Topological quantum phase,transitions in superconductivity on lattices

Topological quantum phase transitions in superconductivity are discussed on two-dimensional lattices. The main focus is on the Chem number for superconducting states. Each superconductivity is characterized by the Chem number, and the quantum phase transition is associated with topological changes of the quasiparticle Bloch function in the Brillouin zone. For the superconducting case, the Chem number has several equivalent but different topological expressions given by vortices, the Dirac monopole, and strings. We demonstrate quantum phase transitions by these topological quantities both for singlet and triplet cases.