arXiv:0805.2678 · Physical Review E

Critical properties of a dilute O(n) model on the kagome lattice

Biao Li, Wenan Guo, Henk W. J. Blöte

Published Aug 21, 2008• 3 citations

Abstract

A critical dilute O(n) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O(n) spin model on the kagome lattice to the exactly solvable critical q-state Potts model on the honeycomb lattice with q=(n+1)2. The intermediate steps involve the random-cluster model on the honeycomb lattice and a fully packed loop model with loop weight n'=sqrt(q) and a dilute loop model with loop weight n , both on the kagome lattice. This mapping enables the determination of a branch of critical points of the dilute O(n) model, as well as some of its critical properties. These properties differ from those of the generic O(n) critical points. For n=0, our model reproduces the known universal properties of the theta point describing the collapse of a polymer. For n not equal 0 it displays a line of multicritical points, with the same universal behavior as a branch of critical points that was found earlier in a dilute O(n) model on the square lattice. These findings are supported by a finite-size-scaling analysis in combination with transfer-matrix calculations.

DOI: 10.1103/physreve.78.021128

Related papers

Phase Transitions of a Dilute O(n) Model

Wenan Guo, Henk W. J. Blöte, Yuanyuan Liu

We investigate tricritical behavior of the O(n) model in two dimensions by means of transfer-matrix and finite-size scaling methods. For this purpose we consider an O(n) symmetric spin model on the honeycomb lattice with vacancies; the tricritical behavior is associated with the percolation threshold of the vacancies. The vacancies are represented by face variables on the elementary hexagons of the lattice. We apply a mapping of the spin degrees of freedom model on a non-intersecting-loop model, in which the number n of spin components assumes the role of a continuously variable parameter. This loop model serves as a suitable basis for the construction of the transfer matrix. Our results reveal the existence of a tricritical line, parametrized by n , which connects the known universality classes of the tricritical Ising model and the theta point describing the collapse of a polymer. On the other side of the Ising point, the tricritical line extends to the n = 2 point describing a tricritical O(2) model.

Communications in Theoretical PhysicsJun 15, 20044 citations

Fully packed loop model on the honeycomb lattice

Henk W. J. Blöte, Bernard Nienhuis

We investigate the O(n) model on the honeycomb lattice, using its loop representation in the limit of full packing. The universal properties, which we calculate by means of finite-size scaling and transfer-matrix techniques, are different from the branches of O(n) critical behavior known thus far. The conformal anomaly of the model varies between -1 and 2 in the interval 0\ensuremath{\le}n\ensuremath{\le}2. The universality class of the model is characterized as a superposition of a low-temperature O(n) phase, and a solid-on-solid model at a temperature independent of n.

Physical Review LettersFeb 28, 199493 citations

FIRST AND SECOND ORDER TRANSITIONS IN DILUTE O (n) MODELS

Wenan Guo, Henk W. J. Blöte, Bernard Nienhuis

We explore the phase diagram of an O (n) model on the honeycomb lattice with vacancies, using finite-size scaling and transfer-matrix methods. We make use of the loop representation of the O (n) model, so that n is not restricted to positive integers. For low activities of the vacancies, we observe critical points of the known universality class. At high activities the transition becomes first order. For n=0 the model includes an exactly known theta point, used to describe a collapsing polymer in two dimensions. When we vary n from 0 to 1, we observe a tricritical point which interpolates between the universality classes of the theta point and the Ising tricritical point.

arXiv:cond-mat/9812270International Journal of Modern Physics CFeb 1, 19995 citations

Critical behaviour and conformal anomaly of the O(n) model on the square lattice

Henk W. J. Blöte, Bernard Nienhuis

Finite-size scaling and transfer-matrix techniques are used to determine the conformal anomaly and critical exponents of O(n) models on the square lattice. These calculations were performed on five branches of critical points parametrised by n. The results for two of the branches agree with the known universal properties of the O(n) model as derived for the honeycomb lattice. Finite-size convergence is poor for most of the third branch, so that in this case the classification in terms of known models is uncertain. The authors' data show that the fourth branch can be interpreted in terms of a combination of a low-temperature O(n) and a critical Ising model. Finally, the numerical results for the O(n) model on the remaining branch agree accurately with the known results for the low-temperature O(n+1) model. They prove this equivalence by means of an exact mapping.

Journal of Physics A Mathematical and GeneralMay 7, 1989177 citations

Critical point of the Kagomé Potts model: a Monte Carlo renormalization group and scaling determination

Jau-Ann Chen, Chin‐Kun Hu, Fa Yueh Wu

The determination of the critical point of the q -state Potts model on the Kagomé lattices has been an outstanding unsolved problem. Here we study this problem by regarding the Potts model as generating correlated bond percolations. By applying a combination of Monte Carlo renormalization group and finite-size scaling analyses to the percolation problem, numerical estimates with an accuracy of are obtained for the Kagomé critical point for q = 1,2,3,4. Our results for q = 1 are consistent with a recent highly accurate numerical estimate by Ziff and Suding, and results for q = 2 agree with the known exact result within the numerical accuracy. Compared with results obtained in a recent series analysis by Jensen et al , our numbers differ from theirs slightly for q = 3, and agree with theirs with a slightly better accuracy for q = 4. Our numbers also confirm that a conjecture due to Wu is extremely accurate.

Journal of Physics A Mathematical and GeneralOct 2, 199811 citations

Percolation and critical O(n) loop configurations

Chengxiang Ding, Youjin Deng, Wenan Guo +1 more

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.

arXiv:0903.2359Physical Review EJun 22, 20099 citations