Phase transition in a four-dimensional fermion model

We study the four-dimensional self-coupled massless fermion model with a dynamical breakdown of chiral symmetry through an emergence of collective excitations and discuss a phase transition under the influence of both temperature $T$ and fermion-number density $n$. We give the self-consistency equation which determines an amount of dynamically generated fermion mass as a function of $T$ and $n$. Using the numerical solutions of the self-consistency equation, we derive the effective potential of the collective excitations. We show that the model has a second-order phase transition at a critical temperature and also a critical density and that the dynamical symmetry violation vanishes above these critical values. We also draw the phase diagram in the $T-n$ plane for several values of the generated fermion mass.