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Relaxation of the order-parameter statistics in the Ising quantum chain

by Mario Collura

This Submission thread is now published as SciPost Phys. 7, 072 (2019)

Submission summary

As Contributors: Mario Collura
Arxiv Link: (pdf)
Date accepted: 2019-11-26
Date submitted: 2019-10-17 02:00
Submitted by: Collura, Mario
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


We study the out-of-equilibrium probability distribution function of the local order parameter in the transverse field Ising quantum chain. Starting from a fully polarised state, the relaxation of the ferromagnetic order is analysed: we obtain a full analytical description of the late-time stationary distribution by means of a remarkable relation to the partition function of a 3-states classical model. Accordingly, depending on the phase whereto the post-quench Hamiltonian belongs, the probability distribution may locally retain memories of the initial long-range order. When quenching deep in the broken-symmetry phase, we show that the stationary order-parameter statistics is indeed related to that of the ground state. We highlight this connection by inspecting the ground-state equilibrium properties, where we propose an effective description based on the block-diagonal approximation of the $n$-point spin correlation functions.

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Ferromagnetic phases Ising quantum chain Transverse-field Ising chain

Published as SciPost Phys. 7, 072 (2019)

Author comments upon resubmission

Dear Editor, dear referees,
I’m glad to know the work has been appreciated by the referees and very grateful for their suggestions.
In the following, I will reply point by point to the referee’s concerns as well as indicate all changes:


1 - I revised the manuscript in order to correct as many spelling mistakes as possible,
and I removed a number of improper logical connectors. I hope now the readability of the manuscript is largely improved.

2 - I rephrased the first paragraph of the introduction as suggested by the referee, thus explicitly referring to the Ising model.
I slightly reformulated some passages of the introduction as well, in order to make the text more fluent.

3 - As pointed out by the referee, there is no connection with a S3 symmetry,
and nowhere in the paper I mentioned any sort of connection to S3.
However, there is no doubt that, in the stationary state, the generating function of the moments does coincide
with the partition function of a 1D classical model with 3 states;
it can be written as it was the 3-states Potts chain partition function, provided the classical couplings
are tuned in a very specific way (as carefully explained in the Appendix D).
Notwithstanding, I may understand that this wording could be “unhappy” thus leading to a possible misunderstanding;
therefore, I complied with the referee’s request and removed the word “Potts” from the text.
However, I do not think the Appendix D needs to be modified.

4 - Here the GGE has been used in order to exploit the extensive behaviour of the overlap with the ground state
in Eq. (78) and extract the scaling regime, namely the scaling of $\gamma$, and thereafter to get Eq. (82).
As pointed out by the referee, the Ground-State (GS) contribution to the stationary PDF has been already put forward in Ref. [33],
where a thermal stationary ensemble has been considered, since this phenomenon is not driven in any means by integrability.
However, in Ref. [33] only the large $\Delta$ expansion of the stationary PDF
has been performed (which is the counterpart of the small $h$ expansion here);
in addition, in this work, I considered the subsystem $\ell$ getting larger with $h\sqrt{\ell}$ kept fixed.
The GGE is crucial to extract in this regime the proper scaling of the stationary
PDF and connect it to both the ground-state PDF plus Gaussian fluctuations.
A couple of sentences have been added to this section to clarify this point.


a) As suggested by both referees, the word “Potts” has been removed.
As well as, “proof” (spelling mistake) became “show”.

b) Typos corrected.

c) In order to avoid confusion, I decided to remove the local identity matrix, thus just keeping the operator notation,
where $(\sigma^{x}_{j})^2 = 1$.

d) Those calculation easily come from series expanding the Fourier coefficient in
Eq.s (34) (and (50)) and therefore integrating (i.e. Fourier transforming) term by term.
Now, after eq. (34), a sentence has been added to make this clear.
Regarding the notation of Eq.s (35), (36) and (51), as pointed out by the referee, it is finite when sin(pi * n) gets cancelled by the poles
of the series. A sentence has been added to point it out.

e) Maybe I did not fully understand the referee's comment: first, how much “remarkable” is the connection, in my very personal opinion,
is a matter of taste; second, has the referee perhaps forgotten that he suggested (point (a)) removing the word “Potts”?

f) I thank the referee for having pointed out the mistake: eq (69) comes from Szego theorem; the other from Fischer-Hartwig conjecture;
I also added proper references.

g1) As a matter of fact, the quantum Ising chain is a very well known model which has been solved more than fifty years ago.
Every times a section of the manuscript is based on some already known results (e.g. in Sec. 2, part of Sec. 3, few equations in Sec.s 4 and 5,
part of Sec. 6), I usually refer to them by citing the appropriate references at the very beginning.
My policy is to avoid to repeat every time in a while the same bibliographic citations.
Moreover, simply manipulation of well known formulae (e.g. Sec.3.1, Sec 3.2) as well as basic expansions (e.g. in Sec. 4.1, 5.1)
are such that they do not need any citation at all, independently whether this is the first time they have been written or not.

g2) The gaussian behaviour has been checked against the numerics (as provided by many figures in the paper),
and it is expected to be rigorously valid in the specific scaling regime explained in Sec. 3.1.
In other words, Eq. (44) is exact when $\ell\to\infty$ and $m$ is rescaled with $\sqrt\ell$.
The same in the ferromagnetic regime, when Gaussian restoration is mentioned after Eq. (48).
Similar reasoning applies for the Ground-State PDF.

h) There is no discussion because it is not special at all. The stationary generating function of the moment reduces
to the partition function of the classical Ising chain. In other words, the quench to $h=1$ is already included in the Sec. 4.1.1.
However, if the referee here is referring to the Ground-State properties at the critical point, well, the scaling of the
order parameter PDF has been already obtained by A. Lamacraft and P. Fendley in Ref. [56].

List of changes

see resubmission letter

Reports on this Submission

Anonymous Report 3 on 2019-10-31 (Invited Report)


The author has answered my questions in a satisfactory manner. I recommend publication in Scipost.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 2 on 2019-10-18 (Invited Report)


In view of the changes made by the author I recommend it for publication in SciPost.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2019-10-18 (Invited Report)


In view of the changes made by the author I recommend it for publication in SciPost.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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