LONG-RANGE ENERGY-LEVEL INTERACTION IN SMALL METALLIC PARTICLES

We consider the energy level statistics of non-interacting electrons w hich diffuse in a d-dimensional disordered metallic conductor of chara cteristic Thouless energy E(c). We assume that the level distribution can be written as the Gibbs distribution of a classical one-dimensiona l gas of fictitious particles with a pairwise additive interaction pot ential f(epsilon). We show that the interaction which is consistent wi th the known correlation function of pairs of energy levels is a logar ithmic repulsion for level separations epsilon < E(c), in agreement wi th the random matrix theory. When epsilon > E(c), f(epsilon) vanishes as a power law in epsilon/E(c) with exponents - 1/2, - 2, and - 3/2 fo r d = 1, 2, and 3, respectively. While for d = 1, 2 the energy level i nteraction is always repulsive, in three dimensions there is long-rang e level attraction after the short-range logarithmic repulsion.