Worm Monte Carlo study of the honeycomb-lattice loop model

We present a Markov-chain Monte Carlo algorithm of worm   type that correctly simulates the O(n)$O(n)$ loop model on any (finite and connected) bipartite cubic graph, for any real n>0$n>0$, and any edge weight, including the fully-packed limit of infinite edge weight. Furthermore, we prove rigorously that the algorithm is ergodic and has the correct stationary distribution. We emphasize that by using known exact mappings when n=2$n=2$, this algorithm can be used to simulate a number of zero-temperature Potts antiferromagnets for which the Wang–Swendsen–Kotecký cluster algorithm is non-ergodic, including the 3-state model on the kagome lattice and the 4-state model on the triangular lattice. We then use this worm algorithm to perform a systematic study of the honeycomb-lattice loop model as a function of n?2$n?2$, on the critical line and in the densely-packed and fully-packed phases. By comparing our numerical results with Coulomb gas theory, we identify a set of exact expressions for scaling exponents governing some fundamental geometric and dynamic observables. In particular, we show that for all n?2$n?2$, the scaling of a certain return time in the worm dynamics is governed by the magnetic dimension of the loop model, thus providing a concrete dynamical interpretation of this exponent. The case n>2$n>2$ is also considered, and we confirm the existence of a phase transition in the 3-state Potts universality class that was recently observed via numerical transfer matrix calculations.