Ensemble Dependent Holographic Phase Transitions in 4$D$ Dyonic AdS Black Holes

This work studies the holographic thermodynamics of $4D$ dyonic Anti--de Sitter black holes within the AdS/CFT correspondence. By allowing variations of the cosmological constant $\Lambda$ and Newton's constant $G_N$, the gravitational thermodynamics is extended to include the CFT central charge $C$ and its conjugate chemical potential $\mu$, together with the standard thermodynamic pair $(T,S)$. A further conjugate pair $(p,\mathcal{V})$, interpreted as the CFT pressure and volume, is also introduced. The presence of both electric and magnetic charges, described by $(\tilde{\Phi}_e,\tilde{Q}_e)$ and $(\tilde{\Phi}_m,\tilde{Q}_m)$, leads to a significantly enriched phase structure. A systematic analysis of all sixteen thermodynamic ensembles reveals a strong dependence of phase behavior on ensemble choice. In the ensembles $(\tilde{Q}_m,\tilde{\Phi}_e,\mathcal{V},C)$ and $(\tilde{Q}_e,\tilde{\Phi}_m,\mathcal{V},C)$, the system exhibits Van der Waals transitions, superfluid $\lambda$ transitions, and Davies criticality, while the ensemble $(\tilde{Q}_m,\tilde{Q}_e,\mathcal{V},C)$ supports similar transitions except for the Davies transitions. When the chemical potential $\mu$ is fixed, only Davies--type transitions occur. Ensembles involving the CFT pressure $p$ show no critical behavior, although Davies transitions persist. Notably, the ensemble $(\tilde{\Phi}_m,\tilde{\Phi}_e,\mathcal{V},C)$ uniquely displays a confined/deconfined transition alongside Davies criticality. These results highlight the crucial role of dyonic charges and CFT variables in shaping holographic phase transitions.