Vortex rings in ferromagnets: Numerical simulations of the time-dependent three-dimensional Landau-Lifshitz equation

Vortex ring solutions are presented for the Landau-Lifshitz equation, which models the dynamics of a three-dimensional ferromagnet. The vortex rings propagate at constant speed along their symmetry axis and are characterized by the integer-valued Hopf charge. They are stable to axial perturbations, but it is demonstrated that an easy axis anisotropy results in an instability to perturbations, which breaks the axial symmetry. The unstable mode corresponds to a migration of spin flips around the vortex ring that leads to a pinching instability and, ultimately, the collapse of the vortex ring. It is found that this instability does not exist for an isotropic ferromagnet. Similarities between vortex rings in ferromagnets and vortons in cosmology are noted.

Figure 1

(Color online) Tubular surfaces indicating the preimages of the points $\mathbf{\varphi }=(0,0,-1)$ (light surface) and $\mathbf{\varphi }=(-1,0,0)$ (dark surface), at times $t=0$, 39, 78, and 109, for a vortex ring with Hopf charge $Q=0$.

Figure 2

(Color online) Same as Fig. 1, but with Hopf charge $Q=5$.

Figure 3

The position of the vortex ring as a function of time, for Hopf charge $Q=0$ (upper curve) and $Q=5$ (lower curve).

Figure 4

(Color online) The isosurface ${\varphi }_3=0$ at times $t=0$, 37.5, 50, and 55.625, for a $Q=0$ vortex ring with an initial 10% squashing in the $x_1$ direction.

Figure 5

(Color online) The isosurface ${\varphi }_3=0$ during increasing stages of a simulated annealing relaxation of a $Q=1$ vortex ring in the anisotropic system with $A=1$. The spin flips and momentum are constrained to be $N=2186$ and $P_3=5163$. The initial condition is squashed by 10% in the $x_1$ direction and leads to a pinching instability.

Figure 6

(Color online) The isosurface ${\varphi }_3=0$ during increasing stages of a simulated annealing relaxation of a $Q=1$ vortex ring in the isotropic system, i.e., $A=0$. The spin flips and momentum are constrained to be $N=2186$ and $P_3=5163$. The initial condition is a slightly pinched and squashed vortex ring, but leads to an axially symmetric stable solution.