Quasi-particle band structure and gapless superconductivity in the vortex-lattice state

The quasi-particle spectrum for s-wave superconductors in the Vortex-lattice state remains gapless in magnetic fields much below the upper critical field H-c2 where the cyclotron frequency is smaller than the amplitude of the pair potential. This is shown through self-consistent solutions of the Bogoliubov-de Gennes equations for the attractive Hubbard model on the square lattice. It has not been predicted in earlier theoretical works based on a continuous model because it is due to the dispersion of Landau subbands in the lattice model.