大槻 東巳

A review of recent progress in numerical studies of the Anderson transition in three dimensional systems is presented. From high precision calculations the critical exponent nu for the divergence of the localization length is estimated to be nu = 1.57 +/- 0.02 for the orthogonal universality class, which is clearly distinguished from nu = 1.43 +/- 0.03 for the unitary universality class. The boundary condition dependences of some quantities at the Anderson transition are also discussed.

The nature of the critical point of the Anderson transition in high magnetic fields is discussed with an emphasis on scale invariance and universality of the critical exponent. Special attention is paid to the distribution function of the conductance which becomes size and model independent at the critical point. The fractal properties of the wave function which are related to scale invariance are also discussed.

The Anderson transitions in a random magnetic field in three dimensions are investigated in detail by means of the transfer-matrix method with high accuracy. Both systems with and systems without an additional random scalar potential are considered. We find the critical exponent nu for the localization length to be 1.45 +/- 0.09 with a strong random scalar potential. Without it, the exponent is smaller but increases with the system size and extrapolates to the above value within the error bars. These results support the conventional classification of universality classes according to symmetry. The mobility edge trajectory in the random magnetic field is also obtained.

Ballistic transport properties in a two dimensional electron gas are studied numerically, where magnetic fields are perpendicular to the plane of two dimensional electron systems and periodically modulated both in x and y directions. We show that there are three types of trajectories of classical electron motions in this system; chaotic, pinned and runaway trajectories. Tt is found that the runaway trajectories can explain the peaks of magnetoresistance as a function of external magnetic fields, which is believed to he related to the commensurability effect between the classical cyclotron diameter and the period of magnetic modulation. The similarity with and difference from the results in the antidot lattice are discussed.

The Anderson transition in three dimensions in a randomly varying magnetic flux is investigated in detail by means of the transfer-matrix method with high accuracy. Systems both with and without an additional random scalar potential are considered. We find a critical exponent of nu=1.45+/-0.09 with random scalar potential. Without it, nu is smaller bur increases with the system size and extrapolates within the error bars to a value close to the above. The present results support the conventional classification of universality classes due to symmetry. [S0163-1829(98)09219-4].

The shape analysis of the energy spacing distribution P(s) obtained from numerical simulation of two-dimensional disordered electron systems subject to strong magnetic fields is performed. In the present work we reanalyze the data obtained in a previous publication. Special moments of the P(s) function related to Renyi entropy differences show a novel scale invariant relation that is attributed to the presence of one-parameter scaling. This relation seems to show both deviations and universality depending on which Landau band is considered and whether the disorder is correlated or uncorrelated. Furthermore, our analysis shows the existence of an huge, however, irrelevant length scale in the case of the second lowest Landau band and no disorder correlations that completely disappear with the introduction of disorder correlations in the range of one magnetic length.

We report a finite size scaling study of the Anderson transition. Different scaling functions and different values for the critical exponent have been found, consistent with the existence of the orthogonal and unitary universality classes which occur in the field theory description of the transition. The critical conductance distribution at the Anderson transition has also been investigated and different distributions for the orthogonal and unitary classes obtained.

Diffusion of electrons in three-dimensional disordered systems is investigated numerically for all three universality classes, namely, orthogonal, unitary and symplectic ensembles. The second moment of the wave packet (r(2)(t)) at the Anderson transition is shown to behave as similar to t(a)(a approximate to 2/3). From the temporal autocorrelation function C(t), the fractal dimension D-2 is deduced, which is almost half the value of the space dimension for all of the universality classes.

Diffusion of electrons in a two-dimensional system with time-dependent random potentials is investigated numerically. The correction to the conductivity due to inelastic scatterings by oscillating potentials is shown to be a universal function of the frequency ω. which is consistent with the weak localization prediction[formula omitted] log ω. © 1997, THE PHYSICAL SOCIETY OF JAPAN. All rights reserved.

金属や半導体中の電子は不規則ポテンシャルにより散乱を受け, 散乱波どうしが干渉し電流が流れにくくなる. この現象は不規則性に起因しているにも拘わらず, 系の対称性に応じたユニバーサルな性質を示すことが知られている. 不規則ポテンシャルがさらに強くなると電流は全く流れなくなり, 系は絶縁体へと転移する. これがAnderson転移である. 最近, 数値計算によりこのAnderson転移点近傍での系のユニバーサルな振舞が明らかになろうとしている.

The Anderson transition in a 3D system with symplectic symmetry is investigated numerically. From a one-parameter scaling analysis the critical exponent nu of the localization length is extracted and estimated to be nu 1.3 +/- 0.2. The level statistics at the critical point are also analyzed and shown to be scale independent. The form of the energy level spacing distribution P(s) at the critical point is found to be different from that for the orthogonal ensemble, suggesting that the breaking of spin rotation symmetry is relevant at the critical point.

The scaling property of level statistics in the quantum Hall regime, i.e. 2D disordered electron systems subject to strong magnetic fields. is analyzed numerically in the light of the random matrix theory. The energy dependences of the effective level repulsion parameter, the two level correlation, the GUE-GOE crossover parameter, and the rigidity (or Delta(3)-statistics) of the level distributions are investigated for different system sizes by unfolding the original data and by dividing the unfolded spectrum into small regions. It is shown that the critical exponent of the localization length as a function of energy can be determined through the energy dependence of the level statistics. The analyses are carried out not only for the lowest Landau band (LB) but also for the second lowest LB. Furthermore the effect of finite range of disordered potential is studied. The short-ranged potential case in the second lowest LB is found to be pathological as in other studies of critical behavior, and it is confirmed that this pathological behavior is improved in the case of disordered potential with finite ranges.

Diffusion of electrons in two-dimensional disordered systems with spin-orbit interactions is investigated numerically. Asymptotic behaviors of the second moment of the wave packet and of the temporal autocorrelation function are examined. At the critical point, the autocorrelation function exhibits the power-law decay with a nonconventional exponent alpha, which is related to the fractal structure in the energy spectrum and in the wave functions. In the metallic regime, the present results imply that transport properties can be described by the diffusion equation for normal metals.

Diffusion of electrons in two-dimensional disordered systems with spin-orbit interactions is investigated numerically. Asymptotic behaviors of the second moment of the wave packet and of the temporal autocorrelation function are examined. At the critical point, the autocorrelation function exhibits the power-law decay with a nonconventional exponent α, which is related to the fractal structure in the energy spectrum and in the wave functions. In the metallic regime, the present results imply that transport properties can be described by the diffusion equation for normal metals. © 1996 The American Physical Society.

The level statistics in two dimensional disordered electron systems in magnetic fields (unitary ensemble) or in the presence of strong spin-orbit scattering (symplectic ensemble) are investigated at the Anderson transition points. The level spacing distribution functions P(s) are found to be independent of the system size or of the type of the potential distribution, suggesting universality. They behave as s(2) in the small s region in the former case, while increase proportional to s4 is seen in the latter.

Electronic states of a two-dimensional tight-binding lattice with finite size in the presence of both uniform electric and magnetic fields are studied. Numerical solutions for eigenenergies are presented. The magnetic subbands at zero electric field known as the Hofstadter butterfly are modified by the electric field and the eigenenergies for high electric fields are represented by the Stark ladder states associated with each of the magnetic subbands. When the electric potential drop across the system becomes comparable to the bandwidth of zero field, the density of states becomes the pyramid shape with steps, the steps being induced by the finiteness of the lattice. The influence on the density of states by the change of the direction of the electric field is also discussed.

The localization problem in a three dimensional tight binding system in the presence of a spatially random vector potential is investigated. The density of states exhibits tails of localized states that are induced by large regions in space where the vector potential is accidentally constant. In the center of the band the states are shown to be extended by investigating the system size dependence of the inverse participation number. The existence of a metal-insulator transition at a critical energy with non-vanishing density of states is demonstrated. The system size and energy dependent exponential decay length of the modulus of the Green's function satisfies a one-parameter scaling law. The value of the critical exponent is estimated, v almost-equal-to 1, which is different from those found previously for the disordered tight binding (Anderson) model with and without homogeneous magnetic field.

The conductance of a two-dimensional system subject to a random magnetic field is calculated within the Landauer-Buttiker formalism. Although there is no potential disorder the random fluctuating magnetic field introduces a random vector potential. Electrons in bulk states are found to be strongly scattered by the fluctuating field. In contrast, edge states are found to be quite stable even if the fluctuation of the field is of the order of a uniform background field. In a fully random field, the U(1) model, clear universal conductance fluctuations (UCF) are observed. The magnitude of the fluctuations is compared with that in the time reversal symmetric Z2 model.

The spacing distributions of unfolded energy levels in the lowest Landau band formed in disordered two-dimensional electron systems are numerically studied for different energy regions. By fitting the distribution function to an appropriate interpolation formula, the level repulsion parameter beta is obtained as a function of the unfolded energy x. The functional forms of beta(x) for different system sizes are found to exhibit a certain scaling behavior. From this scaling property, it is possible to estimate the exponent of the localization length as a function of energy, which is found to be 2.4 +/- 0.2, consistent with previously reported values. Possible applications to other Anderson transitions are also discussed.

Electronic states of a two-dimensional tight-binding model in a uniform electric field are studied. Numerical solutions for eigenenergies and eigenfunctions are presented as functions of the angle between the electric field and the symmetry axis of the lattice. When the direction of the electric field is [M, N], where M and N are mutually prime each other and MN not-equal 0, the eigenenergies are shown to be quantized with an interval equal to the potential drop between the nearest neighbor net lines. Though the level separation varies discontinuously with the change of the direction, the density of states is shown to be independent of the direction of the electric field, except for the direction [1 0] where the motions parallel and perpendicular to the electric field directions are separable. Unexpected gaps open near the band edges for appropriate electric fields, the magnitudes and the positions of which are smooth functions of the angle for a fixed electric field. The effects of the system edges are also discussed.

A stack of two-dimensional (2D) disordered layers in quantizing magnetic fields is considered, where the electrons are allowed to hop from one layer to another. By using numerical scaling analysis, the mobility edges as well as the critical exponent for the divergence of the localization length are obtained. It is shown that a small amount of hopping between the layers is sufficient to delocalize the states near the centre of the Landau band. The critical exponent is almost independent of the hopping energy and is equal to 1.3 +/- 0.2. The results are compared with those obtained for the 2D limit, the 3D Anderson model without a magnetic field and for a 3D system with a random magnetic field applied.

The energy spectra of a two-dimensional tight-binding model in the presence of both magnetic and electric fields are studied. When the electric potential through the system becomes comparable to the bandwidth of zero field, the step-like structure is seen in the density of states. The physical origin is discussed.

The Anderson transition in three dimensional layered systems is investigated in the presence of a quantizing magnetic field perpendicular to the layers. The density of states is calculated. By computing numerically the modulus of the one-body Green's function, the system size dependence of the exponential decay length of the transmission probability is calculated. It satisfies a one-parameter scaling law. The existence of a metal insulator transition is demonstrated. The mobility edge trajectory, and the critical parameters are obtained. The value of the critical exponent, nu = 1.35 +/- 0.15, is found to be close to that reported previously for the Anderson model without magnetic field.

Electronic states in the lowest Landau subband of a disordered double layered system in a quantizing magnetic field are investigated. The random potential is assumed to be short-ranged. When the transfer probability of electrons between the layers is small, the plateau of the filling factor nu = 1 seems to vanish, while it is clearly observed when the transfer is large. The energetic position of the extended states is also investigated using the finite size scaling method.

The density of states, and the mobility edge are calculated numerically for a new model of a three-dimensional disordered system in the presence of a quantizing magnetic field. The critical exponent of the disorder induced metal-insulator transition is found to be nu = 1.3 +/- 0.2 close to the value reported previously for the Anderson model without magnetic field.

Electronic states in a two dimensional random system with a perpendicular strong magnetic field are analyzed by using a model Hamiltonian matrix which has random elements reflecting the characteristics of the random potential such as the correlation length. The random matrix model is suitable for numerical calculations and has advantages compared to preparing the random potential by distributing impurities particularly when discussing correlated potential. As an example, the fractal dimensionality of extended states in the lowest Landau band is considered. It is found that the extended states have multifractal properties. The results are compared with those obtained through the random potential model.

It is argued that the extended states in two-dimensional electronic systems subject to a strong perpendicular magnetic field must not be "area-filling". Their fractal dimensionality d* is calculated from the inverse participation number for the ideal Landau model, and the Teller model. For disordered systems the fractal dimensionality is obtained in the limit of strong magnetic field from percolation theory. The result is d*=1.75. The quantum mechanical Teller model with disorder is treated numerically. © 1987.