  Using complex Langevin dynamics we examine the phase structure of complex
unitary matrix models and compare the numerical results with analytic results
found at large $N$. The actions we consider are manifestly complex, and thus
the dominant contribution to the path integral comes from the space of
complexified gauge field configuration. For this reason, the eigenvalues of
unitary matrix lie off the unit circle and venture out in the complex plane.
One example of a complex unitary matrix model, with Polyakov line as the
unitary matrix, is an effective description of a QCD at finite density and
temperature with $N$ number of colors and $N_f$ number of quark flavors defined
on the manifold $S^1 \times S^3$. A distinct feature of this model, the
occurrence of a series of Gross-Witten-Wadia transitions, as a function of the
quark chemical potential, is reproduced using complex Langevin simulations. We
simulate several other observables including Polyakov lines and quark number
density, for large $N$ and $N_f$ and found excellent match with the analytic
results.
