Small Bipolarons in the 2-dimensional Holstein-Hubbard Model. I The   Adiabatic Limit
  The spatially localized bound states of two electrons in the adiabatic two-dimensional Holstein-Hubbard model on a square lattice are investigated both numerically and analytically. The interplay between the electron-phonon coupling g, which tends to form bipolarons and the repulsive Hubbard interaction $\upsilon \geq 0$, which tends to break them, generates many different ground-states. There are four domains in the $g,\upsilon$ phase diagram delimited by first order transition lines. Except for the domain at weak electron-phonon coupling (small g) where the electrons remain free, the electrons form bipolarons which can 1) be mostly located on a single site (small $\upsilon$, large g); 2) be an anisotropic pair of polarons lying on two neighboring sites in the magnetic singlet state (large $\upsilon$, large g); or 3) be a "quadrisinglet state" which is the superposition of 4 electronic singlets with a common central site. This quadrisinglet bipolaron is the most stable in a small central domain in between the three other phases. The pinning modes and the Peierls-Nabarro barrier of each of these bipolarons are calculated and the barrier is found to be strongly depressed in the region of stability of the quadrisinglet bipolaron. 
Laurent Proville and Serge Aubry