Quasiparticle Hall Transport of d-wave Superconductors in Vortex State

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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 dₓ₂₋y₂ superconductors in the vortex state from a general analysis of the Bogoliubov-de Gennes equation. The spin Hall conductivity σˢxy 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 Hc₁ ≪ B ≪ Hc₂. We then derive the Wiedemann-Franz law for the spin and thermal Hall conductivity in the vortex state. In the limit of T → 0, the thermal Hall conductivity satisfies κₓ y=4π²/3(kB/ℏ)² T σˢxy. The transitions between different quantized values of σˢxy as well as relation to conventional superconductors are discussed.