Potts-model critical manifolds revisited
Abstract
We compute the critical polynomials for the q-state Potts model on all Archimedean lattices, using a parallel implementation of the algorithm of Ref. [1] that gives us access to larger sizes than previously possible. The exact polynomials are computed for bases of size 6 6 unit cells, and the root in the temperature variable v = e K-1 is determined numerically at q = 1 for bases of size 8 8. This leads to improved results for bond percolation thresholds, and for the Potts-model critical manifolds in the real (q; v) plane. In the two most favourable cases, we find now the kagome-lattice threshold to eleven digits and that of the (3; 12 2) lattice to thirteen. Our critical manifolds reveal many interesting features in the antiferromagnetic region of the Potts model, and determine accurately the extent of the Berker-Kadano phase for the lattices studied.
- Authors:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- PSL Research Univ., Paris (France); Sorbonne Univ., Paris (France)
- Publication Date:
- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1399717
- Report Number(s):
- LLNL-JRNL-679200
Journal ID: ISSN 1751-8113
- Grant/Contract Number:
- AC52-07NA27344
- Resource Type:
- Journal Article: Accepted Manuscript
- Journal Name:
- Journal of Physics. A, Mathematical and Theoretical
- Additional Journal Information:
- Journal Volume: 49; Journal Issue: 12; Journal ID: ISSN 1751-8113
- Publisher:
- IOP Publishing
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING
Citation Formats
Scullard, Christian R., and Jacobsen, Jesper Lykke. Potts-model critical manifolds revisited. United States: N. p., 2016.
Web. doi:10.1088/1751-8113/49/12/125003.
Scullard, Christian R., & Jacobsen, Jesper Lykke. Potts-model critical manifolds revisited. United States. https://doi.org/10.1088/1751-8113/49/12/125003
Scullard, Christian R., and Jacobsen, Jesper Lykke. Thu .
"Potts-model critical manifolds revisited". United States. https://doi.org/10.1088/1751-8113/49/12/125003. https://www.osti.gov/servlets/purl/1399717.
@article{osti_1399717,
title = {Potts-model critical manifolds revisited},
author = {Scullard, Christian R. and Jacobsen, Jesper Lykke},
abstractNote = {We compute the critical polynomials for the q-state Potts model on all Archimedean lattices, using a parallel implementation of the algorithm of Ref. [1] that gives us access to larger sizes than previously possible. The exact polynomials are computed for bases of size 6 6 unit cells, and the root in the temperature variable v = eK-1 is determined numerically at q = 1 for bases of size 8 8. This leads to improved results for bond percolation thresholds, and for the Potts-model critical manifolds in the real (q; v) plane. In the two most favourable cases, we find now the kagome-lattice threshold to eleven digits and that of the (3; 122) lattice to thirteen. Our critical manifolds reveal many interesting features in the antiferromagnetic region of the Potts model, and determine accurately the extent of the Berker-Kadano phase for the lattices studied.},
doi = {10.1088/1751-8113/49/12/125003},
url = {https://www.osti.gov/biblio/1399717},
journal = {Journal of Physics. A, Mathematical and Theoretical},
issn = {1751-8113},
number = 12,
volume = 49,
place = {United States},
year = {2016},
month = {2}
}
Web of Science
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Percolation transitions in two dimensions
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Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis
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Critical Surfaces for General Bond Percolation Problems
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Critical surfaces for general inhomogeneous bond percolation problems
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Percolation critical polynomial as a graph invariant
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- Physical Review E, Vol. 86, Issue 4
Polynomial sequences for bond percolation critical thresholds
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Some generalized order-disorder transformations
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Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation
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- Scullard, Christian R.
- Physical Review E, Vol. 73, Issue 1
Generalized cell–dual-cell transformation and exact thresholds for percolation
journal, January 2006
- Ziff, Robert M.
- Physical Review E, Vol. 73, Issue 1
Fehlerabsch�tzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus
journal, December 1964
- Bulirsch, Roland; Stoer, Josef
- Numerische Mathematik, Vol. 6, Issue 1
The antiferromagnetic Potts model in two dimensions: Berker-Kadanoff phase, antiferromagnetic transition, and the role of Beraha numbers
journal, August 1991
- Saleur, H.
- Nuclear Physics B, Vol. 360, Issue 2-3
Works referencing / citing this record:
The three-state Potts antiferromagnet on plane quadrangulations
journal, July 2018
- Lv, Jian-Ping; Deng, Youjin; Jacobsen, Jesper Lykke
- Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 36