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The mean-field theory predicts a very complicated structure of the spin glass state as represented by the full RSB scheme for the SK model. Whether or not this picture applies to three-dimensional systems is not a trivial problem, and many theoretical and experimental investigations are still going on. The reader will find summaries of recent activities in the same volumes as above (Young 1997; Miyako et al. 2000). See also Dotsenko (2001) for the renormalization group analyses of finite-dimensional systems.
The AT line lies below the Nishimori line and the RS solution is stable. 3 that the structure of the phase space is always simple on the Nishimori line. 18) by the same method as above (Nishimori 1986a). 14) is derived when K = Kp : P (E) = 1 2N (2 cosh K)NB τ TrS δ(E − Jij Si Sj ) exp(K τlm Sl Sm ). 19) Summing over bond variables other than the specific one (ij), which we are treating, can be carried out. The result cancels out with the corresponding factors in the denominator. The problem then reduces to the sum over the three variables τij , Si , and Sj , which is easily performed to yield P (E) = pδ(E − J) + (1 − p)δ(E + J).
Susceptibilities to all other external fields (such as a field with random sign at each site) do not diverge at any temperature. 100), various modes continue to diverge one after another below the transition point Tf where the mode with the largest eigenvalue shows a divergent susceptibility. In this sense there exist continuous phase transitions below Tf . This fact corresponds to the marginal stability of the Parisi solution with zero eigenvalue of the Hessian and is characteristic of the spin glass phase of the SK model.