On the renormalization of periodic potentials

TL;DR

This paper studies how periodic potentials are renormalized in scalar field theory using the differential RG approach, revealing phases, symmetry breaking, and the flattening of the effective potential.

Contribution

It provides a simple derivation of known results for the sine-Gordon model and discusses the effects of periodicity and convexity constraints on the effective potential.

Findings

Two phases: ordered with asymptotic freedom and disordered with triviality.

Spontaneous symmetry breaking indicated by winding number.

Effective potential becomes flat due to periodicity and convexity constraints.

Abstract

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.