Simulation of Scalar Field Theories with Complex Actions

We develop a method for the simulation of scalar field theories with complex actions which is local, simple to implement and can be used in any number of space-time dimensions. For model systems satisfying the $\mathcal{PT}$ symmetry condition $L^{*}(\phi)=L(-\phi)$, the complex weight problem is reduced to a sign problem. The sign problem is eliminated completely for a large subclass of these models; this class includes models within the $i\phi^{3}$ universality class, and also models with nonzero chemical potential. Simulations of models from this subclass show a rich set of behaviors. Propagators may exhibit damped oscillations, indicating a clear violation of spectral positivity. Modulated phases occur in some models, exhibiting striping and other pattern-forming behaviors. These field theory models are connected to complex systems where pattern formation occurs because of competition between interactions at two different length scales.