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

Dyson-Schwinger equation approach to Lorentz Symmetry Breaking with finite temperature and chemical potential