Fluctuations near the nematic-smectic- C phase transition

We study the properties of a Ginzburg-Landau-Wilson Hamiltonian in which an order parameter is associated with each wave vector which points from the origin to one of two rings in reciprocal space. The existence of the order parameter breaks the two-dimensional rotational symmetry about the axis of the rings. The Hamiltonian may be interpreted as a simple model of the nematic-smectic-C phase transition of a liquid crystal, if we identify the order parameter with the density of the liquid crystal and the axis with the average direction of the director field. There are no cubic terms present in the Hamiltonian, and the phase transition predicted by mean-field theory is second order. Fluctuations are taken into account by means of a method due to Brasovsky, and the phase transition is found to be of first order.