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Physical review. B, Solid state

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, Eytan Barouch

Published Jan 1, 1976• 694 citations
Publisher

Abstract

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 $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.

DOI
10.1103/physrevb.13.316

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We compute exactly the spin-spin correlation functions 〈\ensuremathσ0,0\ensuremathσM,N〉〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉〈\ensuremathσ0,0​\ensuremathσM,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→Tc,M2+N2\ensuremath→\ensuremath∞T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}T\ensuremath→Tc​,M2+N2\ensuremath→\ensuremath∞ such that (T\ensuremath−Tc)(T\ensuremath{-}{T}_{c})(T\ensuremath−Tc​) is fixed. In this scaling limit 〈\ensuremathσ0,0\ensuremathσM,N〉=R\ensuremath−14F\ifmmode±\else\textpm\fi(t)+R\ensuremath−54F1\ifmmode±\else\textpm\fi(t)+o(R\ensuremath−54)〈{\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}})〈\ensuremathσ0,0​\ensuremathσM,N​〉=R\ensuremath−41​F\ifmmode±\else\textpm\fi​(t)+R\ensuremath−45​F1\ifmmode±\else\textpm\fi​(t)+o(R\ensuremath−45​), where ttt is the scaling variable R\ensuremathξ\frac{R}{\ensuremath{\xi}}\ensuremathξR​ and F\ifmmode±\else\textpm\fi(t){F}_{\ifmmode\pm\else\textpm\fi{}}(t)F\ifmmode±\else\textpm\fi​(t) and F1\ifmmode±\else\textpm\fi(t){F}_{1\ifmmode\pm\else\textpm\fi{}}(t)F1\ifmmode±\else\textpm\fi​(t) are the scaling functions (\ensuremathξ\ensuremath{\xi}\ensuremathξ 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±\else\textpm\fi(t){F}_{\ifmmode\pm\else\textpm\fi{}}(t)F\ifmmode±\else\textpm\fi​(t) (good to ten significant digits) is also given. As an application we computer the 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 kBT\ensuremathχ(T)=C0\ifmmode±\else\textpm\fi∣1\ensuremath−TcT∣\ensuremath−74+C1\ifmmode±\else\textpm\fi∣1\ensuremath−TcT∣\ensuremath−34+O(1){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)kB​T\ensuremathχ(T)=C0\ifmmode±\else\textpm\fi​∣1\ensuremath−TTc​​∣\ensuremath−47​+C1\ifmmode±\else\textpm\fi​∣1\ensuremath−TTc​​∣\ensuremath−43​+O(1) of the zero-field susceptibility \ensuremathχ(T)\ensuremath{\chi}(T)\ensuremathχ(T) as T\ensuremath→Tc\ifmmode±\else\textpm\fiT\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}T\ensuremath→Tc\ifmmode±\else\textpm\fi​.
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​.
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 the four-spin correlation function $〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{N}{\ensuremath{\sigma}}_{N+M}{\ensuremath{\sigma}}_{N+M+L}〉$ along the diagonal of an Ising square lattice. We find that it can be written in the scaling form $〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{N}{\ensuremath{\sigma}}_{N+M}{\ensuremath{\sigma}}_{N+M+L}〉\ensuremath{\sim}〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{N}〉〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{L}〉+{〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{N}〉}^{c}{〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{L}〉}^{c}{f}_{4}(tN,tL,|t|(M+\frac{1}{2N}+\frac{1}{2L}))$, where ${|t|}^{\ensuremath{-}1}$ is the correlation length. The scaling function f4{f}_{4}f4​ is obtained as a asymptotic series with the variables tN\mathrm{tN}tN and tL\mathrm{tL}tL small. The remarkably simple form of this function f4{f}_{4}f4​ suggests the reduction formula ${\ensuremath{\sigma}}_{R\ensuremath{-}\frac{r}{2}}{\ensuremath{\sigma}}_{R+\frac{r}{2}}\ensuremath{-}〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{r}〉={〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{r}〉}^{c}[{c}_{1}(r){\ensuremath{\epsilon}}_{R}+{c}_{2}(r){t}_{R}+{c}_{3}(r){u}_{R}+\ensuremath{\cdots}]$, in which ${\ensuremath{\epsilon}}_{R}={\ensuremath{\sigma}}_{R}{\ensuremath{\sigma}}_{R+1}\ensuremath{-}〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{1}〉$ is the energy-density operator, while tR{t}_{R}tR​ and uR{u}_{R}uR​ denote some less singular operators. The coefficients of this expansion are independent of temperature.
Abstract This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.
For both T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​, we report exact expressions for the nnn-spin correlation function of the two-dimensional Ising model suitable for studying large separations between spins. In particular, the scaling limit of these correlation functions can be shown to exist and yields nnn-point Schwinger functions of some relativistic quantum field theories.
We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin-SSS Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.