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Universal Entanglement Dynamics following a Local Quench
by Romain Vasseur, Hubert Saleur
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Submission summary
| Authors (as registered SciPost users): | Romain Vasseur |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/1701.08866v1 (pdf) |
| Date submitted: | Feb. 13, 2017, 1 a.m. |
| Submitted by: | Vasseur, Romain |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the time dependence of the entanglement between two quantum wires after suddenly connecting them via tunneling through an impurity. The result at large times is given by the well known formula $S(t) \approx {1\over 3}\ln {t}$. We show that the intermediate time regime can be described by a universal cross-over formula $S=F(tT_K)$, where $T_K$ is the crossover (Kondo) temperature: the function $F$ describes the dynamical "healing" of the system at large times. We discuss how to determine $F$ analytically for integrable quantum impurity problems using the exact expression of the matrix elements (Form Factors) of twist and boundary condition changing operators in the corresponding integrable quantum field theory. Our results are confirmed by density matrix renormalization group calculations and exact free fermion numerics.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2017-3-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1701.08866v1, delivered 2017-03-17, doi: 10.21468/SciPost.Report.97
Strengths
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.
Weaknesses
Report
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.
Requested changes
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.
Report #2 by Anonymous (Referee 2) on 2017-3-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1701.08866v1, delivered 2017-03-15, doi: 10.21468/SciPost.Report.96
Strengths
1) Manuscript deals with an interesting problem 2) Interesting analytical approach to calculation of entropy
Weaknesses
1) Some assumptions of the form-factor calculation should be properly clarified 2) Discussion/comparison of results to relevant previous work is missing
Report
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.
Requested changes
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)
Report #1 by Olalla Castro-Alvaredo (Referee 1) on 2017-2-27 (Invited Report)
- Cite as: Olalla Castro-Alvaredo, Report on arXiv:1701.08866v1, delivered 2017-02-27, doi: 10.21468/SciPost.Report.89
Strengths
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.
Weaknesses
Report
Requested changes
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