On the renormalization of periodic potentials

The renormalization of the periodic potential is investigated in the framework of the Euclidean one-component scalar field theory by means of the differential RG approach. Some known results about the sine-Gordon model are recovered in an extremely simple manner. There are two phases, an ordered one with asymptotical freedom and a disordered one where the model is non-renormalizable and trivial. The order parameter of the periodicity, the winding number, indicates spontaneous symmetry breaking in the ordered phase where the fundamental group symmetry is broken and the solitons acquire dynamical stability. It is argued that the periodicity and the convexity are so strong constraints on the effective potential that it always becomes flat. This flattening is reproduced by integrating out the RG equation.