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We derive the exact longitudinal plasmon dispersion relations, $\omega(k)$ ofclassical one and two dimensional Wigner crystals at T=0 from the real spaceequations of motion, of which properly accounts for the full unscreened Coulombinteractions. We make use of the polylogarithm function in order to evaluatethe infinite lattice sums of the electrostatic force constants. From our exactresults we recover the correct long-wavelength behavior of previous approximatemethods. In 1D, $\omega(k) \sim | k |\log ^{1/2} (1/k)$, validating the knownRPA and bosonization form. In 2D $\omega(k) \sim \sqrt k$, agreeing remarkablywith the celebrated Ewald summation result. Additionally, we extend thisanalysis to calculate the band structure of tight-binding models ofnon-interacting electrons with arbitrary power law hopping.

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