Ultracold atoms offer a unique platform to perform quantum simulations of quantum materials and many-body systems. When atoms with spin are arranged in an optical lattice in form of a Mott insulator, they realize paradigmatic Heisenberg spin models, where only neighboring spins interact. Until very recently, all experimental studies with cold atoms addressed the special case of an isotropic Heisenberg model. Using lithium-7 atoms and Feshbach resonances to tune the interactions, we have created spin ½ Heisenberg models with adjustable anisotropy, including the special XX-model which can be exactly solved by mapping it to non-interacting fermions. Spin transport changes from ballistic to diffusive depending on the anisotropy. Special transverse spin patterns in the form of helices are exact eigenstates of the Heisenberg model and are called phantom states since they carry momentum, but no energy. We have realized these phantom helix states and used them to accurately measure the interaction anisotropy. We ﬁnd a major contribution from short-range oﬀ-site interactions of the underlying Hubbard model which had not been observed before.