Frustration effects in antiferromagnets on planar random graphs

We consider the effect of geometric frustration induced by the random distribution of loop lengths in the "fat" graphs of the dynamical triangulations model on coupled antiferromagnets. While the influence of such connectivity disorder is rather mild for ferromagnets in that an ordered phase persists and only the properties of the phase transition are substantially changed in some cases, any finite-temperature transition is wiped out due to frustration for some of the antiferromagnetic models. A wealth of different phenomena is observed: While for the annealed average of quantum gravity some graphs can adapt dynamically to allow the emergence of a Néel ordered phase, this is not possible for the quenched average, where a zero-temperature spin-glass phase appears instead. We relate the latter to the behavior of conventional spin-glass models coupled to random graphs. © 2007 The American Physical Society.