Two minimal-variable symplectic integrators for stochastic spin systems

We present two symplectic integrators for stochastic spin systems, based on the classical implicit midpoint method. The spin systems are identified with Lie-Poisson systems in matrix algebras, after which the numerical methods are derived from structure-preserving Lie-Poisson integrators for isospectral stochastic matrix flows. The integrators are thus geometric methods, require no auxiliary variables, and are suited for general Hamiltonians and a large class of stochastic forcing functions. Conservation properties and convergence rates are shown for several single-spin and multispin systems.