Universality of the metal-insulator transition in three-dimensional disordered systems

Abstract

The universality of the metal-insulator transition in three-dimensional disordered system is confirmed by numerical analysis of the scaling properties of the electronic wave functions. We prove that the critical exponent ν and the multifractal dimensions dq are independent on the microscopic definition of the disorder and universal along the critical line which separates the metallic and the insulating regime. 71.30., One of the main problem of the disorder induced metalinsulator transition (MIT) is the proof of its universality. In the pioneering work [1], it was conjectured that if the sample size exceeds all the length parameters of the model, then the conductance, g, is the only parameter which governs MIT. This scaling hypothesis has been confirmed by various numerical analysis, with the help of the finite-size scaling [2,3].Generally accepted scenario of the Anderson localization is that disorder broadens the conductance band. Electron states in the tail of the band become localized, separated from delocalized (metallic) states by the mobility edges, E c . System exhibits the MIT if the Fermi energy, E F , crosses the mobility edge. With increased disorder, E c moves towards the band center. There is a critical value of the disorder, W c , for which E c reaches band center, E c = 0. For disorder W > W c , all electronic states inside the band become localized. Phase diagram in the energy-disorder plane was calculated in [4] and is schematically shown in the upper panel of Fig. 1.At the band center, E = 0, the universality of the MIT was confirmed by detailed numerical analysis of the disorder and system size dependence of Lyapunov exponents in quasi-one dimensional systems [3,5], mean conductance [6], conductance distribution [7], and level statistics [8,9]. These studies determined the value of the critical exponent ν, which determines the divergence of the correlation length, ξ ∼ |W − W c | −ν , as ν = 1.57 ± 0.02 [5,6]. The analysis of MIT along the critical line (non-zero energy E) is more difficult because the critical region is narrower and finite size effects are stronger [10]. Critical exponent, ν, was obtained only in models with random hopping [11], and very recently in [12].In this paper, we present numerical proof of the universality of MIT. By scaling analysis of the electronic wave functions in the vicinity of two critical points, shown in the upper panel of Fig. 1, we prove that the critical exponent ν and fractal dimensions d q of the wave function (defined below) are universal along the critical line.Electron eigenenergies and wave functions are calculated for three-dimensional Anderson Hamiltonian,

Multifractality and quantum diffusion from self-consistent theory of localization

“…(73) If the function F 2 (x) is calculated (Fig.3), one can compare (72) with results by Brndiar and Markos for n = 5 [35] in three dimensions (Fig.4,a). Due to the presence of large parameter n(n − 1) = 20, all numerical data lie in the critical region x < ∼ 1, where the dependence F n (x) is practically linear in accordance with ν = 1 in the Vollhardt and Wölfle theory.…”

“…Accepting r satisfying the condition ǫ ln(L/r) ≪ 1 and having in mind that a value of g at the Anderson transition is g c ∼ 1/ǫ, one can see that the expansion parameter u = ln(L/r)/g is small for the interval L exp(−1/ǫ) < ∼ r ≤ L both in the metallic and critical region. The limiting value (29) is not attained and the two-cooperon expression is valid for correlator (35). This expression is not affected by variation of the correlation length ξ, which runs in the metallic phase from the minimal value ξ min till infinity, and hence ξ is not manifested as a significant length scale.…”

“…Due to existence of the small parameter n(n − 1)/ ln(1/λ) = 0.11, the main body of data corresponds to large values of x = L/ξ, so the lower branch 10 is determined by its logarithmic asymptotics, while the upper branch remains in the linear [35] were used. A difference between ln P n and ln P n was neglected.…”

Character of eigenstates of the three-dimensional disordered Hamiltonian

“…25,26,38 Scaling of the IPR in the critical region was proved in Ref. 39 Here, we discuss how the probability distribution of the IPR depends on the system size and the distance E − E c of the energy from the mobility edge. Our data show that the probability to find the localized states in the metallic phase decreases exponentially when the size of the system increases.…”

“…The eigenenergies were calculated by the Lanczos algorithm. 39 A comparison of the obtained density of states for L = 16 and 40 ͓͑E =7͒ = 0.0076 for L = 16, and ͑E =7͒ = 0.0087 for L =40͔ indicates that the convergence of the density of states is very slow in the band tail.…”