  In this work, we investigate the dynamical breakdown of Lorentz symmetry in 4
dimensions by the condensation of a fermionic field described by a Dirac
Lagrangian with a four-fermion interaction. Using the Keldysh formalism we show
that the Lorentz symmetry breaking modifies the Dyson-Schwinger equations of
the fermionic propagator. We analyze the nonperturbative solutions for the
Dyson-Schwinger equations using the combination of the rainbow and quenched
approximations and show that, in equilibrium, the Lorentz symmetry breakdown
can occur in the strong coupling regime and new features arise from this
approach. Finally, we analyze the contributions of temperature and chemical
potential and find the respective phase diagram of the model and analyze the
dependence of the critical temperature and chemical potential as functions of
the coupling constant.
