Boundary mutual information in double holography

TL;DR

This paper analyzes the boundary mutual information between two disjoint subregions in a doubly holographic setup where AdS$_3$ gravity is coupled to a flat heat bath. BMI is computed via quantum extremal surfaces and Surface Evolver, revealing a phase transition as subregion separation grows and a robust decomposition into a geometric (surface-area) term and a negative bulk correction from brane/bulk quantum fields. The semiclassical limit shows universal logarithmic behavior with coefficients tied to central charges, while numerical results confirm the phase structure and the sign of the bulk term. A RTN toy model corroborates the negativity and provides an intuitive information-theoretic explanation in terms of volume-law entanglement of the bulk with the bath. The findings illuminate how geometry and bulk entanglement jointly shape BMI in double holography and suggest directions for finite-temperature extensions and other information measures.

Abstract

We consider a composite system where AdS$_3$ gravity is coupled to a flat heat bath and investigate the mutual information between two subregions on the intersection of the AdS$_3$ and bath, referred to as the boundary mutual information (BMI). The corresponding entanglement entropy is captured via quantum extremal surfaces (QES), which holographically be computed by a surface optimization algorithm based on ``Surface Evolver''. We focus on both connected and disconnected configurations of the quantum entanglement wedge (Q-EW) in the AdS$_3$ bulk and analyze the finite corrections to the BMI. Our numerical results reveal a phase transition of the BMI as the separation between two subregions increases. Furthermore, we find that the BMI can naturally be decomposed into two distinct components: a geometric term arising from the areas of the quantum extremal surfaces, and a correction term resulting from bulk quantum fields within the Q-EW. Interestingly, the geometric contribution always exceeds the total BMI, indicating a negative correction from the bulk matter fields. This negativity can be understood as the result of subtracting a greater contribution from quantum fields in the connected Q-EW than in the disconnected one. We also reproduce the negative contribution of bulk quantum fields to BMI within a random tensor network (RTN) toy model of double holography. Modeling the bulk as a highly mixed state entangled with a large bath leads to a volume-law bulk entropy. In the large bond-dimension limit, the geometric part of the BMI remains non-negative, while the bulk entropy contribution becomes non-positive when the Q-EWs merge.

Figures (9)

• Figure 1: The time slice of the AdS$_4$ spacetime with two separated branes.
• Figure 2: (a) Boundary perspective: $\bm{A}$ is a region on $\partial\mathcal{B}$, located at the boundary of the heat bath $\partial$. (b) Brane perspective: $\gamma_{\bm{A}}$ and $\mathcal{W}$ represent the QES and the corresponding Q-EW on the brane. Curved arrows represent the contributions from quantum fields within $\mathcal{W}$. (c) Bulk gravity perspective: $\Gamma_{\bm{A}}$ represents the classical extremal surface bounded by $\gamma_{\bm{A}} \cup \bm{A}$; $l$ denotes the length of $\bm{A}$.
• Figure 3: (a): The initial trial surface corresponding to a simply connected boundary region $\bm{A}$, anchored on $z=\epsilon$, constructed as a triangulated mesh. (b): The final minimal surface obtained after the optimization of the triangular facets via gradient descent in Surface Evolver. (c): A connected configuration of the RT surface for a disconnected boundary region $\bm{A}=\bm{A_1}\cup\bm{A_2}$ shown in red, also anchored on $z=\epsilon$. To properly account for all boundary degrees of freedom, the region $\bm{A}$ is regularized as a rectangular strip with nonzero width $x^*\simeq\epsilon\ll \textit{any other length scales}$, following the prescription outlined in Almheiri:2019hni.
• Figure 4: (a): The entanglement entropy of a single region $\bm{A}$ depicted in blue as a function of $l$, where $l$ is the length of the region. The UV cutoff is fixed to be $\epsilon=0.01$, and the dihedral angle is set to $\theta_0=\pi/34$. The fitted geometric and correction contributions are overlaid in yellow and green, respectively. (b): The fitting coefficient $b_a$ associated with the logarithmic divergence is shown in blue, as a function of the dihedral angle $\theta_0$. The red curve represents the theoretical prediction for the ratio of central charges $c'/c$ from (\ref{['eq:ratio_of_central_charges']}). The difference between the fitting coefficient and the theoretical ratio is illustrated by the yellow dots.
• Figure 5: (a): The entanglement entropy of two adjacent subregions $\bm{A}=\bm{A_1} \cup \bm{A_2}$ (blue curve) as a function of the dimensionless ratio $l/a$, with the UV cutoff fixed to be $\epsilon=0.01$, and the dihedral angle $\theta_0=\pi/26$. The numerically extracted geometric contribution (yellow) and correction term (green) are shown separately to illustrate their respective behaviors. (b): The fitting coefficient $b_a$ (blue dots) before the logarithmic divergence term is plotted as a function of the dihedral angle $\theta_0$, while the theoretical ratio of the central charges (red curve) is obtained from (\ref{['eq:ratio_of_central_charges']}). The discrepancy between the coefficient and the ratio is illustrated by the yellow dots, confirming convergence as $\theta_0\to0$.