A Scheme to Classify Topological Property of Band Insulator Based On One-band U(1) Chern Number

TL;DR

This paper proves that the one-band U(1) Chern number (OBChN) determines the Z2 topological invariant in time-reversal invariant band insulators, providing a new criterion for classifying topological properties and linking 3D TIs to magnetic monopoles.

Contribution

It establishes the equivalence of OBChN and the Z2 invariant, offering a natural classification criterion for topological insulators and connecting 3D TIs with magnetic monopoles.

Findings

OBChN determines the Z2 invariant in TI.

OBChN is a useful tool for classifying topological phases.

A field in 3D TI can be identified with magnetic monopole.

Abstract

Topological insulator(TI) is a phase of matter discovered recently. Kane and Mele proposed this phase is distinguished from the ordinary band insulator by a Z2 topological invariant.2 Several authors have try to related this Z2 invariant to Chern numbers. Roy find a way to calculate Z2 by Chern Number of one of the two degenerate Bands or one-band Chern number(OBChN). However, he give no concrete concrete proof of the equivalence of his Z2 and the Z2 in ref[2] beside \from the topological considerations of K theory". So the importance of OBChN hasn't been recognized by the community. In this letter we prove OBChN determines the Z2 in ref[2]. Then we illustrate OBChN is not only an useful tool to identify TI but also a natural criterion to classify topological property of all time-reversal invariant band insulators. More importantly we find a field in three dimensional TI can be identified with magnetic field with magnetic monopole.