Flat-band Ratio and Quantum Metric in the Superconductivity of Modified Lieb Lattices

Flat bands may offer a route to high critical temperatures of superconductivity. It has been predicted that the quantum geometry of the bands as well as the ratio of the number of flat bands to the number of orbitals determine flat band superconductivity. However, such results have assumed at least one of the following: an isolated flat band, zero temperature, mean-field theory, and/or uniform pairing. Here, we explore flat band superconductivity when these assumptions are relaxed. We consider an attractive Hubbard model for different extensions of the Lieb lattice. The superconducting order parameter, critical temperature, superfluid weight, and Berezinskii-Kosterlitz-Thouless (BKT) temperature are calculated within dynamical mean-field theory. We find that for isolated flat bands, the flat-band ratio and quantum geometry are in general good indicators of superconductivity even at finite temperatures. For non-isolated flat bands, a good guideline of the BKT temperature is provided by the zero-temperature superfluid weight and the flat-band ratio. Quantum geometry has emerged as a key notion in the theory of superconductivity, and its effects become prominent, especially for flat bands. The authors show that the flat-band ratio and quantum geometry serve as good indicators for superconductivity by comprehensively investigating the series of extended Lieb lattices based on the dynamical mean-field theory.