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Exact relaxation to Gibbs and nonequilibrium steady states in the quantum cellular automaton Rule 54
by Katja Klobas, Bruno Bertini
This is not the current version.
Submission summary
As Contributors:  Bruno Bertini · Katja Klobas 
Preprint link:  scipost_202104_00016v1 
Date submitted:  20210414 10:25 
Submitted by:  Klobas, Katja 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study the outofequilibrium dynamics of the quantum cellular automaton Rule 54 using a timechannel approach. We exhibit a family of (nonequilibrium) 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 oneparameter 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 nonequilibrium 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.
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Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 2021916 (Invited Report)
Strengths
1) Formally correct.
2) In some cases it makes a direct connection between exact calculations of the dynamics and the generalized hydrodynamics in an interacting model.
Weaknesses
1) Not easy to read for non specialists.
2) General discussion of Rule 54 model is missing.
Report
In the work 'Exact relaxation to Gibbs and nonequilibrium steady states in the quantum cellular automaton Rule 54', K. Klobas and B. Bertini analyze the the outofequilibrium dynamics of the quantum cellular automaton Rule 54.
The paper presents in a detailed form the technical calculations, mostly of diagrammatic nature, some aspects of the dynamics of the socalled 'Rule 54' model. The authors consider a class of initial states for which they are able to provide an explicit construction of the fixedpoints of the space transfer matrix.
This is used as a tool to investigate their timeevolution and relaxation in terms of finitedimensional quantum maps. For the class of initial states considered, relaxation takes place (somewhat unexpectedly) exponentially fast to Gibbs states. For most of the cases, these results agree with the predictions of generalized hydrodynamics, providing a confirmation of the GHD for an inhomogeneous quench in an interacting system.
The paper is technical and it requires a detailed reading to understand the diagrammatic approach.
I have three main comments:
 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).
 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.
 A minor point: ref. [79] is incorrect. It cites Bobenko et al. but it links to a different PRL.
I would also recommend the authors to check the English as I found a few misspelled words.
In summary, the results are sound, the proofs are detailed and the findings are clear. After a consideration of the suggestions above, I would recommend this work for publication.
Requested changes
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).
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.
3) A minor point: ref. [79] is incorrect. It cites Bobenko et al. but it links to a different PRL.
Anonymous Report 1 on 2021728 (Invited Report)
Strengths
1 Rigorous arguments
2 Use of diagrammatic representation
3 Novelty of results
Report
The authors consider a specific case of out of equilibrium dynamics, modeled by the QCA rule 54, to identify a class of states for which a non equilibrium steady state is reached and an exact characterization can be provided. The analysis is perfored by adopting a general timechannel description for the dynamics of local operators. The authors introduce and intensively exploit a diagrammatic state representation to demostrate the oneparameter family of Gibbs states reached after relaxation.
The analysis is rigorously performed and all the statements are extensively explained.
I only point out a typo on page 19 (pugging >plugging)
In my opinion this work fully meets the general acceptance criteria. Moreover, it presents a breakthrough on a previouslyidentified and longstanding research stumbling block, namely the characterization of dynamics originating non equilibrium steady states. It also could bridge between studies about thermalization in out of equilibrium systems and matrix product states numerical simulations. For these reasons, I suggest publication in the current form of the draft.