Mean-field theory and fluctuations in Potts spin glasses. I

Abstract

A short-range Potts spin glass is studied. A stable, non-marginal, mean-field theory is found with one level of replica symmetry breaking and a discontinuous transition for p>4. A complete stability analysis is provided, and two different correlation lengths are found above eight dimensions. The fluctuations in the ordered phase around this solution are incorporated in a renormalisation group approach and it is found that for a small range of the parameters they restore scaling close to the upper critical dimension. For the three-state case it appears that fluctuations destroy the stability of the solution for d<or=8 and cause the system to undergo a first-order phase transition. Non-universal corrections to the equation of state above the upper critical dimension are discussed.