Quantized Berry phase in twisted Bloch momentum space as a topological order parameter for spin chains

We show that a quantized Berry phase in Bloch momentum space can serve as a topological order parameter to the quantum phases of a gapped spin chain system with time-reversal invariance. Specifically, we study this approach analytically in a class of XY spin-1/2 chain with multiple sites interactions in a transverse field. In order to derive a proper definition of the Berry curvature in a two-dimensional parameter space, we performed a local gauge transformation to the spin chain system by a twist operator, which endows the Hamiltonian of the system with the topology of a torus T2 without changing its energy spectrum. We show that a topological Z2 order parameter can be obtained as a quantized Berry phase by a loop integral of the Berry gauge potential along quarter of the Brillouin zone, which determines the zero-temperature phase diagram of the system. •We show a topological Z2 order parameter as a quantized Berry phase in Bloch momentum space.•Phase diagram of transverse field XY spin chain with multiple sites interactions can be determined by Z2 numbers.•A general approach to 1D gapped spin (fermionic) system with time-reversal invariance.