zbMATH — the first resource for mathematics

Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function. (English) Zbl 1405.82006
Summary: A streamlined derivation of the Kac-Ward formula for the planar Ising model’s partition function is presented and applied in relating the kernel of the Kac-Ward matrices’ inverse with the correlation functions of the Ising model’s order-disorder correlation functions. A shortcut for both is facilitated by the Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also extended here to produce a family of non planar interactions on \(\mathbb{Z}^2\) for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature.

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
Full Text: DOI
[1] Aizenman, M.; Barsky, DJ; Fernández, R., The phase transition in a general class of Ising-type models is sharp, J. Stat. Phys., 47, 343-374, (1987)
[2] Aizenman, M.; Lainz Valcazar, M.; Warzel, S., Pfaffian correlation functions of planar dimer covers, J. Stat. Phys., 166, 1078-1091, (2017) · Zbl 1360.82016
[3] Aizenman, M., Duminil-Copin, H., Tassion, V., Warzel, S.: Emergent planarity in two-dimensional Ising models with finite-range interactions. arXiv:1801.04960
[4] Burgoyne, PN, Remarks on the combinatorial approach to the Ising problem, J. Math. Phys., 4, 1320, (1963) · Zbl 0151.46602
[5] Bowen, R.; Lanford, O., Zeta functions of restrictions of the shift transformation, Proc. Symp. Pure Math., 14, 43-50, (1970) · Zbl 0211.56501
[6] Cimasoni, D., A generalized Kac-Ward formula, J. Stat. Mech., 2010, p07023, (2010)
[7] Cimasoni, D., The critical Ising model via Kac-Ward matrices, Commun. Math. Phys., 316, 99-126, (2012) · Zbl 1253.82011
[8] Chelkak, D.; Cimasoni, D.; Kassel, A., Revisiting the combinatorics of the 2D Ising model, Ann. Inst. Henri Poincaré D, 4, 309-385, (2016) · Zbl 1380.82017
[9] Chelkak, D.; Smirnov, S., Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math., 189, 515-580, (2012) · Zbl 1257.82020
[10] Dotsenko, VS; Dotsenko, VS, Critical behaviour of the phase transition in the 2D Ising model with impurities, Adv. Phys., 32, 129-172, (1983)
[11] Dubedat, J.: Exact bosonization of the Ising model. arXiv:1112.4399 · Zbl 1204.60079
[12] Duminil-Copin, H.: Graphical representations of lattice spin models. In: Lecture Notes (Spartacus-idh, 2015)
[13] Duminil-Copin, H.; Raoufi, A.; Tassion, V., A new computation of the critical point for the planar random-cluster model with \(q \ge 1\)., Ann. Inst. Henri Poincaré, Probab. Stat., 54, 422-436, (2016) · Zbl 1395.82043
[14] Duminil-Copin, H., Raoufi, A., Tassion, V.: Sharp phase transition for the random-cluster and Potts models via decision trees. arXiv:1705.03104 (2017) · Zbl 1395.82043
[15] El-Showk, S.; Paulos, MF; Poland, D.; Rychkov, S.; Simmons-Duffin, D.; Vichi, A., Solving the 3d Ising model with the conformal bootstrap II. c-Minimization and precise critical exponents, J. Stat. Phys., 157, 869-914, (2014) · Zbl 1310.82013
[16] Feynman, R.P.: Statistical Mechanics. A Set of Lectures. Benjamin Publishing, Reading (1972)
[17] Fisher, ME, On the dimer solution of planar Ising models, J. Math. Phys., 7, 1776-1781, (1966)
[18] Grimmett, G.: The Random Cluster Model. Springer, New York (2006) · Zbl 1122.60087
[19] Groeneveld, J.; Boel, RJ; Kasteleyn, PW, Correlation-function identities for general planar Ising systems, Physica, 93A, 138-154, (1978)
[20] Hurst, CA; Green, HS, New solution of the Ising problem for a rectangular lattice, J. Chem. Phys., 33, 1059-1062, (1960)
[21] Kadanoff, LP, Spin-spin correlation in the two-dimensional Ising model, Nuovo Cimento, 44, 276-305, (1966)
[22] Kadanoff, LP; Ceva, H., Determination of an operator algebra for the two-dimensional Ising model, Phys. Rev. B, 3, 3918-3939, (1971)
[23] Kasteleyn, PW, Dimer statistics and phase transitions, J. Math. Phys., 4, 287-293, (1963)
[24] Kaufman, B., Crystal statistics. II. Partition function evaluated by spinor analysis, Phys. Rev., 76, 1232-1243, (1949) · Zbl 0035.42801
[25] Kac, M.; Ward, JC, A combinatorial solution of the two-dimensional Ising model, Phys. Rev., 88, 1332-1337, (1952) · Zbl 0048.45804
[26] Kager, W.; Lis, M.; Meester, R., The signed loop approach to the Ising model: foundations and critical point, J. Stat. Phys., 152, 353-387, (2013) · Zbl 1276.82009
[27] Lis, M., The fermionic observable in the Ising model and the inverse Kac-Ward operator, Ann. H. Poincaré, 15, 1945-1965, (2014) · Zbl 1305.82017
[28] Lis, M., A short proof of the Kac-Ward formula, Ann. H. Poincaré D, 3, 45-53, (2015) · Zbl 1331.05021
[29] McCoy, B., Wu, T.T.: The two-dimensional Ising model, 1973. Dover Pub, Mineola, NY (2014)
[30] McCoy, B.; Perk, JH; Wu, TT, Ising field theory: quadratic difference equations for the \(n\)-point Green’s functions on the lattice, Phys. Rev. Lett., 46, 757-760, (1981)
[31] Onsager, L., Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev., 65, 117-149, (1944) · Zbl 0060.46001
[32] Oren, I.; Godel, A.; Smilansky, U., Trace formulae and spectral statistics for discrete Laplacians on regular graphs (I), J. Phys. A. Math. Theor., 42, 415101, (2009) · Zbl 1179.81083
[33] Palmer, J.: Planar Ising Correlations. Birkhauser, Boston, MA (2007) · Zbl 1136.82001
[34] Potts, RB; Ward, JC, The combinatorial method and the two-dimensional Ising model, Prog. Theor. Phys., 13, 38-46, (1955) · Zbl 0065.24001
[35] Ruelle, D.: Statistical Mechanics: Rigorous Results. (W.A. Benjamin 1969, reprinted World Scientific, Singapore, 1999) · Zbl 0177.57301
[36] Smirnov, S., Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model, Ann. Math., 172, 1435-1467, (2010) · Zbl 1200.82011
[37] Schultz, T.; Mattis, D.; Lieb, E., Two dimensional Ising model as a soluble problem of many fermions, Rev. Mod. Phys., 36, 856, (1964)
[38] Sherman, S., Combinatorial aspects of the Ising model for ferromagnetism. I. A conjecture of Feynman on paths and graphs, J. Math. Phys., 1, 202-217, (1960) · Zbl 0123.45501
[39] Stark, HM; Terras, AA, Zeta functions of finite graphs and coverings, Adv. Math., 121, 124-165, (1996) · Zbl 0874.11064
[40] Whitney, H., On regular closed curves in the plane, Compos. Math., 4, 276-284, (1937) · Zbl 0016.13804
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.