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Second Rényi entropy and annulus partition function for onedimensional quantum critical systems with boundaries
by Benoit Estienne, Yacine Ikhlef, Andrei Rotaru
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Submission summary
Authors (as Contributors):  Andrei Rotaru 
Submission information  

Arxiv Link:  https://arxiv.org/abs/2112.01929v1 (pdf) 
Date submitted:  20211217 16:05 
Submitted by:  Rotaru, Andrei 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider the entanglement entropy in critical onedimensional quantum systems with open boundary conditions. We show that the second Rényi entropy of an interval away from the boundary can be computed exactly, provided the same conformal boundary condition is applied on both sides. The result involves the annulus partition function. We compare our exact result with numerical computations for the critical quantum Ising chain with open boundary conditions. We find excellent agreement, and we analyse in detail the finitesize corrections, which are known to be much larger than for a periodic system.
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Reports on this Submission
Anonymous Report 2 on 2022117 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.01929v1, delivered 20220117, doi: 10.21468/SciPost.Report.4197
Report
The paper deals with computation the entanglement entropy in a bipartite system where the subregion of interest is a single interval away from the boundary. The authors report an exact computation of the second Re’nyi entropy in the ground state of a 1+1 dimensional critical system, which is based on the evaluation of a twopoint function of twist operators on the infinite strip. Choosing suitable boundary conditions, the problem is reduced to the derivation of the twotwist correlation function on the unit disk D without insertions of boundary operators. The final result is expressed in terms of the annulus partition function of the starting CFT. Some numerical checks for the special case of the Ising spin chain are also presented. Fermionic chains of system size up to103 sites are analysed and the finitesize effects are discussed.
The main result of the paper is new and deserves a publication. The exposition is clear, all statements are carefully formulated and supported by explicit computations. The appendix B contains an alternative derivation of the main result. I recommend the publication of the paper in its present form.
Anonymous Report 1 on 2022115 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.01929v1, delivered 20220115, doi: 10.21468/SciPost.Report.4186
Strengths
1 The paper finds the solution for a well defined old open problem.
2 The organization of the paper is good.
3 The derived equation is supported with numerical calculations
Weaknesses
1 The solution is found just for n=2 Renyi entropy.
2 Numerical calculations are done for just one type of BC.
Report
This paper find a clean and compact formula for the n=2 Renyi entanglement entropy of an interval in an open CFT. This problem has been an open problem for more than 15 years. The paper exploits the methods of the references [49] and [77] to find the main result, i.e. equation 1.8.
Then the paper support the formula by redriving the previous results that were valid for especial cases. They also check the validity of the formula using numerical calculations on the transverse field Ising chain with open BC.
Although the main problem of calculating Renyi entropy for general n and the von Neumann entropy still remains an open problem, due to the level of the difficulty of the project, I would not consider it as a major weakness.
The paper is written clearly and follows a good logic. I think the paper has enough merit to be published in Scipost after a few minor additions and changes.
Requested changes
1: I guess a factor 2 is missing in equation 1.4 plus some nonuniversal constant.
2: Fig 2 is not cited in the main text. In addition the caption does not seem self explanatory. In general I think the mapping can be probably explained better by improving this Figure.
3: It would be nice to see limiting cases of equation 2.52.
Author: Andrei Rotaru on 20220221 [id 2229]
(in reply to Report 1 on 20220115)
We thank the referee for their thorough reading of the manuscript, and their useful comments and suggestions. We have adressed them as follows:
1: I guess a factor 2 is missing in equation 1.4 plus some nonuniversal constant.
 The factor of 2 was corrected, and we have addressed the origin of the nonuniversal constant
2: Fig 2 is not cited in the main text. In addition the caption does not seem self explanatory. In general I think the mapping can be probably explained better by improving this Figure.
 The figure is now cited in the main text, and we have modified it to show the preimages of the points $0$ and $x$ under the map g(t)
3: It would be nice to see limiting cases of equation 2.52.
 We have added two subsections, discussing two interesting limits of the Schatten distance
Anonymous on 20220223 [id 2243]
(in reply to Andrei Rotaru on 20220221 [id 2229])The authors successfully took care of my comments. I recommend the paper for publication in its current form.
Author: Andrei Rotaru on 20220221 [id 2228]
(in reply to Report 2 on 20220117)We thank the referee for their careful reading of the manuscript and their appreciations.