Integer Quantum Hall Transition and Random SU(N) Rotation

We reduce the problem of integer quantum Hall transition to a random rotation of an N-dimensional vector by an su(N) algebra, where only N specially selected generators of the algebra are nonzero. The group-theoretical structure revealed in this way allows us to obtain a new series of conservation laws for the equation describing the electron density evolution in the lowest Landau level. The resulting formalism is particularly well suited to numerical simulations, allowing us to obtain the critical exponent \nu numerically in a very simple way. We also suggest that if the number of nonzero generators is much less than N, the same model, in a certain intermediate time interval, describes percolating properties of a random incompressible steady two-dimensional flow. In other words, quantum Hall transition in a very smooth random potential inherits certain properties of percolation.Comment: 4 pages, 1 figur

Bose-Einstein condensate of kicked rotators with time-dependent interaction

A modification of the quantum kicked rotator is suggested with a time-dependent delta-kicked interaction parameter which can be realized by a pulsed turn-on of a Feshbach resonance. The mean kinetic energy increases exponentially with time in contrast to a merely diffusive or linear growth for the first few kicks for the quantum kicked rotator with a constant interaction parameter. A recursive relation is derived in a self-consistent random phase approximation which describes this superdiffusive growth of the kinetic energy and is compared with numerical simulations. Unlike in the case of the quantum rotator with constant interaction, a Lax pair is not found. In general the delta-kicked interaction is found to lead to strong chaotic behaviour.Comment: 4 pages, 3 figure

Integer Quantum Hall Effect for Lattice Fermions

A two-dimensional lattice model for non-interacting fermions in a magnetic field with half a flux quantum per plaquette and $N$ levels per site is considered. This is a model which exhibits the Integer Quantum Hall Effect (IQHE) in the presence of disorder. It presents an alternative to the continuous picture for the IQHE with Landau levels. The large $N$ limit can be solved: two Hall transitions appear and there is an interpolating behavior between the two Hall plateaux. Although this approach to the IQHE is different from the traditional one with Landau levels because of different symmetries (continuous for Landau levels and discrete here), some characteristic features are reproduced. For instance, the slope of the Hall conductivity is infinite at the transition points and the electronic states are delocalized only at the transitions.Comment: 9 pages, Plain-Te

Two-Dimensional Electrons in a Strong Magnetic Field with Disorder: Divergence of the Localization Length

Electrons on a square lattice with half a flux quantum per plaquette are considered. An effective description for the current loops is given by a two-dimensional Dirac theory with random mass. It is shown that the conductivity and the localization length can be calculated from a product of Dirac Green's functions with the {\it same} frequency. This implies that the delocalization of electrons in a magnetic field is due to a critical point in a phase with a spontaneously broken discrete symmetry. The estimation of the localization length is performed for a generalized model with $N$ fermion levels using a $1/N$--expansion and the Schwarz inequality. An argument for the existence of two Hall transition points is given in terms of percolation theory.Comment: 10 pages, RevTeX, no figure

Mesoscopic Effects in the Quantum Hall Regime

We report results of a study of (integer) quantum Hall transitions in a single or multiple Landau levels for non-interacting electrons in disordered two-dimensional systems, obtained by projecting a tight-binding Hamiltonian to corresponding magnetic subbands. In finite-size systems, we find that mesoscopic effects often dominate, leading to apparent non-universal scaling behaviour in higher Landau levels. This is because localization length, which grows exponentially with Landau level index, exceeds the system sizes amenable to numerical study at present. When band mixing between multiple Landau levels is present, mesoscopic effects cause a crossover from a sequence of quantum Hall transitions for weak disorder to classical behaviour for strong disorder. This behaviour may be of relevance to experimentally observed transitions between quantum Hall states and the insulating phase at low magnetic fields.Comment: 13 pages, 6 figures, Proceedings of the International Meeting on Mesoscopic and Disordered Systems, Bangalore December 2000, to appear in Pramana, February 200

Multifractality of the quantum Hall wave functions in higher Landau levels

To probe the universality class of the quantum Hall system at the metal-insulator critical point, the multifractality of the wave function $\psi$ is studied for higher Landau levels, $N=1,2$, for various range $(\sigma )$ of random potential. We have found that, while the multifractal spectrum $f(\alpha)$ (and consequently the fractal dimension) does vary with $N$, the parabolic form for $f(\alpha)$ indicative of a log-normal distribution of $\psi$ persists in higher Landau levels. If we relate the multifractality with the scaling of localization via the conformal theory, an asymptotic recovery of the single-parameter scaling with increasing $\sigma$ is seen, in agreement with Huckestein's irrelevant scaling field argument.Comment: 10 pages, revtex, 5 figures available on request from [email protected]

Liouvillian Approach to the Integer Quantum Hall Effect Transition

We present a novel approach to the localization-delocalization transition in the integer quantum Hall effect. The Hamiltonian projected onto the lowest Landau level can be written in terms of the projected density operators alone. This and the closed set of commutation relations between the projected densities leads to simple equations for the time evolution of the density operators. These equations can be used to map the problem of calculating the disorder averaged and energetically unconstrained density-density correlation function to the problem of calculating the one-particle density of states of a dynamical system with a novel action. At the self-consistent mean-field level, this approach yields normal diffusion and a finite longitudinal conductivity. While we have not been able to go beyond the saddle point approximation analytically, we show numerically that the critical localization exponent can be extracted from the energetically integrated correlation function yielding $\nu=2.33 \pm 0.05$ in excellent agreement with previous finite-size scaling studies.Comment: 9 pages, submitted to PR

Single electron magneto-conductivity of a nondegenerate 2D electron system in a quantizing magnetic field

We study transport properties of a non-degenerate two-dimensional system of non-interacting electrons in the presence of a quantizing magnetic field and a short-range disorder potential. We show that the low-frequency magnetoconductivity displays a strongly asymmetric peak at a nonzero frequency. The shape of the peak is restored from the calculated 14 spectral moments, the asymptotic form of its high-frequency tail, and the scaling behavior of the conductivity for omega -> 0. We also calculate 10 spectral moments of the cyclotron resonance absorption peak and restore the corresponding (non-singular) frequency dependence using the continuous fraction expansion. Both expansions converge rapidly with increasing number of included moments, and give numerically accurate results throughout the region of interest. We discuss the possibility of experimental observation of the predicted effects for electrons on helium.Comment: RevTeX 3.0, 14 pages, 8 eps figures included with eps

Universal Scaling of Strong-Field Localization in an Integer Quantum Hall Liquid

We study the Landau level localization and scaling properties of a disordered two-dimensional electron gas in the presence of a strong external magnetic field. The impurities are treated as random distributed scattering centers with parameterized potentials. Using a transfer matrix for a finite-width strip geometry, we calculate the localization length as a function of system size and electron energy. The finite-size localization length is determined by calculating the Lyapunov exponents of the transfer matrix. A detailed finite-size scaling analysis is used to study the critical behavior near the center of the Landau bands. The influence of varying the impurity concentration, the scattering potential range and its nature, and the Landau level index on the scaling behavior and on the critical exponent is systematically investigated. Particular emphasis is put on studying the effects of finite range of the disorder potential and Landau level coupling on the quantum localization behavior. Our numerical results, which are carried out on systems much larger than those studied before, indicate that pure $\delta$-function disorder in the absence of any Landau level coupling gives rise to non-universal localization properties with the critical exponents in the lowest two Landau levels being substantially different. Inclusion of a finite potential range and/or Landau level mixing may be essential in producing universality in the localization.Comment: 28 pages, Latex, 17 figures (available upon request), #phd0