Field Theory of Quantum Hall Effects, Composite Bosons, Vortices and Skyrmions

TL;DR

This paper develops a field theory for quantum Hall effects based on composite bosons, describing excitations as topological solitons like vortices and Skyrmions, and compares theoretical predictions with experimental data.

Contribution

It introduces a new field-theoretic framework that directly relates to microscopic wave functions and captures topological excitations in quantum Hall systems.

Findings

All excitations are nonlocal topological solitons in spinless systems.

Spontaneous quantum coherence leads to a Goldstone mode and topological solitons in spinful systems.

Skyrmion energies are evaluated and match experimental observations.

Abstract

A field theory of quantum Hall effects is constructed based on the \CB picture. It is tightly related with the microscopic wave-function theory. The characteristic feature is that the field operator describes solely the physical degrees of freedom representing the deviation from the Laughlin state. It presents a powerful tool to analyze excited states within the \LLL. It is shown that all excitations are nonlocal topological solitons in the spinless quantum Hall system. On the other hand, in the presence of the spin degree of freedom it is shown that a quantum coherence develops spontaneously, where excitations include a Goldstone mode besides nonlocal topological solitons. Solitons are vortices and Skyrmions carrying the U(1) and SU(2) topological charges, respectively. Their classical configurations are derived from their microscopic wave functions. The Skyrmion appears merely as a low-energy excitation within the \LLL and not as a solution of the effective nonlinear sigma model. We use it as a consistency check of the Skyrmion theory that the Skyrmion is reduced to the vortex in the vanishing limit of the Skyrmion size. We evaluate the activation energy of one Skyrmion and compare it with experimental data.