First-order phase-transition on dynamical Lorentz symmetry breaking system

A model of N 4-component massless fermions in a quartic self-interaction based on ref. [19] is investigated in the presence of chemical potential and temperature via optimized perturbation theory that accesses finite-N contributions. We use the generating functional approach to calculate the corrections to the effective potential of the model. The model introduces an auxiliary pseudo-vector field with a nontrivial minimum and is influenced by temperature (T) and chemical potential (μ). These thermodynamic quantities are introduced through Matsubara formalism. Thereby, the integrals are modified, and via the principle of minimum sensitivity, we obtain the gap equations of the model. The correspondent finite-N solutions of these equations define the vacuum states of the model associated with the background pseudo-vector field. In particular, one focuses on its temporal component that acts as an effective chiral chemical potential. We discuss the solutions of the four cases in which $(T=0,\mu =0)$, $(T\ne 0,\mu \ne 0)$, $(T\ne 0,\mu =0)$ and $(T=0,\mu \ne 0)$, where the effective potential is so obtained as a function of the background vector field, the chemical potential, and the temperature. The model shows the finite-N corrections generate first-order phase transitions on the self-interacting fermions for the case $N=1$ and the persistence of a second-order phase transition for $N\ge 2$.