大槻 東巳

An overview of the random network model invented by Chalker and Coddington, and its generalizations, is provided. After a short introduction into the physics of the Integer Quantum Hall Effect, which historically has been the motivation for introducing the network model, the percolation model for electrons in spatial dimension 2 in a strong perpendicular magnetic field and a spatially correlated random potential is described. Based on this, the network model is established, using the concepts of percolating probability amplitude and tunneling. Its localization properties and its behavior at the critical point are discussed including a short survey on the statistics of energy levels and wave function amplitudes. Magneto-tran sport is reviewed with emphasis on some new results on conductance distributions. Generalizations are performed by establishing, equivalent Hamiltonians. In particular, the significance of mappings to the Dirac model and the two-dimensional Ising model is discussed. A description of renormalization group treatments is given. The classification of two-dimensional random systems according to their symmetries is outlined. This provides access to the complete set of quantum phase transitions like the thermal Hall transition and the spin quantum Hall transition in two dimensions.The supersymmetric effective field theory for the critical properties of network models is formulated. The network model is extended to higher dimensions including remarks on the chiral metal phase at the surface of a multi-layer quantum Hall system. (c) 2005 Published by Elsevier B.V.

We investigate numerically the spin polarization of the current in the presence of Rashba spin-orbit interaction in a T-shaped conductor proposed by Kiselev and Kim [Appl. Phys. Lett. 78, 775 (2001)]. The recursive Green function method is used to calculate the three terminal spin-dependent transmission probabilities. We focus on single-channel transport and show that the spin polarization becomes nearly 100% with a conductance close to e(2)/h for sufficiently strong spin-orbit coupling. This is interpreted by the fact that electrons with opposite spin states are deflected into an opposite terminal by the spin-dependent Lorentz force. The influence of the disorder on the predicted effect is also discussed. Cases for multichannel transport are studied in connection with experiments.

A spin filtering in a two-dimensional electron system with nonuniform spin-orbit interactions (SOI) is theoretically studied. The strength of SOI is modulated perpendicular to the charge current. A spatial gradient of effective magnetic field due to the nonuniform SOI causes the Stern-Gerlach-type spin separation. The direction of the polarization is perpendicular to the current and parallel to the spatial gradient. Almost 100% spin polarization can be realized even without applying any external magnetic fields and without attaching ferromagnetic contacts. The spin polarization persists even in the presence of randomness.

The quantum phase diagram of disordered wires in a strong magnetic field is studied as a function of wire width and energy. The two-terminal conductance shows zero-temperature discontinuous transitions between exactly integer plateau values and zero. In the vicinity of this transition, the chiral metal-insulator transition (CMIT), states are identified that are superpositions of edge states with opposite chirality. The bulk contribution of such states is found to decrease with increasing wire width. Based on exact diagonalization results for the eigenstates and their participation ratios, we conclude that these states are characteristic for the CMIT, have the appearance of nonchiral edges states, and are thereby distinguishable from other states in the quantum Hall wire, namely, extended edge states, two-dimensionally (2D) localized, quasi-1D localized, and 2D critical states.

Spin polarized current in a three-terminal system in the presence of spin-orbit interaction is numerically studied. The third invasive voltage probe is attached. Our results show that the invasive voltage probe induces not only the dephasing effect but also the polarization of the current. By changing the height of the potential barrier located at the interface between the system and the invasive voltage probe, the polarization can be controlled. The polarization survives even in the presence of impurities.

Quantum transport in disordered magnetic fields is investigated numerically in two-dimensional systems. In particular, the case where the mean and the fluctuation of disordered magnetic fields are of the same order is considered. It is found that in the limit of weak disorder the conductivity exhibits a qualitatively different behavior from that in the conventional random magnetic fields with zero mean. The conductivity is estimated by the equation of motion method and by the two-terminal Landauer formula. It is demonstrated that the conductance stays on the order of e(2)/h even in the weak disorder limit. The present behavior can be interpreted in terms of the Drude formula. The Shubnikov-de Haas oscillation is also observed in the weak disorder regime.

Quantum transport in disordered magnetic fields is investigated numerically in two-dimensional systems. In particular, the case where the mean and the fluctuation of disordered magnetic fields are of the same order is considered. It is found that in the limit of weak disorder the conductivity exhibits a qualitatively different behavior from that in the conventional random magnetic fields with zero mean. The conductivity is estimated by the equation of motion method and by the two-terminal Landauer formula. It is demonstrated that the conductance stays on the order of e 2/h even in the weak disorder limit. The present behavior can be interpreted in terms of the Drude formula. The Shubnikov-de Haas oscillation is also observed in the weak disorder regime. ©2005 The American Physical Society.

We study the Anderson transition in the SU(2) model and the Ando model. We report a new precise estimate of the critical exponent for the symplectic universality class of the Anderson transition. We also report numerical estimation of the beta function.

We report a numerical study of Anderson localization in a two-dimensional (2D) system of noninteracting electrons with spin-orbit coupling. We analyze the scaling of the renormalized localization length for the 2D SU(2) model and estimate its β function over the full range from the localized to the metallic limits.

We report a numerical study of Anderson localization in a two-dimensional (2D) system of noninteracting electrons with spin-orbit coupling. We analyze the scaling of the renormalized localization length for the 2D SU(2) model and estimate its beta function over the full range from the localized to the metallic limits.

We investigate numerically the spin polarization of the current in the presence of spin-orbit interaction with three terminal geometry proposed by Kiselev and Kim (Appl. Phys. Lett. 78 (2001) 775). We obtained high spin polarization at the energy just before a new channel opens. To obtain higher polarization, we then consider the geometry which consists of the combination of two T-shaped blocks. This two blocks configuration results in almost 100% polarization. We also investigate the effect of disorder by including a random potential and taking ensemble average. This not only makes the reduction of the spin polarization but also shifts the peak of the polarization towards the lower energy regime. © Published by Elsevier B.V.

The conductance distribution at the Quantum Hall localization-delocalization transition and the Anderson transition in the two-dimensional SU(2) model are determined numerically. In the quantum Hall case, the distribution is found to be well described by the largest transmission eigenvalue tau(max), whose distribution we relate to a Poisson distribution. This is in contrast to the distribution of the critical conductance for the SU(2) model. (C) 2003 Elsevier B.V. All rights reserved.

Sharp localization transitions of chiral edge states in disordered quantum wires subject to a strong magnetic field are shown to be driven by crossovers from two- to one-dimensional localization of bulk states. As a result, the two-terminal conductance is found to exhibit discontinuous transitions at zero temperature between exactly integer plateau values and zero, reminiscent of first-order phase transitions. We discuss the corresponding phase diagram. The spin of the electrons is shown to result in a multitude of phases when the spin degeneracy is raised by the Zeeman energy. The width of conductance plateaus is found to depend sensitively on the spin flip rate 1/tau(s). (C) 2004 MAIK "Nauka / Interperiodica".

We report on the complete scaling analysis of low temperature electron transport properties with and without magnetic field in the critical regime for the metal-insulator transition in two series of homogeneously doped p-type Ge samples: i) nominally uncompensated neutron-transmutation-doped (NTD) Ge-70:Ga samples with the technological compensation ratio K < 0.001, and ii) intentionally compensated NTD Ge-nat:Ga,As samples with K = 0.32. For the case of the uncompensated series in zero magnetic field, the critical exponents mu, v, and zeta determined for the electrical conductivity (sigma), localization length (xi), and impurity dielectric susceptability (chi(imp)), respectively, change at the very vicinity of the critical Ga concentration (N similar to N-c) Namely, the anomalous critical exponents, e.g. mu approximate to 0.5, change to mu approximate to 1 only within the region 0.99N(c) < N < 1.01N(c). On the other hand, the same critical behavior, mu approximate to 1, was found for the K = 0.32 series in much larger region 0.25N(c) < N < 2.4N(c) This finding suggests that the it mu approximate to 1 critical behavior observed for the nominally uncompensated series in the extremely narrow region is due to the presence of the self-compensation of acceptors by native defects and/or technologically unavoidable very small amount of doping compensation (K < 0.001). Therefore, the width of the concentration that can be fitted with mu approximate to 1 around N-c is likely to scale with the degree of compensation (K), and disappears in the limit K --> 0, i.e., only the region with the anomalous exponent mu approximate to 0.5 remains for the case of K = 0. An externally applied magnetic field to nominally uncompensated samples also broadens the width of mu approximate to 1 around N-c, but with a mechanism clearly different from that of compensation. The unified description of our experimental results unambiguously establishes the values of the critical exponents mu, v, zeta and for doped semiconductors with and without compensation and magnetic field.

The localization properties of electron states in the quantum Hall regime are reviewed. The random Landau model, the random matrix model, the tight-binding Peierls model, and the network model of Chalker and Coddington are introduced. Descriptions in terms of equivalent tight-binding Hamiltonians, and the 2D Dirac model, are outlined. Evidences for the universal critical behavior of the localization length are summarized. A short review of the supersymmetric critical field theory is provided. The interplay between edge states and bulk localization properties is investigated. For a system with finite width and with short-range randomness, a sudden breakdown of the two-point conductance from ne(2)/h to 0 (n integer) is predicted if the localization length exceeds the distance between the edges. (C) 2003 Published by Elsevier B.V.

Electron transport through disordered systems that include spin scatterers is studied numerically. We consider three kinds of magnetic impurities: the Ising, the XY, and the Heisenberg. By extending the transfer matrix method to include the spin degree of freedom, the two terminal conductance is calculated. The variance of conductance is halved as the number of spin components of the magnetic impurities increases. Application of the Zeeman field in the lead causes a further halving of the variance under certain conditions. © 2003 The American Physical Society.

We report an analysis of the Anderson transition in an SU(2) model with chiral symmetry. Clear single-parameter scaling behaviour is observed. We estimate the critical exponent for the divergence of the localization length to be ν = 2.72 ± 0.02 indicating that the transition belongs to the symplectic universality class. © 2003 Elsevier Science B.V. All rights reserved.

The conductance distribution based on scaling hypothesis was investigated near the Anderson transition region of mesoscopic semiconductors. The distribution function showed single parameter scaling in 3D orthogonal systems. The single parameter scaling reconciled the phenomenon of universal conductance fluctuation with the scaling theory of localization.

Quantum transport in inhomogeneous magnetic fields is investigated numerically in two-dimensional systems using the equation of motion method. In particular, the diffusion of electrons in random magnetic fields in the presence of additional weak uniform magnetic fields is examined. It is found that the conductivity is strongly suppressed by the additional uniform magnetic field and saturates when the uniform magnetic field becomes on the order of the fluctuation of the random magnetic field. The value of the conductivity at this saturation is found to be insensitive to the magnitude of the fluctuation of the random field. The effect of random potential on the magnetoconductance is also discussed. © 2003 The American Physical Society.

The scaling hypothesis is the foundation of our understanding of the Anderson transition. We present a direct numerical demonstration of the scaling of the conductance distribution of a disordered system in the critical regime. This complements a previous demonstration of the scaling of certain averages of the conductance distribution [K. Slevin et al., Phys. Rev. Lett. (formula presented) 3594 (2001)]. © 2003 The American Physical Society.

Quantum transport in inhomogeneous magnetic fields is investigated numerically in two-dimensional systems using the equation of motion method. In particular, the diffusion of electrons in random magnetic fields in the presence of additional weak uniform magnetic fields is examined. It is found that the conductivity is strongly suppressed by the additional uniform magnetic field and saturates when the uniform magnetic field becomes on the order of the fluctuation of the random magnetic field. The value of the conductivity at this saturation is found to be insensitive to the magnitude of the fluctuation of the random field. The effect of random potential on the magnetoconductance is also discussed.

Electron transport in a two-dimensional symplectic system with time-dependent perturbations is investigated numerically by the equation of motion method. The effect of time-dependent perturbations on the conductivity is examined in the critical regime as well as in the metallic regime. The universal correction to the conductivity, which is consistent with the weak localization theory, is indeed observed in the metallic regime. In the critical regime, it is found that the dependence of the conductivity on the frequency of perturbations can be described by the one-parameter scaling.

We have studied quantum transports in two dimensional electron gas (2DEG) numerically, where magnetic fields with sine-type modulation are applied perpendicular to the plane of 2DEG. Due to the non-homogeneous magnetic fields, the system becomes chaotic. We use tight-binding model and investigate the energy level statistics and the two terminal conductance. Depending on the spatial symmetry of the magnetic fields, the system belongs to the unitary class or the orthogonal, which is reflected by the energy level statistics. Attaching leads further destroys the symmetry, and the universal conductance fluctuation for the unitary class is observed. The fractal behavior of the magnetoconductance is also observed.

We report a numerical investigation of the Anderson transition in two-dimensional systems with spin-orbit coupling. An accurate estimate of the critical exponent nu for the divergence of the localization length in this universality class has to our knowledge not been reported in the literature. Here we analyze the SU(2) model. We find that for this model corrections to scaling due to irrelevant scaling variables may be neglected permitting an accurate estimate of the exponent nu=2.73+/-0.02.

We reconcile the phenomenon of mesoscopic conductance fluctuations with the single parameter scaling theory of the Anderson transition. We calculate three averages of the conductance distribution, exp(< lng >), <g >, and 1/<R >, where g is the conductance in units of e(2)/h and R = 1/g is the resistance, and demonstrate that these quantities obey single parameter scaling laws. We obtain consistent estimates of the critical exponent from the scaling of ail these quantities.

We analyze the scaling behavior of the higher Lyapunov exponents at the Anderson transition. We estimate the critical exponent and verify its universality and that of the critical conductance distribution for box, Gaussian and Lorentzian distributions of the random potential. © 2001 The American Physical Society.

We analyze the scaling behavior of the higher Lyapunov exponents at the Anderson transition. We estimate the critical exponent and verify its universality and that of the critical conductance distribution for box, Gaussian and Lorentzian distributions of the random potential.

The effect of boundary conditions on the critical level statistics has been discussed recently. To clarify the effect of boundary conditions at the Anderson transition, we have performed a numerical finite-size scaling analysis with Dirichlet boundary conditions. By taking into account corrections to scaling due to the surface effect, we have succeeded in extracting estimates for the critical points and the critical exponents consistent with those obtained with periodic boundary conditions. The implication for the boundary condition dependence of critical level statistics and for critical conductance distributions is also discussed. (C) 2000 Elsevier Science B.V. All rights reserved.

The boundary condition dependence of the critical behavior for the three dimensional Anderson transition is investigated. A strong dependence of the scaling function and the critical conductance distribution on the boundary conditions is found, while the critical disorder and critical exponent are found to be independent of the boundary conditions.

Three-dimensional quantum percolation problems are studied by analyzing energy level statistics of electrons on maximally connected percolating clusters. The quantum percolation threshold p(q), which is larger than the classical percolation threshold p(c), becomes smaller when magnetic fields are applied, i.e., p(q)(B = 0) > p(q)(B not equal 0) > p(c). The critical exponents are found to be consistent. with the recently obtained values of the Anderson model, supporting the conjecture that the quantum percolation is classified onto the same universality classes of the Anderson transition. Novel critical level statistics at the percolation threshold is also reported.

We report a numerical analysis of corrections to finite size scaling at the Anderson transition due to irrelevant scaling variables and nonlinearities of the scaling variables. By taking proper account of these corrections, the universality of the critical exponent for the orthogonal universality class for three different distributions of the random potential is convincingly demonstrated.

The Anderson transitions in a random magnetic field in three dimensions are investigated numerically. The critical behavior near the transition point is analyzed in detail by means of the transfer matrix method with high accuracy for systems both with and without an additional random scalar potential. We find the critical exponent v 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 sizes and extrapolates to the above value within the error bars. These results support the conventional classification of universality classes due to symmetry. Fractal dimensionality of the wave function at the critical point is also estimated by the equation-of-motion method.