The square lattice Ising model on the rectangle III: Hankel and Toeplitz determinants

TL;DR

This paper expresses the partition function of the anisotropic square lattice Ising model on a rectangle with open boundaries as a Hankel determinant or a Toeplitz Pfaffian, facilitating analysis of finite-size effects.

Contribution

It demonstrates that the partition function can be represented by Hankel and Toeplitz matrices derived from Fourier coefficients of a symbol function, extending previous results and enabling finite-size scaling analysis.

Findings

Partition function expressed as Hankel determinant and Toeplitz Pfaffian.

Matrix elements are Fourier coefficients of a ratio of characteristic polynomials.

Results applicable to various boundary conditions and critical scaling analysis.

Abstract

Based on the results obtained in [Hucht, J. Phys. A: Math. Theor. 50, 065201 (2017)], we show that the partition function of the anisotropic square lattice Ising model on the $L \times M$ rectangle, with open boundary conditions in both directions, is given by the determinant of a $M/2 \times M/2$ Hankel matrix, that equivalently can be written as the Pfaffian of a skew-symmetric $M \times M$ Toeplitz matrix. The $M-1$ independent matrix elements of both matrices are Fourier coefficients of a certain symbol function, which is given by the ratio of two characteristic polynomials. These polynomials are associated to the different directions of the system, encode the respective boundary conditions, and are directly related through the symmetry of the considered Ising model under exchange of the two directions. The results can be generalized to other boundary conditions and are well suited for the analysis of finite-size scaling functions in the critical scaling limit using Szeg\H{o}'s theorem.