Skyrmion lattices in chiral ferromagnets

Ferromagnets are materials made up of molecules each of which behaves like a tiny magnet, which interact with one another so that neighbouring molecules want their magnets to align. The state of such a material can be represented by a unit length vector field called the magnetization vector, which records the direction of the magnet at each point in space. If the underlying molecules lack reflexion symmetry, the ferromagnet is said to be chiral. In this case, the lowest energy configuration of the ferromagnet, when exposed to an external magnetic field, may not be the obvious state in which the magnetization vector aligns with the field, but a rather more mysterious state called a skyrmion. This is a configuration in which the magnetization vector ties itself up in a two-dimensional analogue of a knot, ranging through all possible orientations as one moves in space. The knottedness cannot be undone by any continuous deformation of the system - in mathematical language, this is a topological soliton. One dimensional arrays of magnetic skyrmions are the basis of proposed next-generation data storage devices (so-called race track memory). Their mathematical study has focussed almost exclusively on single isolated skyrmions, or small isolated clusters of skyrmions. But in real ferromagnetic systems, skyrmions occur spontaneously in regular two dimensional arrays - skyrmion lattices. Remarkably, there has been no systematic mathematical study of such lattices. We propose to study the existence, stability, genericity, and geometry of skyrmion lattices, focussing in particular on how these properties depend on the strength and direction of the applied magnetic field. To do this, we will adapt mathematical ideas developed originally in models of nuclear physics and superconductivity, implementing these in large-scale computer simulations.