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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_00002v2
Date accepted: 2021-11-12
Date submitted: 2021-10-01 19:40
Submitted by: Granet, Etienne
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical

Abstract

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:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)



Author comments upon resubmission

Dear Editor and Referees,

We appreciate the very positive reports about our manuscript. Please find below our answers to the referees with the corresponding changes.

Report1: We thank the referee for their very positive report and recommendation to publish our manuscript in its present form.

Report2: We thank the referee for their very positive report, constructive comments, and spotting a number of typos (which we have corrected). We indeed consider even lattice lengths L and now state this explicitly above equation (1). Our replies to the referee’s points (1) and (2) are as follows:

(1) "Coherent state" usually refers to states of the form $e^{b^\dagger}|0\rangle$ with $b^\dagger$ bosonic creation operators. We refer to states (10) as coherent states as they can be viewed as coherent states of bosonic zero-momentum pairs of fermions. We have added a comment below eqn (10) to make this clear. The states defined in this way are indeed a specific class.

(2) As we show in Section 2.4 the state of the system initialised in a ground state of H(h,\gamma) following and then driven out of equilibrium by any variation of the magnetic field and/or the anisotropy can always be written as a coherent state at all times. So their physical relevance is not merely that they relax to GGE: they exactly describe the out-of-equilibrium time evolution of the system at all times in such protocols. Also, the actual experimental realizability of these coherent states is thus equivalent to that of quantum quenches.

Report3: We thank the referee for their positive report and helpful comments. Our replies to the referee’s points (1)-(3) are as follows:

(1) In contrast to the references quoted by the referee we are concerned with a homogeneous system here. Hence the local physics at late times is described by a generalised Gibbs ensemble, which describes an equilibrium state in the sense of Jaynes (under the constraints that the values of the extensive number of conserved quantities are fixed by the choice of initial state). In this view “equilibrium” refers to the fact that local physical observables cease to be time dependent.

In order to avoid any confusion we have added a short comment on p.6 and added a references to the work of Jaynes, and Rigol et al:
E. T. Jaynes, Information Theory and Statistical Mechanics. II, Phys. Rev. 108, 171 (1957)

Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons Marcos Rigol,1 Vanja Dunjko,2,3 Vladimir Yurovsky,4 and Maxim Olshanii2

(2) The referee states that "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.” Our understanding of the situation is at variance with this statement. We are of course well aware that the doubling technique is useful in the continuum limit, but as far as we know the latter is in fact crucial, because Jordan-Wigner string operators reduce to bosonic vertex operators and become simple. On the lattice they are as far as we know difficult to deal with, see e.g. (3.22) in JB Zuber, C Itzykson - Physical Review D15, 2875 (1977). We are not aware of works in the literature that use the doubling trick on the lattice to obtain the kind of results the referee alludes to and we would be grateful for any specific references.

(3) The proposed expectation value would require more work. However we could compute the expectation value of $e^{i\sum_{j>J}\sigma^z_j(t)}e^{-i\sum_{j>J}\sigma^z_j(0)}$ exactly along the same lines as for the dynamical two-point function of \sigma^x. Indeed, the form factor of $e^{i\sum^z_{j>J}\sigma^z_j(0)}$ is a determinant similarly to \sigma^x, so exactly the same calculations would apply.

Best regards,
The authors

Reports on this Submission

Anonymous Report 3 on 2021-10-31 (Invited Report)

Report

I thank the authors for briefly commenting. I disagree partly with the understanding in the authors' answers, perhaps I was not clear in my comments.

First, the state obtained in the XY model by Aschbacher and Pillet is a GGE, in the sense the authors discuss. The model, and the resulting state, are homogeneous, and all local observables in the state are independent of time (not: only the initial partioning is inhomogeneous; the homogeneous stationary state obtained is that describing local observables, resulting at long tiems from local relaxation, as considered by the authors). This falls within the set of GGEs the authors discuss. Yet, the state is also out of equilibrium. This is because the GGE is not time-reversal invariant. In the language of Jaynes or Rigol, it contains charges which are not time-reversal invariant. Thus it is out of equilibrium. The more general situation, the nonequilibirum GGEs obtained from the partitioning protocols in integrable models, is discussed in

O.A. Castro-Alvaredo, B.Doyon, and T.Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6, 041065 (2016)

B. Bertini, M. Collura, J. De Nardis, and M. Fagotti, Transport in
out-of-equilibrium $XXZ$ chains: exact profiles of charges and currents, Phys. Rev. Lett. 117, 207201 (2016)

Of course, Jaynes' or Rigol's description is insufficient, as it does not properly address convergence problems in the infinite sum of charges in integrable models, and this is why one usually resort to other descriptions of GGEs, such as in terms of density of quasiparticles; but in any case, a notion of extensivity, as correctly mentioned by the authors, is retained and is part of the fundamental definition of GGEs.

Second, in my understanding there is no obstruction in using the doubling trick in quantum chains. This is simply about the fact that a U(1) symmetry appears in the doubled model as soon as there is a free fermion description. The free fermion description holds sector by sector, and thus the doubling trick can be used in each sector. It allows one to re-write JW strings of the original model, in terms of U(1) strings of the doubled model. There is no need for bosonisation. But indeed, I do not know how much of this has been done in the literature; I believe this has been at least partly considered in

J. H. H. Perk, Equations of motion for the transverse correlations of the one-dimensional XY -model at finite temperature, Phys. Lett. A 79 (1980) 1–2.

If the authors would like to make adjustments to account for the above this would be good. In any case, these are matters of general understanding, and I believe the paper can be published as it is.

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Anonymous Report 2 on 2021-10-4 (Invited Report)

Strengths

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

Weaknesses

No weaknesses.

Report

The authors have corrected typos mentioned in my report. I suggest publishing the paper in its present form.

Requested changes

No changes.

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

Anonymous Report 1 on 2021-10-2 (Invited Report)

Report

I am happy with the comments and small revisions by the authors. The manuscript was of good quality before. Now it should be published as it stands.

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