CFT Phase Transition Analysis of Charged, Rotating Black Holes in $D=4$: A Holographic Thermodynamics Approach

We investigate the holographic thermodynamics of 4-D Kerr-Newman AdS black holes, focusing on the conformal thermal states that are dual to these black holes. We explore the thermodynamic behavior within specific ensembles characterized by fixed sets of variables: $(\mathcal{Q},\mathcal{J},\mathcal{V},C)$, $(\mathcal{Q},\mathrm{\Omega },\mathcal{V},C)$, $(\phi ,\mathrm{\Omega },\mathcal{V},C)$, $(\phi ,\mathcal{J},\mathcal{V},C)$, $(\mathcal{Q},\mathrm{\Omega },p,C)$, and $(\phi ,\mathrm{\Omega },p,C)$. Here, $\phi$, $\mathcal{Q}$, $\mathrm{\Omega }$, $\mathcal{J}$, $p$, $\mathcal{V}$, and $C$ represent the electric potential, electric charge, angular velocity, angular momentum, CFT pressure, CFT volume, and central charge, respectively. The inclusion of both charge and momentum significantly enriches the regime of phase transitions, leading to a variety of phenomena including first-order Van der Waals-type phase transitions, (de)confinement phase transitions, Davies-type phase transitions, and second-order superfluid $\lambda$-type phase transitions. Notably, the introduction of the CFT pressure variable allows us to identify phase transitions and critical behavior in the $(\mathcal{Q},\mathrm{\Omega },p,C)$ and $(\phi ,\mathrm{\Omega },p,C)$ ensembles, which had not been previously observed. This study underscores the complexity and richness of phase transitions in these systems due to the inclusion of both charge and angular momentum.