We present a rigorous proof of an ordering transition for a two-component two-dimensional antiferromagnet with nearest and next-nearest neighbor interactions. The low-temperature phase contains two states distinguished by local order among columns or, respectively, rows. Overall, there is no magnetic order in accord with the classic Mermin-Wagner theorem. The method of proof employs a rigorous version of "order by disorder," whereby a high degeneracy among the ground states is lifted according to the differences in their associated spin-wave spectra.
@article{arxiv.math-ph/0310001,
title = {Order by disorder, without order, in a two-dimensional spin system with O(2) symmetry},
author = {Marek Biskup and Lincoln Chayes and Steven A. Kivelson},
journal= {arXiv preprint arXiv:math-ph/0310001},
year = {2007}
}