Chern character and Fermi point

TL;DR

This paper links the Chern character in topological K-theory to Fermi points, providing new insights into spectral flow and proving key properties of topological insulators with time-reversal symmetry.

Contribution

It introduces a novel formulation of the Chern character using Fermi points and offers elementary proofs of fundamental topological properties in four-dimensional insulators.

Findings

Odd Chern character as a generalization of spectral flow

Elementary proof of the evenness of the edge index

Bulk-edge correspondence for 4D topological insulators

Abstract

This paper expresses the Chern character for topological K-theory based on the formulation of the family of Fredholm operators, by using the points at which the Fredholm operator becomes singular (Fermi points). In particular, we explain that the odd Chern character can be thought of as a generalization of the spectral flow. As applications, we give elementary proofs of the evenness of the edge index and the bulk-edge correspondence for four-dimensional topological insulators with time-reversal symmetry of class AI.