Entanglement spreading and quasiparticle picture beyond the pair structure

- quench results for genuinely interacting initial Hamiltonians
- very good agreement of perturbation theory with numerics

- quench example considered is somewhat artificial

The authors study the growth of entanglement after a quench
where the initial state does not allow for a description
in terms of simple quasiparticle-pair excitations.
In particular, they consider the case with a special multiplet
structure, constructed in a translational invariant fashion,
such that more than two entangled particles are emitted
at every spatial location. A generalized quasi-particle
picture is then developed and applied for the calculation
of the Renyi entropy in a perturbative manner, i.e. in a
power series expansion of the multiplet creation amplitudes.
Unfortunately, the von Neumann entropy can not be obtained
in this way, since the perturbation expansion does not
commute with the analytic continuation of the Renyi index.

The leading order result for the Renyi entropy is obtained
explicitly and tested on a specific lattice model, namely a
weakly perturbed Ising chain. The quench is performed from
the ground state of the weakly interacting chain towards the
free-fermion point. The authors carry out numerical iTEBD
simulations which nicely confirm the first order result,
shown to be clearly distinct from the pair-structure ansatz.

The manuscript deals with an interesting problem and the
results are novel and sound, I believe that they deserve
publication in Scipost Physics. I have only some minor
issues which the authors should address.

I.
In order to test the prediction of the generalized quasiparticle
picture, the authors use a lattice Hamiltonian that is heavily
fine-tuned in order to suppress the pair-creation amplitude.
I believe this is chosen such that the pair/multiplet ansatz
curves for the entropy production be better distinguishable.
However, I'm missing some comment about the generic situation,
i.e. when there are simultaneous pair and quadruplet production.
Is the effect of the quadruplets very small in this case?
Or otherwise, can the authors think of a realistic quench
scenario where the pair-production can be suppressed without
fine-tuning of the couplings?

II.
The manuscript is full of typos and sentences with incorrect grammar.
Below only a few examples I noticed. However, I would strongly
recommend a thorough proofreading of the full text.

1. ansätz --> ansatz (throughout entire manuscript)
2. Von Neumann --> von Neumann (throughout entire manuscript)
3. "being it either classical or quantum" ?? (in introduction)
4. "provides a net and clear framework" ?? (in introduction)
5. "obtained tracing out" --> obtained by tracing out (after Eq. (1))
6. "this is in not true in" (after Eq. (9))
7. Axis labels of Fig. 2 left: the (N) should be in superscript

1- Provides some new analytical results concerning a difficult and interesting problem (entanglement entropy growth after a quantum quench in a 1d interacting problem).
2- The above results go beyond the simple "pairwise-created quasiparticle" picture, and are checked with iTEBD numerics (on an Ising spin chain).

1- The numerical part is quite short and could have been developed further.

This paper presents a theoretical study of the growth of quantum entanglement after a quantum quench. The specific situation that is treated is that of a weakly-interacting one-dimensional system that is quenched to a free-fermion Hamiltonian. It is generally useful to describe the initial state in terms of the elementary excitations of the post-quench Hamiltonian. In the simplest situations (such as free-to-free quenches) the initial state can be described in terms of excitations created (only) in pairs. In such situations where the pairs are uncorrelated, the use of the Wick theorem allows to compute the entanglement entropies. However, in more general situations, higher "multiplets" of excitations are also created (as described in Ref. [71]), and the calculation of the entropy becomes very nontrivial. This manuscript deals with this more general case, by focusing on the limit where the pre-quench Hamiltonian is weakly interacting. In that limit the authors obtain perturbatively an expression for the Rényi entanglement entropy of a subsystem, as a function of time and of the initial state decomposition in terms of quasiparticles (Eq. 36). The authors then consider a specific quench in a 1d quantum Ising chain and compute the associated entropy rate (Eq. 38). This results is then checked with direct numerical simulations, using the iTEBD algorithm.

This paper is well writte and contains some new and interesting results. I would recommend it for publication in SciPost after the authors have addressed the questions below (as well as the "Requested changes").

- The entropies shown in Fig. 2 (as well as in Fig. 3) appear to be extremely small (at most of the order or 10^(-5) or 10^(-4)). Is the present perturbative approach limited to such tiny entropy values ? If not, the authors could try to illustrate their approach with examples where the entropy growth is quantitatively more larger.

- The right panel of Fig. 2 shows that the calculations using the generalized quasi-particle picture are correct, but it also indicates that the simpler pair-ansatz is a relatively good approximation. Given the fact that the parameters of the quench were precisely chosen to suppress the production of pairs of quasiparticles, we might expect that in generic situations the pair-ansatz would be an even better approximation. Can the author comment/elaborate on this ?

- The Eq. (5) does not seem to be an extensive energy. Missing system-size factor ?
- A few details about the derivation of Eq. (29) would be useful.
- The Section 3 explains how to compute entanglement entropies using the quasi-particle picture. The title of this section could therefore be a bit more explicit, for instance by mentioning "entanglement entropy".
- Going from Eq. 36 to eq. 38 certainly requires a few intermediate steps. Giving a few additional calculation details would be useful.
- Due to the tiny linewidths in the right panels of Fig. 2 the colored lines are difficult to distinguish. It is also not very aesthetic to have large fonts in the right panels, and small ones in the left one.
- No details are given about the numerics. It would be useful to provide the readers with some informations about the iTEBD implementation and the simulations parameters (bond dimension and/or discarded weight, time step, possible convergence checks). These precision issues seem all the more important as the data displayed in Fig. 2 are obtained by substracting O(1) entropies (at time t and time 0) which are very close to each other.