Quantum Hall Resistance and Quantum Hall Plateaus from Edge State Quantization

TL;DR

This paper derives the quantum Hall resistance and plateaus directly from first principles by quantizing edge states through boundary conditions, providing a clear theoretical foundation matching experimental observations.

Contribution

It offers a novel derivation of the quantum Hall effect from first principles using boundary conditions, avoiding phenomenological assumptions.

Findings

Derives the Hall resistance directly from edge state quantization.

Reproduces quantum Hall plateaus matching experimental data.

Provides explicit relation for filling factor in terms of physical parameters.

Abstract

Despite the extensive literature on the quantum Hall effect (QHE), a direct derivation of the phenomenological formula $\rho_{xy} = h/e^2\nu$ from first principles has remained elusive. In this work, we revisit the Landau and Landauer-B\"uttiker formalisms and impose hard-wall boundary conditions on the wavefunction, an essential but often overlooked constraint. This condition quantizes the guiding center position and the longitudinal wave number $k_x$, leading naturally to a discrete number of edge states without invoking energy bending. We derive the Hall resistance directly and recover the standard result $\rho_{xy} = h/e^2\nu$, along with an explicit expression for the filling factor $\nu$ in terms of the Fermi energy and magnetic field. The resulting resistance steps reproduce the observed QHE plateaus and match experimental data without fitting parameters.