Quantum Hall Effect on the Hyperbolic Plane

In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between K-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case. Received: 17 March 1997 / Accepted: 24 April 1997     

Hall viscosity and electromagnetic response

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 q(2) term in the Hall conductivity is universal in the limit of strong magnetic field.     

Topological investigation of the fractionally quantized Hall conductivity

Using the fiber bundle concept developed in geometry and topology, the fractionally quantized Hall conductivity is discussed in the relevant many-particle configuration space. Electronmagnetic field and electron-electron interactions under FQHE conditions are treated as functional connections over the torus, the torus being the underlying two-dimensional manifold. Relations to the (2 + 1)-dimensional Chern-Simons theory are indicated. The conductivity being a topological invariant is given as e²/h times a linking number which is the quotient of the winding numbers of the self-consistent field and the magnetic field, respectively. Odd denominators are explained by the two spin structures which have been considered for the FQHE correlated electron system.     

The quantum hall effect for electrons in a random potential

We assume the existence of sufficiently localised states, near the edges of each Landau band. We then prove that the Hall conductivity is quantised and the parallel conductivity vanishes, when the filling factor stays close to an integer. The Hall integer is a topological invariant, given by the Landau band index. We also prove that at weak disorder, the localisation length diverges in each Landau band.     

Vacuum Nodes and Anomalies in Quantum Theories

We show that nodal points of ground states of some quantum systems with magnetic interactions can be identified in simple geometric terms. We analyse in detail two different archetypical systems: i) the planar rotor with a non-trivial magnetic flux Φ and ii) the Hall effect on a torus. In the case of the planar rotor we show that the level repulsion generated by any reflection invariant potential V is encoded in the nodal structure of the unique vacuum for θ=π. In the second case we prove that the nodes of the first Landau level for unit magnetic charge appear at the crossing of the two non-contractible circles α₋, β₋ with holonomies h _α-(A)=h _β-(A)=−1 for any reflection invariant potential V. This property illustrates the geometric origin of the quantum translation anomaly. Received: 6 April 1999 / Accepted: 21 October 2000     

Effects of disorder on modulated quantum Hall systems

The Hall conductivity and the localization length are calculated for weakly modulated two-dimensional systems within the lowest Landau level approximation. We find that the fractal character of the Hofstadter butterfly is reflected on the coincidence in the localization and the Hall conductivity among a series of fluxes φ+2n with integers n.     

Hall Conductivity as the Topological Invariant in the Phase Space in the Presence of Interactions and a Nonuniform Magnetic Field

JETP Letters - The quantum Hall conductivity in the presence of constant magnetic field may be represented as the topological TKNN invariant. Recently, the generalization of this expression has...     

On the Singular Effects in the Relativistic Landau Levels in Graphene with a Disclination

The effect of a pseudo Aharonov-Bohm (AB) magnetic field generated by a disclination on a two-dimensional electron gas in graphene is addressed in the continuum limit within the geometric approach. The influence of the coupling between the spinor fields and the singular conical curvature is investigated, which shows that singularities have pronounced impact in the Hall conductivity. Moreover, the degeneracy related to the Dirac valleys is broken for negative values of the angular momentum quantum numbers, l, includingl ≡ 0. In this case, a Hall plateau develops at the null filling factor. Obtaining the Hall conductivity by summing over the positive and the negative l's, the null Landau level is recovered and the plateau at the null filling factor disappears. In any case, the standard plateaus, which are seen in a flat graphene are not obtained with these curvature and singular effects.     

Topological invariants for fractional quantum hall states

We calculate a topological invariant, whose value would coincide with the Chern number in the case of integer quantum Hall effect, for fractional quantum Hall states. In the case of Abelian fractional quantum Hall states, this invariant is shown to be equal to the trace of the K-matrix. In the case of non-Abelian fractional quantum Hall states, this invariant can be calculated on a case by case basis from the conformal field theory describing these states. This invariant can be used, for example, to distinguish between different fractional Hall states numerically even though, as a single number, it cannot uniquely label distinct states.     

A Pairing Between Super Lie-Rinehart and Periodic Cyclic Homology

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K⁰-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology. The author was partially supported by the KBN grant 1P03A 036 26.     

Gravitational Instability of a Rotating Viscous Thermally Conducting Plasma

This paper deals with the gravitational instability of an infinite homogeneous viscous rotating plasma of finite electrical conductivity in the combined presence of effects of Hall currents, finite Larmor radius (FLR) and thermal conductivity. The ambient magnetic field is assumed to be uniform and acting along the vertical direction. Both longitudinal and transverse modes of wave propagation have been studied. It is shown that Jean's criterion determines the gravitational instability even in the presence of the effects of thermal conductivity, viscosity, finite electrical conductivity, FLR, rotation and Hall currents. Further it is found that while FLR, viscosity and rotation have a stabilizing influence, both the thermal and the electrical conductivities have a destabilizing influence on the gravitational instability of a plasma.     

Anomalous Hall effect in magnetic sandwiches with a dielectric spacer

Quantum-statistical calculations are presented for the anomalous Hall effect in a magnetic sandwich with a tunnel junction across a thin dielectric spacer. The tunneling current flows across the junction perpendicular to the plane of the layers while the Hall component of the current lies in this plane. The Kubo formalism and the Green’s functions are used to calculate the contribution of skew scattering to the Hall conductivity. The classical size effect in the Hall conductivity of this structure is studied and two new effects are observed. One is associated with the dependence of the effective electric field in the magnet on the transparency of the dielectric potential barrier for electrons when the current flows perpendicular to the layers of the structure and may be called “ geometric”. The other occurs as a result of the influence of the strong electric field in the dielectric on the electron motion in the adjacent magnetic layers.     

Quantum Hall Effect on the Hyperbolic Plane in the Presence of Disorder

We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in Comm. Math. Phys. 190 (1998), 629–673, to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.     

Magnetic Bloch states and Hall conductivity of a two-dimensional electron gas in a periodic potential without inversion center

Quantum states of 2D electrons are studied in a periodic potential without inversion center in the presence of a magnetic field. It is shown that the energy spectrum in magnetic subbands is not symmetric about the center of magnetic Brillouin zone E(k)≠E(?k). Singularities (phase branching points) of the electron wave function, which determine the quantization law of Hall conductivity σ_xy, are studied in the k space. It is found that a sharp change takes place in the number of points in the magnetic Brillouin zone and in the corresponding values of topological invariants determining the Hall conductivity of filled subbands. It is noted that the longitudinal conductivity of a lattice without inversion center placed in a magnetic field is not invariant with respect to a change in sign of the electric field, and a photovoltaic effect must arise in an ac electromagnetic field.     

Renormalized vertex modification of Hall conductivity for dilute magnetic alloys

By means of the renormalized vertex procedure for the motion of Green's function developed by the authors, the vertex function of magnetic alloys, based on thes-d exchange interaction, is solved exactly and the corresponding Hall conductivity tensors are obtained. It is found that the value of the renormalized Hall conductivity is (1+h ²)^–1 times less than that before the renormalization (hereh is a reduced magnetic field). It is shown that the renormalized modification of the conductivity is very important in the cases with not too weak external magnetic field and slow relaxation time.     

Effect of the Landau level broadening on the quantum Hall conductance

Summary In a previous paper, Kliroset al. presented a model calculation of the Hall conductivity as a function of the Landau level broadening Γ for finite temperatures. In this paper, the effect of Landau-level broadening on the structure of the Hall conductivity is investigated. The experimental data regarding the Si-MOSFET and GaAs-heterostructure experiments are reproduced including a functional dependence of Γ on the magnetic field. The influence of the effectiveg-factor is considered as well.     

Fermionic Magnons, non-Abelian spinons, and the spin quantum Hall effect from an exactly solvable spin-1/2 Kitaev model with SU(2) symmetry

We introduce an exactly solvable SU(2)-invariant spin-1/2 model with exotic spin excitations. With time reversal symmetry (TRS), the ground state is a spin liquid with gapless or gapped spin-1 but fermionic excitations. When TRS is broken, the resulting spin liquid exhibits deconfined vortex excitations which carry spin-1/2 and obey non-Abelian statistics. We show that this SU(2) invariant non-Abelian spin liquid exhibits the spin quantum Hall effect with quantized spin Hall conductivity σ(xy)(s)=?/2π, and that the spin response is effectively described by the SO(3) level-1 Chern-Simons theory at low energy. We further propose that a SU(2) level-2 Chern-Simons theory is the effective field theory describing the topological structure of the non-Abelian SU(2) invariant spin liquid.     

The Hall effect in La0.67Ba0.33MnO3

The Hall effect in polycrystalline barium-substituted lanthanum manganite La_0.67Ba_0.33MnO₃ has been investigated in the temperature interval 298<T<355 K. It is found that the anomalous Hall coefficient in this material is two orders of magnitude greater than the normal coefficient. At T ₀=333 K the normal Hall coefficient changes sign, which indicates a change in the type of conductivity. The temperature dependence of the normal Hall coefficient, electrical conductivity, and magnetoresistance is explained on the basis of the concept of motion of the mobility edge attendant as the temperature changes. Zh. éksp. Teor. Fiz. 113, 981–987 (March 1998)     

Geometric Phase in the Interaction of a Time-Dependent Light Field with <Emphasis Type="Italic">ℑ</Emphasis><Superscript>(3)</Superscript> System

By using of the invariant theory, we have studied the geometric phase in the interaction of a time-dependent light field with ℑ ⁽³⁾ system, the dynamical and geometric phases are given, respectively. The disappearing condition of the geometric phase is given.     

Electronic-transport properties of unhydrogenated amorphous gallium arsenide

Summary Hall mobility, μ_H, and electrical conductivity, σ, of unhydrogenated amorphous-gallium-arsenide films, prepared by r.f. sputtering, have been measured. Conductivity as a function of temperature shows a variable-range hopping mechanism atT<260 K, while at high temperature, conductivity and Hall mobility are both thermally activated. The results are interpreted in terms of the presence of defect complexes due to an excess of Ga. The stoichiometry and the structure of the films are used to explain the behaviour and the values of μ_H. The values of the activation energy of the conductivity seem in agreement with theoretical calculations on the position of electronic states created by defect complexes in the mobility-gap of a-GaAs.