  We derive the general continuum model for a bilayer system of staggered-flux
square lattices, with arbitrary elastic deformation in each layer. Applying
this general continuum model to the case where the two layers are rigidly
rotated relative to each other by a small angle, we obtain the band structure
of the twisted bilayer staggered-flux square lattice. We show that this band
structure exhibits a "magic continuum" in the sense that an exponential
reduction of the Dirac velocity and bandwidths occurs in a large parameter
regime. We show that the continuum model of the twisted bilayer system
effectively describes a massless Dirac fermion in a spatially modulating
magnetic field, whose renormalized Dirac velocity can be exactly calculated. We
further give an intuitive argument for the emergence of flattened bands near
half filling in the magic continuum and provide an estimation of the large
number of associated nearly-zero-energy states. We also show that the entire
band structure of the twisted bilayer system is free of band gaps due to
symmetry constraints.
