Weak-Field Thermal Hall Conductivity in the Mixed State of d-Wave Superconductors

Thermal transport in the mixed state of a d-wave superconductor is considered within the weak-field regime. We express the thermal conductivity, $\kappa_{xx}$, and the thermal Hall conductivity, $\kappa_{xy}$, in terms of the cross section for quasiparticle scattering from a single vortex. Solving for the cross section (neglecting the Berry phase contribution and the anisotropy of the gap nodes), we obtain $\kappa_{xx}(H,T)$ and $\kappa_{xy}(H,T)$ in surprisingly good agreement with the qualitative features of the experimental results for YBa$_{2}$Cu$_{3}$O$_{6.99}$. In particular, we show that the simple, yet previously unexpected, weak-field behavior, $\kappa_{xy}(H,T) \sim T\sqrt{H}$, is that of thermally-excited nodal quasiparticles, scattering primarily from impurities, with a small skew component provided by vortex scattering.