Generalized Spin Systems and $\sigma$--Models

TL;DR

This paper generalizes $SU(2)$ spin systems to arbitrary compact groups, deriving continuum $\sigma$-models on homogeneous spaces, analyzing their equations of motion, dispersion relations, and topological solutions, with implications for both ferromagnetic and antiferromagnetic cases.

Contribution

It introduces a broad framework for $G$-spin systems and their continuum limits, extending known models to arbitrary groups and exploring their dynamic and topological properties.

Findings

Continuum limits are non-relativistic $\sigma$-models on $G/H$.

Dispersion relations differ for ferromagnetic and antiferromagnetic cases.

Existence of instanton solutions in antiferromagnetic models.

Abstract

A generalization of the $SU(2)$--spin systems on a lattice and their continuum limit to an arbitrary compact group $G$ is discussed. The continuum limits are, in general, non--relativistic $\sigma$--model type field theories targeted on a homogeneous space $G/H$, where $H$ contains the maximal torus of $G$. In the ferromagnetic case the equations of motion derived from our continuum Lagrangian generalize the Landau--Lifshitz equations with quadratic dispersion relation for small wave vectors. In the antiferromagnetic case the dispersion law is always linear in the long wavelength limit. The models become relativistic only when $G/H$ is a symmetric space. Also discussed are a generalization of the Holstein--Primakoff representation of the $SU(N)$ algebra, the topological term and the existence of the instanton type solutions in the continuum limit of the antiferromagnetic systems.