  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 dispersion (band touchings) at special points in the
Brillouin zone. They are protected by a symmetry of the linearized Hamiltonian
that we call T_Dirac. Moreover, for vortex lattices that posess 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 T_Dirac), the
Dirac cone dispersions are found to acquire small gaps (0.5 K/Tesla 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. kappa_{xy}/T =
n[(pi k_B)^2/(3h)] with the integer `n' taking on values n=+2, -2, 0. This
quantization could be seen in low temperature thermal transport measurements of
clean d-wave superconductors with good vortex lattices.
