Integer Quantum Hall Effect with Realistic Boundary Condition : Exact Quantization and Breakdown

TL;DR

This paper presents a theoretical analysis of the integer quantum Hall effect using von Neumann lattice, demonstrating exact quantization of Hall conductance, the role of edge states, and the conditions for breakdown under electric fields.

Contribution

It introduces a realistic system model based on von Neumann lattice, showing that Hall conductance is a topological invariant unaffected by edge states, and derives the breakdown electric field dependence.

Findings

Hall conductance is exactly quantized and topologically invariant.

Edge states do not alter the quantization or topological properties.

Breakdown electric field scales as B^{3/2} and matches experimental observations.

Abstract

A theory of integer quantum Hall effect(QHE) in realistic systems based on von Neumann lattice is presented. We show that the momentum representation is quite useful and that the quantum Hall regime(QHR), which is defined by the propagator in the momentum representation, is realized. In 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 $\sigma_{xy}$ in QHR. We next compute distribution of current based on effective action and find a finite amount of current in the bulk and the edge, generally. 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 the recent experiments.