Zeros of Green Functions in Topological Insulators

TL;DR

This paper reveals that the zeros of Green functions are universal indicators of topological phases in insulators, providing a new method to detect and analyze various topological materials.

Contribution

It introduces the concept that Green function zeros traverse the band gap in topological phases and proves their relation to band inversions using eigenvector-eigenvalue identity.

Findings

Zeros traverse the band gap in all six classes of topological insulators.

Zeros' surfaces in the band gap are guaranteed by band inversions.

Zeros can detect higher-order topological insulators and relate to edge states.

Abstract

This study demonstrates that the zeros of the diagonal components of Green functions are key quantities that can detect non-interacting topological insulators. We show that zeros of the Green functions traverse the band gap in the topological phases. The traverses induce the crosses of zeros, and the zeros' surface in the band gap, analogous to the Fermi surface of metals. By calculating the zeros of the microscopic models, we show the traverses of the zeros universally appear in all six classes of conventional non-interacting topological insulators. By utilizing the eigenvector-eigenvalue identity, which is a recently rediscovered relation in linear algebra, we prove that the traverses of the zeros in the bulk Green functions are guaranteed by the band inversions, which occur in the topological phases. The relevance of the zeros to detecting the exotic topological insulators such as the higher-order topological insulators is also discussed. For the Hamiltonians with the nearest-neighbor hoppings, we also show that the gapless edge state guarantees the zeros' surfaces in the band gap. The analysis demonstrates that the zeros can be used to detect a wide range of topological insulators and thus useful for searching new topological materials.