The superconducting T_c and the metal-insulator transition in the Hubbard-Holstein model

We present that the maximum superconducting T_c is on the phase boundary between metallic and insulating states of the normal state in the Hubbard-Holstein model. It is a prototype model including the local Coulomb interaction U and the electron-phonon coupling g. We performed the dynamical mean field theory (DMFT) calculations employing the numerical renormalization group (NRG) technique and identified the normal state phase diagram for the metal-insulator transition. Interestingly, local maximum T_c occurs along the lower critical phase boundary g_c1 of the 1st order metal-insulator transition in the normal state where the quantum fluctuation becomes maximum. Implications of this finding will be discussed in comparison with the quantum critical superconductivity. In addition, we obtained the electron-phonon coupling spectra α²F(ω) and examined whether the maximum T_c calculated from the Eliashberg theory lies on the g_c1.