  The paramagnetic phase diagrams of the half-filled Hubbard model on a twisted
bilayer square lattice are investigated using coherent potential approximation.
Besides the conventional metallic, band insulating, and Mott insulating phases,
we find two site-selective insulating phases where certain sites exhibit band
insulating behaviors while the others display Mott insulating behaviors. These
phases are identified by the band gap, the double occupancy, the density of
states, as well as the imaginary part of self-energy. Furthermore, we examine
the effect of on-site potential on the stability of the site-selective
insulating phases. Our results indicate that fruitful site-selective phases can
be engineered by twisting.
