Higgs spectroscopy, the study of Higgs bosons of a superconductor, is an emerging field in studying superconductivity. Here we show that the Berry curvature and the quantum metric of bands play a central role in the Higgs mode generation. They allow the detection of Higgs bosons even when the conventional contribution from the band curvature vanishes. Furthermore, we show that the Higgs mode can couple to the external electromagnetic field linearly when mediated by the Berry connection. As a result, we predict the existence of a second-harmonic generation, in addition to the well-known third-harmonic generation. We elucidate our theory further by considering a flat band superconductor realized by the Harper-Hubbard model and show that the geometric Higgs mode is lower bounded by the band Chern number. We demonstrate in twisted bilayer graphene the existence of the geometrically induced Higgs modes when superconductivity is realised in the nearly flat band at the magic angle.

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• Superconducting fluctuations
• Superconducting order parameter
• Graphene
• Strongly correlated systems
• Two-dimensional electron system
• BCS theory
• Bloch wave theory
• Bogoliubov-de Gennes equations