Statistical, collective, and critical phenomena of classical one-dimensional disordered Wigner lattices

Abstract

We derive the exact longitudinal plasmon dispersion relations, ω(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, ω(k) ∼ |k| log 1/2 (1/k), validating the known RPA and bosonization form. In 2D ω(k) ∼ √ 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.

Disorder Induced Transition into a One-Dimensional Wigner Glass

“…Recently, the plasmons of a disordered 1D WC have been reported to exhibit a delocalization transition [8]. Subsequent investigations have clarified the statistical arrangement of the electrons at equilibrium for different types and strengths of disorder [9].…”

“…The charges were numerically relaxed to their equilibrium configuration with the use of methods are outlined in [9] and briefly described below. Let us introduce the various physical quantities of interest that are needed to interpret the results of our investigations.…”

“…Since we are interested in quasi long-range crystalline order(due to zero-point motion), it is sufficient to study large finite systems with periodic boundary conditions. The charges were numerically relaxed to their equilibrium configuration with the use of methods are outlined in [9] and briefly described below. Let us introduce the various physical quantities of interest that are needed to interpret the results of our investigations.…”

“…Apparently, all physical quantities calculated in this article require the determination of the equilibrium R's, therefore an effective numerical relaxation method is absolutely indispensable. We have employed a Newton-Raphson procedure by which one recursively applies the inverse of D(R) to the out-of equilibrium particle coordinates R until the simultaneous total forces on the charges are sufficiently close to zero [9] and consequently the system approaches equilibrium.…”

Perturbative criteria for Anderson localization in long-ranged 1D tight-binding models

“…Eq. (2.4) can be evaluated explicitly in terms of polylogarithms as shown in the reference 10 . Using the definition:…”