Holographic entanglement entropy for hollow cones and banana shaped regions

TL;DR

This paper computes holographic entanglement entropy for complex regions with conical singularities, revealing divergence structures, anomalies, and phase transitions between connected and disconnected minimal surfaces.

Contribution

It extends holographic entanglement entropy calculations to hollow cone regions with singular boundaries, analyzing divergence coefficients, anomalies, and solution transitions.

Findings

Coefficient of squared logarithmic divergence matches circular cone results.

Anomaly observed under exceptional conformal transformations.

Transition from connected to disconnected minimal surfaces as boundary angles vary.

Abstract

We consider banana shaped regions as examples of compact regions, whose boundary has two conical singularities. Their regularised holographic entropy is calculated with all divergent as well as finite terms. The coefficient of the squared logarithmic divergence, also in such a case with internally curved boundary, agrees with that calculated in the literature for infinite circular cones with their internally flat boundary. For the otherwise conformally invariant coefficient of the ordinary logarithmic divergence an anomaly under exceptional conformal transformations is observed. The construction of minimal submanifolds, needed for the entanglement entropy of cones, requires fine-tuning of Cauchy data. Perturbations of such fine-tuning leads to solutions relevant for hollow cones. The divergent parts for the entanglement entropy of hollow cones are calculated. Increasing the difference between the opening angles of their outer and inner boundary, one finds a transition between connected solutions for small differences to disconnected solutions for larger ones.