The Test of a New Critical Exponent Ϙ by Using Ising Model on the Creutz Cellular Automaton

Merdan, Ziya; Gokbel-Keklikoglu, D.

doi:10.12693/aphyspola.133.1200

Cited by 2 publications

(2 citation statements)

References 12 publications

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“…The new exponent already enabled predictions to be more naturally expressed for models such as the nearest-neighbour Ising model above 4 dimensions, percolation above its critical dimension d uc = 6 and for LRIM's with various dimensions above d uc (σ) (with periodic boundary conditions) [48,49]. The extension to general values of n is obvious and we collect the predictions for arbitrary d > d uc for quantities which have been discussed above:…”

Section: Dangerous Irrelevancy In the Correlation Sectormentioning

confidence: 99%

Phase transitions above the upper critical dimension

Berche

Ellis

Holovatch

et al. 2022

SciPost Phys. Lect. Notes

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These lecture notes provide an overview of the renormalization group (RG) as a successful framework to understand critical phenomena above the upper critical dimension d_{uc}duc. After an introduction to the scaling picture of continuous phase transitions, we discuss the apparent failure of the Gaussian fixed point to capture scaling for Landau mean-field theory, which should hold in the thermodynamic limit above d_{uc}duc. We recount how Fisher’s dangerous-irrelevant-variable formalism applied to thermodynamic functions partially repairs the situation but at the expense of hyperscaling and finite-size scaling, both of which were, until recently, believed not to apply above d_{uc}duc. We recall limitations of various attempts to match the RG with analytical and numerical results for Ising systems. We explain how the extension of dangerous irrelevancy to the correlation sector is key to marrying the above concepts into a comprehensive RG scaling picture that renders hyperscaling and finite-size scaling valid in all dimensions. We collect what we believe is the current status of the theory, including some new insights and results. This paper is in grateful memory of Michael Fisher who introduced many of the concepts discussed and who, half a century later, contributed to their advancement.

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Section: Dangerous Irrelevancy In the Correlation Sectormentioning

confidence: 99%

Phase transitions above the upper critical dimension

Berche

Ellis

Holovatch

et al. 2022

SciPost Phys. Lect. Notes

View full text Add to dashboard Cite

show abstract

“…The new exponent already enabled predictions to be more naturally expressed for models such as the nearest-neighbour Ising model, percolation above its critical dimension d uc = 6 and for LRIM's with various dimensions above d uc (σ) (with periodic boundary conditions) [50,51]. The extension to general values of n is obvious and we collect the predictions for arbitrary d > d uc for quantities which have been discussed above:…”

Section: Dangerous Irrelevancy In the Correlation Sectormentioning

confidence: 99%

Phase transitions above the upper critical dimension

Berche,

Ellis,

Holovatch

et al. 2022

Preprint

View full text Add to dashboard Cite

These lecture notes provide an overview of the renormalization group (RG) as a successful framework to understand critical phenomena above the upper critical dimension d uc . After an introduction to the scaling picture of continuous phase transitions, we discuss the apparent failure of the Gaussian fixed point to capture scaling for Landau mean-field theory, which should hold in the thermodynamic limit above d uc . We recount how Fisher's dangerous-irrelevant-variable formalism applied to thermodynamic functions partially repairs the situation but at the expense of hyperscaling and finite-size scaling, both of which were, until recently, believed not to apply above d uc . We recall limitations of various attempts to match the RG with analytical and numerical results for Ising systems. We explain how the extension of dangerous irrelevancy to the correlation sector is key to marrying the above concepts into a comprehensive RG scaling picture that renders hyperscaling and finite-size scaling valid in all dimensions. We collect what we believe is the current status of the theory, including some new insights and results. This paper is in grateful memory of Michael Fisher who introduced many of the concepts discussed and who, half a century later, contributed to their advancement. Contents1 Prelude: Definitions and notations 2 2 Introduction: Second-order phase transitions and the scaling picture 6 3 Ginzburg-Landau mean-field theory 11

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The Test of a New Critical Exponent Ϙ by Using Ising Model on the Creutz Cellular Automaton

Abstract: Above the upper critical dimension dc the Ising model is simulated on the Creutz cellular automaton. The values of a new critical exponent are obtained by using the simulations for the order parameter and the magnetic susceptibility.

Cited by 2 publications

References 12 publications

Phase transitions above the upper critical dimension

Phase transitions above the upper critical dimension

Phase transitions above the upper critical dimension

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