Complexification of the Kerr-AdS black hole thermodynamic phase transition

In this paper, we establish a systematic complex-analytical framework for investigating the thermodynamics of high-dimensional singly spinning Kerr-AdS black holes. By performing an analytical continuation of the black hole horizon radius into the complex plane, we map the thermodynamic behavior for the black hole to the zeros distribution of the corresponding analytical function. In the region below the critical point, this approach provides the criterion for identifying phase transitions and reveals that the zeroth-order phase transition does not alter the underlying thermodynamic topological structure of the black hole. In the supercritical region, based on the Lee-Yang theory of phase transition, we construct the Widom line to characterize the supercritical crossover behavior for the Kerr-AdS black holes. This study provides a new perspective on the thermodynamic analysis of black holes.