Recently I’ve spent some time thinking about domains in some magnetic materials, trying to understand how they influence and interact with topological electronic structure and Hall transport.  In the process, I played around a bit with simple models of the domain formation and dynamics.  This is a kind of classic problem in statistical mechanics.  You have a system which in equilibrium spontaneously orders below some temperature, for example a ferromagnet.  There are several possible directions to the magnetization, all with the same energy.  If you suddenly lower the temperature from above the ordering temperature, where there is no magnetization, to below it, magnetization starts to form.  This is called a “quench”.  But at different points in space, the orientation is different, and random.  Then over time after the quench, nearby regions begin to align with one another, and larger regions, called domains, develop, within which the magnetization is uniform.  These grow with time, and this process is called “coarsening”.

The simplest example is an Ising magnet, in which there are only two types of domains.  But there are more complex situations.  Here I was considering a case in which the magnetization orients within a plane, and has 6 preferred directions, say multiples of 60 degrees.  This is modeled by a simple Langevin dynamics

\gamma \partial_t \theta_i = -\frac{\partial H}{\partial \theta_i} + \eta_i(t)

where \gamma>0 is a damping constant, \eta_i(t) is a stochastic Gaussian delta-function correlated “white” noise, and H is a Hamilonian

H = -J \sum_{\langle ij\rangle} \cos(\theta_i-\theta_j) - \lambda \sum_i \cos 6\theta_i,

where J gives an energy for neighboring spins to align, and \lambda >0 is a six-fold anisotropy that favors the 6 domains.  This is what would be called “model A” dynamics of an XY model with 6-fold anisotropy.

It is surprisingly easy to simulate this thing – I did it with Julia’s DifferentialEquations package. The animation shows the evolution with time starting from a completely random initial condition, on a 90×90 square lattice.  It is beautiful how the domains develop.  You also see some “vortices”, which are points around which the angle winds by \pm 2\pi, which due to the anisotropy become the intersection point of 6 domain walls.  Over time the domains grow – you can see the “walls” between domains moving – and the vortices slowly meander and eventually will find a way to annihilate one another.  The size of the domains, the distance between vortices, etc., all show power-law dependence on time in this type of problem.  Can you spot the different kinds of vortices (clockwise versus counter-clockwise winding), and see some of the annihilation events?

In a real material eventually the domain motions will be affected by disorder, which slows and finally stops them in some frozen or nearly stationary state. Lots of other effects can also come in, like extra forces at the boundaries of the sample, applied fields, strains, etc., that can make the eventual structure quite complex and hard to predict.  But still it is cool.