3D loop models and the CP^{n-1} sigma model

TL;DR

This paper studies three-dimensional loop models, mapping them to $CP^{n-1}$ sigma models, and identifies phase transition types for different loop fugacities using Monte Carlo simulations, with implications for various physical systems.

Contribution

It introduces a mapping of 3D loop models to $CP^{n-1}$ sigma models and characterizes the nature of phase transitions for different $n$ values.

Findings

Continuous transitions for n=1,2,3

First order transitions for n≥5

Relevance to line defects, localization, and quantum magnets

Abstract

Many statistical mechanics problems can be framed in terms of random curves; we consider a class of three-dimensional loop models that are prototypes for such ensembles. The models show transitions between phases with infinite loops and short-loop phases. We map them to $CP^{n-1}$ sigma models, where $n$ is the loop fugacity. Using Monte Carlo simulations, we find continuous transitions for $n=1,2,3$, and first order transitions for $n\geq 5$. The results are relevant to line defects in random media, as well as to Anderson localization and $(2+1)$-dimensional quantum magnets.