Podolsky quantum electrodynamics for strongly coupled Dirac fermions in (2+1)D

We investigate generalized quantum electrodynamics (GQED), a higher-derivative extension of QED in (3+1)D. We perform its dimensional reduction to (2+1)D by confining the Dirac current to a plane while allowing the gauge field to propagate out of the plane. The resulting model, which we call Pseudo Generalized QED (PGQED), is minimally coupled to massless Dirac fermions. In the strong-coupling regime, we show that a dynamical mass is generated through approximate solutions of the Schwinger-Dyson equations, leading to chiral symmetry breaking and modifications of the fermion dispersion relation. We derive an analytic critical coupling $\alpha_{c}(\mu)$ and flavors critical number $N_{c}(\mu)$, dependent on the Podolsky parameter $\mu$ and the ultraviolet cutoff $\Lambda$. These analytical results are found to be consistent with the numerical analysis. Finally, we connect our results to graphene, estimating a range for $\mu$ in the ultrarelativistic limit and highlighting implications for two-dimensional materials.