Phase transitions in 3D loop models and the $CP^{n-1}$ $\sigma$ model

TL;DR

This paper investigates phase transitions in 3D loop models, showing their connection to $CP^{n-1}$ sigma models, and finds continuous and first order transitions depending on the parameter n through simulations and renormalization group analysis.

Contribution

It establishes the link between 3D loop models and $CP^{n-1}$ sigma models, analyzing phase transitions and their dependence on n, with both numerical and theoretical insights.

Findings

Continuous transitions for n=1,2,3

First order transitions for n≥4

Loop properties relate to sigma model correlators

Abstract

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 discretisations of $CP^{n-1}$ $\sigma$ models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the $\sigma$ model, and we discuss the relationship between loop properties and $\sigma$ model correlators. On large scales, loops are Brownian in an ordered phase and have a non-trivial 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 $n\geq 4$. We also give a renormalisation group treatment of the $CP^{n-1}$ model that shows how a continuous transition can survive for values of $n$ larger than (but close to) two, 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 localisation in symmetry class C, and the statistics of random curves in three dimensions.