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Title: Transfer matrix computation of critical polynomials for two-dimensional Potts models

Abstract

We showed, In our previous work, that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial PB(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic two-dimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = eK — 1 of PB(q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, PB(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of PB(q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible.We present results for the critical polynomial on the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The criticalmore » temperatures vc obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain vc(4, 82) = 3.742 489 (4), vc(kagome) = 1.876 459 7 (2), and vc(3, 122) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.« less

Authors:
 [1];  [2]
  1. LPTENS, Ecole Normale Superieure, Paris (France); Univ. Pierre et Marie Curie, Paris (France)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1240068
Report Number(s):
LLNL-JRNL-610232
Journal ID: ISSN 1751-8121
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Physics. A, Mathematical and Theoretical (Online)
Additional Journal Information:
Journal Name: Journal of Physics. A, Mathematical and Theoretical (Online); Journal Volume: 46; Journal Issue: 7; Journal ID: ISSN 1751-8121
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE

Citation Formats

Jacobsen, Jesper Lykke, and Scullard, Christian R. Transfer matrix computation of critical polynomials for two-dimensional Potts models. United States: N. p., 2013. Web. doi:10.1088/1751-8113/46/7/075001.
Jacobsen, Jesper Lykke, & Scullard, Christian R. Transfer matrix computation of critical polynomials for two-dimensional Potts models. United States. doi:10.1088/1751-8113/46/7/075001.
Jacobsen, Jesper Lykke, and Scullard, Christian R. Mon . "Transfer matrix computation of critical polynomials for two-dimensional Potts models". United States. doi:10.1088/1751-8113/46/7/075001. https://www.osti.gov/servlets/purl/1240068.
@article{osti_1240068,
title = {Transfer matrix computation of critical polynomials for two-dimensional Potts models},
author = {Jacobsen, Jesper Lykke and Scullard, Christian R.},
abstractNote = {We showed, In our previous work, that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial PB(q, v), henceforth referred to as the critical polynomial. This polynomial may be defined on any periodic two-dimensional lattice. It depends on a finite subgraph B, called the basis, and the manner in which B is tiled to construct the lattice. The real roots v = eK — 1 of PB(q, v) either give the exact critical points for the lattice, or provide approximations that, in principle, can be made arbitrarily accurate by increasing the size of B in an appropriate way. In earlier work, PB(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of PB(q, v), which facilitates its computation, using the transfer matrix, on much larger B than was previously possible.We present results for the critical polynomial on the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162, and 243 edges, compared to the limit of 36 edges with contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. The critical temperatures vc obtained for ferromagnetic (v > 0) Potts models are at least as precise as the best available results from Monte Carlo simulations or series expansions. For instance, with q = 3 we obtain vc(4, 82) = 3.742 489 (4), vc(kagome) = 1.876 459 7 (2), and vc(3, 122) = 5.033 078 49 (4), the precision being comparable or superior to the best simulation results. More generally, we trace the critical manifolds in the real (q, v) plane and discuss the intricate structure of the phase diagram in the antiferromagnetic (v < 0) region.},
doi = {10.1088/1751-8113/46/7/075001},
journal = {Journal of Physics. A, Mathematical and Theoretical (Online)},
number = 7,
volume = 46,
place = {United States},
year = {2013},
month = {2}
}

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Works referenced in this record:

Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities
journal, May 1984

  • Andrews, George E.; Baxter, R. J.; Forrester, P. J.
  • Journal of Statistical Physics, Vol. 35, Issue 3-4
  • DOI: 10.1007/bf01014383

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


Common structures between finite systems and conformal field theories through quantum groups
journal, January 1990


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


The antiferromagnetic transition for the square-lattice Potts model
journal, May 2006


A bond percolation critical probability determination based on the star-triangle transformation
journal, May 1984


Some generalized order-disorder transformations
journal, January 1952

  • Potts, R. B.
  • Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 48, Issue 1
  • DOI: 10.1017/s0305004100027419

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


Potts model at the critical temperature
journal, November 1973


Critical point of planar Potts models
journal, September 1979


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


On the random-cluster model
journal, February 1972


Critical behaviour of random-bond Potts models: a transfer matrix study
journal, April 1998


Spanning Forests and the q-State Potts Model in the Limit q →0
journal, June 2005

  • Jacobsen, Jesper Lykke; Salas, Jesús; Sokal, Alan D.
  • Journal of Statistical Physics, Vol. 119, Issue 5-6
  • DOI: 10.1007/s10955-005-4409-y

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
  • DOI: 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
  • DOI: 10.1088/1751-8113/45/49/494004

Critical behaviour of the two-dimensional Potts model with a continuous number of states; A finite size scaling analysis
journal, June 1982

  • Blöte, H. W. J.; Nightingale, M. P.
  • Physica A: Statistical Mechanics and its Applications, Vol. 112, Issue 3
  • DOI: 10.1016/0378-4371(82)90187-x

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


The Potts model
journal, January 1982


Some generalized order-disorder transformations
journal, January 1952

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

The Potts model
journal, January 1982


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


Percolation critical polynomial as a graph invariant
journal, October 2012


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


Common structures between finite systems and conformal field theories through quantum groups
journal, January 1990


    Works referencing / citing this record:

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


    High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials
    journal, March 2014


    Potts-model critical manifolds revisited
    journal, February 2016

    • Scullard, Christian R.; Jacobsen, Jesper Lykke
    • Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 12
    • DOI: 10.1088/1751-8113/49/12/125003

    On bond percolation threshold bounds for Archimedean lattices with degree three
    journal, June 2017


    On the growth constant for square-lattice self-avoiding walks
    journal, November 2016

    • Jacobsen, Jesper Lykke; Scullard, Christian R.; Guttmann, Anthony J.
    • Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 49
    • DOI: 10.1088/1751-8113/49/49/494004