$Z_N$ Berry Phases in Symmetry Protected Topological Phases

We show that the $Z_N$ Berry phase (Berry phase quantized into $2\pi/N$) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size. The $Z_N$ Berry phase is defined in a $N-1$ dimensional parameter space of local gauge twists, which we call "synthetic Brillouin zone", and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase. We demonstrate the usefulness of the $Z_N$ Berry phase by studying two 1D models of bosons, SU(3) and SU(4) AKLT models, where topological phase transitions are captured by $Z_3$ and $Z_4$ Berry phases, respectively. we find that the exact quantization of the $Z_N$ Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone.