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Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

Oliver Redner, Michael Baake

Published Jan 1, 1999• 4 citations
OpenAlex

Abstract

. We demonstrate that the invaded cluster algorithm, recently introduced by Machta et al, is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits. On two quasiperiodic graphs which were not investigated in this respect before, the twelvefold symmetric square-triangle tiling and the tenfold symmetric Tubingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from t...

Related papers

Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models

Oliver Redner, Michael Baake

We demonstrate that the invaded cluster algorithm, introduced by Machta et al (1995 Phys. Rev. Lett. 75 2792-5), is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits, but only modest computational effort. On two quasiperiodic graphs which were not investigated in this respect before, the 12-fold symmetric square-triangle tiling and the 10-fold symmetric Tübingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from the Onsager value. But notable exceptions to the criterion are found which consist not only of the exactly solvable cases, in agreement with a recent exact result, but also of the self-dual ones and maybe more.

arXiv:cond-mat/9910136•Journal of Physics A Mathematical and General•Apr 28, 2000•6 citations

Invaded cluster simulations of theXYmodel in two and three dimensions

Ilija Dukovski, Jonathan L. Machta, Lincoln Chayes

The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 2000(2) and 120(3), respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are K(c)=0.454 12(2) for the three-dimensional XY model and eta=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are K(c)=1.120(1) and eta=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.

arXiv:cond-mat/0105143•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jan 3, 2002•24 citations

Equilibriumlike invaded cluster algorithm: Critical exponents and dynamical properties

Ivan Balog, Katarina Uzelac

We present a detailed study of the equilibrium-like invaded cluster algorithm, recently proposed as an extension of the invaded cluster algorithm, designed to drive the system to criticality while still preserving the equilibrium ensemble. We perform extensive simulations on two special cases of the Potts model and examine the precision of critical exponents by including the leading corrections. We show that both thermal and magnetic critical exponents can be obtained with high accuracy compared to the best available results. The choice of the auxiliary parameters of the algorithm is discussed in context of dynamical properties. We also discuss the relation to the Li-Sokal bound for the dynamical exponent z.

arXiv:1004.1509•Physical Review E•Apr 15, 2010•3 citations

Invaded Cluster Simulations of the XY Model in Two and Three

Ilija Dukovski, Jonathan L. Machta, Lincoln Chayes

The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 20002 and 1203, respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are Kc=0.45412(2) for the three-dimensional XY model and η=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are Kc=1.120(1) and η=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.

•Jan 1, 2002•0 citations

Generalization of the invaded cluster algorithm to the tricritical point

Ivan Balog, Katarina Uzelac

The invaded cluster algorithm [1] is generalized [2] to the tricritical point on the example of 2d Potts model with annealed dilution. Self-regulating procedure that locates the tricritical point in the two-parameter space spanned by temperature and chemical potential of vacancies is constructed based on geometrical arguments. The tricritical point is identified as a simultaneous percolation of the Fortuin-Kastelyn cluster and the geometrical cluster consisting of vacancies and isolated spins. The tricritical values of parameters and concentration are presented for q=1, 2, 3 and found to be in a good agreement with the best known results [3]. Scaling properties of the percolating scaling cluster and related critical exponents are also derived. Based on the idea that effective correlation of vacancies is important at the tricritical point, we also examine alternative stopping rules within generalized IC algorithm, and possible extension to higher dimensions. [1] J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, L. V. Chayes, Phys. Rev. Lett. 75, 2792 (1995) ; J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, L. M. Chayes, Phys. Rev. E 54, 1332 (1996). [2] I. Balog, K.Uzelac, preprint cond-mat/0703759. [3] X. Qian, Y. Deng, H. W. J. Bl¨ote, Phys. Rev. E 72, 056132 (2005).

•Jan 1, 2007•0 citations

Cluster Monte Carlo and network flow algorithms for critical phenomena

Ilija Dukovski

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.

•Jan 1, 2001•0 citations
. We demonstrate that the invaded cluster algorithm, recently introduced by Machta et al, is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits. On two quasiperiodic graphs which were not investigated in this respect before, the twelvefold symmetric square-triangle tiling and the tenfold symmetric Tubingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from t...
We demonstrate that the invaded cluster algorithm, introduced by Machta et al (1995 Phys. Rev. Lett. 75 2792-5), is a fast and reliable tool for determining the critical temperature and the magnetic critical exponent of periodic and aperiodic ferromagnetic Ising models in two dimensions. The algorithm is shown to reproduce the known values of the critical temperature on various periodic and quasiperiodic graphs with an accuracy of more than three significant digits, but only modest computational effort. On two quasiperiodic graphs which were not investigated in this respect before, the 12-fold symmetric square-triangle tiling and the 10-fold symmetric Tübingen triangle tiling, we determine the critical temperature. Furthermore, a generalization of the algorithm to non-identical coupling strengths is presented and applied to a class of Ising models on the Labyrinth tiling. For generic cases in which the heuristic Harris-Luck criterion predicts deviations from the Onsager universality class, we find a magnetic critical exponent different from the Onsager value. But notable exceptions to the criterion are found which consist not only of the exactly solvable cases, in agreement with a recent exact result, but also of the self-dual ones and maybe more.
The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 2000(2) and 120(3), respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are K(c)=0.454 12(2) for the three-dimensional XY model and eta=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are K(c)=1.120(1) and eta=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.
We present a detailed study of the equilibrium-like invaded cluster algorithm, recently proposed as an extension of the invaded cluster algorithm, designed to drive the system to criticality while still preserving the equilibrium ensemble. We perform extensive simulations on two special cases of the Potts model and examine the precision of critical exponents by including the leading corrections. We show that both thermal and magnetic critical exponents can be obtained with high accuracy compared to the best available results. The choice of the auxiliary parameters of the algorithm is discussed in context of dynamical properties. We also discuss the relation to the Li-Sokal bound for the dynamical exponent z.
The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 20002 and 1203, respectively. A soft spin O(2) model, in the same universality class as the three-dimensional XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are Kc=0.45412(2) for the three-dimensional XY model and η=0.037(2) for the three-dimensional XY universality class. For the two-dimensional XY model the results are Kc=1.120(1) and η=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the two-dimensional or three-dimensional XY models.
The invaded cluster algorithm [1] is generalized [2] to the tricritical point on the example of 2d Potts model with annealed dilution. Self-regulating procedure that locates the tricritical point in the two-parameter space spanned by temperature and chemical potential of vacancies is constructed based on geometrical arguments. The tricritical point is identified as a simultaneous percolation of the Fortuin-Kastelyn cluster and the geometrical cluster consisting of vacancies and isolated spins. The tricritical values of parameters and concentration are presented for q=1, 2, 3 and found to be in a good agreement with the best known results [3]. Scaling properties of the percolating scaling cluster and related critical exponents are also derived. Based on the idea that effective correlation of vacancies is important at the tricritical point, we also examine alternative stopping rules within generalized IC algorithm, and possible extension to higher dimensions. [1] J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, L. V. Chayes, Phys. Rev. Lett. 75, 2792 (1995) ; J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, L. M. Chayes, Phys. Rev. E 54, 1332 (1996). [2] I. Balog, K.Uzelac, preprint cond-mat/0703759. [3] X. Qian, Y. Deng, H. W. J. Bl¨ote, Phys. Rev. E 72, 056132 (2005).
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.