Physics – Condensed Matter – Strongly Correlated Electrons
Scientific paper
2007-04-17
Phys. Rev. B 76, 165129 (2007)
Physics
Condensed Matter
Strongly Correlated Electrons
4 pages, 1 figure. Important typos and errors fixed, 2D dispersion added
Scientific paper
10.1103/PhysRevB.76.165129
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.
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