Quantum Oscillations in the mixed state of d-wave superconductors

We show that the low-energy density of quasiparticle states in the mixed state of ultra-clean $d$-wave superconductors is characterized by pronounced quantum oscillations in the regime where the cyclotron frequency $\hbar\omega_c\ll \Delta_0$, the $d-$wave pairing gap. Such oscillations as a function of magnetic field $B$ are argued to be due to the internodal scattering of the $d$-wave quasiparticles near wavevectors $(\pm k_D,\pm k_D)$ by the vortex lattice as well as their Zeeman coupling. The periodicity of the oscillations is set by the condition $k_D [hc/(eB)]^{1/2} \equiv k_D' [hc/(eB')]^{1/2}\pmod {2\pi}$. We find that there is additional structure within each period which grows in complexity as the Dirac node anisotropy increases.