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

TL;DR

This paper introduces an efficient algorithm combining worm techniques and a new data structure to simulate the O(N) loop model on the square lattice, accurately determining the critical line for 0<N≤2.

Contribution

The paper presents a novel algorithm that effectively handles loop crossings and loop counting, enabling precise determination of the critical line for the O(N) model on the square lattice.

Findings

Determined the critical line of the O(N) model for 0<N≤2.

Developed an algorithm that manages loop crossings and counts loops efficiently.

Validated the algorithm by accurately locating phase transition points.

Abstract

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.