Edge spectrum for truncated $\mathbb{Z}_2$-insulators

TL;DR

This paper proves that in two-dimensional fermionic topological insulators with different $$-indices, the combined system's spectrum fills the bulk gap, linking topological invariants to spectral properties through a local trace formula.

Contribution

It introduces a local trace formula for the $$-index and demonstrates that different topological phases lead to spectral filling in combined systems.

Findings

Spectrum fills the bulk gap when combining insulators with different indices.

The $$-index can be determined from bulk information in large enough regions.

A new local trace formula for the $$-index is established.

Abstract

Fermionic time-reversal-invariant insulators in two dimensions -- class AII in the Kitaev table -- come in two different topological phases. These are characterized by a $\mathbb{Z}_2$-index: the Fu-Kane-Mele index. We prove that if two such insulators with different indices occupy regions containing arbitrarily large balls, then the spectrum of the resulting operator fills the bulk spectral gap. Our argument follows a proof by contradiction developed in an earlier work by two of the authors for quantum Hall systems. It boils down to showing that the $\mathbb{Z}_2$-index can be computed only from bulk information in sufficiently large balls. This is achieved via a result of independent interest: a local trace formula for the $\mathbb{Z}_2$-index.