  We show how strongly correlated ultracold bosonic atoms loaded in specific
orbital angular momentum states of arrays of cylindrically symmetric potentials
can realize a variety of spin-1/2 models of quantum magnetism. We consider
explicitly the dependence of the effective couplings on the geometry of the
system and demonstrate that several models of interest related to a general
$XYZ$ Heisenberg model with external field can be obtained. Furthermore, we
discuss how the relative strength of the effective couplings can be tuned and
which phases can be explored by doing so in realistic setups. Finally, we
address questions concerning the experimental read-out and implementation and
we argue that the stability of the system can be enhanced by using ring-shaped
trapping potentials.
