We present a computationally efficient general first-principles based method for spin-lattice simulations for solids and clusters. The method is based on a coupling of atomistic spin dynamics and molecular dynamics simulations, expressed through a spin-lattice Hamiltonian, where the bilinear magnetic term is expanded up to second order in displacement. The effect of first order spin-lattice coupling on the magnon and phonon dispersion in bcc Fe is reported as an example, and we observe good agreement with previous simulations. In addition, we also illustrate the coupled spin-lattice dynamics method on a more conceptual level, by exploring dissipation-free spin and lattice motion of small magnetic clusters (a dimer, trimer and quadmer). The here discussed method opens the door for a quantitative description and understanding of the microscopic origin of many fundamental phenomena of contemporary interest, such as ultrafast demagnetization, magnetocalorics, and spincaloritronics. arXiv:1804.03119v2 [cond-mat.mtrl-sci] 9 Jul 2018

The incompressible Euler equations on a sphere are fundamental in models of oceanic and atmospheric dynamics. The long-time behaviour of solutions is largely unknown; statistical mechanics predicts a steady vorticity configuration, but detailed numerical results in the literature contradict this theory, yielding instead persistent unsteadiness. Such numerical results were obtained using artificial hyperviscosity to account for the cascade of enstrophy into smaller scales. Hyperviscosity, however, destroys the underlying geometry of the phase flow (such as conservation of Casimir functions), and therefore might affect the qualitative long-time behaviour. Here we develop an efficient numerical method for long-time simulations that preserve the geometric features of the exact flow, in particular conservation of Casimirs. Long-time simulations on a non-rotating sphere then reveal three possible outcomes for generic initial conditions: the formation of either 2, 3, or 4 coherent vortex structures. These numerical results contradict both the statistical mechanics theory and previous numerical results suggesting that the generic behaviour should be 4 coherent vortex structures. Furthermore, through integrability theory for point vortex dynamics on the sphere, we present a theoretical model which describe the mechanism by which the three observed regimes appear. We show that there is a correlation between a first integral γ (the ratio of total angular momentum and the square root of enstrophy) and the long-time behaviour: γ small, intermediate, and large yields most likely 4, 3, or 2 coherent vortex formations. Our findings thus suggest that the likely long-time behaviour can be predicted from the first integral γ.

The theory of isospectral flows comprises a large class of continuous dynamical systems, particularly integrable systems and Lie-Poisson systems. Their discretization is a classical problem in numerical analysis. Preserving the spectra in the discrete flow requires the conservation of high order polynomials, which is hard to come by. Existing methods achieving this are complicated and usually fail to preserve the underlying Lie-Poisson structure.Here we present a class of numerical methods of arbitrary order for Hamiltonian and non-Hamiltonian isospectral flows, which preserve both the spectra and the Lie-Poisson structure. The methods are surprisingly simple, and avoid the use of constraints or exponential maps. Furthermore, due to preservation of the Lie-Poisson structure, they exhibit near conservation of the Hamiltonian function. As an illustration, we apply the methods to several classical isospectral flows.