Exact longitudinal plasmon dispersion relations for one and two dimensional Wigner crystals

TL;DR

This paper derives exact formulas for the longitudinal plasmon dispersion relations in one and two-dimensional Wigner crystals at zero temperature, confirming known behaviors and extending analysis to electron band structures with arbitrary hopping.

Contribution

It provides the first exact derivation of plasmon dispersion relations for 1D and 2D Wigner crystals, including full Coulomb interactions, using polylogarithm functions.

Findings

1D plasmon dispersion: ω(k) ∼ |k| log^{1/2}(1/k)

2D plasmon dispersion: ω(k) ∼ √k

Extension to tight-binding models with arbitrary power law hopping

Abstract

We derive the exact longitudinal plasmon dispersion relations, $\omega(k)$ of classical one and two dimensional Wigner crystals at T=0 from the real space equations of motion, of which properly accounts for the full unscreened Coulomb interactions. We make use of the polylogarithm function in order to evaluate the infinite lattice sums of the electrostatic force constants. From our exact results we recover the correct long-wavelength behavior of previous approximate methods. In 1D, $\omega(k) \sim | k |\log ^{1/2} (1/k)$, validating the known RPA and bosonization form. In 2D $\omega(k) \sim \sqrt k$, agreeing remarkably with the celebrated Ewald summation result. Additionally, we extend this analysis to calculate the band structure of tight-binding models of non-interacting electrons with arbitrary power law hopping.