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

## TL;DR

This paper presents a novel method for calculating magnetization in a zig-zag layered 2D Ising model using orthogonal polynomials and Hankel determinants, providing explicit formulas and new proofs of classical theorems.

## Contribution

It introduces an explicit determinant representation of magnetization via spectral measures and Jacobi matrices, extending classical results with new proofs and formulas.

## Key findings

- Explicit magnetization formulas using Hankel determinants.
- Representation of $M_m$ as Toeplitz+Hankel determinants.
- New proofs of Kaufman-Onsager-Yang and McCoy-Wu theorems.

## Abstract

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.