Self - Dual Chern - Simons Solitons and Generalized Heisenberg Ferromagnet Models

We consider the (2+1)-dimensional gauged Heisenberg ferromagnet model coupled with the Chern-Simons gauge fields. Self-dual Chern-Simons solitons, the static zero energy solution saturating Bogomol’nyi bounds, are shown to exist when the generalized spin variable is valued in the Hermitian symmetric spaces G/H . By gauging the maximal torus subgroup of H , we obtain self-dual solitons which satisfy vortex-type nonlinear equations thereby extending the two dimensional instantons in a nontrivial way. An explicit example for the CP (N) case is given. 1 E-mail address; [email protected] 2 E-mail address; [email protected] Recently, there appeared an action principle of the generalized Heisenberg ferromagnet model in terms of a nonrelativistic nonlinear sigma model defined on a Lie group G [1]. This action possesses a local H subgroup symmetry so that the physical spin variables take value on the coadjoint orbit of the Hermitian symmetric space G/H . The symplectic structure on each orbit also allows a direct first order action in terms of generalized spin variables. The use of Hermitian symmetric space made possible a systematic generalization of the Heisenberg ferromagnet model according to the Cartan’s classification of symmetric spaces [2] and led to the infinite conservation laws of the model [1]. In this Letter, we consider the generalized Heisenberg ferromagnet model in (2+1)dimensions. The motivations are twofold. Firstly, the model itself can be used in describing generalized planar ferromagnetisms where the generality coming from the large degrees of freedom of symmetric spaces can be used to handle various physical situations. Secondly, by gauging and coupling the model with the Chern-Simons gauge fields, we are led to the Chern-Simons self-dual solitons [3] which attracted an upsurge of recent interests in regard of the application to the quantum Hall effect and the high-Temperature superconductivity [4, 5, 6]. Here, we focus on the second motivation and show that, using the properties of Hermitian symmetric spaces, the Hamiltonian of the model is bounded below by a topological charge. The resulting Chern-Simons solitons satisfy a vortex-type equation when the model is gauged with the maximal torus subgroup of H and added by a gauge invariant term which induces vacuum symmetry breaking. To our knowledge, this vortex-type equation is new and is likely to possess similar properties to those of the vortex equation of Abelian Higgs model [7] or gauged non-linear Schrödinger model [8]. As an example, we present an explicit expression for the vortex-type equation in the case of CP (N). We first recall that the action principle for the (1+1)-dimensional generalized Heisenberg ferromagnet model defined on the Hermitian symmetric space G/H can be given by [1]