Potts-model critical manifolds revisited

doi:https://doi.org/10.1088/1751-8113/49/12/125003

Title: 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:

Scullard, Christian R. ^[1]; Jacobsen, Jesper Lykke ^[2]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
PSL Research Univ., Paris (France); Sorbonne Univ., Paris (France)

Publication Date:: 2016-02-11

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.

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                    Scullard, Christian R., & Jacobsen, Jesper Lykke. Potts-model critical manifolds revisited.  United States.  https://doi.org/10.1088/1751-8113/49/12/125003

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                    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.

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@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}

}

Copy to clipboard

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https://doi.org/10.1088/1751-8113/49/12/125003

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Cited by: 6 works

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Works referenced in this record:

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
https://doi.org/10.1007/bf01386092

Critical points of Potts and O( N ) models from eigenvalue identities in periodic Temperley–Lieb algebras
journal, October 2015

Jacobsen, Jesper Lykke
Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue 45
https://doi.org/10.1088/1751-8113/48/45/454003

Critical Surfaces for General Bond Percolation Problems
journal, May 2008

Scullard, Christian R.; Ziff, Robert M.
Physical Review Letters, Vol. 100, Issue 18
https://doi.org/10.1103/physrevlett.100.185701

Phase diagram of the chromatic polynomial on a torus
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Jacobsen, Jesper Lykke; Salas, Jesús
Nuclear Physics B, Vol. 783, Issue 3
https://doi.org/10.1016/j.nuclphysb.2007.04.023

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis
journal, June 2010

Ding, Chengxiang; Fu, Zhe; Guo, Wenan
Physical Review E, Vol. 81, Issue 6
https://doi.org/10.1103/physreve.81.061111

Percolation critical polynomial as a graph invariant
journal, October 2012

Scullard, Christian R.
Physical Review E, Vol. 86, Issue 4
https://doi.org/10.1103/physreve.86.041131

Determination of the bond percolation threshold for the Kagomé lattice
journal, August 1997

Ziff, Robert M.; Suding, Paul N.
Journal of Physics A: Mathematical and General, Vol. 30, Issue 15
https://doi.org/10.1088/0305-4470/30/15/021

Some generalized order-disorder transformations
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Potts, R. B.
Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 48, Issue 1
https://doi.org/10.1017/s0305004100027419

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
https://doi.org/10.1016/0550-3213(91)90402-j

Exact bond percolation thresholds in two dimensions
journal, November 2006

Ziff, Robert M.; Scullard, Christian R.
Journal of Physics A: Mathematical and General, Vol. 39, Issue 49
https://doi.org/10.1088/0305-4470/39/49/003

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models: IV. Chromatic Polynomial with Cyclic Boundary Conditions
journal, February 2006

Jacobsen, Jesper Lykke; Salas, Jesús
Journal of Statistical Physics, Vol. 122, Issue 4
https://doi.org/10.1007/s10955-005-8077-8

Critical percolation in finite geometries
journal, February 1992

Cardy, J. L.
Journal of Physics A: Mathematical and General, Vol. 25, Issue 4
https://doi.org/10.1088/0305-4470/25/4/009

Percolation on hypergraphs with four-edges
journal, September 2015

Damavandi, Ojan Khatib; Ziff, Robert M.
Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue 40
https://doi.org/10.1088/1751-8113/48/40/405004

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation
journal, January 2006

Scullard, Christian R.
Physical Review E, Vol. 73, Issue 1
https://doi.org/10.1103/physreve.73.016107

Transfer matrix computation of critical polynomials for two-dimensional Potts models
journal, February 2013

Jacobsen, Jesper Lykke; Scullard, Christian R.
Journal of Physics A: Mathematical and Theoretical, Vol. 46, Issue 7
https://doi.org/10.1088/1751-8113/46/7/075001

On the random-cluster model
journal, February 1972

Fortuin, C. M.; Kasteleyn, P. W.
Physica, Vol. 57, Issue 4
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Polynomial sequences for bond percolation critical thresholds
journal, September 2011

Scullard, Christian R.
Journal of Statistical Mechanics: Theory and Experiment, Vol. 2011, Issue 09
https://doi.org/10.1088/1742-5468/2011/09/p09022

Critical manifold of the kagome-lattice Potts model
journal, November 2012

Jacobsen, Jesper Lykke; Scullard, Christian R.
Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 49
https://doi.org/10.1088/1751-8113/45/49/494003

Transfer matrix computation of generalized critical polynomials in percolation
journal, November 2012

Scullard, Christian R.; Jacobsen, Jesper Lykke
Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 49
https://doi.org/10.1088/1751-8113/45/49/494004

Site percolation thresholds for Archimedean lattices
journal, July 1999

Suding, Paul N.; Ziff, Robert M.
Physical Review E, Vol. 60, Issue 1
https://doi.org/10.1103/physreve.60.275

Extrapolation and the Bulirsch-Stoer algorithm
journal, June 2002

Monroe, James L.
Physical Review E, Vol. 65, Issue 6
https://doi.org/10.1103/physreve.65.066116

Pseudo-random-number generators and the square site percolation threshold
journal, September 2008

Lee, Michael J.
Physical Review E, Vol. 78, Issue 3
https://doi.org/10.1103/physreve.78.031131

Generalized cell–dual-cell transformation and exact thresholds for percolation
journal, January 2006

Ziff, Robert M.
Physical Review E, Vol. 73, Issue 1
https://doi.org/10.1103/physreve.73.016134

Critical surfaces for general inhomogeneous bond percolation problems
journal, March 2010

Scullard, Christian R.; Ziff, Robert M.
Journal of Statistical Mechanics: Theory and Experiment, Vol. 2010, Issue 03
https://doi.org/10.1088/1742-5468/2010/03/p03021

Percolation transitions in two dimensions
journal, September 2008

Feng, Xiaomei; Deng, Youjin; Blöte, Henk W. J.
Physical Review E, Vol. 78, Issue 3
https://doi.org/10.1103/physreve.78.031136

Pseudo-random-number generators and the square site percolation threshold
journal, September 2008

Lee, Michael J.
Physical Review E, Vol. 78, Issue 3
https://doi.org/10.1103/PhysRevE.78.031131

Site percolation thresholds for Archimedean lattices
journal, July 1999

Suding, Paul N.; Ziff, Robert M.
Physical Review E, Vol. 60, Issue 1
https://doi.org/10.1103/PhysRevE.60.275

Percolation transitions in two dimensions
journal, September 2008

Feng, Xiaomei; Deng, Youjin; Blöte, Henk W. J.
Physical Review E, Vol. 78, Issue 3
https://doi.org/10.1103/PhysRevE.78.031136

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis
journal, June 2010

Ding, Chengxiang; Fu, Zhe; Guo, Wenan
Physical Review E, Vol. 81, Issue 6
https://doi.org/10.1103/PhysRevE.81.061111

Critical Surfaces for General Bond Percolation Problems
journal, May 2008

Scullard, Christian R.; Ziff, Robert M.
Physical Review Letters, Vol. 100, Issue 18
https://doi.org/10.1103/PhysRevLett.100.185701

Critical surfaces for general inhomogeneous bond percolation problems
journal, March 2010

Scullard, Christian R.; Ziff, Robert M.
Journal of Statistical Mechanics: Theory and Experiment, Vol. 2010, Issue 03
https://doi.org/10.1088/1742-5468/2010/03/P03021

Percolation critical polynomial as a graph invariant
journal, October 2012

Scullard, Christian R.
Physical Review E, Vol. 86, Issue 4
https://doi.org/10.1103/PhysRevE.86.041131

Polynomial sequences for bond percolation critical thresholds
journal, September 2011

Scullard, Christian R.
Journal of Statistical Mechanics: Theory and Experiment, Vol. 2011, Issue 09
https://doi.org/10.1088/1742-5468/2011/09/P09022

Some generalized order-disorder transformations
journal, January 1952

Potts, R. B.
Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 48, Issue 1
https://doi.org/10.1017/S0305004100027419

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation
journal, January 2006

Scullard, Christian R.
Physical Review E, Vol. 73, Issue 1
https://doi.org/10.1103/PhysRevE.73.016107

Generalized cell–dual-cell transformation and exact thresholds for percolation
journal, January 2006

Ziff, Robert M.
Physical Review E, Vol. 73, Issue 1
https://doi.org/10.1103/PhysRevE.73.016134

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
https://doi.org/10.1007/BF01386092

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
https://doi.org/10.1016/0550-3213(91)90402-J

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
https://doi.org/10.1088/1751-8121/aad1fe

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Similar Records

Title: Potts-model critical manifolds revisited

Abstract

Citation Formats

Fehlerabsch�tzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus journal, December 1964

Critical points of Potts and O( N ) models from eigenvalue identities in periodic Temperley–Lieb algebras journal, October 2015

Critical Surfaces for General Bond Percolation Problems journal, May 2008

Phase diagram of the chromatic polynomial on a torus journal, November 2007

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis journal, June 2010

Percolation critical polynomial as a graph invariant journal, October 2012

Determination of the bond percolation threshold for the Kagomé lattice journal, August 1997

Some generalized order-disorder transformations journal, January 1952

The antiferromagnetic Potts model in two dimensions: Berker-Kadanoff phase, antiferromagnetic transition, and the role of Beraha numbers journal, August 1991

Exact bond percolation thresholds in two dimensions journal, November 2006

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models: IV. Chromatic Polynomial with Cyclic Boundary Conditions journal, February 2006

Critical percolation in finite geometries journal, February 1992

Percolation on hypergraphs with four-edges journal, September 2015

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation journal, January 2006

Transfer matrix computation of critical polynomials for two-dimensional Potts models journal, February 2013

On the random-cluster model journal, February 1972

Polynomial sequences for bond percolation critical thresholds journal, September 2011

Critical manifold of the kagome-lattice Potts model journal, November 2012

Transfer matrix computation of generalized critical polynomials in percolation journal, November 2012

Site percolation thresholds for Archimedean lattices journal, July 1999

Extrapolation and the Bulirsch-Stoer algorithm journal, June 2002

Pseudo-random-number generators and the square site percolation threshold journal, September 2008

Generalized cell–dual-cell transformation and exact thresholds for percolation journal, January 2006

Critical surfaces for general inhomogeneous bond percolation problems journal, March 2010

Percolation transitions in two dimensions journal, September 2008

Pseudo-random-number generators and the square site percolation threshold journal, September 2008

Site percolation thresholds for Archimedean lattices journal, July 1999

Percolation transitions in two dimensions journal, September 2008

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis journal, June 2010

Critical Surfaces for General Bond Percolation Problems journal, May 2008

Critical surfaces for general inhomogeneous bond percolation problems journal, March 2010

Percolation critical polynomial as a graph invariant journal, October 2012

Polynomial sequences for bond percolation critical thresholds journal, September 2011

Some generalized order-disorder transformations journal, January 1952

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation journal, January 2006

Generalized cell–dual-cell transformation and exact thresholds for percolation journal, January 2006

Fehlerabsch�tzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus journal, December 1964

The antiferromagnetic Potts model in two dimensions: Berker-Kadanoff phase, antiferromagnetic transition, and the role of Beraha numbers journal, August 1991

The three-state Potts antiferromagnet on plane quadrangulations journal, July 2018

Fehlerabsch�tzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus
journal, December 1964

Critical points of Potts and O( N ) models from eigenvalue identities in periodic Temperley–Lieb algebras
journal, October 2015

Critical Surfaces for General Bond Percolation Problems
journal, May 2008

Phase diagram of the chromatic polynomial on a torus
journal, November 2007

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis
journal, June 2010

Percolation critical polynomial as a graph invariant
journal, October 2012

Determination of the bond percolation threshold for the Kagomé lattice
journal, August 1997

Some generalized order-disorder transformations
journal, January 1952

The antiferromagnetic Potts model in two dimensions: Berker-Kadanoff phase, antiferromagnetic transition, and the role of Beraha numbers
journal, August 1991

Exact bond percolation thresholds in two dimensions
journal, November 2006

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models: IV. Chromatic Polynomial with Cyclic Boundary Conditions
journal, February 2006

Critical percolation in finite geometries
journal, February 1992

Percolation on hypergraphs with four-edges
journal, September 2015

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation
journal, January 2006

Transfer matrix computation of critical polynomials for two-dimensional Potts models
journal, February 2013

On the random-cluster model
journal, February 1972

Polynomial sequences for bond percolation critical thresholds
journal, September 2011

Critical manifold of the kagome-lattice Potts model
journal, November 2012

Transfer matrix computation of generalized critical polynomials in percolation
journal, November 2012

Site percolation thresholds for Archimedean lattices
journal, July 1999

Extrapolation and the Bulirsch-Stoer algorithm
journal, June 2002

Pseudo-random-number generators and the square site percolation threshold
journal, September 2008

Generalized cell–dual-cell transformation and exact thresholds for percolation
journal, January 2006

Critical surfaces for general inhomogeneous bond percolation problems
journal, March 2010

Percolation transitions in two dimensions
journal, September 2008

Pseudo-random-number generators and the square site percolation threshold
journal, September 2008

Site percolation thresholds for Archimedean lattices
journal, July 1999

Percolation transitions in two dimensions
journal, September 2008

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis
journal, June 2010

Critical Surfaces for General Bond Percolation Problems
journal, May 2008

Critical surfaces for general inhomogeneous bond percolation problems
journal, March 2010

Percolation critical polynomial as a graph invariant
journal, October 2012

Polynomial sequences for bond percolation critical thresholds
journal, September 2011

Some generalized order-disorder transformations
journal, January 1952

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation
journal, January 2006

Generalized cell–dual-cell transformation and exact thresholds for percolation
journal, January 2006

Fehlerabsch�tzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus
journal, December 1964

The antiferromagnetic Potts model in two dimensions: Berker-Kadanoff phase, antiferromagnetic transition, and the role of Beraha numbers
journal, August 1991

The three-state Potts antiferromagnet on plane quadrangulations
journal, July 2018