Progress of Theoretical Physics

On the Spin-Spin Correlation Function in the Ising Square Lattice

Keiji Yamada

Published Apr 1, 1983• 14 citations

Abstract

The spin-spin correlation function X(m, n) in the square lattice of Ising model is given by a determinant with large dimension on the basis of the Kaufman solution for the transfer matrix. In the limit of infinitely large lattice, it is reduced to an eigenvalue problem in the form of an integral equation.

DOI: 10.1143/ptp.69.1295

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Progress of Theoretical Physics

On the Spin-Spin Correlation Function in the Ising Square Lattice

Keiji Yamada

Published Apr 1, 1983• 14 citations

Publisher

Abstract

DOI: 10.1143/ptp.69.1295

Related papers

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T<{T}_{c}

and

T>{T}_{c}

. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case

T>{T}_{c}

, but much more complicated for

T<{T}_{c}

. 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

{T}_{c}

T>{T}_{c}

and

T<{T}_{c}

. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where

t

is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-

t

behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.