Counting topological interface modes using simplicial characteristic classes

TL;DR

This paper introduces a robust, noise-resistant computational method using simplicial characteristic classes to accurately predict topological interface modes in hermitian systems, with potential applications in experimental wave analysis.

Contribution

It presents a novel, gauge-invariant algorithm that computes Chern numbers via simplicial complexes, enabling precise prediction of TIMs in complex systems.

Findings

Successfully predicts TIMs in fluid wave models

Reproduces expected TIM counts in topological Langmuir waves

Demonstrates robustness to noise and applicability to experimental data

Abstract

A computational approach for predicting the number of topological interface modes (TIMs) in hermitian systems using the spectral flow - monopole (SFM) correspondence is presented. The number of TIMs is determined by calculating the Chern number of a complex line bundle of local polarisation vectors over a phase space sphere surrounding a Weyl point. The Chern number is computed by constructing the simplicial first Chern class of a discrete vector bundle on a simplicial mesh. This approach is gauge invariant, derivative free, structure preserving, and robust to noise. The algorithm is shown to reproduce the expected number of TIMs for the case of equatorial fluid waves and the topological Langmuir cyclotron wave. The possibility of using this algorithm to analyse experimental measurements of bulk wave polarisations and predict the associated number of TIMs is explored in a synthetic example.