On the quantum Hall effect

We consider a particle in a 2 dimensional plane in a periodic potential and a homogeneous magnetic field perpendicular to the plane. Kubo's expression for conductivity of the Hall current is an integer.

This result of Thouless, Kohomoto, Nightingale and den Nijs is interpreted geometrically.     

Time-reversal-symmetry-broken quantum spin Hall effect

The quantum spin Hall (QSH) state of matter is usually considered to be protected by time-reversal (TR) symmetry. We investigate the fate of the QSH effect in the presence of the Rashba spin-orbit coupling and an exchange field, which break both inversion and TR symmetries. It is found that the QSH state characterized by nonzero spin Chern numbers C(±) = ±1 persists when the TR symmetry is broken. A topological phase transition from the TR-symmetry-broken QSH phase to a quantum anomalous Hall phase occurs at a critical exchange field, where the bulk band gap just closes. It is also shown that the transition from the TR-symmetry-broken QSH phase to an ordinary insulator state cannot happen without closing the band gap.     

Spin-orbit scattering in the quantum Hall effect

We study dynamics of electrons in a magnetic field using a network model with two channels per link with random mixing, while the intrachannel potential is periodic (non-random); the channels represent two spin states. We consider channel mixing as function of the energy separation of the two extended states, and show that the phase diagram is different from the standard quantum Hall diagram for random intrachannel potential.     

Disorder and the fractional quantum Hall effect

The fractional quantum Hall effect is studied in the 2D electron gas of four GaAs-Al_xGa_1?xAs heterostructures. Localization due to disorder, known to give rise to the wide integral quantum Hall plateaus, is demonstrated to inhibit the fractional effect, which is observed only in the higher mobility samples.     

Ohm's law in the quantum Hall effect

The linearity of the Hall currents as functions of the voltage is investigated. We consider interacting electrons on a torusT ₂. They are subjected to a strong perpendicular magnetic field and a smooth but disordered substrate potential. In a moving frame of reference the Hamiltonian is time-dependent due to the constantly shifted substrate. Assuming a small energy gap separating the lowlying subspace, adiabatic expansion is used to construct a solution. Its general properties allow to calculate the Hall currents. Any power-law corrections to linearity are absent. The reliability of the result is assessed by estimates of the parameter controlling the expansion which is asymptotic.     

Geometric invariants of the quantum Hall effect

We study the two-dimensional Hall effect with a random potential. The Hall conductivity is identified as a geometric invariant associated with an algebra of observables. Using the pairing betweenK-theory and cyclic cohomology theory, we identify this geometric invariant with a topological index, thereby giving the Hall conductivity a new interpretation.Supported in part by the National Science Foundation under Grant No. DMS-8717185     

Vortex drag in the quantum Hall effect

A new model of momentum and electric field transfer between two adjacent 2D electron systems in the quantum Hall effect is proposed. The drag effect is due to momentum transfer from the vortex system of one layer to the vortex system of another layer. The remarkable result of this approach is a periodic change of sign of the dragged electric field as a function of the difference between the layer filling factors. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 4, 276–279 (25 February 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.