Critical Line of the O(N) Loop Model on the Square Lattice

An efficient algorithm is presented to simulate the O(N) loop model on the square lattice for arbitrary values of $N>0$. The scheme combines the worm algorithm with a new data structure to resolve both the problem of loop crossings and the necessity of counting the number of loops at each Monte Carlo update. With the use of this scheme, the line of critical points (and other properties) of the O($N$) model on the square lattice for $0<N\le 2$ have been determined.