Thermodynamic curvature of charged black holes with $AdS_2$ horizons

Abstract

Sign and magnitude of the thermodynamic curvature provides empirical information about the nature of microstructures of a general thermodynamic system. For charged black holes in AdS, thermodynamic curvature is positive for large charge or chemical potential, and diverges for extremal black holes, indicating strongly repulsive nature. We compute the thermodynamic curvature at low temperatures, for charged black holes with AdS$_2$ near horizon geometry, and containing a zero temperature horizon radius $r_h$, in a spacetime which asymptotically approaches $AdS_D$ (for $D>3$). In the semi-classical analysis at low temperatures, the curvature shows a novel crossover from negative to positive side, indicating the shift from attraction to repulsion dominated regime near $T=0$, before diverging as $1/(γT)$, where $γ$ is the coefficient of leading low temperature correction to entropy. Accounting for quantum fluctuations, the curvature computed in the canonical ensemble is positive, whereas the one in the grand canonical ensemble, continues to show a crossover from negative to positive side. Moreover, the divergence of curvature at $T=0$ is cured irrespective of the ensemble used, resulting in a universal constant.