Topological Signatures and Geometrothermodynamics of Critical Phenomena in Regularized Maxwell Black Holes

TL;DR

The paper studies charged AdS black holes in Regularized Maxwell (RegMax) gravity, using Duan's φ-mapping thermodynamic topology and Ruppeiner geometry to connect phase structure with microscopic interactions. A central finding is that the RegMax coupling $\alpha$ controls a sharp transition in topology and can induce second-order critical behavior via an intermediate black hole branch, while also driving violations of classical energy conditions at macroscopic scales. The work links topological defect dynamics (via $\alpha^{2}|Q|$) to the sign and magnitude of the Ruppeiner curvature, revealing repulsive microinteractions for large, thermally stable configurations and strong, possibly attractive correlations in small black holes, with EC violations correlating with richer thermodynamic topology. Collectively, this provides a framework to classify black hole thermodynamics in RegMax gravity and suggests observational or analogue avenues to probe the underlying microstructure and modified electromagnetism.

Abstract

We study the thermodynamic topology and microscopic interaction properties of charged black holes in RegMax gravity, focusing on the role of the coupling parameter $α$. Using the Duan topological current method together with Ruppeiner geometry, we show that $α$ controls a sharp change in phase structure. Above a certain critical threshold, we find that the Duan defect curve develops an intermediate branch and vertical tangency points, producing continuous (second-order) critical behaviour. Furthermore, the Ruppeiner curvature becomes negative at very small horizon radii before turning positive and progressively vanishing at larger radii. By contrast, below the critical value of the coupling, the intermediate black hole phase disappears, and the system shows a simpler small/large first-order/coexistence behaviour driven by free-energy competition. In this regime, the Ruppeiner curvature remains predominantly positive. Overall, increasing $α$ enriches the thermodynamic topology (allowing for second-order criticality) while simultaneously reducing the domain in which classical energy conditions (ECs) are satisfied, thus linking exotic thermodynamic behaviour to more severe violations of standard energy conditions.

Figures (11)

• Figure 1: Embedding-style surfaces $z=f(r)$ for $Q=0.1$, $M=0.2$, $\Lambda=-0.001$. Each panel corresponds to one value of $\alpha\in\{0.2,0.6,1.0,1.2,1.6,2.0\}$ (panels arranged left-to-right, top-to-bottom).
• Figure 2: Plot of WEC using the fixed benchmark $\alpha=1$ and $Q=1$.
• Figure 3: Plot of NEC using the fixed benchmark $\alpha=1$ and $Q=1$.
• Figure 4: Plot of SEC using the fixed benchmark $\alpha=1$ and $Q=1$.
• Figure 5: Defect curve $\tau$ versus $r_h$ for various values of the pressure $P$, the electric charge $Q$ and the coupling parameter $\alpha$.