Phase transitions in generalized XY models

We associate height function models to a range of generalized two-dimensional XY models and prove that delocalisation of the height function model rules out exponential decay of the nematic order parameter in the primal spin model. The argument is based on a generalized loop representation which relates the variance of a height difference between two faces to the nematic order parameter. This generalizes the results by van Engelenburg and Lis (2023). The range of models also includes the much-studied generalized XY model as proposed by Korshunov and independently by Lee and Grinstein, where neighbouring spins interact ferromagnetically favouring parallel alignment and nematically favouring parallel or antiparallel alignment. Moreover, we recover the existence of a nematic region by a classical comparison argument of Pfister (1982).