On the topological explanation of the integer quantum hall effect

We investigate the static and dynamic Kubo Hall conductivity of a non-interacting electron system in a random potential on a torus. Considering the universal covering space of the torus the Bloch theorem is applied for rational values of the filling factor. The localisation is simulated by the assumption of bound states. The Hall conductivity at zero temperatur is shown to be topologically quantized, if the Fermi energy lies in a spectral gap or in a localisation regime. The relation to previous formulations of the topological approach to the integer quantum Hall effect (QHE) is discussed.     

On the phase boundaries of the integer quantum hall effect

It has been pointed out that, according to the two-parameter scaling theory, the magnetic-field position of the phases of the integer quantum Hall effect (IQHE) at ω_cτ ? 1 is not determined by the filling factor ν = nh/eB. The position of the IQHE phases is given by the bare Hall conductivity σ _xy ⁰ . In this regard, it has been shown that the diagonal resistivity in the magnetic field measured by Sakr et al. [Phys. Rev. B 64, 161308 (2001)] does not exhibit transitions between the σ _xy = 3, 4 and 6 IQHE states on the one hand and the dielectric state on the other hand in contrast to the assertion by Sakr et al.     

Hofstadter butterfly and integer quantum hall effect in three dimensions

For a three-dimensional (3D) lattice in magnetic fields we have shown that the hopping along the third direction, which normally smears out the Landau quantization gaps, can rather give rise to a Hofstadter's butterfly specific to 3D when a criterion is fulfilled by anisotropic (quasi-one-dimensional) systems. In 3D the angle of the magnetic field plays the role of the field intensity in 2D, so that the butterfly can occur in much smaller fields. We have also calculated the Hall conductivity in terms of the topological invariant in the Kohmoto-Halperin-Wu formula, and each of sigma(xy),sigma(zx) is found to be quantized.     

Kubo Hall conductivity on a finite cylinder and the integer quantum hall effect

An additive disordered electron system on a finite cylinder with perpendicular magnetic field is considered. The inequivalent representations of the aximuthal translational group are used to derive conductivity expressions from the Kubo formula and to formulate a phenomenological criterion for localization. These conductivity expressions are shown to be equivalent to the Steda, formula and applied to find spectral conditions for the integer quantum Hall effect to occur on a finite cylinder. The confinement does not give rise to corrections to the quantized values in and, order in the ratio magnetic length/system length. The present approach to the integer quantum Hall effect is compared with several others developed so far.     

A new network model for the integer quantum Hall effect

A Landauer–Büttiker-type formulation of backscattering between pairs of opposite directed channels is used to describe the coupling at the nodes of a network. Physically, these nodes correspond to saddle points of a slowly varying lateral potential modulation in a 2D electron system in the high magnetic field regime. We show that the network can be solved without needing a transfer matrix as used by Chalker and Coddington. We use an exponential dependence of the coupling on the filling factor of the associated Landau level. We demonstrate that our network representation allows a quantitative modeling of almost every realistic sample geometry in the quantum Hall regime, including the effect of gate electrodes across a Hall bar.     

Role of electron spin in integer quantum hall photoluminescence

We investigate numerically the photoluminescence (PL) spectrum in the integer quantum Hall regime and find that the electron spins play important roles. The spectra for the left circularly polarized light show peak splittings when the Fermi levels lies in the excited Landau level, which is caused by the inter Landau level scattering between electrons with anti-parallel spins. At around ν_e∼1 the PL energy is strongly affected by the interplay between the screening and multiple spin flipping (skyrmion) effects.     

Spin-dependent resistivity at transitions between integer quantum hall states

The longitudinal resistivity at transitions between integer quantum Hall states in two-dimensional electrons confined to AlAs quantum wells is found to depend on the spin orientation of the partially filled Landau level in which the Fermi energy resides. The resistivity can be enhanced by an order of magnitude as the spin orientation of this energy level is aligned with the majority spin. We discuss possible causes and suggest a new explanation for the spikelike features observed at the edges of quantum Hall minima near Landau level crossings.     

The lattice gas model of fractional quantum hall effect

The two-dimensional interacting electron gas under a strong homogeneous magnetic field is approximated by the one-dimensional classical lattice gas. We show that in our model the eigenstates of the system approach single-particle Slater determinants in the large particle number limit. Cusps in the energy of our model are found at simple fractional occupations.     

On the fractional quantum hall effect

We show in the Hartree-Fock approximation that the formation of a two dimensional electron lattice allows for a natural explanation of the anomalous fractional quantum Hall effect. Landau levels are broadened and split in a number of bands in such a way that if the number of electrons per unit cell is a half odd integer the Fermi energy is in a gap for an odd filling fraction denominator, and at the center of a band if the denominator is even.     

Adiabatic theorems and applications to the quantum hall effect

We study an adiabatic evolution that approximates the physical dynamics and describes a natural parallel transport in spectral subspaces. Using this we prove two folk theorems about the adiabatic limit of quantum mechanics: 1. For slow time variation of the Hamiltonian, the time evolution reduces to spectral subspaces bordered by gaps. 2. The eventual tunneling out of such spectral subspaces is smaller than any inverse power of the time scale if the Hamiltonian varies infinitly smoothly over a finite interval. Except for the existence of gaps, no assumptions are made on the nature of the spectrum. We apply these results to charge transport in quantum Hall Hamiltonians and prove that the flux averaged charge transport is an integer in the adiabatic limit.     

Doubly graded sigma model with torsion

Using the Hull—Witten construction it is shown how to introduce torsion to the doubly graded sigma model. This construction enables one to find a link between this model and the ten-dimensional supergravity theory in superspace.     

Noise-driven nonlinear sigma model

We present a noise-driven model for obtaining the gap and line-width as functions of the temperature in the nonlinear sigma model. The method is phenomenological and rests on the following physical idea: a classical external stochastic field is introduced representing the coupling of the sigma field with a noise source. Moreover, we assume that the inelastic scattering length is much longer than the elastic one, justifying the neglect of dissipation for temperatures such that the nonlinear sigma model is a good approximation for antiferromagnetic spin chains. This phenomenological approach is justified by comparison with other theoretical predictions and with experiment.     

The lattices with the continuous vorticity as a model for fractional quantum hall effect

It was shown that the including spin of 2d electrons at high magnetic field is possible to remove the divergences in the cores of the vortex lattice and construct the topologically stable states. These states can be considered as the lattices of skyrmions where the unit cell is mapped on the whole sphere of spin directions. That gives the gapped ground state for electrons and can be used as a model for fractional quantum Hall effect.     

Nonabelions in the fractional quantum hall effect

Applications of conformal field theory to the theory of fractional quantum Hall systems are discussed. In particular, Laughlin's wave function and its cousins are interpreted as conformal blocks in certain rational conformal field theories. Using this point of view a hamiltonian is constructed for electrons for which the ground state is known exactly and whose quasihole excitations have nonabelian statistics; we term these objects “nonabelions”. It is argued that universality classes of fractional quantum Hall systems can be characterized by the quantum numbers and statistics of their excitations. The relation between the order parameter in the fractional quantum Hall effect and the chiral algebra in rational conformal field theory is stressed, and new order parameters for several states are given.