Interacting electrons in a random medium: a simple one-dimensional model

TL;DR

This paper analyzes a one-dimensional model of interacting electrons in a random medium, deriving asymptotic behavior of the ground state energy and describing how interactions influence particle distribution across disjoint segments.

Contribution

It provides a novel asymptotic analysis of the ground state energy for interacting electrons in a random medium, highlighting the effects of repulsive interactions on particle localization.

Findings

Two-term asymptotic for ground state energy per particle in low density regime

Interactions cause particles to redistribute into new segments

Ground state described via one- and two-particle reduced density matrices

Abstract

The present paper is devoted to the study of a simple model of interacting electrons in a random background. In a large interval $\Lambda$, we consider $n$ one dimensional particles whose evolution is driven by the Luttinger-Sy model, i.e., the interval $\Lambda$ is split into pieces delimited by the points of a Poisson process of intensity $\mu$ and, in each piece, the Hamiltonian is the Dirichlet Laplacian. The particles interact through a repulsive pair potential decaying polynomially fast at infinity. We assume that the particles have a positive density, i.e., $n/|\Lambda|\to\rho>0$ as $|\Lambda|\to+\infty$. In the low density or large disorder regime, i.e., $\rho/\mu$ small, we obtain a two term asymptotic for the thermodynamic limit of the ground state energy per particle of the interacting system; the first order correction term to the non interacting ground state energy per particle is controlled by pairs of particles living in the same piece. The ground state is described in terms of its one and two-particles reduced density matrix. Comparing the interacting and the non interacting ground states, one sees that the effect of the repulsive interactions is to move a certain number of particles living together with another particle in a single piece to a new piece that was free of particles in the non interacting ground state.