Superconducting Hall effect from a two-particle geometric phase

We investigate the impact of a two-particle geometric phase, which is the Aharanov-Bohm phase associated with the kinetic motion of electron pairs, on the transport and optical properties of superconductors. Our findings reveal that this geometric phase drives a nonlinear superconducting Hall effect (SHE), which is a divergence of the Hall conductivity in the zero-frequency limit. By introducing a model analogous to the Haldane model for electron pairs, we demonstrate both numerically and analytically that this system exhibits the SHE in its second-order response. Additionally, we discover that higher-order responses related to collective modes, such as the Higgs and Leggett modes, also contribute to the SHE. Finally, within the Ginzburg-Landau framework, we show that these responses are governed by the higher-order version of the Lifshitz invariants, which is a symmetry invariant that contains odd-order spatial derivatives. Our work broadens the understanding of geometric effects on transport and optical phenomena in superconductors.