  We present new results on the Gross-Neveu model at finite temperature and at
next-to-leading order in the 1/N expansion. In particular, a new expression is
obtained for the effective potential which is explicitly invariant under
renormalization group transformations. The model is used as a playground to
investigate various features of field theory at finite temperature. For example
we verify that, as expected from general arguments, the cancellation of
ultraviolet divergences takes place at finite temperature without the need for
introducing counterterms beyond those of zero-temperature. As well known, the
discrete chiral symmetry of the 1+1 dimensional model is spontaneously broken
at zero temperature and restored, in leading order, at some temperature T_c; we
find that the 1/N approximation breaks down for temperatures below T_c: As the
temperature increases, the fluctuations become eventually too large to be
treated as corrections, and a Landau pole invalidates the calculation of the
effective potential in the vicinity of its minimum. Beyond T_c, the 1/N
expansion becomes again regular: it predicts that in leading order the system
behaves as a free gas of massless fermions and that, at the next-to-leading
order, it remains weakly interacting. In the limit of large temperature, the
pressure coincides with that given by perturbation theory with a coupling
constant defined at a scale of the order of the temperature, as expected from
asymptotic freedom.
