Topological Classification of a 4D AdS Black Hole with Non-Minimal Maxwell Coupling

TL;DR

This work addresses how the phase structure of a four-dimensional AdS black hole with non-minimal Maxwell coupling can be understood in a universal, model-independent way. It applies a topological framework, built on a generalized free energy and Duan's phi-mapping, to assign winding numbers to black hole branches and compute a global invariant $W$. The analysis reveals a dual thermodynamic character controlled by the Maxwell charge $Q$: large $Q$ yields a van der Waals–type topology with $W=+1$ (three branches with windings $(+1,-1,+1)$), while small $Q$ yields a Hawking–Page–type topology with $W=0$ (two branches with windings $(-1,+1)$). Importantly, the non-minimal coupling $\lambda$ stabilizes the $W=0$ Hawking–Page class for charged black holes, a phenomenon absent in RN–AdS, thereby tying microscopic couplings to macroscopic topological universality and illustrating the power of topological methods in modified gravity and holography.

Abstract

We perform a topological classification of the phase structure of a four-dimensional AdS black hole with non-minimal Maxwell coupling. Critical points are treated as topological defects, allowing us to assign a winding number to each black hole branch and compute the global topological invariant W. The system exhibits a duality governed by its Maxwell charge Q: for large Q it falls into the class W = 1, displaying van der Waals-type behavior with a first-order small-large black hole transition. For small Q, it shifts to W = 0, characteristic of a Hawking-Page transition. This topological classification provides a model-independent validation of the conventional thermodynamic analysis. Crucially, we find that the non-minimal coupling lambda stabilizes the Hawking-Page universality class W=0 for black holes with non-zero charge, a phenomenon absent in the standard Reissner-Nordstrom-AdS case. This establishes a direct link between the microscopic coupling and the macroscopic topological class, demonstrating the power of topological methods in decoding thermodynamic universality across modified gravity theories.

Figures (9)

• Figure 2: Thermodynamic potentials demonstrating first-order phase transition signatures in the van der Waals regime. The swallowtail structure in $F(T)$ and the discontinuity in $S(T)$ occur at subcritical pressure $P = 0.01$ for $Q = 0.5$, $\lambda = 0.001$.
• Figure 3: Heat capacity at constant pressure, $C_P$, versus horizon radius $r_h$ in the van der Waals regime. For subcritical pressure ($P < P_c$), the system progresses through two stable phases (positive $C_P$) separated by an intermediate unstable phase (negative $C_P$).
• Figure 5: Thermodynamic potentials in the small-charge (Hawking-Page) regime. Both panels are for $Q = 0.01$ and $\lambda = 0.001$, illustrating (a) the global stability condition via free energy and (b) the local stability via heat capacity.
• Figure 6: (a) Vector field topology showing zero points (black dots) at $(r_h,\Theta) = (0.500,\pi/2)$, $(0.959,\pi/2)$, and $(3.381,\pi/2)$ for ZP1, ZP2, and ZP3, respectively. Red contours $C_i$ enclose these points for $Q=0.5, P=0.01, \tau=11$. The behavior of the unit vector field at the boundaries shows the system lies within Case III. (b)$r_h$--$\tau$ diagram showing generation and annihilation points in the large-$Q$ regime. Points $a$ and $b$ mark the generation and annihilation of black holes at $\tau=10.04$ and $\tau=12.75$, respectively. For each colored rectangular region, $W=1$.
• Figure 7: Unit vector field diagrams: (a) Zero points at $(r_h,\Theta) = (0.42,\pi/2)$, $(1.97,\pi/2)$, and $(2.67,\pi/2)$. (b) Zero points at $(r_h,\Theta) = (0.78,\pi/2)$, $(1.62,\pi/2)$, and $(1.84,\pi/2)$. Red contours $C_i$ enclose these points. The boundary behavior of the vector field places the system within Case III of the topological classification.