Finite-temperature behavior of a relativistic field theory with dynamical symmetry breaking

Recently, Gross and Neveu have studied a two-dimensional field theory of an $N$-component fermion in the large-$N$ limit. This theory is asymptotically free and has dynamical spontaneous symmetry breaking. In this paper we study certain finite-temperature properties of this theory, especially those related to the survival of the "condensate," or symmetry breaking. Within the mean-field approximation, we find that the symmetry breaking disappears at a finite temperature $T_0$, which is of the same order of magnitude as the physical mass of a fermion. However, the mean-field approximation is not good for any finite $N$. At any nonzero temperature, however small, the system prefers to be in space-dependent field configurations such that the condensate vanishes. The critical temperature is thus zero.