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

A lattice model of critical dense polymers O(n) is considered for finite cylinder geometry. Due to the presence of noncontractible loops with a fixed fugacity ξ, the model at n=0 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 ρ of noncontractible loops is obtained for N→∞ and large L. The results are compared with those found for the anisotropic quantum chain with twisted boundary conditions. Using the latter method, we derived ρ for any O(n) model and an arbitrary fugacity.