Partial Flatbands in a Spin-Orbit Coupled Twisted Bilayer Lieb Lattice

Topological classification of band-structure in various solid state materials has become one of the centerpieces of research in modern condensed matter physics. Typically, the specific topological nature of a dispersive band in a lattice system originates from a particular discrete symmetry present in the crystal. In this work, we analyze the topological properties of a partial/complete flatband in an apparently not so well known system -- a bilayer Lieb lattice. We demonstrate the emergence of flatbands with large Chern numbers in the presence of spin-orbit coupling. Our results are then contrasted with the flatbands in bilayer graphene and bilayer kagome lattice. We also consider a multi-layer Lieb lattice with even larger Chern numbers and speculate on the possibility of a superfluid phase in the presence of certain types of interactions.