Beyond the Sherrington-kirkpatrick Model

The state of art in spin glass field theory is reviewed. We start from an Edwards-Anderson-type model in finite dimensions, with finite but long range forces, construct the effective field theory that allows one to extract the long wavelength behaviour of the model, and set up an expansion scheme (the loop expansion) in the inverse range of the interaction. At the zeroth order we recover mean field theory. We evaluate systematic corrections to this around Parisi’s replica symmetry broken solution. At the level of quadratic fluctuations we derive a set of coupled integral equations for the free propagators of the theory and show how they can be solved for short, intermediate and extreme long distances. To reveal the physical meaning of these results, we relate the various propagator components to overlaps of spin-spin correlation functions inside a single phase space valley resp. between different valleys. Next we calculate the first loop corrections to the theory above 8 dimensions, where we find that it maps back onto mean field theory, with basically temperature independent renormalization of the coupling constants, thereby demonstrating that Parisi’s mean field theory is, at least perturbatively, stable against finite range corrections. In the range between six and eight dimensions various physical quantities pick up nontrivial temperature dependences which can, however, still be determined exactly. Upon approaching the upper critical dimension (d = 6) of the model, scaling which is badly violated in Parisi’s mean field theory is gradually restored. Below 6 dimensions one should apply renormalization group methods. Unfortunately, the structure of RG is not completely understood in spin glass theory. Nevertheless, the first corrections in 6 − d to e.g. the exponent of the order parameter can still be calculated, moreover exponentiation to this power can be checked at the next order. The theory is, however, plagued by infrared divergences due to the presence of zero modes and soft modes. Systematic methods (like those developed in the O(n) model) to handle these infrared singularities are not yet available in spin glass theory.