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Chinese Physics B

Monte Carlo study of the universal area distribution of clusters in the honeycomb O( n ) loop model

Yadong Xu, Qingquan Liu, Youjin Deng

Published Jul 1, 2012• 2 citations
Publisher

Abstract

We investigate the area distribution of clusters (loops) in the honeycomb O( n ) loop model by means of the worm algorithm with n = 0.5, 1, 1.5, and 2. At the critical point, the number of clusters, whose enclosed area is greater than A , is proportional to A −1 with a proportionality constant C . We confirm numerically that C is universal, and its value agrees well with the predictions based on the Coulomb gas method.

DOI
10.1088/1674-1056/21/7/070211

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We investigate the area distribution of clusters (loops) in the honeycomb O( n ) loop model by means of the worm algorithm with n = 0.5, 1, 1.5, and 2. At the critical point, the number of clusters, whose enclosed area is greater than A , is proportional to A −1 with a proportionality constant C . We confirm numerically that C is universal, and its value agrees well with the predictions based on the Coulomb gas method.
We study the fractal geometry of O( n ) loop configurations in two dimensions by means of scaling and a Monte Carlo method, and compare the results with predictions based on the Coulomb gas technique. The Monte Carlo algorithm is applicable to models with noninteger n and uses local updates. Although these updates typically lead to nonlocal modifications of loop connectivities, the number of operations required per update is only of order 1. The Monte Carlo algorithm is applied to the honeycomb O( n ) model for several values of n , including noninteger ones. We thus determine scaling exponents that describe the fractal nature of O( n ) loops at criticality. The results of the numerical analysis agree with the theoretical predictions.
We study the O(N) loop model on the Honeycomb lattice with real value N≥1N \geq 1N≥1 by means of a cluster algorithm. The formulation of the algorithm is based on the equivalence of the O(N) loop model and the low-temperature graphical representation of a NNN-color Ashkin-Teller model on the triangular lattice. The latter model with integer NNN can be simulated by means of an embedding Swendsen-Wang-type cluster method. By taking into account the symmetry among loops of different colors, we develop another version of the Swendsen-Wang-type method. This version allows the number of colors NNN to take any real value N≥1N \geq 1N≥1. As an application, we investigate the N=1.25,1.50,1.75N=1.25, 1.50, 1.75N=1.25,1.50,1.75, and 2 loop model at criticality. The determined values of various critical exponents are in excellent agreement with theoretical predictions. In particular, from quantities associated with half of the loops, we determine some critical exponents that corresponds to those for the tricritical q=N2q=N^2q=N2 Potts model but have not been observed yet. Dynamic scaling behavior of the algorithm is also analyzed. The numerical data strongly suggest that our cluster algorithm {\it hardly} suffers from critical slowing down.
We study a percolation problem based on critical loop configurations of the O(n) loop model on the honeycomb lattice. We define dual clusters as groups of sites on the dual triangular lattice that are not separated by a loop, and investigate the bond-percolation properties of these dual clusters. The universal properties at the percolation threshold are argued to match those of Kasteleyn-Fortuin random clusters in the critical Potts model. This relation is checked numerically by means of cluster simulations of several O(n) models in the range 1<or=n<or=2. The simulation results include the percolation threshold for several values of n, as well as the universal exponents associated with bond dilution and the size distribution of the diluted clusters at the percolation threshold. Our numerical results for the exponents are in agreement with existing Coulomb-gas results for the random-cluster model, which confirms the relation between both models. We discuss the renormalization flow of the bond-dilution parameter p as a function of n, and provide an expression that accurately describes a line of unstable fixed points as a function of n, corresponding with the percolation threshold. Furthermore, the renormalization scenario indicates the existence, in a p versus n diagram, of another line of fixed points at p=1, which is stable with respect to p.
The loop-gas approach to statistical physics provides an alternative, geometrical description of phase transitions in terms of line-like objects. The resulting statistical random-graph ensemble composed of loops and (open) chains can be e_ciently generated by Monte Carlo simulations using the so-called “worm” update algorithm. Concepts from percolation theory and the theory of self-avoiding random walks are used to derive estimators of physical observables that utilize the nature of the worm algorithm. The fractal structure of random loops and chains as well as their scaling properties encode the critical behavior of the statistical system. The general approach is illustrated for the hightemperature series expansion of the Ising model, or O(1) loop model, on a honeycomb lattice, with its known exact results as valuable benchmarks.