Revisiting thermodynamic topology of Hawking-Page and Davies type phase transitions

In this work, we propose a common vector field to study the thermodynamic topology of the Davies type and Hawking-Page phase transitions. Existing literature has shown that studying these two types of phase transitions typically requires defining two separate vector fields. In our approach, we adopt Duan's ϕ-mapping topological current theory to define a novel vector field, denoted as ϕ, whose critical points exactly correspond to the Davies point and the Hawking-Page phase transition point. More importantly, we can differentiate between these two points by their topological charge. While, the topological charge for the critical point corresponding to the Davies-type phase transition is found to be −1, the same for the Hawking-Page phase transition point, it is +1. Although our analysis is applicable to all black hole systems where both types of phase transitions are found, we illustrate it using three simple systems as examples: the Schwarzschild AdS black hole, the Reissner-Nordström AdS black hole in the grand canonical ensemble, and finally the Kerr AdS black holes in the grand canonical ensemble. It is well-known that these black holes exhibit both Davies and Hawking-Page phase transitions. With our proposed vector ϕ, the critical points obtained for these three systems exactly match the Davies-type and Hawking-Page phase transition points, and the associated topological charges are found to be −1 for the Davies point and +1 for the Hawking-Page phase transition point.