Higgs criticality of Dirac spin liquids on depleted triangular lattices

We investigate Higgs criticality in candidate U(1) Dirac spin liquids across a family of depleted triangular lattices: the triangular, kagome, and maple-leaf geometries. For each, we identify the symmetry-allowed spinon-pairing channel connecting the U(1) state to a proximate $\mathbb{Z}_2$ spin liquid, deriving the corresponding quantum electrodynamics (QED$_3$)-Higgs theory. While the triangular and kagome lattices share a low-energy description with $N_f=4$ Dirac fermions, the maple-leaf lattice yields an analogous theory with $N_f=12$ and a distinct nodal structure where the Dirac cones can move along high-symmetry lines in momentum space. Using a large-$N_{f,b}$ expansion, we compute critical exponents and the scaling dimensions of the symmetry-allowed Yukawa couplings. We find that while Higgs-field fluctuations and a large fermion flavor number both act to suppress the relevance of the Yukawa coupling -- pushing the maple-leaf lattice closer to stability than its counterparts -- the coupling remains weakly relevant in all three cases. This rendering of the Higgs critical point as asymptotically unstable is partly driven in the maple-leaf case by an additional coupling associated with the momentum-space mobility of the Dirac cones. Ultimately, our results provide a unified framework demonstrating how the interplay between fermion flavor count and nodal geometry dictates the fate of the QED$_3$-Higgs transition.