Phase diagram of the three-orbital Hubbard-Holstein model simulating Superconducting Tungsten Bronze AxWO3

We study superconductivity in the three-dimensional three-orbital Hubbard-Holstein model which consists of the W-t2g orbitals and is extracted from the first-principles band calculation of the superconducting tungsten bronze Na_{x}WO_3. Effects of the on-site coulomb interaction U and the Jahn-Teller electron-phonon coupling g are taken into account within the random phase approximation (RPA). We obtain the phase diagram on the U-g plane, where the instability towards the magnetic order is observed for a huge value of U~28eV almost independent of g while the instability towards the ferro-orbital order accompanied by the cubic-tetragonal structural phase transition is observed for a realistic value of g=0.21~0.28eV depending on U.