General Construction and Topological Classification of Perfectly Flat Bands*

Systems harboring flat bands are excellent testbeds for strongly interacting physics, having generated much excitement in the condensed matter community. We present a generic technique to construct perfectly flat bands (FBs) from bipartite crystalline lattices (BCLs). Our prescription generalizes the line- and split-graph constructions encapsulating many of the various other models from literature. Using Topological Quantum Chemistry, we create a full topological classification in terms of symmetry eigenvalues of all (gapped and gapless) BCL FBs, in all Magnetic Space Groups (MSGs). We argue that the BCL FBs can be understood as formal differences of band representations. This allows us to find criteria for the existence of (and fully classify) unitary symmetry-protected touching points between the flat and dispersive bands, and identify the gapped FBs as prime candidates for fragile topological bands. Finally, we show that the set of BCL FBs is finitely generated and construct the corresponding bases in all MSGs, providing a comprehensive list of BCLs realizable in real materials.