Fine structure in holographic entanglement and entanglement contour

TL;DR

The paper investigates the fine structure of holographic entanglement entropy in $AdS_3/CFT_2$ by exploiting boundary and bulk modular flows to slice the entanglement wedge with modular planes. It establishes a one-to-one boundary-to-RT correspondence and defines an entanglement contour $s_{\mathcal{A}}(\mathcal{A}_2)$ whose bulk quantity is the length of the corresponding RT subinterval $\mathcal{E}_2$, supported by a simple linear prescription. A key result is a general, additive, positive contour function $s_{\mathcal{A}}(\mathcal{A}_2)=\tfrac12\big(S_{\mathcal{A}_1\cup\mathcal{A}_2}+S_{\mathcal{A}_2\cup\mathcal{A}_3}-S_{\mathcal{A}_1}-S_{\mathcal{A}_3}\big)$ that matches known CFT$_2$ cases, including finite-temperature and spatial-circle geometries. The work offers a holographic definition of the entanglement contour and paves the way for generalizations to higher dimensions and nonlocal modular Hamiltonians, linking spacetime structure to entanglement distribution in a concrete and testable way.

Abstract

We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS$_3$/CFT$_{2}$. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are co-dimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval $\mathcal{A}$ and the points on the Ryu-Takayanagi (RT) surface $\mathcal{E}_{\mathcal{A}}$. In the same sense an arbitrary subinterval $\mathcal{A}_2$ of $\mathcal{A}$ will correspond to a subinterval $\mathcal{E}_2$ of $\mathcal{E}_{\mathcal{A}}$. This fine correspondence indicates that the length of $\mathcal{E}_2$ captures the contribution $s_{\mathcal{A}}(\mathcal{A}_2)$ from $\mathcal{A}_2$ to the entanglement entropy $S_{\mathcal{A}}$, hence gives the contour function for entanglement entropy. Furthermore we propose that $s_{\mathcal{A}}(\mathcal{A}_2)$ in general can be written as a simple linear combination of entanglement entropies of single intervals inside $\mathcal{A}$. This proposal passes several non-trivial tests.

Figures (6)

• Figure 1: The left figure shows the modular flow in the causal development $\mathcal{D}_{\mathcal{A}}$ on $\mathcal{B}$. The brown line is the interval $\mathcal{A}$. The right figure shows the modular plane $\mathcal{P}(u_0)$ which is the bulk extension of the boundary modular flow line $\mathcal{L}_{u_0}$. The red and orange arrows depict the direction of the boundary and bulk modular flows respectively.
• Figure 2: This figure shows a typical modular plane $\mathcal{P}(u_0)$ in the entanglement wedge. We depict $\mathcal{P}(u_0)$ as the blue surface that intersect with $\mathcal{B}$ and $\mathcal{N}_{\pm}$ on $\mathcal{L}_{u_0},\bar{\mathcal{L}}_{\bar{u}_0}^{\pm}$, which are depicted as the three red lines, respectively. The green line is $\mathcal{R}_{\mathcal{A}}^{u_0}$ where the modular plane $\mathcal{P}(u_0)$ intersect with $\mathcal{R}_{\mathcal{A}}$.
• Figure 3: The replica story on the modular plane $\mathcal{P}(u_0)$ with $n=2$. The left and right figures are the first and second copies of $\mathcal{P}(u_0)$, and the green line is the $\mathcal{R}_{\mathcal{A}}^{u_0}$ which is cut open and glued cyclically. The gluing is depicted by the two dashed arrows. Through $\mathcal{R}_{\mathcal{A}}^{u_0}$, the modular flow $\tau_1$ flows from one subregion of the first copy to a subregion on the second copy (see the blue arrows). The subregion on the second copy should have the same Im$[\tau]$, thus also denoted as $\tau_1$. It is easy to see that on the cyclically glued $\mathcal{P}_2(u_0)$, the thermal circle becomes $\tau_1\to\tau_2\cdots \to \tau_8\to\tau_1$ or in other words $\tau\sim \tau+4\pi i$.
• Figure 4: The left figure shows the subinterval to subinterval correspondence, where the brown line is the boundary interval $\mathcal{A}$ while the blue line is the RT surface $\mathcal{E}_{\mathcal{A}}$. Here $\mathcal{A}$ is divided into three subintervals $\mathcal{A}_1,\mathcal{A}_2$ and $\mathcal{A}_3$ and the two green lines are $\mathcal{R}_{\mathcal{A}}^{u_1}$ and $\mathcal{R}_{\mathcal{A}}^{u_2}$ respectively. The right figure depicts another interval $\mathcal{A}'$ which is homologous to $\mathcal{A}$, and divided into three subintervals $\mathcal{A}'_1,\mathcal{A}'_2$ and $\mathcal{A}'_3$. We require that the endpoints of $\mathcal{A}'_2$ and $\mathcal{A}_2$ are anchored on the same boundary modular flow lines.
• Figure 5: The RT surfaces associated to $S_{\mathcal{A}_1\cup\mathcal{A}_2}$ and $S_{\mathcal{A}_2\cup\mathcal{A}_3}$ intersect at the point P. We divide these two RT surfaces by P, then the combination of their left parts is a surface homological to $\mathcal{A}_1$ and its length should be larger than the RT surface associated to $S_{\mathcal{A}_1}$ as $S_{\mathcal{A}_1}$ is minimal. The same logic applies to the combination of the right parts. Then we get \ref{['positivity']}.