Extraction of Bose-Hubbard parameters from a 1D microscopic model

The Bose-Hubbard model is a powerful tool to understand the many-body physics of cold atoms in lattices. The link between its parameters and the underlying microscopic model is therefore of outstanding importance. The standard Bose-Hubbard model assumes that (i) the excited energy bands are neglected, (ii) tunneling is allowed only between nearest neighbors and (iii) the interaction only acts on-site. However it has been shown in Ref.~[1] from an exact 2-body 1D calculation that the effective interaction of two cold atoms in a lattice strongly depends on the center-of-mass motion, a behavior not predicted by the standard Bose-Hubbard model. We present here an approach to extract effective Bose-Hubbard parameters from a microscopic two-body model that is based on the solution of the Schrödinger equation in a lattice without approximations. As a crucial intermediate we compute the two-body interacting Green function, expressed in terms of regular and irregular solutions. In order to avoid solution linear-dependence problems, we adapt the algorithm of Ref.~[2] to our spectral-element solution approach. [1] H. Terrier \textit{et al.}, Phys. Rev. A {\bf 93}, 032703 (2016) [2] S. J. Singer \textit{et al.}, J. Chem. Phys. {\bf 87}, 4762 (1987)