Quantitative disorder effects in low-dimensional spin systems

Understand

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.