$Z_N$ Berry Phases in Symmetry Protected Topological Phases

TL;DR

This paper introduces the $Z_N$ Berry phase as a practical numerical tool for identifying symmetry protected topological phases in 1D models, demonstrating its quantization and relation to gapless band structures.

Contribution

It defines the $Z_N$ Berry phase in a synthetic Brillouin zone and shows its exact quantization at topological transitions in bosonic SU(3) and SU(4) models.

Findings

$Z_N$ Berry phase is quantized and useful for characterizing topological phases.

Topological transitions correspond to gapless band structures in the synthetic zone.

The method applies to small systems and captures phase transitions effectively.

Abstract

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.