Lattice ℤ3 theories with complex actions share many key features with finite- density QCD including a sign problem and \( \mathcal{CK} \) symmetry. Complex ℤ3 spin and gauge models exhibit a generalized Kramers-Wannier duality mapping them onto chiral ℤ3 spin and gauge models, which are simulatable with standard lattice methods in large regions of parameter space. The Migdal-Kadanoff real-space renormalization group (RG) preserves this duality, and we use it to compute the approximate phase diagram of both spin and gauge ℤ3 models in dimensions one through four. Chiral ℤ3 spin models are known to exhibit a Devil’s Flower phase structure, with inhomogeneous phases that can be thought of as ℤ3 analogues of chiral spirals. Out of the large class of models we study, we find that only chiral spin models and their duals have a Devil’s Flower structure with an infinite set of inhomogeneous phases, a result we attribute to Elitzur’s theorem. We also find that different forms of the Migdal-Kadanoff RG produce different numbers of phases, a violation of the expectation for universal behavior from a real-space RG. We discuss extensions of our work to ℤN models, SU(N) models and nonzero temperature.
