Designer flat bands: Topology and enhancement of superconductivity

Abstract

We construct quasi-one-dimensional topological and nontopological three-band lattices with a tunable band gap and winding number of the flat band. Using full multiband mean field (MF) and exact density matrix renormalization group (DMRG) calculations, we show explicitly how the band gap affects pairing and superconductivity (SC) in a Hubbard model with attractive interactions. We obtain excellent agreement between MF and DMRG calculations. The SC weight ${D}_{s}$ on the gapped topological, $W\ensuremath{\ne}0$, flat band increases linearly with interaction strength $U$ for low values and with a slope that depends on the details of the compact localized state at $U=0$. As $U\ensuremath{\rightarrow}0$ for the gapped nontopological flat band ($W=0$), ${D}_{s}$ decays with a power law faster than quadratically but slower than exponentially. In the gapless case (flat band touching the band above it), we find at low $U$ (for both $W=0$ and $W\ensuremath{\ne}0$) that ${D}_{s}\ensuremath{\propto}{U}^{\ensuremath{\varphi}}$, with $\ensuremath{\varphi}<1$. In other words, ${D}_{s}$ increases faster than linearly for low $U$, thus favoring SC at weak interaction more than the gapped case. Our results reestablish that the BCS mean field and quantum metric alone are insufficient to characterize SC at weak coupling.