Magnetic order

• Materials have magnetic structure down to the atomic level.
• Magnetic order is due to the relative alignment of atomic magnetic moments $\mathbf{\mu}$. Aligned moments form magnetic domains.
• At the microscopic level, magnetic moments order to form structures.
• We are interested in the magnetic order at the nanometer to micrometer size.
• (Note, the magnetic field of a magnet is only a small subtopic of this field of science.)

A disc-shaped magnetic particle. Dark means magnetic moment pointing up and light colour means down. [Moutafis et al, PRB, 2007]

Vectors show the magnetic moments. Dark colour means that $\bm{\mu}$ is pointing out-of-plane. [Tonomura et al, NanoLett. 2012]

Microscopic magnetic domains

Magnetic recording, using microscopic magnetic domains, is an application of magnetic order.

The interest across sciences

• The model for ferromagnets, the Landau-Lifshitz equation, presents an interesting challenge for Theoretical Physics. (1) It has significant differences from standard models in Sciences (such as the Nonlinear Schrödinger Equation). (2) A large amount of data are produced in experimental laboratories.
• In Applied Mathematics, research has focused on just a few models of Physics and the Natural Sciences. The Landau-Lifshitz equation is a further paradigm that has the potential to open areas in Mathematics. (1) It studies a field defined on the sphere $S^2$. (2) It presents challenges for Numerical Analysis. (3) It draws a lot of motivation from a physical system. (4) Nanoscopic systems may be tractable mathematically.