  We show that Zeeman excitations of ultracold Dy atoms trapped in an optical
lattice can be used to engineer extended Hubbard models with tunable inter-site
and particle number-non-conserving interactions. We show that the ratio of the
hopping amplitude and inter-site interactions in these lattice models can be
tuned in a wide range by transferring the atoms to different Zeeman states. We
propose to use the resulting controllable models for the study of the effects
of direct particle interactions and particle number-non-conserving terms on
Anderson localization.
