Long-Range Energy-Level Interaction in Small Metallic Particles

TL;DR

This paper models the energy level statistics of disordered metallic conductors using a classical gas analogy, revealing dimension-dependent interactions including long-range attraction in three dimensions.

Contribution

It introduces a novel classical gas model with dimension-dependent energy level interactions consistent with random matrix theory and extends understanding of spectral correlations in disordered metals.

Findings

Logarithmic repulsion for level separations below E_c

Power-law decay of interaction for separations above E_c

Long-range attraction in three-dimensional systems

Abstract

We consider the energy level statistics of non-interacting electrons which diffuse in a $ d $-dimensional disordered metallic conductor of characteristic Thouless energy $ E_c. $ We assume that the level distribution can be written as the Gibbs distribution of a classical one-dimensional gas of fictitious particles with a pairwise additive interaction potential $ f(\varepsilon ). $ We show that the interaction which is consistent with the known correlation function of pairs of energy levels is a logarithmic repulsion for level separations $ \varepsilon <E_c, $ in agreement with Random Matrix Theory. When $ \varepsilon >E_c, $ $ f(\varepsilon ) $ vanishes as a power law in $ \varepsilon /E_c $ with exponents $ -{1 \over 2},-2, $ and $ -{3 \over 2} $ for $ d=1,2, $ and 3, respectively. While for $ d=1,2 $ the energy-level interaction is always repulsive, in three dimensions there is long-range level attraction after the short-range logarithmic repulsion.