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—which share two peculiar properties: each of their band structures exhibits a flat band with a nontrivial winding number. The difference, however, is that only the Creutz lattice is genuinely topological, due to a chiral (sublattice) symmetry, resulting in a quantized winding number and zero energy edge modes for open boundary conditions. We use a multiband mean field and exact density matrix renormalization group in our work. Our three main results are as follows: (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. We have therefore extended the approach to this more complex situation and found an excellent agreement with the numerical results. (c) At moderate and large $U$, the conventional BCS mean field is no longer 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 multiband mean-field method where the pairing parameters and the local particle densities on the inequivalent sublattices are variational mean-field parameters.