We study the nonlinear Hall effect in superconductors without magnetic fields induced by a quantum geometric phase (i.e., the Aharonov-Bohm phase) carried by single or pair particles. We find that the second-order nonlinear Hall conductivity diverges in the dc limit in a robust way against dissipation when the system is superconducting, suggesting that the supercurrent flows perpendicular to the direction of the applied electric field. This superconducting nonlinear Hall effect (SNHE) is demonstrated for the Haldane model with attractive interaction and its variant with pair hoppings. In the Ginzburg-Landau theory, the SNHE can be understood as those arising from a higher-order Lifshitz invariant, that is, a symmetry invariant constructed from order parameters that contains an odd number of spatial derivatives. We perform real-time simulations including the effect of collective modes for the models driven by a multi-cycle pulse, and show that the SNHE leads to large rectification of the Hall current under light driving.
Takasan et al. (Tue,) studied this question.