New magnetic-field-induced macroscopic quantum phenomenon in a   superconductor with gap nodes
  High-$T_c$ superconductivity is unconventional because the gap is not isotropic as in simple metals but has $d_{x^2-y^2}$ symmetry with lines of nodes. In a fascinating thermal transport experiment on a high-$T_c$ superconductor, Krishana et al \cite{Krishana} have reported mysterious magnetic field induced first order transitions from a superconducting state with gap nodes to a state without gap nodes. We show here that this is an experimental manifestation of a novel {\it macroscopic} quantum phenomenon induced by the magnetic field, qualitatively different from the usual quantum Hall effects. It corresponds to the {\it quantization of the superfluid density} in a superconductor with gap nodes due to the generation of Confined Field Induced Density Waves (CFIDW) in the node regions of the Fermi surface. The Landau numbers $L$ are not sufficient to index these macroscopic quantum states and the addition of a new quantum number $\zeta$ is necessary. Distinct qualitative implications of this non-integer $|L,\zeta>$ quantization are also evident in a number of recent unexplained experimental reports in the cuprates. 
Georgios Varelogiannis and Michel Heritier