Fractional Aharonov-Bohm oscillation of a two-layer ring with two electrons

When a circular ring suffers a special topological transformation, it becomes a two-layer ring. Due to the special topology of the two-layer ring, orbital angular momenta are allowed to be a half-integer, this would affect the traditional Aharonov-Bohm oscillation (ABO). In this paper the fractional ABO of the ground state energy, persistent current, and dipole transition of a two-layer ring with two electrons has been studied. Collective and internal coordinates $\theta_{C}$ and $\phi$ have been introduced. Based on them a very simple formula for the current has been obtained, the symmetry constraint imposed on the dipole transition has been clarified, a strict relation between the photon energies of the dipole radiation and the persistent current of the ground state has been found. Comparing with the one-layer rings, the period of the fractional ABO of the two-layer rings becomes much shorter.