We show how spatial light modulation allows one to optically compute an Ising Hamiltonian. This enables the use of optical radiation to accelerate algorithms to find the ground state and hence the optimal solution of a combinatorial optimization problem. The phase matrix on a spatial light modulator acts as a lattice of spins whose interaction is ruled by the constrained optical intensity in the far field. Feedback from the detection plane allows the spatial phase distribution to evolve toward the minimum of the selected spin model. Related topics such as annealing, adiabatic evolution, the XY model, and decomposition methods are reviewed.

Negative exchange J < 0 leads to magnetic order that depends on the lattice topology. Structures with more than one magnetic sublattice include antiferromagnets and ferrimagnets. An antiferromagnet has two equal but oppositely directed sub-lattices, where the sublattice magnetization disappears above the Néel point TN. Two unequal oppositely directed magnetic lattices constitute a ferrimagnet. The molecular field theory is extended to cover these cases. A wealth of more complex noncollinear magnetic structures exist. The subtle effects of a non-crystalline structure are manifest in amorphous magnets, where spins sometimes freeze in random orientations. Magnetic model systems highlight the influence of some particular feature on collective magnetic order, such as reduced space or spin dimensionality, a particular distribution of exchange interactions, special topology or lack of crystal structure. Examples include the two-dimensional Ising model, frustrated antiferromagnets and canonical spin glasses.