  We demonstrate how the spin-1/2 XYZ quantum Heisenberg model can be realized
with bosonic atoms loaded in the p band of an optical lattice in the Mott
regime. The combination of Bose statistics and the symmetry of the p-orbital
wave functions leads to a nonintegrable Heisenberg model with antiferromagnetic
couplings. Moreover, the sign and relative strength of the couplings
characterizing the model are shown to be experimentally tunable. We display the
rich phase diagram in the one-dimensional case and discuss finite size effects
relevant for trapped systems. Finally, experimental issues related to
preparation, manipulation, detection, and imperfections are considered.
