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Register a new userThe State(s) of Replica Symmetry Breaking: Mean Field Theories vs. Short-Ranged Spin Glasses
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C.M. Newman
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We prove the impossibility of recent attempts to decouple the Replica Symmetry Breaking (RSB) picture for finite-dimensional spin glasses from the existence of many thermodynamic (i.e., infinite-volume) pure states while preserving another signature RSB feature --- space filling relative domain walls between different finite-volume states. Thus revisions of the notion of pure states cannot shield the RSB picture from the internal contradictions that rule out its physical correctness in finite dimensions at low temperature in large finite volume.
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In this topical review we discuss the nature of the low-temperature phase in both infinite-ranged and short-ranged spin glasses. We analyze the meaning of pure states in spin glasses, and distinguish between physical, or ``observable, states and (probably) unphysical, ``invisible states. We review replica symmetry breaking, and describe what it would mean in short-ranged spin glasses. We introduce the notion of thermodynamic chaos, which leads to the metastate construct. We apply these tools to short-ranged spin glasses, and conclude that replica symmetry breaking, in any form, cannot describe the low-temperature spin glass phase in any finite dimension. We then discuss the remaining possible scenarios that_could_ describe the low-temperature phase, and the differences they exhibit in some of their physical properties -- in particular, the interfaces that separate them. We also present rigorous results on metastable states and discuss their connection to thermodynamic states. Finally, we discuss some of the differences between the statistical mechanics of homogeneous systems and those with quenched disorder and frustration.
The fully-connected Ising $p$-spin model has for $p >2$ a discontinuous phase transition from the paramagnetic phase to a stable state with one-step replica symmetry breaking (1RSB). However, simulations in three dimension do not look like these mean-field results and have features more like those which would arise with full replica symmetry breaking (FRSB). To help understand how this might come about we have studied in the fully connected $p$-spin model the state of two-step replica symmetry breaking (2RSB). It has a free energy degenerate with that of 1RSB, but the weight of the additional peak in $P(q)$ vanishes. We expect that the state with full replica symmetry breaking (FRSB) is also degenerate with that of 1RSB. We suggest that finite size effects will give a non-vanishing weight to the FRSB features, as also will fluctuations about the mean-field solution. Our conclusion is that outside the fully connected model in the thermodynamic limit, FRSB is to be expected rather than 1RSB.
We study chaotic size dependence of the low temperature correlations in the SK spin glass. We prove that as temperature scales to zero with volume, for any typical coupling realization, the correlations cycle through every spin configuration in every fixed observation window. This cannot happen in short-ranged models as there it would mean that every spin configuration is an infinite-volume ground state. Its occurrence in the SK model means that the commonly used `modified clustering notion of states sheds little light on the RSB solution of SK, and conversely, the RSB solution sheds little light on the thermodynamic structure of EA models.
Simulational studies of spin glasses in the last decade have focussed on the so-called replicon exponent $alpha$ as a means of determining whether the low-temperature phase of spin glasses is described by the replica symmetry breaking picture of Parisi or by the droplet-scaling picture. On the latter picture, it should be zero, but we shall argue that it will only be zero for systems of linear dimension $L > L^*$. The crossover length $L^*$ may be of the order of hundreds of lattice spacings in three dimensions and approach infinity in 6 dimensions. We use the droplet-scaling picture to show that the apparent non-zero value of $alpha$ when $L < L^*$ should be $2 theta$, where $theta$ is the domain wall energy scaling exponent, This formula is in reasonable agreement with the reported values of $alpha$.
Critical slowing down dynamics of supercooled glass-forming liquids is usually understood at the mean-field level in the framework of Mode Coupling Theory, providing a two-time relaxation scenario and power-law behaviors of the time correlation function at dynamic criticality. In this work we derive critical slowing down exponents of spin-glass models undergoing discontinuous transitions by computing their Gibbs free energy and connecting the dynamic behavior to static in-state properties. Both the spherical and Isi
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