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: When U<U_{c1}, there is a perturbative mixing between a few Slater determinants (Rabi oscillations) and g_N \propto |U|^P <1, where P=N/2 (even N) or (N+1)/2 (odd N). When U=U_{c1}, the level spacing distribution exhibits a crossover from Poisson to Wigner, related to the crossover between weak perturbative mixing and effective golden-rule decay, and g_N \approx 1. Moreover, we show that the 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 ~ U^{N/(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.