Recent sign-problem-free quantum Monte Carlo simulations of (2+1)-dimensional lattice quantum electrodynamics (QED$_3$) with $N_f$ flavors of fermions on the square lattice have found evidence of continuous quantum phase transitions between a critical phase and a gapped valence-bond-solid (VBS) phase for flavor numbers $N_f=4$, $6$, and $8$. We derive the critical theory for these transitions, the chiral $O(2)$ QED$_3$-Gross-Neveu model, and show that the latter is equivalent to the gauged Nambu--Jona-Lasinio model. Using known large-$N_f$ results for the latter, we estimate the order parameter anomalous dimension and the correlation length exponent for the transitions mentioned above. We obtain large-$N_f$ results for the dimensions of fermion bilinear operators, in both the gauged and ungauged chiral $O(2)$ Gross-Neveu models, which respectively describe the long-distance power-law decay of two-particle correlation functions at the VBS transition in lattice QED$_3$ and the Kekule-VBS transition for correlated fermions on the honeycomb lattice.
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