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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 | |
|---|---|
| Preprint Link: | http://arxiv.org/abs/1701.08866v2 (pdf) |
| Date submitted: | 2017-05-09 02:00 |
| Submitted by: | Vasseur, Romain |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| 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 obtain analytic information about $F$ in the case of an integrable quantum impurity problem using the massless Form-Factors formalism for twist and boundary condition changing operators. Our results are confirmed by density matrix renormalization group calculations and exact free fermion numerics.
Author comments upon resubmission
We apologize for the delay in getting back to you: both authors were traveling until recently.
We are now happy to submit a modified version of our manuscript which, we hope, will be acceptable for publication.
First, we thank the referees for their careful reading of the manuscript and their constructive criticism.
Since the three referees seemed essentially in agreement about the qualities and weaknesses of our paper, we will take the liberty to paraphrase their comments rather than cite each of them in turn.
In a nutshell, the referees found the paper interesting and timely, found that it contained interesting results, but complained about the ``brutal'' way we obtained our analytical curve by performing a seemingly arbitrary renormalization by a numerical factor ($4/3$) of the result of a form-factor calculation.
Before discussing this factor in detail, we would like to emphasize that we did not consider that the analytical calculation was the main point of the paper. Rather, we felt the most interesting result was the existence of a well defined scaling function describing the crossover of the entanglement entropy from $S=0$ to the conformal asymptotic behavior $S\sim {c\over 3}\ln t$ after a local quench involving a weakly coupled impurity. The existence of this scaling function is argued on general grounds in our paper, and amply verified by high quality computer simulations.
Coming to the analytical calculation presented in the paper, we also would like to emphasize that it is the {\bf only} calculation we are aware of that can produce any usable information about the scaling curve for ${dS\over d\ln L}$. Perturbative approaches have been shown, in this kind of problem, to either converge extremely slowly (and thus to be unable to produce useful results), or to be plagued with uncontrollable divergences. See for instance the paper arXiv:1305.1482 where some of these aspects are discussed in the related context of crossovers involving sizes instead of time. Meanwhile, particle propagation pictures ``\`a la Cardy Calabrese'' do not seem to be able to recover the kind of fine structure present in the crossover either. Hence, we believe our calculation, however unsatisfactory, has the merit of existing, and, by a strange coincidence which may well {\sl not be} a coincidence, does provide remarkable accurate results. This is why we decided to publish it, despite the shortcomings mentioned by the referees.
Now the main shortcoming is that, at the order of form-factors expansion we are working, we get a satisfactory result only if we multiply the result of our calculation, by a numerical factor $4/3$. While this may seem horribly ad hoc, there is in fact a rationale behind this. It originates from early calculations performed by F. Lesage and H. Saleur (J. Phys. A30 (1997) L457), themselves inspired by calculations of F. Smirnov. In a nutshell, what happens in many problems involving ``massless form-factors'' or form-factors in the UV limit, is that a) the integrals over rapidities are diverging and b) once these integrals are properly regulated, the expansion itself is divergent. The way to cure this second divergence is to focus on the {\sl ratio} of two quantities, for instance, in the case of the paper by Lesage and Saleur, the ratio of a correlation function evaluated at finite value of the impurity coupling and the same correlation function evaluated in the conformal limit. To put it schematically:
%
$$
R(T_B)\equiv {\hbox{FF expansion of} \langle ...\rangle (T_B)\over
\hbox{FF expansion of} \langle ...\rangle (\hbox{CFT limit})}=\hbox{well defined and convergent}
$$
%
The trick used in the paper of Lesage and Saleur is thus to calculate $R(T_B)$ using form-factors, and then multiply the result by the (known) result in the conformal limit to obtain the searched for result at finite $T_B$.
There are, to our knowledge, no strong results as to why this should work, especially because the form-factors involved in this kind of calculation are extremely complicated.
What we did in our paper is, in spirit at least, identical. Instead of multiplying by the numerical factor $4/3$, we could say that what we have done is perform a calculation of the ratio
%
$$
R(T_B)\equiv {\hbox{FF expansion of } S(T_B)\over
\hbox{FF expansion of } S(\hbox{CFT limit})}
$$
and then multiplied by the known CFT result. In this case, the form-factors expansion of the entanglement is itself in fact well defined (at the price of taking a derivative wrt t). The numerator goes as $S_{FF}\sim {1\over 4}\ln t$ at leading order, while $S_{CFT} \sim {1\over 3}\ln t$. Hence the overall ``renormalization'' by a factor $4/3$.
It would certainly be better to investigate in more detail the form-factors expansion, in order to see whether higher orders render this renormalization unnecessary indeed (by correcting the denominator into ${1\over 4}\ln t$). But this is an extremely technical endeavor, and, as we pointed out already, we felt it was not the main point of the paper.
We have, in the present modified version, explained the ``historical'' origin of the renormalization, and toned down our claim of doing an analytical calculation some more, so as not to confuse the reader into believing we accomplished more that we did. We have also, to answer the additional criticism of one referee, discussed a little more the behavior of the extra terms we had initially discarded. We have added comments emphasizing how more difficult the time dependent case is, compared with the equilibrium cases we had studied so far. Despite one of the referees' suggestion, we have preferred not to put the form-factors calculation in the main text, since a) it is not our main point and b) it is not that satisfactory anyhow. We have otherwise been happy to follow all the minor changes suggested by the referees.
We hope our manuscript will be accepted for publication.
Sincerely,
H. Saleur and R. Vasseur.
List of changes
1) We updated the abstract
2) We have explained the origin of the renormalization in detail in the main text and in appendix, and commented on the discarded terms
3) We made many minor changes and added references following the referees' recommendations
Current status:
Reports on this Submission
Anonymous Report 3 on 2017-5-24 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1701.08866v2, delivered 2017-05-24, doi: 10.21468/SciPost.Report.147
Strengths
Interesting problem and attempt of an analytical calculation of entropy
Weaknesses
4/3 deficiency in the form-factor calculation
Report
The authors have provided an explanation for the 4/3-renormalization, based on earlier work, where similar behaviour was observed. The argument is essentially that the form-factor calculation could only yield the ratio w.r.t the result in the conformal limit. Although there is no clear analytical justification behind this trick, it has also been used in earlier studies and miraculously yields rather accurate results. Clearly, it would be nice to obtain a better understanding of the mechanism behind this trick. However, the authors have now given an honest account of the state of affairs in the text, without overclaiming the utility of their method for the calculation of entropy. I believe that, despite the deficiency of the form-factor method, this is still a decent work with an interesting result and I recommend the publication of the manuscript.
Requested changes
Typo in the definition of Renyi entropy in the beginning of paragraph before Eq. (4) is still there and should be corrected
Anonymous Report 2 on 2017-5-21 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1701.08866v2, delivered 2017-05-21, doi: 10.21468/SciPost.Report.142
Strengths
The strengths of the paper are exactly the same as in my original report. The subject is timely, and interesting partial analytical results are provided on a difficult problem.
Weaknesses
1) Several not so small terms are dropped to make the agreement with the numerics better, and the authors fail to mention that important fact in the main text. This problem should be fixed.
2) The "renormalization" by a factor 4/3 is still a weakness, but it is discussed clearly and honestly now.
Report
In my previous report, which is quoted below, I requested the following 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.
----------------------------------------------------------------------
Neither of the two changes has been made, I cannot recommend publication in the present form.
We are not talking about a minor detail here, but about the main statement of the paper. Equation (7), which is shown in figure 2, is the result of clear cherry-picking. The result to lowest order, as defined in the last paragraph on page 4, is a sum of five terms. Four (!) of them are simply discarded, as admitted to shortly after equation (38), (39), (40) at the end of the appendix on page 15. The procedure, designed to get better agreement with the numerical data, is not acknowledged in the main text. Hiding this crucial piece of information in the appendix is of course unacceptable, especially since the authors mention in the new version of the appendix that the discarded terms are not so small after all. [This is a separate issue from the "renormalization" by 4/3, the discussion of this prescription is now satisfactory.]
As a result, several statements made in the main text are misleading. For example, the authors explain the various physical processes contributing to lowest order on page 4, name the next paragraph "the result at lowest order", but show a result which is not the result at lowest order. At the beginning of the discussion several statements are also misleading, including again the use of "lowest order" which does not mention the discarded terms.
I also have a few comments on the part of the appendix mentioned above:
"The only process which remains non-zero in the conformal limit is the sin term in (38): it is the
contribution we have used to obtain the curve on Fig. 2. The other contributions are extremely
tedious to evaluate numerically, because of the less favorable behavior of the integrals involved
that are naively divergent without additional regularization. We have checked however that,
while they do not add up to zero any longer, they remain small ($\lesssim$ 10%) and can be essentially
neglected for our purpose."
1) The integrals are convergent, and do not require additional regularization. Please clarify.
2) If the authors can guarantee the contributions are of the order of 10%, it means they are able to evaluate them numerically, at least to some reasonable degree of approximation. Therefore, incorporating them in figure 2 as I requested in my previous report is possible. Otherwise please clarify, and mention these contributions in the main text.
3) 10% is not small, it is more than enough to significantly worsen the agreement shown in figure 2. Especially if these contributions happen to move the location of the maximum.
4) Is it 10% before, or after renormalization by 4/3?
Requested changes
The list of requested changes is exactly the same as in my previous report.
1) Show in figure 2 the full lowest order result, not just equation (7).
2) State clearly in the main text what is equation (7), and what is shown in figure 2. Failing to mention the discarded terms is not acceptable.
Author: Romain Vasseur on 2017-06-19 [id 147]
(in reply to Report 2 on 2017-05-21)
Referee 2 is concerned about the "leading contribution" plotted in the main text being "cherry-picked" among other terms. We do agree with the referee that the proliferation of "subleading" first order terms is one of the unsatisfactory aspects of this non-equilibrium massless Form Factor approach which would have to be clarified in future works. However, it is important to notice that the massless Form Factor expansion is not controlled by a small parameter anyway (in sharp contrast with massive theories for which the FF program is much more controlled). In that sense, it is equally (un)satisfactory to discard terms at first or second order in the expansion. Moreover, there is a clear sense in which the contribution we plot is the "leading" one, even though it is not the only first-order term: it is the only term that does not vanish in the IR limit (and is clearly "dominant" in that limit). We also checked that the contribution of the other terms is roughly one order of magnitude smaller. To be more accurate, we changed "lowest order" to "leading contribution" in the main text.
Incidentally, this contribution is the only integral that we were able to evaluate numerically in a satisfactory way (the other ones are less well-behaved in the IR, and/or are highly oscillatory), which is why we did not include the other contributions in Fig. 2. For the few points where we were able to compute these other contributions, we checked that they are relatively small as mentioned above, but are large enough to worsen the agreement with the numerical results. Although this is obviously unpleasant, the only way to clarify the situation will be to compute higher order contributions, and to find a stable way to evaluate these contributions numerically. Given that the first order calculations were already quite involved, we defer these investigations to future work.
Contrary to what Referee 2 seems to imply, we feel we have been especially honest about this point from the first version of our paper. We commented on these issues only in the appendix as we thought this would only be of interest to Form Factor experts, but following the referee’s recommendations, we were happy to add a few sentences in the main text as well. We also improved and clarified the discussion in the appendix.
Regarding the specific comments:
“1) The integrals are convergent, and do not require additional regularization. Please clarify.”
Some of these integrals are divergent in the IR. We clarified this sentence and added the explicit form of the IR integrals to illustrate this.
“2) If the authors can guarantee the contributions are of the order of 10%, it means they are able to evaluate them numerically, at least to some reasonable degree of approximation. Therefore, incorporating them in figure 2 as I requested in my previous report is possible. Otherwise please clarify, and mention these contributions in the main text.”
As we mentioned above, we were not able to evaluate these integrals in a satisfactory way, which is why we decided not to plot them. We were happy to clarify this point in the main text and appendix.
"3) 10% is not small, it is more than enough to significantly worsen the agreement shown in figure 2. Especially if these contributions happen to move the location of the maximum."
We agree, and as we clearly stated in the first version of our draft, these contributions do seem to worsen the agreement with the numerical results — even though we reiterate we could not find a way to evaluate these integrals accurately. We added a sentence to emphasize this point in the main text. However, we feel that one order of magnitude is enough to justify calling eq 7 the ``leading’’ contribution.
“4) Is it 10% before, or after renormalization by 4/3?”
10% refers to a relative error and is the same after and before renormalization.
Report 1 by Olalla Castro-Alvaredo on 2017-5-10 (Invited Report)
- Cite as: Olalla Castro-Alvaredo, Report on arXiv:1701.08866v2, delivered 2017-05-10, doi: 10.21468/SciPost.Report.135
Strengths
The strengths of the paper are the same as I had already pointed out in my original report. The results are timely and interesting and there is very good agreement between the non-trivial analytical calculations and the numerics performed by the authors.
Weaknesses
In my original report I had found one main weakness relating the re-normalisation factor that was introduced seemingly ad hoc to achieve good agreement between a form factor calculation, the CFT prediction and the numerics. This weakness remains but the authors have made an effort to better explain where this prescription comes from and why it is expected to work.
Report
The authors have engaged constructively with all my original comments and they have made an effort to explain the origin of their form factor calculation normalisation, while being honest about the lack of a strong rational for this choice. I find both their answer and the changes they have introduced satisfactory. I consider that they have successfully addressed the points I had raised.
Requested changes
None
Author: Romain Vasseur on 2017-06-19 [id 146]
(in reply to Report 1 by Olalla Castro-Alvaredo on 2017-05-10)We thank the Referee for their useful comments and for recommending our work for publication in SciPost.
Author: Romain Vasseur on 2017-06-19 [id 145]
(in reply to Report 3 on 2017-05-24)We thank the Referee for their useful comments and for recommending our work for publication in SciPost. Concerning the "typo in the definition of Renyi entropy in the beginning of paragraph before Eq. (4)", we do not see a typo in this definition: would the referee prefer we define the Renyi entropy in a more standard way with a logarithm? We would be very grateful to the referee if he/she could clarify what specific aspect of this definition is a "typo".