Entanglement dynamics in Rule 54: exact results and quasiparticle picture

1- First exact calculation of R\'enyi entropies in an out-of-equilibrium interacting system.
2- Clear exposition of technical details about the rule 54 and its solution.
3- Rigorous and clearly written.

1-No major weaknesses.

The paper by Klobas and Bertini study the out-of-equilibrium dynamics of the R\'enyi entropies after a quantum quench in the rule 54 chain, which in recent years emerged as a valuable toy model for generic out-of-equilibrium interacting integrable systems.

This is an excellent paper that deserves publication in Scipost. The most important result of the paper is the calculation of the dynamics of the Renyi entropies in an interacting integrable system. This is a challenging problem. The authors find that while a hydrodynamic description of the dynamics of the Renyi entropies is possible, it is difficult to reconcile it with the standard quasiparticle picture for entanglement spreading, in agreement with previous observations in the literature. I believe that the exact results derived by Klobas and Bertini could be useful to shed light on this problem.

I have only some minor remarks:

1) In the introduction the authors discuss the applicability of the quasiparticle picture to describe the entanglement dynamics. They mention that it has not been applied in systems where the dynamics is not unitary. This is not entirely correct, as it has been extended recently for free fermions and free bosons models in

https://arxiv.org/ct?url=https%3A%2F%2Fdx.doi.org%2F10.1103%2FPhysRevB.103.L020302&v=1136ae07

and

https://arxiv.org/abs/2106.11997

and in a related setting in

https://scipost.org/10.21468/SciPostPhys.7.2.024

2) The authors observe that the dynamics of the Renyi entropies is seemingly not compatible with the quasiparticle picture. As they stress this is in accord with previous observations in the literature (Ref. 73).
However, they should mention that it is still possible to derive the steady
state R\'enyi entropies (as done in Ref. 73) for interacting integrable systems.

3) The the most important results of the paper is the calculation of the Renyi entropies. However, in the introduction they are barely discussed. I would suggest to move part of the discussion on the Renyi entropies in the introduction, for instance mentioning why they are important and what has been done already in the literature (i.e., the calculation of their steady-state value in integrable systems).

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?