J. Phys. I France

**7**(1997) 1559-1581

## Thouless Numbers for Few-Particle Systems with Disorder and Interactions

**Dietmar Weinmann**

^{1, 2}, Jean-Louis Pichard^{1}and Yoseph Imry^{3}^{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)

** Abstract **

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
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
*g*_{N} 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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*Les Editions de Physique 1997*