Crossover Critical Behavior in the Three-Dimensional Ising Model

Young C. Kim; Mikhail A. Anisimov; Jan V. Sengers; Erik Luijten

doi:10.1023/A:1022199516676

Crossover Critical Behavior in the Three-Dimensional Ising Model

Young C. Kim^*, Mikhail A. Anisimov, Jan V. Sengers, Erik Luijten

^*Corresponding author for this work

Materials Science and Engineering

Research output: Contribution to journal › Article › peer-review

45 Scopus citations

Abstract

The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the three-dimensional Ising universality class. Recent numerical studies of Ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation-induced critical behavior and mean-field-type critical behavior. In this work, we compare these numerical results with a crossover Landau model based on renormalization-group matching. For this purpose we consider an application of the crossover Landau model to the three-dimensional Ising model without fitting to any adjustable parameters. The crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range (ten orders) of the scaled distance to the critical temperature. The dependence of the coupling constant on the interaction range, governing the crossover critical behavior, is discussed.

Original language	English (US)
Pages (from-to)	591-609
Number of pages	19
Journal	Journal of Statistical Physics
Volume	110
Issue number	3-6
DOIs	https://doi.org/10.1023/A:1022199516676
State	Published - Mar 2003

Funding

The authors have benefited from many stimulating discussions with M. E. Fisher, to whom this article is dedicated with great appreciation. Kh. S. Abdulkadirova and V. A. Agayan provided assistance in the initial stage of this research. The research of M. A. Anisimov and J. V. Sengers is supported by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, Department of Energy under Grant DE-FG02-95-ER-14509. The research of Y. C. Kim and E. Luijten has been supported by Grant CHE99-81772 from the National Science Foundation to M. E. Fisher at the University of Maryland.

Keywords

Critical phenomena
Crossover Landau model
Ising model
Order parameter
Susceptibility

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1023/A:1022199516676

Cite this

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title = "Crossover Critical Behavior in the Three-Dimensional Ising Model",

abstract = "The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the three-dimensional Ising universality class. Recent numerical studies of Ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation-induced critical behavior and mean-field-type critical behavior. In this work, we compare these numerical results with a crossover Landau model based on renormalization-group matching. For this purpose we consider an application of the crossover Landau model to the three-dimensional Ising model without fitting to any adjustable parameters. The crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range (ten orders) of the scaled distance to the critical temperature. The dependence of the coupling constant on the interaction range, governing the crossover critical behavior, is discussed.",

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author = "Kim, {Young C.} and Anisimov, {Mikhail A.} and Sengers, {Jan V.} and Erik Luijten",

note = "Funding Information: The authors have benefited from many stimulating discussions with M. E. Fisher, to whom this article is dedicated with great appreciation. Kh. S. Abdulkadirova and V. A. Agayan provided assistance in the initial stage of this research. The research of M. A. Anisimov and J. V. Sengers is supported by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, Department of Energy under Grant DE-FG02-95-ER-14509. The research of Y. C. Kim and E. Luijten has been supported by Grant CHE99-81772 from the National Science Foundation to M. E. Fisher at the University of Maryland.",

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AU - Luijten, Erik

N1 - Funding Information: The authors have benefited from many stimulating discussions with M. E. Fisher, to whom this article is dedicated with great appreciation. Kh. S. Abdulkadirova and V. A. Agayan provided assistance in the initial stage of this research. The research of M. A. Anisimov and J. V. Sengers is supported by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, Department of Energy under Grant DE-FG02-95-ER-14509. The research of Y. C. Kim and E. Luijten has been supported by Grant CHE99-81772 from the National Science Foundation to M. E. Fisher at the University of Maryland.

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N2 - The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the three-dimensional Ising universality class. Recent numerical studies of Ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation-induced critical behavior and mean-field-type critical behavior. In this work, we compare these numerical results with a crossover Landau model based on renormalization-group matching. For this purpose we consider an application of the crossover Landau model to the three-dimensional Ising model without fitting to any adjustable parameters. The crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range (ten orders) of the scaled distance to the critical temperature. The dependence of the coupling constant on the interaction range, governing the crossover critical behavior, is discussed.

AB - The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the three-dimensional Ising universality class. Recent numerical studies of Ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation-induced critical behavior and mean-field-type critical behavior. In this work, we compare these numerical results with a crossover Landau model based on renormalization-group matching. For this purpose we consider an application of the crossover Landau model to the three-dimensional Ising model without fitting to any adjustable parameters. The crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range (ten orders) of the scaled distance to the critical temperature. The dependence of the coupling constant on the interaction range, governing the crossover critical behavior, is discussed.

KW - Critical phenomena

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KW - Ising model

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KW - Susceptibility

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Crossover Critical Behavior in the Three-Dimensional Ising Model

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