Numerical Studies of Quantum Oscillations in the Superconducting vortex mixed state

We studied the quantum oscillations in superconducting vortex mixed states with d-wave pairing. We showed that the Onsager relation does not always apply. Furthermore, at mean field level the quantum oscillations from electron pockets are suppressed by the pairing. We conclude that an interpretation of the experimental results asccoming from the four hole pockets created by a $(\pi ,\pi )$ folding cannot be ruled out.