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Cluster Simulations of Loop Models on Planar Lattices

Youjin Deng, Wenan Guo, Timothy M. Garoni, Alan D. Sokal, Henk W. J. Blöte

Published Aug 21, 2006• 0 citations
OpenAlex

Abstract

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

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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≥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 have developed a cluster algorithm for Monte Carlo simulations of the fully frustrated Ising model on the square lattice. The method does not suffer from problems of metastability, and is extremely efficient even at T=0, as we demonstrated previously. Here we describe results of more extensive simulations of the model at T=0 which allow us to extract the effective dynamical critical exponent of the algorithm, z\ensuremath{\approxeq}0.55, as well as the static exponent \ensuremath{\eta}=0.5. We also provide an argument that explains why our method works.
Phase transitions and critical phenomena in several spin models are studied. These models include a two and three dimensional XY model and Ising models in structured and random field. The methods of study are cluster Monte Carlo simulations and network flow optimization algorithms. The XY model and a related soft spin model belonging to the O(2) universality class are studied using a modified version of the Invaded Cluster algorithm. The static critical properties of the models and the dynamic characteristics of the algorithm are studied. The critical exponent η for the O(2) universality class in two and three dimensions is measured. The Invaded Cluster algorithm for the XY model does not show any critical slowing for global variables. Benzene adsorption in a nanoporous solid is modeled with an Ising model with both internal and external magnetic fields. The first order phase transition and the critical properties of the system are studied. The system is simulated using the two replica cluster algorithm. The critical temperature of the system is found to decrease with increasing difference of the values of the internal fields. It is found that the system undergoes a phase transition for a relatively wide, experimentally obtainable, region of values of its parameters. The random field Ising model in three dimensions is studied at zero temperature. The nature of randomness induced phase transition is studied using a numerical network flow optimization method. It is found that the latent heat at the phase transition vanishes in the limit of infinite system size which can be taken as a proof of a second order phase transition.
We determine the phase diagram of the O(n) loop model on the honeycomb lattice, in particular, in the range n>2, by means of a transfer-matrix method. We find that, contrary to the prevailing expectation, there is a line of critical points in the range between n = 2 and infinity. This phase transition, which belongs to the three-state Potts universality class, is unphysical in terms of the O(n) spin model, but falls inside the physical region of the n-component corner-cubic model. It can also be interpreted in terms of the ordering of a system of soft particles with hexagonal symmetry.