Hermitian formulation of multiple scattering induced topological phases in metamaterial crystals

In this article we develop an analog to the SSH model in tight-binding chains of resonators and an innovative Hermitian matrix formulation to describe the topological phases induced by multiple scattering at subwavelength scales in one-dimensional structured and locally resonant metamaterial crystals. We first start from a set of coupled dipole equations capturing the nature of the wave-matter interactions, i.e., hybridization between locally dispersive resonances and a continuum as well as infinite long range multiple scattering coupling, to analytically derive a matrix operator H MS , which is found to be Hermitian when evaluated on propagative bands. This new operator straightforwardly highlights how the composition, structure, scattering resonance, and Bloch periodicity together set in the chain macroscopic properties, in particular its topology. We analytically confirm the existence of a structure based topological transition in chiral-symmetric biperiodic metamaterial crystal chains, characterized by a winding number. We further demonstrate that chiral symmetry breaking in metamaterial crystal chains prevents us from defining a proper topological invariant. We finally numerically confirm, in the microwave domain, the existence of the topological transition in a chiral symmetric metamaterial crystal chain through the study of topological interface modes and their robustness to disorder.