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In recent years, two-dimensional Dirac materials patterned with asuperlattice structure have emerged as a rich platform for exploring correlatedand topological quantum matter. In this work, we propose that by subjectingDirac electrons to a periodic magnetic field with triangular lattice symmetryit is possible to realize a quantum critical phase of $N_f=3$ Dirac fermionspecies strongly coupled to an emergent gauge field, or 2+1-D quantumelectrodynamics (QED$_3$). We demonstrate explicitly that the QED$_3$ phasenaturally arises from a Dirac composite fermion (CF) picture, where theperiodic magnetic field manifests as a periodic CF potential and transforms theCF Fermi surface into gapless Fermi points. We further show that by breakingthe particle-hole symmetry of the TI surface -- either by doping or byintroducing a periodic electrostatic potential with zero mean -- our quantumcritical phase gives way to a sequence of fractional Chern insulator phases.Our theory illustrates the rich menagerie of quantum phases possible aroundhalf filling of a flat Chern band.

Emergent QED$_3$ from half-filled flat Chern bands