Plasmons and excitons in insulators: dielectric treatment

A dielectric theory of collective excitations in insulating crystals is formally developed, including exchange and correlation effects by a mean field approximation and lattice defects through a dielectric matrix formalism based on wavefunctions in the Wannier representation. Specific application is made to a simple two-band model, for which an explicit secular equation for the longitudinal collective modes is derived with overlap between Wannier functions on first-neighbour sites being treated perturbatively. The response matrix of the model is easily inverted when overlap is neglected, and an explicit expression follows for the macroscopic dielectric function which also displays the existence of transverse collective modes.