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

TL;DR

The paper investigates how $T\bar{T}$-deformed CFTs behave under holographic entanglement measures, comparing single-trace (linear-dilaton UV) and double-trace (finite cutoff AdS) realizations. Using the RT formula, $E_W$, and reflected entropy, it shows that the single-trace case exhibits a finite-distance divergence in mutual information and reflected entropy, signaling a breakdown of the split property at the inverse Hagedorn temperature, while the double-trace case yields UV-finite entanglement measures with a natural geometric cutoff. Conformal perturbation theory is employed to cross-check bulk results, revealing formal similarities but quantitative discrepancies that point to necessary refinements in the holographic dictionary or nonperturbative effects. The work also discusses $T\bar{J}$ and $J\bar{T}$ deformations and their influence on the entanglement structure, highlighting how nonlocal UV physics reconfigures information-theoretic regulators in holographic duals and informing future directions in non-AdS holography and the split-property framework.

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.

Figures (10)

• Figure 1: There are two phases of the entanglement wedge for disjoint intervals. In the connected regime (left), the Ryu-Takayanagi surfaces (red) stretch between the boundary two subregions (purple and orange). This phase manifestly has nontrivial mutual information and entanglement wedge cross section (green). Alternatively, when the intervals are sufficiently distant ($d > d_*$), the disconnected regime (right) dominates and the entanglement wedge becomes the union of the two individual entanglement wedges. This phase has manifest zero mutual information and entanglement wedge cross section.
• Figure 2: The entanglement wedge cross section and half the mutual information per central charge are plotted as a function of the conformally invariant cross-ratio \ref{['conf_cross']} in the vacuum state of a 2D CFT. The bound $E_W > I/2$ is manifest. Notably, $E_W$ discontinuously jumps to zero at $x=1/2$. At this point, the mutual information is continuous but its first derivative is discontinuous. These analytic breakdowns are thought to be artifacts of the $c\rightarrow \infty$ limit.
• Figure 3: The geometric regularization scheme for the von Neumann entropy. We take $A$ and $B$ to be disjoint and separated by $\epsilon$. Then, the regulated "entropy" is $S_R(A:B)/2$ or $I(A:B)/2$ as $L_A/\epsilon \rightarrow 0$.
• Figure 4: The single-trace $T \bar{T}$ deformation changes the bulk AdS$_3$ geometry to asymptote in the UV to a linear dilaton regime represented in yellow. This smoothly crosses over to the undeformed AdS$_3$ regime in the IR. For further generality, we have placed a black hole in the IR regime which sets the boundary theory to finite temperature. The IR region is thus more generally that of the BTZ black hole 1992PhRvL..69.1849B.
• Figure 5: Left: the mutual information per central charge for disjoint intervals in the vacuum. Each subregion is of length $5$ and we plot $\sqrt{\frac{\pi c \mu}{6}} = \{0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} \}$ (blue to red respectively). We mark, with vertical dashed lines, the corresponding values of $l_{min}$ where the mutual information intriguingly diverges. Right: the area of the entanglement wedge cross section for the same parameters. We find the same divergences at $d= l_{min}$.