Quasiparticle spectrum of d-wave superconductors in the mixed state

The quasiparticle spectrum of a two-dimensional d-wave superconductor in th e mixed state, Ii,, much less than H much less than H-c2, is studied both a nalytically and numerically using the linearized Bogoliubov-de Gennes equat ion. We consider various values of the ''anisotropy ratio'' v(F)/v(Delta) f or the quasiparticle velocities at the Dirac points, and we examine the imp lications of symmetry. For a Bravais lattice of vortices, we find there is always an isolated energy zero (Dirac point) at the center of the Brillouin zone, but for a non-Bravais lattice with two vortices per unit cell there is generally an energy gap. In both of these cases, the density of states s hould vanish at zero energy, in contrast with the semiclassical prediction of a constant density of states, though the latter may hold down to very lo w energies for large anisotropy ratios. This result is closely related to t he particle-hole symmetry of the band structures in lattices with two vorti ces per unit cell. More complicated non-Bravais vortex lattice configuratio ns with at least four vortices per unit cell can break the particle-hole sy mmetry of the linearized energy spectrum, and lead to a finite density of s tates at zero energy.