Theory of the critical state of low-dimensional spin glass

TL;DR

This paper develops a theoretical framework for understanding the critical behavior of low-dimensional Ising spin glasses near the lower critical dimension, introducing the concept of surface-organized active degrees of freedom and multifractal correlations.

Contribution

It proposes a novel surface-based picture of criticality in spin glasses and explains critical chaos, with scaling laws for correlation functions.

Findings

Active degrees of freedom are surfaces enclosing spin clusters.

Correlation functions are multifractal with specific scaling laws.

Results are testable via Monte Carlo simulations in three dimensions.

Abstract

We analyse the critical region of finite-($d$)-dimensional Ising spin glass, in particular the limit of $d$ closely above the lower critical dimension $d_\ell$. At criticality the thermally active degrees of freedom are surfaces (of width zero) enclosing clusters of spins that may reverse with respect to their environment. The surfaces are organised in finite interacting structures. These may be called {\em protodroplets}\/, since in the off-critical limit they reduce to the Fisher and Huse droplets. This picture provides an explanation for the phenomenon of critical chaos discovered earlier. It also implies that the spin-spin and energy-energy correlation functions are multifractal and we present scaling laws that describe them. Several of our results should be verifiable in Monte Carlo studies at finite temperature in $d=3$.