Exact solution of a two-dimensional random Ising model

Abstract

The model considered is a $d=2$ layered random Ising system on a square lattice with a nearest-neighbor interaction. It is assumed that all the vertical couplings are equal and take the positive value $J,$ while the horizontal couplings are quenched random variables that are equal in the same row but can take the two possible values $J$ and $J\ensuremath{-}K$ in different rows. The exact solution is obtained in the limit case $K\ensuremath{\rightarrow}\ensuremath{\infty}$ for any distribution of the horizontal couplings. The model that corresponds to this limit can be seen as an ordinary Ising system where the spins of some rows, chosen at random, are frozen in an antiferromagnetic order. No phase transition is found if the horizontal couplings are independent random variables, while for correlated disorder one finds a low temperature phase with some glassy properties.