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Entanglement entropies of minimal models from null-vectors

by Thomas Dupic, Benoit Estienne, Yacine Ikhlef

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Submission summary

Authors (as registered SciPost users): Thomas Dupic · Benoit Estienne
Submission information
Preprint Link:  (pdf)
Date accepted: 2018-05-17
Date submitted: 2018-03-08 01:00
Submitted by: Dupic, Thomas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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.

Author comments upon resubmission

We thank the three referees for their careful reading of the manuscript and for their comments.

List of changes

- 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).

Published as SciPost Phys. 4, 031 (2018)

Reports on this Submission

Anonymous Report 3 on 2018-4-3 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1709.09270v5, delivered 2018-04-03, doi: 10.21468/SciPost.Report.407


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.

  • validity: high
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: excellent

Report 2 by Olalla Castro-Alvaredo on 2018-3-21 (Invited Report)

  • Cite as: Olalla Castro-Alvaredo, Report on arXiv:1709.09270v5, delivered 2018-03-21, doi: 10.21468/SciPost.Report.387


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.

  • validity: ok
  • significance: good
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: good

Author:  Thomas Dupic  on 2018-03-22  [id 232]

(in reply to Report 2 by Olalla Castro-Alvaredo on 2018-03-21)

We really did appreciate the detailed comments you made on our paper, it was a great help in clarifying what we were trying to say and we did try to answer it thoroughly.

You may have missed the detailed answer to your first comment which was posted as a reply in the version 4 ( when the new version (v5) was added. It may not have been the right place to post it, as the new version now covers it by default, this is my first test of the Scipost submission system, I may have made a mistake and I apologize for it.

Sorry also for the short list of changes, it was just thought as a complement to the more detailed answers (and was not meant to be considered on its own).

We do take into account your new comments, and we will post an answer to them specifically as soon as possible

I repost here, for convenience, the first answer we made (which may answer some of your remarks) , sorry again for the misunderstanding.

Thank you for your careful reading, and in-depth comment,

  1. done.
  2. We have added the relevant references.
  3. done.
  4. done.
  5. done.
  6. done.

  7. We have added a substantial discussion addressing this question (section 2.3) and an appendix (E).
    7.1. Some results of arXiv:1405.2804 are also in finite volume, see Figure 2 therein. 7.2. We added a section (2.3) discussing this point. We argue that the choice of density matrix ($| \Psi_L \rangle \langle \Psi_R |$ or $| \Psi_R \rangle \langle \Psi_R |$) is to some extent a matter of definition. We also note that most papers considering entanglement entropy for non-unitary models end up working with the same choice we made, namely $| \Psi_L \rangle \langle \Psi_R |$, whether they compute partition functions (as in arXiv:1509.04601), or they claim that $| \Psi_R \rangle = | \Psi_L \rangle$ (as in arXiv:1405.2804), or they simply make this choice (arXiv:1611.08506).

7.3 -- 7.6. It could be argued that non-positivity is not really an issue when dealing with non-unitary models. Underlying this is the choice of an inner product that this not positive-definite but such that the Hamiltonian density is self-adjoint. This is now discussed at length in section 2.3.

7.7. We have two issues with reference arXiv:1405.2804. First we disagree that left and right eigenvectors coincide (in finite size they clearly don't). Moreover if it were true, then our entropy would be positive, which is clearly not the case.

Second, the mapping of the R\'enyi entropy to a ratio of a two-point function of twist operators and $Z_1^N= \langle \phi(0) \phi(\ell) \rangle^N$ is incorrect. The issue is that $\langle \phi(0) \phi(\ell) \rangle$ is \emph{not} the norm of the ground state. In the CFT formalism, the state $| \phi \rangle$ has norm $1$ by definition. Only when doing a lattice calculation does one need to normalize states. The correct quantity is our equation (2.18). The collapse of the numerical data with different system sizes and the excellent agreement with our numerical calculations leaves very little doubt about this (see Figure 3). This is to be compared with the inadequate agreement in figure 2 of arXiv:1405.2804. First the authors did not demonstrate the collapse of numerical data using different system sizes, thus failing to establish that the data shown is indeed in the critical regime of the Yang-Lee model. Second, the agreement between the theoretical and numerical results is quite poor in the regime $\ell/L\sim 1/2$, i.e. the middle of the curve. However this is the regime in which the fit has to work best : for $\ell/L$ small the finite size effects are drastic (region $A$ being too small, only a few sites).

7.8. The $\ell \ll L$ behaviour of the Yang-Lee groundstate entropy for $N=2$ is all contained in the limit $x \to 1$ of function $G$ (5.12). The exponents associated to conformal blocks of the related function $F$ are given in (5.19). For generic integer $N \geq 1$, when $u \to v$ in (2.18), the dominant behaviour is determined by the OPE

$$ \tau_\phi(u,\bar u) \tau_\phi(v, \bar v) \sim |u-v|^{-4\widehat{h}_\phi+2Nh_\phi} \, \Phi(v,\bar v) \,, $$

which gives

$$ S_N \sim \left( \frac{N+1}{6N} c_{\rm eff} + \frac{2N}{N-1} h_\phi \right) \log|u-v| \,. $$

For instance, in the Yang-Lee model at $N=2$, one gets a negative prefactor $-11/5$, consistently with our numerical results.

7.9. Indeed in arXiv:1509.04601 the entropy of a semi-infinite interval within an infinite-volume system was studied for RSOS models in the massive regime. There it was found that the entanglement entropy diverges with the correlation length $\xi$ according to

$$S_N \sim \frac{c_{\rm eff}}{12} \frac{N+1}{N} \log \xi \,,$$

This calculation was done by mapping the R\'enyi entropy to the trace of some power of the corner transfer matrix, \emph{i.e.} a partition function on some multi-sheeted surface. In view of the above discussion, this calculation corresponds to our choice of density matrix\footnote{It was claimed in arXiv:1405.2804 that for RSOS models one always has $r_0 = l_0$ simply based on the reality of the transfer matrix. However this argument if flawed : while the transfer matrix is indeed real, it is not symmetric. Through a gauge transformation it can be made symmetric, but of course it will no longer be real. So there is actually no reason to expect that left and right eigenvectors coincide, and it is a simple matter of checking numerically to see that they do not agree in finite size.}, namely $\rho = r_0 l_0^{\dagger}$.

However the striking similarity between this result and the claim of arXiv:1405.2804 should not be taken as an argument in favor of the validity of the latter. It is true that for a unitary system the scaling of the entanglement entropy of a finite subsystem at criticality and that of an infinite subsystem with a small but finite mass gap are related. However the argument underlying this connection relies strongly on unitarity, and there is no reason to expect that this result can be extended to non-unitary systems, as was pointed out in arXiv:1509.04601.

  1. Corrected.
  2. Corrected.
  3. In the first equation, there are no Fourier modes, because it deals with the mother theory. In the second equation, the choice of $r$ is obvious from (3.18). Added a comment.
  4. Corrected.
  5. The normalisation is implemented by taking $\langle \psi_L|\psi_R \rangle=1$.

Anonymous Report 1 on 2018-3-8 (Invited Report)


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.

  • validity: top
  • significance: top
  • originality: top
  • clarity: good
  • formatting: good
  • grammar: good

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