Level Statistics and Localization for Two Interacting Particles in a Random Potential

Abstract

We consider two particles with a local interaction U in a random potential at a scale L1 (the one particle localization length). A simplified description is provided by a Gaussian matrix ensemble with a preferential basis. We define the symmetry breaking parameter µ ∝ U −2 associated to the statistical invariance under change of basis. We show that the Wigner-Dyson rigidity of the energy levels is maintained up to an energy Eµ. We find that Eµ ∝ 1/ √ µ when Γ (the inverse lifetime of the states of the preferential basis) is smaller than ∆2 (the level spacing), and Eµ ∝ 1/µ when Γ > ∆2. This implies that the two-particle localization length L2 first increases as |U | before eventually behaving as U 2 .PACS numbers: 72.15, 73.20For a single particle diffusing in a disordered system of size L smaller than the one particle localization length L 1 , there are two characteristic energies: the Thouless energy E c =hD/L 2 and the level spacing, D and d are the band width, the diffusion constant and the system dimension, respectively). If one writes the distribution of energy levels as a Gibbs factor of a fictituous Coulomb gas, the corresponding pairwise interaction for levels with separation ǫ < E c coincides [1] with the logarithmic repulsion characteristic of the matrix ensembles which are statistically invariant under change of basis, e. g. the Gaussian Orthogonal Ensemble (GOE). For ǫ > E c , the level repulsion vanishes more or less quickly, depending on the system dimension. The dimensionless conductance g 1 is given by E c /∆ 1 . This ratio is the single relevant parameter in the scaling theory of localization. In quasi-one dimension, the size where g 1 ≈ 1 defines L 1 . In three dimensions, the mobility edge is characterized by g 1 ≈ g c where g c is of order 1.We shall generalize those concepts to two particles with a local (repulsive or attractive) interaction. This two interacting particle (TIP) problem has received a particular attention since Shepelyansky [2] pointed out that certain TIP states may extend over a scale L 2 much larger than L 1 . Shepelyansky's original reasoning consists in mapping the problem for L ≫ L 1 onto a random band matrix model with a superimposed diagonal matrix (SBRM-model). Imry [3] used later the Thouless scaling block picture to arrive at precisely the same results as Shepelyansky. The smearing due to the interaction of the energy levels within L 1 was estimated using Fermi's golden rule, yielding L 2 ∝ U 2 . This delocalization effect has been confirmed by transfer matrix studies [4,5], and unambiguously illustrated from numerical studies [6] of rings threaded by an AB-flux. However, in one dimension, for system sizes which can be numerically investigated, one obtains To understand those contradictory results, we study the TIP energy level statistics at a scale L 1 in order to identify the energy which plays the role of E c in this case, and to determine its dependence on U . For the original TIP-problem, we assume a tight-binding model1 sites p where the random potential is ...

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