TT¯ , the entanglement wedge cross section, and the breakdown of the split property

Abstract

We consider fine-grained probes of the entanglement structure of two-dimensional conformal field theories deformed by the irrelevant double-trace operator TT 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 TJ and JT deformations.

On the flow of states under $T\overline{T}$

“…Finally, using ∂ λ H = dθ O T T , the operator U in (2.24) can be rewritten as 25) where M − = − × Σ, with Σ = or S 1 . Note that if we try to naively take (2.24) to be true even in the cases where the energy spectrum complexifies, then the e iθ n factor would either diverge or decay.…”

“…One of the most interesting observables to consider in T T deformed theories is the entanglement entropy of a spatial region [21][22][23][24][25]. In ordinary QFT this is already hard to compute and one has to resort to various techniques like the replica trick to do the calculation.…”

“…In the past few years, much effort has gone into understanding various apsects of T T deformed quantum field theories, such as the spectrum on the circle and its complexification, sphere and torus partition functions [12][13][14][15], the holographic aspect of the T T deformation, correlation functions on the Euclidean plane [9,16,17] and higher-dimensional generalization [18][19][20]. Furthermore, a particularly interesting direction is the study of the entanglement structure of states in these (non-local) theories [21][22][23][24][25]. However, it would be fair to say that beyond the deformed energy spectrum and partition functions, many of these aspects are not fully understood.…”

Strings in irrelevant deformations of AdS3/CFT2

“…In this letter, we continue the study [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] of a class of solvable irrelevant deformations of AdS 3 /CFT 2 duality. These models touch on a number of important issues in quantum field theory and string theory.…”

Reflected entropy for free scalars

“…(A, B) = 2E W (A, B), at leading in order in Newton's constant [10]. Subsequent work studying aspects of reflected entropy building up on the results of [10] includes [11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Candidates for multipartite versions of reflected entropy have also been proposed in [25][26][27].…”