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Canadian Journal of Physics

Three-site correlation functions of the two-dimensional Ising model

David A. Pink

Published Nov 1, 1968• 9 citations
Publisher

Abstract

Three-site correlation functions of a square lattice of the form [Formula: see text] have been analytically calculated for all temperatures. It has been found necessary to evaluate the inverse of t × t Toeplitz matrix as t → ∞, which has been done by using the Wiener–Hopf technique. The results are used to give an expression for the magnetization at any site in a row when one of the bonds in that row is a defect bond, and comments are made upon applications to decorated lattices.

DOI
10.1139/p68-598

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Three-site correlation functions of a square lattice of the form [Formula: see text] have been analytically calculated for all temperatures. It has been found necessary to evaluate the inverse of t × t Toeplitz matrix as t → ∞, which has been done by using the Wiener–Hopf technique. The results are used to give an expression for the magnetization at any site in a row when one of the bonds in that row is a defect bond, and comments are made upon applications to decorated lattices.
Expressions for many-site correlation functions of a square-lattice Ising model have been derived for the case in which all sites lie in the same row. The results involve expressions for minors of Toeplitz matrices which are obtained by the Wiener-Hopf technique. We derive an expression for the magnetization at any site in a row if a finite number of defect bonds are distributed at given positions in that row; this expression can be used to study the effect of a dislocation. Some remarks about the decoupling of correlation functions are made.
The calculation of two- and four-spin correlations of the [Formula: see text] ferromagnet has been extended to an N = 21 site triangular lattice. By fitting a quadratic in 1/N to the nearest neighbour transverse pair correlations, we have estimated the ground-state energy per bond on the infinite lattice to be E0/3NJ = −0.5326 ± 0.003. We conjecture that the square of the magnetization per site vanishes very sharply as N−0.06. The nearest neighbour longitudinal correlation per bond [Formula: see text] for all two-dimensional lattices.
We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases T<TcT<{T}_{c}T<Tc​ and T>TcT>{T}_{c}T>Tc​. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case T>TcT>{T}_{c}T>Tc​, but much more complicated for T<TcT<{T}_{c}T<Tc​. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature Tc{T}_{c}Tc​.
We study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the same row of the two-dimensional Ising model with the particular value of a purely imaginary magnetic field $H=\frac{1}{2}i\ensuremath{\pi}\mathrm{kT}$. We derive the partition function and the magnetization per spin first proposed by Lee and Yang and find that the correlation function approaches its limiting value exponentially. In contrast to the case of no magnetic field, the approach is shown to be oscillatory rather than monotonic. The first three terms of the asymptotic behavior are explicitly given. The correlation function is obtained by expressing it as a block Toeplitz determinant whose behavior for large separation is studied by solving a pair of coupled Wiener-Hopf sum equations.
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.