Characterization of topological insulators based on the electronic polarization with spiral boundary conditions

TL;DR

This paper extends the concept of electronic polarization to two-dimensional topological insulators using spiral boundary conditions, providing a unified way to characterize topological phases across dimensions.

Contribution

It introduces a novel approach to characterize 2D topological insulators via electronic polarization with spiral boundary conditions, linking topological transitions to polarization sign changes.

Findings

Polarization changes sign at topological transition points.

Polarization acts as an order parameter for topological phases.

Method unifies the study of topological phases across dimensions.

Abstract

We introduce the electronic polarization originally defined in one-dimensional lattice systems to characterize two-dimensional topological insulators. The main idea is to use spiral boundary conditions which sweep all lattice sites in one-dimensional order. We find that the sign of the polarization changes at topological transition points of the two-dimensional Wilson-Dirac model (the lattice version of the Bernevig-Hughes-Zhang model) in the same way as in one-dimensional systems. Thus the polarization plays the role of "order parameter" to characterize the topological insulating state and enables us to study topological phases in different dimensions in a unified way.