Long-range energy-level interaction in small metallic particles

We consider the energy level statistics of non-interacting electrons which diffuse in a d-dimensional disordered metallic conductor of characteristic Thouless energy Ec. 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(ε). 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 ε < Ec, in agreement with Random Matrix Theory. When ε > Ec, f(ε) vanishes as a power law in ε/Ec with exponents −12 ,−2, and −32 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.