On planar fermions with quartic interaction at finite temperature and density

We study the breaking of parity symmetry in the 2+1 Gross-Neveu model at finite temperature with chemical potential $\mu$, in the presence of an external magnetic field. We find that the requirement of gauge invariance, which is considered mandatory in the presence of gauge fields, breaks parity at any finite temperature and provides for dynamical mass generation, preventing symmetry restoration for any non-vanishing $\mu$. The dynamical mass becomes negligibly small as temperature is raised. We comment on the relevance of our observation for the gap generation of nodal quasi-particles in the pseudo-gap phase of high $T_c$ superconductors.