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Entanglement gap, corners, and symmetry breaking
by Vincenzo Alba
This Submission thread is now published as
Submission summary
As Contributors:  Vincenzo Alba 
Arxiv Link:  https://arxiv.org/abs/2010.00787v2 (pdf) 
Date accepted:  20210215 
Date submitted:  20201209 10:29 
Submitted by:  Alba, Vincenzo 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the finitesize scaling of the lowest entanglement gap $\delta\xi$ in the ordered phase of the twodimensional quantum spherical model (QSM). The entanglement gap decays as $\delta\xi=\Omega/\sqrt{L\ln(L)}$. This is in contrast with the purely logarithmic behaviour as $\delta\xi=\pi^2/\ln(L)$ at the critical point. The faster decay in the ordered phase reflects the presence of magnetic order. We analytically determine the constant $\Omega$, which depends on the lowenergy part of the model dispersion and on the geometry of the bipartition. In particular, we are able to compute the corner contribution to $\Omega$, at least for the case of a square corner.
Published as SciPost Phys. 10, 056 (2021)
Author comments upon resubmission
Dear Editor,
I would like to thank you and the referee for your work and for accepting my manuscript.
Here below I answer the comments of the referee:
REFEREE:
In this paper the author studies the finitesize scaling of the lowest entanglement gap in the ordered phase of the 2D quantum spherical model n a square lattice which is exactly solvable. The main result of the paper is that asymptotically the gap decays as \Omega / \sqrt{L \log L} as opposed to the ~1/sqrt{\log L} scaling at the critical point, and the author also determines the corner contribution to \Omega.
In recent years the study of the entanglement properties in quantum manybody systems has become a central topic. This works provides an interesting and valuable contribution to the field, and builds also an interesting basis for future works. The paper is written in a clear way, and the results have been carefully checked also numerically.
For these reasons I can recommend this paper to be accepted in SciPost Physics. I have only minor comments, listed below, which the author may want to consider when revising the manuscript.
ANSWER:
I would like to thank the referee for the accurate summary of my work and for considering it interesting.
REFEREE:
It would be interesting to extend the discussion why a different scaling behavior was predicted in Ref.[29], in order to better understand the difference with the result in this paper. A brief comment is made around Eq.(48) and in the conclusions, but if there is any more information that could be added here, that would be useful (e.g. based on what approach/assumptions the result in Ref.[29] was obtained).
ANSWER:
I would like to thank the referee for raising this important point. I now discuss at the end of the introduction the origin of the discrepancy with Ref 29. It is clear that the goal of Ref. 29 is to study the effects on the entanglement spectrum of a continuous symmetry that is broken in the thermodynamic limit. The crucial observation is that for a finite system the symmetry is preserved. The results of Ref. 29 are derived within this scenario. On the other hand, in the quantum spherical model that I consider the spherical symmetry is imposed only on averave even for finite systems. In the scenario of Ref. 29 the entanglement spectrum is sensitive to the fluctuations (for instance of the magnetization) between the two subsystems. If the symmetry is not preserved, however, the total magnetization has also fluctuations.
REFEREE:
In the conclusions the author mention the importance to understand how the scaling of the entanglement gap depends on dimensionality and the range of interactions. I was wondering if there is "natural" conjecture that could be made here, or if it is completely unknown.
ANSWER:
I thank the referee for this question. Clearly, there is a natural scenario. As it is clear from formula (41) the entanglement gap, or at least some of the ingredients needed to understand its scaling, can be related to more standard quantities. For instance the first term in (41) is the contribution of the zero mode to the spin susceptibility. Its scaling should be L^d. In order to derive the behaviour of e_1 one has to understand the scaling of the momentum correlator in (41). This requires a full calculation. However, it is interesting to observe that in d=1 the correlator has the same behavior as ln(L)/L as in d=2. Assuming that this remain true for generic d one would guess that e_1\propto L^(d1) ln(L). There are, however, some interesting remarks to make. For instance, it is not clear a priory that the scaling above remains true at the upper critical dimension, where additional logarithmic corrections are expected.
REFEREE:
Here is a list of minor typos I spotted while reading:  p3 first line of Sec.2: "cubic lattice" > "square lattice" (since it is 2D)  Eq.(9) A point "." is missing  p4, 4th line after Eq (17): remove either "finite" or "nonzero". Same line: "at the critical point"  p8, 3d line after Eq (39): "vecto" > "vector"  p10, Sec 6, there is an extra new line before "(see Figure 1(a))" which should be removed.  p14, 3 lines after Eq (57): "These be calculated" > "These can be calculated"  please check the arxiv references in the bibliography, especially the links which do not seem to work properly (e.g. [29], [39]).
ANSWER:
I thank the referee for the useful comments that I implemented in the revised version of the manuscript.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 202129 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2010.00787v2, delivered 20210209, doi: 10.21468/SciPost.Report.2522
Strengths
Interesting observation concerning the finite size
scaling of the entanglement gap
Report
The most important observation of the paper is
the finite size scaling of the $\delta \xi$ decays
faster than the the same at the quantum critical point
in an exactly solvable twodimensional quantum spherical model. This is an interesting observation. The faster decay is attributed to the ordered phase. The parameter $\Omega$ is analytically derived.
The paper indeed represents a new and interesting
result in the context. The authors have responded
to comments of the first referee thoroughly. However, it would be interesting in future works what happens in finite $N$ which also authors mention in the conclusion. In short, I recommend the present version of the paper for publication.
Anonymous Report 1 on 20201230 (Invited Report)
Report
The author has addressed all points and has revised the manuscript accordingly. I can thus recommend publication of this paper in Scipost Physics in its present form.