Pairing and superconductivity in quasi one-dimensional flat band systems: Creutz and sawtooth lattices

We study the pairing and superconducting properties of the attractive Hubbard model in two quasi one-dimensional topological lattices: the Creutz and sawtooth lattices. They share two peculiar properties: each of their band structures exhibits a flat band with a non-trivial winding number. The difference, however, is that only the Creutz lattice is genuinely topological, owing to a chiral (sub-lattice) symmetry, resulting in a quantized winding number and zero energy edge modes for open boundary conditions. We use mean field and exact density matrix renormalization group in our work. Our three main results are: (a) For both lattice systems, the superconducting weight, $D_s$, is linear in the coupling strength, $U$, for low values of $U$; (b) for small $U$, $D_s$ is proportional to the quantum metric for the Creutz system but not for the sawtooth system because its sublattices are not equivalent; (c) conventional BCS mean field is not appropriate for such systems with inequivalent sublattices. We show that, for a wide range of densities and coupling strengths, these systems are very well described by a full multi-band mean field method where the pairing parameters and the local particle densities on the inequivalent sublattices are variational mean field parameters.