Coexistence of superconductivity and charge-density waves in a two-dimensional Holstein model at half-filling

On the one hand, in one dimension the coupling of electrons to phonons leads to a transition from a metallic to a Peierls distorted insulated state if the coupling exceeds a critical value. On the other hand, in two dimensions the electron-phonon interaction may also lead to the formation of Cooper pairs. In this letter, we study for two dimensions the competing influence of superconductivity and charge order (in conjunction with a lattice distortion) by means of the projector-based renormalization method (PRM). In this way, we can not only approach correlation functions of superconductivity and charge density wave but also have direct access to the order parameters. Increasing the electron-phonon interaction, we find a crossover behavior between a purely superconducting state and a charge-density wave where a well-defined parameter range of coexistence of superconductivity and lattice distortion exists.