Thermally Activated Deviations from Quantum Hall Plateaus

The Hall conductivity $\sigma_{\rm xy}$ of a two-dimensional electron system is quantized in units of $e^2/h$ when the Fermi level is located in the mobility gap between two Landau levels. We consider the deviation of $\sigma_{\rm xy}$ from a quantized value caused by the thermal activation of electrons to the extended states for the case of a long range random potential. This deviation is of the form $\sigma_{\rm xy}^*\exp(-\Delta/T)$. The prefactor $\sigma_{\rm xy}^*$ is equal to $e^2/h$ at temperatures above a characteristic temperature $T_2$. With the temperature decreasing below $T_2$, $\sigma_{\rm xy}^*$ decays according to a power law: $\sigma_{\rm xy}^* = \frac{e^2}{h}(T/T_2)^\gamma$. Similar results are valid for a fractional Hall plateau near filling factor $p/q$ if $e$ is replaced by the fractional charge $e/q$.Comment: 4 pages in PostScript (figures included

Continuous Elastic Phase Transitions in Pure and Disordered Crystals

We review the theory of second--order (ferro--)elastic phase transitions, where the order parameter consists of a certain linear combination of strain tensor components, and the accompanying soft mode is an acoustic phonon. In three--dimensional crystals, the softening can occur in one-- or two--dimensional soft sectors. The ensuing anisotropy reduces the effect of fluctuations, rendering the critical behaviour of these systems classical for a one--dimensional soft sector, and classical with logarithmic corrections in case of a two--dimensional soft sector. The dynamical critical exponent is $z = 2$, and as a consequence the sound velocity vanishes as $c_s \propto | T - T_c |^{1/2}$, while the phonon damping coefficient is essentially temperature--independent. Disorder may lead to a variety of precursor effects and modified critical behaviour. Defects that locally soften the crystal may induce the phenomenon of local order parameter condensation. When the correlation length of the pure system exceeds the average defect separation $n_{\rm D}^{-1/3}$, a disorder--induced phase transition to a state with non--zero average order parameter can occur at a temperature $T_c(n_{\rm D})$ well above the transition temperature $T_c^0$ of the pure crystal. Near $T_c^0$, the order--parameter curve, susceptibility, and specific heat appear rounded. For $T < T_c(n_{\rm D})$ the spatial inhomogeneity induces a static central peak with finite $q$ width in the scattering cross section, accompanied by a dynamical component that is confined to the very vicinity of the disorder--induced phase transition.Comment: 26 pages, Latex (rs.sty now IS included), 11 figures can be obtained from U.C. T\"auber ([email protected]); will appear in Phil. Trans. Roy. Soc. Lond. A (October 1996

Continuous Versus First Order Transitions in Compressible Diluted Magnets

The interplay between disorder and compressibility in Ising magnets is studied. Contrary to pure systems in which a weak compressibility drives the transition first order, we find from a renormalization group analysis that it has no effect on disordered systems which keep undergoing continuous transition with rigid random-bond Ising model critical exponents. The mean field calculation exhibits a dilution-dependent tricritical point beyond which, at stronger compressibility the transition is first order. The different behavior of XY and Heisenberg magnets is discussed.Comment: 16 pages, latex, 2 figures not include

Transport and Dephasing in a Quantum Dot: Multiply Connected Graph Model

Using the theory of diffusion in graphs, we propose a model to study mesoscopic transport through a diffusive quantum dot. The graph consists of three quasi-1D regions: a central region describing the dot, and two identical left- and right- wires connected to leads, which mimic contacts of a real system. We find the exact solution of the diffusion equation for this graph and evaluate the conductance including quantum corrections. Our model is complementary to the RMT-models describing quantum dots. Firstly, it reproduces the universal limit at zero temperature. But the main advantage compared to RMT-models is that it allows one to take into account interaction-induced dephasing at finite temperatures. Besides, the crossovers from open to almost closed quantum dots and between different regimes of dephasing can be described within a single framework. We present results for the temperature dependence of the weak localization correction to the conductance for the experimentally relevant parameter range and discuss the possibility to observe the elusive 0D-regime of dephasing in different mesoscopic systems.Comment: 12 page

Peak Values of Conductivity in Integer and Fractional Quantum Hall Effect

The diagonal conductivity $\sigma_{xx}$ was measured in the Corbino geometry in both integer and fractional quantum Hall effect (QHE). We find that peak values of $\sigma_{xx}$ are approximately equal for transitions in a wide range of integer filling factors $3<\nu<16$, as expected in scaling theories of QHE. This fact allows us to compare peak values in the integer and fractional regimes within the framework of the law of corresponding states.Comment: 8 pages (revtex format), 3 postscript figure

Self-averaging in the random 2D Ising ferromagnet

We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size $L$ like $\sim L \ln\ln(L)$. In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a $\delta$-peak in the thermodynamic limit $L \to \infty$. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.Comment: 12 pages, accepted versio

Quantum Hall effect at low magnetic fields

The temperature and scale dependence of resistivities in the standard scaling theory of the integer quantum Hall effect is discussed. It is shown that recent experiments, claiming to observe a discrepancy with the global phase diagram of the quantum Hall effect, are in fact in agreement with the standard theory. The apparent low-field transition observed in the experiments is identified as a crossover due to weak localization and a strong reduction of the conductivity when Landau quantization becomes dominant.Comment: 4 pages, 2 figures, minor corrections, to appear in PR

Dynamical scaling at the quantum Hall transition: Coulomb blockade versus phase breaking

We argue that the finite temperature dynamics of the integer quantum Hall system is governed by two independent length scales. The consistent scaling description of the transition makes crucial use of two temperature critical exponents, reflecting the interplay between charging effects and interaction-induced dephasing. Experimental implications of the two-scale picture are discussed.Comment: 4 pages, RevTeX, 1 figure included, minor changes, accepted in PR

A Unified Model for Two Localisation Problems: Electron States in Spin-Degenerate Landau Levels, and in a Random Magnetic Field

A single model is presented which represents both of the two apparently unrelated localisation problems of the title. The phase diagram of this model is examined using scaling ideas and numerical simulations. It is argued that the localisation length in a spin-degenerate Landau level diverges at two distinct energies, with the same critical behaviour as in a spin-split Landau level, and that all states of a charged particle moving in two dimensions, in a random magnetic field with zero average, are localised.Comment: 7 pages (RevTeX 3.0) plus 4 postscript figure