Topological Mod(A)Max AdS Black Holes

• Topological Mod(A)Max AdS Black Holes are four-dimensional solutions characterized by fixed horizon topology and influenced by nonlinear electrodynamics with deformation parameters.
• They display rich phase structures including van der Waals–like transitions, critical phenomena, and a robust global winding number, impacting thermodynamic stability.
• Their extended thermodynamics framework, which treats the cosmological constant as pressure, unifies varied horizon geometries and nonlinearities under universal topological classifications.

Topological Mod(A)Max AdS Black Holes are a class of four-dimensional anti-de Sitter (AdS) black hole solutions of Einstein gravity coupled to the ModMax class of nonlinear electrodynamics, with fixed horizon topology. These solutions generalize the familiar Reissner–Nordström–AdS family by introducing both a nonlinear deformation parameter (γ, ModMax) or its phantom analog (ModAMax, η = –1), along with the conventional topological index k ∈ {+1, 0, –1} controlling the horizon’s constant curvature. Their phase structure, thermodynamic stability, critical phenomena, and global classification are deeply influenced by the interplay between the nonlinear electrodynamics sector and horizon topology. Recent advances in black hole thermodynamics have also uncovered a robust topological invariant—the global winding number W—characterizing the critical and equilibrium structures of these systems (Panah et al., 27 Dec 2025, Panah et al., 2024, Panah et al., 2024).

1. ModMax and ModAMax Nonlinear Electrodynamics in AdS

The canonical setting is four-dimensional gravity with cosmological constant Λ < 0, minimally or nonminimally coupled to a nonlinear gauge field. The action is

$$\mathcal{I} = \frac{1}{16\pi} \int d^4x\sqrt{-g} \left[R - 2\Lambda - 4\eta \mathcal{L}(F, \widetilde{F})\right].$$

For ModMax (η = +1), the Lagrangian is

$$\mathcal{L} = \mathcal{S}\cosh\gamma - \sqrt{\mathcal{S}^2+\mathcal{P}^2}\sinh\gamma,$$

with electromagnetic invariants $\mathcal{S} = \tfrac14 F_{\mu\nu} F^{\mu\nu}$, $$\mathcal{P} = \tfrac14 F_{\mu\nu} \widetilde{F}^{\mu\nu}$$, and $\gamma \ge 0$ the "ModMax deformation parameter." ModAMax (η = –1) corresponds to the phantom branch. The linear Maxwell case is recovered at γ → 0. The horizon geometry is characterized by the index $k$:

• $k=+1$: $S^2$ (spherical)
• $k=0$: $\mathbb{R}^2$ (planar)
• $k=-1$: $H^2$ (hyperbolic).

The ansatz for the black hole metric is

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2\, d\Omega_k^2,$$

with $d\Omega_k^2$ appropriate to the horizon topology. The electromagnetic solution for a static, purely electric ansatz yields

$h(r) = -\frac{q}{r},$

and the complete line element satisfies

$$f(r) = k - \frac{m}{r} - \frac{\Lambda}{3} r^2 + \eta \frac{q^2 e^{-\gamma}}{r^2}.$$

The integration constants are related to the ADM mass $M = m/8\pi$ and charge $Q = q/4\pi$. The Kretschmann scalar diverges at $r\to 0$ indicating a typical curvature singularity (Panah et al., 27 Dec 2025, Panah et al., 2024).

2. Thermodynamics and Extended Phase Structure

The horizon radius $r_+$ is the largest real root of $f(r_+)=0$. The black hole thermodynamics is governed by:

• Mass: $$m = r_+ k - \frac{\Lambda}{3}r_+^3 + \eta q^2 e^{-\gamma}/r_+$$
• Bekenstein–Hawking entropy: $S = \frac{r_+^2}{4}$
• Hawking temperature:

$$T = \frac{1}{4\pi} \left( \frac{k}{r_+} - \Lambda r_+ - \eta \frac{q^2 e^{-\gamma}}{r_+^3}\right)$$

• Electric potential: $\Phi = q e^{-\gamma}/r_+$
• First law: $dM = T dS + \eta \Phi dQ$

In extended ("black hole chemistry") thermodynamics, $\Lambda$ is promoted to a pressure $P=-\Lambda/(8\pi)$, and the thermodynamic volume is $V = r_+^3/3$. The equation of state becomes

$$P = \frac{T}{2r_+} - \frac{k}{8\pi r_+^2} + \eta \frac{q^2 e^{-\gamma}}{8\pi r_+^4},$$

displaying van der Waals–like oscillations for suitable parameter regions and revealing critical points, multiple branches, and swallowtail structures in the free energy (Panah et al., 27 Dec 2025, Panah et al., 2024).

3. Topological Classification and Universal Winding Number

The phase diagram and stability structure are encoded by a topological invariant—the global winding number $W$—computed from the zeroes of a two-component vector field $(\partial_{r_+}\mathcal{F}, -\cot\Theta\csc\Theta)$ constructed from an off-shell free energy functional $\mathcal{F}(r_+, \tau)$ (with fictitious inverse temperature $\tau$). Each zero (defect) in the $(r_+, \Theta)$ plane carries a local winding $w_i\in\{-1, 0, +1\}$ (with sign corresponding to thermodynamic stability/instability): $W = \sum_i w_i.$

Typical topological classes for these black holes (canonical ensemble, fixed $Q$) (Panah et al., 27 Dec 2025, Panah et al., 2024):

• $W=+1$: three branches ($+1$, $-1$, $+1$), corresponding to van der Waals–type first-order transitions,
• $W=0$: two branches ($-1$, $+1$), Hawking–Page–type transitions (thermal AdS $\leftrightarrow$ large black hole),
• $W=-1$: all unstable/no equilibrium (generally not realized for physical ModMax/ModAMax AdS solutions).

This invariant is robust under continuous deformations of parameters; it shifts only at critical points where defects coalesce or annihilate.

4. Influence of Nonlinear Electrodynamics and Topology

The ModMax parameter $\gamma$ exponentially suppresses the $q^2$ term, tuning the effective electric hair and shifting phase boundaries. For large $\gamma$, the charged solution approaches the neutral Schwarzschild–(A)dS limit. Topology index $k$ alters not only the presence and nature of horizons but also the critical phenomena:

• $k=+1$: supports van der Waals transitions, multicritical structure for suitable $\gamma$,
• $k=0,-1$: typically single branch, globally locally stable (no Hawking–Page transition for $k=0$).

The phantom sector (ModAMax, $\eta = -1$) can enhance thermal stability and efficiency in certain entropy regimes, and may eliminate some critical points present in the standard case.

5. Thermal Stability, Joule–Thomson Expansion, and Heat Engine Efficiency

Stability is analyzed via the heat capacity at constant charge,

$$C_Q = T \left(\frac{\partial S}{\partial T}\right)_Q = \frac{2S\left(4\Lambda S^2 - kS + 4\eta \pi^2 Q^2 e^{-\gamma}\right)}{kS + 4\Lambda S^2 - 12\eta \pi^2 Q^2 e^{-\gamma}}$$

with zeros marking physical boundaries and divergences indicating second-order phase transitions. For $k=+1$ and $\gamma < \gamma_c$, two divergences indicate three-phase structure (small/unstable, intermediate/stable, large/stable); $k=0,-1$ present only one divergence and one stable phase.

The Joule–Thomson expansion exhibits topology- and $\gamma$-dependent inversion curves, separating heating and cooling regions. The efficiency of black hole heat engines in rectangular $P$–$V$ cycles exhibits a nontrivial dependence on $\gamma$ and $\eta$: for ModMax ($\eta=+1$), efficiency generically decreases with increasing $\gamma$, while for ModAMax ($\eta=-1$), efficiency can be enhanced for certain cycles and topology (Panah et al., 27 Dec 2025).

6. Comparison to Related Topological and Thermodynamic Frameworks

The topological winding classification aligns with and extends the universal schemes found for Einstein–Maxwell, Gauss–Bonnet, and Lovelock black holes, where the global number $W$ organizes phase structure irrespective of coordinates or thermodynamic potential (Panah et al., 2024, Panah et al., 2024, Jeon et al., 2024, Wang et al., 2024). ModMax/ModAMax AdS black holes provide a tunable testbed bridging linear and nonlinear electrodynamics within this framework.

The introduction of nonlinearities (via $\gamma$) and nontrivial horizon topology $k$ creates a more flexible thermodynamic landscape, leading to the possibility of switching between universal classes by continuous deformations of couplings. Phantom sectors further expand the phenomenology by stabilizing phases otherwise unstable in standard Maxwell–AdS theory (Panah et al., 27 Dec 2025, Rahmani et al., 23 Dec 2025).

7. Broader Implications and Universality

Topological invariants such as $W$ encode fundamental aspects of black hole microphysics, universality in phase transitions, and universality classes. The non-minimal and nonlinear couplings in ModMax/ModAMax–AdS black holes directly manifest as shifts in $W$, providing a model-independent method to distinguish between different macroscopic phase structures arising from distinct microscopic couplings. The robustness of this classification under generalizations such as gravity’s rainbow and higher curvature corrections suggests that topological universality is a foundational organizing principle in black hole thermodynamics across broad classes of gravitational theories (Panah et al., 27 Dec 2025, Panah et al., 2024, Rahmani et al., 23 Dec 2025).