Flat bands at the Fermi level offer a promising platform to study a variety of intriguing correlated phase of matter. Here we present band engineering to realize both 2D and 1D band flatterning in twisted gamma valley semiconductor bilayers, which lead to topological and correlated phases in two dimensions.

The first example is the 2D topological flat bands. We find the line graph or biparticle sublattice of moire pattern emerge with a minimal C3 symmetry, which exhibit isolated electronic flat bands with nontrivial topology. The band flatness is insensitive to the twist angle since they come from the interference effect. Armed with this guiding principle, we predict that twisted bilayers of 2H-PbS2 and CdS realize the salient physics to engineer two-dimensional topological quantum phases.

The second example is the anisotropic band flattering to realize 1D flat bands in 2D. We find that in twisted anisotropic two-dimensional crystals with a rectangular unit cell of C2z or mirror symmetries, a larger effective mass anisotropy has a stronger tendency to be further enhanced compared to that of monolayer, which leads to correlated physics in one dimension effectively. We predict twisted bilayer black phosphorus (tBBP) has giant anisotropic flattened moire bands, where the low energy physics is described by the weakly coupled array of one-dimensional wires. We further calculate the phase diagram based on sliding Luttinger liquid by including the screened Coulomb interactions in tBBP, and find a large parameter space may host the non-Fermi liquid phase.