain
φ(t) =
1− k4cn(K4t/τ0, 1− k4) , (86)
that, contrarily to all the other cases we have considered, is oscillating and not exponentially decreasing.
To our knowledge there are no result in the ǫ-expansion for the magnetization profile. However, using “local-
functional methods” [46] it has been obtained an approximate profile in D = 3 that involves, as in mean-field theory,
Jacobi elliptic functions. It can be easily continued to real time via z → τ0 + it and again one finds an oscillating
behavior with time. The method exploited in Ref. [46] can be used in any dimensions, obtaining an always an
oscillating behavior with a period that diverges as D approaches 2, recovering Eq. (7). All these calculations are
essentially mean field and we do not know how the inclusion of fluctuations changes them. It can possible that for the
extraordinary transition (i.e. fixed initial conditions), the exponential decay founds at D = 2 is more an exception
rather than a rule, because of the simple analytic structure of the trigonometric functions in the complex plane.
Another possibility is that fluctuations destroy these oscillations. Only a complete analytical calculation (e.g. in large
N) can help in understanding this point.
D. A real-time solvable model
As for the one-dimensional case, it is worth to check the results coming from the analytical continuation of large
imaginary-time with exactly solvable models. The simplest (and probably one of the few) model solvable in generic
dimension is the generalization of the hamiltonian (38) to a d-dimensional hypercubic lattice. The solution of such
model proceeds via Fourier transform as in one dimension. The final result is simply give by Eq. (53) with the
replacement dk → ddk, i.e.
〈ϕr(t)ϕ0(t)〉 − 〈ϕr(0)ϕ0(0)〉 =
(2π)d
eik·r
(Ω20k − Ω2k)(1 − cos(2Ωkt))
Ω2kΩ0k
. (87)
First of all let us note that this expression in virtually identical to Eq. (75) if we take τ0 ∝ m−10 → 0. In fact, taking
τ0 → 0 in Eq. (75), only the third term matters, since it is O(τ−10 ) relative to the first two. Taking z = r + it and
b = Ωp we get G(p) ∝ τ−10
Ω−2p (1− cos 2Ωpt) that is Eq. (87) for m0 → ∞.
To understand the general features let us consider in details the conformal evolution (Ωk = v|k|) from a disordered
state (Ω0k = m0). The derivative of the two-point function is (k = |k|)
∂t〈ϕr(t)ϕ0(t)〉 = 2m0Im
(2π)d
ei(k·r−2kvt)
. (88)
Except for d = 1, this can be done analytically only in d = 3, where we can write it as
dθ sin θeik(r cos θ−2vt) ∼
sin(kr)/r
e2ikvt ∼ (1/r)δ(vt− r/2) . (89)
Integrating with respect to t we get zero for t < r/2v and 1/r for t > r/2v.
For general d we have
∂t〈ϕr(t)ϕ0(t)〉 ∝ 2m0Im
kd−2dk
dθ(sin θ)d−2eik(r cos θ−2vt) . (90)
By a saddle point argument, we can assume that the dominant behavior comes from θ close to zero, so the θ integral
gives
dθθd−2e−ikrθ
2/2 ∼ (kr)−(d−1)/2 , (91)
leading to
(kr)(d−1)/2
ei(k(r−2vt)) ∼ r−(d−1)/2(2vt− r)−(d−1)/2Θ(vt− r/2) . (92)
Integrating with respect to t, we get zero for t < r/2v, as expected by causality, and
r−(d−1)/2(2vt− r)(3−d)/2 , (93)
for t > r/2v. It is interesting that this gaussian correlation function blows up at large t only for d < 3, when we
expect the fluctuations to become important. It would be interesting to study this in the φ4 theory for large N , by
replacing φ4 by 3〈φ2〉φ2 in the usual way, where now 〈φ2〉 depends on t and is calculated self-consistently.
VI. PHYSICAL INTERPRETATION AND DISCUSSION
In this paper we studied in general the non-equilibrium unitary dynamics that follows a sudden quantum quench.
We showed that if the hamiltonian H governing the time evolution is at a critical point, while H0 (i.e. the one for
t < 0) is not, the expectation value of a class of operators (primary ones in CFT) relaxes to the ground-state value
exponentially in time with universal ratio of decaying constants. We also found that connected two-point functions
of operators at distance r are vanishing for t < r/2v, while for t > r/2v reach exponentially fast a value that depends
exponentially on the separation, in contrast with the power laws typical of equilibrium configuration.
We also considered the real-time dynamics of simple exactly solvable models and we found that several of the typical
characteristics of the critical points still hold. Roughly speaking, critical points are not special as far as quenching
dynamics is concerned. In fact, also for gapped systems, connected correlation functions vanish (or are strongly
suppressed) for t < r/2v and for asymptotic large times resemble those at finite temperature despite the fact that the
whole system is in a pure state. Several other examples in the recent literature (see e.g. [9, 15, 24, 26, 39, 41]) gives
further evidence that these two effects are actually true in general, at least in the realm of exactly solvable models
considered so far.
In the following we give a simple interpretation of these two features separately trying to understanding their
physical origin.
A. The horizon effect
The qualitative, and many of the quantitative, features found for the time evolution of correlation functions may
be understood physically on the basis of a picture we first introduced in Ref. [39] to describe the time evolution of the
entanglement entropy. Later we generalized it to correlation functions in Ref. [33] and it has been largely adopted
thereafter [2, 24, 26, 41].
We emphasize that such scheme is not an ab initio calculation but rather a simplified picture which allows us to
explain physically our findings. The initial state |ψ0〉 has an (extensively) high energy relative to the ground state
of the hamiltonian H which governs the subsequent time evolution, and therefore acts as a source of quasiparticle
excitations. Those quasi-particles originating from closely separated points (roughly within the correlation length
ξ0 = m
0 of the ground state of H0) are quantum entangled and particles emitted from far different points are
incoherent. If the quasiparticle dispersion relation is E = Ωk, the classical velocity is vk = ∇kΩk. We assume that
there is a maximum allowed speed vm = maxk |vk|. A quasiparticle of momentum k produced at r is therefore at
r+vkt at time t, ignoring scattering effects. This is the only physical assumption of the argument. Scattering effects
are not present in the theories considered so far, but as evident from the argument outlined below they can play a
role for only t > r/2vm (allowing, perhaps, for a renormalization of vm by the interactions).
These free quasi-particles have two distinct effects. Firstly, incoherent quasi-particles arriving a given point r from
well-separated sources cause relaxation of (most) local observables at r towards their ground state expectation values.
(An exception is the local energy density which of course is conserved.) Secondly, entangled quasi-particles arriving
at the same time t at points with separation |r| ≫ ξ0 induce quantum correlations between local observables. In the
case where they travel at a unique speed v (as in CFT), therefore, there is a sharp “horizon” or light-cone effect: the
connected correlations do not change from their initial values until time t ∼ |r|/2v. In the CFT case this horizon effect
is rounded off in a (calculable) manner over the region t−|r|/2v ∼ τ0, since quasi-particles remain entangled over this
distance scale. After this they rapidly saturate to time-independent values. For large separations (but still ≪ 2vt),
these decay exponentially ∼ exp(−πx|r|/2τ0). Thus, while the generic one-point functions relax to their ground-state
values (we recall in CFT this relaxation is exponential ∼ exp(−πxvt/τ0)), the correlation functions do not, because,
at quantum criticality, these would have a power law dependence. Of course, this is to be expected since the mean
energy is much higher than that of the ground state, and it does not relax.
This simple argument also explains why for the case of a semi-infinite chain the relevant time scale is r/v rather
than r/2v, since one of the two particles arriving in r has been reflected from the end of the chain. This has been
also stressed in Ref. [41], in the study of the time dependence of entanglement entropy of finite chains with open
boundary conditions.
All our results are consistent with this picture as long as the quasi-particles are assumed to all propagate at the
same speed, resulting from a “conformal” dispersion relation Ωk = v|k|. However, it is very simple to generalize this
picture to different dispersion relations, taking into account that each particle propagates at group velocity vk ≡ Ω′k
appropriate to the wave number k. In this case the horizon effect first occurs at time t ∼ |r|/2vm, where vm is
the maximum group velocity. If vm occurs at a non-zero wave number, it gives rise to spatial oscillations in the
correlation function. Thus again connected correlation function are expected to be strongly suppressed for t < t∗
and start developing only after t∗. In the case of a general dispersion relation we do not have a proof, beyond the
stationary phase approximation, that the connected correlation functions remain constant up to this time, but one
would expect it on the grounds of causality. (The proof in the case of a relativistic dispersion relation uses Lorentz
invariance in an essential way.) However, because there are also quasi-particles moving at speeds less than vm, the
approach to the asymptotic behavior at late times is less abrupt. In fact, for a lattice dispersion relation where Ω′k
vanishes at the zone boundary, the approach to the limit is slow, as an inverse power of t. A similar result applies to
the 1-point functions. This is consistent with the exact results obtained here and elsewhere.
B. The large time limit and the generalized Gibbs ensemble
The existence and the understanding of the asymptotic state resulting from the evolution from an arbitrary state is
one of the most-interesting problem in statistical mechanics. A robust theory able to predict this state ab-initio still
does not exist. A currently popular idea is that for late times the system (or rather macroscopically large subsystems)
‘look like’ they are in a thermal state, despite the fact that the actual state of the whole system is pure. A common
belief is that a region of dimension r can be thermalized by the infinitely large rest of the system which acts as a bath
(see e.g. [24, 39, 47]). But this intriguing idea is not sufficient to give the value of the resulting effective temperature.
A major step toward the clarification of the properties of the asymptotic state has been made by Rigol et al. [15].
In fact, it was conjectured that if the asymptotic stationary state exists, it is given by a generalized Gibbs ensemble
obtained by maximizing the entropy S = −Tr ρ log ρ, subject to all the constraints imposed by the dynamics [15].
Consequently, denoting with Im a maximal set of commuting and linearly independent integrals of motion, the density
matrix is
ρ = Z−1e−
λmIm , Z = Tr e−
λmIm . (94)
We note that such a density matrix describes a pure state only if the model under consideration is integrable, i.e. if
the number of integral of motions equals the number degrees of freedom. If there are not enough integrals of motion,
ρ corresponds to a mixed state and it is not clear to us to which extent it can describe the pure state resulting from
the time-evolution. The values of the Lagrange multipliers λm are fixed by the initial conditions:
Tr Imρ = 〈Im〉t=0 . (95)
In the following we will take this generalized Gibbs ensemble as a postulate and we will show how it nicely and
naturally explains the “effective temperature” effect observed for large times. However we stress that there is still no
proof for this assumption that, to our opinion, cannot be considered on the same fundamental level as the thermal
Gibbs ensemble.
Let us consider the chain of harmonic oscillators of the previous section as a typical example. We will soon see that
most of the features are quite general. In this case the natural choice for an infinite set of integral of motion is the
number of particles with momentum k, i.e. nk = A
kAk. Most other observables can be written in terms of these, i.e.
k Ωknk. Consequently the expectation value is given by
〈O〉t=∞ = TrOρ = TrOZ−1e−
λknk , (96)
that can be seen as a thermal density matrix with a k dependent effective temperature given by
βeff(k)Ωk = λk . (97)
Thus an effective temperature already appeared. Note that this state can still be pure because such a temperature is
k dependent.
To fix λk we need 〈nk〉t=0. From Eq. (50) we get
〈nk〉t=0 = 〈A†kAk〉t=0 = d
. (98)
The calculation then proceeds as for a thermal distribution
〈nk〉ρ = Trnkρ = −
lnZ , with Z = Tr e−
λknk =
e−λknk =
1− e−λk
, (99)
so that
〈nk〉ρ =
ln(1− e−λk) = 1
eλk − 1
. (100)
From this eλk = 1 + d−2k and
βeff(k) =
ln(1 + d−2k )
. (101)
At finite temperature the correlation function in momentum space is
〈ϕkϕ−k〉β = 〈
(Ak +A
k)(A−k +A
−k)〉β =
〈A†kA−k〉β + 〈AkA
1 + e−βΩk
1− e−βΩk
, (102)
that substituting the previous result for βeff gives
ρ(k)t=∞ = 〈ϕkϕ−k〉t=∞ =
(1 + 2d2k) =
Ω2k +Ω
Ω2kΩ0k
, (103)
that is exactly the time-average of Eq. (54) reproducing, after Fourier transforming, the well defined correlation in
real space for t→ ∞.
So this generalized Gibbs ensemble correctly reproduces the exact diagonalization of the model and gives also
few insights more. In fact, in the limit m0 → ∞ the effective temperature is independent from k and m obtaining
βeff = 4/m0, explaining a posteriori the simplicity of the results in this case. Note instead that for arbitrary m0 and
m = 0, i.e. conformal evolution, βeff(k) is a function of k. In this case, the large distance properties of correlation
functions are described by the mode with k = 0, for which, independently of m0 we get βeff(k = 0,m = 0) = 4/m0,
consistently with the previous findings and general expectations. Furthermore we can conclude that the large r
asymptotic behavior is always governed by the effective temperature βeff(k = 0) = 2(log(|m0 − m|/(m0 + m))/m.
Another interesting feature is that βeff(k = 0) gives the asymptotic behavior of the correlation functions of all those
observables that are effectively coupled with the zero-mode. In the opposite case the relevant temperature is the
largest βeff(k) with the k mode coupled with the observable. Another interesting feature is that on a very rough basis
one can be tempted to assume that that βeff(k) is directly related to the excess of energy of the mode k generated by
the quench, but this is not the case. In fact 〈H〉t=0 =
k Ωkd
k that is different from β
eff , being the same only in the
limit d−2k → 0, i.e. m0 → ∞.
Clearly the same reasoning of before applies every time we consider a model that can be diagonalized in momentum
space with a proper choice of the quasi-particles, i.e. every time that H =
k ΩkA
kAk for some Ak. This means that
the excitations are non-interacting. As far as we are aware all the applications of the generalized Gibbs ensemble to
the date only concern this kind of models [15, 26], but there are few numerical hints suggesting that can be true more
generically [29].
Now we outline how we imagine the generalized Gibbs ensemble given by Eq. (94) could be used to justify the
effective temperature scenario for large time for any integrable system, i.e. with a complete set of integrals of motion.
The hamiltonian can be written as H =
m amIm, with some am eventually zero. Thus one can think to an m-
dependent effective temperature βeff(m) = λm/am, but this temperature does not give directly the behavior of the
correlation function for large distance because in general the integral of motions are not diagonal in k-space. Thus we
can only conjecture that the correlation functions of a given operator O(r) are governed by the largest βeff(m) with
m among the integral of motions to which O(r) effectively couples. We stress that this is only a crude argument and
we are still not able to put it on a firmer basis.
C. Discussion and open questions
We presented a quite complete picture on the time evolution of a quantum system after a sudden quench of one
hamiltonian parameter. Despite the fact that a lot of work has been done, still more is left for future investigation.
The first problem that must be addressed is the real-time dynamics of effectively interacting systems. In this
direction Bethe Ansatz solvable models like the Lieb-Liniger gas and Heisenberg spin chains are among the best
candidates for an analytical approach. This would clarify how a non-trivial two-body scattering matrix can modify the
time-evolution (only inside the light-cone) of the correlation functions and whether the crude argument we outlined
for an “effective temperature” for the long-time state is valid. To clarify this point also the study of (boundary)
integrable massive field theories can be of some help.
Another approach, currently under investigation[48], is the direct analysis of the perturbative expansion for the
correlation functions in a λφ4 field theory. This has some simplifying features in the large m0 limit which may allow
it to be resummed to all orders. It is very important to get non-trivial results for such a non-integrable interacting
theory.
Numerical computations for non-integrable systems should also be performed to understand if and eventually to
which extent the generalized Gibbs ensemble picture is valid beyond integrability. Some numerics concerning this
point are already available [22, 23, 29] but still a clear scenario is not emerged. Among non-integrable models special
care must be given to disordered systems because of the non existence of a speed of sound as a consequence of
Anderson localization [49]. Some insights can be obtained from those models whose equilibrium behavior can be
analytically obtained by means of the strong-disorder renormalization-group [50]. Even in this case time-dependent
DMRG can help a general understanding. The time-evolution of the entanglement entropy [41] already revealed that
in these systems there is no sign of the light-cone (as easily predictable because of the absence of speed of sound) and
the effective motion of “quasi-particles” is more diffusive rather than ballistic. Furthermore the results at the largest
available times does not look thermal at all, but this can be also due to the time window accessible with numerics.
Another very interesting question is what happens at finite temperature and under what conditions a system can
equilibrate. It should be relatively simple to generalize the results for all the “quasi-free” models already considered
at T = 0. Work in this direction is in progress, but it will be almost impossible to consider the same problem for
more complicated models.
Finally it is also important to understand the role played by the initial state. We assumed always a translational
invariant one with short range correlations. Thus one natural modification consists in taking a state that is only
locally different from the actual ground-state. This problem is known in the literature as a “local quench” and it
has already been considered to some extent (see e.g. [51–53]), but still a general picture as the one outlined here
for global quenches does not exist (this is also interesting for eventual connections with quantum impurity problems,
see [54] as a review). Another natural modification of the initial state is one with long-range correlations, such as a
critical one and let it evolve with another critical hamiltonian. This has been done for the Luttinger liquid [26] and
the results show the typical functional dependence of a light-cone scenario (i.e. everything depends only on x± 2vt)
but the long-time correlations decays as power-laws, with exponents that are different from equilibrium ones and can
be predicted by the generalized Gibbs ensemble.
Acknowledgments
This work was supported in part by EPSRC grants GR/R83712/01 and EP/D050952/1. This work has been
finished when PC was a guest of the Institute for Theoretical Physics of the Universiteit van Amsterdam. This stay
was supported by the ESF Exchange Grant 1311 of the INSTANS activity.
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