We investigate the different large N phases of a deformed Gross-Witten-Wadia U(N) matrix model. The deformation, which leads to a solvable model, corresponds to the addition of characteristic polynomial insertions and mimicks the one-loop determinant of fermion matter. In one version of the model, the GWW phase transition is smoothed out and it becomes a crossover. In another version, the phase transition occurs along a critical line in the two-dimensional parameter space spanned by the 't~Hooft coupling $\lambda $ and the Veneziano parameter $\tau $. A calculation of the $\beta $ function shows the existence of an IR stable fixed point. ---- Part of London Integrability Journal Club. If you are a new participant, please register filling the form at integrability-london.weebly.com. The link will be emailed.