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Synthetic spin-orbit coupling mediated by a bosonic environment

We study a mobile quantum impurity, possessing internal rotational degrees of freedom, confined to a ring in the presence of a many-particle bosonic bath. By considering the recently introduced rotating polaron problem, we define the Hamiltonian and examine the energy spectrum. The weak-coupling regime is studied by means of a variational ansatz in the truncated Fock space. The corresponding spectrum indicates that there emerges a coupling between the internal and orbital angular momenta of the impurity as a consequence of the phonon exchange. We interpret the coupling as a phonon-mediated spin-orbit coupling and quantify it by using a correlation function between the internal and orbital angular momentum operators. The strong-coupling regime is investigated within the Pekar approach and it is shown that the correlation function of the ground state shows a kink at a critical coupling, that is explained by a sharp transition from the non-interacting state to the states that exhibit strong interaction with the surroundings. The results might find applications in such fields as spintronics or topological insulators, where spin-orbit coupling is of crucial importance.