Worm Monte Carlo study of the honeycomb-lattice loop model

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.

33 pages, 12 figures

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• The O(n) loop model on a three-dimensional lattice
• Worm-algorithm-type Simulation of Quantum Transverse-Field Ising Model
• An integrable modification of the critical Chalker-Coddington network model
• Spin-liquid behaviour and the interplay between Pokrovsky-Talapov and Ising criticality in the distorted, triangular-lattice, dipolar Ising antiferromagnet
• The Hintermann-Merlini-Baxter-Wu and the Infinite-Coupling-Limit Ashkin-Teller Models
• Lifted Worm Algorithm for the Ising Model
• Stability and Loop Models from Decohering Non-Abelian Topological Order
• Backbone and shortest-path exponents of the two-dimensional $Q$-state Potts model
• Simulating the All-Order Strong Coupling Expansion V: Ising Gauge Theory
• Worm-type Monte Carlo simulation of the Ashkin-Teller model on the triangular lattice
• Control of probability flow in Markov chain Monte Carlo -- Nonreversibility and lifting
• Phase diagram and the strong-coupling fixed point in the disordered O(n) loop model
• Geometric allocation approach to accelerating directed worm algorithm
• Lifted directed-worm algorithm
• Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation
• Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions
• Computational Study of a Multistep Height Model
• Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories
• Phase transitions in 3D Ising model with cluster weight by Monte Carlo method
• Flux fractionalization transition in anisotropic $S=1$ antiferromagnets and dimer-loop models