Multi-gap topology of the Wilson loop operator in mirror symmetric insulators

TL;DR

This paper introduces new topological invariants for Wilson loop operators in mirror symmetric insulators, revealing complex multi-gap topological properties and their relation to higher-order boundary phenomena.

Contribution

It develops two novel topological invariants for Wilson loop spectra, expanding understanding of multi-gap topology and boundary correspondence in mirror symmetric topological insulators.

Findings

New invariants characterize multi-gap topology in Wilson loops.

These invariants explain higher-order boundary states.

Application to anomalous cases beyond nested Wilson loops.

Abstract

We study the multi-gap topology of the periodic spectra of Wilson loop operators (WLOs) in mirror symmetric insulators. We develop two topological invariants each associated with a mirror-invariant gap in the Wilson loop spectrum. We propose that both topological invariants in combination determine the general higher-order bulk-boundary correspondence in 2D mirror symmetric, boundary-obstructed topological insulators. Finally, we demonstrate that these new multi-gap topological invariants apply to anomalous cases beyond those captured by the nested Wilson loop, and we subsequently develop an understanding of the correlation between WLOs along two orthogonal directions.