Anomalous Scaling and Gapless Fermions of d-Wave Superconductors in a Magnetic Field

The problem of d_x²-y²-wave quasiparticles in a weakly disordered Abrikosov vortex lattice is studied. Starting with a periodic lattice, the topological structure of the magnetic crystal momenta of gapless fermions is found for the particle-hole symmetric case. If in addition the site centered inversion symmetry is present, both the location and the number of the gapless fermions can be determined using an index theorem. In the case of spatially aperiodic vortex array, Simon and Lee scaling is found to be violated due to a quantum anomaly. The electronic density of states is found to scale with the root-mean-square vortex displacement as √Hf(u_rms²/ξ²), while thermal conductivity is H independent, but different from the H=0 case.