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 d_x²-y² superconductors in the vortex state from a general analysis of the Bogoliubov–de Gennes equation. The spin Hall conductivity σ_xy^s 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 H_c1≪B≪H_c2. We then derive the Wiedemann–Franz law for the spin and thermal Hall conductivity in the vortex state. In the limit of T⃗0, the thermal Hall conductivity satisfies κ_xy=(4π²/3)(k_B/ħ)²Tσ_xy^s. The transitions between different quantized values of σ_xy^s as well as relation to conventional superconductors are discussed.