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Spin glass phase at zero temperature in the Edwards-Anderson model

Sourav Chatterjee

TL;DR

This work provides the first rigorous demonstration of glassy behavior in a short-range spin-glass model by analyzing the Edwards–Anderson model at zero temperature on finite graphs. It proves disorder chaos for the ground state under small Gaussian perturbations, establishes fractal lower bounds on the overturned-region boundary, and shows the existence of large regions overturnable with negligible energy cost relative to their boundary. The results also give a lower bound on the size of the critical droplets and demonstrate polynomial, not exponential, decay of correlations with boundary conditions, contrasting with RFIM behavior. Collectively, these theorems substantiate a chaotic, multi-valley glassy phase in finite-dimensional EA models and relate to broader questions about the Parisi picture, droplet theory, and metastates in lattice spin glasses.

Abstract

While the analysis of mean-field spin glass models has seen tremendous progress in the last twenty years, lattice spin glasses have remained largely intractable. This article presents the solutions to a number of questions about the Edwards-Anderson model of short-range spin glasses (in all dimensions) that were raised in the physics literature many years ago. First, it is shown that the ground state is sensitive to small perturbations of the disorder, in the sense that a small amount of noise gives rise to a new ground state that is nearly orthogonal to the old one with respect to the site overlap inner product. Second, it is shown that one can overturn a macroscopic fraction of the spins in the ground state with an energy cost that is negligible compared to the size of the boundary of the overturned region - a feature that is believed to be typical of spin glasses but clearly absent in ferromagnets. The third result is that the boundary of the overturned region in dimension $d$ has fractal dimension strictly greater than $d-1$, confirming a prediction from physics. The fourth result is that the expected size of the critical droplet of a bond grows at least like a power of the volume. The fifth result is that the correlations between bonds in the ground state can decay at most like the inverse of the distance. This contrasts with the random field Ising model, where it has been shown recently that the correlation decays exponentially in distance in dimension two. Taken together, these results comprise the first mathematical proof of glassy behavior in a short-range spin glass model.

Spin glass phase at zero temperature in the Edwards-Anderson model

TL;DR

This work provides the first rigorous demonstration of glassy behavior in a short-range spin-glass model by analyzing the Edwards–Anderson model at zero temperature on finite graphs. It proves disorder chaos for the ground state under small Gaussian perturbations, establishes fractal lower bounds on the overturned-region boundary, and shows the existence of large regions overturnable with negligible energy cost relative to their boundary. The results also give a lower bound on the size of the critical droplets and demonstrate polynomial, not exponential, decay of correlations with boundary conditions, contrasting with RFIM behavior. Collectively, these theorems substantiate a chaotic, multi-valley glassy phase in finite-dimensional EA models and relate to broader questions about the Parisi picture, droplet theory, and metastates in lattice spin glasses.

Abstract

While the analysis of mean-field spin glass models has seen tremendous progress in the last twenty years, lattice spin glasses have remained largely intractable. This article presents the solutions to a number of questions about the Edwards-Anderson model of short-range spin glasses (in all dimensions) that were raised in the physics literature many years ago. First, it is shown that the ground state is sensitive to small perturbations of the disorder, in the sense that a small amount of noise gives rise to a new ground state that is nearly orthogonal to the old one with respect to the site overlap inner product. Second, it is shown that one can overturn a macroscopic fraction of the spins in the ground state with an energy cost that is negligible compared to the size of the boundary of the overturned region - a feature that is believed to be typical of spin glasses but clearly absent in ferromagnets. The third result is that the boundary of the overturned region in dimension has fractal dimension strictly greater than , confirming a prediction from physics. The fourth result is that the expected size of the critical droplet of a bond grows at least like a power of the volume. The fifth result is that the correlations between bonds in the ground state can decay at most like the inverse of the distance. This contrasts with the random field Ising model, where it has been shown recently that the correlation decays exponentially in distance in dimension two. Taken together, these results comprise the first mathematical proof of glassy behavior in a short-range spin glass model.
Paper Structure (16 sections, 14 theorems, 88 equations, 1 figure)

This paper contains 16 sections, 14 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model
  3. Chaotic nature of the ground state
  4. Fractal dimension of the droplet boundary
  5. Existence of multiple valleys
  6. Size of the critical droplet
  7. Polynomial decay of correlations
  8. Related literature and open problems
  9. Proofs
  10. Proof of Theorem \ref{['mainthm']}
  11. Proof of Theorem \ref{['chaosthm']}
  12. Proof of Theorem \ref{['boundarythm']}
  13. Proof of Theorem \ref{['fractalthm']}
  14. Proof of Theorem \ref{['oldthm']}
  15. Proof of Theorem \ref{['dropletthm']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let all notations be as above. Let $d$ denote the graph distance on $G$. Suppose that there are positive constants $\alpha$, $\beta$, $\gamma$ and $\delta$ such that for any $i\in V^\circ$ and $r\ge 1$, the number of $j$ such that $d(i,j)\le r$ is at most $\alpha r^\beta$, and the number of $j$ such where $C$ is a constant depending only on $\alpha$, $\beta$, $\gamma$ and $\delta$.

Figures (1)

  • Figure 1: Schematic illustration of the sets $S_1, S_2$ and the paths $P_1, P_2$.

Theorems & Definitions (18)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 2
  • Theorem 5
  • Theorem 6
  • Corollary 3
  • Corollary 4
  • ...and 8 more