Resonant pairing isotope effect in polaronic systems

The intermediate coupling regime in polaronic systems, situated between the adiabatic and the anti-adiabatic limit, is characterized by resonant pairing between quasi-free electrons which is induced by an exchange interaction with localized bipolarons. The onset of this resonant pairing takes place below a characteristic temperature T* and is manifest in the opening of a pseudogap in the density of states of the electrons. The variation of T* is examined here as a function of (i) the typical frequency \omega_0 of the local lattice modes, which determines the binding energy of the bipolarons, and (ii) the doping, which amounts to a relative change of the bipolaron concentration n_B to that of the free electrons n_F. We concentrate on a doping regime, where small changes in doping give rise to a large change in T*, which is the case when n_B is small (< 0.1 per site). For finite values of n_B we find negative and practically doping independent values of the isotope coefficient \alpha^* which characterizes the formation of resonating electron pairs. Upon decreasing the total particle density such that n_B becomes exponentially small, we find a rapid change in sign of \alpha^*. This is related to the fact that the system approaches a state which is more BCS-like, where electron pairing occurs via virtual excitations into bipolaronic states and where T* coincides with the onset of superconductivity.