The bound states of two electrons in the adiabatic Holstein-Hubbard model are studied numerically in one and two dimensions from the anticontinuous limit. This mode1 involves acompetition between a local electron-phonon coupling (with a classical lattice) which tends to form pairs ofelectrons and the repulsive Hubbard interaction I/ z 0 which tends to break them.
In one-dimensional ( ID). the ground-state always consists of a pair of localized polarons in a singlet state. They are located at the same site for I/ = 0. Increasing U, there is a first-order transition at which the bipolaron becomes a spin singlet pair of two polurons bounded by a magnetic interaction. The pinning mode of the bipolaron soften in the vicinity of this transition leading to a higher mobility of the bipolaron which is tested numerically.
In 2D. and for any U. the electron-phonon coupling needs to be large enough in order to form small polarons or bipolarona instead of extended electrons. We calculate the phase diagram of the bipolaron involving first-order transition lines with a triple point. A pair of polarons can form three types of bipolarons: (a) on a single site at small U, (b) a spin singlet state on two nearest neighbor sites for larger U as in ID, and (c) a new intermediate state obtained as the resonant combination of four ?-site singlet states sharing a central site, called quadrisinglet.
The breathing and pinning internal modes of bipolarons in 2D generally only weakly soften and thus. they are practically not mobile. On the contrary. in the vicinity of the triple point involving the quadrisinglet. both modes exhibit a significant softening. However, it was not sufficient for allowing the existence of a classical mobile bipolar-on (at least in that model).