Thouless numbers for few-particle systems with disorder and interactions

Considering N spinless Fermions in a random potential, we study how a short range pairwise interaction delocalizes the N-body states in the basis of the one-particle Slater determinants, and the spectral rigidity of the N-body spectrum. The maximum number g(N) of consecutive levels exhibiting the universal Wigner-Dyson rigidity (the Thouless number) is given as a function of the strength U of the interaction for the bulk of the spectrum. In the dilute limit, one finds two thresholds U-c1 and U-c2. When U < U-c1, there is a perturbative mixing between a few Slater determinants (Rabi oscillations) and g(N) proportional to \U\(P) < 1, where P = N/2 (even N) or (N + 1)/2 (odd N). When U = U-c1, the matrix element of a Slater determinant to the "first generation" of determinants directly coupled to it by the interaction is of the order of the level spacing of the latter determinants, g(N) approximate to 1 and the level spacing distribution exhibits a crossover from Poisson to Wigner, related to the crossover between weak perturbative mixing and effective golden-rule decay. Moreover, we show that thle same U-c1 signifies also the breakdown of the perturbation theory in U. For U-c1 < U < U-c2, the states are extended over the energetically nearby Slater determinants with a non-ergodic hierarchical structure related to the sparse form of the Hamiltonian. Above a second threshold U-c2, the sparsity becomes irrelevant, and the states are extended more or less ergodically over g(N) consecutive Slater determinants. A self-consistent argument gives g(N) proportional to UN/(N-1). We compare our predictions to a numerical study of three spinless Fermions in a disordered cubic lattice. Implications for the interaction-induced N-particle delocalization in real space are discussed. The applicability of Fermi's golden rule for decay in this dilute gas of "real" particles is compared with the one characterizing a finite-density Fermi gas. The latter is related to the recently suggested Anderson transition in Fock space.