Several physical phenomena in superconductors, such as helical superconductivity and the diode effect, rely on breaking time-reversal symmetry. This symmetry breaking is usually accounted for via the Lifshitz invariant, a contribution to the free energy which is linear in the phase gradient of the order parameter. In dispersive single-band superconductors with conventional pairing, the Lifshitz invariant can be computed from the asymmetries of the spectrum near the Fermi surface. We show that in multiband superconductors, the quantum geometry also contributes to the Lifshitz invariant, and this is the dominant contribution when the low-energy bands are flat. We also analogously demonstrate quantum-geometry-driven commensurate-incommensurate transitions in charge and pair density waves.

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• Flat bands
• Multiband superconductivity
• Superconductivity
• Superconducting devices
• Twisted heterostructures
• Landau-Ginzburg theory