A geometrical picture for finite dimensional spin glasses

– A controversial issue in spin glasses is whether mean field theory 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 energy barriers whose heights diverge in the infinite volume limit. The droplet model should be valid for length scales smaller than the size of the system (θ > 0), but θ may be zero for these system-size excitations without destroying the spin glass phase. The picture we then propose combines droplet-like behavior at finite length scales with a potentially mean field behavior at the system-size scale. Introduction. – The solution of the Sherrington-Kirkpatrick mean field model of spin glasses shows that its equilibrium states are organised in a hierarchy generated by continuous replica symmetry breaking (RSB) [1]. A working paradigm for some years has been that such a sequence of replica symmetry breakings also applies to finite dimensional spin glasses above the lower critical dimension (2 < dl < 3); we will call this school of thought the mean field picture. The question of whether this paradigm is correct is still the subject of an active debate (see [2] and references therein). The mean field hierarchical organisation of states in phase space corresponds to valleys within valleys ... within valleys. Such a structure is appealing to many, and to us it seems necessary to describe how it can arise for spins lying in Euclidean space. As an example, consider the many nearly degenerate ground states predicted by mean field; what is the nature of the clusters of spins that flip when going from one such state to another? It is not clear a priori that mean field has much predictive power here for the following reason. In any finite 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 (infinite range) Sherrington-Kirkpatrick model or mean field 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 difference is very important in spin glass models having up-down symmetry: when flipping a Typeset using EURO-LTEX 2 EUROPHYSICS LETTERS cluster, the change in energy comes from the surface only, but the change in quantities like the overlap goes as the volume of the cluster. Another reason for insisting on having a geometrical (sometimes called “real space”) picture is that the dynamics of a real spin glass is local in space, leading to coherence effects that build up from small to large length scales. Realistic theories of spin glasses should then allow for these scales by incorporating the Euclidean geometry in which the spins are embedded. Clusters in Euclidean space have generated much interest since the early 80’s [3] and have been studied in detail by Fisher and Huse [4, 5]. These authors focused on understanding the properties of low lying excitations above the ground state, and their objects of study were localised compact clusters of spins (droplets). However, their goal was not to come up with a picture compatible with RSB, and on the contrary their conclusions are in direct conflict with the mean field picture. Unfortunately, there have been very few other works based on geometrical points of view. The possibility that droplets may be fractal rather than compact has been considered several times [6, 7] and has gained renewed interest recently via the hierarchical droplet model [8, 9]. In our work, we want to deepen this type of approach and provide a coherent geometrical picture of valleys in finite dimensional spin glasses. We hope to convince the reader that appropriately constructed clusters that are neither compact nor localised (i) have low energies, and (ii) are separated by energy barriers that diverge in the thermodynamic limit. These clusters occur on the size of the whole system which is assumed to be finite but arbitrarily large, so we call them system-size clusters. From the spongy nature of these clusters we can see why the spin glass stiffness exponent θ can be positive in spite of the presence of system-size excitations of energy O(1). Within our approach, the droplet model can be valid at finite length scales (corresponding to properties within one valley) while the mean field picture may be valid at the scale of the whole system. (The two scales of validity of course do not overlap.) We will provide several plausibility arguments for this kind of a mixed picture. No numerical evidence will be presented here, rather we will ask the reader to imagine what happens when one searches for low lying states; we believe that this point of view can uncover the essential qualitative properties of the energy landscape. θ exponents in the droplet and scaling pictures. – For definiteness, we consider the EdwardsAnderson (EA) model on an L×L×L cubic lattice, but our reasoning can be applied to any short range spin glass model in dimension d ≥ 3. The Hamiltonian has nearest neighbors couplings and no external field: