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Quantitative disorder effects in low-dimensional spin systems

Paul Dario, Matan Harel, Ron Peled

Abstract

The Imry-Ma phenomenon, predicted in 1975 by Imry and Ma and rigorously established in 1989 by Aizenman and Wehr, states that first-order phase transitions of low-dimensional spin systems are `rounded' by the addition of a quenched random field to the quantity undergoing the transition. The phenomenon applies to a wide class of spin systems in dimensions $d\le 2$ and to spin systems possessing a continuous symmetry in dimensions $d\le 4$. This work provides quantitative estimates for the Imry--Ma phenomenon: In a cubic domain of side length $L$, we study the effect of the boundary conditions on the spatial and thermal average of the quantity coupled to the random field. We show that the boundary effect diminishes at least as fast as an inverse power of $\log\log L$ for general two-dimensional spin systems and for four-dimensional spin systems with continuous symmetry, and at least as fast as an inverse power of $L$ for two- and three-dimensional spin systems with continuous symmetry. Specific models of interest for the obtained results include the two-dimensional random-field $q$-state Potts and Edwards-Anderson spin glass models, and the $d$-dimensional random-field spin $O(n)$ models ($n\ge 2$) in dimensions $d\le 4$.

Quantitative disorder effects in low-dimensional spin systems

Abstract

The Imry-Ma phenomenon, predicted in 1975 by Imry and Ma and rigorously established in 1989 by Aizenman and Wehr, states that first-order phase transitions of low-dimensional spin systems are `rounded' by the addition of a quenched random field to the quantity undergoing the transition. The phenomenon applies to a wide class of spin systems in dimensions and to spin systems possessing a continuous symmetry in dimensions . This work provides quantitative estimates for the Imry--Ma phenomenon: In a cubic domain of side length , we study the effect of the boundary conditions on the spatial and thermal average of the quantity coupled to the random field. We show that the boundary effect diminishes at least as fast as an inverse power of for general two-dimensional spin systems and for four-dimensional spin systems with continuous symmetry, and at least as fast as an inverse power of for two- and three-dimensional spin systems with continuous symmetry. Specific models of interest for the obtained results include the two-dimensional random-field -state Potts and Edwards-Anderson spin glass models, and the -dimensional random-field spin models () in dimensions .

Paper Structure

This paper contains 39 sections, 19 theorems, 256 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two-dimensional disordered spin systems
  3. The general setup
  4. Examples
  5. Results
  6. Main result
  7. Uniqueness conjecture and additional results
  8. Spin systems with continuous symmetry
  9. Background
  10. Strategy of the proof
  11. Outline of the proof of Theorem \ref{['thm1prop2.31708main']}
  12. Outline of the proofs of Theorem \ref{['prop:subcritical']} and Theorem \ref{['thm.thm2']}.
  13. Notations, assumptions and preliminaries
  14. General notation
  15. Structure of the random field
  16. ...and 24 more sections

Key Result

Theorem 1

Let $\beta > 0$ be the inverse temperature and $\lambda>0$ be the disorder strength. There exist constants $C,c>0$ depending only on $\lambda$, $C_H$, $m$ and $R$ such that, for each integer $L\ge 3$, and moreover, in a translation-invariant setup, where $\alpha\in\mathbb{R}^m$ depends only on the spin system considered, on the inverse temperature $\beta$ and on the disorder strength $\lambda$ (

Figures (2)

  • Figure 1: The figure represents a $\lambda$-Lipshitz convex function $F_1$. The area in purple represents the surface where functions in the set $N_{\lambda,r}(F_1)$ must lie. An example of a function $F_2 \in N_{\lambda,r}(F_1)$ is drawn in red, and the set $\mathrm{Stab}(\lambda, \delta, r , g)$ is drawn in orange.
  • Figure 2: A realization of the Mandelbrot percolation with the values $k = 3$ and $l_{\max} = 3$. The bad cubes are drawn in black.

Theorems & Definitions (44)

  • Theorem 1
  • Conjecture 2.1
  • Theorem 2
  • Corollary 2.2
  • Theorem 3: Uniqueness for translation-covariant Gibbs states
  • Theorem 4
  • Theorem 5
  • Proposition 6.1: Domain subadditivity property
  • Definition 6.2: Finite-volume free energy
  • Proposition 6.3: Properties of the free energy
  • ...and 34 more