Critical Wave Functions Research Articles

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 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 $< (\delta N)^{2} > = \chi <N>$ for $<N> >> 1$. The compressibility, $\chi=\eta/2d$, is given ``exactly'' in terms of the multifractal exponent $\eta=d-D_2$ at the mobility edge in a $d$-dimensional system.

Distribution of level curvatures for the Anderson model at the localization-delocalization transition.

We compute the distribution function of single-level curvatures, $P(k)$, for a tight binding model with site disorder, on a cubic lattice. In metals $P(k)$ is very close to the predictions of the random-matrix theory (RMT). In insulators $P(k)$ has a logarithmically-normal form. At the Anderson localization-delocalization transition $P(k)$ fits very well the proposed novel distribution $P(k)\propto (1+k^{\mu})^{3/\mu}$ with $\mu \approx 1.58$, which approaches the RMT result for large $k$ and is non-analytical at small $k$. We ascribe such a non-analiticity to the spatial multifractality of the critical wave functions.

Physical nature of critical wave functions in Fibonacci systems.

We report on a new class of critical states in the energy spectrum of general Fibonacci systems. By introducing a transfer matrix renormalization technique, we prove that the charge distribution of these states spreads over the whole system, showing transport properties characteristic of electronic extended states. Our analytical method is a first step to find out the link between the spatial structure of these critical wave functions and the quasiperiodic order of the underlying lattice.

Energy Spectrum and the Critical Wavefunctions of the Quasiperiodic Harper Equation –the Silver Mean Case–

The one-dimentional quasiperiodic tight-binding model is -ψ_n+1-ψ_n-1+λV(nω)ψ_n=Eψ_n, where ω is an irrational number and V is a periodic function, i.e., V ( x +1)= V ( x ). In the Harper model V ( n ω)=cos (2π n ω), all the states are critical at the critical coupling λ c =2. The critical properties of the spectrum and the wavefunctions for \(\omega=\sqrt{2}-1\) (the inverse of the silver mean) are numerically studied by the scaling and multifractal methods. The results are compared with the golden mean case \(\omega=(\sqrt{5}-1)/2\). Some universal properties are proposed.

14a-H-12 磁場中次近接hoppingを持つ正方格子tight-binding modelの波動関数の臨界性

14a-H-12 磁場中次近接hoppingを持つ正方格子tight-binding modelの波動関数の臨界性

ELECTRONIC SPECTRAL AND WAVEFUNCTION PROPERTIES OF ONE-DIMENSIONAL QUASIPERIODIC SYSTEMS: A SCALING APPROACH

We review the results of the scaling and multifractal analyses for the spectra and wave-functions of the finite-difference Schrödinger equation: [Formula: see text] Here V is a function of period 1 and ω is irrational. For the Fibonacci model, V takes only two values (it is constant except for discontinuities) and the spectrum is purely singular continuous (critical wavefunctions). When V is a smooth function, the spectrum is purely absolutely continuous (extended wavefunctions) for λ small and purely dense point (localized wavefunctions) for λ large. For an intermediate λ, the spectrum is a mixture of absolutely continuous parts and dense point parts which are separated by a finite number of mobility edges. There is no singular continuous part. (An exception is the Harper model V (x) = cos (2πx), where the spectrum is always pure and the singular continuous one appears at λ = 2.)

Spectral density correlations and eigenfunction fluctuations in one-dimensional quasi-periodic systems

The authors study the scaling and correlation-fluctuation properties for the spectra and wavefunctions of a simple one-dimensional quasi-periodic system that displays an Anderson metal-insulator transition. This system and its extensions are models for studying localization phenomena, and their usefulness for the description of the Anderson transition as well as insulating and metallic phases in disordered systems is exploited. They present numerical work on the critical behaviour of the spectrum and wavefunctions, which display multifractal fluctuations. Appropriate probability densities are studied, and it is demonstrated that: (i) the subband energy width statistics, which may express spectral correlations, is consistent with linear level repulsion at the critical point and a Poisson distribution in the insulating regime: and (ii) the critical wavefunction probability amplitude distributions approach a universal function as the system size increases. It is concluded that certain critical properties of quasi-periodic models are similar to what is expected for electronic states in weakly disordered metals, and others show a striking similarity to mobility edge behaviour in three dimensions.

Multifractal analysis of wave functions in generalized Fibonacci-chains

The f(α) singularity spectrum of wave functions in a 1d generalized. Fibonacci tight binding electron model is investigated. Localized, extended and critical wave functions are found in contrast to the pure Fibonacci chain.

Critical exponents for Anderson localization

We perform numerical calculations on a simple cubic lattice for a standard diagonally disordered tight-binding Hamiltonian, whose random site energies are chosen from a Gaussian distribution with variance ∑2. From phenomenological renormalization group studies of the localization length, we determine that the critical disorder is σc≡∑c/J=6.00±0.17, which is in good agreement with previous results (J is the nearest neighbor transfer matrix element). From our calculations we can also determine the mobility edge trajectory, which appears to be analytic at the band center. Defining an order parameter exponent β, which determines how the fraction of extended states vanishes as the critical point is approached from below, this implies that β=1/2, in agreement with a previous study. From a finite-size scaling analysis we find that π2/ν=1.43±0.10, where π2 and ν are the inverse participation ratio and localization length critical exponents, respectively. This ratio of exponents can also be interpreted as the fractal dimension (also called the correlation dimension) D2 of the critical wave functions. Generalizations of the inverse participation ratio lead to a whole set of critical exponents πk, and corresponding generalized fractal dimensions Dk=πk/ν(k−1). From finite-size scaling results we find that D3=1.08±0.10 and D4=0.87±0.09. The inequality of the three dimensions D2, D3, and D4 shows that the critical wave functions have a multifractal structure. Using a generalized phenomenological renormalization technique on the participation ratios, we find that ν=0.99±0.04. This result is in agreement with experiments on compensated or amorphous doped semiconductors.

Multifractal, Localized, and Extended Wave Functions in Generalized Fibonacci Chains

AbstractThe ƒ(α) singularity spectrum of wave functions in a 1d generalized Fibonacci tight binding electron model are investigated. Localized, extended, and critical wave functions are found in contrast to the pure Fibonacci chain.

The Effect of an Interelectron Interaction on Electronic Properties in One-Dimensional Quasiperiodic Systems

The effect of an interelectron interaction ( U ) on electronic properties in one-dimensional quasiperiodic systems (the incommensurate Harper model and the Fibonacci model) is studied within the Hartree-Fock (HF) approximation. For the Fibonacci model, the HF one-body spectrum remains singular continuous (critical wave functions) for small | U |. For the Harper model, on the other hand, the singular continuous spectrum at critical coupling (λ c =2) disappears as soon as U is added.

Multifractal wave functions on a Fibonacci lattice.

Wave functions on a Fibonacci lattice are analyzed from the multifractal point of view. An entropy function [f(\ensuremath{\alpha})] which represents the distribution of a particle probability density is obtained exactly for the state at the center of the spectrum. Numerical calculations for other states are also presented. A finite-size scaling analysis shows that the ordered critical wave functions are multifractals.

Correlations in multifractals

We present an analysis of the implicit assumptions that lie at the basis of the scaling relations for fractal measures proposed by CATES et. al. [1]. Furthermore,the scaling relations which have been shown to hold for trivial multifractals like the generalized Cantor set [2] are tested numerically for critical wave functions of an incommensurate system.

Critical electronic wave functions on quasiperiodic lattices: Exact calculation of fractal measures.

Critical electronic wave functions on quasiperiodic lattices: Exact calculation of fractal measures.

Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model.

The electronic properties of a tight-binding model which possesses two types of hopping matrix element (or on-site energy) arranged in a Fibonacci sequence are studied. The wave functions are either self-similar (fractal) or chaotic and show ``critical'' (or ``exotic'') behavior. Scaling analysis for the self-similar wave functions at the center of the band and also at the edge of the band is performed. The energy spectrum is a Cantor set with zero Lebesque measure. The density of states is singularly concentrated with an index ${\ensuremath{\alpha}}_{E}$ which takes a value in the range [${\ensuremath{\alpha}}_{E}^{\mathrm{min}}$,${\ensuremath{\alpha}}_{E}^{\mathrm{max}}$]. The fractal dimensions f(${\ensuremath{\alpha}}_{E}$) of these singularities in the Cantor set are calculated. This function f(${\ensuremath{\alpha}}_{E}$) represents the global scaling properties of the Cantor-set spectrum.