  The Hubbard model underlies our understanding of strongly correlated
materials. While its standard form only comprises interaction between particles
at the same lattice site, its extension to encompass long-range interaction,
which activates terms acting between different sites, is predicted to
profoundly alter the quantum behavior of the system. We realize the extended
Bose-Hubbard model for an ultracold gas of strongly magnetic erbium atoms in a
three-dimensional optical lattice. Controlling the orientation of the atomic
dipoles, we reveal the anisotropic character of the onsite interaction and
hopping dynamics, and their influence on the superfluid-to-Mott insulator
quantum phase transition. Moreover, we observe nearest-neighbor interaction,
which is a genuine consequence of the long-range nature of dipolar
interactions. Our results lay the groundwork for future studies of novel exotic
many-body quantum phases.
