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

TL;DR

This paper demonstrates that the replica symmetry breaking (RSB) theory for short-range spin glasses can be understood through the metastate framework, revealing power-law correlations and fractal structures in the low-temperature phase.

Contribution

It provides a rigorous connection between RSB mean-field theory and the metastate approach, clarifying the structure of pure states and correlations in short-range spin glasses.

Findings

Metastate-averaged state exhibits power-law correlations with exponent 4.

Number of distinguishable pure states scales as W^{d-ζ}.

Arguments against the non-standard RSB scenario are inconclusive.

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.