  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".

Strongly Correlated Electrons