Equivalence of Chern bands and Landau levels from projected Interactions

We introduce a universal formulation of the generalized real space interactions with translational invariance, when projected into a single Landau level, can be equivalent to density-density interaction projected into any Chern bands (e.g. Landau levels, continuous moire Chern bands and discrete lattice Chern bands). By constructing a complete basis of generalized pseudopotentials, we define the projected interaction range that can be used to characterize important features of different Chern bands relevant to the robustness of interacting topological phases. The analytical construction allows us to study general Chern bands beyond the torus geometry, for example on the disk or spherical geometry where the edge dynamics and curvature effects can be explored. For moire systems (and lattice Chern bands with large incompressibility gap) one can see transparently the topological Hall viscosity for the fractional states are identical to those in the Landau levels. The physical unprojected interaction also provides a natural embedding for lattice systems, shedding light on the concerns of relying on quantum geometric tensor (varying with lattice embedding in real space) to understand the properties of interacting phases of matters.