Size scaling of the exchange interaction in the quantum Hall effect regime

Daniel Werner,Josef Oswald

doi:10.1103/physrevb.102.235305

Abstract

We use the self-consistent Hartree-Fock approximation for numerically addressing the integer quantum Hall effect (IQHE) regime in terms of many-body physics at higher Landau levels (LLs). We investigate the dependence of many-particle interactions on the lateral size of the electron system. We use the exchange enhancement of the $g$-factor for spin-polarized Landau levels as an indicator for the strength of the exchange interaction. The driving force for the $g$-factor enhancement is a Hund's rule behavior for the occupation of spin-split Landau levels that lowers the many-particle ground state energy by arranging as many spins in parallel as possible. By increasing the total number of electrons and total number of available states per LL, it can therefore be expected that the exchange-enhanced spin gap should increase as well. In contrast to the dependence on the magnetic field, an increase of the total number of states by simply increasing the system size at constant magnetic field shows a clear saturation behavior above a lateral system size of 1000 nm. The importance of this result is underlined by an extended introduction, which demonstrates the permanent dominance of many-body interactions in all transport regimes of the IQHE. A modeling of IQHE systems therefore has to include many-body interactions, and our results open a pathway towards many-body modeling of quantum Hall systems of macroscopic size.

Highlights

Almost 40 years after its discovery, the quantum Hall effect (QHE) [1] was formally included among the select group of high-precision experiments to form the basis of a new SI system
Three major research fields can be identified within the integer quantum Hall effect (IQHE) that pinpoint the milestones for theoretical approaches in the past
As research field I, we identify the famous scaling theory that is based on the so-called localization picture of the integer QHE (IQHE)

Summary

INTRODUCTION

Almost 40 years after its discovery, the quantum Hall effect (QHE) [1] was formally included among the select group of high-precision experiments to form the basis of a new SI system. Our approach allows us to study a smooth crossover between these transport regimes from dominating narrow noninteracting channels (field I, scaling theory) to the creation of wide compressible stripes (field II, so far understood on the basis of single-particle interactions) to the generation bubbles and stripes (field III, many-body interactions) in low-disorder systems. The latter has been completed very recently [26] and completes a unifying picture for all transport regimes of the IQHE [27]. On this background it might be sufficient to model a macroscopic electron system by dividing it into plaquettes of 1000 × 1000 nm and solving each separately by the HF procedure, and subsequently applying a suitable matching technique for joining them to get the complete system

Self-consistent Hartree-Fock in Landau basis

Data representation

RESULTS

DISCUSSION AND OUTLOOK