Phase transitions in Spin glasses

the greatest theoretical success in spin glasses to date has to do with the analysis and solution of a spin glass in an (effectively) infinite number of dimensions, where we have the only theoretical results that there is a phase transition

It is easy to prove theoretically that an EA spin glass in one dimension has no phase transition (but this is true for any one-dimensional system with short-range interactions only)

From numerical simulations, the current consensus is that, for the EA Ising model, there is a phase transition (at nonzero temperature) in four dimensions and higher, probably one in three dimensions as well, and none in two.

See Edwards-Anderson Hamiltonian for a description of the order parameter, and Replica symmetry breaking.