Holographic Geometric Entropy at Finite Temperature from Black Holes in Global Anti de Sitter Spaces

TL;DR

This paper investigates holographic geometric entropy in Schwarzschild black holes within global AdS spaces, revealing phase transitions and multiple solution branches at finite temperature, with universal features across different dimensions.

Contribution

It provides a detailed analysis of the behavior of holographic geometric entropy at finite temperature in various AdS backgrounds, uncovering phase transitions and solution multiplicity.

Findings

Reproduces geometric entropy for AdS3 and 2D CFTs.

Identifies sign changes indicating phase transitions in AdS4 and AdS5.

Discovers multiple solution branches and universal behavior across dimensions.

Abstract

Using a holographic proposal for the geometric entropy we study its behavior in the geometry of Schwarzschild black holes in global $AdS_p$ for $p=3,4,5$. Holographically, the entropy is determined by a minimal surface. On the gravity side, due to the presence of a horizon on the background, generically there are two solutions to the surfaces determining the entanglement entropy. In the case of $AdS_3$, the calculation reproduces precisely the geometric entropy of an interval of length $l$ in a two dimensional conformal field theory with periodic boundary conditions. We demonstrate that in the cases of $AdS_{4}$ and $AdS_{5}$ the sign of the difference of the geometric entropies changes, signaling a transition. Euclideanization implies that various embedding of the holographic surface are possible. We study some of them and find that the transitions are ubiquitous. In particular, our analysis renders a very intricate phase space, showing, for some ranges of the temperature, up to three branches. We observe a remarkable universality in the type of results we obtain from $AdS_4$ and $AdS_5$.