Twisted square lattices: emergent symmetries and strong correlations

We study the electronic properties of square lattice moire homobilayers with band extrema located at the M point. At low twist angles an approximate layer-exchange symmetry emerges, that does not reduce to any crystalline symmetries of the twisted bilayer. For bands emerging from the original Brillouin zone corner, this symmetry enforces absence of nearest-neighbor hopping (t=0) on the moire lattice for the isolated flat band. Application of interlayer voltage or in-plane magnetic field allows to tune the value of t, realizing the Hubbard model with widely tunable t'/t ratio - a rich playground for high-temperature superconductivity and frustrated spin physics. Closed-form expressions for t, t' and U are obtained in the low-twist angle limit using WKB approximation and compared with exact results using Mathieu functions. We discuss the effects of band degeneracies and spin-orbit coupling, and present several candidate materials.