Quantitative disorder effects in low-dimensional spin systems

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 $d\le 4$.