Renormalization of the vacuum angle for a particle on a ring
  We analyze the vacuum (topological) angle $\theta$ renormalization for the quantum mechanical model of a particle moving around a ring, where $\theta$ is the magnetic flux through the ring. We construct a renormalization group (RG) transformation for this model and derive exact RG equations which lead to the flow diagram similar to that of the Quantum Hall effect. Renormalization of $\theta$ is seen to follow from the loss of information about the initial topological charge in the course of the RG procedure. 
S.M. Apenko