An analytic approach for holographic entanglement entropy at (quantum) criticality

TL;DR

The paper presents a systematic large-$D$ framework to compute holographic entanglement entropy (HEE) for strip regions in AdS black hole spacetimes, by splitting the bulk into near-boundary and near-horizon patches and matching analytic NB and NH solutions. This yields fully analytic expressions for HEE in neutral, charged, extremal, and soliton geometries, with a consistent large-width expansion where the leading term is the thermal (volume) contribution and the subleading term encodes $sT+\mu Q$-type thermodynamics via the area of the entangling surface. A key result is that, at large $D$, the NB region controls the subleading $1/L$ correction and that a universal NB-dominated formula $S_E(A)\approx \mathrm{Vol}(A)s+\mathrm{Vol}(\partial A)\frac{2\pi}{d}(sT+\mu Q)$ emerges for sufficiently large regions, connecting holographic entanglement to thermodynamics. The approach demonstrates analytic tractability across a family of geometries and suggests broader applicability to nearly critical theories and other observables, while clarifying how extremality and soliton backgrounds affect the entanglement structure.

Abstract

We consider holographic entanglement entropy in AdS black hole backgrounds by using the limit of large number of dimensions. By dividing the geometry to two patches (with one patch covering the vicinity of the black hole horizon and another covering the other regions), we are able to obtain fully analytic expressions for the entropy when the entanglement region is a strip. We argue that apart from conformal field theories at finite temperature in high number of dimensions, our method works for nearly critical theories in 3+1 dimensions. In the case of extremal black holes, dual to quantum critical systems, the results take a particularly simple form. We also comment on the case of soliton geometries. Finally, we analyze entanglement entropy for wide strips, and propose a general formula for the first subleading term in the expansion of the entropy in (inverse) system size for generic entanglement regions.

Figures (9)

• Figure 1: Entanglement entropy for strips with $d=10$ (i.e., AdS$_{11}$ black holes) using the direct expansion \ref{['eq:BHexactser']}. Top left: Length of the strip as a function of the turning point $r_*$. Top right: Regulated area of the embedding as a function of $r_*$. Bottom: Area as a function of the length.
• Figure 2: Sketch of the large $D$ method.
• Figure 3: The critical length $L_c$ at which near-horizon geometry starts to contribute as a function of $d$, compared to an approximation obtained by treating the black hole as a membrane, Eqs. \ref{['eq:Lmemb0']} and \ref{['eq:Lmemb1']}.
• Figure 4: Breakdown of the analytical approximation of the Entanglement entropy to near-boundary and near-horizon contributions for neutral black holes. Top left: Length of the strip as a function of the turning point $w_*$. Top right: Regulated area of the embedding as a function of $w_*$. Bottom: Area as a function of the length. We use $d=25$, $50$, and $100$ shown as the blue, orange, and green curves, respectively, as indicated in the Legend. The thick solid curves show the full result, labeled as "Tot". The dotted (dotdashed) curves show the NB (NH) contribution to the full result, labeled as "NBc" ("NH"). Finally, the dashed curve shows the result if the NH terms are ignored and only the NB expression is used (labeled as "NB").
• Figure 5: Comparison of the analytic expressions (curves) to exact numerical results (data points) at relatively low $d$. The black circles denote the transition of the turning point of the embedding from the near-boundary to the near-horizon region.