The scaling theory of the transitions between plateaus of the Hall conductivity in the integer Quantum Hall effect is reviewed. In the model of twodimensional 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 finitesize 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 localizationdelocalization transition are discussed in terms of multifractal measures. Finally, the results of extensive numerical investigations are compared with experimental findings.

The localization properties of noninteracting electrons in the lowest Landau band of a disordered twodimensional system are investigated using a new version of the numerical finite-size scaling method which includes a quantitative statistical procedure of data evaluation. In contrast to other suggestions, universal one-parameter scaling behavior is obtained. The localization length diverges at the center of the band with a critical exponent v='2.34±0.04. The critical region is determined to be about half of the bandwidth. Comparison with experimental quantum-Hall-effect data yields the temperature exponent of the phase-breaking time /? »*2.0 ± 0.2. PACS numbers: 71.50.+t, 71.30.+h, 71.55.Jv The one-parameter scaling theory of localization'"^ indicates that in two dimensions all states are localized in the absence of a magnetic field. On the other hand, the quantum Hall effect requires states which can carry a current"^ and which must, therefore, extend throughout the entire system.Ono was the first to suggest the presence of singularities in the localization length ^ near the centers of the Landau subbands £'".^ Extremely large localization lengths in the band centers were also indicated by numerical calculations.^'^ The nature of the singularity was studied by Aoki and Ando^ by adapting the scaling method proposed earlier by MacKinnon and Kramer^ to the disordered Landau model. They found that ^ oc lE-En I ~\ with v<2 and v<4 for AZ=0 and 1, respectively. Hikami^ estimated v= 1.9 ± 0.2 by perturbational calculation of the inverse participation ratio for the lowest Landau band.Ando and Aoki interpreted their data as showing a magnetic-field-induced destruction of the one-parameter scaling behavior of the localization length. A breakdown of one-parameter scaling of the conductance and a singular behavior in the centers of the Landau bands was also proposed by Levine, Libby, and Pruisken'^ using field theory, but no critical exponent was obtained.For slowly varying random potentials, the quantummechanical problem is equivalent to a classical percolation problem.'' While the percolation critical exponent is j,'^ the inclusion of quantum tunneling leads to v= J. '^ A numerical study, taking into account quantum tunneling and interference near the percolation threshold, yielded v = 2.5 ±0.5 and one-parameter scaling,'"* in contrast to the results mentioned previously.One-parameter scaling of transport is also consistent with recent experimental investigations of the temperature behavior of the magnetoresistivity pxx and the Hall resistivity pxy near the centers of different Landau bands.'^ The maxima of dpxy/dB, d^pxx/dB^, and d^pxy/dB^ diverge as T~"\ with K:=p/2v=0.42 and for /t = l, 2, and 3, respectively, which is consistent withwhere L-.^CCT'P^'^ is the phase coherence length.This paper resolves the discrepancies in the scaling behavior discussed in the various approaches mentioned above. Results of a numerical finite-size scaling investigation of the normalized exponential decay length XM/M in the lowest Land...

We study the time evolution of wavepackets of non-interacting electrons in a two-dimensional disordered system in strong magnetic field. For wavepackets built from states near the metal-insulator transition in the center of the lowest Landau band we find that the return probability to the origin p(t) decays algebraically, p(t) ∼ t −D 2 /2 , with a non-conventional exponent D 2 /2. D 2 is the generalized dimension describing the scaling of the second moment of the wavefunction. We show that the corresponding spectral measure is multifractal and that the exponent D 2 /2 equals the generalized dimension D 2 of the spectral measure. 71.50.+t,71.30.+h,71.55.Jv Typeset using REVT E X 1

We study transport through a two-dimensional billiard attached to two infinite leads by numerically calculating the Landauer conductance and the Wigner time delay. In the generic case of a mixed phase space we find a power law distribution of resonance widths and a power law dependence of conductance increments apparently reflecting the classical dwell time exponent, in striking difference to the case of a fully chaotic phase space. Surprisingly, these power laws appear on energy scales below the mean level spacing, in contrast to semiclassical expectations.PACS numbers: 05.45.+b,05.60.Gg,72.20.Dp,73.23.Ad Advances in the fabrication of semiconductor heterostructures and metal films have made it possible to produce two dimensional nanostructures with a very low amount of disorder [1]. At low temperatures, scattering of the electrons happens mostly at edges of the structures with the electrons moving ballistically between collisions with the boundary. Theoretical and experimental investigations have shown that the spectral and transport properties of such quantum coherent cavities, commonly called "billiards", depend strongly on the nature of their classical dynamics. In particular, integrable and chaotic systems were found to behave quite differently [2,3].Generic billiards are neither integrable nor ergodic [4], but have a mixed phase space with regions of regular as well as chaotic dynamics [5]. Their dynamics is much richer than in either of the extreme cases, as phase space has a hierarchical structure at the boundary of regular and chaotic motion. In particular, this leads to a trapping of chaotic trajectories close to regular regions with a probability P (t) ∼ t −β for t > t 0 , to be trapped longer than a time t, with t 0 of the order of a few traversal times [6]. The exponent β > 1 depends on system and parameters with typically β ≈ 1.5 [6]. This power-law trapping in mixed systems is in contrast to the typical exponentially decaying staying probability of fully chaotic systems (see Fig. 1).Recently, it was shown semiclassically employing the diagonal approximation that the variance of conductance increments (for a small dc bias voltage) over small energy intervals ∆E grows as [7,8] for mixed systems if β < 2. This is in strong contrast to an increase as (∆E) 2 in the case of fully chaotic systems [3]. The semiclassical approximation requires ∆E to be larger than the mean level spacing ∆, corresponding to the picture that quantum mechanics can follow the classical power law trapping at most until the Heisenberg time. In the semiclassical approximation the graph of g vs. E has the statistical properties of fractional Brownian motion with a fractal dimension. Fractal conductance fluctuations have indeed been found in experiments on gold wires [10] and semiconductor nanostructures [11] and numerically for the quantum separatrix map [12]. In this Letter, we numerically study quantum transport through a simple cavity, the cosine billiard [13] (see insets of Fig. 1). Although we observe completely different...

The scaling theory of the transitions between plateaus of the Hall conductivity in the integer Quantum Hall effect is reviewed. In the model of twodimensional 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 finitesize 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 localizationdelocalization transition are discussed in terms of multifractal measures. Finally, the results of extensive numerical investigations are compared with experimental findings.