Superconductivity on a Möbius strip: Numerical studies of order parameter and quasiparticles

Superconducting states of an anisortopic s-wave superconductor on a M\"obius strip are studied numerically based on the Ginzburg-Landau theory and the Bogoliubov-de Gennes theory. In both, the equations are solved numerically on discitized lattice and the nonlinearity and the self-consistency are fully taken into account. First, we study the superconducting states on the M\"obius strip in the presence of the Aharonov-Bohm flux threading the ring by employing the Ginzburg-Landau theory, and confirm the phase diagram previously proposed by Hayashi and Ebisawa [J. Phys. Soc. Jpn. {\bf 70}, 3495 (2002)]. The metastable states as well as the equilibrium energy state are studied and the nonequiriblium processes when the magnetic field is varied at a fixed temperature are discussed. Next, we study the microscopic superconducting states on the M\"obius strip based on the Bogoliubov-de Gennes theory, especially focusing on the state with a real-space node in the superconducting gap, which is expected to appear when the flux threading the ring is half the superconducting flux quantum. The local density of states in this {\it nodal state} is calculated in detail and the existence of the zero-energy bound states is shown.