Crystalline Motion of Interfaces Between Patterns

We consider the dynamical problem of an antiferromagnetic spin system on a two-dimensional square lattice εZ2 with nearest-neighbour and next-to-nearest neighbour interactions. The key features of the model include the interaction between spatial scale ε and time scale τ, and the incorporation of interfacial boundaries separating regions with microstructures. By employing a discrete-time variational scheme, a limit continuous-time evolution is obtained for a crystal in R2 which evolves according to some motion by crystalline curvatures. In the case of anti-phase boundaries between striped patterns, a striking phenomenon is the appearance of some “non-local” curvature dependence velocity law reflecting the creation of some defect structure on the interface at the discrete level.