Phase diagram of the one dimensional Hubbard-Holstein Model at 1/2 and   1/4 filling
  The Hubbard-Holstein model is one of the simplest to incorporate both electron-electron and electron-phonon interactions. In one dimension at half filling the Holstein electron-phonon coupling promotes onsite pairs of electrons and a Peierls charge density wave while the Hubbard onsite Coulomb repulsion U promotes antiferromagnetic correlations and a Mott insulating state. Recent numerical studies have found a possible third intermediate phase between Peierls and Mott states. From direct calculations of charge and spin susceptibilities, we show that (i) As the electron-phonon coupling is increased, first a spin gap opens, followed by the Peierls transition. Between these two transitions the metallic intermediate phase has a spin gap, no charge gap, and properties similar to the negative-U Hubbard model. (ii) The transitions between Mott/intermediate and intermediate/Peierls states are of the Kosterlitz-Thouless form. (iii) For larger U the two transitions merge at a tritical point into a single first order Mott/Peierls transition. In addition we show that an intermediate phase also occurs in the quarter-filled model. 
R.P. Hardikar and R.T. Clay