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

1- Interesting results on a not so well studied problem.
2- Careful numerical checks of all claims made.

1- Clarity could be improved a bit.
2- No new analytical results for the critical point.

This paper deals with the probability distribution of the order parameter in the one dimensional quantum Ising chain. Particular emphasis is put on the long time behavior after a quench from a polarized state, where analytical results are obtained using the mapping onto free fermion.

Overall the paper contains valuable results on a not so well studied problem, where the free fermion machinery is not so easy to apply. All analytical insights are supplemented by convincing numerical checks. The derivations of some of the results, as well as the writing, could be improved, however.

I list below a few issues I have with the paper, some of which are minor. The paper should be publishable in Scipost, provided those are addressed.

a) Abstract: The reference, here and elsewhere, to the three state Potts model should be removed. 'We proof' should read 'we prove'. Also, there are no rigorous proofs in the paper, so the author might consider removing such claims, and use the more neutral 'we show' instead.

b) Various typos. Page 1 'let us suppose to prepare' --> 'let us prepare'. Page 7, top: 'does not applies' --> 'does not apply'. Page 8, top 'move in the interval' --> 'are in the interval'. 'Previous equation'--> 'The previous equation'. Page 18: 'able to verified' --> 'able to verify'. There is an exclamation mark in equation (62).

c) The notation $\mathbb{I}_j$ in section 3.2 is confusing, and even leads the author to make statements that are, strictly speaking, incorrect. For a chain of length $L$ the matrix $\sigma_j^x$ is a $2^L\times 2^L$ matrix, which acts as a $2\times 2$ Pauli matrix at site $j$, and identity elsewhere. Its square or exponential is a $2^L\times 2^L$ matrix and cannot equal a $2\times 2$ matrix. It would be preferable to use $\mathbb{I}$ in formula (21) and (22), since the $2^L \times 2^L$ identity acts as identity everywhere.

d) The calculations in 4.1 are quite cryptic, and some of the expressions showed are not very readable. For example in equation (35) \sin \pi n is zero. While I understand this gets cancelled by the poles on the rhs, there surely must be a better way to present it. It might also be useful to rewrite (34) in a way that makes it clear why most Fourier coefficients are zero. Same comment around equation (51).

e) Near equation (48), it should be made clear that this is the partition function of the 1d classical Potts model, which makes the connection not particularly remarkable.

f) A reference is needed near equation (69). Also, is it the Fischer-Hartwig conjecture, or Szego's theorem? A reference is also needed around equation (74) and on page 23.

g) In several places in the paper, it is difficult to tell which results are well known, and which are new. Any improvement in that direction would be welcome.

g) It is not clear to me whether the gaussian behavior discussed in several places in the paper is an assumption that is checked, or a result that is derived.

h) There is no real discussion of the quench to the critical point $h=1$.

1-tackles a not so well studied, but physically interesting problem that is the relaxation of the order-parameter statistics after quantum quenches in models with a broken-symmetry phase
2- exact results using free fermions, complemented by extensive numerics
3-nice phenomenological discussion in sec. 6

1- Many typos, syntax or spelling mistakes
2- Some sloppy statements ("remarkable connection" with the 3-states Potts model sounds inexact and misleading to me... se below)

The author studies the relaxation of the order parameter probability distribution function (PDF) after a quantum quench in a paradigmatic model with a spontaneously symmetry-broken phase, the transverse field Ising model.
While there is a huge amount of literature on quantum quenches, most results to this date are devoted to the study of local correlation functions or entanglement measures, and the broader question of full counting statistics has been tackled only recently. Using the free fermionic representation of the transverse field Ising model, the author re-expresses the PDF both at finite time or at equilibrium in terms of 2-point functions, which are known from previous works. This yields exact expressions for the PDFs in the late time stationary states, which allow to observe interesting physical features among which a memory of the initial order when quenching within the symmetry broken phase. The PDF within a subsystem of size $\ell$ indeed retains a bimodal structure whenever $\ell$ is not too large, while it becomes gaussian in the $\ell \to \infty$ limit.
These cute observations, supported by an interesting phenomenological comparison with equilibrium ground state order, make up for an good quality paper which deserves publication in SciPost.
However, there are several aspects (including some rather important) that I feel would need to be corrected before acceptance.

1 - There is a huge number of spelling or syntax mistakes, for instance :
- in the abstract : "we *proof* that the stationary order-parameter..."
- in the intro : "doublY degenerate ground state"
- later in the intro : "let us suppose *to* prepare" does not feel quite right
- "in particular analytical findingS"
- section 2, 1st sentence : "transfer field" -> "transverse field"
- "Bogoliuobov" is not the most common spelling of "Bogoliubov"
- before eq. (6), no spacing between "a part"
- after equation (9), the use of "Indeed" seems quite improper. There is no clear logical connection between what's before and what follows
- etc....
In general there are many such mistakes throughout the paper, as well as improper uses of logical connectors (some abuse of "indeed" or "However"). This should clearly be improved.

2- The introduction needs to be reorganized a little.
For instance, the first paragraph deals with models with broken symmetry phases in general, so it is not fully appropriate to mention doubly degenerate ground states : either the author should mention the Ising chain (which is the focus of the paper) at this stage, or talk more generally about multiply degenerate ground states.
Furthermore, in paragraph 5, the sentence "In general, after a global quantum quench, we expect long-range ferromagnetic order disappears...." sounds like a repetition of paragraph 3, and should therefore be fixed..

3- On a more physical level, it sounds quite incorrect and misleading to mention "a striking connection with the partition function of the 3-states Potts model", as is done in several places in the paper. What the authors refers to here, is the possibility to rephrase the moments generating function as the partition function of a 1D model, which can be computed through a transfer matrix approach.
While this is very practical, I don't think that there is a $S_3$ symmetry here, and the only common feature with a Potts model is that one deals with a 3-states system.
If the author agrees with this, I propose to remove any reference to the Potts model, which may confuse the reader into thinking that there exists a connection between Ising and 3 states Potts models.

4- In section 6, the author spends a paragraph discussing the GGE in the transverse field Ising chain, however it is not completely clear to which extent this is used in the following. In particular, it seems that the features described here, namely a crossover as $\ell$ is increased between ordered and Gaussian behaviours, have been related in ref. [33] to features of thermal equilibrium states in a similarly integrable model. It would be useful if the author could discuss a bit the comparison of the present results and those of ref [33], and what additional features the GGE might bring.