Geometry of renormalization group flows constrained by discrete global symmetries

Abstract

The connection between the phase structure and the geometry of the renormalization group (RG) flow in systems with discrete parameter space symmetries is studied. These are symmetries of the partition function, and therefore determine the geometry of the phase diagram for the system. The C-function, which is the potential for the RG flow, inherits the symmetries of the partition function, so that the RG flow must respect the geometry of the phase diagram. It is suggested that if the symmetry group is sufficiently large, i.e. an infinite discrete non-abelian group, then it may constrain the C-function so much that global (topological) information about the RG can be obtained. A systematic taxonomy of possible phase and flow diagrams associated with subgroups of the modular group at level two is given, and applications to the quantum Hall effect and high-temperatire superconductivity are discussed.