Thermodynamic phase transition of the AdS black holes from the perspective of the complex analysis

Black hole thermodynamics is usually carried out over a span of the real variables. In this study, we extend this process in the complex domain. Often resorting to the complex plane, we find that the Hawking-Page phase transition is the switch of the combination of two vortices of the heat flow, while the van der Waals-type phase transition is predicted by the local maximum winding number without lengthy conventional thermodynamic calculations. As a by-product, we also give the Riemann surface description of the black hole thermodynamics phase transition in the complex domain, so as to make the thermodynamics phase transition intuitive. This method is expected to be extended to various black hole thermodynamics systems and even to general thermodynamic statistical physical system.