Integral Quantum Hall Effect Research Articles

Precise coincidence of effective potentials in the integral and fractional quantum Hall effects

The exact n-body distribution functions are calculated for a two-dimensional, noninteracting quantum electron gas in an external magnetic field for any temperature and density. At low temperature and filled lowest-Landau-level, these functions are identical to the exact distribution functions obtained by Jancovici [Phys. Rev. Lett. 46 (1981) 386] for the classical two-dimensional one-component plasma at the special plasma parameter Γ=2, establishing that the quantum state with filling factor ν=1, associated with the integral quantum Hall effect, is precisely described by an effective classical potential φ( r)=−2ln r. Further, this Boltzmann factor exactly matches that constructed by Laughlin [Phys. Rev. Lett. 50 (1983) 1395] to account for the fractional quantum Hall effect.

On the structure of strips exhibiting the integral quantum Hall effect in an inhomogeneous 2D electron system

A formalism is developed to generalize the results obtained for “incompressible” strips exhibiting the integral quantum Hall effect in a spatially inhomogeneous 2D electron system to the cases of finite temperatures, significant electron density gradients, etc. Specifically, the concept of the “quality” of a given integer quantum Hall effect strip (channel) is introduced; the quality is proportional to the derivative dn(x)/dx in the central part of the channel [n(x) is the electron density distribution over the channel]. For a well-defined channel, this derivative tends to zero. If a noticeable gradient arises in the n(x) distribution, the channel does not exhibit the quantum Hall effect and ceases to exist. The conditions are determined under which a channel exhibiting the integral quantum Hall effect breaks down. The results of calculations are used to interpret the available experimental data.

Fractional quantum Hall effect of composite fermions.

In a GaAs/AlGaAs quantum well of density 1 x 10(11) cm(-2) we observed a fractional quantum Hall effect (FQHE) at nu = 4/11 and 5/13, and weaker states at nu = 6/17, 4/13, 5/17, and 7/11. These sequences of fractions do not fit into the standard series of integral quantum Hall effects of composite fermions (CF) at nu = p/(2mp +/- 1). They rather can be regarded as the FQHE of CFs attesting to residual interactions between these composite particles. In tilted magnetic fields the nu = 4/11 state remains unchanged, strongly suggesting it to be spin polarized. The weak nu = 7/11 state vanishes quickly with tilt.

Theoretical search for the nested quantum Hall effect of composite fermions

Almost all quantum Hall effect to date can be understood as {\em integral} quantum Hall effect of appropriate particles, namely electrons or composite fermions. This paper investigates theoretically the feasibility of nested states of composite fermions which would lead to a quantum Hall effect that cannot be understood as integral quantum Hall effect of composite fermions. The weak residual interaction between composite fermions will play a crucial role in the establishment of such quantum Hall states by opening a gap in a partially filled composite-fermion level. To treat the problem of interacting composite fermions, we develop a powerful method that allows us to obtain the low energy spectra at composite fermion fillings of $\nu^*=n+\bar \nu$ without making any assumption regarding the structure of composite fermions in the topmost partially filled level. The method is exact aside from neglecting the composite-fermion Landau level mixing, and enables us to study rather large systems, for example, 24 particles at a total flux of 62 $hc/e$, for which the dimension of the lowest Landau level Hilbert space is $\sim 10^{17}$. We have investigated, for fully spin polarized composite fermions, several filling factors between 1/3 and 2/5 using this approach. The results indicate that any possible incompressibility at these fractions is likely to have a fundamentally different origin than that considered earlier.

Mean Field Theory and Composite Fermions in the Fractional Quantum Hall Regime

We show that within mean field theory it is possible to construct a model for a 2DEG that unifies the fractional and integral quantum Hall effects, much as the composite fermion model does.

Hysteresis and spikes in the quantum Hall effect

We observe sharp peaks and strong hysteresis in the electronic transport of a two-dimensional electron gas (2DEG) in the region of the integral quantum Hall effect. The peaks decay on time scales ranging from several minutes to more than an hour. Momentary grounding of some of the contacts can vastly modify the strength of the peaks. All these features disappear under application of a negative bias voltage to the backside of the specimen. We conclude, that a conduction channel parallel to the high mobility 2DEG is the origin for the peaks and their hysteretic behavior.

(Mis-)handling gauge invariance in the theory of the quantum Hall effect. II. Perturbative results

The concept of F invariance, which previously arose in our analysis of the integral and half-integral quantum Hall effects, is studied in $2+2\ensuremath{\varepsilon}$ spatial dimensions. We report the results of a detailed renormalization-group analysis, and establish the renormalizability of the (Finkelstein) action to two-loop order. We show that the infrared behavior of the theory can be extracted from gauge-invariant (F-invariant) quantities only. For these quantities (conductivity, specific heat) we derive explicit scaling functions. We identify a bosonic quasiparticle density of states which develops a Coulomb gap as one approaches the metal-insulator transition from the metallic side. We discuss the consequences of F invariance for the strong-coupling, insulating regime.

Quantum response at finite fields and breakdown of Chern numbers

We show that the response to an electric field, in models of the integral quantum Hall effect, is analytic in the field and has isolated essential singularity at zero field. We also study the breakdown of Chern numbers associated with the response of Floquet states. We argue, and give evidence, that the breakdown of Chern numbers in Floquet states is a discontinuous transition at zero field. This follows from an observation, of independent interest, that Chern numbers for finite-dimensional Floquet operators are generically zero. These results rule out the possibility that the breakdown of the Hall conductance is a phase transition at finite fields for a large class of models.

Two‐Dimensional Electron Gases Continue to Exhibit Intriguing Behavior

The startling discoveries of the integral and fractional quantum Hall effects were made in two‐dimensional electron gases subjected to very high magnetic fields (see the story on page 17). Now, it appears, other surprises await us at lower fields. In a recent study, the longitudinal resistivity exhibited a strong anisotropy in a certain range of temperature and magnetic field: When plotted as a function of the magnetic field, the resistivity has a dip when the current flows in one direction and a strong peak when it flows in an orthogonal direction. There's no a priori reason to think that these two directions are different.

Topological Aspects of Quantum Hall Fluid and Berry Phase

We have analyzed here the recent development towards our understanding of the Integral and Fractional Quantum Hall effect. It has been pointed out that the chiral anomaly and Berry phase approach embraces in a unified way the whole spectrum of quantum Hall systems with their various characteristic features. This formalism also helps us to understand the edge states observed in Hall fluids. It is argued that Hall fluids with even denominator filling factor leads to the non-Abelian Berry phase.

The evolution of the topological structure of the integral quantum Hall effect

The topological structure of the integral Hall conductance is determined by making use of the decomposition of the gauge potential on a U(1) principal fibre bundle over k-space. The integral Hall conductance density is found to take the form of a delta-function δ(ψ). Only the zeroes of Bloch waves contribute to the Hall conductance. The topological structure of the intergal Hall conductance is labeled by the Hopf index β and the Brouwer degree η. The Hall conductane topological current and its evolution is discussed. The Hall topological charges are found generating or annihilating at the limit points of wave function and splitting and merging at the bifurcation points of the wave function. Different directions of the bifurcation are given.

Localization bounds for an electron gas

Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies, or weak disorder away from the unperturbed spectrum. The present work establishes on this basis exponential decay for the modulus of the two--point function, at all temperatures as well as in the ground state, for a Fermi gas within the one-particle approximation. Different implications, in particular for the Integral Quantum Hall Effect, are reviewed.

Universal scaling and diagonal conductivity in the integral quantum Hall effect

We perform a numerical finite-size study for the static homogeneous diagonal conductivity σxx at the critical filling factor 3/2 for different microscopic realizations of the random impurity potential. The variation of σxx with the system size defines a scaling function. It turns out to be independent of the particular realization of disorder and also of the Landau-level index. However, the diagonal conductivity in the second-lowest Landau level varies strongly with disorder. The universal critical conductivity is recovered only asymptotically when the correlation length of the potential is increased.

Determination of the von Klitzing constant and the fine-structure constant through a comparison of the quantized Hall resistance and the ohm derived from the NIST calculable capacitor

This paper describes a recent determination of the von Klitzing constant and the fine-structure constant by comparisons of values of the ohm as defined in the International System of Units (SI), derived from the National Institute of Standards and Technology (NIST) calculable cross-capacitor, and values of the international practical unit of resistance derived from the integral quantum Hall effect. In this determination, the comparisons were made in a series of measurements lasting three years. A small difference is observed between this determination and an earlier comparison carried out in this laboratory and reported in 1988. The most recent value of the fine-structure constant based on the experimental value and theoretical expression for the magnetic moment anomaly of the electron, which has the smallest uncertainty of any value currently available, is consistent with both of these results. The new value exceeds the 1990 conventional value of the von Klitzing constant RK-90 by slightly more than twice the relative standard uncertainty of the present measurement, which is 2.4 × 10-8.

Low-frequency and long-wavelength anomalies of the integral quantum Hall effect

The static, homogeneous, diagonal conductivityfor the normal quantum Hall effect at half filling of the lowest Landau level under certain conditions seems to approach the value e 2/2h. The deviations from this value in dynamical, inhomogeneous situations show anomalies with a nonanalytical behavior of σ(ω, k) in ω and k. Existing results concerning this behavior are briefly reviewed and improved, taking screening effects into account. The combination of low-frequency anomalies and screening effects turns out to have dramatic consequences suppressing the low-frequency conductivity very strongly. From our considerations we conclude that it should be possible to observe all these effects.

Optical determination of the spin polarization of a quantum Hall ferromagnet

Abstract Recent experimental and theoretical investigations have resulted in a shift in our understanding of the ν =1 quantum Hall state. There now exists a wealth of evidence that the excitation gap and the resulting quasi-particle spectrum at ν =1 are due predominately to the ferromagnetic many-body exchange interaction. A great variety of experimentally observed correlations at ν =1 cannot be incorporated into a perturbative expansion around the single-particle model, a scheme long thought to describe the integral quantum Hall effect (IQHE) at filling factor 1. Theorists now refer to the ν =1 state as the quantum Hall ferromagnet. In this paper, we review recent theoretical and experimental progress and detail our own optical investigations of the ν =1 quantum Hall regime.

Noise spectroscopy in the breakdown of the IQHE

The noise spectrum of the magnetoresistivity in the breakdown of the Integral Quantum Hall Effect (IQHE) at high currents is studied in wide high-quality GaAs/AlGaAs heterostructures on thei=2 Hall plateau, in particular for values of current and/or magnetic field at which pronounced steps and striking temporal patterns showing switching between different levels are observed. The noise spectra obtained from data which display switching are compared with the spectra obtained in the absence of switching. By an analysis based on time series simulation, Lorentzian noise components are separated and found to have similar characteristics in both types of data.

Paired states in the even integer quantum Hall effect.

We argue that a new type of quantum Hall state requiring non-perturbative Landau level mixing arises at low electron density. In these states, up and down spin electrons pair to form spinless bosons that condense into a bosonic quantum Hall state. We describe a wavefunction for a paired quantum Hall state at $\nu=2$ and argue that it is stabilized by a BCS instability arising in flux attachment calculations. Based on this state, we derive a new global phase diagram for the integral quantum Hall effect with spin. Additional experimental implications are discussed.

Topological Aspects of the Berry Phase

We shall explore here the relationship between chiral anomaly and Berry phase from the view-point of the topological investigations of anomaly. This will be extended in the coherent state representation of a quantized spinor. The relevance of Berry phase will also be studied in an unified formalism of integral and fractional quantum Hall effect. Finally we shall study the role of this topological phase in polarised light.

Thermopower of composite fermions.

Measured diffusion thermopower of a low-disorder two-dimensional hole system in the extreme quantum limit is used to probe the thermal properties of the recently proposed particle-flux composite fermions (CF's). The data are consistent with the CF's exhibiting the integral quantum Hall effect away from filling factor \ensuremath{\nu}=1/2. The magnitudes of the thermopower maxima between the fractional quantum Hall states yield an estimate for the CF Landau-level broadening ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{CF}}$, which agrees well with ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{CF}}$ deduced from the analysis of the excitation energy gaps for the fractional liquid states.