Intermediate phase of the one dimensional half-filled Hubbard-Holstein   model
  We present a detailed numerical study of the Hubbard-Holstein model in one dimension at half filling, including full finite-frequency quantum phonons. At half filling, the effects of the electron-phonon and electron-electron interactions compete, with the Holstein phonon coupling acting as an effective negative Hubbard onsite interaction U that promotes on-site electron pairs and a Peierls charge-density wave state. Most previous work on this model has assumed that only Peierls or U>0 Mott insulator phases are possible at half filling. However, there has been speculation that a third metallic phase exists between the Peierls and Mott phases. We present results confirming the intermediate metallic phase, and show that the Luttinger liquid correlation exponent K_rho>1 in this region, indicating dominant superconducting pair correlations. We explore the full phase diagram as a function of onsite Hubbard U, phonon coupling constant, and phonon frequency. 
R.T. Clay and R.P. Hardikar