Integer quantum Hall effect with realistic boundary condition: Exact quantization and breakdown

Abstract

A theory of the integer quantum Hall effect ͑QHE͒ in realistic systems based on a von Neumann lattice is presented. We show that the momentum representation is quite useful and that the quantum Hall regime ͑QHR͒, which is defined by the propagator in the momentum representation, is realized. In the QHR, the Hall conductance is given by a topological invariant of the momentum space and is quantized exactly. The edge states do not modify the value and topological property of xy in the QHR. We next compute the distribution of current based on the effective action and generally find a finite amount of current in the bulk and the edge. Due to the Hall electric field in the bulk, breakdown of the QHE occurs. The critical electric field of the breakdown is proportional to B 3/2 and the proportional constant has no dependence on Landau levels in our theory, in agreement with recent experiments.

Field Theory of Anisotropic Quantum Hall Gas: Metrology and a Novel Quantum Hall Regime

“…41 Hence the Hall resistance measured in the present situation agrees with the exactly quantized value. Thus resistivities at low temperature are given in our theory as,…”

“…As was shown in Ref. 41 , this topological invariant is determined by the magnetic field and has no dependence on spatial component of the momentum. Hence this topological invariant does not change the value even when the integration variables are replaced with the discrete values.…”

“…Thus these states depend on the Hall electric field. 41,42 The other line (solid line) stands for those of strip states around Fermi energy which are disconnected from current contacts and are not carrying current in the lowest order. The dashed line stands for the Coulomb interaction.…”

“…E eff H is an effective Hall electric field and is given as E eff H = E H /γ, which is enhanced by the strong localization effect. 41,42 In experiments, 43 1/γ is on the order of 10. ∆E 2 (p) comes from the band electrons which is computed in the previous section.…”

Phase Diagram for the Breakdown of the Quantum Hall Effect

“…An understanding of the physical origin of breakdown and its implications for the resistance standard are of fundamental importance [3]. From a theoretical point of view a number of models have been proposed [6][7][8][9][10][11] but the exact mechanism for the breakdown remains controversial. What is clear is that the breakdown is driven by the Hall voltage V H I h ne 2 for a current I at integer filling factor n. In addition, for the rather high critical currents typically observed, the current flows through both edge states and the bulk of the sample [5,12].…”