Topology of Holographic Thermodynamics within Non-extensive Entropy

TL;DR

This work investigates the topology of holographic thermodynamics for AdS Reissner-Nordström black holes using nonextensive entropies, specifically Rényi and Sharma-Mittal, within bulk-boundary and restricted phase space frameworks. By applying Duan's phi-mapping topological current approach to the generalized free energy $\mathcal{F}$, the authors compute topological defects via zeros of a two-component vector and extract winding numbers that classify local stability. In the bulk-boundary setting, Rényi statistics yield a single stable on-shell black hole with $W=+1$, while Sharma-Mittal statistics reveal that increasing $\alpha$ adds topological charges $(+1,-1,+1)$ and increasing $\beta$ reduces them, still with $W=+1$. In contrast, the restricted phase space shows a universal topological class with $\omega = +1$ and $W = +1$ for both Rényi and Sharma-Mittal, a feature that persists even when reducing to Bekenstein-Hawking entropy. Overall, the results demonstrate a robust, framework-spanning topological structure for holographic black hole thermodynamics under nonextensive entropy, with implications for stability and phase behavior in AdS/CFT contexts.

Abstract

In this paper, we delve into the thermodynamic topology of AdS Reissner-Nordstr$\ddot{o}$m (R-N) black holes by employing nonextensive entropy frameworks, specifically R$\acute{e}$nyi (with nonextensive parameter $λ$) and Sharma-Mittal entropy (with nonextensive parameter $α, β$). Our investigation spans two frameworks: bulk boundary and restricted phase space (RPS) thermodynamics. In the bulk boundary framework, we face singular zero points revealing topological charges influenced by the free parameter $(λ)$ with a positive topological charge $(ω= +1)$ and the total topological charge $(W = +1)$, indicating the presence of a single stable on-shell black hole. Further analysis shows that when $(λ)$ is set to zero, the equations align with the Bekenstein-Hawking entropy structure, demonstrating different behaviors with multiple topological charges $(ω= +1, -1, +1)$. Notably, increasing the parameter $α$ in Sharma-Mittal entropy results in multiple topological charges $(ω= +1, -1, +1)$ with the total topological charge $(W = +1)$. Conversely, increasing $(β)$ reduces the number of topological charges, maintaining the total topological charge $(W = +1)$. Extending our study to the restricted phase space, we observe consistent topological charges $(ω= +1)$ across all conditions and parameters. This consistency persists even when reducing to Bekenstein-Hawking entropy, suggesting similar behaviors in both non-extended and Hawking entropy states within RPS.

Figures (4)

• Figure 1: The curve described by Eq. (\ref{['BBR3']}) is illustrated in Figs. (\ref{['1a']}), (\ref{['1c']}), and (\ref{['1e']}). In Figs. (\ref{['1b']}), (\ref{['1d']}), and (\ref{['1f']}), the zero points (ZPs) are located at coordinates $(r, \theta)$ on the circular loops, corresponding to the free parameter ($\lambda, q, G$), $P=0.9$ and $\tau=6, 1, 17$
• Figure 2: The curve described by Eq. (\ref{['BBSM3']}) is illustrated in Figs. (\ref{['2a']}), (\ref{['2c']}), and (\ref{['2e']}). In Figs. (\ref{['2b']}), (\ref{['2d']}), and (\ref{['2f']}), the zero points (ZPs) are located at coordinates $(r, \theta)$ on the circular loops, corresponding to the free parameter ($\alpha$, $\beta$$G$), $P=0.9$ and $\tau=3, 13.5, 7$
• Figure 3: The curve described by Eq. (\ref{['RPSR3']}) is illustrated in Figs. (\ref{['3a']}), (\ref{['3c']}), and (\ref{['3e']}). In Figs. (\ref{['3b']}), (\ref{['3d']}), and (\ref{['3f']}), the zero points (ZPs) are located at coordinates $(r, \theta)$ on the circular loops, corresponding to the free parameter ($\lambda, C$), $q=1$, $l=1$ and $\tau=3, 0.8, 5$
• Figure 4: The curve described by Eq. (\ref{['RPSSM3']}) is illustrated in Figs. (\ref{['4a']}), (\ref{['4c']}), and (\ref{['4e']}). In Figs. (\ref{['4b']}), (\ref{['4d']}), and (\ref{['4f']}), the zero points (ZPs) are located at coordinates $(r, \theta)$ on the circular loops, corresponding to the free parameter ($\alpha$, $\beta$, $C$), $q=1$, $l=1$ and $\tau=3, 6, 5$