  Non-standard Bose-Hubbard models can exhibit rich ground state phase
diagrams, even when considering the one-dimensional limit. Using a
self-consistent Gutzwiller diagonalisation approach, we study the mean-field
ground state properties of a long-range interacting atomic gas in a
one-dimensional optical lattice. We first confirm that the inclusion of
long-range two-body interactions to the standard Bose-Hubbard model introduces
density wave and supersolid phases. However, the introduction of pair and
density-dependent tunnelling can result in new phases with two-site periodic
density, single-particle transport and two-body transport order parameters.
These staggered phases are potentially a mean-field signature of the known
novel twisted superfluids found via a DMRG approach [PRA \textbf{94}, 011603(R)
(2016)]. We also observe other unconventional phases, which are characterised
by sign staggered order parameters between adjacent lattice sites.
