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A longstanding open question in the theory of disordered systems is whethershort-range models, such as the random field Ising model or theEdwards-Anderson model, can indeed have the famous properties that characterizemean-field spin glasses at nonzero temperature. This article shows that this isat least partially possible in the case of the random field Ising model.Consider the Ising model on a discrete $d$-dimensional cube under free boundarycondition, subjected to a very weak i.i.d. random external field, where thefield strength is inversely proportional to the square-root of the number ofsites. It turns out that in $d\ge 2$ and at subcritical temperatures, thismodel has some of the key features of a mean-field spin glass. Namely, (a) thesite overlap exhibits one step of replica symmetry breaking, (b) the quencheddistribution of the overlap is non-self-averaging, and (c) the overlap has theParisi ultrametric property. Furthermore, it is shown that for Gaussiandisorder, replica symmetry does not break if the field strength is taken to bestronger than the one prescribed above, and non-self-averaging fails if it isweaker, showing that the above order of field strength is the only one thatallows all three properties to hold. However, the model does not have two otherfeatures of mean-field models. Namely, (a) it does not satisfy theGhirlanda-Guerra identities, and (b) it has only two pure states instead ofmany.

Features of a spin glass in the random field Ising model