Lorentz-violating ModMax black holes in phantom-enhanced Kalb–Ramond gravity: Thermodynamics and topological charges

We derive exact static, spherically symmetric black hole (BH) solutions within the framework of four-dimensional Einstein gravity, coupled to ModMax nonlinear electrodynamics and a Kalb–Ramond two-form. This coupling is governed by a discrete sign parameter, denoted as $\zeta =\pm 1$. In the ”ordinary” branch ($\zeta =+1$), the metric features both Cauchy and event horizons, while the electromagnetic stress–energy adheres to all classical energy conditions. In contrast, the ”phantom” branch ($\zeta =-1$) accommodates only a single horizon and simultaneously violates the weak, null, and strong conditions, which reflects genuine ghost-matter behaviour. The thermodynamic behaviour for $\zeta =+1$ results in multiple divergences in heat capacity and presents a distinctive swallowtail pattern in the AdS Gibbs free energy, indicative of a first-order phase transition that encompasses small, intermediate, and large phases. In contrast, for $\zeta =-1$, there is only a single divergence observed, accompanied by a smooth Gibbs curve with no van der Waals-type oscillatory behaviour, which precludes the possibility of true coexistence. Topological analysis using Duan’s $\phi$-mapping in AdS further differentiates the branches by examining their winding numbers. For $\zeta =+1$, the winding numbers are $(+1,-1,+1)$, resulting in a total topological charge (net winding number) of $W=+1$. In contrast, for $\zeta =-1$, the winding numbers are $(-1,+1)$, leading to a total topological charge of $W=0$. This distinction assigns a clear topological charge, which differentiates between the first-order transitions associated with $\zeta =+1$ and the smooth, latent-heat-free phase changes linked to $\zeta =-1$. Overall, $\zeta$ serves as the primary determinant influencing causal structure, energy condition profiles, critical phenomena, and topological charge in ModMax-Kalb-Ramond BH.