Dynamics of the Heisenberg model and a theorem on stability

摘要

We consider the general discrete classical Heisenberg model (HM) with z axis anisotropy and external magnetic field and show that its phase space is foliated into a family of invariant manifolds (the leaves) diffeomorphic to (S~2)λ, where λ is the number of spins. We also show that the flow on each leaf S is Hamiltonian. Subsequently, we focus on the isotropic HM in the absence of external field. We discuss the rotational symmetry of the model and further analyze its phase space structure. We prove that the manifold F of longitudinal fixed points intersects each leaf S orthogonally. For a real local flow with a continuous symmetry, we show that the Lyapunov stability of invariant sets on an invariant subspace can be extended to the whole phase space. This general theorem is the main result of the article. We use it to show that, in the case of the isotropic HM, the ferromagnetic state and the antiferromagnetic state with non-zero total spin are both stable fixed points. The theorem does not apply at total spin zero, and indeed the AF state on an equal-spins leaf is found to be unstable.