On the replica symmetric Ising spin glasses
Confronted with the contradiction between mounting evidence for a phase transition in three dimension with only two states (in zero field) and the necessity for a symmetry breaking à la PARISI when treating an Ising spin glass with a gaussion bond distribution, we look, as an escape, for possible stable replica symmetric solutions, in the tree approximation, associated with non-gaussian distributions.
We first consider a toy model where the bond distribution has one more non vanishing cumulant than the gaussian. We show that there is some theoretical range of stability provided the second cumulant is negative enough. We then turn to a general bond distribution and derive closed expressions for the “masses” that allow to determine a stability region (in dilution/temperature space). We also show that instead of using an infinite number of order parameters qr (generalizing the q2 of EDWARDS ANDERSON one may use a function as MEZARD and PARISI have done for optimization problems. This is illustrated on simpler models due to VIANA and BRAY, and to FU and ANDERSON. Finally we give some hints for the treatment of the general stability problem.
KeywordsSpin Glass Tree Approximation Closed Expression Replica Symmetry Breaking Bond Distribution
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