Symmetry in Micromagnetics

Hysteresis occurs in a ferromagnet because the magnetization can become unstable and jump to a different state. One of the challenges for micromagnetics is finding all the solution curves for a given variable such as the magnetic field. In this talk I show that the symmetries of magnetic states can be used to analyze existing solutions and obtain new ones. A given particle has a magnetic and a nonmagnetic symmetry group. The nonmagnetic group is determined by the crystallographic point group and the shape of the particle. In response to a given operation in this group, the magnetization and magnetic field transform as pseudovectors. In zero field, every symmetry that leaves the body invariant also maps one equilibrium state onto itself or another state of the same energy. Thus, symmetry operations can quickly generate new solutions. Comparisons between equivalent solutions provide a good estimate of their precision. Magnetic symmetry groups are a combination of spatial operations and time reversal (which changes the sign of the magnetization). They leaves the magnetic field and moment invariant, so the highest symmetry the magnetization can have is determined by the physical symmetry of the particle and the field direction. The single-domain (SD) or ``flower'' state has the highest symmetry. Other states can be traced back to the SD state through bifurcations. The most important such bifurcation, a generalization of curling-mode nucleation, breaks inversion symmetry. The result is that the SD state splits into two states, each of which can be obtained from the other by inversion. For many materials (including magnetite), this bifurcation can occur even when the magnetic field is in an arbitrary direction. An important feature of bifurcations is that the pre-bifurcation state continues as an unstable state. Numerical micromagnetic algorithms can easily be fooled into following the unstable state. Several modelers have obtained a remanent state called the double vortex state. This is the result of a second bifurcation off the unstable state, and is itself unstable. The incorrect choice of continuation at a bifurcation is a major source of error in micromagnetic models, and it is difficult to spot without some understanding of the symmetry.