Exotic phases in finite-density ℤ₃ theories

Lattice ℤ₃ theories with complex actions share many key features with finite- density QCD including a sign problem and 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 ℤ_N models, SU(N) models and nonzero temperature.