$T\bar{T}$, the entanglement wedge cross section, and the breakdown of the split property

TL;DR

This paper investigates how $Tar{T}$ and related deformations affect entanglement measures in holographic conformal field theories, revealing divergences and breakdowns of the split property linked to UV modifications of bulk geometry.

Contribution

It provides a detailed analysis of entanglement measures under $Tar{T}$ deformations, highlighting differences between single- and double-trace cases and their holographic duals.

Findings

Mutual information diverges at finite separation for single-trace deformation.

Divergences disappear in double-trace deformation with a finite radial cutoff.

Quantitative differences between bulk and boundary computations of reflected entropy.

Abstract

We consider fine-grained probes of the entanglement structure of two dimensional conformal field theories deformed by the irrelevant double-trace operator $T\bar{T}$ and its closely related but nonetheless distinct single-trace counterpart. For holographic conformal field theories, these deformations can be interpreted as modifications of bulk physics in the ultraviolet region of anti-de Sitter space. Consequently, we can use the Ryu-Takayanagi formula and its generalizations to mixed state entanglement measures to test highly nontrivial consistency conditions. In general, the agreement between bulk and boundary quantities requires the equivalence of partition functions on manifolds of arbitrary genus. For the single-trace deformation, which is dual to an asymptotically linear dilaton geometry, we find that the mutual information and reflected entropy diverge for disjoint intervals when the separation distance approaches a minimum, finite value that depends solely on the deformation parameter. This implies that the mutual information fails to serve as a geometric regulator which is related to the breakdown of the split property at the inverse Hagedorn temperature. In contrast, for the double-trace deformation, which is dual to anti-de Sitter space with a finite radial cutoff, we find all divergences to disappear including the standard quantum field theory ultraviolet divergence that is generically seen as disjoint intervals become adjacent. We furthermore compute reflected entropy in conformal perturbation theory. While we find formally similar behavior between bulk and boundary computations, we find quantitatively distinct results. We comment on the interpretation of these disagreements and the physics that must be altered to restore consistency. We also briefly discuss the $T{\bar J}$ and $J{\bar T}$ deformations.