The persistent current in a chain of two quantum rings threaded by an Aharonov-Bohm flux is studied in the presence of electron-phonon interactions and Rashba spin-orbit coupling. The chain is modeled by the Holstein-Hubbard-Rashba Hamiltonian, the phonon's degrees of freedom were eliminated by the conventional Lang-Firsov transformation, the effective electronic Hamiltonian was diagonalized by using the Hartree-Fock approximation. The equations for ground state energy, persistent current and Drude weight were also obtained. The persistent current was calculated by differentiating the GS energy. The dependence of ground state energy, persistent current and Drude weight as a functions of flux for different values of Rashba spin-orbit interaction was numerically shown. The effects of Aharonov-Bohm flux, temperature, chemical potential spin-orbit interaction and electron-phonon interaction on the persistent current were also investigated.

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The electron transmission through a system of two orthogonal quantum rings in a magnetic field with one input and two output wires is studied. It is shown that the transmission to only one of the two output leads can be governed by a change in the construction orientation in respect to the field. A model of nonplane quantum graph embedded in three‐dimensional space is used.

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The spin-dependent electron transport in two coupled quantum rings with Rashba spin-orbit interaction and magnetic flux was studied by using quantum graph approach. We show the dependence of spin-polarization on system parameters such as the Rashba coupling constant, the radius of the rings, the angles between the leads and the attachment point of the rings. The spin-polarization can be controlled and changed from -1 to +1 by using a magnetic flux. Also it was shown that this model is a limiting case of double Rashba quantum ring when the length of the middle lead between rings vanishes.