Worm Monte Carlo study of the honeycomb-lattice loop model

TL;DR

This paper introduces a new worm Monte Carlo algorithm for simulating the O(n) loop model on bipartite cubic graphs, rigorously proves its ergodicity, and uses it to analyze the model's critical behavior and phase transitions.

Contribution

The paper presents a novel worm algorithm applicable to the O(n) loop model on bipartite graphs, with rigorous proof of ergodicity and stationary distribution, enabling detailed study of critical phenomena.

Findings

Identified exact scaling exponents matching Coulomb gas theory.

Demonstrated the algorithm's effectiveness in simulating zero-temperature Potts antiferromagnets.

Confirmed phase transition in the 3-state Potts universality class for n>2.

Abstract

We present a Markov-chain Monte Carlo algorithm of "worm"type that correctly simulates the O(n) loop model on any (finite and connected) bipartite cubic graph, for any real 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, this algorithm can be used to simulate a number of zero-temperature Potts antiferromagnets for which the Wang-Swendsen-Kotecky 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, on the critical line and in the densely-packed and fully-packed phases. By comparing our numerical results with Coulomb gas theory, we identify the exact scaling exponents governing some fundamental geometric and dynamic observables. In particular, we show that for all 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 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.