Phase transitions in 3D gravity and fractal dimension

Understand

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.