Short-range Ising spin glasses: the metastate interpretation of replica symmetry breaking

Abstract

Parisi's formal replica-symmetry--breaking (RSB) scheme for mean-field spin glasses has long been interpreted in terms of many pure states organized ultrametrically. However, the early version of this interpretation, as applied to the short-range Edwards-Anderson model, runs into problems because as shown by Newman and Stein (NS) it does not allow for chaotic size dependence, and predicts non-self-averaging that cannot occur. NS proposed the concept of the metastate (a probability distribution over infinite-size Gibbs states in a given sample that captures the effects of chaotic size dependence) and a non-standard interpretation of the RSB results in which the metastate is non-trivial and is responsible for what was called non-self-averaging. Here we use the effective field theory of RSB, in conjunction with the rigorous definitions of pure states and the metastate in infinite-size systems, to show that the non-standard picture follows directly from the RSB mean-field theory. In addition, the metastate-averaged state possesses power-law correlations throughout the low temperature phase; the corresponding exponent $\zeta$ takes the value $4$ according to the field theory in high dimensions $d$, and describes the effective fractal dimension of clusters of spins. Further, the logarithm of the number of pure states in the decomposition of the metastate-averaged state that can be distinguished if only correlations in a window of size $W$ can be observed is of order $W^{d-\zeta }$. These results extend the non-standard picture quantitatively; we show that arguments against this scenario are inconclusive.

32 pages. v2: various small changes and additional references. v3: yet more small changes; published version

References in corpus (3)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• Evidence of non-mean-field-like low-temperature behavior in the Edwards-Anderson spin-glass model
• The cumulative overlap distribution function in realistic spin glasses

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• Low-temperature behavior of the statistics of the overlap distribution in Ising spin-glass models
• Ground State Stability and the Nature of the Spin Glass Phase
• Spin-glass dynamics: experiment, theory and simulation
• Nontrivial maturation metastate-average state in a one-dimensional long-range Ising spin glass: above and below the upper critical range
• On the superposition principle and non-linear response in spin glasses
• Proof of single-replica equivalence in short-range spin glasses
• Free Energy Difference Fluctuations in Short-Range Spin Glasses
• Critical Droplets and Replica Symmetry Breaking
• Overlap Structure and Free Energy Fluctuations in Short-Range Spin Glasses
• Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman
• Deepening the knowledge of Spin Glasses: Metastate, Off-equilibrium phenomena and Temperature Chaos