Doping dependent isotope effects of the quasi-1D electron-phonon system: comparison with the high-temperature superconductors

The weak-coupling quantum phase diagrams of the one-dimensional (1D) Holstein-Hubbard and Peierls-Hubbard models are computed near half-filling, using a multi-step renormalization group technique. If strong enough, the electron-phonon interaction induces a spin gap. The spin gap, which determines the superconducting pairing energy, depends strongly on the band filling and decreases monotonically as the system is doped away from half-filling. However, the superconducting susceptibility exhibits a different doping dependence; it can vary non- monotonically with doping and exhibit a maximum at an "optimal" value of the doping. For a quasi-1D array of weakly coupled, fluctuating 1D chains, the superconducting transition temperature $T_c$ exhibits a similar non-monotonic doping dependence. The effect of changing the ion mass (isotope effect) on $T_c$ is found to be largest near half-filling and to decrease rapidly upon doping away from half-filling. The isotope effect on the spin gap is the opposite sign as the isotope effect on $T_c$. We discuss qualitative similarities between these results and properties of the high-temperature superconductors.