The nearest-neighbor level spacing distribution is numerically investigated by directly diagonalizing disordered Anderson Hamiltonians for systems of sizes up to 100×100×100 lattice sites. The scaling behavior of the level statistics is examined for large spacings near the delocalization-localization transition and the correlation length exponent is found. By using high-precision calculations we conjecture a new interpolation of the critical cumulative probability, which has size-independent asymptotic form ln I(s) ∝ −s α with α = 1.0 ± 0.1. The statistical fluctuations in energy spectra of disordered quantum systems attract at present much attention [1,2,3,4,5]. It is known that by increasing the fluctuations of a random potential the one-electron states undergo a localization transition, which is the origin of the Anderson metal-insulator transition (MIT) [6]. The influence of the disorder on the wave functions is reflected by the mutual correlations between the corresponding energy levels, so that the statistics of energy levels is sensitive to the MIT. In the metallic limit the statistics of energy spectra can be described by the random-matrix theory (RMT) developed by Wigner and Dyson [7,8]. This was shown by solving the zero-mode nonlinear σ-model using the supersymmetric formalism [9]. Later, perturbative corrections to the two-level correlation function obtained in the RMT were evaluated in the diffusive regime by the impurity diagram technique [10]. In the insulating regime, when the degree of disorder W is much larger than the critical value W c , the energy levels of the strongly localized eigenstates fluctuate as independent random variables.An important quantity for analyzing the spectral fluctuations is the nearest-neighbor level spacing distribution P (s). It contains information about all of the n−level correlations. In the metallic regime P (s) is very close to the Wigner surmise P W (s) = π s/2 exp −π s 2 /4 [11] (s is measured in units of the mean level spacing ∆). In the localized regime the spacings are distributed according to the Poisson law, P P (s) = exp(−s), because the levels are completely uncorrelated. The study of the crossover of P (s) between the Wigner and the Poissonian limits which accompanies the disorder-induced MIT in threedimensional system (3D) was started in Ref. [12] and became the subject of several subsequent investigations [1,2,5,13,14].It was suggested earlier [1] that P (s) exhibits critical behavior and should be size-independent at the MIT. Investigating the finite-size scaling properties of P (s) provides not only an alternative method for locating the transition [2], but allows also to determine the critical behavior of the correlation length [14]. A technical advantage of the method is that one needs to compute only energy spectra and not eigenfunctions and/or the conductivity. On the other hand, a large number of realizations of the random potential has to be considered. In comparison with the well-established transfer-matrix method [15] by which one approache...

We report a novel scale-independent, Aharonov-Bohm flux controlled crossover from critical orthogonal to critical unitary statistics at the disorder induced metal insulator transition. Our numerical investigations show that at the critical point the level statistics are definitely distinct and determined by fundamental symmetries. The latter is similar to the behavior of the metallic phase known from random matrix theory. The Aharonov-Bohm flux dependent crossover is characteristic of the critical ensemble.

Rapid Communications are intended for the accelerated publication of important new results and are therefore given priority treatment both in the editorial office and in production AR.apid Communication in Physical Review B should be no longer than four printed pages and must be accompanied by an abstract P. age proofs are sent to authors The distribution of energy-level separations for lattices of sizes up to 28&&28X28 sites is numerically calculated for the Anderson model. The results show one-parameter scaling. The size-independent universality of the critical level-spacing distribution allows one to detect with high precision the critical disorder W, = 16.35. The scaling properties yield the critical exponent, v= 1.45~0.08, and the disorder dependence of the correlation length.

The crossover between a free magnetic moment phase and a Kondo phase in low dimensional disordered metals with dilute magnetic impurities is studied. We perform a finite size scaling analysis of the distribution of the Kondo temperature as obtained from a numerical renormalization group calculation of the local magnetic susceptibility and from the solution of the self-consistent Nagaoka-Suhl equation. We find a sizable fraction of free (unscreened) magnetic moments when the exchange coupling falls below a disorder-dependent critical value Jc. Our numerical results show that between the free moment phase due to Anderson localization and the Kondo screened phase there is a phase where free moments occur due to the appearance of random local pseudogaps at the Fermi energy whose width and power scale with the elastic scattering rate 1/τ .The Kondo problem is of central importance for understanding low-temperature anomalies in low-dimensional disordered metals [1,2,3,4,5,6,7,8,9,10,11,12], such as the saturation of the dephasing rate [13] and the non-Fermi-liquid behavior of certain magnetic alloys [1,14]. For a clean metal, the screening of a spin-1/2 magnetic impurity is governed by a single energy scale, the Kondo temperature T K . Thermodynamic observables and transport properties obey universal functions which scale with T K . Thus, in a metal where nonmagnetic disorder is also present, two fundamental questions naturally arise: (i) Is the Kondo temperature modified by nonmagnetic disorder? (ii) Is the one-parameter scaling behavior still valid? It is well known [1,2,5,7,11] that magnetic moments remain unscreened when conduction electrons are localized due to disorder. However, in weakly disordered two-dimensional electron systems, the localization length is macroscopically large, and so is the number N c of eigenstates with a finite amplitude at the position of the magnetic impurity. In this case, one does not expect to find unscreened magnetic moments for experimentally relevant values of the exchange coupling J.Another situation where magnetic moments remain free in metals at low temperatures occurs when the density of states has a global pseudogap at the Fermi energy E F , namely,. In clean metals, the pseudogap quenches the Kondo screening when J falls below a critical value J c (α). So far, only a few values of α have been realized experimentally: α = 1 in graphene and in d-wave superconductors and α = 2 in p-wave superconductors. In this Letter we examine the quantum phase diagram of magnetic moments diluted in two-dimensional disordered metals using a modified version of the numerical renormalization group (NRG) method. We find a free moment phase which we attribute to the random occurrence of local pseudogaps. The existence of free moments is confirmed directly with NRG by the Curie-like behavior of the the local magnetic susceptibility at low temperatures. Finite-size scaling is performed to demonstrate the robustness of our finding. Furthermore, the distribution of Kondo temperatures obtained numeric...

The energy level statistics of 2D electrons with spin-orbit scattering are considered near the disorder induced metal-insulator transition. Using the Ando model, the nearest-level-spacing distribution is calculated numerically at the critical point. It is shown that the critical spacing distribution is size independent and has a Poisson-like decay at large spacings as distinct from the Gaussian asymptotic form obtained by the random-matrix theory.