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

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.

4 pages, 1 figure. Important typos and errors fixed, 2D dispersion added

References in corpus (9)

• Generic properties of a quasi-one dimensional classical Wigner crystal
• Pinned Wigner Crystals
• Thermodynamic density of states of two-dimensional GaAs systems near the apparent metal-insulator transition
• Low-frequency dynamics of disordered XY spin chains and pinned density waves: from localized spin waves to soliton tunneling
• Zero field Wigner crystal
• Normal modes of a quasi-one-dimensional multi-chain complex plasma
• Critical wave-packet dynamics in the power-law bond disordered Anderson Model
• Specific heat of classical disordered elastic systems
• Anderson transition of the plasma oscillations of 1D disordered Wigner lattices

Cited by in corpus (2)

• Phase diagram of a quantum Coulomb wire
• Disorder induced transition into a one-dimensional Wigner glass