Higher Chern Numbers and Quadratic Band Crossing Lines in Bilayer Lieb Lattice

We consider a bilayer Lieb lattice which undergoes an unusual topological transition in the presence of intra-layer spin-orbit coupling (SOC). The specific configuration induces an effective non-symmorphic 2D lattice structure, even though the constituent monolayer Lieb lattice is characterized by a symmorphic space group. This emergent non-symmorphicity leads to multiple doubly-degenerate bands extending over the edge of the Brillouin zone, i.e. Quadratic Band Crossing Lines. In the presence of intra-layer SOC, these doubly-degenerate bands typically form three 2-band subspaces, mutually separated by two band gaps. We analyze the topological properties of these multi-band subspaces, using specially devised Wilson loop operators to compute non-abelian Berry phases, in order to show that they carry a higher Chern number, 2.