The State(s) of Replica Symmetry Breaking: Mean Field Theories vs. Short-Ranged Spin Glasses

TL;DR

This paper demonstrates that recent theories attempting to reconcile Replica Symmetry Breaking with finite-dimensional spin glasses are fundamentally flawed, showing that such models cannot simultaneously have multiple pure states and space-filling domain walls.

Contribution

It proves the incompatibility of certain RSB models with the existence of multiple pure states and space-filling domain walls in finite-dimensional spin glasses.

Findings

Recent RSB attempts are incompatible with multiple pure states in finite dimensions.

Space-filling domain walls cannot coexist with the RSB picture in finite-dimensional spin glasses.

Revisions to the pure state concept do not resolve the contradictions in RSB models.

Abstract

We prove the impossibility of recent attempts to decouple the Replica Symmetry Breaking (RSB) picture for finite-dimensional spin glasses from the existence of many thermodynamic (i.e., infinite-volume) pure states while preserving another signature RSB feature --- space filling relative domain walls between different finite-volume states. Thus revisions of the notion of pure states cannot shield the RSB picture from the internal contradictions that rule out its physical correctness in finite dimensions at low temperature in large finite volume.