Entanglement dynamics after quantum quenches in generic integrable systems

1- very well explained
2- quite exhaustive in the models and effects it checks
3- excellent numerical verifications

1- not a lot new, as it is mainly a long version of another paper proposing the formula
2- slightly repetitive reading

In this paper the authors study the evolution of entanglement entropy in homogeneous quantum quenches. The main point is to study and verify numerically formula (4) which was proposed in [10], which gives full precision, in the case of Bethe integrable systems, on the "general structure" formula (3) proposed in [6]. Formula (3) was based on physically counting particle pairs creating entanglement between the region and its outside, as they evolve after a quench. Formula (4) gives meaning to these pairs via the quasiparticles of the thermodynamic Bethe ansatz. In the present paper, formula (4) is fully and exhaustively analyzed in a variety of models, including free models (in the free boson it seems there does not exist derivations of it), in the XXZ chain and in the Lieb-Liniger model (both attractive and repulsive). Numerical verification is giving in free models and in the XXZ chain, which are very convincing. The physics underlying the actual behaviour of the entanglement entropy as predicted by formula (4) is also analyzed, with contributions from various types of quasiparticles highlighted.

I believe this is a very good paper. Of course, it does not present fundamental new ideas, but its main purpose is to provide full analysis of ideas proposed previously, which it does well. It is a little bit boring to read, but this is in a sense by construction as per its main purpose. It is not too complicated, with emphasis given on the intuitive explanations instead of the technical details (although all technical details seem to be appropriately presented).

I do not have any proposed changes (only typo I found: "a part" -> "apart" top of page 5).

no change requested

1. Quantitative picture of entanglement dynamics in interacting systems
2. Good agreement with numerical results
3. Well-written paper, results easy to follow

Extended version of Ref. [10], with only a few new results

In this paper, the authors study the evolution of the entanglement entropy after a quantum quench in integrable systems using a quasi-particle picture. This paper is essentially an extended version of Ref 10, including technical details and some new results. The main idea is to combine the quasi-particle picture of entanglement propagation put forward by Calabrese and Cardy in [6] with more quantitative ingredients accessible using Bethe ansatz results (quasi-particle velocities and Yang-Yang entropy in the steady state) to arrive at a quantitative prediction for the dynamics of entanglement. Remarkably, this simple picture that entanglement is generated by EPR pairs of quasi-particles emitted at the same point in the initial state appears to be quantitatively accurate. The authors explore this idea in detail and compare to numerical results for non-interacting and interacting systems. In the case of non-interacting systems, the approach is especially simple and the results can be compared to existing ab-initio results, and to numerical results. For interacting systems such as the XXZ spin chain, the authors explain review how the steady-state thermodynamic entropy and the group velocities can be computed using Bethe ansatz techniques, and they compare the predictions of the quasi-particle approach to numerical matrix product state calculations. This part of the paper follows closely Ref 10, but it includes more technical details and some new interesting discussions including a breakdown of the quasi-particle contributions to entanglement, and results on the dynamics of the mutual information. The authors also study the entanglement dynamics in the Lieb-Liniger model using their approach: it will be interesting to see if their results can be compared to numerics or to other approaches in the future.

I think this is an interesting, well-written paper that is suitable for publication in SciPost. In my opinion, it is quite remarkable that the quasi-particle picture of Ref 6 is quantitatively correct. I recommend publication as is.

In the introduction, the authors write:
``A major breakthrough in this respect has been achieved in [10] where it has been shown that, at least for certain classes of quenches in integrable models, the function $s(\lambda)$ can be conjectured from the equivalence between the entanglement and the thermodynamic entropy in the stationary state.’’
While I find the idea of Ref 10 to use the picture of Ref 6 more quantitatively very nice, I find it odd to put the emphasis on the relation between the entanglement and the thermodynamic entropy in the stationary state, which seems like the most natural part of this result. For any thermalizing system (in a generalized sense for integrable systems, which thermalize to GGEs), one would expect the entanglement entropy to coincide with the thermodynamic entropy in the steady state. In generic systems, this follows from ETH or from the very idea of thermalization. While it’s remarkable that (4) can be used to capture the entanglement dynamics, I see no reason to doubt that at long times, the entanglement entropy should coincide with the steady state entropy. Do the authors have any reason to question this?

1- Analytical results on a timely subject.
2- The numerical verifications are rather convincing.
3- Easy to read.

1- Most of the results rely heavily on another paper, Ref[10], by the same authors.

In this paper, the two authors study the large time entanglement dynamics following a quantum quench in several Bethe Ansatz integrable models. Explicit analytical formulas are derived in various quench setups. Those are then verified numerically using the DMRG algorithm.

The starting point is equation (4), which was derived in Ref.[10] by the same authors. The result is then applied to several interesting quench situations, where it can be used to derive explicit analytical formulas. Most of those protocols were considered before by other authors, but the entanglement results are new.

Overall the paper is interesting and timely. The derivations are explained in great detail, which makes the manuscript rather easy to follow. Given the interest in entanglement growth and effective descriptions of quantum systems out of equilibrium, the results presented here will be quite useful to researchers working in this field.

For these reasons I recommend publication, provided the authors address the minors points listed below.

a) The logic in the introduction could be improved a bit, I think. The authors spend a great deal of time discussing equation (3), and where it comes from. Then, suddenly, they switch to (4), a multi-species generalization with very little explanation. It should be at least mentioned, already around (4) that (i) in general there are more than one species (ii) the different species do not interact.

Another confusing aspect is the equality in (3), (4). It is not clear whether those are meant as strict equalities, or as an asymptotic expansion as in (106). It seems to me there could be order one corrections to (3) and (4). Those should be mentioned at the beginning, to avoid confusion.

b) Pages 2-3. The sentence "Remarkably, in recent years [...]" seems a little bit out of place, and should perhaps appear earlier in the introduction.

c) Pages 5-6, section 2.1.1. Mention that periodic boundary conditions are assumed.

d) In section 3.1, it's not completely clear starting from where $\Delta>1$ is assumed. Could the author also comment somewhere on the more complicated "gapless" case?

e) The expression "quench action" appears for the first time in section 5. The terminology used in sections 4 and 5 should perhaps be made more consistent, however the authors see fit.

f) Page 27. This is mentioned in the conclusion, but it might be useful to add a quick comment regarding the difficulties in numerical simulations in the Lieb-Liniger model.

Typos, etc.

a) Page 1, second paragraph. "i.e. , at $t=0$ a parameter". i. e. would be advantageously replaced by e. g.

b) Beginning of page 5. " a part" should read "apart"

c) Page 30. A word is missing at the end of the sentence "but it would be very interesting to do."