Quasi-Particle States with Topological Quantum Numbers in the Mixed State of d-wave Superconductors

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We investigate the extended quasi-particle states in the mixed state of d-wave superconductors on the basis of the Bogoliubov-de Gennes equation. We prove that the quasi-particle eigen-states can be classified in terms of new topological quantum numbers which are related to the topological nature of the non-trivial phases of the quasi-particles. Numerical results for the quasi-particle eigen-states reveal the crossover behavior from gapless to gapped states as the flux density B increases. In the strong field region quantum oscillations appear in the excitation energy of the quasi-particles.