Narrow Results

The von Neumann lattice representation is a convenient representation for studying several intriguing physics of quantum Hall systems. In this formalism, electrons are mapped to lattice fermions. A topological invariant expression of the Hall conductance is derived and is used for the proof of the integer quantum Hall effect in the realistic situation. Anisotropic quantum Hall gas is investigated based on the Hartree–Fock approximation in the same formalism. Thermodynamic properties, transport properties, and unusual response under external modulations are found. Implications for the integer quantum Hall effect in the finite systems are also studied and a new quantum Hall regime with non-zero longitudinal resistance is shown to exist.

We report on the calculation of the frequency-dependent conductivity of the quantum Hall stripes and bubble crystals that form in high Landau level. We use the replica and Gaussian variational methods (GVM) with a dynamical matrix obtained from the time-dependent Hartree-Fock approximation. In the stripe state, we go beyond the semiclassical approximation of the saddle point equations obtained with the GVM and demonstrate the existence of a quantum depinning transition as a function of filling factor. Below a critical filling factor, the pinned state is described by a replica symmetry breaking (RSB) solution that gives resonant peaks in the frequency-dependent conductivity in both directions, parallel and perpendicular to the stripes orientation. These peaks shift to zero frequency as the critical filling is approached. Above the critical filling, we find a depinned stripe state described by a partial replica symmetry breaking solution in which there is free sliding only along the stripe direction. The transition has a Kosterlitz-Thouless character and includes a jump in the low-frequency exponent of the dynamical conductivity. In the bubble crystals a semiclassical approximation yields a pinning peak frequency and resonance width that generally decrease with increasing filling factor, in accordance with recent microwave absorption experiments.

Energy gaps in the fractional quantum Hall regime are measured with phonon spectroscopy for various electron densities and filling factors ν=1/3, 2/5, 4/7, 3/5, 2/3, 4/3, and 5/3. At a given filling factor nearly all gaps measured show a square-root dependence on the magnetic field. The measured gaps can be described well with composite fermion theory: They are transitions between composite fermion Landau levels that are not affected by disorder on phonon wavelength. All transitions can be described by one single fit parameter related to the composite fermion effective mass. In contrast, the gaps at filling factor ν=2/3 in the high fields regime show a linear field dependence. This hints onto an only partially polarized ground state of 2/3.

By strictly adhering to the microscopic theory of composite fermions (CFs) for the Landau-level filling fractions ν_e=p/(2p+1), we reproduce, with remarkable accuracy, the surface-acoustic-wave (SAW)-based experimental results by Willett and co-workers concerning two-dimensional electron systems with ν_e close to ½. Our results imply that the electron band mass m_b, as distinct from the CF mass m_⋆, must undergo a substantial increase under the conditions corresponding to ν_e ≈ ½. In view of the relatively low aerial electronic densities n_e to which the underlying SAW experiments correspond, our finding conforms with the experimental results by Shashkin et al. [Phys. Rev. B 66, 073303 (2002)], concerning two-dimensional electrons in silicon, that signal sharp increase in m_b for n_e decreasing below approximately 2×10¹¹cm^-2. We further establish that a finite mean-free path ℓ₀ is essential for the observed linearity of the longitudinal conductivity σ_xx(q) as deduced from the SAW velocity shifts.

Since 1999, experiments have shown a plethora of surprising results in the low-temperature magnetotransport in intermediate regions between quantum Hall (QH) plateaus: the extreme anisotropies observed for half-filling, or the re-entrant integer QH effects at quarter filling of high Landau levels (LL); or even an apparent melting of a Wigner Crystal (WC) at filling factor ν = 1/7 of the lowest LL. A large body of seemingly distinct experimental evidence has been successfully interpreted in terms of liquid crystalline phases in the two-dimensional electron system (2DES). In this paper, we present a review of the physics of liquid crystalline states for strongly correlated two-dimensional electronic systems in the QH regime. We describe a semi-quantitative theory for the formation of QH smectics (stripes), their zero-temperature melting onto nematic phases and ultimate anisotropic-isotropic transition via the Kosterlitz–Thouless (KT) mechanism. We also describe theories for QH-like states with various liquid crystalline orders and their excitation spectrum. We argue that resulting picture of liquid crystalline states in partially filled LL-s is a valuable starting point to understand the present experimental findings, and to suggest new experiments that will lead to further elucidation of this intriguing system.

Contrary to common belief, the current emitted by a contact embedded in a two-dimensional electron gas (2DEG) is quantized in the presence of electric and magnetic fields. This observation suggests a simple, clearly defined model for the quantum current through a Hall device that does not invoke disorder or interactions as the cause of the integer quantum Hall effect (QHE), but is based on a proper quantization of the classical electron drift motion. The theory yields a quantitative description of the breakdown of the QHE at high current densities that is in agreement with experimental data. Furthermore, several of its key points are in line with recent findings of experiments that address the dependency of the QHE on the 2DEG bias voltage and results that are not easily explained within the framework of conventional QHE models.

Precise scanning of the (B_⊥,B_‖)-plane while measuring magnetoresistance of the n-InGaAs/GaAs double quantum well (DQW) reveals a number of peculiarities connected with intricate DQW energy spectrum, which are analyzed on the basis of quasiclassical calculations. Magnetic breakdown effects are also considered. Peaks due to the latter mechanism reveal spin-splittings (in spite of lower mobilities as compared with the traditional n-GaAs/AlGaAs DQWs) corresponding to an enhanced effective Lande g-factor.

We report on the calculation of the frequency-dependent conductivity of the quantum Hall stripes and bubble crystals that form in high Landau level. We use the replica and Gaussian variational methods (GVM) with a dynamical matrix obtained from the timedependent Hartree-Fock approximation. In the stripe state, we go beyond the semiclassical approximation of the saddle point equations obtained with the GVM and demonstrate the existence of a quantum depinning transition as a function of filling factor. Below a critical filling factor, the pinned state is described by a replica symmetry breaking (RSB) solution that gives resonant peaks in the frequency-dependent conductivity in both directions, parallel and perpendicular to the stripes orientation. These peaks shift to zero frequency as the critical filling is approached. Above the critical filling, we find a depinned stripe state described by a partial replica symmetry breaking solution in which there is free sliding only along the stripe direction. The transition has a Kosterlitz-Thouless character and includes a jump in the low-frequency exponent of the dynamical conductivity. In the bubble crystals a semiclassical approximation yields a pinning peak frequency and resonance width that generally decrease with increasing filling factor, in accordance with recent microwave absorption experiments.

Energy gaps in the fractional quantum Hall regime are measured with phonon spectroscopy for various electron densities and filling factors ν = 1/3, 2/5, 4/7, 3/5, 2/3, 4/3, and 5/3. At a given filling factor nearly all gaps measured show a square-root dependence on the magnetic field. The measured gaps can be described well with composite fermion theory: They are transitions between composite fermion Landau levels that are not affected by disorder on phonon wavelength. All transitions can be described by one single fit parameter related to the composite fermion effective mass. In contrast, the gaps at filling factor ν = 2/3 in the high fields regime show a linear field dependence. This hints onto an only partially polarized ground state of 2/3.

The electron reservoir model for the integer quantum Hall effects, the magnetoplasmon dispersion plateaus, and the radiation-induced magnetoresistance oscillations, are briefly reviewed.

We show that, for Galilean invariant quantum Hall states, the Hall viscosity appears in the electromagnetic response at finite wave numbers q. In particular, the leading q dependence of the Hall conductivity at small q receives a contribution from the Hall viscosity. The coefficient of the q² term in the Hall conductivity is universal in the limit of strong magnetic field.