Semiclassical analysis of two- and three-spin antiferromagnets and anyons on a sphere

TL;DR

This paper performs a semiclassical analysis of small antiferromagnetic spin systems, revealing their connection to anyons on a sphere and ensuring correct permutation symmetry through the Wess-Zumino term.

Contribution

It introduces a method to derive semiclassical wave functions from spin wave functions, accounting for the Wess-Zumino term's role in spin permutation symmetry.

Findings

Wess-Zumino term ensures correct permutation symmetry of wave functions.

Semiclassical problem maps to anyons on a sphere with specific statistics parameters.

Wess-Zumino term does not affect the energy spectrum.

Abstract

We do a semiclassical analysis for two or three spins which are coupled antiferromagnetically to each other. The semiclassical wave functions transform correctly under permutations of the spins if one takes into account the Wess-Zumino term present in the path integral for spins. The Wess-Zumino term here is a total derivative which has no effect on the energy spectrum. The semiclassical problem is related to that of anyons moving on a sphere with the statistics parameter $\theta$ being $2 \pi S$ for two spins and $3 \pi S$ for three spins. Finally, we present a novel way of deriving the semiclassical wave functions from the spin wave functions.