Chern - Simons Gauge Field Theory of Two - Dimensional Ferromagnets

TL;DR

This paper develops a Chern-Simons gauge field theory for two-dimensional ferromagnets, linking magnetic vortices to anyon theory, and deriving equations describing vortex configurations and their topological charges.

Contribution

It introduces a Chern-Simons gauged nonlinear Schrödinger equation from the Heisenberg model, connecting magnetic vortices to self-dual solitons and anyon electric charges.

Findings

Magnetic vortices correspond to self-dual Chern-Simons solitons.

Vortex-antivortex configurations are described by the sinh-Gordon equation.

Topological charge relates to the electric charge of anyons.

Abstract

A Chern-Simons gauged Nonlinear Schr\"odinger Equation is derived from the continuous Heisenberg model in 2+1 dimensions. The corresponding planar magnets can be analyzed whithin the anyon theory. Thus, we show that static magnetic vortices correspond to the self-dual Chern - Simons solitons and are described by the Liouville equation. The related magnetic topological charge is associated with the electric charge of anyons. Furthermore, vortex - antivortex configurations are described by the sinh-Gordon equation and its conformally invariant extension. Physical consequences of these results are discussed.