Sparse random matrix configurations for two or three interacting electrons in a random potential

We investigate the random matrix configurations for two or three interacting electrons in one-dimensional disordered systems. In a suitable non-interacting localized electron basis we obtain a sparse random matrix with very long tails which is different from a superimposed random band matrix usually thought to be valid. The number of non-zero off-diagonal matrix elements is shown to decay very weakly from the matrix diagonal and the non-zero matrix elements are distributed according to a Lorentzian around zero with also very weakly decaying parameters. The corresponding random matrix for three interacting electrons is similar but even more sparse. 1 There is a great current interest in the localization weakening effect due to the interaction of two electrons in one-dimensional mapped this problem onto a class of random banded matrices with strongly fluctuating diagonal elements, being the eigenenergies of the non-interacting problem, and independent Gaussian random off-diagonal matrix elements of zero average and typical strength U ξ 3/2 1 , lying in a band of width the one particle localization length ξ 1 with U the strength of the interaction. Moreover, by the mapping to a superimposed banded random matrix ensemble (SBRM) a fraction of ξ 1 2L states with a considerable enhancement ξ ∼ U 2 ξ 2 1 of the localization length along the center of mass coordinate is predicted due to coherent propagation of the electronic pair. It was also shown that the interaction has no effect for the majority of other states with the two particles localized in isolated spatial positions which do not allow overlapping. This conclusion was confirmed and extended to higher dimensions via Thouless block scaling picture by Imry [2]. The subsequent numerical verified the main qualitative results concerning the presence of Shepelyansky states, mostly by supressing single particle transport via efficient Green function or bag model methods which examine pair propagation [8]. The deviations from the predicted behavior of the two-particle localization length ξ found were usually attributed to the oversimplified statistical assumptions concerning the band random matrix model of the original Shepelyansky construction. However, there is an ongoing debate whether coherent pair propagation actually exists for two interacting electrons in infinite disordered systems [11, 12], which began by a recent transfer matrix study where no propagation enhancement is found at E = 0 for an infinite chain [11]. Moreover, it was pointed out that the reduction to a SBRM relies on questionable assumptions …