Momentum space Z 2 number, quantized Berry phase and the quantum phase transitions in spin chain systems

We characterize the quantum phase transitions of gapped time-reversal invariant spin chain systems by nontrivial Z2 numbers derived from a quantized Berry phase in a (1 + 1)-dimensional Bloch momentum space. We study this approach analytically in a general two-band model and give a concrete example in a transverse field XY spin chain system. In our scheme, an extra local gauge transformation is performed to the spin system by a time-dependent twist operator, which endows the Hamiltonian of the system with the topology of a torus without changing its energy spectrum. The Z2 number is obtained by a loop integral of the Berry gauge potential along a quarter of the Brillouin zone. We show that the different phases of the XY spin chain system are distinguished by the Z2 topological order in momentum space.