Effects of quantum geometry on the Higgs mode in flat-band superconductors

Flat-band systems are of great interest due to their strong electron correlations and unique band geometry. In this paper, we investigate quantum geometric effects on a collective mode of the order parameter, the Higgs mode. We derive the Higgs-mode correlation length and further answer two key questions: whether the third-harmonic generation (THG) of the Higgs mode persists in a flat-band system and the relative contributions of quantum-metric-mediated quasiparticle fluctuations and Higgs fluctuations to THG. We numerically calculated the correlation length and THG in the Lieb lattice. In contrast to traditional single-band results, we find that the correlation length determined by the quantum metric plays the role of the effective mass to determine the dispersion of the Higgs mode, and Higgs fluctuations contribute almost entirely to THG, with the quasiparticle contribution being negligible. This finding provides valuable guidance for detecting Higgs modes in flat-band systems through optical methods and reveals the profound influence of quantum geometry in flat-band systems.