Thermodynamics in 2+1 Dimensional QED with Chern-Simons Term

It is known that in the 2+1 dimensional quantum electrodynamics with Chern-Simons term, spontaneous magnetic field induces Lorentz symmetry breaking. In this paper, thermodynamical characters, especially the phase structure of this model are discussed. To see the behavior of the spontaneous magnetic field at finite temperature, the effective potential in the finite temperature system is calculated within the weak field approximation and in the fermion massless limit. We found that the spontaneous magnetic field never vanishes at any finite temperature. This result doesn't change even when the chemical potential is introduced. We also investigate the consistency condition and the probability that fermion stays in a lowest Landau level at finite temperature.