Crossover phenomena involving the dense O($n$) phase

TL;DR

This study investigates the low-temperature phase of the 2D O(n) loop model, analyzing its stability and phase transitions under various perturbations using transfer-matrix and finite-size scaling methods.

Contribution

It provides a detailed analysis of how different perturbations affect the stability and phase behavior of the O(n) loop model, confirming Coulomb gas predictions.

Findings

Cubic anisotropy and crossing bonds induce crossover to different behaviors.

Loop-loop attractions and double bonds are generally irrelevant perturbations.

Strong perturbations of attractions and double bonds can cause Ising-type phase transitions.

Abstract

We explore the properties of the low-temperature phase of the O($n$) loop model in two dimensions by means of transfer-matrix calculations and finite-size scaling. We determine the stability of this phase with respect to several kinds of perturbations, including cubic anisotropy, attraction between loop segments, double bonds and crossing bonds. In line with Coulomb gas predictions, cubic anisotropy and crossing bonds are found to be relevant and introduce crossover to different types of behavior. Whereas perturbations in the form of loop-loop attractions and double bonds are irrelevant, sufficiently strong perturbations of these types induce a phase transition of the Ising type, at least in the cases investigated. This Ising transition leaves the underlying universal low-temperature O($n$) behavior unaffected.