Phase transitions in 3D gravity and fractal dimension

TL;DR

This paper explores phase transitions in 3D gravity with higher genus boundaries, revealing a new scalar condensation transition linked to the Hausdorff dimension of certain limit sets, impacting holographic entropies.

Contribution

It identifies a second order phase transition in 3D gravity related to scalar fields and connects it to the Hausdorff dimension of limit sets, providing analytical and numerical insights.

Findings

Discovery of a second order phase transition involving scalar condensation.

Relation of the critical dimension to the Hausdorff dimension of limit sets.

Good agreement between gravity calculations and CFT conformal block bounds.

Abstract

We show that for three dimensional gravity with higher genus boundary conditions, if the theory possesses a sufficiently light scalar, there is a second order phase transition where the scalar field condenses. This three dimensional version of the holographic superconducting phase transition occurs even though the pure gravity solutions are locally AdS$_3$. This is in addition to the first order Hawking-Page-like phase transitions between different locally AdS$_3$ handlebodies. This implies that the R\'enyi entropies of holographic CFTs will undergo phase transitions as the R\'enyi parameter is varied, as long as the theory possesses a scalar operator which is lighter than a certain critical dimension. We show that this critical dimension has an elegant mathematical interpretation as the Hausdorff dimension of the limit set of a quotient group of AdS$_3$, and use this to compute it, analytically near the boundary of moduli space and numerically in the interior of moduli space. We compare this to a CFT computation generalizing recent work of Belin, Keller and Zadeh, bounding the critical dimension using higher genus conformal blocks, and find a surprisingly good match.