Phase transitions in three-dimensional loop models and the CPn-1 sigma model

We consider the statistical mechanics of a class of models involving close-packed loops with fugacity n on three-dimensional lattices. The models exhibit phases of two types as a coupling constant is varied: in one, all loops are finite, and in the other, some loops are infinitely extended. We show that the loop models are discretizations of CPn-1 sigma models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the s model, and we discuss the relationship between loop properties and s model correlators. On large scales, loops are Brownian in an ordered phase and have a nontrivial fractal dimension at a critical point. We simulate the models, finding continuous transitions between the two phases for n = 1,2,3 and first order transitions for n >= 4. We also give a renormalization-group treatment of the CPn-1 model that shows how a continuous transition can survive for values of n larger than (but close to) 2, despite the presence of a cubic invariant in the Landau-Ginzburg description. The results we obtain are of broader relevance to a variety of problems, including SU(n) quantum magnets in (2 + 1) dimensions, Anderson localization in symmetry class C, and the statistics of random curves in three dimensions.