  We show how a broad class of lattice spin-1/2 models with angular- and
distance-dependent couplings can be realized with cold alkali atoms stored in
optical or magnetic trap arrays. The effective spin-1/2 is represented by a
pair of atomic ground states, and spin-spin interactions are obtained by
admixing van der Waals interactions between fine-structure split Rydberg states
with laser light. The strengths of the diagonal spin interactions as well as
the "flip-flop", and "flip-flip" and "flop-flop" interactions can be tuned by
exploiting quantum interference, thus realizing different spin symmetries. The
resulting energy scales of interactions compare well with typical temperatures
and decoherence time-scales, making the exploration of exotic forms of quantum
magnetism, including emergent gauge theories and compass models, accessible
within state-of-the-art experiments.
