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

Abstract

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.PACS numbers: 72.15. Rn, 71.30.+h, 74.25.Fy 1 There is a great current interest in the localization weakening effect due to the interaction of two electrons in one-dimensional (1D) disordered systems [1,2,3,4,5,6,7,8,9,10]. Shepelyansky [1] 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 , 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 fractionstates 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 studies [3,4,5,6,7,8,9,10] 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 regarding chaoticity of the non-interacting electron localized states within ξ 1 , so that the relevant matrix model could be prob-2 ably different [13,14]. Although the reported absence of propagation enhancement[11] can be critisized, since the transfer matrix method may not measure the actual pair localization length, it is correct that...

Quantum diffusion and scaling of localized interacting electrons

“…An appropriate random matrix model gave similar results as the one proposed by Shepelyansky [6]. Xiong and Evangelou derived similar results for the two-and threeparticle interaction matrix, the latter was only more sparse than the former [158].…”

“…The most detailed analysis was performed by Ro ¨mer et al [159]. As also observed in [93,101,158], all diagonal elements were positive and depended on the sign of the interaction. The distribution of off-diagonal elements was symmetric around zero but strongly non-Gaussian, in contrast to the assumption by most authors which had mapped the two-particle problem onto random matrix models.…”