We develop cluster algorithms for a broad class of loop models on two-dimensional lattices, including several standard O(n) loop models at n> or =1. We show that our algorithm has little or no critical slowing-down when 1< or =n< or =2. We use this algorithm to investigate the honeycomb-lattice O(n) loop model, for which we determine several new critical exponents, and a square-lattice O(n) loop model, for which we obtain new information on the phase diagram.
We develop cluster algorithms for a broad class of loop models on two-dimensional lattices, including several standard O(n) loop models at n> or =1. We show that our algorithm has little or no critical slowing-down when 1< or =n< or =2. We use this algorithm to investigate the honeycomb-lattice O(n) loop model, for which we determine several new critical exponents, and a square-lattice O(n) loop model, for which we obtain new information on the phase diagram.
Making use of the duality relation of the Ising model, we develop cluster algorithms for several loop models on planar lattices. We then investigate the O(n) model on the honeycomb lattice and face- and corner-cubic models on the square lattice. Several new physical phenomena are observed. For instance, critical exponents, which have not been reported for the O(n) model in literature, are determined; both ‘ferromagnetic ” and “antiferromagnetic” phase transitions are located for the cubic models. Dynamical behavior is also studied. It is shown that, for 1 < n ≤ 2, our cluster algorithms hardly suffer from critical slowing down. Typeset using REVTEX
We study the O(N) loop model on the Honeycomb lattice with real value N≥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 N-color Ashkin-Teller model on the triangular lattice. The latter model with integer N 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 N to take any real value N≥1. As an application, we investigate the N=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=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 present a new type of cluster algorithm that strongly reduces critical slowing down in simulations of vertex models. Since the clusters are closed paths of bonds, we call it the loop algorithm. The basic steps in constructing a cluster are the breakup and the freezing of vertices. We concentrate on the case of the F model, which is a subset of the six-vertex model exhibiting a Kosterlitz-Thouless transition. The loop algorithm is also applicable to simulations of other vertex models and of one- and two-dimensional quantum spin systems.
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.
We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional Q-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents dB=1.7320(3) and 1.794(2) for Q=2,3, respectively. However, for large Q, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the O(n) loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with Q=n2. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops dB=1.64339(5),1.73227(8),1.7938(3),1.8384(5),1.8753(6) for Q=1,2,3,2+3,4, respectively, and dmin=1.0945(2),1.0675(3),1.0475(3),1.0322(4) for Q=2,3,2+3,4, respectively. Our estimates significantly improve over the existing results for both dB and dmin. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of n for the leading correction exponent.