Loop models with crossings

TL;DR

This paper investigates two-dimensional loop models with crossings, revealing new phase transitions, universal properties, and connections to gauge theories, polymers, and electronic systems through analytical and large-scale numerical methods.

Contribution

It introduces a generalized loop model with crossings, characterizes its phase diagram, and links it to sigma models and gauge theories, advancing understanding of critical phenomena in complex systems.

Findings

Identified new continuous phase transitions separating Goldstone and short-loop phases.

Mapped the model to a Z_2 lattice gauge theory and a sigma model on RP^{n-1}.

Performed large-scale simulations up to L=10^6 to analyze critical behavior.

Abstract

The universal behaviour of two-dimensional loop models can change dramatically when loops are allowed to cross. We study models with crossings both analytically and with extensive Monte Carlo simulations. Our main focus (the 'completely packed loop model with crossings') is a simple generalisation of well-known models which shows an interesting phase diagram with continuous phase transitions of a new kind. These separate the unusual 'Goldstone' phase observed previously from phases with short loops. Using mappings to Z_2 lattice gauge theory, we show that the continuum description of the model is a replica limit of the sigma model on real projective space (RP^{n-1}). This field theory sustains Z_2 point defects which proliferate at the transition. In addition to studying the new critical points, we characterise the universal properties of the Goldstone phase in detail, comparing renormalisation group (RG) calculations with numerical data on systems of linear size up to L=10^6 at loop fugacity n=1. (Very large sizes are necessary because of the logarithmic form of correlation functions and other observables.) The model is relevant to polymers on the verge of collapse, and a particular point in parameter space maps to self-avoiding trails at their \Theta-point; we use the RG treatment of a perturbed sigma model to resolve some perplexing features in the previous literature on trails. Finally, one of the phase transitions considered here is a close analogue of those in disordered electronic systems --- specifically, Anderson metal-insulator transitions --- and provides a simpler context in which to study the properties of these poorly-understood (central-charge-zero) critical points.