Exact diagonalization study of rare events in disordered conductors

Abstract

We determine the statistical properties of wave functions in disordered quantum systems by exact diagonalization of one-, two-and 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, Typeset using REVT E X 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][2][3][4][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][8][9][10][11][12][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][17][18] We use the Anderson model of localization [23] which is a tight-binding model on aHere r = (x, y, . . .) denotes sites on the lattice, c † r and c r are the usual creation and annihilation operators, the hopping amplitudes are t rr ′ = 1 for nearest neighbour sites and zero otherwise. The on-site potential υ r is taken to be uncorrelated white noise, with zero mean and variance υ r υ r ′ = δ rr ′ W 2 /12. The parameter W characterizes the disorder strength.As is well-known (see for instance [24,25] We refer to these two cases by assigning, as usual, the parameter β = 1 to the former and β = 2 to the latter. The metallic regime is characterized by g ≫ 1 whereis the dimensionless conductance (we take = 1). Here · · · W denotes an average over disorder realisations. The wave functions are normalizedis a window function of width η, centered around E = 0 and normalized to unity. In the following we describe the results of our calculations and compare them to the predictions of Refs. [8][9][10][11][12][13]19].1D case. The eigenstates in a disordered chain are localized with localization length ξ = 4ℓ. According to Ref.[7] the distribution of wave-function amp...

Localization–delocalization transition for disordered cubic harmonic lattices

“…55 Upon increasing the disorder, corrections to GOE have been studied which we expect to see present also in the case of our vibrational disorder. 56,57 We determine the The dotted grey and red lines in the base denote the phase boundaries and the band edges respectively as in Fig. 1.…”

Resonance width distribution for high-dimensional random media

“…These states are precursors of the Anderson localization and were termed prelocalized states 26,27,28,29 . In 3D conductors they have sharp amplitude peaks on the top of a homogeneous background 26,27 .…”

Spatial structure of anomalously localized states in disordered conductors

“…As far as the numerical work is concerned, the situation may be summarised as follows. While it can be concluded that the DNLSM appears to be adequate in the quasione-dimensional (Q1D) Anderson model under certain conditions 10,11 , the DNLSM may not correctly describe ALS in two and three dimensions 10,12 , at least for the parameter values considered in these studies. On the other hand, it was found recently 14 that the DNLSM appears to describe the statistics of rare events adequately in a 2D kicked rotor.…”

“…In Ref. 11 it was confirmed that as the disorder is reduced to reach the weakly disordered regime, the distribution function remains log-normal. It has, however, not been possible to resolve a discrepancy (between the DNLSM 5,6 and the so-called direct optimal fluctuation method 8 ) in the prediction of the parameters of this distribution.…”