Noncontractible loops in the dense O(n) loop model on the cylinder

Area of Science:

• Statistical mechanics
• Polymer physics
• Condensed matter theory

Background:

• The study considers a lattice model of critical dense polymers, specifically the O(n) model.
• It focuses on finite cylinder geometry, incorporating noncontractible loops with a fixed fugacity (ξ).
• The model at n=0 is a generalization of critical dense polymers previously solved by Pearce, Rasmussen, and Villani.

Purpose of the Study:

• To investigate the free energy of the O(n) polymer model in a finite cylinder geometry.
• To determine the density (ρ) of noncontractible loops in the thermodynamic limit (N→∞) and large circumference (L).
• To generalize findings for arbitrary O(n) models and fugacities using a quantum chain method.

Main Methods:

• Utilizing a lattice model approach for critical dense polymers in a cylinder.
• Applying techniques from the anisotropic quantum chain with twisted boundary conditions.
• Calculating free energy and loop density through analytical methods.

Main Results:

• The free energy was determined for any height (N) and circumference (L) of the cylinder.
• The density (ρ) of noncontractible loops was obtained for N→∞ and large L.
• A generalized formula for ρ was derived for any O(n) model and arbitrary fugacity.

Conclusions:

• The study successfully derived key thermodynamic properties for a generalized critical dense polymer model in a cylinder.
• The findings extend previous results and provide a unified framework for analyzing such models.
• The quantum chain method proved effective for deriving loop densities across various model parameters.