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Multientropy at low Renyi index in 2d CFTs
by Jonathan Harper, Tadashi Takayanagi, Takashi Tsuda
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Submission summary
Authors (as registered SciPost users):  Jonathan Harper 
Submission information  

Preprint Link:  https://arxiv.org/abs/2401.04236v2 (pdf) 
Date submitted:  20240123 05:25 
Submitted by:  Harper, Jonathan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
For a static time slice of AdS$_3$ we describe a particular class of minimal surfaces which form trivalent networks of geodesics. Through geometric arguments we provide evidence that these surfaces describe a measure of multipartite entanglement. By relating these surfaces to RyuTakayanagi surfaces it can be shown that this multipartite contribution is related to the angles of intersection of the bulk geodesics. A proposed boundary dual, the multientropy, generalizes replica trick calculations involving twist operators by considering monodromies with finite group symmetry beyond the cyclic group used for the computation of entanglement entropy. We make progress by providing explicit calculations of Renyi multientropy in two dimensional CFTs and geometric descriptions of the replica surfaces for several cases with low genus. We also explore aspects of the free fermion and free scalar CFTs. For the free fermion CFT we examine subtleties in the definition of the twist operators used for the calculation of Renyi multientropy. In particular the standard bosonization procedure used for the calculation of the usual entanglement entropy fails and a different treatment is required.
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Reports on this Submission
Report
This manuscript concerns multientropy, which is a recently proposed measure of multipartite entanglement in CFTs with a definite conjectured holographic dual (Steiner trees in AdS3/CFT2). This quantity is generally difficult to calculate on the CFT side. The present manuscript manages to calculate it in several cases of interest. This is a technical feat, which will aid other researchers interested in multientropy to get off the ground.
The paper is clearly written, valid, useful, and on a timely subject. I recommend publication.
Requested changes
I have the following recommendations for the text:
(1) In equation (22), the authors wish to write $\sigma_{a^3 b^3}$ but the quantity that is actually given is $\sigma_{a b}$, i.e. the inverse of $\sigma_{a^3 b^3}$.
(2) I apologise for bringing this up but I recommend another proofreading of the text at the level of grammar and vocabulary. Some sentences are not easy to understand, for example "Along the four remaining intervals are shown the corresponding copy it will be glued to to form the replica manifold."
(3) Optional: I think it would be easier to understand definition (20) if the indices corresponded better with the letters. It is strange to associate $\alpha, \beta, \alpha', \beta'$ with $A$ and $a,b,a', b'$ with $B$. A far clearer notation would be to associate $a,a',\alpha, \alpha'$ with $A$ and $b,b', \beta, \beta'$ with $B$. I think this change does not impose a huge burden on the authors because it can be done locally (these indices do not show up in many other places in the text). On the other hand, it will make the manuscript much clearer to absorb for readers.
Report 1 by Matthew Headrick on 2024313 (Invited Report)
Report
Multientropy is an interesting mutliparty generalization of entanglement entropy recently proposed by Gadde et al, with a conjectured holographic dual in terms of networks of bulk minimal surfaces. The present paper further develops the computation of multientropies both in CFTs and holographically. Unfortunately, with current CFT technology, multientropies are harder to calculate from first principles even than entanglement entropies. Nonetheless, the authors are able to compute (Renyi) multientropies in a number of cases, both in a general CFT and in specific ones such as free bosons and fermions, while importantly pointing out subtleties and pitfalls concerning for example the correct definition of the relevant twist operators in the fermionic case.
Investigations of the kind in this paper are essential for furthering our understanding of multientropy, a promising tool for understanding the structure of multiparty entanglement in quantum field theories and holography. The paper is generally clearly written. I recommend its publication in SciPost.
I have a few minor recommendations concerning the presentation.
First, a definition of multientropy in terms of the density matrix is not given until eqs. (19) and (20). So the formulas before that may appear somewhat arbitrary. It might be useful instead to start with the definition, then derive the twist operator formula. A small figure illustrating the tensor contractions in (20) would also be helpful for the reader’s intuition. A full review of multientropy isn’t necessary, but a quick reminder for a reader who hasn’t looked at the previous papers in a while would be useful.
Shortly after that, in (21), it wasn’t clear to me if the particular group representation given there was somehow unique (up to trivial equivalences like relabelling), or if some choice was being made.
A few even more minor corrections: In a few place, the cyclic group of order $n$ is written $Z^n$, where it would conventionally (in the physics literature) be written $Z_n$. Above (16), the $(q1)$fold power of $Z_n$ is written with a tensor product symbol, whereas it is actually just a (Cartesian) group product. On p. 19, just below (79), “Even though we are expecting” should be “Even though we are NOT expecting”. Also on p. 19, “fig. 14” should be “fig. 11” in a couple of place. In several places throughout the paper, “explicate” should be “explicit”.
Requested changes
1) A definition of multientropy in terms of the density matrix is not given until eqs. (19) and (20). So the formulas before that may appear somewhat arbitrary. It might be useful instead to start with the definition, then derive the twist operator formula. A small figure illustrating the tensor contractions in (20) would also be helpful for the reader’s intuition. A full review of multientropy isn’t necessary, but a quick reminder for a reader who hasn’t looked at the previous papers in a while would be useful.
2) In (21), it wasn’t clear to me if the particular group representation given there was somehow unique (up to trivial equivalences like relabelling), or if some choice was being made.
3) In a few place, the cyclic group of order $n$ is written $Z^n$, where it would conventionally (in the physics literature) be written $Z_n$.
4) Above (16), the $(q1)$fold power of $Z_n$ is written with a tensor product symbol, whereas it is actually just a (Cartesian) group product.
5) On p. 19, just below (79), “Even though we are expecting” should be “Even though we are NOT expecting”.
6) Also on p. 19, “fig. 14” should be “fig. 11” in a couple of place.
7) In several places throughout the paper, “explicate” should be “explicit”.