Spatial Decay of the Single-Particle Density Matrix in Insulators: Analytic Results in Two and Three Dimensions

Analytic results for the asymptotic decay of the electron density matrix in insulators have been obtained in all three dimensions $(D=1,2,3\mathrm{})$ for a tight-binding model defined on a simple cubic lattice. The anisotropic decay length is shown to be dependent on the energy parameters of the model. The existence of the power-law prefactor, $\propto r^{-D/2\mathrm{}}$, is demonstrated.