  Coupling spin models to complex external fields can give rise to interesting
phenomena like zeroes of the partition function (Lee-Yang zeroes, edge
singularities) or oscillating propagators. Unfortunately, it usually also leads
to a severe sign problem that can be overcome only in special cases; if the
partition function has zeroes, the sign problem is even
representation-independent at these points. In this study, we couple the
N-state Potts model in different ways to a complex external magnetic field and
discuss the above mentioned phenomena and their relations based on analytic
calculations (1D) and results obtained using a modified cluster algorithm
(general D) that in many cases either cures or at least drastically reduces the
sign-problem induced by the complex external field.
