Monte Carlo evaluation of trial wavefunctions for the fractional quantized Hall effect: Spherical geometry

Monte Carlo methods have been employed to evaluate trial wave-functions for quantized Hall states and excitations using up to 92 electrons on a sphere. Results are presented for the energy of quasiparticle and quasihole excitations at filling factorv=1/3. These are consistent with extrapolations of exact small system calculations and earlier Monte Carlo results. A trial wavefunction for neutral excitations atv=1/m (m odd integer) is proposed. Results for its excitation energy in the long wavelength limit, atv=1/3, are consistent with those based on the single mode approximation. We have also studied a trial wavefunction representing a quasiparticle excitation involving an electron with reversed spin. Results are presented for the effects on excitation energies of the finite extent of the electron wave function perpendicular to the surface. We have also evaluated the energy of a microscopic trial wave-function for the ground state atv=2/5.Dedicated to Professor Harry Thomas on the occasion of his 60th birthday     

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.     

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.     

Topology and fractional quantum Hall effect

Starting from Laughlin-type wave functions with generalized periodic boundary conditions describing the degenerate ground state of a quantum Hall system we explicitly construct r-dimensional vector bundles. It turns out that the filling factor ν is given by the topological quantity c₁/r where c₁ is the first Chem number of these vector bundles. In addition, we managed to proof that under physical natural assumptions the stable vector bundles correspond to the experimentally dominating series of measured fractional filling factors ν = n/(2pn±1). Most remarkably, due to the very special form of the Laughlin wave functions the fluctuations of the curvature of these vector bundles converge to zero in the limit of infinitely many particles which shows a new mathematical property. Physically, this means that in this limit the Hall conductivity is independent of the boundary conditions which is very important for the observability of the effect. Finally, we discuss the relation of this result to a theorem of Donaldson.     

Theory of the quantized Hall effect (III)

In the previous paper, we have demonstrated the need for a phase transition as a function of θ in the non-linear σ-model describing the quantized Hall effect. In this work, we present arguments for the occurrence of exactly such a transition. We make use of a dilute gas instanton approximation as well as present a more rigorous duality argument to show that the usual scaling of the conductivity to zero at large distances is altered whenever $\text{σ}{}_{\mathrm{xy}}{}^{\mathrm{(0)}}\text{≈}\text{1}\text{2}\text{ne}{}^2\text{/h}$, n integer. This then completes our theory of the quantized Hall effect.     

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.     

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.     

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.