Eur. Phys. J. B (2016) 89: 195
https://doi.org/10.1140/epjb/e2016-70325-x

Regular Article

Topological invariants for phase transition points of one-dimensional Z₂ topological systems

¹ Beijing Computational Science Research Center, Beijing 100089, P.R. China
² CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
³ Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P.R. China
⁴ Collaborative Innovation Center of Quantum Matter, Beijing, P.R. China

^a e-mail: rubilacxelee@gmail.com

Received: 23 May 2016
Received in final form: 4 July 2016
Published online: 12 September 2016

Abstract

We study topological properties of phase transition points of two topologically non-trivial Z₂ 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 Z₂ system is generally characterized by a Z₂ 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.