From independent particle towards collective motion in two electron square lattices

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 L× L 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 ∆r of the spacing r between the two particles is larger than the lattice spacing a, there is a scaling length L0 = √ 8π(t/U) such that the relative fluctuation ∆r/〈r〉 is a universal function of the dimensionless ratio L/L0, up to finite size corrections of order L . L < L0 and L > L0 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, ∆r becomes smaller than a, 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. PACS. 71 10-w Theories and models for many-electron systems – 71 27+a Strongly correlated electron systems – 73.20.Qt Electron solids