The renormalization group for a slab model of the spin reorientation transition

In 1975, Polyakov invented a renormalization group algorithm for treating the problem of a classical isotropic planar Heisenberg ferromagnet. Here we define a tool - “parallel transport of frames” - that extends Polyakov’s original renormalization group method to local magnetic anisotropies and, most significantly, the non-local dipolar interaction. We apply this tool to the problem of the spin reorientation transition in an ultrathin ferromagnetic slab with finite thickness. We find that integrating out the short wavelength spin wave fluctuations generates two local anisotropies that are absent from the microscopic dipolar energy Hamiltonian. Taking them into account yields the correct topology of the spin reorientation transition line, as observed experimentally and in numerical simulations.