Theory of the quantized Hall effect (I)

This is the first of a series of three papers presenting a field theoretic approach to the (integrally) quantized Hall effect. The basic idea is that the transverse conductivity σ_xy directly couples to a topological quantum number characterizing the phase relationship between advanced and retarted electron propagators. This allows us to present a reformulation of the Laughlin quantization argument as well as a direct demonstration of the breakdown of the two-dimensional scaling theory of localization. This paper summarizes all our results and discusses a physical picture of the emergence of extended states.     

Theory of the vertical Hall effect in a dimensionally quantized system

The transverse redistribution of carriers that occurs in a 2D system under the effect of a tangential electric field and a magnetic field possessing a tangential component is studied. It is shown that the redistribution of carriers gives rise to a Hall voltage across isolated electrodes positioned above and under the quantum film. This voltage is determined by the 2D conductivity tensor and the transverse static electric polarizability of the 2D layer. The additional contribution that appears in the vertical Hall voltage because of the electron spin orientation induced by magnetic field and the spin-orbit interaction of electrons with the quantum well potential is determined.     

Theory of the quantized hall effect (II)

We review the derivation that the effective lagrangian describing the critical fluctuations in a two-dimensional disordered electronic system in a transverse magnetic field contains a novel, topological term. We extend this result in several directions. We show how the importance of topological concepts can be seen by examining in detail the nature of the boundary current whenever the Fermi energy lies within a localized state region. This insight allows us to construct a field theoretic quantization argument. Our argument is reminiscent of the Laughlin-Halperin quantization approach, in that we make use of the response of the system to sources with nontrivial gauge topology. This then leads to a discussion of how to use the effective field theory to actually compute the response, and of why localization must break down somewhere within the Landau band. Our methodology unifies the results of Laughlin, Halperin and Thouless with the field theoretic approach to localization pioneered by Wegner.     

Fermionization of grassmannians and the quantized Hall effect

The rules of fermionization of grassmannian sigma models with instanton terms are derived. Their application to the quantized Hall effect is discussed.     

On the theory of the integral quantized Hall effect

Summary In the present paper the integral quantum Hall effect is studied using the Schrauben functions which are suitable eigenfunctions to describe the quantum transport in uniform electric and magnetic fields. The effect of Landau band structure on the Hall quantization is investigated. A model calculation of the conductivities σ_xy and σ_yy is presented and the onset of a Hall current dissipation is discussed. Also, the quantum oscillations of a free-electron gas into the quantum Hall regime are studied, including the electric-field effect.     

Disorder and the theory of the fractionally quantized Hall effect

In a recent article we showed that weak disorder does not change the ground states of a system of interacting electrons on a torus in a strong magnetic field at a filling factorf ₀, if the denominator off ₀ is odd and small. At filling factors near but not equalf ₀ disorder becomes important. We show that the ground states and the low excited states of such a system may be constructed from the ground states atf ₀ through adding localized electrons or holes. These states have the same degeneracy as the ground states atf ₀ and lead to the same value of the Hall conductance.     

Nondissipative spin Hall effect via quantized edge transport

The spin Hall effect in a two-dimensional electron system on honeycomb lattice with both intrinsic and Rashba spin-orbit couplings is studied numerically. Integer quantized spin Hall conductance is obtained at the zero Rashba coupling limit when electron Fermi energy lies in the energy gap created by the intrinsic spin-orbit coupling, in agreement with recent theoretical prediction. While nonzero Rashba coupling destroys electron spin conservation, the spin Hall conductance is found to remain near the quantized value, being insensitive to disorder scattering, until the energy gap collapses with increasing the Rashba coupling. We further show that the charge transport through counterpropagating spin-polarized edge channels is well quantized, which is associated with a topological invariant of the system.     

The dip effect under integer quantized Hall conditions

In this work we investigate an unusual transport phenomenon observed in two-dimensional electron gas under integer quantum Hall effect conditions. Our calculations are based on the screening theory, using a semi-analytical model. The transport anomalies are dip and overshoot effects, where the Hall resistance decreases (or increases) unexpectedly at the quantized resistance plateaus intervals. We report on our numerical findings of the dip effect in the Hall resistance, considering GaAs/AlGaAs heterostructures in which we investigated the effect under different experimental conditions. We show that, similar to overshoot, the amplitude of the dip effect is strongly influenced by the edge reconstruction due to electrostatics. It is observed that the steep potential variation close to the physical boundaries of the sample results in narrower incompressible strips, hence, the experimental observation of the dip effect is limited by the properties of these current carrying strips. By performing standard Hall resistance measurements on gate defined narrow samples, we demonstrate that the predictions of the screening theory is in well agreement with our experimental findings.     

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.     

Chern semimetal and the quantized anomalous Hall effect in HgCr2Se4

In 3D momentum space, a topological phase boundary separating the Chern insulating layers from normal insulating layers may exist, where the gap must be closed, resulting in a "Chern semimetal" state with topologically unavoidable band crossings at the Fermi level. This state is a condensed-matter realization of Weyl fermions in (3+1)D, and should exhibit remarkable features, such as magnetic monopoles and Fermi arcs. Here we predict, based on first principles calculations, that such a novel quantum state can be realized in a known ferromagnetic compound HgCr2Se4, with a single pair of Weyl fermions separated in momentum space. The quantum Hall effect without an external magnetic field can be achieved in its quantum-well structure.     

The plateau widths of the quantized Hall conductance

A possible mechanism for the normal quantized Hall conductance is suggested from a viewpoint different from Laughlin's theory, and a formula for the widths of the Hall conductance plateaus is presented, which shows good agreement with the recent experimental data by Störmer et al.