Deformable quantum inductance (ring) interacting with a phonon field: Persistent electrical currents re-visited ((q_e/ℏ) versus (2qe/ℏ))

The Hamiltonian of a flexible purely inductive quantum circuits, with discrete charge (q_e), coupled to the vibrational degrees of a lattice (phonons) is presented. In the static regime, when the inductance is threaded by an external fixed magnetic flux, the usual periodic behavior in the persistent electrical current (frequency q_e/ℏ) becomes altered. A new harmonic appears which is twice that of the fundamental (2q _e/ℏ). All this is the consequence of the geometric static deformation in the inductance produced by the persistent electrical current itself (Lorentz force).