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Entanglement dynamics in Rule 54: exact results and quasiparticle picture

by Katja Klobas, Bruno Bertini

Submission summary

As Contributors: Bruno Bertini · Katja Klobas
Preprint link: scipost_202104_00017v1
Date submitted: 2021-04-14 10:30
Submitted by: Klobas, Katja
Submitted to: SciPost Physics
Academic field: Physics
  • Mathematical Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


We study the entanglement dynamics generated by quantum quenches in the quantum cellular automaton Rule 54. We consider the evolution from a recently introduced class of solvable initial states. States in this class relax (locally) to a one-parameter family of Gibbs states and the thermalisation dynamics of local observables can be characterised exactly by means of an evolution in space. Here we show that the latter approach also gives access to the entanglement dynamics and derive exact formulas describing the asymptotic linear growth of all Rényi entropies in the thermodynamic limit and their eventual saturation for finite subsystems. While in the case of von Neumann entropy we recover exactly the predictions of the quasiparticle picture, we find no physically meaningful quasiparticle description for other Rényi entropies. Our results apply to both homogeneous and inhomogeneous quenches.

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Submission scipost_202104_00017v1 on 14 April 2021

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Anonymous Report 1 on 2021-5-26 (Invited Report)


In this work the authors study the quench dynamics in a quantum model for the discrete unitary evolution, the so-called Rule-54. This is an interesting model, which has recently received renewed attention due to the fact that, in some sense, it represents one of the simplest, interacting integrable models. The authors expand the result recently presented in Ref. [31]. They certainly go much beyond what was discussed therein, and the manuscript contains many new and non-trivial results.

The research presented is timely, as it explores an approach to the quench dynamics based on a ``transverse evolution”, which is currently of interest for different groups. In fact, the authors provide a series of exact results which are expected to be appreciated even beyond the field of integrability.

In summary, the main results of the present manuscript are: 1) the derivation of a new family of initial states for which the quench dynamics can be solved exactly; 2) a rigorous derivation of a formula for the asymptotic growth of the Renyi entropies after the quench; 3) the first analytical test of a conjecture due to Alba and Calabrese for the growth of the Von Neumann entanglement entropy; 4) finally, the authors exhibited evidence of the impossibility of establishing a quasi-particle picture for the growth of Renyi entropies.

All these results are of physical significance. In addition, I believe that the paper is very clear and well written.

For the reasons above, I recommend publication.

I have, however, one question for the authors. Although I believe the answer could be of interest, I do not expect them to comment on this on the manuscript.

My question is the following. It is known that Rule 54 can be solved using the Bethe Ansatz method (as shown for instance in Ref. [67]). Therefore, in principle one could expect that the folded transfer matrix could also be diagonalized using Bethe Ansatz. Have the authors explored this direction? How does this relate to the tensor-network approach presented by the authors? In fact, a similar approach could presumably be applied to more general (Floquet) integrable evolutions, such as the Heisenberg XXZ chain, where transfer matrices can be also diagonalized via Bethe Ansatz. Has this approach been explored or do the authors believe there is some fundamental obstacle?

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