Geometrical Picture for Spin Glasses

{ A controversial issue in spin glass theory is whether mean eld correctly describes 3-dimensional spin glasses. If it does, how can replica symmetry breaking arise in terms of spin clusters in Euclidean space? Here we argue that there exist system-size low energy excitations that are \sponge-like", generating multiple valleys separated by diverging energy barriers. The droplet model should be valid for length scales smaller than the size of the system (> 0), but nevertheless there can be system-size excitations of constant energy without destroying the spin glass phase. The picture we propose then combines droplet-like behavior at nite length scales with a potentially mean eld behavior at the system-size scale. Introduction. { The solution of the Sherrington-Kirkpatrick mean eld model of spin glasses shows that its equilibrium states are organised in a hierarchy associated with continuous replica symmetry breaking (RSB) 1]. A working paradigm for some years has been that this type of replica symmetry breaking also occurs in nite dimensional spin glasses above the lower critical dimension (2 < d l < 3); we will call this school of thought the mean eld picture. The question of whether this paradigm is correct is still the subject of an active debate (see 2] and references therein). The mean eld hierarchical organisation of states corresponds to valleys within valleys ... within valleys. Though such a structure is appealing to many, it seems to us necessary to describe how it can possibly arise for spins lying in Euclidean space. As an example, consider the many nearly degenerate ground states predicted by mean eld; what is the nature of the clusters of spins that ip when going from one such state to another? It is not clear a priori that mean eld has much predictive power here for the following reason. In any nite dimension, there are clusters whose surface to volume ratio is arbitrarily small. However this kind of object does not arise in models without geometry such as the (innnite range) Sherrington-Kirkpatrick model or mean eld diluted models (such as the Viana-Bray model); any cluster in those models has a surface growing essentially as fast as its volume. This key diierence is very important in spin glass models having up-down symmetry: when ipping a cluster, the change in energy comes from the surface only, but the change in quantities like the