We report measurements of Little-Parks oscillation on the hierarchical honeycomb-superconducting network for investigating possible effects of hierarchical structure in terms of spatial symmetry, parity and duality. We observed an asymmetric Little-P
arks oscillation about $Phi/Phi_0 = 1/2$, although spatial symmetry was kept in the network. In comparison with a regular honeycomb network, the asymmetric oscillation is attributed to hierarchy which induces mixture of commensurate and incommensurate regions. The asymmetric oscillation is found to indicate breaking of the duality of vortex configuration.
We observed an unconventional parity-violating vortex in single domain Sr2RuO4 single crystals using a transport measurement. The current-voltage characteristics of submicron Sr2RuO4 shows that the induced voltage has anomalous components which are e
ven function of the bias current. The results may suggest that the vortex itself has a helical internal structure characterized by a Hopf invariant (a topological invariant). We also discuss that the hydrodynamics of such a helical vortex causes the parity violation to retain the topological invariant.
