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We derive the general continuum model for a bilayer system of staggered-fluxsquare lattices, with arbitrary elastic deformation in each layer. Applyingthis general continuum model to the case where the two layers are rigidlyrotated relative to each other by a small angle, we obtain the band structureof the twisted bilayer staggered-flux square lattice. We show that this bandstructure exhibits a "magic continuum" in the sense that an exponentialreduction of the Dirac velocity and bandwidths occurs in a large parameterregime. We show that the continuum model of the twisted bilayer systemeffectively describes a massless Dirac fermion in a spatially modulatingmagnetic field, whose renormalized Dirac velocity can be exactly calculated. Wefurther give an intuitive argument for the emergence of flattened bands nearhalf filling in the magic continuum and provide an estimation of the largenumber of associated nearly-zero-energy states. We also show that the entireband structure of the twisted bilayer system is free of band gaps due tosymmetry constraints.

Magic continuum in twisted bilayer square lattice with staggered flux