Topological Quantum Phase Transition between Quantum Spin Hall State and Quantum Anomalous Hall State

In this paper, starting from a lattice model of topological insulators, we study the quantum phase transitions among different quantum states, including quantum spin Hall state, quantum anomalous Hall state and normal band insulator state by calculating their topological properties (edge states, quantized spin Hall conductivities and the number of zero mode on a Pi-flux). We find that there exist universal features for the topological quantum phase transitions (TQPTs) in different cases : the emergence of nodal fermions at high symmetry points, the non-analytic third derivative of ground state energy and the jumps of the topological order parameters. In particular, the relation between TQPTs and symmetries of the systems are explored : different TQPTs are protected by different (global) symmetries and then described by different topological order parameters.