Unconventional Hall Effect in Gapless Superconductors: Transverse Supercurrent Converted from Normal Current

A normal metal proximitized by a superconductor can exhibit a gapless superconducting state characterized by segmented Fermi surfaces, as confirmed experimentally. In this state, quasiparticle states remain gapless along one direction, while a superconducting gap opens in the perpendicular direction. We show that this anisotropic phase enables an unconventional Hall effect in gapless superconductors, termed the superconducting Hall effect (ScHE), in which a longitudinal quasiparticle current is converted into a dissipationless transverse supercurrent without a phase transition. Using both the thermodynamic approach for bulk systems and quantum transport theory for a four-probe setup, we demonstrate the existence of this effect and reveal its intrinsic origin as the quasiparticle Berry curvature. The predicted ScHE can be experimentally verified via angular-dependent Hall measurements performed on gapless superconductors, such as Bi_{2}Te_{3} proximitized by superconducting NbSe_{2} as well as altermagnetic materials coupled to superconductors.