Restoration of $hc/2e$ Magnetic Flux Periodicity in a Hollow $d$-Wave   Superconducting Cylinder
  The magnetic flux dependence of order parameter and supercurrent is studied in a hollow d-wave superconducting cylinder. It is shown that the existence of line nodal quasiparticles in a pure $d_{x^2-y^2}$ pairing state gives rise to an $hc/e$ periodicity in the order parameter and a first-order quantum phase transition for a large system size. We demonstrate that the flux periodicity in the supercurrent is sensitive to the detailed electronic band structure and electron filling factor. In particular, we find that, in cooperation with the increase of the cylinder circumference, the $hc/2e$ periodicity can be restored significantly in the supercurrent by avoiding the particle-hole symmetry point. A similar study of a $d_{x^2-y^2}+id_{xy}$ pairing state verifies the peculiarity of unconventional superconductors with nodal structure. 
Jian-Xin Zhu