Nonlinear $\sigma$-Model in (2+1) dimensions

TL;DR

This paper explores the (2+1)-dimensional nonlinear sigma model, highlighting how skyrmions relate to fermionic properties and how the Hopf term induces exotic spin and fractional statistics.

Contribution

It demonstrates the connection between skyrmion topological charge and fermionic number, and shows how the Hopf term leads to fractional statistics in the model.

Findings

Skyrmions carry fermionic charge

Hopf term induces fractional statistics

Topological configurations influence quantum properties

Abstract

The nonlinear $\sigma$-model in (2+1) dimensions admits topological configurations called skyrmions. The topological charge of skyrmions turn out to be the fermionic number and the fermionic current is dictated by the skyrmion field configuration. The peculiar feature of this model is that a Hopf term introduced into the Lagrangian can lead to exotic spin and fractional statistics.