Novel topological subclass in Hourava-Lifshitz black holes

TL;DR

The paper addresses whether existing thermodynamic topological classifications adequately describe charged black holes in $z=3$ Horava-Lifshitz gravity. It employs the Duan phi-mapping topological current framework, analyzing both canonical and grand canonical ensembles to map black-hole states onto a two-component vector field and extract winding numbers that classify stability. A central contribution is the identification of a new topological subclass $\ddot{W}^{1-}$, which exhibits distinct low- and high-temperature phase patterns (e.g., in hyperbolic-horizon cases) and can differ between ensembles, signaling ensemble-dependence in the topological classification. The work broadens the thermodynamic topology framework, revealing richer phase structures in HL gravity and suggesting the existence of additional, hitherto unidentified topological categories.

Abstract

This work explores the universal classification of thermodynamic topology for charged static black holes within the $z=3$ Hourava-Lifshitz gravity theory, considering both canonical and grand canonical ensembles. We introduce a new topological subclass, denoted as $\ddot{W}^{1-}$. This finding expands the existing topological classification, going beyond the five previously defined classes and their respective subclasses. The $\ddot{W}^{1-}$ subclass presents a distinct and previously unobserved stability profile: In the low-temperature regime, an unstable small black hole appears in the phase space, whereas, while in the high temperature regime, two unstable small black holes exist together with a stable large black hole. Our study underscores the dependence of charged black hole stability on the selection of the ensemble. These results contribute to refining and expanding the topological framework in black hole thermodynamics, providing key perspectives on the underlying nature of black holes and gravity.

Figures (7)

• Figure 1: In the $r_h-\beta$ plane, the zero points of the vector $\phi^{r_h}$ for the charged HL black hole, with the parameters $k / r_0 = 1$, $q / r_0 = 1$, and $P r_0^2 = 0.01$. The stable branch is represented by the red curve.
• Figure 2: In the $r_h-\beta$ plane, the zero points of the vector $\phi^{r_h}$ for the charged HL black hole, using parameters $k / r_0 = 0$, $q / r_0 = 1$, and $P r_0^2 = 0.01$. The stable branch is represented by the red curve.
• Figure 3: In the $r_h-\beta$ plane, the zero points of the vector $\phi^{r_h}$ for the charged HL black hole, with parameters $k / r_0 = -1$, $q / r_0 = 1$, and $P r_0^2 = 0.01$. The blue unstable branch ($w = -1$) and the red stable branch ($w = +1$) meet at the annihilation point (AP) marked by the pink dot. The count of stable and unstable states yields two stable and one unstable state, leading to a total topological number of $W = -1 + 1 + 1 = 1$.
• Figure 4: In the $r_h-\beta$ plane, the zero points of the vector $\phi^{r_h}$ for the charged HL black hole, with the parameters $k / r_0 = 1$, $q / r_0 = 1$, and $P r_0^2 = 0.01$. The stable branch is represented by the red curve.
• Figure 5: In the $r_h-\beta$ plane, the zero points of the vector $\phi^{r_h}$ for the charged HL black hole, with parameters $k / r_0=0$, $q / r_0=1$, and $P r_0^2=0.01$. At the intersection of the blue unstable ($w=-1$) and red stable ($w=+1$) branches, denoted by the pink annihilation point (AP).