Exotic phases in finite-density $\mathbb{Z}_3$ theories

TL;DR

This paper investigates the phase structure of finite-density $ ext{Z}_3$ lattice theories, revealing a unique Devil's Flower in chiral models and analyzing the effects of different RG approaches on phase predictions.

Contribution

It demonstrates the existence of a Devil's Flower phase structure in chiral $ ext{Z}_3$ models and explores how different Migdal-Kadanoff RG implementations affect phase counts.

Findings

Chiral $ ext{Z}_3$ models exhibit a Devil's Flower phase structure.

Different RG schemes predict varying numbers of phases.

Only chiral models and their duals show inhomogeneous phases.

Abstract

Lattice $\mathbb{Z}_3$ theories with complex actions share many key features with finite-density QCD including a sign problem and $CK$ symmetry. Complex $\mathbb{Z}_3$ spin and gauge models exhibit a generalized Kramers-Wannier duality mapping them onto chiral $\mathbb{Z}_3$ 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 $\mathbb{Z}_3$ models in dimensions one through four. Chiral $\mathbb{Z}_3$ spin models are known to exhibit a Devil's Flower phase structure, with inhomogeneous phases which can be thought of as $\mathbb{Z}_3$ 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 $\mathbb{Z}_N$ models, SU($N$) models and nonzero temperature.