Superconductivity and Gutzwiller correlations in a two band Hubbard-Fr\"{o}hlich model

We studied the two-band Hubbard-Fr\"{o}hlich model with the hopping parameters from the first-principle calculation for La-Phenanthrene. The mean-field approximation shows that two-band superconductivity with opposite signs is possible even with the presence of a large Hubbard $U$. Using Gutzwiller approximation, we show that the Hubbard $U$ can help the superconductivity in this system. When $U$ is too large, it will go into the insulator state. Roughly, we determined the critical $U$ for this superconductor-insulator (SI) transition. We further shows that before the SI transition, the antiferromagnetic order will come into being. However, there is still a large range of $U$ for the superconductivity to win. We also discussed the possibility to apply our model to other polycyclic aromatic hydrocarbons.