Tunable t-t^{'}-U Hubbard Models in Twisted Square Homobilayers.

Square lattice Hubbard model with tunable hopping ratio t^{'}/t is highly promising for realizing a variety of quantum phases, including higher-temperature superconductivity. We show that twisted square lattice homobilayers offer such tunability when the flat bands originate from the corner of the Brillouin zone. An emergent symmetry, generically present at low twist angles, enforces t=0 for these bands. Breaking it with interlayer displacement field or in-plane magnetic field introduces t and anisotropy, tunable in a wide range for correlated electrons on the moiré square lattice.