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Massive Corrections to Entanglement in Minimal E8 Toda Field Theory
by Olalla A. Castro-Alvaredo
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|Authors (as registered SciPost users):||Olalla Castro-Alvaredo|
|Preprint Link:||http://arxiv.org/abs/1610.07040v1 (pdf)|
|Date submitted:||2016-11-09 01:00|
|Submitted by:||Castro-Alvaredo, Olalla|
|Submitted to:||SciPost Physics|
In this letter we study the exponentially decaying corrections to saturation of the second R\'enyi entropy of one interval of length L in minimal E8 Toda field theory. It has been known for some time that the entanglement entropy of a massive quantum field theory in 1+1 dimensions saturates to a constant value for m1 L <<1 where m1 is the mass of the lightest particle in the spectrum. Subsequently, results by Cardy, Castro-Alvaredo and Doyon have shown that there are exponentially decaying corrections to this behaviour which are characterised by Bessel functions with arguments proportional to m1 L. For the von Neumann entropy the leading correction to saturation takes the precise universal form -K0(2m1 L)/8 whereas for the R\'enyi entropies leading corrections which are proportional to K0(m1 L) are expected. Recent numerical work by P\'almai for the second R\'enyi entropy of minimal E8 Toda has found next-to-leading order corrections decaying as exp(-2m1 L) rather than the expected decay as exp(-m1 L). In this paper we investigate the origin of this result and show how it might be reconciled with analytical results based on the computation of correlators of branch point twist fields.
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:1610.07040v1, delivered 2016-12-18, doi: 10.21468/SciPost.Report.52
1. The scaling of entanglement entropy with subsystem size in massive quantum field theories is an interesting and timely topic.
2. The general exposition and the theoretical computation in Sections 1-3 is clear and well described, and gives a very good example of the application of the twist field form factor method to compute Renyi entropies.
3. The results are also novel and interesting.
1. The discussion of the comparison between the form factor expansion and numerical TCSA data is quite superficial in many respects, some important points are glossed over.
2. The author does not state sufficiently clearly her conclusion regarding the problems considered in the paper.
3. There are some English grammar mistakes/misprints in the text, some of them also affecting its comprehensibility by the reader. There are also some formatting issues in the figures.
The paper presents a form factor based calculation of the second Renyi entropy in the E8 (Ising) model. The aim is to compare it to recent TCSA results obtained by T. Palmai, and clarify the role played by one-particle contributions. The main problems with the text concern Sections 4 and 5 and correspond to items 1 and 2 in the list of weaknesses above.
1. The author points out that the fit (60) does not agree well with the higher volume (m1L=8) TCSA data, while the agreement with the form factor result (51) improves with increasing volume. However, it is not clear what was the volume L for which the fit (60) was obtained. Since it is a numerical fit, it needs to be re-evaluated for m1L=7 and m1L=8, however, Fig 6. suggests this was not done.
Also, the significance of deviations between either (51) or (60) on the one hand, and the TCSA on the other, is predicated on the understanding of the truncation error inherent in TCSA. The author mentions in passing “some oscillations due to numerical inaccuracy”. While it is clear that here the author is relying on data from T. Palmai, it is unclear what this means and also what is the justification for assigning these “oscillations” to TCSA error since no estimate of truncation errors is given.
The author also states “The reason for this is that although m1L = 8 may not seem a large value, it has been observed numerically that higher values produce comparable results.” While it may very well be true that m1L=8 is sufficiently close to infinite volume, I do not see the evidence, and no details are given concerning the numerical observations mentioned here.
2. By considering the fact that the form factor calculations have a firm theoretical basis, and that it clearly agrees quite well with TCSA data (despite all the issues I mentioned above), I would have expected the author to draw a clear and unambiguous conclusion, especially since the fit (60) is much more heuristic.
The author also discusses some points in separate places, which makes it harder to put together her statement regarding the issue.
For example, in Section 4:
“Therefore, prior to this study, it was not possible to know if the leading term proportional to K0(m1l) was going to be suppressed by, for instance, an extremely small value of the one-particle form factor.”
Then in Section 5 the 2nd to 4th paragraphs seem to be devoted to precisely this problem.
It would be better to decide on where to discuss this point, and rewrite the relevant part of the text in a more focused way, clearly stating the conclusion and the evidence/considerations supporting it.
3. Examples of English grammar mistakes/misprints/badly formulated sentences:
p. 11 “This obviously posses the question”
p. 13 “may well also contribute” (may or well? Possibility or a certainty?)
p. 16 “as volume in increased”
p. 17 “wee know from”
p. 17 “to be large of small”
p. 17 “It is therefore desirable that the TCSA approach may be extended to” (may??)
4. Figures (especially axis labels) are not well formatted; some of the text eventually overlaps, due to too large fonts used.
1. A revision of Sections 4&5 in order for the presentation be more focused and clear, with the conclusions stated clearly.
2. Clarify the issues surrounding the comparison to TCSA, stating the significance of the observed agreement/deviations more clearly.
3. Correct the grammar mistakes/misprints, especially those influencing the comprehensibility of the text.
4. Correct font sizes in figures, eliminate text overlaps.
- Cite as: Anonymous, Report on arXiv:1610.07040v1, delivered 2016-12-08, doi: 10.21468/SciPost.Report.49
1- The paper presents a simple introduction to the IMMF model.
2- The FF analysis is clear and to my level of understanding appears correct.
3- The paper sets out to answer a question arising from a controversial conclusion of a recent publication.
See general report. The main problem is in the qualitative and overly discursive nature of the comparison with TCSA results.
This paper presents a form-factor (FF) expansion calculation of the the 2nd Renyi entropy of the so called IMMF model. This main stated purpose is to compare with a recent TCSA calculation of Palmai  that came to the surprising conclusion that the sub-leading contribution appeared to be fitted by a function exp(-2 m_1 l) as opposed to the anticipated exp(-m_1 l). The explanation of the FF expansion for the correlation function of twist fields is generally clear and follows a fairly well-established argument that draws on and extends the results of  and . Up to the end of Section 3 the results are convincing.
The paper has weaknesses however in Sections 4 and 5. The paper is supposed to be carrying out a comparison of two approximate methods (the FF method is still an approximation due to the truncation involved) but the comparison carried out is qualitative and sometimes the conclusions are unclear. In particular:
(1) A key difference between the two techniques is that TCSA method works for finite m_1 L and the FF for infinite m_1 L. In order to compare them in any way, some attempt at extrapolation of the TCSA results to L infinite needs to be made, but this is currently lacking.
(2) Further to (1), it is stated that `it has been observed numerically that higher [m_1 L] values produce comparable results', but no data is presented to explain or back this statement. Is this a private communication from Palmai, or an independent TCSA calculation? Either way, it would be very helpful to show this higher L data and to include it in the comparison.
(3) The comparison of fits described in the penultimate paragraph of Section 4 is just qualitative. There are of course many quantitative ways of comparing fits and I think that it is reasonable to expect that such an analysis should be carried out (after carrying out the extrapolation described in (1)).
(4) The statement is made 'there are some oscillations due to numerical inaccuracy'. I'm not sure what this means and no error bars are given. Is the numerical accuracy resulting from poor numerical evaluation of truncated TCSA results, or something to do with the TCSA truncation level itself (in which case the error is
not really numerical). Also is there any evidence of the dependence of the TCSA results on the truncation level?
(5) Section 5 is again overly discursive and sometimes seems self-contradictory. For example, 'a priori the results of  seemed to disagree with a prediction based on branch point twist fields', 'the TCSA data are not precise enough to differentiate [the two behaviours]', 'the agreement between numerical data and the form factor data [..] is still remarkable', 'the [two approaches] complement each other and lead to consistent results'.
It is hard to extract a clear message from this discussion. Is the final conclusion simply that the L-extrapolated TCSA data fits (51) well and better than (60)? If so, this should be clearly stated clearly and ideally backed by quantitative evidence.
For the paper to be accepted, I would suggest that changes have to be made to Sections 4 and 5 so that the comparison is quantitative and clear, otherwise, in my opinion, the stated purpose of the paper is not met.
1- A reworking of Sections 4 & 5 to address the points raised in the general report.