Chern-Simons-Fermion Vector Model with Chemical Potential

We study SU(N) level k Chern-Simons theories coupling to a fundamental fermion with chemical potential for U(1) flavor symmetry. We solve this system exactly in the 't Hooft limit, N,k\to\infty with \lambda = N/k fixed. The solution exists up to |\lambda|=1 for a fixed value of chemical potential. We study the system beyond |\lambda|=1 by considering U(N) gauge group and taking the diagonal U(1) holonomy into account. The result suggests the system becomes unstable and a phase transition happens at |\lambda|=1.