  Recent experiments have realized a twisted bilayer-like optical potential for
ultra-cold atoms, which in contrast to solid-state set ups may allow for an
arbitrary ratio between the inter- and intra-layer couplings. For commensurate
Moir\'e twistings a large-enough inter-layer coupling results in particle
transport dominated by channel formation. For incommensurate twistings, the
interlayer coupling acts as an effective disorder strength. Whereas for weak
couplings the whole spectrum remains ergodic, at a critical value part of the
eigenspectrum transitions into multifractal states. A similar transition may be
observed as well as a function of an energy bias between the two layers. Our
study reveals atoms in optical twisted-bilayer lattices as an interesting new
platform for the study of ergodicity breaking and multifractality.
