Emergent Flat band lattices in spatially periodic magnetic fields

Motivated by the recent discovery of Mott insulating phase and unconventional superconductivity due to the flat bands in twisted bilayer graphene, we propose more generic ways of getting two-dimensional (2D) emergent flat band lattices using either 2D Dirac materials or ordinary electron gas subject to moderate periodic magnetic fields with zero spatial average. We provide simple formulas for the "magic ratios" between the field strength and its wavenumber for getting flat bands, and give an intuitive explanation for their origin by constructing coarse-grained lattice models. Our work provides new, flexible platforms for exploring interaction-driven phases in 2D systems with arbitrary superlattice symmetries.