Dimensional crossover in dipolar magnetic layers

We investigate the static critical behaviour of a uniaxial magnetic layer, with finite thickness L in one direction, yet infinitely extended in the remaining d dimensions. The magnetic dipole-dipole interaction is taken into account. We apply a variant of Wilson’s momentum shell renormalisation group approach to describe the crossover between the critical behaviour of the 3-D Ising, 2-d Ising, 3-D uniaxial dipolar, and the 2-d uniaxial dipolar universality classes. The corresponding renormalisation group fixed points are in addition to different effective dimensionalities characterised by distinct analytic structures of the propagator, and are consequently associated with varying upper critical dimensions. While the limiting cases can be discussed by means of dimensional ǫ expansions with respect to the appropriate upper critical dimensions, respectively, the crossover features must be addressed in terms of the renormalisation group flow trajectories at fixed dimensionality d.