Quasiparticle Hall transport of d-wave superconductors in the vortex state

We present a theory of quasiparticle Hall transport in strongly type-II superconductors within their vortex state. We establish the existence of integer quantum spin Hall effect in clean unconventional (formula presented) superconductors in the vortex state from a general analysis of the Bogoliubov–de Gennes equation. The spin Hall conductivity (formula presented) is shown to be quantized in units of ħ/8π. This result does not rest on linearization of the BdG equations around Dirac nodes and therefore includes inter-nodal physics in its entirety. In addition, this result holds for a generic inversion-symmetric lattice of vortices as long as the magnetic field B satisfies (formula presented). We then derive the Wiedemann–Franz law for the spin and thermal Hall conductivity in the vortex state. In the limit of (formula presented), the thermal Hall conductivity satisfies (formula presented). The transitions between different quantized values of (formula presented) as well as relation to conventional superconductors are discussed.