Topological changes and response singularities in black hole thermodynamic branch structure

Abstract

In this paper, we investigate how different diagnostics characterize the thermodynamic branch structure of black holes. In three charged AdS black hole examples, the thermodynamic topological transition point $\tau_a$, where the total topological number changes, is separated from the point $\tau_b$ where the branch response considered here, evaluated along a path with fixed charge parameters, becomes singular. Using the zero-point curves $\tau(r_h)$, we identify the local mechanisms behind this separation. At $\tau_a$, a zero point with nonzero winding enters or leaves the positive-radius physical branch set through the boundary of the counted domain, changing $W$ without making the corresponding positive-radius branch singular. At $\tau_b$, the zero-point curve develops a finite-radius turning point, where two branches with opposite winding numbers annihilate and the branch response diverges. Thus, in the analyzed examples, a change in the topological number records a change in branch membership, whereas a response singularity records a loss of local invertibility along an admitted branch. The two diagnostics therefore probe different local events of the same zero-point branch geometry.