Thermodynamic Geometry, Heat Engines, and Topology of Sharma--Mittal ModMax-dRGT Black Holes

Technical Summary: Thermodynamic Geometry, Heat Engines, and Topology of Sharma–Mittal ModMax-dRGT Black Holes

Problem Statement
This study investigates the thermodynamic structure of charged Anti-de Sitter (AdS) black holes within a theoretical framework that unifies three distinct modifications to general relativity: ModMax nonlinear electrodynamics, dRGT-like massive gravity, and Sharma–Mittal generalized entropy corrections. The authors aim to characterize the microscopic interactions, macroscopic energy conversion capabilities, and global phase stability of these black holes. Specifically, the work addresses how the interplay between nonlinear electromagnetic fields, massive graviton couplings, and nonextensive entropy deformations influences thermodynamic curvature, heat engine efficiency, and the topological classification of critical points.

Methodology
The research employs a multi-faceted analytical approach within the extended phase-space formalism, where the cosmological constant is treated as thermodynamic pressure ($P$) and black hole mass as enthalpy ($H$).

1. Thermodynamic Setup: The authors utilize a static, spherically symmetric metric incorporating the ModMax parameter ($\gamma$), graviton mass ($m_g$), and massive gravity coupling constants ($c_1, c_2$). They derive the modified Hawking temperature ($T_{SM}$) by applying the Sharma–Mittal entropy formalism ($S_{SM}$) to the standard Bekenstein-Hawking area law.
2. Thermodynamic Geometry: Using the Weinhold metric construction ($g_{ij} = T^{-1} \partial^2 M / \partial X_i \partial X_j$) in the parameter space of horizon radius ($r_+$) and electric charge ($q$), the authors calculate the thermodynamic Ricci scalar ($R_{th}$). This scalar is used to probe the nature of effective microscopic correlations (attractive vs. repulsive).
3. Heat Engine Analysis: A rectangular thermodynamic cycle in the $P-V$ plane is constructed to evaluate the efficiency ($\eta$) of converting absorbed heat into mechanical work. The study derives exact expressions for work output and heat input, comparing the results against the Maxwell limit and the Sharma–Mittal corrected Carnot bound.
4. Topological Analysis: Following the $\phi$-mapping topological current theory, the authors treat critical points as topological defects in vector fields derived from the isobaric corrected temperature and the off-shell generalized free energy. They calculate topological charges by analyzing the winding numbers of these vector fields around zero points.

Key Contributions and Results

• Thermodynamic Geometry and Microscopic Interactions:
  The study establishes that the thermodynamic Ricci scalar ($R_{th}$) diverges at extremal boundaries and metric degeneracies. The sign of $R_{th}$ reveals a transition in microscopic interaction regimes:

  • At small horizon radii, the charge sector dominates, yielding $R_{th} < 0$, which indicates predominantly attractive statistical correlations.
  • At larger radii, the massive-gravity sector becomes dominant. Positive values of the coupling $c_1$ drive $R_{th} > 0$, signaling repulsive correlations.
  • The ModMax parameter $\gamma$ suppresses the effective charge ($q_{eff}^2 = q^2 e^{-\gamma}$), thereby weakening the attractive regime and shifting the system toward repulsive correlations. For $\gamma = 0.8$, the effective charge contribution is reduced by approximately 55.1% compared to the Maxwell case.

• Heat Engine Efficiency:
  The authors derive an exact expression for the efficiency of a rectangular heat engine. Key findings include:

  • The ModMax parameter $\gamma$ reduces the engine efficiency by suppressing the electromagnetic contribution to the enthalpy difference. For a representative cycle, efficiency drops from $\approx 0.5465$ (Maxwell limit) to $\approx 0.5316$ when $\gamma = 0.8$.
  • The massive gravity couplings ($c_1, c_2$) directly modify the enthalpy and heat input, altering the efficiency.
  • The Sharma–Mittal parameters ($R, \delta$) do not affect the mechanical efficiency of a fixed $P-V$ cycle (as work and heat are enthalpy/volume dependent) but significantly modify the corrected temperature ($T_{SM}$), thereby altering the Carnot bound ($\eta_C$).

• Topological Classification:
  The topological analysis yields two distinct results based on the potential used:

  • Corrected Temperature: Analysis of the isobaric $T_{SM}$ reveals two critical points with topological charges of $-1$ (conventional) and $+1$ (novel), resulting in a conserved total charge of zero.
  • Generalized Free Energy: Analysis of the off-shell free energy $F$ at specific inverse temperatures reveals three branches (small, intermediate, large black holes). The topological charges are $+1$, $-1$, and $+1$, respectively. This sums to a total topological charge of $+1$, classifying the black hole within the universal thermodynamic topological class $W_{1+}$.

Significance and Claims
The paper claims that the combined application of thermodynamic geometry, heat engine analysis, and topological methods provides a coherent and quantitative framework for distinguishing the specific thermodynamic roles of different theoretical sectors. The authors assert that:

1. Thermodynamic geometry effectively probes the effective microscopic interaction structure of the horizon, distinguishing between charge-dominated attractive regimes and massive-gravity-dominated repulsive regimes.
2. The heat engine framework serves as a diagnostic tool to quantify how nonlinear electrodynamics and massive gravity influence energy conversion, while generalized entropy corrections specifically reshape the thermal limits (Carnot bound) without altering the mechanical work output of fixed cycles.
3. The topological analysis confirms the stability and phase structure of the system, identifying the black hole as belonging to a specific universal topological class ($W_{1+}$) despite the complex interplay of nonlinear and massive gravity corrections.

The work does not propose experimental verification or future applications beyond the theoretical characterization of these specific black hole solutions, maintaining a focus on the internal consistency and thermodynamic properties of the Sharma–Mittal ModMax-dRGT model.