$SU(N)\to SU(2)$ symmetry breaking in quantum antiferromagnets

TL;DR

This paper investigates how symmetry breaking from $SU(4)$ to $SU(2)$ in a quantum antiferromagnet leads to a novel effective field theory, revealing that skyrmions bind into triplets due to topological charge relations.

Contribution

It introduces the $O(3) imes O(2)$ nonlinear sigma model as the effective theory for symmetry-broken $SU(4)$ systems and establishes a topological charge relation linking $CP^{3}$ and $O(3)$ skyrmions.

Findings

Skyrmions in the $CP^{3}$ model bind into triplets under symmetry breaking.

The effective theory is an $O(3) imes O(2)$ nonlinear sigma model, not the usual $O(3)$ model.

General relation $2S Q_{O(3)}=q_{CP^{2S}}$ for spin-$S$ systems.

Abstract

We study a $SU(2)$-symmetric spin-${3}/{2}$ system on a bipartite lattice close to the antiferromagnetic $SU(4)$-symmetric point, which can be described by the $CP^{3}$ model with a perturbation breaking the symmetry from $SU(4)$ down to $SU(2)$ and favoring the N\'eel ordering. We show that the effective theory of the perturbed model is not the usual $O(3)$ nonlinear sigma model (NLSM), but rather the $O(3)\times O(2)$ NLSM. We show that in the presence of perturbation, the topological charge $q$ of the $CP^{3}$ field is connected to the $O(3)$-NLSM type topological charge of the spin texture $Q$ (defined in a usual way via the unit N\'eel vector) by the relation $q=3Q$, thus under the influence of the perturbation unit-charge skyrmions of $CP^{3}$ model bind into triplets. We also show that in the general spin-$S$ case, symmetry breaking from $SU(2S+1)$ to $SU(2)$ results in the general relation $2S Q_{O(3)}=q_{CP^{2S}}$ between $CP^{2S}$ and $O(3)$ charges, so one can expect $2S$-multiplet binding of skyrmions.