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First and Second Order Phase Transitions in Maxwell--Chern-Simons Theory Coupled to Fermions

In the Maxwell--Chern-Simons theory coupled to $N_f$ flavors of 4-component fermions (or even number of 2-component fermions) we construct the gauge-covariant effective potential written in terms of two order parameters which are able to probe the breakdown of chiral symmetry and parity. In the absence of the bare Chern-Simons term, we show that the chiral symmetry is spontaneously broken for fermion flavors $N_f$ below a certain finite critical number $N_f^c$, while the parity is not broken spontaneously. This chiral phase transition is of the second order. In the presence of the bare Chern-Simons term, on the other hand, the chiral phase transition associated with the spontaneous breaking of chiral symmetry is shown to continue to exist, although the parity is explicitly broken. However it is shown that the existence of the bare Chern-Simons term changes the order of the chiral transition into the first order, no matter how small the bare Chern-Simons coefficient may be. This gauge-invariant result is consistent with that recently obtained by the Schwinger-Dyson equation in the non-local gauge.