Casimir effects in monatomically thin insulators polarizable perpendicularly: nonretarded approximation

A flat monatomically thin insulating sheet is modelled initially as a square lattice and then as an amorphous distribution of harmonic oscillators, polarizable only perpendicularly to the sheet. In an approximation neglecting dissipation and retardation, we calculate the polarizability X(ω,k) per unit area as a function of the frequency ω and surface-parallel wave-vector k of an externally applied electric field. To find the underlying so-called local fields one must first replace the familiar three-dimensional Lorenz–Lorentz accounts of dielectric functions with their well-established and very different two-dimensional analogues. Image fields are given by weighted integrals over X(0,k); the poles of X(ω,k) identify the normal-mode frequencies ω-bar (k). The Hamiltonian version of the theory is quantized via the normal modes; from it we determine the van der Waals interaction of the sheet with a nearby atom, and between two dynamically identical parallel sheets. (paper)