  An exciting development in the field of correlated systems is the possibility
of realizing two-dimensional (2D) phases of quantum matter. For a systems of
bosons, an example of strong correlations manifesting themselves in a 2D
environment is provided by helium adsorbed on graphene. We construct the
effective Bose-Hubbard model for this system which involves hard-core bosons
$(U\approx\infty)$, repulsive nearest-neighbor $(V>0)$ and small attractive
$(V'<0)$ next-nearest neighbor interactions. The mapping onto the Bose-Hubbard
model is accomplished by a variety of many-body techniques which take into
account the strong He-He correlations on the scale of the graphene lattice
spacing. Unlike the case of dilute ultracold atoms where interactions are
effectively point-like, the detailed microscopic form of the short range
electrostatic and long range dispersion interactions in the helium-graphene
system are crucial for the emergent Bose-Hubbard description. The result places
the ground state of the first layer of $^4$He adsorbed on graphene deep in the
commensurate solid phase with $1/3$ of the sites on the dual triangular lattice
occupied. Because the parameters of the effective Bose-Hubbard model are very
sensitive to the exact lattice structure, this opens up an avenue to tune
quantum phase transitions in this solid-state system.
