Thermodynamic curvature of AdS black holes with dark energy

Abstract: In this paper, we study the effect of dark energy on the extended thermodynamic structure and interacting microstructures of black holes in AdS, through an analysis of thermodynamic geometry. Considering various limiting cases of the novel equation of state obtained in charged rotating black holes with quintessence, and taking enthalpy $H$ as the key potential in the extended phase space, we scrutinize the behavior of the Ruppeiner curvature scalar $R$ in the entropy-pressure $(S,P)$-plane (or equivalently in the temperature-volume ($T,V$)-plane). Analysis of $R$ empirically reveals that dark energy parameterized by $\alpha$, significantly alters the dominant interactions of neutral, charged and slowly rotating black hole microstructures. In the Schwarzschild-AdS case: black holes smaller than a certain size continue to have attractive interactions whereas larger black holes are completely dominated by repulsive interactions which arise to due dark energy. For charged or rotating AdS black holes with quintessence, $R$ can change sign at multiple points depending upon the relation between $\alpha$ and charge $q$ or angular momentum $J$. In particular, above a threshold value of $\alpha$, $R$ is never negative at all, suggesting heuristically that the repulsive interactions due to quintessence are long ranged as opposed to the previously known short ranged repulsion in charged AdS black holes. A mean field interaction potential is proposed whose extrema effectively capture the points where the curvature $R$ changes sign.