Classification scheme for statistical theories with unusual symmetries

Abstract

A large class of theories with unusual symmetries is constructed and discussed from a unified viewpoint. These are statistical models with symmetries which correspond to an interpolation between standard globally-invariant theories and theories with a local symmetry. The models describe the dynamics of U(1) (or Z(N)) variables, ei[pi], located at the sited of a lattice of arbitrary dimension and geometry. We describe a classification of these theories based on what we call their symmetry index: the value of n for which the hamiltonian of a d-dimensional system is invariant under the transformation [pi](x1,...,xd) --> [pi] (x1,...,xd) + [Lambda] (x1,...,xd), where [Lambda] satisfies a set of n linearly-independent constraints. We discuss how the critical properties and the structure of topological defects of a model are determined by d and n.