Holographic entanglement entropy in AdS4/BCFT3 and the Willmore functional

TL;DR

This paper analyzes the holographic entanglement entropy in AdS4/BCFT3, deriving an analytic formula related to the Willmore functional and numerically exploring entangling regions with arbitrary shapes, including corners.

Contribution

It provides an explicit analytic expression for the finite term of holographic entanglement entropy in AdS4/BCFT3, linking it to the Willmore functional and extending understanding to regions with corners.

Findings

Derived an analytic formula for the finite entanglement entropy term.

Reproduced known results for smooth domains and corner contributions.

Numerically computed entanglement entropy for arbitrary-shaped regions, including ellipses.

Abstract

We study the holographic entanglement entropy of spatial regions having arbitrary shapes in the AdS4/BCFT3 correspondence with static gravitational backgrounds, focusing on the subleading term with respect to the area law term in its expansion as the UV cutoff vanishes. An analytic expression depending on the unit vector normal to the minimal area surface anchored to the entangling curve is obtained. When the bulk spacetime is a part of AdS4, this formula becomes the Willmore functional with a proper boundary term evaluated on the minimal surface viewed as a submanifold of a three dimensional flat Euclidean space with boundary. For some smooth domains, the analytic expressions of the finite term are reproduced, including the case of a disk disjoint from a boundary which is either flat or circular. When the spatial region contains corners adjacent to the boundary, the subleading term is a logarithmic divergence whose coefficient is determined by a corner function that is known analytically, and this result is also recovered. A numerical approach is employed to construct extremal surfaces anchored to entangling curves with arbitrary shapes. This analysis is used both to check some analytic results and to find numerically the finite term of the holographic entanglement entropy for some ellipses at finite distance from a flat boundary.