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Physical Review Letters

Two-Series Approach to Partial Differential Approximants: Three-Dimensional Ising Models

Martin J. George, John J. Rehr

Published Nov 26, 1984• 48 citations
Publisher

Abstract

A two-series approach to partial differential approximant analysis of power series is presented. Instead of double series, $f(x,y)=\ensuremath{\Sigma}{c}_{\mathrm{ij}}{x}^{i}{y}^{j}$, our approach uses two one-variable series in $x$, $f$ and $\frac{\ensuremath{\partial}f}{\ensuremath{\partial}y}$, and has the efficiency and stability of one-variable methods. 21-term high-temperature series are analyzed for the susceptibility and correlation length squared for double-Gaussian Ising models on the bcc lattice. Critical exponents are $\ensuremath{\gamma}=1.2378(6)$, $2\ensuremath{\nu}=1.2623(6)$, and $\ensuremath{\eta}=0.0375(5)$; correction-to-scaling exponents are ${\ensuremath{\theta}}_{\ensuremath{\chi}}=0.52(3)$ and ${\ensuremath{\theta}}_{{\ensuremath{\xi}}^{2}}=0.49(4)$; and the subdominant critical amplitude ratio is $\frac{{a}_{\ensuremath{\xi}}}{{a}_{\ensuremath{\chi}}}=0.83(5)$.

DOI
10.1103/physrevlett.53.2063

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•Nov 4, 2004•0 citations
A two-series approach to partial differential approximant analysis of power series is presented. Instead of double series, f(x,y)=\ensuremathΣcijxiyjf(x,y)=\ensuremath{\Sigma}{c}_{\mathrm{ij}}{x}^{i}{y}^{j}f(x,y)=\ensuremathΣcij​xiyj, our approach uses two one-variable series in xxx, fff and \ensuremath∂f\ensuremath∂y\frac{\ensuremath{\partial}f}{\ensuremath{\partial}y}\ensuremath∂y\ensuremath∂f​, and has the efficiency and stability of one-variable methods. 21-term high-temperature series are analyzed for the susceptibility and correlation length squared for double-Gaussian Ising models on the bcc lattice. Critical exponents are \ensuremathγ=1.2378(6)\ensuremath{\gamma}=1.2378(6)\ensuremathγ=1.2378(6), 2\ensuremathν=1.2623(6)2\ensuremath{\nu}=1.2623(6)2\ensuremathν=1.2623(6), and \ensuremathη=0.0375(5)\ensuremath{\eta}=0.0375(5)\ensuremathη=0.0375(5); correction-to-scaling exponents are \ensuremathθ\ensuremathχ=0.52(3){\ensuremath{\theta}}_{\ensuremath{\chi}}=0.52(3)\ensuremathθ\ensuremathχ​=0.52(3) and \ensuremathθ\ensuremathξ2=0.49(4){\ensuremath{\theta}}_{{\ensuremath{\xi}}^{2}}=0.49(4)\ensuremathθ\ensuremathξ2​=0.49(4); and the subdominant critical amplitude ratio is a\ensuremathξa\ensuremathχ=0.83(5)\frac{{a}_{\ensuremath{\xi}}}{{a}_{\ensuremath{\chi}}}=0.83(5)a\ensuremathχ​a\ensuremathξ​​=0.83(5).
High-temperature series for two bcc lattice models which interpolate between the Gaussian (or free-field) model and the S=12S=\frac{1}{2}S=21​ Ising model are analyzed by partial differential approximants. Series to order 21 in both $x\ensuremath{\propto}\frac{1}{T}$ and the interpolation parameter, yyy, yield unbiased estimates for the correction-to-scaling exponent, $\ensuremath{\theta}=0.54\ifmmode\pm\else\textpm\fi{}5$, and the susceptibility exponent, $\ensuremath{\gamma}=1.2385\ifmmode\pm\else\textpm\fi{}15$. The results are universal and agree tolerably with field-theoretic estimates and well with biased, one-variable analyses of general spin Ising models.
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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).
We investigate the Ising model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interactions on the square and body-centered-cubic lattices. For 0<J1J2<320<\frac{{J}_{1}}{{J}_{2}}<\frac{3}{2}0<J2​J1​​<23​ on the bcc lattice the model has two nontrivially degenerate ground states and has been conjectured to have XY\mathrm{XY}XY (n=2n=2n=2) critical exponents. For J1J2=32\frac{{J}_{1}}{{J}_{2}}=\frac{3}{2}J2​J1​​=23​ there are three degenerate ground states and Heisenberg critical exponents (n=3n=3n=3) are expected. We have obtained nine terms in the high-temperature expansion of the staggered susceptibility. For 0<J1J2< 10<\frac{{J}_{1}}{{J}_{2}}<~10<J2​J1​​< 1 we find $\ensuremath{\gamma}=1.25$ (n=1n=1n=1). At J1J2=32\frac{{J}_{1}}{{J}_{2}}=\frac{3}{2}J2​J1​​=23​ the results are not conclusive, but $\ensuremath{\gamma}=1.33$ (n=2n=2n=2) is not ruled out by the series analysis whereas $\ensuremath{\gamma}=1.4$ (n=3n=3n=3) is not probable. Results for the square lattice are inconclusive.
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Square lattice Ising model susceptibility: Series expansion method and differential equation for chi(3)\\chi^{(3)}chi(3)
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