Many features predicted by mean field spin glass models, such as the behaviour of susceptibilities and correlation functions or the occurrence of aging and off-equilibrium dynamics, are qualitatively observed in experiments, suggesting that the mean field scenario may hold for finite dimensional systems also. To investigate this hypothesis a field theory for the fluctuations around the mean field solution has been developed. This theory is formally rather intricate, due to the non trivial nature of the order parameter, which is a whole function rather than simply the magnetization, as it would be in an ordinary ferromagnet.
The analysis of Gaussian fluctuations indicate that the low temperature Replica solution is stable. The validity of the mean field scenario in finite dimension is a hot topic in spin glasses, and a great effort has been devoted in the last years to tackle this problem with analytic approaches and numerical simulations. In our research we have contributed to develop such a field theory. In particular, we recently provided analytical predictions for a series of correlation functions, and compared them to the results of numerical simulations in the Edwards Anderson model (spin glass in 3D).
Spatial correlation functions in three-dimensional Ising Spin Glasses, C. De Dominicis, I. Giardina, E. Marinari, O. Martin and F. Zulliani, Phys. Rev. B 72, 014443 (2005)
Random Fields and Spin Glasses – A Field Theory Approach. C. de Dominicis and I. Giardina. Cambridge University Press, Cambridge, 2006.