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Physical review. B, Solid state

Equation of State near the Critical Point. I. Calculation of the Scaling Function for S=12 and S=∞ Heisenberg Models Using High-Temperature Series Expansions

S. Milošević, Eugene Stanley

Published Aug 1, 1972• 51 citations
Publisher

Abstract

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 $h(x)$ that determines the scaling-law equation of state $H={M}^{\ensuremath{\delta}}h(x)$. Previously, $h(x)$ has been calculated only for the $S=\frac{1}{2}$ 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 $h(x)$ for the Ising model (bcc, fcc, and simple cubic lattices) in order to assess the utility and credibility of our method. Our Ising model $h(x)$ agrees well with the previous calculation that used both high-and low-temperature expansions. Next we calculate $h(x)$ in its entire region of definition for the $S=\frac{1}{2}$ Heisenberg model (fcc and bcc lattices) and the $S=\ensuremath{\infty}$ Heisenberg models (fcc lattice), where $S$ 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.

DOI
10.1103/physrevb.6.986

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Gerald Paul, Eugene Stanley

High-temperature series expansions are used to examine the dependence of critical-point exponents upon lattice anisotropy (different interaction strengths in different directions in the lattice). The two-spin correlation function ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is calculated to tenth order in ($\frac{1}{{k}_{B}T}$) for the Ising Hamiltonian ${\mathcal{H}}_{l \mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}$ for a wide range of anisotropy parameters $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{\mathrm{ij}}$ has no $z$ component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for $R=0$, both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are then used to obtain series of corresponding lengths for the specific heat, susceptibility, and second moment. Analysis of these series yields results consistent with the universality hypothesis of critical-point exponents. Specifically, it is found that when lattice anisotrophy is introduced, the critical-point exponents studied (the susceptibility exponent $\ensuremath{\gamma}$ and the correlation length exponent $\ensuremath{\nu}$) do not appear to change from their values for an isotropic lattice. The problem of next-nearest-neighbor interactions is treated using similar methods in Paper II of this series (and briefly discussed in this paper).

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Rigorous Scaling-Law Equation of State near the Critical Point

Julio A. Gonzalo

Making use of the scaling properties of $H(\ensuremath{\epsilon}, M)={M}^{\ensuremath{\delta}}h(\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}})$, it is shown that the "scaling function" $h(x)$ can be expressed rigorously as $\frac{h(x)}{h(0)}={[\frac{({x}_{0}+x)}{{x}_{0}}]}^{\ensuremath{\beta}(\ensuremath{\delta}\ensuremath{-}1)}$ in the neighborhood of the critical point. This result is in agreement with the known mean-field-model prediction, directly obtainable from power-series expansion, and with accurate numerical results for the Ising and Heisenberg models.

Physical review. B, Solid state•Oct 1, 1973•5 citations
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\ensuremathδh(x)H={M}^{\ensuremath{\delta}}h(x)H=M\ensuremathδh(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∞S=\ensuremath{\infty}S=\ensuremath∞ 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.
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 $\ensuremath{\eta}\ensuremath{\gtrsim}0.05$.
We calculate the scaled constant-magnetization and constant magnetic field specific heats $\frac{[{C}_{M}(\ensuremath{\epsilon},M)\ensuremath{-}{C}_{M}(\ensuremath{\epsilon},0)]}{{M}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}}}}$ and $\frac{[{C}_{H}(\ensuremath{\epsilon},H)\ensuremath{-}{C}_{H}(\ensuremath{\epsilon},0)]}{{H}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}\ensuremath{\delta}}}}$ as functions of the scaled variable $x\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}}$ and $y\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{H}^{\frac{1}{\ensuremath{\beta}\ensuremath{\delta}}}}$, respectively [where $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and Tc{T}_{c}Tc​ is the critical temperature],for the spin-\textonehalf{} Ising model (bcc lattice) and the spin-\textonehalf{}, $\ensuremath{\infty}$ Heisenberg models (fcc lattice). Our calculations are based on previously calculated scaling functions for the M(H,T)M(H,T)M(H,T) equation of state. We also calculate the amplitudes of the zero-field specific heat for T>TcT>{T}_{c}T>Tc​ and T<TcT<{T}_{c}T<Tc​. Thus, we obtain the functions ${C}_{M}(\ensuremath{\epsilon},M)$ and ${C}_{H}(\ensuremath{\epsilon},H)$, though for the Heisenberg models we cannot obtain the finite nonzero constant CM(0,0)=CH(0,0){C}_{M}(0,0)={C}_{H}(0,0)CM​(0,0)=CH​(0,0). We compare our calculated functions with the data of Connelly, Loomis, and Mapother on nickel.
Using a Monte Carlo procedure with a self-consistent-field boundary condition, the magnetization of a simple cubic classical Heisenberg ferromagnet with nearest-neighbor interactions only is calculated. For reduced temperatures and reduced fields in the range $0.02<~|\frac{T}{{T}_{c}}\ensuremath{-}1|$, HJ< 0.5\frac{H}{J}<~0.5JH​< 0.5 the data of this computer experiment can be described in terms of the "effective" critical exponents $0.31<~\ensuremath{\beta}<~0.35$, $1.33<~\ensuremath{\gamma}<~1.39$, and $4.9<~\ensuremath{\delta}<~5.3$. Although the present data are not very close to the critical point they do obey the homogeneity requirement and determine the scaling function rather precisely. This function agrees very well with the scaling function for the face-centered-cubic classical Heisenberg magnet derived recently by Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley using high-temperature---series-expansion techniques. This agreement supports their hypothesis that neither critical exponents nor the scaling function depend on the lattice structure in the Heisenberg model.
Much attention has been paid to the d=3, s= 1 / 2 Ising model on the BCC lattice in recent years because the best high-temperature series expansions were available for this lattice. However, in order to compare series estimates of critical exponents and temperatures with independent evaluations via the Monte Carlo method, lattices of lower coordination number must be considered, since Monte Carlo studies are usually made on these. A study of extant susceptibility series on the FCC, SC, diamond and tetrahedron lattices is reported on and some comments are made on aspects of recent Monte Carlo analyses of the d=3, s= 1 / 2 Ising model on the SC and diamond lattices.
High-temperature series expansions are used to examine the dependence of critical-point exponents upon lattice anisotropy (different interaction strengths in different directions in the lattice). The two-spin correlation function ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is calculated to tenth order in (1kBT\frac{1}{{k}_{B}T}kB​T1​) for the Ising Hamiltonian ${\mathcal{H}}_{l \mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}$ for a wide range of anisotropy parameters $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{\mathrm{ij}}$ has no zzz component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for R=0R=0R=0, both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are then used to obtain series of corresponding lengths for the specific heat, susceptibility, and second moment. Analysis of these series yields results consistent with the universality hypothesis of critical-point exponents. Specifically, it is found that when lattice anisotrophy is introduced, the critical-point exponents studied (the susceptibility exponent $\ensuremath{\gamma}$ and the correlation length exponent $\ensuremath{\nu}$) do not appear to change from their values for an isotropic lattice. The problem of next-nearest-neighbor interactions is treated using similar methods in Paper II of this series (and briefly discussed in this paper).
Making use of the scaling properties of $H(\ensuremath{\epsilon}, M)={M}^{\ensuremath{\delta}}h(\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}})$, it is shown that the "scaling function" h(x)h(x)h(x) can be expressed rigorously as $\frac{h(x)}{h(0)}={[\frac{({x}_{0}+x)}{{x}_{0}}]}^{\ensuremath{\beta}(\ensuremath{\delta}\ensuremath{-}1)}$ in the neighborhood of the critical point. This result is in agreement with the known mean-field-model prediction, directly obtainable from power-series expansion, and with accurate numerical results for the Ising and Heisenberg models.