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The Critical Ising Model via Kac-Ward Matrices

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Abstract

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 22g matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer \({\overline{\partial}}\)-torsions of the Riemann surface in which G embeds.

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References

  1. Alvarez-Gaumé L., Moore G., Vafa C.: Theta functions, modular invariance, and strings. Commun. Math. Phys. 106(1), 1–40 (1986)

    Article  ADS  MATH  Google Scholar 

  2. Atiyah M.F.: Riemann surfaces and spin structures. Ann. Sci. École Norm. Sup. (4) 4, 47–62 (1971)

    MathSciNet  MATH  Google Scholar 

  3. Bass H.: The Ihara-Selberg zeta function of a tree lattice. Internat. J. Math 3(6), 717–797 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baxter R.J.: Free-fermion, checkerboard and Z-invariant lattice models in statistical mechanics. Proc. Roy. Soc. London Ser. A 404(1826), 1–33 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  5. Baxter, R.J.: Exactly solved models in statistical mechanics. London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1989. Reprint of the 1982 original

  6. Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: the periodic case. Probab. Th. Rel. Fields 147(3–4), 379–413 (2010)

    Article  MATH  Google Scholar 

  7. Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: locality property. Commun. Math. Phys. 301(2), 473–516 (2011)

    Article  ADS  MATH  Google Scholar 

  8. Cimasoni D.: A generalized Kac-Ward formula. J. Stat. Mech 2010, P07023 (2010)

    Article  Google Scholar 

  9. Cimasoni D.: Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices. J. Eur. Math. Soc. (JEMS) 14(4), 1209–1244 (2012)

    Article  MATH  Google Scholar 

  10. Cimasoni D., Reshetikhin N.: Dimers on surface graphs and spin structures. I. . Comm. Math. Phys. 275(1), 187–208 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Costa-Santos R.: Geometrical aspects of the Z-invariant ising model. Euro. Phys. J. B - Cond. Matt. Com. Sys. 53, 85–90 (2006)

    ADS  Google Scholar 

  12. Costa-Santos R., McCoy B.M.: Dimers and the critical Ising model on lattices of genus > 1. Nuclear Phys. B 623(3), 439–473 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Costa-Santos R., McCoy B.M.: Finite size corrections for the Ising model on higher genus triangular lattices. J. Statist. Phys. 112(5–6), 889–920 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. de Tilière, B.: Critical ising model and cycle rooted spanning forests: an explicit correspondence. Preprint, http://arxiv.org/abs/1012.4836v2 [math-ph], 2010

  15. Dolbilin, N.P., Zinov’ev, Y.M., Mishchenko, A.S., Shtan’ko, M.A., Shtogrin, M.I.: Homological properties of two-dimensional coverings of lattices on surfaces. Funkt. Anal. i Pril., 30(3),19–33, 95 (1996)

    Google Scholar 

  16. Dolbilin N.P., Zinov’ev Y.M., Mishchenko A.S., Shtan’ko M.A., Shtogrin M.I.: The two-dimensional Ising model and the Kac-Ward determinant. Izv. Ross. Akad. Nauk Ser. Mat. 63(4), 79–100 (1999)

    MathSciNet  Google Scholar 

  17. Eckmann B.: Harmonische Funktionen und Randwertaufgaben in einem Komplex. Comment. Math. Helv. 17, 240–255 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fisher M.E.: Statistical mechanics of dimers on a plane lattice. Phys. Rev. 124(6), 1664–1672 (1961)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Fisher M.E.: On the dimer solution of planar Ising models. J. Mathe. Physi. 7(10), 1776–1781 (1966)

    Article  ADS  Google Scholar 

  20. Foata D., Zeilberger D.: A combinatorial proof of Bass’s evaluations of the Ihara-Selberg zeta function for graphs. Trans. Amer. Math. Soc. 351(6), 2257–2274 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Forman R.: Determinants of Laplacians on graphs. Topology 32(1), 35–46 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Galluccio, A., Loebl, M.: On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electron. J. Combin 6, Research Paper 6, 18 pp. (electronic) (1999)

  23. Hurst C.A., Green H.S.: New solution of the Ising problem for a rectangular lattice. J. Chem. Phys. 33(4), 1059–1062 (1960)

    Article  MathSciNet  ADS  Google Scholar 

  24. Johnson D.: Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2) 22(2), 365–373 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Kac M., Ward J.C.: A combinatorial solution of the two-dimensional Ising model. Phys. Rev. 88, 1332–1337 (1952)

    Article  ADS  MATH  Google Scholar 

  26. Kasteleyn P.W.: The statistics of dimers on a lattice. Physica 27, 1209–1225 (1961)

    Article  ADS  MATH  Google Scholar 

  27. Kenyon R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150(2), 409–439 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kenyon R., Okounkov A.: Planar dimers and Harnack curves. Duke Math. J. 131(3), 499–524 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kenyon, R., Schlenker, J.-M.: Rhombic embeddings of planar quad-graphs. Trans. Amer. Math. Soc. 357(9), 3443–3458 (electronic) (2005)

    Google Scholar 

  30. Kirchhoff G.: Über die Auflsung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72, 497–508 (1847)

    ADS  Google Scholar 

  31. Kramers H.A., Wannier G.H.: Statistics of the two-dimensional ferromagnet. I. Phys Rev. (2) 60, 252–262 (1941)

    Article  MathSciNet  ADS  Google Scholar 

  32. Loebl, M.: A discrete non-Pfaffian approach to the Ising problem. In: Graphs, Morphisms and Statistical Physics, Volume 63 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Providence, RI: Amer. Math. Soc., 2004, pp. 145–154

  33. Loebl M., Masbaum G.: On the optimality of the Arf invariant formula for graph polynomials. Adv. Math. 226(1), 332–349 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. McCoy, B.M., Wu, T.T.: The two-dimensional Ising model. Cambridge, MA: Harvard University Press, 1973

  35. Ray D.B., Singer I.M.: Analytic torsion for complex manifolds. Ann. of Math. (2) 98, 154–177 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sarnak P.: Determinants of Laplacians. Commun. Math. Phys. 110(1), 113–120 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. Serre J.-P.: Arbres, amalgames SL2. Paris: soc. Math. de France, 1977, Avec un sommaire anglais, édigé avec la collaboration de H. Bass, Astérisque, No. 46

  38. Tesler G.: Matchings in graphs on non-orientable surfaces. J. Comb. Th. Ser. B 78(2), 198–231 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Troyanov M.: Les surfaces euclidiennes à singularités coniques. Enseign. Math. (2) 32(1–2), 79–94 (1986)

    MathSciNet  MATH  Google Scholar 

  40. van der Waerden B.L.: Die lange Reichweite der regelmassigen Atomanordnung in Mischkristallen. Z. Physik 118, 473–488 (1941)

    Article  ADS  Google Scholar 

  41. Wannier G.H.: The statistical problem in cooperative phenomena. Rev. Mod. Phys. 17(1), 50–60 (1945)

    Article  ADS  Google Scholar 

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  1. Section de Mathématiques, Université de Genève, 2-4 Rue du Lièvre, 1211, Genève 4, Switzerland

    David Cimasoni

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  1. David Cimasoni
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Correspondence to David Cimasoni.

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Communicated by H. Spohn

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Cimasoni, D. The Critical Ising Model via Kac-Ward Matrices. Commun. Math. Phys. 316, 99–126 (2012). https://doi.org/10.1007/s00220-012-1575-z

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  • DOI: https://doi.org/10.1007/s00220-012-1575-z

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