Magnetization in the zig-zag layered Ising model and orthogonal polynomials

Chelkak, Dmitry; Hongler, Clément; Mahfouf, Rémy

doi:10.5802/aif.3605

Magnetization in the zig-zag layered Ising model and orthogonal polynomials
[Magnétisation du modèle d’Ising par couches en zig-zag et polynômes orthogonaux]

Chelkak, Dmitry ^1,² ; Hongler, Clément ³ ; Mahfouf, Rémy ⁴

¹ ENS–MHI Chair, Département de mathématiques et applications, École Normale Supérieure, CNRS, PSL University, 45 rue d’Ulm, 75005 Paris (France)
² St. Petersburg Dept. of Steklov Mathematical Institute RAS, Fontanka 27, 191023 St. Petersburg (Russia)
³ Chair of Statistical Field Theory, MATHAA Institute, École Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne (Switzerland)
⁴ Département de mathématiques et applications, École Normale Supérieure, CNRS, PSL University, 45 rue d’Ulm, 75005 Paris (France)

Annales de l'Institut Fourier, Online first, 56 p.

Résumé
Abstract

Nous étudions la magnétisation $M_{m}$ de la m-ième colonne du modèle d’Ising planaire par couches en zig-zag sur le demi-plan en utilisant les fermions de Kadanoff–Ceva et les polynômes orthogonaux. Notre résultat principal exprime $M_{m}$ comme un déterminant de Hankel $m \times m$ construit à partir de la mesure spectrale d’un certain opérateur de Jacobi encodant les interactions entre colonnes successives. Nous illustrons aussi notre approche en donnant des preuves courtes des résultats classiques pour le modèle homogène de Kaufman–Onsager–Yang et McCoy–Wu, et exprimons $M_{m}$ comme un déterminant Toeplitz+Hankel dans le cadre du modèle homogène sous-critique en présence d’un champ magnétique extérieur au bord.

We discuss the magnetization $M_{m}$ in the $m$ -th column of the zig-zag layered 2D Ising model on a half-plane using Kadanoff–Ceva fermions and orthogonal polynomials techniques. Our main result gives an explicit representation of $M_{m}$ via $m \times m$ Hankel determinants constructed from the spectral measure of a certain Jacobi matrix which encodes the interaction parameters between the columns. We also illustrate our approach by giving short proofs of the classical Kaufman–Onsager–Yang and McCoy–Wu theorems in the homogeneous setup and expressing $M_{m}$ as a Toeplitz+Hankel determinant for the homogeneous sub-critical model in presence of a boundary magnetic field.

Reçu le : 2021-03-05
Révisé le : 2022-10-04
Accepté le : 2022-12-16
Première publication : 2024-05-17

DOI : 10.5802/aif.3605

Classification : 82B20, 47B36, 33C47
Keywords: Planar Ising model, magnetization, discrete fermions, orthogonal polynomials, Hankel determinants, Toeplitz+Hankel determinants.
Mot clés : Modèle d’Ising planaire, fermions discrets, polynômes orthogonaux, déterminants de Hankel, déterminants Toeplitz+Hankel.

Affiliations des auteurs :

Chelkak, Dmitry ^{1, 2} ; Hongler, Clément ³ ; Mahfouf, Rémy ⁴

@unpublished{AIF_0__0_0_A86_0,
     author = {Chelkak, Dmitry and Hongler, Cl\'ement and Mahfouf, R\'emy},
     title = {Magnetization in the zig-zag layered {Ising} model and orthogonal polynomials},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3605},
     language = {en},
     note = {Online first},
}

TY  - UNPB
AU  - Chelkak, Dmitry
AU  - Hongler, Clément
AU  - Mahfouf, Rémy
TI  - Magnetization in the zig-zag layered Ising model and orthogonal polynomials
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3605
LA  - en
ID  - AIF_0__0_0_A86_0
ER  -

%0 Unpublished Work
%A Chelkak, Dmitry
%A Hongler, Clément
%A Mahfouf, Rémy
%T Magnetization in the zig-zag layered Ising model and orthogonal polynomials
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3605
%G en
%F AIF_0__0_0_A86_0

Chelkak, Dmitry; Hongler, Clément; Mahfouf, Rémy. Magnetization in the zig-zag layered Ising model and orthogonal polynomials. Annales de l'Institut Fourier, Online first, 56 p.

Bibliographie
Cité par

[1] Au-Yang, Helen Criticality in alternating layered Ising models. II. Exact scaling theory, Phys. Rev. E, Volume 88 (2013), 032148 | DOI

[2] Au-Yang, Helen; McCoy, Barry M. Theory of layered Ising models. II. Spin correlation functions parallel to the layering, Phys. Rev. B, Volume 10 (1974), pp. 3885-3905 | DOI

[3] Au-Yang, Helen; McCoy, Barry M. Theory of layered Ising models: Thermodynamics, Phys. Rev. B, Volume 10 (1974), pp. 886-891 | DOI

[4] Au-Yang, Helen; Perk, Jacques H. H. Critical correlations in a $Z$ -invariant inhomogeneous Ising model, Physica A, Volume 144 (1987) no. 1, pp. 44-104 | DOI | MR

[5] Basor, Estelle L.; Chen, Yang; Haq, Nazmus S. Asymptotics of determinants of Hankel matrices via non-linear difference equations, J. Approx. Theory, Volume 198 (2015), pp. 63-110 | DOI | MR | Zbl

[6] Baxter, Rodney J. Exactly solved models in statistical mechanics, Academic Press Inc., 1989, xii+486 pages (reprint of the 1982 original) | MR | Zbl

[7] Baxter, Rodney J. Onsager and Kaufman’s calculation of the spontaneous magnetization of the Ising model, J. Stat. Phys., Volume 145 (2011) no. 3, pp. 518-548 | DOI | MR | Zbl

[8] Baxter, Rodney J. Onsager and Kaufman’s calculation of the spontaneous magnetization of the Ising model: II, J. Stat. Phys., Volume 149 (2012) no. 6, pp. 1164-1167 | DOI | MR | Zbl

[9] Baxter, Rodney J.; Enting, Ian G. 399th solution of the Ising model, J. Phys. A. Math. Gen., Volume 11 (1978) no. 12, p. 2463 | DOI

[10] Beffara, Vincent; Duminil-Copin, Hugo Smirnov’s fermionic observable away from criticality, Ann. Probab., Volume 40 (2012) no. 6, pp. 2667-2689 | DOI | MR | Zbl

[11] Boutillier, Cédric; de Tilière, Béatrice; Raschel, Kilian The $Z$ -invariant Ising model via dimers, Probab. Theory Relat. Fields, Volume 174 (2019) no. 1-2, pp. 235-305 | DOI | MR | Zbl

[12] Chelkak, Dmitry 2D Ising model: correlation functions at criticality via Riemann-type boundary value problems, European Congress of Mathematics, European Mathematical Society, 2018, pp. 235-256 | DOI | MR | Zbl

[13] Chelkak, Dmitry Planar Ising model at criticality: state-of-the-art and perspectives, Proceedings of the International Congress of Mathematicians 2018 (ICM 2018), Vol. 3, World Scientific, 2019, pp. 2789-2816 | DOI | Zbl

[14] Chelkak, Dmitry Ising model and s-embeddings of planar graphs (2020) (https://arxiv.org/abs/2006.14559)

[15] Chelkak, Dmitry; Cimasoni, David; Kassel, Adrien Revisiting the combinatorics of the 2D Ising model, Ann. Inst. Henri Poincaré D, Comb. Phys. Interact., Volume 4 (2017) no. 3, pp. 309-385 | DOI | MR | Zbl

[16] Chelkak, Dmitry; Hongler, Clément; Izyurov, Konstantin Correlations of primary fields in the critical Ising model (2021) (https://arxiv.org/abs/2103.10263)

[17] Chelkak, Dmitry; Izyurov, Konstantin; Mahfouf, Rémy Universality of spin correlations in the Ising model on isoradial graphs (2021) (https://arxiv.org/abs/2104.12858)

[18] Chelkak, Dmitry; Smirnov, Stanislav Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math., Volume 189 (2012) no. 3, pp. 515-580 | DOI | MR | Zbl

[19] Cimasoni, David; Duminil-Copin, Hugo The critical temperature for the Ising model on planar doubly periodic graphs, Electron. J. Probab., Volume 18 (2013), 44, 18 pages | DOI | MR | Zbl

[20] Comets, Francis; Giacomin, Giambattista; Greenblatt, Rafael L. Continuum limit of random matrix products in statistical mechanics of disordered systems, Commun. Math. Phys., Volume 369 (2019) no. 1, pp. 171-219 | DOI | MR | Zbl

[21] Deift, Percy; Its, Alexander; Krasovsky, Igor Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher–Hartwig singularities, Ann. Math., Volume 174 (2011) no. 2, pp. 1243-1299 | DOI | MR | Zbl

[22] Deift, Percy; Its, Alexander; Krasovsky, Igor Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results, Commun. Pure Appl. Math., Volume 66 (2013) no. 9, pp. 1360-1438 | DOI | MR | Zbl

[23] Dotsenko, Viktor S.; Dotsenko, Vladimir S. Critical behaviour of the phase transition in the $2$ D Ising model with impurities, Adv. Phys., Volume 32 (1983) no. 2, pp. 129-172 | DOI | MR

[24] Duminil-Copin, Hugo; Smirnov, Stanislav Conformal invariance of lattice models, Probability and statistical physics in two and more dimensions (Clay Mathematics Proceedings), Volume 15, American Mathematical Society, 2012, pp. 213-276 | MR | Zbl

[25] Friedli, Sacha; Velenik, Yvan Statistical mechanics of lattice systems. A concrete mathematical introduction, Cambridge University Press, 2018, xix+622 pages | DOI | MR | Zbl

[26] Fröhlich, Jürg; Pfister, Charles-Edouard Semi-infinite Ising model. II. The wetting and layering transitions, Commun. Math. Phys., Volume 112 (1987) no. 1, pp. 51-74 | DOI | MR | Zbl

[27] Gheissari, Reza; Hongler, Clément; Park, Seong C. Ising model: local spin correlations and conformal invariance, Commun. Math. Phys., Volume 367 (2019) no. 3, pp. 771-833 | DOI | MR | Zbl

[28] Grenander, Ulf; Szegő, Gábor Toeplitz forms and their applications, Chelsea Publishing, 1984, x+245 pages | MR | Zbl

[29] Hongler, Clément; Kytölä, Kalle; Johansson Viklund, Fredrik Conformal Field Theory at the Lattice Level: Discrete Complex Analysis and Virasoro Structure, Commun. Math. Phys., Volume 395 (2022) no. 1, pp. 1-58 | DOI | MR | Zbl

[30] Hongler, Clément; Kytölä, Kalle; Zahabi, Ali Discrete holomorphicity and Ising model operator formalism, Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong (Contemporary Mathematics), Volume 644, American Mathematical Society, 2015, pp. 79-115 | DOI | MR | Zbl

[31] Jimbo, Michio; Miwa, Tetsuji Studies on holonomic quantum fields. XVII, Proc. Japan Acad., Ser. A, Volume 56 (1980) no. 9, pp. 405-410 | DOI | MR

[32] Kadanoff, Leo P.; Ceva, Horacio Determination of an operator algebra for the two-dimensional Ising model, Phys. Rev. B, Volume 3 (1971), pp. 3918-3939 | DOI | MR

[33] Kenyon, Richard The Laplacian and Dirac operators on critical planar graphs, Invent. Math., Volume 150 (2002) no. 2, pp. 409-439 | DOI | MR | Zbl

[34] Kenyon, Richard; Lam, Wai Yeung; Ramassamy, Sanjay; Russkikh, Marianna Dimers and Circle patterns (2018) (https://arxiv.org/abs/1810.05616)

[35] Li, Jhih-Huang; Mahfouf, Rémy Conformal invariance in the quantum Ising model (2021) (https://arxiv.org/abs/2112.04811)

[36] McCoy, Barry M. Theory of a two-dimensional Ising model with random impurities. III. Boundary effects, Phys. Rev., Volume 188 (1969), pp. 1014-1031 | DOI | MR

[37] McCoy, Barry M. Integrable models in statistical mechanics: the hidden field with unsolved problems, Int. J. Mod. Phys. A, Volume 14 (1999) no. 25, pp. 3921-3933 | DOI | MR | Zbl

[38] McCoy, Barry M.; Maillard, Jean-Marie The Importance of the Ising Model, Prog. Theor. Phys., Volume 127 (2012) no. 5, pp. 791-817 | DOI | Zbl

[39] McCoy, Barry M.; Perk, Jacques H. H.; Wu, Tai Tsun Ising field theory: quadratic difference equations for the $n$ -point Green’s functions on the lattice, Phys. Rev. Lett., Volume 46 (1981) no. 12, pp. 757-760 | DOI | MR

[40] McCoy, Barry M.; Wu, Tai Tsun Theory of a two-dimensional Ising model with random impurities. I. Thermodynamics, Phys. Rev., Volume 176 (1968), pp. 631-643 | DOI | MR

[41] McCoy, Barry M.; Wu, Tai Tsun Theory of a two-dimensional Ising model with random impurities. II. Spin correlation functions, Phys. Rev., Volume 188 (1969), pp. 982-1013 | DOI | MR

[42] McCoy, Barry M.; Wu, Tai Tsun The two-dimensional Ising model, Dover Publications, 2014, xvi+454 pages corrected reprint, with a new preface and a new chapter (Chapter XVII) | MR | Zbl

[43] Mercat, Christian Discrete Riemann surfaces and the Ising model, Commun. Math. Phys., Volume 218 (2001) no. 1, pp. 177-216 | DOI | MR | Zbl

[44] Messikh, R. J. The surface tension near criticality of the 2d-Ising model (2006) (https://arxiv.org/abs/math/0610636)

[45] Niss, Martin History of the Lenz-Ising model 1920–1950: from ferromagnetic to cooperative phenomena, Arch. Hist. Exact Sci., Volume 59 (2005) no. 3, pp. 267-318 | DOI | MR | Zbl

[46] Niss, Martin History of the Lenz-Ising model 1920–1950: from ferromagnetic to cooperative phenomena, Arch. Hist. Exact Sci., Volume 59 (2005) no. 3, pp. 267-318 | DOI | MR | Zbl

[47] Niss, Martin History of the Lenz-Ising model 1950–1965: from irrelevance to relevance, Arch. Hist. Exact Sci., Volume 63 (2009) no. 3, pp. 243-287 | DOI | MR | Zbl

[48] Palmer, John Planar Ising correlations, Progress in Mathematical Physics, 49, Birkhäuser, 2007, xx+363 pages | MR | Zbl

[49] Pelizzola, Alessandro Boundary critical behaviour of two-dimensional layered Ising models, Int. J. Mod. Phys. B, Volume 11 (1997) no. 11, pp. 1363-1388 | DOI | Zbl

[50] Perk, Jacques H. H. Nonlinear partial difference equations for Ising model n-point Green’s functions, Proc. II International Symposium on Selected Topics in Statistical Mechanics, Dubna, August 25–29, 1981, 1981, pp. 165-180 (https://perk.okstate.edu/papers/older/Dubna1.pdf)

[51] Perk, Jacques H. H.; Au-Yang, Helen Ising models and soliton equations, III international symposium on selected topics in statistical mechanics, Vol. II (Dubna, 1984) (Ob”ed. Inst. Yadernykh Issled. Dubna, D17-84-850), Ob”ed. Inst. Yadernykh Issled., 1985, pp. 138-151 | MR

[52] Perk, Jacques H. H.; Au-Yang, Helen New results for the correlation functions of the Ising model and the transverse Ising chain, J. Stat. Phys., Volume 135 (2009), pp. 599-619 | DOI | MR | Zbl

[53] Pfister, Charles-Edouard; Velenik, Yvan Mathematical theory of the wetting phenomenon in the 2D Ising model. Papers honouring the 60th birthday of Klaus Hepp and of Walter Hunziker, Part I (Zürich, 1995), Helv. Phys. Acta, Volume 69 (1996) no. 5-6, pp. 949-973 | MR | Zbl

[54] Sato, Mikio; Miwa, Tetsuji; Jimbo, Michio Studies on holonomic quantum fields. I-IV, Proc. Japan Acad., Ser. A, Volume 53 (1977) no. 1, p. 6-10, 147-152, 153-158, 183-185 | MR

[55] Schultz, Theodore D.; Mattis, Daniel C.; Lieb, Elliott H. Two-dimensional Ising model as a soluble problem of many fermions, Rev. Mod. Phys., Volume 36 (1964), pp. 856-871 | DOI | MR

[56] Simon, Barry OPUC on one foot, Bull. Am. Math. Soc., Volume 42 (2005) no. 4, pp. 431-460 | DOI | MR | Zbl

[57] Witte, Nicholas S. Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model, J. Phys. A. Math. Theor., Volume 40 (2007) no. 24, p. f491-f501 | DOI | MR | Zbl

[58] Wu, Tai Tsun; McCoy, Barry M.; Tracy, Craig A.; Barouch, Eytan Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region, Phys. Rev. B, Volume 13 (1976) no. 1, pp. 316-374 | DOI

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT