  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.
