  A longstanding issue in the study of quantum chromodynamics (QCD) is its
behavior at nonzero baryon density, which has implications for many areas of
physics. The path integral has a complex integrand when the quark chemical
potential is nonzero and therefore has a sign problem, but it also has a
generalized $\mathcal PT$ symmetry. We review some new approaches to $\mathcal
PT$-symmetric field theories, including both analytical techniques and methods
for lattice simulation. We show that $\mathcal PT$-symmetric field theories
with more than one field generally have a much richer phase structure than
their Hermitian counterparts, including stable phases with patterning behavior.
The case of a $\mathcal PT$-symmetric extension of a $\phi^4$ model is
explained in detail. The relevance of these results to finite density QCD is
explained, and we show that a simple model of finite density QCD exhibits a
patterned phase in its critical region.
