# Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology

## TL;DR

This paper introduces a Wilson-loop method for accurately calculating topological invariants in classical wave systems, revealing fragile topology in photonic and sonic crystals, and discusses the bulk-edge correspondence with broken symmetries.

## Contribution

The paper presents a novel Wilson-loop approach for exact numerical topological invariant calculation in classical waves, uncovering fragile topology in photonic and sonic crystals.

## Key findings

- Identification of fragile topology in classical wave crystals
- Development of Wilson-loop method for topological invariant calculation
- Analysis of bulk-edge correspondence with symmetry breaking

## Abstract

The rapid development of topological photonics and acoustics calls for accurate understanding of band topology in classical waves, which is not yet achieved in many situations. Here, we present the Wilson-loop approach for exact numerical calculation of the topological invariants for several photonic/sonic crystals. We demonstrate that these topological photonic/sonic crystals are topological crystalline insulators with fragile topology, a feature which has been ignored in previous studies. We further discuss the bulk-edge correspondence in these systems with emphasis on symmetry broken on the edges.