Quasiparticle Hall Transport of d-wave Superconductors in Vortex State

Abstract

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 $d_{x^2-y^2}$ superconductors in the vortex state from a general analysis of the Bogoliubov-de Gennes equation. The spin Hall conductivity $σ^s_{xy}$ is shown to be quantized in units of $\frac{\hbar}{8π}$. 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 $H_{c1} \ll B \ll H_{c2}$. We then derive the Wiedemann-Franz law for the spin and thermal Hall conductivity in the vortex state. In the limit of $T \to 0$, the thermal Hall conductivity satisfies $κ_{x y}=\frac{4π^2}{3}(\frac{k_B}{\hbar})^2 T σ^s_{xy}$. The transitions between different quantized values of $σ^s_{xy}$ as well as relation to conventional superconductors are discussed.