Reflected entropy and timelike entanglement in $\textrm{T}\bar{\textrm{T}}$ deformed CFT$_2$s

TL;DR

This work addresses mixed-state entanglement in TTbar-deformed CFT$_2$s by computing the reflected entropy $S_R$ for boosted subsystems using a covariant conformal perturbation framework and a replica approach. The authors obtain the leading $O(μ)$ corrections to the Rényi reflected entropy on $nm$-sheet replicas and verify the results holographically through entanglement wedge cross sections in cut-off AdS$_3$ geometries (BTZ and global AdS). They systematically analyze thermal and finite-size settings, including disjoint, adjacent, and single subsystems, and extend the results to timelike entanglement via analytic continuation, finding consistent agreement with the holographic duals in the large central charge limit. The findings consolidate the holographic interpretation of reflected entropy as twice the EWCS in TTbar-deformed theories and reveal characteristic negative corrections for spacelike configurations, with rich timelike behavior motivating future explorations of more general entanglement measures and non-perturbative effects.

Abstract

We develop a covariant formalism to investigate the mixed state entanglement structure of time-dependent boosted subsystems in $\textrm{T}\bar{\textrm{T}}$ deformed CFT$_2$s through the reflected entropy. To this end we utilize the conformal perturbation theory to obtain the Rényi reflected entropy through the partition function on replica manifold. The correction to the reflected entropy to the first order in the deformation parameter $μ$ in then obtained in the replica limit for finite temperature and finite sized systems. We verify our field theoretic computations by obtaining the dual EWCS in the corresponding bulk cut-off AdS$_3$ geometries and find perfect agreement between the two.

Figures (4)

• Figure 1: Schematics of the cut-off BTZ geometry dual of $\text{T}\bar{\text{T}}$ deformed CFT$_2$ at a finite temperature. The new cut-off surface is placed at $r_c$. The coordinates of a subsystem $A \equiv [x'_1, x'_2]$ in CFT are scaled to $[\tilde{x}_1, \tilde{x}_2]$ while pushing the cut-off surface inside the bulk. Figure modified from Asrat:2020uib.
• Figure 2: Schematics of two Lorentz boosted disjoint subsystems $A$ and $B$ in the cut-off BTZ geometry. The two shaded planes represent the original and the shifted asymptotic boundary, and the corresponding scaling of the subsystems. Green curve is the EWCS.
• Figure 3: Schematics of two Lorentz boosted adjacent subsystems $A$ and $B$ in the cut-off BTZ geometry.
• Figure 4: Schematics of a Lorentz boosted subsystem $A$ in the cut-off BTZ geometry. $B_1$ and $B_2$ are auxiliary intervals. Union of the green curves gives the upper bound to the EWCS for this configuration.