Simple three-state model with infinitely many phases

A nearest-neighbor three-state model is introduced that has chiral interactions and exhibits spatially modulated order. A Migdal-Kadanoff renormalization group for this model is constructed and analyzed for general dimensionality d. This renormalization group is exact when applied to the model on certain hierarchical or fractal lattices. The resulting phase diagrams are of remarkable complexity: They exhibit an infinite number of distinct ordered phases, each identified by q, the principle wave number of the modulations in the local order. All ordered phases are commensurate with the lattice structure, and for sufficiently large d there is apparently a phase for every rational fraction q.