Chiral symmetry breaking in the pseudo-quantum electrodynamics with non-Abelian four-fermion interactions

In the context of 2+1 dimensional Dirac materials, we consider electromagnetic interactions alongside a type of spin-dependent Hubbard interaction. The former is described by PQED theory, while the latter corresponds to an effective theory represented by the $SU(N_c)$ Thirring model. Employing Hubbard-Stratonovich transformation and large N expansion in the model yields a non-local $SU(N_c)$ Yang-Mills action. Subsequently, we solve Schwinger-Dyson equations to obtain the self-energy function of the fermion propagator, from which we determine the critical fermion flavor number $N^c_f$ and critical fine structure constant $\alpha_c$ indicative of chiral symmetry breaking. Our findings suggest that as the non-Abelian color number $N_c$ increases, the minimum value of the critical fermion flavor number monotonically increases, while the maximum value of the critical fine structure constant decreases accordingly, rendering the system more susceptible to chiral symmetry breaking.