Topological Theory of Classical and Quantum Phase Transition
  We presented the topological current of Ehrenfest definition of phase transition. It is shown that different topology of the configuration space corresponds to different phase transition, it is marked by the Euler number of the interaction potential. The two phases separated by the coexistence curve is assigned with different winding numbers of opposite sign. We also found an universal equation of coexistence curve, from which one can arrive the phase diagram of any order classical and quantum phase transition. The topological quantum phase transition theory is established, and is applied to the Bose-Hubbard model, the phase diagram of the first order quantum PT is in agreement with recent progress. 
Tieyan Si