Spontaneous Chiral-Symmetry Breaking in Three-Dimensional QED with a Chern--Simons Term

In three-dimensional QED with a Chern–Simons term we study the phase structure associated with chiral-symmetry breaking in the framework of the Schwinger–Dyson equation. We give detailed analyses on the analytical and numerical solutions for the Schwinger–Dyson equation of the fermion propagator, where the nonlocal gauge-fixing procedure is adopted to avoid wave-function renormalization for the fermion. In the absence of the Chern–Simons term, there exists a finite critical number of four-component fermion flavors, at which a continuous (infinite-order) chiral phase transition takes place and below which the chiral symmetry is spontaneously broken. In the presence of the Chern–Simons term, we find that the spontaneous chiral-symmetry-breaking transition continues to exist, but the type of phase transition turns into a discontinuous first-order transition. A simple stability argument is given based on the effective potential, whose stationary point gives the solution of the Schwinger-Dyson equation.