Topological interpretation for phase transitions of black holes

In this work, we develop a universal picture from topology in the thermodynamic parameters space to describe first order phase transitions of black holes. By employing an off-shell internal energy, we find two types of topological defects. The first is normal, describing black holes which move with a nonzero velocity and acceleration, in a central force field. The second type of defects are exotic: they are static in the space, not describing black holes but encoding information about first order transitions. For each defect, we assign a winding number and an inertial mass. By studying neutral and charged black holes in asymptotically anti--de Sitter space, we show that first order transitions can be viewed as a once or twice interchange of winding numbers between black holes and the exotic defects, through wired action at a distance. This corresponds to the usual notion: a smaller black hole grows into a larger black hole or vice versa. However, our topological analysis illustrates that the transition can also be locally interpreted as virtual collisions between black holes and the exotic defects. In this interpretation, a smaller black hole first grows into a new exotic defect whereas an original exotic defect grows into a larger black hole. All the defects simply change their positions and momentums, rather than interchanging the winding numbers. The critical point of the transition can be extracted when all the defects meet in the parameters space. Certain quantities, such as the Jacobians and the velocities of normal defects show universal behaviors near the critical point.