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|As Contributors:||Thomas Dupic|
|Submitted by:||Dupic, Thomas|
|Submitted to:||SciPost Physics|
|Subject area:||Mathematical Physics|
We present a new method to compute R\'enyi entropies in one-dimensional critical systems. The null-vector conditions on the twist fields in the cyclic orbifold allow us to derive a differential equation for their correlation functions. The latter are then determined by standard bootstrap techniques. We apply this method to the calculation of various R\'enyi entropies in the non-unitary Yang-Lee model.
We thank the three referees for their careful reading of the manuscript and for their comments.
- the first section was separated in two parts
- A subsection, specifically dedicated to non-unitary model was added in the first section to better explain our choices.
- Two examples were added, both apply the method of the paper to the entropy of a two-intervals subsystem . The first example (in subsection 4.1) concern the Yang-Lee model, while the second (in subsection 5.2 ) focus on the Ising model.
- The figures associated with the numerical results have been reordered.
- There is a new appendix (E) dedicated to the spin-chain representation of the Yang-Lee model .
- A part of the calculation originally explained in section 4. was displaced to the appendix (appendix B).
1- old CFT method employed in a new context
2- both analytic and numerical analysis for a specific simple model
1- difficult for the reader to extract the new results with respect to the previous literature
2- not clearly explained the problems in applying the method to other models
The authors have employed the occurrence of null vectors in a rational conformal field theories (CFT)
to compute the entanglement entropies of an interval.
The analysis requires a highly non trivial understanding of the replica trick and its implementation in CFT.
The technical steps are exposed in a clear way.
Also a numerical analysis in the RSOS model is performed and agreement with theoretical formulas is obtained.
The paper still has the same strengths I had pointed out in my original report. It proposes a method which seems to be new in the context of entanglement and which, as I said in my previous report, I think has merit and potential applications. The work is interesting beyond its application to entanglement as it more generally deals with the computation of multipoint functions of twist operators, some of which play an important role in the study of entanglement.
Unfortunately, my two main criticisms from the original report still remain.
Not only have the authors not acted on my comment that it is intrinsically problematic to use the the name of "entanglement entropies" for functions that have negative values, negative curvature as functions of subsystem size, and are based on a reduced density matrix that is not positive-definite, but instead they have gone out of their way to insist that such functions are indeed entanglement entropies.
My other criticism that their so-called entanglement entropies did not seem to reduce to the expressions for two semi-infinite regions in the appropriate limit seems have been ignored.
My previous report to the authors was very long and detailed and it entailed a considerable amount of work on my part. I am disappointed that the authors have not taken the time to properly and carefully respond to the report. Instead their description of changes made is so generic (and 7 lines long) that the only way for me to find out whether or not they followed my recommendations is to once again read the paper in great detail. I find this rather unprofessional and luckily in my experience, pretty rare.
Despite the above, I have had another good look at the paper and I appreciate that the authors have made an effort to adopt many of my suggestions, especially regarding references. They have also added additional sections to the paper to elaborate further on the special features of non-unitary theories and an appendix where they have done additional numerics on a non-hermitian quantum spin chain whose critical points describe the Lee-Yang edge singularity.
Despite these efforts the authors have not addressed my two main original criticisms. One of them was that it was intrinsically problematic to have an entanglement entropy function which is (in some cases) negative and which has also negative curvature as a function of the ratio of length scales. The authors have responded to this by confirming that indeed in some cases their reduced density matrices will not be positive-definite. Indeed, on page 7 of their paper they have now included the sentence
“The disadvantages of this construction are twofold. The main one is that the reduced density matrix (and hence the entanglement entropy) may not be positive. While this may seem like a pathological property, loss of positivity in a non-unitary system might be acceptable depending on the context and motivations”
It is good that the authors now write this explicitly but I happen to strongly disagree with their statement in the case of the entanglement entropies. Yes, non-unitary systems are different and some of their features seem counterintuitive but nonetheless admit a physical interpretation (for instance, complex energies may be interpreted as a sign of the presence of gain and loss in the system, which is a perfectly physical phenomenon). However, to my knowledge, there is no current physical interpretation of an“entropy” which is constructed from a non positive definite density matrix. Indeed, the whole interpretation of the entanglement entropy as a good measure of entanglement relies critically on the density matrix being positive definite, on it having positive eigenvalues whose sum is 1 and on it producing entanglement entropies whose minimum value is 0, corresponding to separable states. Therefore the authors’ statement that “negative values might be acceptable depending on context and motivation” is, in the case of the entanglement entropy, simply false.
Therefore, I think it is a shame the authors insist in calling their functions entropies. I have no problem with their statements that these functions have nice physical properties or even, as they say on their new appendix, that they may be used to precisely identify the phase transition in a particular physical system. That is all very good. These functions should be investigated. They are just not Rényi entropies.
The second point I had made in my previous report was that if they had any connection to any standard notion of entanglement entropy, then their functions should at least have a limit in which they reproduce the logarithmic behaviour of the entanglement entropy that has been found for two semi-infinite regions in gapped systems. Even if they study critical systems it is known since the work of Calabrese and Cardy that the scaling of the entanglement entropy of two semi-infinite gapped systems should mimic the scaling of the entanglement entropy of an interval of finite length within an infinite critical system. Given that the scaling of entanglement in gapped non-unitary systems has been identified in several publications and seems beyond doubt (it is not disputed by the authors either), it should be possible for the authors to take an infinite volume limit of their formulae and at least recover the correct known behaviour. As far as I can see in the current manuscript this point has not been addressed. I suspect that if it were addressed the results would be that their so-called entropies do not reproduce the correct infinite volume limit which would mean that non only they are not entanglement entropies, but they are not related to other quantities that are known to be.
In summary, although this paper has strengths and merit, the authors have unfortunately insisted on characterizing some of their results in an erroneous way. Although this may seem just a matter of words, I think it is an important matter. It is just the case that entanglement entropies are functions with a very particular set of properties and functions that do not share those properties, as interesting as they may be, cannot be called entanglement entropies. Based on these comments I am afraid my view is that the paper cannot be accepted for publication in SciPost in its current form.
The authors properly took into account my suggestions, as well as the ones of the other referees. Although the list of changes and the answers to the referees should have been more detailed, I believe the paper is now ready for publication. I am sure that in spite of the technicality of the paper, it represents a genuine original and important piece of the literature on entanglement in CFT.