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

Jacobsen, Jesper Lykke; Scullard, Christian R.

doi:10.1088/1751-8113/46/7/075001

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

Journal Article · Mon Feb 04 00:00:00 EST 2013 · Journal of Physics. A, Mathematical and Theoretical (Online)

DOI:https://doi.org/10.1088/1751-8113/46/7/075001· OSTI ID:1240068

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

LPTENS, Ecole Normale Superieure, Paris (France); Univ. Pierre et Marie Curie, Paris (France)
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

We showed, In our previous work, that critical manifolds of the q-state Potts model can be studied by means of a graph polynomial P_B(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 = e^K — 1 of P_B(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, P_B(q, v) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give a probabilistic definition of P_B(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 v_cobtained 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 v_c(4, 8²) = 3.742 489 (4), v_c(kagome) = 1.876 459 7 (2), and v_c(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.

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Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1240068

Report Number(s):: LLNL-JRNL-610232

Journal Information:: Journal of Physics. A, Mathematical and Theoretical (Online), Vol. 46, Issue 7; ISSN 1751-8121

Publisher:: IOP PublishingCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 25 works

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The three-state Potts antiferromagnet on plane quadrangulations 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	journal	July 2018
Three-state Potts model on the centered triangular lattice Fu, Zhe; Guo, Wenan; Blöte, Henk W. J. Physical Review E, Vol. 101, Issue 1 https://doi.org/10.1103/physreve.101.012118	journal	January 2020
Recent advances and open challenges in percolation Araújo, N.; Grassberger, P.; Kahng, B. The European Physical Journal Special Topics, Vol. 223, Issue 11 https://doi.org/10.1140/epjst/e2014-02266-y	journal	October 2014
On bond percolation threshold bounds for Archimedean lattices with degree three Wierman, John C. Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 29 https://doi.org/10.1088/1751-8121/aa76f0	journal	June 2017
On the growth constant for square-lattice self-avoiding walks Jacobsen, Jesper Lykke; Scullard, Christian R.; Guttmann, Anthony J. Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 49 https://doi.org/10.1088/1751-8113/49/49/494004	journal	November 2016
Recent advances and open challenges in percolation Araújo, N.; Grassberger, Peter; Kahng, Byungnam ETH Zurich https://doi.org/10.3929/ethz-b-000097888	text	January 2014
High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials Jacobsen, Jesper Lykke arXiv https://doi.org/10.48550/arxiv.1401.7847	preprint	January 2014
Recent advances and open challenges in percolation Araújo, N. A. M.; Grassberger, P.; Kahng, B. arXiv https://doi.org/10.48550/arxiv.1404.5325	text	January 2014
On the growth constant for square-lattice self-avoiding walks Jacobsen, Jesper Lykke; Scullard, Christian R.; Guttmann, Anthony J. arXiv https://doi.org/10.48550/arxiv.1607.02984	text	January 2016

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