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Ground State Excitations and Energy Fluctuations in Short-Range Spin Glasses

C. M. Newman, D. L. Stein

TL;DR

The paper proves that space-filling critical droplets do not exist in ground states of the Edwards–Anderson Ising spin glass in any finite dimension, under translation-covariant boundary conditions. This absence implies that energy differences between incongruent ground states grow proportionally with the volume, enabling a zero-temperature extension of positive-temperature spin-glass results. Consequently, in two dimensions the metastate is unique and supported on a single spin-reversed ground-state pair, and large-scale, space-filling O(1) excitations predicted by replica symmetry breaking are ruled out. The analysis also shows that RSB interfaces cannot arise from standard perturbation constructions, reinforcing a picture closer to droplet-like or TNT scenarios than to full RSB in finite dimensions. Overall, the work tightly constrains the zero-temperature structure of finite-dimensional spin glasses and clarifies the relationship between metastates, energy fluctuations, and possible ground-state pictures across dimensions.

Abstract

We study the stability of ground states in the Edwards-Anderson Ising spin glass in dimensions two and higher against perturbations of a single coupling. After reviewing the concepts of critical droplets, flexibilities and metastates, we show that, in any dimension, a certain kind of critical droplet with space-filling (i.e., positive spatial density) boundary does not exist in ground states generated by coupling-independent boundary conditions. Using this we show that if incongruent ground states exist in any dimension, the variance of their energy difference restricted to finite volumes scales proportionally to the volume. This in turn is used to prove that a metastate generated by (e.g.) periodic boundary conditions is unique and supported on a single pair of spin-reversed ground states in two dimensions. We further show that a type of excitation above a ground state, whose interface with the ground state is space-filling and whose energy remains O(1) independent of the volume, as predicted by replica symmetry breaking, cannot exist in any dimension.

Ground State Excitations and Energy Fluctuations in Short-Range Spin Glasses

TL;DR

The paper proves that space-filling critical droplets do not exist in ground states of the Edwards–Anderson Ising spin glass in any finite dimension, under translation-covariant boundary conditions. This absence implies that energy differences between incongruent ground states grow proportionally with the volume, enabling a zero-temperature extension of positive-temperature spin-glass results. Consequently, in two dimensions the metastate is unique and supported on a single spin-reversed ground-state pair, and large-scale, space-filling O(1) excitations predicted by replica symmetry breaking are ruled out. The analysis also shows that RSB interfaces cannot arise from standard perturbation constructions, reinforcing a picture closer to droplet-like or TNT scenarios than to full RSB in finite dimensions. Overall, the work tightly constrains the zero-temperature structure of finite-dimensional spin glasses and clarifies the relationship between metastates, energy fluctuations, and possible ground-state pictures across dimensions.

Abstract

We study the stability of ground states in the Edwards-Anderson Ising spin glass in dimensions two and higher against perturbations of a single coupling. After reviewing the concepts of critical droplets, flexibilities and metastates, we show that, in any dimension, a certain kind of critical droplet with space-filling (i.e., positive spatial density) boundary does not exist in ground states generated by coupling-independent boundary conditions. Using this we show that if incongruent ground states exist in any dimension, the variance of their energy difference restricted to finite volumes scales proportionally to the volume. This in turn is used to prove that a metastate generated by (e.g.) periodic boundary conditions is unique and supported on a single pair of spin-reversed ground states in two dimensions. We further show that a type of excitation above a ground state, whose interface with the ground state is space-filling and whose energy remains O(1) independent of the volume, as predicted by replica symmetry breaking, cannot exist in any dimension.

Paper Structure

This paper contains 18 sections, 30 theorems, 30 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of ground states, critical droplets, and metastates
  3. Ground states
  4. Critical droplets and flexibility
  5. Metastates
  6. Properties of critical droplets
  7. Types of critical droplets
  8. Space-filling critical droplets
  9. Positive weights
  10. Zero weights
  11. Case I
  12. Continuity of flexibility
  13. Case II
  14. Fluctuations in ground state energy differences
  15. Discussion
  16. ...and 3 more sections

Key Result

Lemma 2.1

(Newman-Stein NS22). Consider two distinct edges $b_1$ and $b_2$ and an infinite-volume ground state $\sigma$. (a) If $f(J(b_1),\sigma) > f(J(b_2),\sigma)$, then $b_1$ cannot belong to $\partial D(b_2,\sigma)$, while $b_2$ may or may not belong to ${\partial D}(b_1,\sigma)$. (b) If $b_1$ and $b_2$ s

Figures (2)

  • Figure 1: Arrangement of couplings as described in the text.
  • Figure 2: Sketch of SFCD of $b_0$ discussed in text. Here $C_1$ refers to the critical droplet boundary of $b_0$ with the single bond $b_1$ removed and $C_2$ refers to the critical droplet boundary of $b_2$ with the single bond $b_1$ removed.

Theorems & Definitions (30)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • ...and 20 more