Generalized spin systems and σ models

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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, nonrelativistic σ-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.