An equal area law for holographic entanglement entropy of the AdS-RN black hole

TL;DR

This paper demonstrates that the entanglement entropy of AdS-RN black holes obeys Maxwell's equal area law during phase transitions, similar to black hole entropy, across various dimensions and entangling region sizes, but not in dyonic cases.

Contribution

The study provides numerical evidence that entanglement entropy obeys the equal area law in AdS-RN black holes, establishing a deeper connection between entanglement and black hole thermodynamics.

Findings

Entanglement entropy obeys Maxwell's equal area law with ~1% accuracy.

Law holds for various sizes of spherical entangling regions.

Law is valid in 4 and 5 dimensions for AdS-RN.

Abstract

The Anti-de Sitter-Reissner-Nordstrom (AdS-RN) black hole in the canonical ensemble undergoes a phase transition similar to the liquid-gas phase transition, i.e. the isocharges on the entropy-temperature plane develop an unstable branch when the charge is smaller than a critical value. It was later discovered that the isocharges on the entanglement entropy-temperature plane also exhibit the same van der Waals-like structure, for spherical entangling regions. In this paper, we present numerical results which sharpen this similarity between entanglement entropy and black hole entropy, by showing that both of these entropies obey Maxwell's equal area law to an accuracy of around 1 %. Moreover, we checked this for a wide range of size of the spherical entangling region, and the equal area law holds independently of the size. We also checked the equal area law for AdS-RN in 4 and 5 dimensions, so the conclusion is not specific to a particular dimension. Finally, we repeated the same procedure for a similar, van der Waals-like transition of the dyonic black hole in AdS in a mixed ensemble (fixed electric potential and fixed magnetic charge), and showed that the equal area law is not valid in this case. Thus the equal area law for entanglement entropy seems to be specific to the AdS-RN background.