From independent particle towards collective motion in two electron square lattices

TL;DR

This paper investigates the transition from independent particle behavior to collective motion in a two-electron square lattice system, identifying regimes characterized by Coulomb interaction strength and lattice size, with implications for correlated electron states.

Contribution

It introduces a detailed analysis of the crossover regimes in a two-electron lattice system, including a universal scaling law and the effects of disorder on collective electron motion.

Findings

Universal scaling of relative fluctuation with system size and interaction strength.

Identification of a threshold U*(L)/t for transition to correlated lattice regime.

Weak disorder promotes correlated electron motion.

Abstract

The two dimensional crossover from independent particle towards collective motion is studied using 2 spinless fermions interacting via a U/r Coulomb repulsion in a LxL square lattice with periodic boundary conditions and nearest neighbor hopping t. Three regimes characterize the ground state when U/t increases. Firstly, when the fluctuation $\Delta r$ of the spacing r between the two particles is larger than the lattice spacing, there is a scaling length $Lo=\sqrt{8}\pi^2(t/U)$ such that the relative fluctuation $\Delta r/<r>$ is a universal function of the dimensionless ratio L/Lo, up to finite size corrections of order $L^{-2}$. L<Lo and L>Lo are respectively the limits of the free particle Fermi motion and of the correlated motion of a Wigner molecule. Secondly, when U/t exceeds a threshold U*(L)/t, $\Delta r$ becomes smalller than the lattice spacing, giving rise to a correlated lattice regime where the previous scaling breaks down and analytical expansions in powers of (t/U) become valid. A weak random potential reduces the scaling length and favors the correlated motion.