Critical transport and anomalously localized states at the Anderson transition

The effect of anomalously localized states (ALS) for electron transport at the critical point of the Anderson transition is numerically investigated for two-dimensional symplectic systems. Defining ALS quantitatively, it is found that a probability of finding ALS at criticality increases with the system size, and saturates in an infinite system. This remarkable feature influences transport properties at criticality.     

Anomalously localised states at the Anderson transition

Non-multifractal critical wave functions at the Anderson transition are numerically investigated for the SU(2) model belonging to the two-dimensional symplectic class. These states can be regarded as anomalously localised states (ALS) at criticality. Giving a quantitative definition of ALS, it has been revealed that the probability to find ALS increases with the system size and remains at a finite value even in the thermodynamic limit. The most probable, namely typical, critical states have the multifractal nature, while its probability measure is zero. In order to understand how ALS affect critical properties in infinite systems, we studied the distribution of the correlation dimension D₂ and the nearest-neighbour level spacing distribution P(s) by paying attention to ALS. Results show that the influence of ALS to these distribution functions is limited. This is because the spatial distribution of amplitudes in tail regions of ALS exhibits multifractality as in the case of typical critical wave functions.     

Critical dynamics and multifractal exponents at the Anderson transition in 3d disordered systems

We investigate the dynamics of electrons in the vicinity of the Anderson transition in d = 3 dimensions. Using the exact eigenstates from a numerical diagonalization, a number of quantities related to the critical behavior of the diffusion function are obtained. The relation η = d ? D₂ between the correlation dimension D₂ of the multifractal eigenstates and the exponent η which enters into correlation functions is verified. Numerically, we have η ≈? 1.3. Implications of critical dynamics for experiments are predicted. We investigate the long-time behavior of the motion of a wave packet. Furthermore, electron-electron and electron-phonon scattering rates are calculated. For the latter, we predict a change of the temperature dependence for low T due to η. The electron-electron scattering rate is found to be linear in T and depends on the dimensionless conductance at the critical point.     

Scaling law and critical exponent for α0 at the 3D Anderson transition

We use high‐precision, large system‐size wave function data to analyse the scaling properties of the multifractal spectra around the disorder‐induced three‐dimensional Anderson transition in order to extract the critical exponents of the transition. Using a previously suggested scaling law, we find that the critical exponent ν is significantly larger than suggested by previous results. We speculate that this discrepancy is due to the use of an oversimplified scaling relation.     

The Anderson transition in three-dimensional quasiperiodic lattices: finite-size scaling and critical exponent

The influence of quasiperiodicity on the metalinsulator transition (MIT) in the Anderson model of localization is investigated. The eigenstates of a 3D Amman-Kramer lattice are studied in the vertex model. The participation numbers are calculated and evaluated by means of a finitesize scaling procedure to characterize the MIT. The critical disorder W _c = 21.2 ± 0.6 and the exponent υ = 1.4 ± 0.3 are computed.     

Spectral rigidity and eigenfunction correlations at the Anderson transition

The statistics of energy levels for a disordered conductor are considered in the critical energy window near the mobility edge. It is shown that, if the critical wave functions are multifractal, the one-dimensional gas of levels on the energy axis is compressible, in the sense that the variance of the level number in an interval is 〈 (δN)²〉∼χ〈N〉 for 〈N〉≫1. The compressibility, χ=η/2d, is given exactly in terms of the multifractal exponent η =d−D ₂ at the mobility edge in a d-dimensional system. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 5, 355–360 (10 September 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Finite-size scaling of the level compressibility at the Anderson transition

We compute the number level variance Σ ₂ and the level compressibility χ from high precision data for the Anderson model of localization and show that they can be used in order to estimate the critical properties at the metal-insulator transition by means of finite-size scaling. With N, W, and L denoting, respectively, linear system size, disorder strength, and the average number of levels in units of the mean level spacing, we find that both χ(N, W) and the integrated Σ ₂ obey finite-size scaling. The high precision data was obtained for an anisotropic three-dimensional Anderson model with disorder given by a box distribution of width W/2. We compute the critical exponent as ν≈ 1.45±0.12 and the critical disorder as W _c≈ 8.59±0.05 in agreement with previous transfer-matrix studies in the anisotropic model. Furthermore, we find χ≈ 0.28±0.06 at the metal-insulator transition in very close agreement with previous results. Received 1st November 2001 and Received in final form 8 March 2002 Published online 6 June 2002     

Strict parabolicity of the multifractal spectrum at the Anderson transition

Using the well-known “algebra of multifractality,” we derive the functional equation for anomalous dimensions Δ _q , whose solution Δ = χq(q–1) corresponds to strict parabolicity of the multifractal spectrum. This result demonstrates clearly that a correspondence of the nonlinear σ-models with the initial disordered systems is not exact.     

Exact relations between multifractal exponents at the Anderson transition

Two exact relations between mutlifractal exponents are shown to hold at the critical point of the Anderson localization transition. The first relation implies a symmetry of the multifractal spectrum linking the exponents with indices q<1/2 to those with q>1/2. The second relation connects the wave-function multifractality to that of Wigner delay times in a system with a lead attached.     

Universal dynamic exponent at the liquid-gas transition from molecular dynamics

The liquid-gas system is expected to exhibit distinct dynamic behavior in the fluid's critical region (model H). We present molecular dynamics simulations of a Lennard-Jones fluid model starting from specially designed, near-equilibrium, initial conditions. By following the fluid's relaxation towards equilibrium, we calculate the requisite transport coefficients in the critical region. The results yield the scaling behavior of the thermal diffusion coefficient D(T) approximately xi(-1.023+/-0.018) (xi is the correlation length) and a nonconventional divergent heat conductivity, all of which are in accord with mode-coupling and renormalization group predictions, as well as some experimental data.     

Critical fidelity at the metal-insulator transition

Using a Wigner Lorentzian random matrix ensemble, we study the fidelity, F(t), of systems at the Anderson metal-insulator transition, subject to small perturbations that preserve the criticality. We find that there are three decay regimes as perturbation strength increases: the first two are associated with a Gaussian and an exponential decay, respectively, and can be described using linear response theory. For stronger perturbations F(t) decays algebraically as F(t) approximately t(-D2(mu)), where D2(mu) is the correlation dimension of the local density of states.     

Boundary multifractality at the integer quantum Hall plateau transition: implications for the critical theory

We study multifractal spectra of critical wave functions at the integer quantum Hall plateau transition using the Chalker-Coddington network model. Our numerical results provide important new constraints which any critical theory for the transition will have to satisfy. We find a nonparabolic multifractal spectrum and determine the ratio of boundary to bulk multifractal exponents. Our results rule out an exactly parabolic spectrum that has been the centerpiece in a number of proposals for critical field theories of the transition. In addition, we demonstrate analytically exact parabolicity of the related boundary spectra in the two-dimensional chiral orthogonal "Gade-Wegner" symmetry class.     

ε-expansion for the density of states of a disordered system near an Anderson transition

The density of states for the Schrödinger equation with a Gaussian random potential is calculated in a space of dimension d=4?ε in the entire energy range, including the vicinity of an Anderson transition.     

Localized states at the helicoidal phase transition

The appearence of a new type of localized states at the helicoidal transition is predicted. The order parameter decays with an oscillation in the vicinity of the defect provoking the localized transition. The cases of point, linear, and planar defects are considered, and the specific heat jumps are calculated. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 10, 776–781 (25 May 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.