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Entanglement Negativity in Flat Holography
by Debarshi Basu, Ashish Chandra, Himanshu Parihar, Gautam Sengupta
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Authors (as registered SciPost users):  Debarshi Basu · Ashish Chandra · Himanshu Parihar · Gautam Sengupta 
Submission information  

Preprint Link:  scipost_202107_00037v1 (pdf) 
Date submitted:  20210721 08:45 
Submitted by:  Sengupta, Gautam 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We advance holographic constructions for the entanglement negativity of bipartite states in a class of $(1+1)$dimensional Galilean conformal field theories dual to asymptotically flat three dimensional bulk geometries described by Einstein Gravity and Topologically Massive Gravity. The construction involves specific algebraic sums of the lengths of bulk extremal curves homologous to certain combinations of the intervals appropriate to such bipartite states. Our analysis exactly reproduces the corresponding replica technique results in the large central charge limit. We substantiate our construction through a semi classical analysis involving the geometric monodromy technique for the case of two disjoint intervals in such holographic Galilean conformal field theories.
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Reports on this Submission
Anonymous Report 2 on 202197 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202107_00037v1, delivered 20210906, doi: 10.21468/SciPost.Report.3501
Strengths
The authors consider various cases and perform several checks of their results.
Weaknesses
Unclear what new physics the results obtained teach us.
Report
A potentially interesting result of this article is the large central charge negativity of two disjoint intervals. To compute it, the authors use the standard CFT monodromy method applied to the case of Galilean conformal symmetry with the help of [67].
They provide formula (104) for the Galilean conformal block associated with certain four point functions (relevant in the computation eg of entanglement, negativity etc). The result is valid in the large central charge limit, and when the dominant contribution comes from a light operator (and the monodromy is computed around a light operator).
In section 6.1.3, the authors apply (104) for computing the entanglement negativity of disjoint intervals. However, they simultaneously claim that the dominant contribution in this case comes from an operator that remains heavy in the large rental charge limit  like in [25]. I would be grateful if the authors could further clarify this point, before considering the article for publication.
Requested changes
See above.
Anonymous Report 1 on 2021829 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202107_00037v1, delivered 20210829, doi: 10.21468/SciPost.Report.3460
Report
This paper studied extensively the holography negativity in flat holography, ranging form single interval to multiple intervals, from Einstein gravity to TMG. The paper is rather long and contains lots of small technical details. Without going through all the details, I find that the strategies and results look reasonable to me. Therefore, I would recommend the publication of the paper.
Author: Gautam Sengupta on 20211009 [id 1830]
(in reply to Report 1 on 20210829)We would like to thank the referee for the comment and for recommending publication.
Author: Gautam Sengupta on 20211009 [id 1831]
(in reply to Report 2 on 20210907)We would like to thank the referee for the comments and raising the issue described in the report. Our detailed response to the comment and the clarification of the issue is appended. We have included this clarification in our revised manuscript as suggested by the referee.
In this article we consider a four point twist correlator in the GCFT$_{1+1}$ and wish to compute the monodromy in the $t$channel by going around a loop enclosing the light operators situated at $(0,1)$ and $(X,T)$ which fuse together. By performing the $\mathcal{M}$ and $\mathcal{L}$monodromies, we computed the generic conformal block for an exchange operator of conformal dimension $\chi_\alpha$ in eq. (104). The conformal block for the fourpoint function of the twist operators is obtained perturbatively in the parameter $\epsilon_{\alpha}$ which is related to the conformal dimension of the corresponding exchange operator which is hence required to be light. Nevertheless, as described in reference [13, 25] of our revised manuscript, in the large central charge limit the dominant contribution to equation (78) comes from the exchange operator with the lowest conformal dimension which can be obtained from the leading order term in the operator product expansion (OPE) in the $s$ and $t$channels. For our case the dominant contribution to the entanglement negativity for the two disjoint intervals in proximity (for the $t$channel) may be obtained from that of the conformal block corresponding to the exchange of the lightest operator.
Specifically, we consider the OPE of light operators located at the positions $[(x_2,t_2)\,,\,(x_3,t_3)]$ for the $t$channel as shown in equation (105) in our updated submission. The fusion channels include the light operators $\Phi_{n_e}(x_2,t_2)$, $\Phi_{n_e}(x_3,t_3)$ and $\Phi_{n_e}(x_1,t_1)$, $\Phi_{n_e}(x_4,t_4)$ for the $t$channel. It is evident from equation (105) that the leading order contribution arises from the exchange operator $\Phi^2_{\pm n_e}$ for the $t$channel as given in equation (107). In this case the operator $\Phi^2_{\pm n_e}$ is the lightest of the exchange operators with the lowest conformal dimension for the $t$channel. This serves as a justification for the monodromy analysis as described in ref [23, 25] in the framework of the usual relativistic CFT, and the use of equation (104) although $\Phi^2_{\pm n_e}$ actually remains heavy in the replica limit. We have added a discussion of this subtle issue and provided the relevant OPE expansions in equation (105) in the beginning of subsection 6.1.3 of our revised submission.
We would also like to point out that the holographic entanglement negativity for the disjoint intervals (in proximity) has also been obtained from the extremal entanglement wedge cross section in arXiv 2106.14896 recently. The corresponding entanglement negativity matches with our result computed via monodromy analysis which further serves as a consistency check for our computations.