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Out-of-equilibrium dynamics of the XY spin chain from form factor expansion

by Etienne Granet, Henrik Dreyer, Fabian H.L. Essler

Submission summary

As Contributors: Fabian Essler · Etienne Granet
Preprint link: scipost_202107_00002v1
Date submitted: 2021-07-02 13:25
Submitted by: Granet, Etienne
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


We consider the XY spin chain with arbitrary time-dependent magnetic field and anisotropy. We argue that a certain subclass of Gaussian states, called Coherent Ensemble (CE) following [1], provides a natural and unified framework for out-of-equilibrium physics in this model. We show that all correlation functions in the CE can be computed using form factor expansion and expressed in terms of Fredholm determinants. In particular, we present exact out-of-equilibrium expressions in the thermodynamic limit for the previously unknown order parameter 1-point function, dynamical 2-point function and equal-time 3-point function.

Current status:
Editor-in-charge assigned

Reports on this Submission

Anonymous Report 3 on 2021-9-15 (Invited Report)


1- well written
2- interesting technique and results
3- correct


1- results particular to the XY model


In this paper, a technical development is presented for the XY model. Certain states, called Coherent States, are introduced, which are shown to be the correct states to describe quenches in the XY model. In these states, physical quantities of interest are expressed in terms of Fredholm determinants and pfaffians. Such expressions had not been obtained before in the XY model, although other techniques allow one to obtain such types of expressions in the XX model, which possesses U(1) symmetry.

Identifying a family of explicit states that describes the result of quenches, with arbitrary time-dependent model parameters, is something that had been done in the transverse field Ising model before, and this paper adapts the theory to the XY chain.

The expressions obtained for the various physical quantities studied appear to be new, and are certainly very interesting. They open the door for asymptotic analysis.

I think the paper should be published, after the following comments have been briefly addressed.

1. Page 6: the term “equilibrium physics” can be misleading with respect to previous literature. Certain states of this type have been characterised previously as “non-equilibrium steady states”, for instance those occurring in the “partitioning protocol” or with “domain wall initial state”. For the XY model see [Aschbacher W H and Pillet C-A 2003 Non-equilibrium steady states of the XY chain J. Stat. Phys. 
112 1153] (this is older than GGEs, but the result can be interpreted as a GGE), and more generally [32,33]. In fact the inclusion in (18) of any H_n that is not time-reversal invariant makes the state "non-equilibrium", in a common nomenclature.

2. There is a well-known technique by which the XY model is “doubled” in such a way that a U(1) symmetry emerges. Normally, by the doubling trick, in the doubled model, order parameter correlation functions may be evaluated using the standard techniques from U(1) invariant models. Can the authors comment on this?

3. One quantity of interest is the FCS for transport: the exponential of the total transfer of spin, with sum_{j>J} (sigma^z_j(t) - sigma^z_j(0)) instead of the space-sum of the density as in eq 66. Is this accessible by the current techniques? Working this out here would probably be too much, but at least a comment on feasibility would be welcomed.

Requested changes

1- Address the three comments made in the report
2- Typo: “at different at different“ page 5

  • validity: top
  • significance: good
  • originality: good
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Anonymous Report 2 on 2021-9-12 (Invited Report)


original new results, opens a new route of thinking about the subject

clear and detailed presentation of results


no particular weaknesses


The authors consider correlation functions of local operators in a large family of coherent states of the XY chain with time dependent anisotropy $\gamma (t)$ and magnetic field $h(t)$. The coherent states have peculiar properties. They depend on the parameters $\gamma$ and $h$ of the Hamiltonian and on two arbitrary functions $A$ and $f$. Their time evolution can be expressed as an explicit time evolution of these functions. Moreover, Theorem 1 of the manuscript claims that every coherent state with system parameters $\gamma$ and $h$ and functional parameters $A$ and $f$ can be explicitly re-expressed as a coherent states with an arbitrary different set of system parameters $\tilde \gamma$ and $\tilde h$ and new functional parameters $\tilde A$, $\tilde f$ that depend on $\gamma$, $h$, $\tilde \gamma$ and $\tilde h$. This makes it possible to use special values of $\gamma$, $h$, for which the form factors in the Hamiltonian basis are of Cauchy determinant form, in the calculation of matrix elements and overlaps. Building upon these technical achievements the authors are able to express order parameter one-point functions, dynamical two-point functions and equal time three-point function in the thermodynamic limit in terms of Fredholm determinants and Fredholm Pfaffians. The latter objects are numerically efficient and enable to compute the correlation functions at high numerical precision. They will be also amenable to asymptotic analysis in the framework of non-linear steepest descent methods.

For the physical interpretation the authors show that in the long-time limit their coherent state averages relax to a sub-class of so-called generalized Gibbs ensemble averages.

The paper is well-written. The results are presented in sufficient detail such that the readers may verify them on a technical level. As far as I can judge about it all important references are included.

Scope and quality of the paper doubtlessly qualify it for the publication in SciPost Physics.

Here are a few minor remarks on the text. I guess the authors consider even $L$ on page 4. This might be explicitly stated. Below equation (14) on page 6 it should probably read "are given in Lemma 1" instead of "is given in Lemma 1". On the right hand side of equation (60) I would have expected a bold face K under the integral.

Being not so much of an expert on the subject I would have liked the following two point a bit more discussed. 1.) I am familiar with coherent states of Bosons. Are the coherent states of Fermions as used in the paper a standard notion or were they actually introduced in [66,67]? Are they the most general known coherent states in this case, or are they a sub-class? 2.) I wonder in which cases systems can be prepared in a coherent state, in other words, how much physically relevant such coherent state averages are beyond the fact that they can relax to GGE states. I understand that the authors give a nice example in section 4.1, but I wonder, how special this result is and if there are more relevant examples of this kind.

Requested changes

No specific changes required. Please take my comments in the last paragraph of the report as a suggestion of optional amendments.

  • validity: high
  • significance: top
  • originality: top
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Anonymous Report 1 on 2021-8-13 (Invited Report)


1. Important results are obtained.
2. An interesting method to study the nonequilibrium dynamics is proposed.
3. The paper is beautifully written.


No weaknesses.


The article studies the nonequilibrium dynamics of the XY Heisenberg chain. The authors calculate the order parameter expectation value, two-point function (dynamical), and three-point (static) correlation functions. These results certainly deserve attention. The approach based on the use of coherent states is of particular interest. This method strongly simplifies calculations by choosing the most convenient basis. If this approach can be generalized to models with nontrivial interactions, this will be a significant advance in studying the nonequilibrium dynamics of quantum integrable systems.

The paper is beautifully written. All formulas are either deduced in detail or provided with links to original papers. I only noticed two typos (see requested changes).

I highly recommend this article for publication in its present form.

Requested changes

1. Line above Theorem 1. Double `at different'.
2. Paragraph between (153) and (154), 2nd line. \sigma^x_\ell should be \sigma^x_j.

  • validity: high
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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