Noncontractible loops in the dense $\mathrm{O}\left(n\right)$ loop model on the cylinder

A lattice model of critical dense polymers $\mathrm{O}\left(n\right)$ is considered for finite cylinder geometry. Due to the presence of noncontractible loops with a fixed fugacity $\xi$, 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 $\rho$ of noncontractible loops is obtained for $N\to \infty$ 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 $\rho$ for any $\mathrm{O}\left(n\right)$ model and an arbitrary fugacity.

Figure 1

Elementary cells.

Figure 2

Loops on the horizontally periodic lattice. One noncontractible loop joins the left and the right sides of the figure.

Figure 3

A link configuration.

Figure 4

The six link pattern configurations for $L=4$ sites on a cylinder and two circles without sites (two noncontractible loops shown in red). The open arcs and circles meet behind the cylinder.

Figure 5

The action of the $e_2$ generator acting on the bond between the sites 2 and 3 in one of the configurations appearing in Fig. 4. The factor $x$ is due to the contractible loop.

Figure 6

(a) The action of the $e_2$ generator acting on the bond between the sites 2 and 3, which are the end points of an arc of the size of the system. (b) The action of $e_2$ gives a factor $\alpha$ to the new configuration.

Figure 7

The bijection between bonds and elementary cells.

Figure 8

The bond configuration corresponding to the loop configuration from Fig. 2.

Figure 9

Spanning web with a single noncontractible loop (shown in red).

Figure 10

Rotated square lattices with different boundary conditions on the cylinder. The two dashed vertical lines in each of the four panels represent a single line due to periodicity in the horizontal direction. For example, the top-left lattice has $L=10$ and $N=7$.

Figure 11

Spanning digraph generated by a single term in the determinant expansion of the Laplacian matrix (see the text).