Constraints on beta functions from duality

We analyse the way in which duality constrains the exact beta function and correlation length in single-coupling spin systems. We propose a consistency condition which shows very concisely the relation between self-dual points and phase transitions, and implies that the correlation length must be duality invariant. These ideas are then tested on the two-dimensional Ising model, and used towards finding the exact beta function of the q-state Potts model. Finally, a generic procedure is given for identifying a duality symmetry in other single-coupling models with a continuous phase transition.