Generalized Spin Systems and $σ$--Models

Details Generalized Spin Systems and $σ$--Models Generalized Spin Systems and $σ$--Models

1992-10-28

arxiv.org/abs/hep-th/9210145v2

Phys.Rev.B48:3190-3205,1993

Physics

High Energy Physics

High Energy Physics - Theory

35, plain, IC/92/294

Scientific paper

10.1103/PhysRevB.48.3190

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.

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