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Nonlinear Dynamics of Ferromagnets

A ferromagnetic material contains many interacting magnetic moments that, in the groundstate, line up to point in the same direction everywhere within the sample (or within each magnetic domain). If the ferromagnet is excited energetically, disturbing the magnetic moments away from this mutually-parallel conformation, their directions in space will undergo some temporal dynamics or oscillation before eventually relaxing through dissipation.

Understanding this dynamics poses a difficult problem:  there is a large number of coupled degrees of freedom, which evolve according to a non-linear dynamics.  Furthermore, any description is complicated by topological constraints involving the global conformation of magnetic moments.  I am interested in studying the non-linear dynamics of ferromagnets (including insulating, itinerant, and ``quantum Hall'' ferromagnets), and understanding the properties of highly-excited ferromagnets, such as itinerant ferromagnets or quantum Hall ferromagnets under conditions of  high current density.

Surprisingly, it turns out that insulating ferromagnetic materials support excitations that are closely analogous to the vortex rings of fluid dynamics (familiar to us all as ``smoke-rings''). These are spatially-localized non-linear disturbances that propagate without change of shape at a constant velocity through the ferromagnet (compare the ideal motion of a circular smoke ring, which moves at constant velocity through the air without change of radius).  In fact, for reasons related to topology, there exists a sequence of magnetic vortex ring solitary waves that are classified by a topological invariant known as the Hopf invariant. (Two configurations corresponding to different values of the Hopf topological invariant cannot be interconverted by smooth deformations.)
The images below illustrate typical conformations of the two simplest magnetic vortex ring excitations: with Hopf invariant equal to zero and one (this difference in the global topology of the two configurations can be deduced from the different linking properties of the red and green curves). The direction of propagation of each excitation is perpendicular to the plane containing the red circle.

Magnetic Vortex Ring with zero Hopf InvariantMagnetic Vortex Ring with unit Hopf Invariant
The images show the orientations of the magnetic moments within a plane cutting through the centre of the magnetic vortex ring. The vectors represent the orientation of the local magnetic moment  projected onto the x-y plane, while their colours indicate the z component of the magnetic moment (see colour legend). The red and green curves indicate the loci of points in space where the magnetic moment points in the -z direction and the +x direction, respectively.


Propagating Magnetic Vortex Rings in Ferromagnets
N. R. Cooper
Phys. Rev. Lett. 82, 1554 (1999). [PRL, cond-mat/9901037]

Solitary Waves in Planar Ferromagnets and the Breakdown of the Spin-Polarized Quantum Hall Effect
N. R. Cooper
Phys. Rev. Lett. 80, 4554 (1998).  [PRL, cond-mat/9801160]