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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.

Persistent current in a chain of two Holstein-Hubbard rings in the presence of Rashba spin-orbit interaction