On the validity of the tight-binding method for describing systems of subwavelength resonators

TL;DR

This paper investigates the relationship between the capacitance matrix formalism and the tight-binding approximation in subwavelength resonator systems, highlighting limitations of common approximations and implications for metamaterials analysis.

Contribution

It establishes conditions under which the capacitance matrix formalism aligns with the tight-binding model, emphasizing the significance of long-range interactions.

Findings

Tight-binding approximation matches capacitance formalism only for dilute resonators.

Nearest-neighbour approximation often neglects significant long-range interactions.

Long-range interactions are relatively strong even in dilute resonator systems.

Abstract

The goal of this paper is to relate the capacitance matrix formalism to the tight-binding approximation. By doing so, we open the way to the use of mathematical techniques and tools from condensed matter theory in the mathematical and numerical analysis of metamaterials, in particular for the understanding of their topological properties. We firstly study how the capacitance matrix formalism, both when the material parameters are static and modulated, can be posed in a Hamiltonian form. Then, we use this result to compare this formalism to the tight-binding approximation. We prove that the correspondence between the capacitance formulation and the tight-binding approximation holds only in the case of dilute resonators. On the other hand, the tight-binding model is often coupled with a nearest-neighbour approximation, whereby long-range interactions are neglected. Even in the dilute case, we show that long-range interactions between subwavelength resonators are relatively strong and nearest-neighbour approximations are not generally appropriate.