Unconventional superconducting states and flux-dependent current evolution in mesoscopic p-wave loops

The $p$-wave superconducting state and persistent supercurrent in mesoscopic symmetric and asymmetric loops are investigated by numerically solving the Bogoliubov--de Gennes equations self-consistently. For square loops, the spatial variations of the superconducting order parameters are sensitive to the arm width and the superconducting pairing strength as well as temperature. Finite circulating currents emerge even if the threaded magnetic flux is absent, arising from the broken equality of the ${\mathrm{\ensuremath{\Delta}}}_{+}$ and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{-}}$ pairing components. When the flux turns on, novel sawtooth patterns of the current evolution show up due to the appearance of energy jumps in flux for the loop with a thick wall, which is consistent with the experimental detection of unconventional quantum oscillations in thick samples of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ [X. Cai et al., Phys. Rev. B 87, 081104(R) (2013)]. Several spatial patterns of the pairing orders are obtained, and the peaks of the zero-energy local density of states indicate the occurrence of the odd-frequency pairing for some appropriate flux. Particularly, with increasing flux, a peculiar intermediate state with asymmetric profiles is found in our present $p$-wave system. In the case of rectangular loops, the flux-dependent evolution feature behaves in a more complicated manner than in the square case. Nonzero currents can also occur at zero flux by tuning the ratio of the length to height of the loop. Interestingly, a stable state with asymmetric modulations similar to those of the intermediate state presented in the square loop can be easily realized at finite flux in the rectangular system due to the aspect-ratio effect.