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

TL;DR

This paper studies the structure of sparse random matrices representing two or three interacting electrons in disordered one-dimensional systems, revealing unique long-tailed distributions and decay properties.

Contribution

It introduces a novel sparse random matrix model for interacting electrons in disordered systems, highlighting distinctive decay and distribution characteristics.

Findings

Non-zero off-diagonal elements decay very weakly from the diagonal.

Matrix elements follow a Lorentzian distribution around zero.

Three-electron matrices are even more sparse than two-electron matrices.

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.