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 $q$.