Applications of Chebyshev polynomials and Toeplitz theory to topological metamaterials

Abstract

We survey the use of Chebyshev polynomials and Toeplitz theory for the study of topological metamaterials. We consider both Hermitian and non-Hermitian systems of subwavelength resonators and provide a mathematical framework to quantitatively explain and characterise some spectacular properties of metamaterials. Our characterisations are based on translation invariance properties of the capacitance matrices associated to the different investigated systems of resonators together with properties of Chebyshev polynomials. The three-term recurrence relation satisfied by the Chebyshev polynomials is shown to be the key to the mathematical analysis of spectra of tridiagonal (perturbed) both Toeplitz (for monomer systems) and 2-Toeplitz (for dimer systems) capacitance matrices.