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

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.