Quantitative disorder effects in low-dimensional spin systems

TL;DR

This paper provides quantitative estimates on how quenched random fields influence phase transitions in low-dimensional spin systems, showing the boundary effects diminish at specific rates depending on the system's dimension and symmetry.

Contribution

It offers the first detailed quantitative analysis of the Imry-Ma phenomenon, including explicit decay rates of boundary effects in various low-dimensional spin models.

Findings

Boundary effects decay as inverse power of log log L in 2D and 4D continuous symmetry systems.

Boundary effects decay as inverse power of L in 2D and 3D systems with continuous symmetry.

Results apply to random-field Potts, Edwards-Anderson spin glass, and O(n) models in low dimensions.

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 $d\le 4$.