Spectral rigidity and eigenfunction correlations at the Anderson transition (original) (raw)

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

Annalen der Physik, 2010

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 − D 2 between the correlation dimension D 2 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. c

Critical level statistics at the Anderson transition in four-dimensional disordered systems

Annalen der Physik, 1998

The level spacing distribution is numerically calculated at the disorder-induced metal-insulator transition for dimensionality d = 4 by applying the Lanczos diagonalisation. The critical level statistics are shown to deviate stronger from the result of the random matrix theory compared to those of d = 3 and to become closer to the Poisson limit of uncorrelated spectra. Using the finite size scaling analysis for the probability distribution Qn(E) of having n levels in a given energy interval E we find the critical disorder Wc = 34.5 ± 0.5, the correlation length exponent ν = 1.1 ± 0.2 and the critical spectral compressibility κc ≈ 0.5.

Universal Fluctuations in Spectra of Disordered Systems at the Anderson Transition

Japanese Journal of Applied Physics, 1995

Using the level-spacing distribution and the total probability function of the numbers of levels in a given energy interval we analyze the crossover of the level statistics between the delocalized and the localized regimes. By numerically calculating the electron spectra of systems of up to 32 3 lattice sites described by the Anderson Hamiltonian it is shown that the distribution P (s) of neighboring spacings is scale-independent at the metal-insulator transition. For large spacings it has a Poisson-like asymptotic form P (s) ∝ exp(−A s/∆), where A ≈ 1.9. At the critical point we obtain a linear relationship between the variance of the number of levels [δn(ε)] 2 and their average number n(ε)

Anticorrelations from power-law spectral disorder and conditions for an Anderson transition

Physical Review B, 2013

We resolve an apparent contradiction between numeric and analytic results for one-dimensional disordered systems with power-law spectral correlations. The conflict arises when considering rigorous results that constrain the set of correlation functions yielding metallic states to those with non-zero values in the thermodynamic limit. By analyzing the scaling law for a model correlated disorder that produces a mobility edge, we show that no contradiction exists as the correlation function exhibits strong anticorrelations in the thermodynamic limit. Moreover, the associated scaling function reveals a size-dependent correlation with a smoothening of disorder amplitudes as the system size increases.

Anderson transition in one-dimensional systems with spatial disorder

2009

A simple Kronig-Penney model for one-dimensional (1D) mesoscopic systems with delta\delta delta peak potentials is used to study numerically the influence of a spatial disorder on the conductance fluctuations and distribution at different regimes. We use the Levy laws to investigate the statistical properties of the eigenstates. We found the possibility of an Anderson transition even in 1D meaning that the disorder can also provide constructive quantum interferences. We found at this transition that the conductance probability distribution has a system-size independent shape with large fluctuations in good agreement with previous works. In these 1D systems, the metallic phase is well characterized by a Gaussian conductance distribution. Indeed, the results for the conductance distribution are in good agreement with the previous works in 2D and 3D systems for other models.

Anderson transition in 1D systems with spatial disorder

Physica A-statistical Mechanics and Its Applications, 2010

A simple Kronig–Penney model for 1D mesoscopic systems with δδ peak potentials is used to study numerically the influence of spatial disorder on conductance fluctuations and distribution at different regimes. The Lévy laws are used to investigate the statistical properties of the eigenstates. It is found that an Anderson transition occurs even in 1D meaning that the disorder can also provide constructive quantum interferences. The critical disorder WcWc for this transition is estimated. In these 1D systems, the metallic phase is well characterized by a Gaussian conductance distribution. Indeed, the results relative to conductance distribution are in good agreement with the previous works in 2D and 3D systems for other models. At this transition, the conductance probability distribution has a system size independent shape with large fluctuations in good agreement with previous works.

Two-eigenfunction correlation in a multifractal metal and insulator

Physical Review B, 2007

We consider the correlation of two single-particle probability densities |ΨE(r)| 2 at coinciding points r as a function of the energy separation ω = |E−E ′ | for disordered tight-binding lattice models (the Anderson models) and certain random matrix ensembles. We focus on the parameter range close but not exactly at the Anderson localization transition. We show that even away from the critical point the eigenfunction statistics exhibit the remnant of multifractality characteristic of the critical states. This leads to an enhancement of eigenfunction correlations and a corresponding enhancement of matrix elements of the local electron interaction at small energy separations. This enhancement is accompanied by a depression of correlations at large energy separations, both phenomena being a consequence of the stratification of space into densely packed but mutually avoiding resonance clusters. We also demonstrate that the correlation function of localized states in a d-dimensional insulator is logarithmically enhanced at small energy separations provided that d > 1. A simple and general physical picture of all these phenomena is presented. Finally by a combination of numerical results on the Anderson model and analytical and numerical results for the relevant random matrix theories we identified the Gaussian random matrix ensembles that describe the multifractal features both in the metal and in the insulator phases.