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Physical Review Letters

Zero-Field Susceptibility of the Two-Dimensional Ising Model near T c

Eytan Barouch, Barry M. McCoy, Tai Tsun Wu

Published Dec 3, 1973• 238 citations
Publisher

Abstract

We compute exactly the four coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${\ensuremath{\beta}}^{\ensuremath{-}1}\ensuremath{\chi}={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ for the susceptibility of the rectangular two-dimensional Ising model.

DOI
10.1103/physrevlett.31.1409

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Zero-field susceptibility of the antiferromagnetic square Ising lattice

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The first two terms in the expansion near ${\mathit{T}}_{\mathit{C}}$ of the zero-field susceptibility of the antiferromagnetic square Ising lattice are known exactly. We have isolated the next four terms in this expansion by carrying out a differential-approximant analysis of Nickel's 55-term high-temperature ferromagnetic series with the variable v=tanhJ/${\mathit{k}}_{\mathit{B}}$T replaced by -v. The four new terms are found to vary as t, ${\mathit{t}}^{2}$ln\ensuremath{\Vert}t\ensuremath{\Vert}, ${\mathit{t}}^{2}$, and ${\mathit{t}}^{3}$ln\ensuremath{\Vert}t\ensuremath{\Vert} where t=1-${\mathit{T}}_{\mathit{C}}$/T and values for their respective coefficients are obtained. It is established that to order ${\mathit{t}}^{3}$ln\ensuremath{\Vert}t\ensuremath{\Vert}, there are no confluent singularities associated with irrelevant variables for this system. A direct extrapolation of Nickel's series coefficients is also carried out and found to be consistent with the present results.

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The study of the equation of state of the two-dimensional Ising model is initiated by use of the recently calculated $n$-point spin-correlation functions. In the scaling region where $T\ensuremath{\rightarrow}{T}_{c}$ and $h=B{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{15}{8}}(\frac{H}{k{T}_{c}})$ is of order 1 ($B$ is a lattice-dependent constant) the free energy below ${T}_{c}$ is of the form $f(T, h)\ensuremath{-}f(T, 0)=\mathrm{const}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{2}$ $\ensuremath{\Sigma}{n=1}^{\ensuremath{\infty}}(\frac{1}{n!}){C}_{n}^{\ensuremath{-}}{h}^{n}$ where the lattice-dependent constant is chosen so that ${C}_{1}^{\ensuremath{-}}=1$. The constant ${C}_{2}^{\ensuremath{-}}$ has been studied previously. In this paper we study ${C}_{3}^{\ensuremath{-}}$ and ${C}_{4}^{\ensuremath{-}}$ and find as a first approximation $\frac{{C}_{3}^{\ensuremath{-}}}{3!}\ensuremath{\sim}\ensuremath{-}\frac{11}{72{\ensuremath{\pi}}^{2}}$ and $\frac{{C}_{4}^{\ensuremath{-}}}{4!}\ensuremath{\sim}\frac{5}{189\ensuremath{\pi}}+\frac{59}{162{\ensuremath{\pi}}^{3}}$. These are in close agreement with the low-temperature-series values of -0.01538 and 0.0195.

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Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

Tai Tsun Wu, Barry M. McCoy, Craig A. Tracy +1 more

We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for $T>{T}_{c}$ 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 $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.

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Magnetic susceptibility of the two-dimensional Ising antiferromagnet

Yoshiaki Tanaka, Norikiyo Uryû

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.

Physical review. B, Condensed matter•Mar 1, 1980•16 citations
We compute exactly the four coefficients C0\ifmmode±\else\textpm\fi{C}_{0\ifmmode\pm\else\textpm\fi{}}C0\ifmmode±\else\textpm\fi​ and C1\ifmmode±\else\textpm\fi{C}_{1\ifmmode\pm\else\textpm\fi{}}C1\ifmmode±\else\textpm\fi​ in the expansion \ensuremathβ\ensuremath−1\ensuremathχ=C0\ifmmode±\else\textpm\fi∣1\ensuremath−TcT∣\ensuremath−74+C1\ifmmode±\else\textpm\fi∣1\ensuremath−TcT∣\ensuremath−34+O(1){\ensuremath{\beta}}^{\ensuremath{-}1}\ensuremath{\chi}={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)\ensuremathβ\ensuremath−1\ensuremathχ=C0\ifmmode±\else\textpm\fi​∣1\ensuremath−TTc​​∣\ensuremath−47​+C1\ifmmode±\else\textpm\fi​∣1\ensuremath−TTc​​∣\ensuremath−43​+O(1) for the susceptibility of the rectangular two-dimensional Ising model.
The first two terms in the expansion near TC{\mathit{T}}_{\mathit{C}}TC​ of the zero-field susceptibility of the antiferromagnetic square Ising lattice are known exactly. We have isolated the next four terms in this expansion by carrying out a differential-approximant analysis of Nickel's 55-term high-temperature ferromagnetic series with the variable v=tanhJ/kB{\mathit{k}}_{\mathit{B}}kB​T replaced by -v. The four new terms are found to vary as t, t2{\mathit{t}}^{2}t2ln\ensuremath{\Vert}t\ensuremath{\Vert}, t2{\mathit{t}}^{2}t2, and t3{\mathit{t}}^{3}t3ln\ensuremath{\Vert}t\ensuremath{\Vert} where t=1-TC{\mathit{T}}_{\mathit{C}}TC​/T and values for their respective coefficients are obtained. It is established that to order t3{\mathit{t}}^{3}t3ln\ensuremath{\Vert}t\ensuremath{\Vert}, there are no confluent singularities associated with irrelevant variables for this system. A direct extrapolation of Nickel's series coefficients is also carried out and found to be consistent with the present results.
The study of the equation of state of the two-dimensional Ising model is initiated by use of the recently calculated nnn-point spin-correlation functions. In the scaling region where $T\ensuremath{\rightarrow}{T}_{c}$ and $h=B{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{15}{8}}(\frac{H}{k{T}_{c}})$ is of order 1 (BBB is a lattice-dependent constant) the free energy below Tc{T}_{c}Tc​ is of the form $f(T, h)\ensuremath{-}f(T, 0)=\mathrm{const}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{2}$ $\ensuremath{\Sigma}{n=1}^{\ensuremath{\infty}}(\frac{1}{n!}){C}_{n}^{\ensuremath{-}}{h}^{n}$ where the lattice-dependent constant is chosen so that ${C}_{1}^{\ensuremath{-}}=1$. The constant ${C}_{2}^{\ensuremath{-}}$ has been studied previously. In this paper we study ${C}_{3}^{\ensuremath{-}}$ and ${C}_{4}^{\ensuremath{-}}$ and find as a first approximation $\frac{{C}_{3}^{\ensuremath{-}}}{3!}\ensuremath{\sim}\ensuremath{-}\frac{11}{72{\ensuremath{\pi}}^{2}}$ and $\frac{{C}_{4}^{\ensuremath{-}}}{4!}\ensuremath{\sim}\frac{5}{189\ensuremath{\pi}}+\frac{59}{162{\ensuremath{\pi}}^{3}}$. These are in close agreement with the low-temperature-series values of -0.01538 and 0.0195.
We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​. 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 ttt 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-ttt 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 $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.
We review the estimates for the critical indices of the Ising model below TC{T}_{C}TC​ and conclude (for 3 dimensions): ${\ensuremath{\alpha}}^{\ensuremath{'}}=0.066_{\ensuremath{-}0.04}^{}{}_{}{}^{+0.16}$, $\ensuremath{\beta}=0.312_{\ensuremath{-}0.005}^{}{}_{}{}^{+0.002}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.310_{\ensuremath{-}0.05}^{}{}_{}{}^{+0.04}$ for the specific heat, magnetization, and magnetic susceptibility, respectively. For 2 dimensions, we estimate ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.75_{\ensuremath{-}0.00}^{}{}_{}{}^{+0.01}$. In order to explain previous estimates of ${\ensuremath{\alpha}}^{\ensuremath{'}}=0$, we point out that a low power can, in practice, look deceptively like a logarithm. Finally, we discuss the behavior of the specific heat at constant magnetization.
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.