This paper links two different ways of looking at a class of two-dimensional spin models and uses that link to prove a general statement about phases. The authors show that when a related height function model is delocalised — meaning the typical difference between heights at two distant faces keeps growing — then the nematic order in the original spin model cannot decay exponentially with distance. The result applies to a wide family of “generalized XY” models that include both ordinary ferromagnetic couplings (which favour parallel spins) and nematic couplings (which favour parallel or antiparallel spins), such as the model introduced by Korshunov and by Lee and Grinstein.

At a technical level the paper builds an ‘‘enhanced random-current expansion’’ and a generalized loop representation. These constructions give a precise identity that relates the variance of a height difference in the dual height-function model to the nematic two-point correlation in the original spin model. The nematic two-point function is the expectation of cos(2(θ0−θv)), which measures how much the spin orientation at v is correlated with the spin at a reference site up to reversal. The identity generalizes earlier work by van Engelenburg and Lis, and it is the central tool used to transfer information from the height model to the spin model.

Why this matters: the behavior of the dual height model is easier to study in some regimes, and delocalisation there signals the presence of a phase in the spin model where nematic correlations decay slowly rather than exponentially. Concretely, the authors prove (Theorem 1) that for the class of Hamiltonians they consider, delocalisation of the dual height model at a given inverse temperature β implies that the nematic correlation does not have exponential decay at that β. They also show there is a finite threshold β0 such that the height model is delocalised for all β≥β0, so a regime of non-exponential nematic correlations does occur at low temperature. For the specific family HΔ that mixes ferromagnetic and nematic couplings the paper records a corollary that this implication holds across 0≤Δ≤1.