Quantum Geometric Effects 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. Recent studies have linked the properties of Cooper pairs in flat-band superconductors to the quantum metric. Unlike prior studies primarily focused on quasiparticle fluctuations, in this work, we investigate quantum geometric effects on a collective mode of the order parameter-the Higgs mode. We derive the quantum goemetric Higgs-mode correlation length and investigate whether the nonlinear electromagnetic response of the Higgs mode persists in flat band. It turns out that the quantum metric will replaces energy derivatives, playing a crucial role in third-harmonic generation(THG). We further numerically calculated the correlation length and THG in the Lieb lattice. In contrast to traditional single-band results, Higgs mode 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.