Topologically protected interface modes in multi-band damped lattice models

TL;DR

This paper explores topologically protected interface modes in one-dimensional damped lattice models using tridiagonal $k$-Toeplitz operators, revealing their connection to eigenvalues and symmetry properties.

Contribution

It establishes a link between edge modes and eigenvalues of principal submatrices, analyzing symmetric interface operators and demonstrating robustness in disordered systems.

Findings

Topological edge modes are characterized by eigenvalues of submatrix symbols.

Global inversion symmetry influences the existence of interface modes.

Disordered systems can support robust zero-energy interface states.

Abstract

Tridiagonal $k$-Toeplitz operators provide a natural framework for modelling one-dimensional $k$-periodic lattice systems. A fundamental connection is obtained between Coburn's lemma for tridiagonal $k$-Toeplitz operators and the existence of edge modes. We reveal that topological edge modes are characterised by the eigenvalues of the leading principal submatrix of the symbol function. A complete analysis of tridiagonal interface operators satisfying global inversion symmetry is then presented. These results are applied to finite one-dimensional $k$-periodic chains of damped resonators that satisfy both local and global inversion symmetry. Additionally, disordered tight-binding interface operators are shown to support a topologically robust zero-energy interface state. Numerical simulations are conducted to illustrate the theoretical findings.