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Physical review. B, Solid state

Multispin correlation functions for Ising models

Helen Au-Yang

Published Mar 1, 1977• 9 citations
Publisher

Abstract

We calculate the multispin correlation functions away from ${T}_{c}$ of a two-dimensional Ising lattice along the diagonal. We also present the result for a four-spin correlation function in the scaling limit. These results show that with slight modifications, the operator reduction formulae of Kadanoff hold.

DOI
10.1103/physrevb.15.2704

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

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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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Four-spin correlation function of two parallel nonaligned pairs in the planar Ising model

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The four-spin correlation function of two parallel nonaligned pairs in the two-dimensional rectangular Ising lattice is found to have the scaling form $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}{\ensuremath{\sigma}}_{J,J+l}{\ensuremath{\sigma}}_{J+L,J+L+l}〉\ensuremath{\sim}〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}〉〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{LL}}〉+{〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}〉}^{c}{〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{LL}}〉}^{c}{f}_{4}({x}_{1},{x}_{2},X,\ensuremath{\varphi})$, where ${f}_{4}$ is the scaling function. The scaling variables ${x}_{1},{x}_{2}$, and $X$ are, respectively, the lengths of the pairs ($N$ and $L$) and distance between the centers of the pairs scaled by the corresponding correlation lengths. The angle $\ensuremath{\varphi}$ measures the deviation from the diagonal axis of the line joining the centers. The scaling function is given as a series of ${x}_{1}$ and ${x}_{2}$, truncated at the order of ${({x}_{1}{x}_{2})}^{3}$.

Physical review. B, Condensed matter•May 1, 1981•0 citations
We calculate the multispin correlation functions away from Tc{T}_{c}Tc​ of a two-dimensional Ising lattice along the diagonal. We also present the result for a four-spin correlation function in the scaling limit. These results show that with slight modifications, the operator reduction formulae of Kadanoff hold.
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.
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.
The one-dimensional (1D) Ising model is revisited. The integral operator method is used for the model to derive the general statistical identity for multispin correlation functions. As an application of this identity to generate various correlations, recursive formulas are derived for the correlations of pairs, threes, fours, and fives of spins. Both compact clusters and disjoint sets of spins are considered. It is shown that multispin correlations for the 1D Ising model can be exactly expressed in terms of pairwise correlations using recursive procedures. For clusters with different structures, numerical calculations of multispin correlations as a function of temperature and external magnetic field are performed. The results of these calculations are presented in the figures and discussed.
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 $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.
It is noted that expressions for the T=TcT={T}_{c}T=Tc​ nnn-spin correlation functions of a two-dimensional Ising system obtained by continuum methods do agree with exact results for restricted geometries. This leads to the conjecture that the continuum result is true for all nnn and all geometries. Speculation regarding the behavior of these functions away from T=TcT={T}_{c}T=Tc​ are made.
The four-spin correlation function of two parallel nonaligned pairs in the two-dimensional rectangular Ising lattice is found to have the scaling form $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}{\ensuremath{\sigma}}_{J,J+l}{\ensuremath{\sigma}}_{J+L,J+L+l}〉\ensuremath{\sim}〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}〉〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{LL}}〉+{〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{NN}}〉}^{c}{〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{LL}}〉}^{c}{f}_{4}({x}_{1},{x}_{2},X,\ensuremath{\varphi})$, where f4{f}_{4}f4​ is the scaling function. The scaling variables x1,x2{x}_{1},{x}_{2}x1​,x2​, and XXX are, respectively, the lengths of the pairs (NNN and LLL) and distance between the centers of the pairs scaled by the corresponding correlation lengths. The angle $\ensuremath{\varphi}$ measures the deviation from the diagonal axis of the line joining the centers. The scaling function is given as a series of x1{x}_{1}x1​ and x2{x}_{2}x2​, truncated at the order of (x1x2)3{({x}_{1}{x}_{2})}^{3}(x1​x2​)3.