An exact-diagonalization study of rare events in disordered conductors

We determine the statistical properties of wave functions in disordered quantum systems by exact diagonalization of one-, twoand quasi-one dimensional tight-binding Hamiltonians. In the quasi-one dimensional case we find that the tails of the distribution of wave-function amplitudes are described by the non-linear σ-model. In two dimensions, the tails of the distribution function are consistent with a recent prediction based on a direct optimal fluctuation method. 72.15.Rn,71.23.An,05.40.-a Typeset using REVTEX 1 It is well established that disordered quantum systems in the metallic regime (i.e., in the limit of weak disorder) and highly excited classically chaotic quantum systems exhibit universal quantum fluctuations that can be described by random matrix theory (RMT): statistical properties, on the scale of the mean level spacing, of eigenvalues, eigenfunctions, and matrix elements are universal, i.e., they do not depend on the microscopic details of the systems under consideration [1–5]. However, in ballistic, classically chaotic quantum systems, non-hyperbolic phase-space structures may lead to deviations from universal RMT statistics [3]. Similarly fluctuations in disordered, classically diffusive quantum systems may deviate considerably from the RMT predictions due to increased localization. This effect is naturally very significant in the tails of distribution functions [6] (corresponding to rare events) of wave-function amplitudes [7–13], of the local density of states [7,12], of inverse participation ratios [12,13] and of NMR line shapes [7]. In all of these cases (with the exception of Ref. [7] which deals with one-dimensional (1D) systems), the distribution functions have been calculated using the non-linear σ-model (NLSM). Very recently, this approach has been extended to ballistic systems [14,15] (see also [16–18]). In Ref. [19] a direct optimal fluctuation method [20] was used to calculate the tails of distributions of current relaxation times and wave-function amplitudes; and predictions differing from [8–13] were put forward. This led the authors of [19] to question the suitability of the NLSM to describe rare events in disordered conductors. It is thus of great interest to test the predictions of [7–13] and [19] against results of independent calculations. In this letter, we have determined distribution functions of wavefunction amplitudes by exact diagonalization of 1D, 2D and quasi-1D tight-binding Hamiltonians; in this case rare events correspond to unusually high splashes of wave-function amplitudes. We note that wave-function amplitude distributions can be measured in microwave experiments [21,22]. We use the Anderson model of localization [23] which is a tight-binding model on a d-dimensional hyper-cubic lattice 2