Soliton dynamics in 3D ferromagnets
Introduction
In the continuum approximation the state of a ferromagnet is described by a three component unit vector, , which gives the local orientation of the magnetization. The dynamics of the ferromagnet, in the absence of dissipation, is governed by the Landau–Lifshitz equationwhere E is the magnetic crystal energy of the ferromagnet. We have chosen units in which the spin stiffness and magnetic moment density of the ferromagnet are set to 1.
The case we study in this paper is that of a three-dimensional ferromagnet with isotropic exchange interactions and an easy axis anisotropy, in which case the energy is given bywhere A>0 is the anisotropy parameter and we choose the ground state to be .
In this case the Landau–Lifshitz equation becomes
This equation has finite energy, stable, exponentially localized solutions known as magnetic solitons [5]. In Section 2, we review the properties of stationary magnetic solitons and in Section 3, we numerically compute solutions describing moving solitons, display their energy dispersion relation and discuss their structure. Finally, in Section 4, we perform numerical simulations of the time dependent equations of motion to investigate the interaction and scattering of two solitons. We find that the force between two solitons depends on their relative internal phase, and that during a collision two solitons can form an unstable loop of magnons which subsequently decays into solitons.
Section snippets
Stationary solitons
In addition to the energy (1.2), the Landau–Lifshitz equation (1.3) has two other conserved quantities. These are the number of spin reversals, N, and the momentum given by [6]andIn the quantum description, N counts the number of quasi-particles in the magnet, that is, it may be interpreted as the magnon number.
In this section, we consider only stationary solitons, i.e. , whose properties have been discussed in Ref. [5]. There are no static
Moving solitons
To compute moving solitons is a more difficult task than in the stationary case and cannot be reduced to simply solving an ordinary differential equation. There is an ansatz which is consistent with the equations of motion and describes a soliton which moves at constant velocity and rotates in the internal space with frequency ω. Explicitly, the ansatz readsSubstituting this ansatz into the equation of motion (1.3) leads to a partial
Multi-soliton interactions
In this section, we discuss the results of a numerical evolution of the full time-dependent Landau–Lifshitz equation (1.3), in order to investigate the interaction and scattering of two solitons.
Two initially stationary and well-separated solitons have an axial symmetry about the line connecting them, and we make use of this symmetry in our numerical evolution code. If we take two solitons on the x3-axis, each with momentum only in the x3 direction, then both the initial conditions and the
Conclusion
We have used several different numerical techniques to study the dynamics and interaction of magnetic solitons in a three-dimensional ferromagnet with easy axis anisotropy. We have computed moving solitons using a minimization algorithm and compared the results to those of a simple radial ansatz, which we have shown is a good approximation for low momenta. However, for large momenta there is a transition from lump-like solitons to ring-like solitons, where obviously the radial ansatz fails
Acknowledgments
We acknowledge the Nuffield Foundation for an award (TI) and the EPSRC for an Advanced Fellowship (PMS).
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