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 nonstandard interpretation of the RSB results in which the metastate is nontrivial and is responsible for what was called non-self-averaging. In this picture, each state drawn from the metastate has the ultrametric properties of the old theory, but when the state is averaged using the metastate, the resulting mixed state has little structure. This picture was constructed so as to agree both with the earlier RSB results and with rigorous results. 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 nonstandard 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 ζ 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-ζ). These results extend the nonstandard picture quantitatively; we show that arguments against this scenario are inconclusive. More generally, in terms of Parisi's function q(x), if q(0)≠∫(0)(1)dxq(x), then the metastate is nontrivial. In an Appendix, we also prove rigorously that the metastate-averaged state of the Sherrington-Kirkpatrick model is a uniform distribution on all spin configurations at all temperatures.