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Second Rényi entropy and annulus partition function for one-dimensional quantum critical systems with boundaries
by Benoit Estienne, Yacine Ikhlef, Andrei Rotaru
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|Authors (as Contributors):||Andrei Rotaru|
|Arxiv Link:||https://arxiv.org/abs/2112.01929v1 (pdf)|
|Date submitted:||2021-12-17 16:05|
|Submitted by:||Rotaru, Andrei|
|Submitted to:||SciPost Physics|
We consider the entanglement entropy in critical one-dimensional 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 finite-size 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 2022-1-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2112.01929v1, delivered 2022-01-17, doi: 10.21468/SciPost.Report.4197
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 two-point function of twist operators on the infinite strip. Choosing suitable boundary conditions, the problem is reduced to the derivation of the two-twist 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 finite-size 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 2022-1-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2112.01929v1, delivered 2022-01-15, doi: 10.21468/SciPost.Report.4186
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
1- The solution is found just for n=2 Renyi entropy.
2- Numerical calculations are done for just one type of BC.
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  and  to find the main result, i.e. equation 1.8.
Then the paper support the formula by re-driving 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.
1: I guess a factor 2 is missing in equation 1.4 plus some non-universal 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.