Quantum diffusion and scaling of localized interacting electrons

Abstract

A new numerical method is introduced that enables a reliable study of disorder-induced localization of interacting particles. It is based on a quantum mechanical time evolution calculation combined with a finite size scaling analysis. The time evolution of up to four particles in one dimension is studied and localization lengths are defined via the long-time saturation values of the mean radius, the inverse participation ratio and the center of mass extension. A systematic study of finite size effects using the finite size scaling method is performed in order to extract the localization lengths in the limit of an infinite system size. For a single particle, the well-known scaling of the localization length λ1 with disorder strength W is observed, λ1 ∝ W—2. For two particles, an interaction-induced delocalization is found, confirming previous results obtained by numerically calculating matrix elements of the two-particle Green's function: in the limit of small disorder, the localization length increases with decreasing disorder as λ2 ∝ W—4 and can be much larger than <$>\mitlambda λ1. For three and four particles, delocalization is even stronger. Based on analytical arguments, an upper bound for the n-particle localization length λn is derived and shown to be in agreement with the numerical data, λn ∝ λ1. Although the localization length increases superexponentially with particle number and can become arbitrarily large for small disorder, it does not diverge for finite λ1 and n. Hence, no extendedstates exist in one dimension, at least for spinless fermions.