Entanglement Negativity in Flat Holography

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 four-point 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.

We would like to thank the referee for the comment and for recommending publication.