  This work theoretically explores how to emulate twisted double bilayer
graphene with ultracold atoms in multiorbital optical lattices. In particular,
the quadratic band touching of Bernal stacked bilayer graphene is emulated
using a square optical lattice with $p_x$, $p_y$, and $d_{x^2-y^2}$ orbitals on
each site, while the effects of a twist are captured through the application of
an incommensurate potential. The quadratic band touching is stable until the
system undergoes an Anderson like delocalization transition in momentum space,
which occurs concomitantly with a strongly renormalized single particle
spectrum inducing flat bands, which is a generalization of the magic-angle
condition realized in Dirac semimetals. The band structure is described
perturbatively in the quasiperiodic potential strength, which captures miniband
formation and the existence of magic-angles that qualitatively agrees with the
exact numerical results in the appropriate regime. We identify several
magic-angle conditions that can either have part or all of the quadratic band
touching point become flat. In each case, these are accompanied by a diverging
density of states and the delocalization of plane wave eigenstates. It is
discussed how these transitions and phases can be observed in ultracold atom
experiments.
