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

We consider fine-grained probes of the entanglement structure of two-dimensional conformal field theories deformed by the irrelevant double-trace operator $T\overline{T}$ 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 $T\overline{J}$ and $J\overline{T}$ deformations.

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 (1.4) 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 $c\to \infty$ 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 $L_A/\epsilon \to 0$.

Figure 4

The single-trace $T\overline{T}$ deformation changes the bulk ${\mathrm{AdS}}_3$ geometry to asymptote in the UV to a linear dilaton regime represented in yellow. This smoothly crosses over to the undeformed ${\mathrm{AdS}}_3$ 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 Bañados-Teitelboim-Zanelli (BTZ) black hole [81].

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 $d=l_{\min }$.

Figure 6

Left: the mutual information per central charge for disjoint intervals at finite temperature. Each subregion is of length 5 and $\sqrt{\frac{\pi c\mu }{6}}$ is set to $\frac{1}{10}$. The temperature is varied as $\frac{\beta }{2\pi }=\{100,3,2,1\}$ from blue to red respectively. Right: the entanglement wedge cross section per central charge for the same parameters. We observe the divergences at $d=l_{\min }$ for both quantities.

Figure 7

The mutual information (left) and entanglement wedge cross section (right) for $z_c=\{0,\frac{1}{4},\frac{1}{2},\frac{3}{4}\}$ in descending order. When the deformation parameter is finite (solid lines), $E_W$ and $I$ are manifestly finite even as the intervals become adjacent. We use symmetric intervals of length $l=5$.

Figure 8

In the cutoff AdS approach, the standard Dirichlet boundary conditions of holography are moved from the conformal boundary at $r_{\infty }$ to a hard Dirichlet boundary at $r_c$. Because the geometry within the new boundary remains the same, so do the geodesics. In the figure, we show how the entangling surface points of the interval are mapped from the asymptotic boundary to the finite boundary.

Figure 9

The analog of the mapping in Fig. 8, this time with the additional scale, $r_H$, representing the black hole horizon radius.

Figure 10

Left: the mutual information per unit central charge for disjoint intervals of equal length (5). We set the radial cutoff to $r_c=10$ and vary the temperature $r_H=\{{10}^{-2},1,2,3\}$ from blue to red respectively. Right: the same configurations, but for the entanglement wedge cross section.