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Journal of Physics C Solid State Physics

Critical correlation-function approximations for the Ising model

B. Frank, O Mitran

Published May 28, 1978• 27 citations
Publisher

Abstract

In a new theory for the spin one-half Ising ferromagnet, the fourth-order equation is used to provide expressions for the higher-order site-independent correlation functions M 2n (n=3,4,5,...) in terms of the two- and four-spin correlation functions M 2 and M 4 , at the critical point. As an example, the expression for M 6 is M 4 2 /M 2 -9M 2 M 4 -6M 2 3 . These expressions (for up to M 60 ) are applied to the calculation of T c for the SC, BCC, and FCC lattices. The results converge to, at most, 0.755, 0.799, and 0.819 respectively, which are within 2 / 3 % of the accepted results 0.752, 0.794, and 0.816 of Sykes et al. (1972).

DOI
10.1088/0022-3719/11/10/019

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Physical review. B, Solid state•Aug 1, 1972•51 citations
In a new theory for the spin one-half Ising ferromagnet, the fourth-order equation is used to provide expressions for the higher-order site-independent correlation functions M 2n (n=3,4,5,...) in terms of the two- and four-spin correlation functions M 2 and M 4 , at the critical point. As an example, the expression for M 6 is M 4 2 /M 2 -9M 2 M 4 -6M 2 3 . These expressions (for up to M 60 ) are applied to the calculation of T c for the SC, BCC, and FCC lattices. The results converge to, at most, 0.755, 0.799, and 0.819 respectively, which are within 2 / 3 % of the accepted results 0.752, 0.794, and 0.816 of Sykes et al. (1972).
A stringent test is applied to a set of approximations used, in an earlier work (CCFA) see Frank et al. (ibid., vol.11, p.208 (1978) to calculate T c for the spin- 1 / 2 Ising ferromagnet. In CCFA, all cumulant averages M 2n (2n>or=6) were approximated, for all the cubic lattices, and justified a posteriori. Here, a spin operator reduction relation allows one, for the SC lattice, to approximate only M 6 , and moreover, to understand the approximation used. The extreme closeness at every stage (within 0.01%) of the present T c values to those of CCFA gives one confidence method of approximating the M 2n for the BCC and FCC lattices as well. Here also, as in CCFA, the M 60 value of T c is within 0.5% of the series approximation value of Sykes et al. (1972).
The spin-spin correlation functions and the critical-scattering intensity for Heisenberg models of general spin, S=12S=\frac{1}{2}S=21​ to $\ensuremath{\infty}$, on the sc, bcc, and fcc lattices are studied on the basis of high-temperature series expansions along the lines developed in Paper I [M. E. Fisher and R. J. Burford, Phys. Rev. 156, 583 (1967)]. Subject to increased uncertainties for low spin, it is concluded that the exponents $\ensuremath{\gamma}={1.375}_{\ensuremath{-}0.01}^{+0.02}$, $2\ensuremath{\nu}={1.405}_{\ensuremath{-}0.01}^{+0.02}$, and $\ensuremath{\eta}=0.043\ifmmode\pm\else\textpm\fi{}0.014$ describe all lattices and all spin. Explicit formulas are presented for the susceptibility/zero-angle scattering ${\ensuremath{\chi}}_{0}(T)$, for the inverse correlation length ${\ensuremath{\kappa}}_{1}(T)$, for the effective interaction range r1(T){r}_{1}(T)r1​(T), and using the Fisher-Burford approximant, for the total scattering $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)$. The shape parameter ${\ensuremath{\varphi}}_{c}$ attains the "universal" value ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 11 for large spin but shows signs of spin dependence (and lattice dependence) for low spin. At fixed k the scattering is predicted to display a maximum above Tc{T}_{c}Tc​ determined by $\frac{{\ensuremath{\kappa}}_{1}({T}_{max})}{k}\ensuremath{\simeq}0.10 (\mathrm{for} S\ensuremath{\gtrsim}2)$ to 0.15. A detailed study is made of the structure dependence of the critical-point correlations ${〈{S}_{0}^{z}{S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉}_{c}$ for various models. This leads to the revised, universal estimate ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 15 for all three cubic lattice, spin-\textonehalf{} Ising models. The results are compared d briefly with various experiments which support $\ensuremath{\eta}\ensuremath{\gtrsim}0.05$.
A generalised non-uniform magnetic field method is used to derive, from the linearisation assumption of a new theory for the Ising ferromagnet, a set of critical correlation-function approximations (CCFA) which express the higher-order correlations in terms of the second- and fourth-order correlations. This set is found to coincide at least for the first five relations, with an earlier CCFA set which was suggested by a term-by-term comparison of the second- and fourth-order equations of the theory, and which was justified a posteriori by the accurate results for the transition temperature to which it leads.
A class of fourth-order spin correlations whose sum is closely related to the specific heat is calculated exactly by Pfaffian perturbation theory. In the particular case when the four spins lie on a lattice axis, it is shown that the reduced fourth-order correlation function has the asymptotic form 〈σ0, 0σ1, 1σk, kσk+1, k+1〉−〈σ0, 0σ1, 1〉 〈σk, kσk+1, k+1〉≏A′(e−2βk/k2),where β−1 is the (positive) mean range of order. The decay of correlations with spin separation k is symmetric about the critical point β = 0. The amplitude A′ is 1/π2 at the critical point and is 1/(2π) at all other temperatures. Correlations on ferro- and antiferromagnetic triangular and quadratic lattices are discussed in some detail.
The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.
In recent years there have been many measurements of the scaling-law equation of state for different materials, and the "scaling function" so obtained has generally been fit by an empirical equation involving the selection of several adjustable parameters. We propose a method for calculating, directly from high-temperature series expansions, the function h(x)h(x)h(x) that determines the scaling-law equation of state $H={M}^{\ensuremath{\delta}}h(x)$. Previously, h(x)h(x)h(x) has been calculated only for the S=12S=\frac{1}{2}S=21​ Ising model, but the method is not generalizable to the case of the Heisenberg model because it relies upon the use of low-temperature expansions as well, and these are not known for the Heisenberg model. First we calculate h(x)h(x)h(x) for the Ising model (bcc, fcc, and simple cubic lattices) in order to assess the utility and credibility of our method. Our Ising model h(x)h(x)h(x) agrees well with the previous calculation that used both high-and low-temperature expansions. Next we calculate h(x)h(x)h(x) in its entire region of definition for the S=12S=\frac{1}{2}S=21​ Heisenberg model (fcc and bcc lattices) and the $S=\ensuremath{\infty}$ Heisenberg models (fcc lattice), where SSS denotes the spin quantum number. The accuracy of our resulting expressions is limited by the finite number of known terms in the corresponding high-temperature series expansions, but it is generally of the order of a few percent. In Paper II the scaling functions calculated here are compared with experiment and with the predictions of the universality hypothesis.