Ordering and Broken Symmetry in Short-Ranged Spin Glasses

TL;DR

This review critically examines the nature of the low-temperature phase in short-ranged spin glasses, arguing that replica symmetry breaking does not describe their true phase and exploring alternative scenarios.

Contribution

It provides a comprehensive analysis of pure states, replica symmetry breaking, and thermodynamic chaos in short-ranged spin glasses, challenging previous assumptions.

Findings

Replica symmetry breaking cannot describe the low-temperature phase in finite dimensions.

Different physical properties distinguish remaining possible low-temperature phases.

Rigorous results connect metastable states to thermodynamic states.

Abstract

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.