Abstract
A graphical representation based on duplication is developed that is suitable for the study of Ising systems in external fields. Two independent replicas of the Ising system in the same field are treated as a single four-state (Ashkin–Teller) model. Bonds in the graphical representation connect the Ashkin–Teller spins. For ferromagnetic systems it is proved that ordering is characterized by percolation in this representation. The representation leads immediately to cluster algorithms; some applications along these lines are discussed.
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REFERENCES
M. Aizenman, J. T. Chayes, L. Chayes, and C. M. Newman, Discontinuity of the magnetization in one-dimensional 1/∣x − y∣2 Ising and Potts models, J. Stat. Phys. 50:1–40 (1988).
A. Coniglio and W. Klein, Clusters and Ising critical droplets: A renormalization froup approach, J. Phys. A: Math. Gen. 13:2775–2780 (1980).
L. Chayes and J. Machta, Graphical representations and cluster algorithms. Part I: Discrete spin systems, Physica A 239:542–601 (1997).
L. Chayes and J. Machta, Graphical representations and cluster algorithms. Part II, Physica A 254:477–516 (1998).
R. G. Edwards and A. D. Sokal, Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm, Phys. Rev. D 38:2009–2012 (1988).
C. M. Fortuin and P. W. Kasteleyn, On the random cluster model. I. Introduction and relation to other models, Physica 57:536–564 (1972).
H. O. Georgii, Gibbs Measures and Phase Transitions (Walter de Gruyter, Berlin, 1988).
J. R. Heringa and H. W. J. Blöte, The simple cubic lattice gas with nearest neighbor exclusion: Ising universality, Physica A 232:369–374 (1996).
J. R. Heringa and H. W. J. Blöte, Geometric cluster Monte Carlo simulation, Phys. Rev. E 57:4976–4978 (1998).
J. R. Heringa and H. W. J. Blöte, Cluster dynamics and universality of Ising lattice gases, Preprint (1998).
D. Kandel and E. Domany, General cluster Monte Carlo dynamics, Phys. Rev. B 43:8539–8548 (1991).
M. E. J. Newman and G. T. Barkema, Monte Carlo study of the random-field Ising model, Phys. Rev. E 53:393–403 (1996).
O. Redner, J. Machta, and L. F. Chayes, Graphical representations and cluster algorithms for critical points with fields, Phys. Rev. E 58:2749–2752 (1998) and cond-mat/9802063.
Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36:423–439 (1965).
R. H. Swendsen and J. S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58:86 (1987).
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Chayes, L., Machta, J. & Redner, O. Graphical Representations for Ising Systems in External Fields. Journal of Statistical Physics 93, 17–32 (1998). https://doi.org/10.1023/B:JOSS.0000026726.43558.80
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DOI: https://doi.org/10.1023/B:JOSS.0000026726.43558.80