Universal Entanglement Dynamics following a Local Quench

1) Interesting analytical attempt at calculating the entanglement entropy in the crossover regime, which is a difficult problem.
2) Strong evidence for a universal scaling form of the entanglement entropy at intermediate times.
3) Variety of the techniques used: form factors, numerics, perturbed conformal field theory.

I see only one weakness, but it is significant. As pointed out by the other two referees, the authors multiply their "first order" result by a factor 4/3 to correctly reproduce the conformal limit. I find it very hard to justify such a "trick", without knowing the higher order form factor contributions. Even more worrisome is the fact that the actual result of the first order expansion is not really shown in Fig.2. Near the end of the supplemental material, it is stated that several terms of the same order are discarded, because they agree less well with the numerics.

This paper deals with the real time dynamics after connecting two wires through an impurity. In particular, the entanglement generated between the two wires is studied.

Overall the paper is interesting, and the subject is timely. The method used (form factor approach) allows to go beyond the known results of conformal field theory, and to explore the crossover physics at intermediate time scales of the order of the inverse Kondo temperature. The analytical results are also compared to numerical simulations.

My main unease with the manuscript as it stands lies in the comparison between their main result and the numerical simulations (see weaknesses). While reading through the text near figure 2, one gets the impression that the first order form factor calculation is shown, modulo multiplication by 4/3. This is not so if one believes what is written in the supplemental material: out of the five terms generated to first order, four are dismissed because they "give a result, which, in fact, agrees less well with the numerics". That such terms vanish in the conformal limit makes no difference; after all, what the authors are after is precisely the intermediate, non conformal, regime.

1) Show in Figure 2 the full first order result, not just equation (7).
2) State more clearly in the main text what is shown in figure (2).

It is crucial that those two points be successfully addressed. Even though the manuscript is interesting, it lacks clarity in the present form.

1) Manuscript deals with an interesting problem
2) Interesting analytical approach to calculation of entropy

1) Some assumptions of the form-factor calculation should be properly clarified
2) Discussion/comparison of results to relevant previous work is missing

The manuscript deals with entanglement evolution after a local quench
through an impurity, realized by the interacting resonant level model.
Namely, the system is composed of two free-fermion wires in their ground
states that are, at $t=0$, coupled to each other via an extra site.
In equilibrium, the value of the tunneling amplitude sets an energy
scale, the so-called Kondo temperature $T_K$, which governs the physics of
the problem. In the quench setting, the authors argue that the entropy
evolution should be governed by a universal crossover formula, depending
only on the variable $t T_K$. That is, for large times $t \gg T^{-1}_K$,
the chain appears to be healed and one should recover the CFT result
$S \sim 1/3 \ln (t)$. On the contrary, for small times the chains should
appear to be weakly coupled only, leading to a slower entanglement growth.

The problem is studied numerically on one hand, considering a lattice
model, and analytically on the other hand, using a form factor approach.
In particular, the entropy is calculated via the replica trick, rewriting
it as a time-dependent expectation value of a twist-operator, and
obtaining the leading order terms in a form-factor expansion.

In my opinion, the manuscript deals with an interesting problem,
and presents a new approach in handling it.
There are, however, some major issues that should be clarified
before it could be considered for publication. Moreover, the
presentation of the manuscript could also be improved.

1) The foremost concerning issue is the mismatch between the numerics
vs. form-factor result. Namely, the main result (7) gives, in the
$t T_K \gg 1$ limit, the prefactor 1/4, instead of the CFT result 1/3.
The authors solve this problem simply by multiplying Eq. (7) with 4/3.
This is a very questionable way of handling this mismatch, and the
authors do not give any reasonable explanation. Why can one expect
that higher order FF terms give just a multiplicative factor?
How can one at all rely on the first order term, when the correction
is so large? In fact, the agreement in Fig. 2 is rather qualitative.
I believe, the authors should discuss this issue in much more detail.

2) It is somewhat unclear, why all the material about form-factor
calculations is presented as an appendix. In the end, this is what
is essentially new in this paper, numerical calculations of entropy
were already presented in a previous work [7]. In my opinion,
at least some of the RLM FF calculation should be moved to
the corresponding section. In fact, right now this section is rather
difficult to understand, as no details are given there.

3) The list of literature is rather one-sided. There are, in fact,
a number of other works where the evolution of entanglement through
an impurity was considered, see e.g.

EPL 99, 20001 (2012)
J. Phys. A 46, 175001 (2013)
PRB 91, 125406 (2015)

I believe, these results should be cited and discussed against
the findings of the present manuscript.

4) There is a typo in the definition of Renyi entropy in the
text before Eq. (4)

1) The research carried out in this paper is very timely as it addresses a problem of much current interest, namely the time evolution of entanglement following a quantum quench.
2) The present work goes beyond the most commonly studied regime in this context, namely the evolution of entanglement at large times after a local quench. The latter was famously characterised by Calabrese and Cardy (in critical systems) and shown to scale logarithmically with time. In contrast, the present work considers the evolution of the entanglement entropy (a particular measure of entanglement) not only at large times but also for intermediate and small times. In this way it provides a full analysis of the evolution of the entanglement entropy (EE) from t=0 to t=infinity giving several interesting new insights into the evolution of EE a short time after the quench.
3) Related to the previous point, the current work provides both a numerical and an analytical study of the EE as a function of time. Although numerical studies of the EE after a quantum quench are relatively common nowadays there are very few analytical results going beyond the large time logarithmic scaling mentioned in 2). This is therefore a significant contribution even if the analytic results provided are not exact.
4) The analytical and numerical results are in very good agreement.
5) The analytical results although approximate follow from a non-trivial form factor computation which is presented in detail in the appendix.

In my view the paper has one weakness. In order to compare the form factor (FF) calculation to the numerical results the authors use a “trick” which is really only justified by the fact there is good agreement with the numerical results. The authors have used this “trick” in a previous work, again finding good agreement with numerics. Because of this good agreement I think there is some justification in what they do but from the analytical point of view there is really no justification. Essentially by cutting the form factor expansion at the lowest non-trivial order, they find that the coefficient of the log t term that is predicted to describe entanglement at large t is not the expected 1/3 but rather ¼. As they point out, this is not surprising but rather a result of the truncation of the FF series which for t->infinity is slowly convergent. In other words further terms in the expansion would need to be added in order to approach the coefficient 1/3. I understand computing such terms would be very challenging and perhaps not feaseble at all. Instead the authors suggest that they may just multiply their FF result by 4/3 to get the correct large t behaviour. As I mentioned earlier, by doing this they indeed find a result that agrees very well with numerics as seen in Fig. 2. The problem with this is that there is really no reason why this should work so well. Besides, even if multiplying by 4/3 will certainly “correct” the large t behaviour it seems to me it should spoil the low t behaviour that I would expect is well described by the truncated FF expansion. Why not multiplying instead by say 4/3 t/(t+1)+sin(t)/t? This goes to 4/3 when t->infinity and goes to 1 when t->0. I know this would be a rather unusual guess but the point I am trying to make is that there are probably many ways of "renormalizing" the FF results which would have the right long and short time asymptotic and would fit the data well.

I think the paper deserves to be published in SciPost. In my view its strengths are more prominent than its one weakness. Also I admit that dealing with this weakness is difficult and doing it rigorously would mean writing a rather longer and more technical paper. I think the good agreement between the analytical results and the numerical results provides good support for the work, even if some of the analytical results are not rigorously justified. The paper tackles a timely problem by various approaches and gives new insights into the time evolution of entanglement following a local quench. I think these new insights are very important and it would be nice to understand them in a more general context. For instance, a good question is how general are these results? Do they depend strongly on the model? Would the author’s trick of multiplying by 4/3 (or another convenient number, depending on the model) provide a good fit in all cases? I think these are all interesting questions and this paper paves the way to perhaps answering them in the near future. As such it is likely to have impact in the academic community working in this area.

Comments: I have a few comments regarding mainly small typos and one reference.

1) At the end of the first paragraph the authors refer to the special issue [2]. This special issue deals mainly with measures of entanglement rather than quenches. As far as I can remember there are very few papers (if any) that discuss quenches. On the other hand there is a more recent special issue published by JSTAT that focuses on systems out of equilibrium (see http://iopscience.iop.org/journal/1742-5468/page/extraspecial7). I think it would make sense to cite this instead of or in conjunction with [2].
2) I have noticed that the word Rényi appears as “Reny” in at least two places.
3) I think it would make sense to refer to the appendix either just before equations (7)-(8) or just after.
4) In equation (9) the style of the superindices “a” is different in the second and third sum from the first sum. The same applies to the operators c_i, the coupling J’, U_1 etc.
5) After equation (9) I think it would be useful to remind the reader of what the operators c_i are. Similarly, it would be good to remind the reader of what the operator d in equation (2) is.
6) I was a bit confused by the statement in the Discussion section stating that: “Plotting the derivative emphasizes however the intriguing fact that the instant slope (wrt ln t) of the entanglement growth saturates at values greater than c/3 in the intermediate regime” It is obviously true that t S’ is not a monotonic function of t. However the EE itself is a monotonic function (since its derivative is always positive). Is it the case that the derivative of the EE is in general also monotonic? (hence this result is a surprise). Could you say a bit more about why this result is surprising?
7) Relating to my report, could the authors say anything more about their "renormalisation" by 4/3? Did they try other renormalizations? Do they expect other renormalizations to work even better? Do they know why this renormalisation works both for t large and small? Any additional discussion would be useful.
7) In the appendix I found three instances of the use of the word “disappearing” or “disappears” instead of “vanishing” or “vanishes”. For instance in the sentence after equation (12) it should say that “its moment and energy vanish”. In the same sentence “going” should be replaced by “go”.
8) Before or after equation (38) it would be useful to mention that the new variables v_i=u_i/Tk