  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.
