We reveal a novel macroscopic quantum phenomenon induced by a magnetic field. It corresponds to the { \it non-integer quantization of the superfluid density} in a superconductor with gap nodes due to the generation of confined field-induced density waves (CFIDW) over a portion of the Fermi surface (FS). The Landau numbers $L$ are not sufficient to index these macroscopic quantum states and new quanrum numbers $\zeta$ must be added. Distinct qualitative implications of this $|L,\zeta >$ quantization are evident in a number of puzzling experiments in high-$T_c$ cuprates including the plateaus behavior in the field profile of thermal conductivity, field induced magnetic moments, charge textures around the vortices, and field induced vortex-solid to cascade vortex glass transitions.
Categories: cond-mat.supr-con cond-mat.str-el