Coulomb repulsion (Hubbard $U$), the model is reduced to an extended Hubbard model with intersite attraction and suppressed double-occupancy [@alekor2] by applying the Lang-Firsov canonical transformation (section 2). Then the Hamiltonian can be projected onto the subspace of nearest neighbor intersite *crab* bipolarons (sections 4). In contrast with the crawler bipolaron, the crab bipolaron’s mass scales linearly with the polaron mass ($m^{**}=4m^*$ on the staggered chain and $m^{**}=6m^{*}$ on the triangular lattice).

Extending the CTQMC algorithm to systems of two particles with strong EPI and Coulomb repulsion solved the bipolaron problem on a staggered ladder, triangular and anisotropic hexagonal lattices from weak to strong coupling in a realistic parameter range where usual strong and weak-coupling limiting approximations fail. Importantly small but light bipolarons have been found for more realistic intermediate values of EPI, $\lambda \lesssim 1 $ and phonon frequency, $\omega \lesssim T(a)$ [@jim2; @jim3].

![ Bipolaron radius (in units of $a$) for a range of $\bar{\omega}$ and $\lambda$ on the staggered ladder (Reproduced from J. P. Hague et al., Phys. Rev. Lett [**98**]{}, 037002 (2007), (c) American Physical Society, 2007. [@jim2]" ).](fig5.ps){height="105mm"}

Figure 4 shows the ratio of the polaron to bipolaron masses on the staggered ladder as a 