Vortex equations governing the fractional quantum Hall effect

TL;DR

This paper proves the existence and uniqueness of vortex solutions to coupled non-linear equations modeling the fractional quantum Hall effect in double-layered electron systems, providing explicit conditions and decay estimates.

Contribution

It establishes a comprehensive existence theory for vortex equations in both periodic and full-plane settings, including explicit conditions and flux quantization.

Findings

Existence and uniqueness of vortex solutions proven.

Explicit conditions relating domain size and vortex numbers derived.

Quantization of magnetic flux demonstrated.

Abstract

An existence theory is established for a coupled non-linear elliptic system, known as "vortex equations", describing the fractional quantum Hall effect in 2-dimensional double-layered electron systems. Via variational methods, we prove the existence and uniqueness of multiple vortices over a doubly periodic domain and the full plane. In the doubly periodic situation, explicit sufficient and necessary conditions are obtained that relate the size of the domain and the vortex numbers. For the full plane case, existence is established for all finite-energy solutions and exponential decay estimates are proved. Quantization phenomena of the magnetic flux are found in both cases.