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Features of a spin glass in the random field Ising model

Sourav Chatterjee

Abstract

A longstanding open question in the theory of disordered systems is whether short-range models, such as the random field Ising model or the Edwards-Anderson model, can indeed have the famous properties that characterize mean-field spin glasses at nonzero temperature. This article shows that this is at least partially possible in the case of the random field Ising model. Consider the Ising model on a discrete $d$-dimensional cube under free boundary condition, subjected to a very weak i.i.d. random external field, where the field strength is inversely proportional to the square-root of the number of sites. It turns out that in $d\ge 2$ and at subcritical temperatures, this model has some of the key features of a mean-field spin glass. Namely, (a) the site overlap exhibits one step of replica symmetry breaking, (b) the quenched distribution of the overlap is non-self-averaging, and (c) the overlap has the Parisi ultrametric property. Furthermore, it is shown that for Gaussian disorder, replica symmetry does not break if the field strength is taken to be stronger than the one prescribed above, and non-self-averaging fails if it is weaker, showing that the above order of field strength is the only one that allows all three properties to hold. However, the model does not have two other features of mean-field models. Namely, (a) it does not satisfy the Ghirlanda-Guerra identities, and (b) it has only two pure states instead of many.

Features of a spin glass in the random field Ising model

Abstract

A longstanding open question in the theory of disordered systems is whether short-range models, such as the random field Ising model or the Edwards-Anderson model, can indeed have the famous properties that characterize mean-field spin glasses at nonzero temperature. This article shows that this is at least partially possible in the case of the random field Ising model. Consider the Ising model on a discrete -dimensional cube under free boundary condition, subjected to a very weak i.i.d. random external field, where the field strength is inversely proportional to the square-root of the number of sites. It turns out that in and at subcritical temperatures, this model has some of the key features of a mean-field spin glass. Namely, (a) the site overlap exhibits one step of replica symmetry breaking, (b) the quenched distribution of the overlap is non-self-averaging, and (c) the overlap has the Parisi ultrametric property. Furthermore, it is shown that for Gaussian disorder, replica symmetry does not break if the field strength is taken to be stronger than the one prescribed above, and non-self-averaging fails if it is weaker, showing that the above order of field strength is the only one that allows all three properties to hold. However, the model does not have two other features of mean-field models. Namely, (a) it does not satisfy the Ghirlanda-Guerra identities, and (b) it has only two pure states instead of many.
Paper Structure (25 sections, 23 theorems, 133 equations)

This paper contains 25 sections, 23 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Replica symmetry breaking and non-self-averaging
  4. Ultrametricity
  5. Behavior of the magnetization
  6. Two pure states
  7. Failure of the Ghirlanda--Guerra identities
  8. Failure of spin glass behavior at other field strengths
  9. The antiferromagnetic RFIM
  10. Uniformity of correlations in the ordinary Ising model
  11. Proof of Theorem \ref{['diffthmmain']}
  12. The FK-Ising model
  13. Uniformity of connectivities in infinite volume
  14. Uniformity of two-point correlations in finite volume
  15. Uniformity of four-point correlations in finite volume
  16. ...and 10 more sections

Key Result

Theorem 2.1

Take any $d\ge 2$ and $n\ge 1$ and consider the model defined above on $B_n = \{-n,\ldots,n\}^d$ at inverse temperature $\beta>\beta_0$, where $\beta_0$ is the critical inverse temperature for the ordinary Ising model on $\mathbb{Z}^d$. Then there is a deterministic value $q>0$ depending only on $\b then we have that Consequently, as $n\to\infty$, $\langle R_{1,2} \rangle$ converges in law to $q\

Theorems & Definitions (41)

  • Theorem 2.1: Replica symmetry breaking and non-self-averaging
  • Theorem 2.2: Ultrametricity
  • Theorem 2.3: Behavior of the magnetization
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7: Failure of the Ghirlanda--Guerra identities
  • Theorem 2.8: Failure of spin glass behavior at other field strengths
  • Theorem 2.9: Results for the antiferromagnetic RFIM
  • Theorem 2.10: Uniformity of correlations in the Ising model
  • ...and 31 more