Quantum corrections to holographic entanglement entropy

TL;DR

This work extends holographic entanglement entropy beyond the classical Ryu-Takayanagi formula by computing one-loop (order $G_N^0$) quantum corrections in the bulk. The authors formulate a clean decomposition: the total entropy $S(A)$ equals the classical area term plus a bulk entanglement contribution, plus Wald-like and counterterm corrections arising from background shifts and ultraviolet renormalization. Using the replica trick in the bulk, they show how bulk entanglement, geometric backreaction, and local counterterms piece together to produce finite quantum corrections, and they validate the framework through several applications: logarithmic terms from massless bulk fields, bulk contributions to mutual information, entanglement-plateau corrections, and bulk EPR-type correlations. The results provide a detailed, controllable account of quantum corrections in holographic entanglement, with implications for confinement, thermal states, and nonlocal correlations in AdS/CFT. Overall, the paper clarifies how bulk quantum effects refine the geometric RT prescription and enrich the structure of entanglement in holographic theories.

Abstract

We consider entanglement entropy in quantum field theories with a gravity dual. In the gravity description, the leading order contribution comes from the area of a minimal surface, as proposed by Ryu-Takayanagi. Here we describe the one loop correction to this formula. The minimal surface divides the bulk into two regions. The bulk loop correction is essentially given by the bulk entanglement entropy between these two bulk regions. We perform some simple checks of this proposal.

Figures (10)

• Figure 1: The red segment indicates a spatial region, $A$, of the boundary theory. The leading contribution to the entanglement entropy is computed by the area of a minimal surface that ends at the boundary of region $A$. This surface divides the bulk into two, region $A_b$ and its complement. Region $A_b$ lives in the bulk and has one more dimension than region $A$. The leading correction to the boundary entanglement entropy is given by the bulk entanglement entropy between region $A_b$ and the rest of the bulk.
• Figure 2: Slighly deformed disk and angular direction around the boundary.
• Figure 3: Computation of the entropy using the replica trick. $a)$ Original geometry with no $U(1)$ symmetry. $b)$ Replicated smooth geometry $g_4$. $c)$ After a $Z_n$ quotient of the $g_n$ geometry of b) we get the geometry $\hat{g}_n = g_n/Z_n$. It has a conical singularity with opening angle $2\pi/n$. This geometry has the same asymptotic boundary conditions as the original one in a). We can analytically continue this geometry to non-integer values of $n$. $d)$ We use the geometries in c) to construct the density matrix $\hat{\rho}_n$. $\hat{\rho}_n$ is defined as a path integral on this geometry with arbitrary boundary conditions at $\tau =0, 2\pi$. It can be computed using the bulk Hamiltonian for $\tau$ evolution.
• Figure 4: The contribution to $S_{\cdots}$ from the change in the area of the minimal surface, $\delta A$, due to the quantum corrections of the background. We can interpret this diagram as solving \ref{['backh']} for $\bar{h}$ in terms of 1-loop stress tensor. We need to solve for $\bar{h}$ along the minimal surface and integrate the stress tensor over all space.
• Figure 5: Shape of the minimal area surface in the Klebanov-Strassler theory. The yellow region is the interior. The quantum correction is given by the entanglement between the interior and the exterior.