Phases of unitary matrix models and lattice QCD2

We investigate the different large $N$ phases of a generalized Gross-Witten-Wadia $U(N)$ matrix model. The deformation mimics the one-loop determinant of fermion matter with a particular coupling to gauge fields. 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 $\tau$. We compute the expectation value of Wilson loops in both phases, showing that the transition is third-order. A calculation of the $\beta $ function shows the existence of an IR stable fixed point.