Long-Range Energy-Level Interaction in Small Metallic Particles

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 $f(\epsilon ).$ 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 $\epsilon <E_c,$ in agreement with Random Matrix Theory. When $\epsilon >E_c,$ $f(\epsilon )$ vanishes as a power law in $\epsilon /E_c$ with exponents $-\frac{1}{2},-2,$ and $-\frac{3}{2}$ 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.

Saclay-s93/014 Email: [email protected] [2017: missing figure included]

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