Thouless Numbers for Few-Particle Systems with Disorder and Interactions

Dietmar Weinmann; Jean-Louis Pichard; Yoseph Imry

doi:doi:10.1051/jp1:1997100

Numéro		J. Phys. I France Volume 7, Numéro 12, December 1997


Page(s)		1559 - 1581
DOI		https://doi.org/10.1051/jp1:1997100

DOI: 10.1051/jp1:1997100
J. Phys. I France 7 (1997) 1559-1581

Thouless Numbers for Few-Particle Systems with Disorder and Interactions

Dietmar Weinmann^{1, 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)

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 $U_{\rm c1}$ and $U_{\rm c2}$ . When $U<U_{\rm c1}$ , there is a perturbative mixing between a few Slater determinants (Rabi oscillations) and $g_N\propto \vert U\vert^P < 1$ , where P=N/2 (even N) or ( N+1)/2 (odd N). When $U=U_{\rm c1}$ , 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, $g_N\approx 1$ 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 the same $U_{\rm c1}$ signifies also the breakdown of the perturbation theory in U. For $U_{\rm c1}<U<U_{\rm c2}$ , 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 $U_{\rm c2}$ , the sparsity becomes irrelevant, and the states are extended more or less ergodically over g_N consecutive Slater determinants. A self-consistent argument gives $g_N\propto U^{N/(N-1)}$ . 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.