Spectral and Phase Structure of a Unitary Matrix Model with Fisher-Hartwig Singularities

We investigate a unitary matrix model with a complex potential with Fisher-Hartwig singularities. We show that the model exhibits finite-$N$ phase transitions. The order of the phase transition is coupling-dependent. At large-$N$, these transitions are replaced by third-order Gross-Witten-Wadia transitions between multiple ungapped phases and a single gapped phase, with transitions between ungapped phases forbidden. At both finite and large $N$, the phases are characterized by the locations of the Fisher-Hartwig singularities in the complex plane.