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Journal of the Physical Society of Japan

Spin Correlations and Magnetic Susceptibility of the Two-Dimensional Ising Model

Yoshiaki Tanaka

Published May 15, 1995• 1 citation
Publisher

Abstract

For the Ising model on the two-dimensional lattice with M × N spin sites ( M is a finite number; N →∞), arbitrary two-spin correlations are evaluated exactly. We make use of spinor analysis, which was employed by Kaufman and Onsager. The zero-field magnetic susceptibility χ 0 that is equal to the sum of all two-spin correlations, is calculated.

DOI
10.1143/jpsj.64.1726

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For the Ising model on the two-dimensional lattice with M × N spin sites ( M is a finite number; N →∞), arbitrary two-spin correlations are evaluated exactly. We make use of spinor analysis, which was employed by Kaufman and Onsager. The zero-field magnetic susceptibility χ 0 that is equal to the sum of all two-spin correlations, is calculated.
We calculate the two-point correlation function and magnetic susceptibility in the anisotropic 2D Ising model on a lattice with one infinite and the other finite dimension, along which periodic boundary conditions are imposed. Using exact expressions for a part of lattice form factors, we propose the formulas for arbitrary spin matrix elements, thus providing a possibility to compute all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. The scaling limit of the corresponding expressions is also analyzed.
With the use of a rigorous relation of the correlations in the Ising spin system, an approach to the exact magnetic susceptibility of the two-dimensional Ising antiferromagnet has been attempted. The zero-field magnetic susceptibilities for the rectangular as well as the square antiferromagnets are shown to be expressed in terms of finite near-spin correlations. With some decoupling approximations for many-spin correlations, the result for the square antiferromagnet by the present method is proved to be within 0.35% from that of the series expansion.
The form factor representation of the correlation function of the 2D Ising model on a cylinder is generalized to the case of arbitrary arrangement of correlating spins. The magnetic susceptibility on a lattice, one of whose dimensions ( N ) is finite, is calculated in both the ferromagnetic and the paramagnetic region of the parameters of the model. The structure of singularities of susceptibility in the complex temperature plane at finite values of N and the transition to the thermodynamic limit N →∞ are discussed.
In the Ising model on a helical lattice with period M (in the limit M →∞ of this lattice, the model corresponds to the two-dimensional Ising model), the exact simultaneous equations to calculate physical quantities are derived. The zero field magnetic susceptibility χ 0 , and also the specific heat C are obtained numerically for the system of M ≤11. Furthermore, in the Ising model with up to the next nearest neighbor interactions on the helical lattice, χ 0 and C are calculated for the system of M ≤9. The results obtained are an excellent approximation to the exact χ 0 and C of the two-dimensional systems.