Quantum Mechanically Induced Hopf Term in the O(3) Nonlinear Sigma Model

TL;DR

This paper demonstrates that the Hopf term in the 2+1 dimensional O(3) nonlinear sigma model is a quantum-induced topological term, similar to the theta-term in QCD, revealed through quantization ambiguity and topological analysis.

Contribution

It provides a detailed topological analysis showing the Hopf term as a quantum effect arising from quantizing an S^1 degree of freedom in the model.

Findings

The Hopf term is induced quantum mechanically, akin to the QCD theta-term.

The adjoint orbit parametrization effectively handles topological numbers.

Quantization of hidden degrees of freedom reveals the Hopf term.

Abstract

The Hopf term in the $2 + 1$ dimensional O(3) nonlinear sigma model, which is known to be responsible for fractional spin and statistics, is re-examined from the viewpoint of quantization ambiguity. It is confirmed that the Hopf term can be understood as a term induced quantum mechanically, in precisely the same manner as the $\theta$-term in QCD. We present a detailed analysis of the topological aspect of the model based on the adjoint orbit parametrization of the spin vectors, which is not only very useful in handling topological (soliton and/or Hopf) numbers, but also plays a crucial role in defining the Hopf term for configurations of nonvanishing soliton numbers. The Hopf term is seen to arise explicitly as a quantum effect which emerges when quantizing an $S^1$ degree of freedom hidden in the configuration space.