Topological invariants for phase transition points of one-dimensional $\mathbb{Z}_2$ topological systems

TL;DR

This paper introduces a Berry phase-based method to characterize topological phase transition points in one-dimensional $ ext{Z}_2$ topological systems, providing a new way to understand phase changes in classes D and DIII.

Contribution

It defines a Berry phase at phase transition points that captures topological changes, linking it to particle-hole symmetry and extending topological characterization methods.

Findings

Berry phase at transition points reflects topological phase changes

The scheme successfully characterizes phase transitions in models of classes D and DIII

Topological invariants are linked to Berry phases protected by symmetries

Abstract

We study topological properties of phase transition points of two topologically non-trivial $\mathbb{Z}_2$ classes (D and DIII) in one dimension by assigning a Berry phase defined on closed circles around the gap closing points in the parameter space of momentum and a transition driving parameter. While the topological property of the $\mathbb{Z}_2$ system is generally characterized by a $\mathbb{Z}_2$ topological invariant, we identify that it has a correspondence to the quantized Berry phase protected by the particle-hole symmetry, and then give a proper definition of Berry phase to the phase transition point. By applying our scheme to some specific models of class D and DIII, we demonstrate that the topological phase transition can be well characterized by the Berry phase of the transition point, which reflects the change of Berry phases of topologically different phases across the phase transition point.