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

Chelkak, Dmitry; Hongler, Clément; Mahfouf, Rémy

doi:10.5802/aif.3605

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, Volume 74 (2024) no. 6, pp. 2275-2330.

Abstract
Résumé

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.

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.

Received: 2021-03-05
Revised: 2022-10-04
Accepted: 2022-12-16
Published online: 2024-10-11

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.

Author's affiliations:

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

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Magnetization in the zig-zag layered {Ising} model and orthogonal polynomials},
     journal = {Annales de l'Institut Fourier},
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Chelkak, Dmitry; Hongler, Clément; Mahfouf, Rémy. Magnetization in the zig-zag layered Ising model and orthogonal polynomials. Annales de l'Institut Fourier, Volume 74 (2024) no. 6, pp. 2275-2330. doi : 10.5802/aif.3605. https://aif.centre-mersenne.org/articles/10.5802/aif.3605/

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