The Anderson Transition in Two-Dimensional Systems with Spin-Orbit Coupling

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 analyse 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 $\nu=2.73 \pm 0.02$

Diffusion of electrons in two-dimensional disordered symplectic systems

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 auto-correlation function are examined. At the critical point, the auto-correlation function exhibits the power-law decay with a non-conventional 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.Comment: 4 pages RevTeX. Figures are available on request either via fax or e-mail. To be published in Phys. Rev.

Conductance fluctuations at the integer quantum Hall plateau transition

We study numerically conductance fluctuations near the integer quantum Hall effect plateau transition. The system is presumed to be in a mesoscopic regime, with phase coherence length comparable to the system size. We focus on a two-terminal conductance G for square samples, considering both periodic and open boundary conditions transverse to the current. At the plateau transition, G is broadly distributed, with a distribution function close to uniform on the interval between zero and one in units of e^2/h. Our results are consistent with a recent experiment by Cobden and Kogan on a mesoscopic quantum Hall effect sample.Comment: minor changes, 5 pages LaTex, 7 postscript figures included using epsf; to be published Phys. Rev. B 55 (1997

Critical Level Statistics in Two-dimensional Disordered Electron Systems

The level statistics in the 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 $P(s)$'s are found to be independent of the system size or of the type of the potential distribution, suggesting the universality. They behave as $s^2$ in the small $s$ region in the former case, while $s^4$ rise is seen in the latter.Comment: LaTeX, to be published in J. Phys. Soc. Jpn. (Letter) Nov., Figures will be sent on reques

Disordered Electrons in a Strong Magnetic Field: Transfer Matrix Approaches to the Statistics of the Local Density of States

We present two novel approaches to establish the local density of states as an order parameter field for the Anderson transition problem. We first demonstrate for 2D quantum Hall systems the validity of conformal scaling relations which are characteristic of order parameter fields. Second we show the equivalence between the critical statistics of eigenvectors of the Hamiltonian and of the transfer matrix, respectively. Based on this equivalence we obtain the order parameter exponent $\alpha_0\approx 3.4$ for 3D quantum Hall systems.Comment: 4 pages, 3 Postscript figures, corrected scale in Fig.

A topological characterization of delocalization in a spin-orbit coupling system

We show that wavefunctions in a two-dimensional (2D) electron system with spin-orbit coupling can be characterized by a topological quantity--the Chern integer due to the existence of the intrinsic Kramers degeneracy. The localization-delocalization transition in such a system is studied in terms of such a Chern number description, which reproduces the known metal-insulator transition point. The present work suggests a unified picture for various known 2D delocalization phenomena based on the same topological characterization.Comment: RevTex, 12 pages; Two PostScript figure

Statistics of pre-localized states in disordered conductors

The distribution function of local amplitudes of single-particle states in disordered conductors is calculated on the basis of the supersymmetric $\sigma$-model approach using a saddle-point solution of its reduced version. Although the distribution of relatively small amplitudes can be approximated by the universal Porter-Thomas formulae known from the random matrix theory, the statistics of large amplitudes is strongly modified by localization effects. In particular, we find a multifractal behavior of eigenstates in 2D conductors which follows from the non-integer power-law scaling for the inverse participation numbers (IPN) with the size of the system. This result is valid for all fundamental symmetry classes (unitary, orthogonal and symplectic). The multifractality is due to the existence of pre-localized states which are characterized by power-law envelopes of wave functions, $|\psi_t(r)|^2\propto r^{-2\mu}$, $\mu <1$. The pre-localized states in short quasi-1D wires have the power-law tails $|\psi (x)|^2\propto x^{-2}$, too, although their IPN's indicate no fractal behavior. The distribution function of the largest-amplitude fluctuations of wave functions in 2D and 3D conductors has logarithmically-normal asymptotics.Comment: RevTex, 17 twocolumn pages; revised version (several misprint corrected

Point-Contact Conductances at the Quantum Hall Transition

On the basis of the Chalker-Coddington network model, a numerical and analytical study is made of the statistics of point-contact conductances for systems in the integer quantum Hall regime. In the Hall plateau region the point-contact conductances reflect strong localization of the electrons, while near the plateau transition they exhibit strong mesoscopic fluctuations. By mapping the network model on a supersymmetric vertex model with GL(2|2) symmetry, and postulating a two-point correlator in keeping with the rules of conformal field theory, we derive an explicit expression for the distribution of conductances at criticality. There is only one free parameter, the power law exponent of the typical conductance. Its value is computed numerically to be X_t = 0.640 +/- 0.009. The predicted conductance distribution agrees well with the numerical data. For large distances between the two contacts, the distribution can be described by a multifractal spectrum solely determined by X_t. Our results demonstrate that multifractality can show up in appropriate transport experiments.Comment: 18 pages, 15 figures included, revised versio

Scaling Theory of the Integer Quantum Hall Effect

The scaling theory of the transitions between plateaus of the Hall conductivity in the integer Quantum Hall effect is reviewed. In the model of two-dimensional noninteracting electrons in strong magnetic fields the transitions are disorder-induced localization-delocalization transitions. While experimental and analytical approaches are surveyed, the main emphasis is on numerical studies, which successfully describe the experiments. The theoretical models for disordered systems are described in detail. An overview of the finite-size scaling theory and its relation to Anderson localization is given. The field-theoretical approach to the localization problem is outlined. Numerical methods for the calculation of scaling quantities, in particular the localization length, are detailed. The properties of local observables at the localization-delocalization transition are discussed in terms of multifractal measures. Finally, the results of extensive numerical investigations are compared with experimental findings.Comment: 96 pages, REVTeX 3, 28 figures, Figs. 8-24, 26-28 appended as uuencoded compressed tarred PostScript files. Submitted to Rev. Mod. Phys

Between history and values: A study on the nature of interpretation in international law

My thesis discusses the place of evaluative judgements in the interpretation of general international law. It concentrates on two questions. First, whether it is possible to interpret international legal practices without making an evaluative judgement about the point or value that provides the best justification of these practices. Second, whether the use of evaluative judgements in international legal interpretation threatens to undermine the objectivity of international law, the neutrality of international lawyers or the consensual and voluntary basis of the international legal system. I answer both questions in the negative. As regards the first, I argue that international legal practice has an interpretive structure, which combines appeals to the history of international practice with appeals to the principles and values that these practices are best understood as promoting. This interpretive structure is apparent not only in the claims of international lawyers about particular rules of international law (here I use the rule of estoppel as an example) but also in the most basic intuitions of international theorists about the theory and sources of general international law. I then argue that some popular concerns to the effect that the exercise of evaluation in the interpretation of international law will undermine the coherence or the usefulness of the discipline are generally unwarranted. The fact that international legal practice has an interpretive structure does not entail that propositions of international law are only subjectively true, that the interpreter enjoys license to manipulate their meaning for self-serving purposes, or that international law will collapse under the weight of irresolvable disagreements, divisions and conflicts about its proper interpretation