states, indicates that bipolarons repel each other. The system is effectively a charged Bose-gas, which is a superconductor [@ogg; @sha]. This superconducting state requires a rather fine balance between electronic and ionic interactions in cuprates.

All-coupling lattice bipolarons
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The multi-polaron CFM model discussed above is analytically solvable in the strong-coupling nonadibatic ($\omega_0 \gtrsim T(a))$ limit using the Lang-Firsov transformation of the Hamiltonian, Eq.(1), and projecting it on the inter-site pair Hilbert space [@ale5; @alekor2]. To extend the theory for the whole parameter space an advanced continuous time QMC technique (CTQMC) has been recently developed for bipolarons [@jim2; @jim3]. Using CTQMC refs. [@jim2; @jim3] simulated the CFM Hamiltonian on a staggered triangular ladder (1D), triangular (2D) and strongly anisotropic hexagonal (3D) lattices including triplet pairing [@jim3]. On such lattices, bipolarons are found to move with a crab like motion (Fig. 1), which is distinct from the crawler motion found on cubic lattices [@aleran]. Such bipolarons are small but very light for a wide range of electron-phonon couplings and phonon frequencies. EPI has been modeled using the force function in the site-representation as

$$H_{e-ph} =    - \sum_{\mathbf{n}\mathbf{m}\sigma}
f_{\mathbf{m}}(\mathbf{n}) c^{\dagger}_{\mathbf{n}\sigma}
c_{\mathbf{n}\sigma} \xi_{\mathbf{m}} \: . \label{eq:four}$$

Each vibrating ion has one phonon degree 