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Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton Rule 54

by Katja Klobas, Bruno Bertini

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Submission summary

Authors (as registered SciPost users): Bruno Bertini · Katja Klobas
Submission information
Preprint Link: scipost_202104_00016v2  (pdf)
Date accepted: 2021-11-17
Date submitted: 2021-10-08 12:06
Submitted by: Klobas, Katja
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


We study the out-of-equilibrium dynamics of the quantum cellular automaton Rule 54 using a time-channel approach. We exhibit a family of (non-equilibrium) product states for which we are able to describe exactly the full relaxation dynamics. We use this to prove that finite subsystems relax to a one-parameter family of Gibbs states. We also consider inhomogeneous quenches. Specifically, we show that when the two halves of the system are prepared in two different solvable states, finite subsystems at finite distance from the centre eventually relax to the non-equilibrium steady state (NESS) predicted by generalised hydrodynamics. To the best of our knowledge, this is the first exact description of the relaxation to a NESS in an interacting system and, therefore, the first independent confirmation of generalised hydrodynamics for an inhomogeneous quench.

Published as SciPost Phys. 11, 106 (2021)

Author comments upon resubmission

We thank both referees for their careful reading of our manuscript, for their relevant comments, and for their positive assessment. We have made a number of modifications to the manuscript to accommodate the comments of Referee 2 and to follow an interesting comment of Referee 1 of the companion paper. To help identifying the changes we highlighted them in red in the new version.

Response to referee 1

We sincerely thank the referee for their very positive assessment. We corrected the typo that they pointed out.

Response to referee 2

We thank the referee for their time and overall positive assessment. However, we believe that they did not entirely appreciate the novelty and importance of the paper's results and we find some of their marks unreasonably low. We provide a detailed response below.

We begin by commenting on some of the referee's judgements concerning strengths and weaknesses.

S2 In some cases it makes a direct connection between exact calculations of the dynamics and the generalized hydrodynamics in an interacting model.

We believe that this judgment (as well as the analogous one presented in the summary) is not entirely giving justice to the results of the paper. Our work presents the first known instance where some predictions of GHD (as well as some statements concerning relaxation after homogeneous quenches) could be rigorously proven in the presence of interactions. This is of course done in some cases because, for such a microscopic proof, one needs to specify a particular rule for the microscopic dynamics and some particular initial states. Moreover, as discussed in Sec. 6, some regions of the profiles could not be accessed. Nevertheless, we think that this is a very remarkable achievement, which is not entirely conveyed by the above statement.

W1 Not easy to read for non specialists.

As mentioned in the response to S2 this paper aims at providing (unprecedented) microscopic derivations for some conjectures concerning relaxation in interacting quantum many-body systems. This requires some level of technical detail. To make the results of the paper as accessible as possible we decided to use the well known diagrammatic language of tensor network theory, see, e.g., Ref. [62].

Even though the referee probably does not share this opinion, we believe that presenting our results using the aforementioned diagrammatic approach makes the paper easier to read for a sizeable community of theoreticians (this is, for instance, the case of Referee 1). Therefore, we decided to stick to this choice.

To further improve readability we modified the discussion in Sec. 2.

W2 General discussion of Rule 54 model is missing.

In this paper the RCA Rule 54 is used merely as a particularly simple interacting quantum circuit. To avoid overloading the reader with inessential information, we didn't include a detailed description of the many interesting properties of the model. However, we partially took this point of the referee and in the new version we expanded the discussion about Rule 54 (within the reasonable space limits for a paper that is already long (further detail can be found, e.g., in the recent review [57]).

Next, let us briefly answer the referee's main comments.

1 This paper is the first of a series of two works by the same authors. For this reason, I would strongly recommend to add a couple of additional paragraphs at the beginning of Sec.3, or alternatively in the introduction, to discuss Rule 54 model with a broader perspective (not only citing relevant references) which would be of great importance for non experts in this technical field. I am also convinced that this would improve much the
readability of the paper(s).

As discussed in the response of W2, we decided to partially follow the referee’s suggestion concerning this point.

2 In Sec.5 the authors derive relevant relations for inhomogeneous quenches. The author might consider to move the proof of property 3 to the appendices and leave the relevant physical discussion about the exponential relaxation in the main text.

After careful consideration we decided not to follow this suggestion. The proof of Property 3 is one of the main results of the paper and gives important physical insights. Therefore, we believe that it belongs in the main text. Moreover, some of the concepts introduced there are also used in the following subsections.

3 A minor point: ref. [79] is incorrect. It cites Bobenko et al. but it links to a different PRL.

Fixed. We thank the referee for pointing this out.

4 I would also recommend the authors to check the English as I found a few misspelled words.

After a very careful double-check we found 4 typos. They are all corrected in the new version.

Finally, let us briefly comment on the marks given by the referee as we find them unjustifiably low.

Validity: High

This paper is about rigorous results. Unless the referee can point out anything wrong in the derivation we would expect to see the maximal mark in this entry.

Significance: Good

We struggle to understand this mark. This paper provides the first ever proof of relaxation in an inhomogeneous setting. In the words of Referee 1 "it presents a breakthrough on a previously-identified and long-standing research stumbling block".

Originality: Good

We disagree also with this mark as we believe that the proposed discussion of the time-channel approach is, in fact, rather transparent. As discussed in the response to W1, this low mark is probably due to the referee’s lack of appreciation for the diagrammatic representation employed in the paper.

Grammar: Good

Can the referee point out an example of bad English grammar for giving such a low score? Note that "grammar" is not the same as typos.

List of changes

- Improved the presentation in Sec. 2.
- Added a paragraph about time-channel approaches in integrable systems in Sec. 2.
- Added a short discussion of the model at the beginning of Sec. 3.
- Extended Conclusions.
- Minor changes across the paper (typos etc).

All changes are for clarity denoted by red in the new version.

Reports on this Submission

Anonymous Report 2 on 2021-11-7 (Invited Report)


As from my previous report, the work can be published.

  • validity: -
  • significance: -
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Anonymous Report 1 on 2021-11-7 (Invited Report)


I am satisfied with the modifications of the manuscript and with the detailed reply to my comments. The authors provided convincing explanations of the points raised in my first report. I therefore recommend the paper for publication in SciPost.

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

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