J. Phys. I France 7 (1997) 1559-1581
Thouless Numbers for Few-Particle Systems with Disorder and InteractionsDietmar Weinmann1, 2, Jean-Louis Pichard1 and Yoseph Imry3
1 CEA, Service de Physique de l'État Condensé, Centre d'Études de Saclay, 91191 Gif-sur-Yvette Cedex, France
2 Institut für Physik, Universität Augsburg, Memminger Strasse 6, 86135 Augsburg, Germany
3 Weizmann Institute of Science, Department of Condensed Matter Physics, 76100 Rehovot, Israel
(Received 9 May 1997, revised 17 July 1997, accepted 5 August 1997)
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 gN 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 and . When , there is a perturbative mixing between a few Slater determinants (Rabi oscillations) and , where P=N/2 (even N) or ( N+1)/2 (odd N). When , 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, 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 the same signifies also the breakdown of the perturbation theory in U. For , 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 , the sparsity becomes irrelevant, and the states are extended more or less ergodically over gN consecutive Slater determinants. A self-consistent argument gives . 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.
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