Hostname: page-component-cb9f654ff-5jtmz Total loading time: 0 Render date: 2025-08-23T19:42:36.757Z Has data issue: false hasContentIssue false
  • English
  • Français

Lee-Yang zeros of the antiferromagnetic Ising model

Part of: Complex dynamical systems Graph theory

Published online by Cambridge University Press:  08 April 2021

FERENC BENCS ,
PJOTR BUYS ,
LORENZO GUERINI  and
HAN PETERS
FERENC BENCS
Affiliation:
Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13–15, 1053 Budapest, Hungary Department of Mathematics, Central European University, Nádor u. 9, 1051 Budapest, Hungary Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, 1090GE Amsterdam, The Netherlands (e-mail: ferenc.bencs@gmail.com, pjotr.buys@gmail.com, hanpeters77@gmail.com)
PJOTR BUYS
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, 1090GE Amsterdam, The Netherlands (e-mail: ferenc.bencs@gmail.com, pjotr.buys@gmail.com, hanpeters77@gmail.com)
LORENZO GUERINI*
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, 1090GE Amsterdam, The Netherlands (e-mail: ferenc.bencs@gmail.com, pjotr.buys@gmail.com, hanpeters77@gmail.com)
HAN PETERS
Affiliation:
Korteweg de Vries Institute for Mathematics, University of Amsterdam, Science Park 107, 1090GE Amsterdam, The Netherlands (e-mail: ferenc.bencs@gmail.com, pjotr.buys@gmail.com, hanpeters77@gmail.com)
*
e-mail: lorenzo.guerini92@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.

Keywords

Ising modelLee–Yang zeroscomplex dynamics

MSC classification

Primary: 05C31: Graph polynomials 37F10: Polynomials; rational maps; entire and meromorphic functions

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 7 , July 2022 , pp. 2172 - 2206
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Barvinok, A.. Combinatorics and Complexity of Partition Functions (Algorithms and Combinatorics, 30). Springer, Cham, 2016.CrossRefGoogle Scholar
Barata, J. C. A. and Goldbaum, P. S.. On the distribution and gap structure of Lee–Yang zeros for the Ising model: periodic and aperiodic couplings. J. Stat. Phys. 103(5/6) (2001), 857–891.CrossRefGoogle Scholar
Barata, J. C. A. and Marchetti, D. H. U.. Griffiths’ singularities in diluted Ising models on the Cayley tree. J. Stat. Phys. 88(1–2), (1997), 231–268.CrossRefGoogle Scholar
Chio, I., He, C., Ji, A. L. and Roeder, R. K. W.. Limiting measure of Lee–Yang zeros for the Cayley tree. Comm. Math. Phys. 370(3) (2019), 925–957.CrossRefGoogle Scholar
Dujardin, R. and Favre, C.. Distribution of rational maps with a preperiodic critical point. Amer. J. Math. 130(4) (2008), 979–1032.CrossRefGoogle Scholar
Levin, G. M.. Irregular values of the parameter of a family of polynomial mappings. Uspekhi Mat. Nauk 36(6(222)) (1981), 219–220.Google Scholar
Lee, T. D. and Yang, C. N.. Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87(2) (1952), 404–409.CrossRefGoogle Scholar
Lee, T. D. and Yang, C. N.. Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87(2) (1952), 410–419.CrossRefGoogle Scholar
Lyubich, M. Y.. Some typical properties of the dynamics of rational mappings. Uspekhi Mat. Nauk 38(5(233)) (1983), 197–198.Google Scholar
McMullen, C. T.. The Mandelbrot set is universal. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274). Cambridge University Press, Cambridge, 2000, pp. 1–17.Google Scholar
McMullen, C. T.. Complex Dynamics and Renormalization (Annals of Mathematics Studies, 135). Princeton University Press, Princeton, NJ, 1994.Google Scholar
Müller-Hartmann, E.. Theory of the Ising model on a Cayley tree. Z. Phys. B Condens. Matter 27(2) (1977), 161–168.Google Scholar
Müller-Hartmann, E. and Zittartz, J.. Phase transitions of continuous order: Ising model on a Cayley tree. Z. Phys. B Condens. Matter 22(1) (1975), 59–67.Google Scholar
Milnor, J.. On rational maps with two critical points. Exp. Math. 9(4) (2000), 481–522.CrossRefGoogle Scholar
Mañé, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4) 16(2) (1983), 193–217.CrossRefGoogle Scholar
Patel, V. and Regts, G.. Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. SIAM J. Comput. 46(6) (2017), 1893–1919.CrossRefGoogle Scholar
Peters, H. and Regts, G.. On a conjecture of Sokal concerning roots of the independence polynomial. Michigan Math. J. 68(1) (2019), 33–35.CrossRefGoogle Scholar
Peters, H. and Regts, G.. Location of zeros for the partition function of the Ising model on bounded degree graphs. J. Lond. Math. Soc. (2) 101(2) (2020), 795–785.CrossRefGoogle Scholar
Slodkowski, Z.. Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc. 111(2) (1991), 347–355.CrossRefGoogle Scholar
Sumi, H.. On dynamics of hyperbolic rational semigroups. J. Math. Kyoto Univ. 37(4) (1997), 717–733.Google Scholar
Sumi, H.. On Hausdorff dimension of Julia sets of hyperbolic rational semigroups. Kodai Math. J. 21(1) (1998), 10–28.CrossRefGoogle Scholar