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 $E_c.$ 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.