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 ${\sigma }_{xy}^s$ is shown to be quantized in units of $\frac{\hslash }{8\pi }$. 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 ${\kappa }_{xy}=\frac{4{\pi }^2}{3}(\frac{k_B}{\hslash }{)}^{2}T{\sigma }_{xy}^s$. The transitions between different quantized values of ${\sigma }_{xy}^s$ as well as relation to conventional superconductors are discussed.

arXiv comment18 pages REVTex, 3 figures, references added

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