Dynamics of Vortices in Two-Dimensional Magnets

Abstract

Theories, simulations and experiments on vortex dynamics in quasi-two-dimensional magnetic materials are reviewed. These materials can be modelled by the classical two-dimensional anisotropic Heisenberg model with XY (easy-plane) symmetry. There are two types of vortices, characterized by their polarization (a second topological charge in addition to the vorticity): Planar vortices have Newtonian dynamics (evenorder equations of motion) and exhibit strong discreteness effects, while non-planar vortices have non-Newtonian dynamics (odd-order equations of motion) and smooth trajectories. These results are obtained by a collective variable theory based on a generalized travelling wave ansatz which allows a dependence of the vortex shape on velocity, acceleration etc.. An alternative approach is also reviewed and compared, namely the coupling of the vortex motion to certain quasi-local spinwave modes. The influence of thermal fluctuations on single vortices is investigated. Different types of noise and damping are discussed and implemented into the microscopic equations which yields stochastic equations of motion for the vortices. The stochastic forces can be explicitly calculated and a vortex diffusion constant is defined. The solutions of the stochastic equations are compared with Langevin dynamics simulations. Moreover, noise-induced transitions between opposite polarizations of a vortex are investigated. For temperatures above the Kosterlitz-Thouless vortex-antivortex unbinding transition, a phenomenological theory, namely the vortex gas approach, yields central peaks in the dynamic form factors for the spin correlations. Such peaks are observed both in combined Monte Carlo-and Spin Dynamics-Simulations and in inelastic neutron scattering experiments. However, the assumption of ballistic vortex motion appears questionable.

Coarsening of two-dimensional XY model with Hamiltonian dynamics: logarithmically divergent vortex mobility

“…Here we find that the same model can explain the coarsening process only with a modification of the initial condition that accounts for the finite length scale L 0 at time t = 0 due to the special nature of Hamiltonian dynamics and corresponding energy conservation in the present system. Similar phenomenon of logarithmically divergent mobility was also found experimentally in the quasi-two-dimensional diffusion of colloids [40] or diffusion of protein molecules on the membranes [41] under hydrodynamic effect, and also in numerical simulations of the vortex diffusion in the anisotropic Heisenberg system in two dimensions with spin precession [42][43][44].…”

“…We will set the proportionality constant k as k = 1. As for the functional form of the m(R) and µ(R), we use the same forms as that employed in [5] where logarithmic dependences of the vortex mobility µ(R) and m(R) are used [42][43][44] as follows…”

2D Heisenberg model from rotating membrane

“…The excitation in the two-dimensional system is much richer than that in the one-dimensional spin chain. To our particular interests, topological objects such as vortices can be excited [19,20]. The vortices here carry two topological charges under the homotopy group π 1 and π 2 , respectively.…”

“…The vortices here carry two topological charges under the homotopy group π 1 and π 2 , respectively. They are the vorticity, In fact, the shape of vortex can be given by a more generic travelling wave ansatz [20],…”

Geometric space–time integration of ferromagnetic materials

“…This paper addresses the Landau-Lifshitz equation (LLE) as a nonlinear wave equation supporting solitons and stable magnetic vortices, as considered, e.g., in [5,19,23]. The LLE governs the flow of magnetic spin in a ferromagnetic material.…”