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.     

Rational numbers of the fractionally quantized new states and their observation in fractionally quantized hall effect

We have found energy states at a fraction of the frequency of the Landau levels. The fraction is in accord with the experimental measurements of fractionally quantized conductance.     

Capacitively coupled measurements of the quantized hall effect in silicon inversion layers

We report the observation of the quantized Hall effect in silicon inversion layers using capacitors in place of ohmic contacts. Measurements were made at frequencies between 5.00 and 100.0 kHz.     

Fractional quantized hall effect in a model with shortranged interaction on a sphere

We consider a model forN electrons confined to the surface of a sphere in a strong radial magnetic field. The electron-electron interaction is modelled as a simple hardcore like repulsion. Numerical calculations have been carried out for up toN=6, neglecting contributions from higher Landau levels. Spectral properties of low-lying states are discussed within the quasihole-quasiparticle picture.     

Experimental search for dynamic current oscillations in the quantum hall effect

We report on our search for the equivalent of a Josephson effect in the quantum Hall effect regime. The negative result of the experiment allows us to set an upper limit on such effects of ∼2×10⁻⁴ of the predicted magnitude.     

de Haas-van Alphen,quantized hall and Meissner effects

In a general case the two dimensional de Haas-van Alphen effect is related to macroscopic non-dissipative currents at the edge states. The currents exponentially decay inside the sample. At low enough temperatures this is manifested by temperature-independent spikes in differential magnetic susceptibility χ. Diamagnetism yields the deviation in the Quantized Hall filling factor from an exact integer. Temperature independent χ spikes are demonstrated also in relatively thick samples.     

Observation of a quantized hall resistivity in the presence of mesoscopic fluctuations

We present an experimental study of mesoscopic, two-dimensional electronic systems at high magnetic fields. Our samples, prepared from a low-mobility InGaAs/InAlAs wafer, exhibit reproducible, sample specific, resistance fluctuations. Focusing on the lowest Landau level, we find that, while the diagonal resistivity displays strong fluctuations, the Hall resistivity is free of fluctuations and remains quantized at its nu=1 value, h/e(2). This is true also in the insulating phase that terminates the quantum Hall series. These results extend the validity of the semicircle law of conductivity in the quantum Hall effect to the mesoscopic regime.     

On localization in the theory of the quantized hall effect: A two-dimensional realization of the θ-vacuum

It is shown that the localization problem in the theory of the quantized Hall effect is governed by the zero-component grassmannian U(2m) non-linear σ-model with θ-term, a two-dimensional analogue of the θ-vacuum in Yang-Mills theory. In this case, θ is to be interpreted as the “bare” value for the Hall conductivity, determined by an underlying non-critical theory. A detailed derivation is presented starting from the replica method and a delta function distribution for the impurities.     

Theory of the quantized hall conductivity plateaus of two-dimensional electron systems

The width of the plateaus of the quantized Hall conductivity of two- dimensional electron systems is expected to drop sharply as a function of μ_BH/kT from the zero point value of $\text{2mμ}{}_{\mathrm{B}}\text{H/π}\text{h}\text{?}\text{;}{}^2$, when it is expressed in terms of electron density.     

Tunneling hall effect

Electron tunneling in a semiconductor heterostructure with a barrier in a weak magnetic field applied parallel to the barrier interfaces is analyzed theoretically. A novel mechanism of the Hall effect in this structure is suggested. It is shown that the Hall current near the sufficiently wide barrier is determined by the orbital effect of the magnetic field on the electron motion under the barrier, rather than by the electron {ie778-1}-drift and scattering in the conductive regions lying to the left and to the right of the barrier.     

Planar hall effect in ferromagnets

The planar Hall effect in a ferromagnetic conductor is considered within a simple two-liquid hydro-dynamic model. It is shown that, even in the simple case of an isotropic Fermi surface in the absence of thermal spread, the magnitude of the Hall effect is comparable to that in semiconductors because of the presence of two groups of conduction electrons with their spins parallel and perpendicular to the quantization axis, respectively. In addition to the planar Hall field, a spin flux parallel to this field arises, with the consequence that the extent of spin polarization of the conduction electrons varies along the Hall field direction (planar spin Hall effect).     

Optical spin hall effect

A remarkable analogy is established between the well-known spin Hall effect and the polarization dependence of Rayleigh scattering of light in microcavities. This dependence results from the strong spin effect in elastic scattering of exciton polaritons: if the initial polariton state has a zero spin and is characterized by some linear polarization, the scattered polaritons become strongly spin polarized. The polarization in the scattered state can be positive or negative dependent on the orientation of the linear polarization of the initial state and on the direction of scattering. Very surprisingly, spin polarizations of the polaritons scattered clockwise and anticlockwise have different signs. The optical spin Hall effect is possible due to strong longitudinal-transverse splitting and finite lifetime of exciton polaritons in microcavities.     

The hall effect in liquid semiconductors

The proper interpretation of the Hall coefficient R_H of liquid semiconductors is still an unsolved problem, and it is a particularly interesting one in view of the frequently observed discrepancy in sign with the Seebeck coefficient. We present some new data which include measurements of the Hall mobility μ_H in liquid thallium-tellurium through the composition range where the Seebeck coefficient S changes sign, and in the range where S is negative. An abrupt change in magnitude of μ_H occurs at the intrinsic composition (Tl₂Te, where S changes sign), and μ_H is observed to have an appreciable dependence on temperature only at this composition. This is consistent with transport by carriers in two bands. μ_H is lower on the Tl-rich side of Tl₂Te, and the formula n_H = 1/R_He yields a value for n_H which is ten times larger than the electron concentration inferred from the composition, assuming that electrons are derived from the Tl in excess of Tl₂Te. We review the various suggested interpretations of the Hall effect in liquid semiconductors from several points of view, and conclude that the conventional formula n = ± 1/R_He is unreliable for inferring either the sign or concentration of the carriers.     

Diffusion hall effect in compensated metals

Analytical investigation is made of the electron-hole diffusion in compensated media like semimetals, which takes place under crossed electric and magnetic fields. It is shown that the diffusion of carriers causes a transverse field effect different to the usual Hall one, if the conducting medium is bounded. The transverse voltage due to this effect increases almost linearly with the magnetic field (H), and changes its sign according to the polarity of H. This voltage is observable even for bulk specimens for which the usual size effect is negligible, but its appearance is easily masked in the presence of impurities. A detailed discussion is made for bismuth, considering a non-parabolic model for the electron Fermi surface. The transverse voltage is found to be very sensitive to the surface recombination velocity of the carriers as well as to the impurity concentration.     

Casimir effect in quantized R2 gravitation

The Casimir energy is calculated in theories of R² gravitation and conformai gravitation in the space R₁ x T₃, where T₃ is a 3D torus. Cosmological applications of the results are discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 59–62, June, 1989.I whish to thank I. L. Bukhbinder for useful discussions.