Dirac nodes and quantized thermal Hall effect in the mixed state ofd-wave superconductors

Abstract

We consider the vortex state of d-wave superconductors in the clean limit. Within the linearized approximation the quasiparticle bands obtained are found to posess Dirac cone dispersions (band touchings) at special points in the Brillouin zone. They are protected by a symmetry of the linearized Hamiltonian that we call ${\mathcal{T}}_{\mathrm{Dirac}}.$ Moreover, for vortex lattices that possess inversion symmetry, it is shown that there is always a Dirac cone centered at zero energy within the linearized theory. On going beyond the linearized approximation and including the effect of the smaller curvature terms (that break ${\mathcal{T}}_{\mathrm{Dirac}}),$ the Dirac cone dispersions are found to acquire small gaps $(\ensuremath{\approx}0.5 {\mathrm{K}\mathrm{}\mathrm{T}}^{\ensuremath{-}1}$ in YBCO) that scale linearly with the applied magnetic field. When the chemical potential for quasiparticles lies within the gap, quantization of the thermal-Hall conductivity is expected at low temperatures, i.e., ${\ensuremath{\kappa}}_{\mathrm{xy}}/T=n({\ensuremath{\pi}}^{2}{k}_{B}^{2}/3h)$ with the integer n taking on values $n=\ifmmode\pm\else\textpm\fi{}2$ and 0. This quantization could be seen in low-temperature thermal transport measurements of clean d-wave superconductors with good vortex lattices.