Thermodynamic topology of black holes from bulk-boundary, extended, and restricted phase space perspectives

TL;DR

This work develops and applies a unified topological framework for black-hole thermodynamics across bulk-boundary (BB), extended (EPS), and restricted (RPS) phase-space formalisms. Using the generalized off-shell Helmholtz free energy $\mathcal{F}=M-\frac{S}{\tau}$ and Duan's topological current approach, the authors compute zeros of the associated vector field and their winding numbers to assign topological charges to three AdS black holes: RN, EGB, and EPYM. They find that RN and EGB preserve a topological charge $W=+1$ in both BB and RPST (and are consistent with EPST), while EPYM exhibits distinctive behavior with $W=0$ across BB and RPST and a special EPST characterization, likely due to nonlinear YM contributions. The results demonstrate the robustness of thermodynamic topology under Gauss–Bonnet corrections and dimensional extension, offer insight into holographic phase structures, and point to future connections with AdS/CFT and broader physical systems.

Abstract

In this article, we investigate the thermodynamic topology of some black holes, namely AdS Reissner Nordstrom (R-N), AdS Einstein-Gauss-Bonnet (EGB), and AdS Einstein-power-Yang-Mills (EPYM), from different frameworks: bulk-boundary (BB) and restricted phase space (RPS). Using the generalized off-shell Helmholtz free energy method, we calculate the thermodynamic topology of the selected black holes in each space separately and determine their topological classifications. We show that the addition of GB terms, dimensions, and other factors do not affect the topological classes of black holes in both spaces. The calculations and plots indicate that the AdS R-N and AdS EGB black holes show similar behavior and their topological numbers sets in both spaces, i.e., BB and RPS, are similar and equal to ($W=+1$). However, AdS EPYM black holes show an interesting behavior. In addition to BBT and RPS, we also consider the extended phase space thermodynamics (EPST) and evaluate the thermodynamic topology for AdS EPYM black hole. The changing ($r-τ$) in both spaces shows similar behavior. Also, the topological number and the total topological numbers for this black hole in the BB, RPS and EPS thermodynamics are completely same, i,e., $(ω_{BBT}=ω_{RPS}=ω_{EPST}=+1, -1)$ or $W_{BBT}=W_{RPS}=W_{EPST}=0$. An important point is that the Einstein-Yang-Mills black hole has thermodynamic topology equivalence in three spaces. The present result may be due to the non-linear YM charge parameter and the difference between the gauge and gravity corrections in the above black holes

Figures (9)

• Figure 1: The vector field $n$ on a part of the $(r-\Theta)$ plane for the AdS R-N black holes with $(q=1,G=0.04 < G_{critical}=0.0498)$ is shown by the blue arrows in Fig (1a). The ZP are at $(r,\theta)=(0.28,1.57), (0.58,1.57), (1.37,1.57)$. We choose three closed loops (purple loop) and (blue loop) that encircle the ZP. Fig (1b) shows the curve of equation (27).
• Figure 2: The vector field $n$ on a part of the $(r-\Theta)$ plane for the AdS EGB black holes with $(P=0.1, G=0.012 < G_{critical}=0.01406567655,q=10, \alpha=0.1)$ is shown by the blue arrows in Fig (2a). The ZPs are at $(r,\theta)=(1.745506274,1.57), (2.842950665,1.57), (9.002657827,1.57)$. We choose three closed loops (purple loop) and (blue loop) that encircle the ZPs. Fig (2b) shows the curve of equation (34).
• Figure 3: The vector field $n$ on a part of the $(r-\Theta)$ plane for the AdS EPYM black holes with $( q=0.1, G=0.3 < G_{critical}=0.3068,P=0.1, \gamma=3)$ is shown by the blue arrows in Fig (3a). The ZPs are at $(r,\Theta)=(0.88,1.57), (1.49,1.57)$. We choose two closed loops (blue loop) that encircle the ZPs. Fig (3b) shows the curve of equation (41).
• Figure 4: The vector field $n$ on a part of the $(r-\Theta)$ plane for the AdS R-N black holes with $(\hat{q}=1, l=1,C=5 < C_{critical}=6)$ is shown by the blue arrows in Fig (4a). The ZP is at $(r,\Theta)=(8.37,1.57)$. We choose two closed loops $Z_1$ (purple loop) and $Z_2$ (blue loop), where $Z_2$ encircles the ZP but $Z_1$ does not. Fig (4b) shows the curve of equation (48).
• Figure 5: The vector field $n$ on a part of the $(r-\Theta)$ plane for the AdS EGB black holes with $(C=0.5 < C_{critical}=0.9447273930,\hat{q}=0.1, \alpha=0.9, l=10)$ is shown by the blue arrows in Fig (5a). The ZPs are at $(r,\Theta)=(419.6558504,1.57)$. We choose two closed loops $Z_1$ (purple loop) and $Z_2$ (blue loop), where $Z_2$ encircles the ZPs but $Z_1$ does not. Fig (5b) shows the curve of equation (55).