Finite-density QCD, $\mathcal{PT}$ symmetry, and dual algorithms

Finite-density QCD and many other field theories with sign problems have a $\mathcal{PT}$-type symmetry. After a brief introduction to $\mathcal{PT}$-symmetric field theories, a real dual representation for $\mathcal{PT}$-symmetric scalar field theories with complex actions is derived. We show that $\mathcal{PT}$-symmetric field theories can exhibit exotic behavior, including sinusoidally modulated propagators, disorder lines, and spatially inhomogeneous pattern-forming phases. We discuss the interplay of duality, $\mathcal{PT}$-symmetry and pattern formation using a $\phi^4$ model and $Z(N)$ spin model with sign problems as examples. These behaviors may occur in finite-density QCD and related models.