Non-contractible loops in the dense O(n) loop model on the cylinder

TL;DR

This paper analyzes the dense O(n) loop model on a finite cylinder, deriving free energy and non-contractible loop density, and compares results with quantum chain models, extending understanding of loop configurations in statistical physics.

Contribution

It introduces a generalized model with non-contractible loops on a cylinder and computes free energy and loop density for various parameters, extending prior dense polymer models.

Findings

Derived free energy for any cylinder dimensions.

Calculated non-contractible loop density in the large size limit.

Extended analysis to arbitrary O(n) models with any fugacity.

Abstract

A lattice model of critical dense polymers $O(0)$ is considered for the finite cylinder geometry. Due to the presence of non-contractible loops with a fixed fugacity $\xi$, the model is a generalization of the critical dense polymers solved by Pearce, Rasmussen and Villani. We found the free energy for any height $N$ and circumference $L$ of the cylinder. The density $\rho$ of non-contractible loops is found for $N \rightarrow \infty$ and large $L$. The results are compared with those obtained for the anisotropic quantum chain with twisted boundary conditions. Using the latter method we obtained $\rho$ for any $O(n)$ model and an arbitrary fugacity.