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Physical Review

Effect of Change of Spin on the Critical Properties of the Ising and Heisenberg Models

Cyril Domb, M F Sykes

Published Oct 1, 1962• 213 citations
Publisher

Abstract

High-temperature series expansions of the partition functions for the Ising and Heisenberg models are analyzed for various values of the spin $s$. The fcc lattice is used for which successive coefficients are sufficiently regular for estimates of critical behavior to be made with confidence. It is suggested that the magnetic susceptibility above the Curie temperature is of the form $A{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{\ensuremath{-}\frac{4}{3}}$ for the Heisenberg model for all $s$, instead of ${A}^{\ensuremath{'}}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{\ensuremath{-}\frac{5}{4}}$ for the Ising model. Critical estimates of energy and entropy show that the magnitude of the "tail" of the specific heat anomaly is insensitive to the value of $s$, and is about 2.5 times larger for the Heisenberg than for the Ising model. The sharpness of the anomaly at the Curie point increases as $s$ increases, and on passing from the Heisenberg to the Ising model. A brief reference is made to experimental results.

DOI
10.1103/physrev.128.168

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High-temperature series expansions of the partition functions for the Ising and Heisenberg models are analyzed for various values of the spin sss. The fcc lattice is used for which successive coefficients are sufficiently regular for estimates of critical behavior to be made with confidence. It is suggested that the magnetic susceptibility above the Curie temperature is of the form A(1\ensuremath−TcT)\ensuremath−43A{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{\ensuremath{-}\frac{4}{3}}A(1\ensuremath−TTc​​)\ensuremath−34​ for the Heisenberg model for all sss, instead of A\ensuremath′(1\ensuremath−TcT)\ensuremath−54{A}^{\ensuremath{'}}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{\ensuremath{-}\frac{5}{4}}A\ensuremath′(1\ensuremath−TTc​​)\ensuremath−45​ for the Ising model. Critical estimates of energy and entropy show that the magnitude of the "tail" of the specific heat anomaly is insensitive to the value of sss, and is about 2.5 times larger for the Heisenberg than for the Ising model. The sharpness of the anomaly at the Curie point increases as sss increases, and on passing from the Heisenberg to the Ising model. A brief reference is made to experimental results.
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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 $\ensuremath{\eta}\ensuremath{\gtrsim}0.05$.
Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.
A detailed series-extrapolation investigation of some critical properties ($T\ensuremath{\rightarrow}{T}_{c}+$) of the Heisenberg model is presented. Systematic use is made of ratio and Pad\'e-approximant techniques, some of the latter being new. Attention is confined to those nearest-neighbor and order-two equivalent-neighbor models which are based on the simple cubic, body-centered cubic, and face-centered cubic lattices. For the face-centered cubic nearest-neighbor classical Heisenberg model, the following estimates are obtained for familiar critical exponents: $\ensuremath{\alpha}=0, \ensuremath{-}\frac{1}{8}<~{\ensuremath{\alpha}}_{s}<~\ensuremath{-}\frac{1}{16}$, $\ensuremath{\gamma}=1.375\ifmmode\pm\else\textpm\fi{}0.002$, $0.6875<~\ensuremath{\nu}<~0.7125$, $0<~\ensuremath{\eta}<~0.07$. Estimates for $\ensuremath{\gamma}$ are also obtained for the other nearest-neighbor and equivalent-neighbor classical Heisenberg models, and it is conjectured that $\ensuremath{\gamma}=1\frac{3}{8}$ may be the exact result in all the cases considered. For the face-centered cubic order-two equivalent-neighbor spin-\textonehalf{} Heisenberg ferromagnet, it is estimated that $\ensuremath{\gamma}=1.3744\ifmmode\pm\else\textpm\fi{}0.0008$. A similar but less precise result is found for the corresponding simple cubic model, and the spin-\textonehalf{} nearest-neighbor susceptibility series are briefly examined. It is suggested that consideration should be given to the possibility that for ferromagnetic Heisenberg interactions $\ensuremath{\gamma}$ has the same value 1381\frac{3}{8}183​ irrespective of spin and the particular three-dimensional nearest-neighbor or finite-order equivalent-neighbor model considered. Other quantities estimated include various Curie points and singularity amplitudes and the critical values of certain thermodynamic functions. Some attention is given to the simple classical Heisenberg antiferromagnet.
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.