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Subdominant Critical Singularities in the bcc

Pierre Moussa

Published Jan 1, 1982• 0 citations
OpenAlex

Abstract

In order to analyze coalescing singularities using series expansion, a modification of the method introduced by Baker and Hunter is proposed. The Pad~ approximants on the Mellin transform are computed simultaneously for the initial series and its derivatives, allowing unbiased estimates for the critical parameters. The method is applied to the series generated by Nickel for various values of the spin in the bcc Ising model. We show that even with these longer series, the subdominant indices display large variation with the spin, and remain too small compared to the universal renormalization group prediction. However, the method does not detect other singularities in the temperature plane which are responsible for the observed discrepancy. Therefore the discrepancy is not significant.

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Power-series expansions of the spin-pair correlation functions of the square, sc, and bcc Ising lattices have been obtained using the semi-invariant approach for general field $H$ and temperature $T$. In three dimensions P\'ade-approximant analyses indicate $2{v}^{\ensuremath{'}}=1.285\ifmmode\pm\else\textpm\fi{}0.020$, in agreement with scaling, but $2{v}^{c}\ensuremath{\simeq}0.835\ifmmode\pm\else\textpm\fi{}0.02$ which is some (1-3)% above the scaling prediction ${v}^{c}=\frac{v}{\ensuremath{\beta}\ensuremath{\delta}}\ensuremath{\simeq}0.823$. However, ratio techniques reveal that this discrepancy can be attributed to significant confluent critical singularities. Cubic and quintic parametric representations of the critical equation of state, and corresponding expressions for the correlation length, ${\ensuremath{\xi}}_{1}(H, T)$, are developed, which are considerable improvements over the linear model. The universality and spherical symmetry of the critical scattering intensity $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ is confirmed to within (1-2)% by estimating suitable invariant combinations of amplitudes. The deviations from Ornstein-Zernike theory for general $H$ and $T$ are found to be considerably greater than for $H=0$, $T>{T}_{c}$. Complete parametric scaling representations of $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ are developed for three dimensions; corresponding scaling approximants are constructed in two dimensions but only for $T={T}_{c}$ and for $T\ensuremath{\gtrless}{T}_{c}$ with $H=0$.

Physical review. B, Solid state•Feb 1, 1975•237 citations
In order to analyze coalescing singularities using series expansion, a modification of the method introduced by Baker and Hunter is proposed. The Pad~ approximants on the Mellin transform are computed simultaneously for the initial series and its derivatives, allowing unbiased estimates for the critical parameters. The method is applied to the series generated by Nickel for various values of the spin in the bcc Ising model. We show that even with these longer series, the subdominant indices display large variation with the spin, and remain too small compared to the universal renormalization group prediction. However, the method does not detect other singularities in the temperature plane which are responsible for the observed discrepancy. Therefore the discrepancy is not significant.
Starting from the high-temperature series for the susceptibility of the spin- 1 / 2 ferromagnetic Ising models-square, planar triangular, SC, BCC, FCC, diamond, and HSC in four, five and six dimensions-the authors analyse the analytical structure near the critical point of the susceptibility by writing it as a Laplace transform in the variable log (1-w/w c ), where w=tanh( beta J). The interest of the method is that the coalescing singularities which sit at w=w c are spread out and can be analysed separately, the first subdominant critical indices, appearing as stable poles of Pade approximants. They first recover the results for the two-dimensional models, with a high accuracy: gamma =1.74995 and gamma s =0.757. The most stable results in three dimensions are obtained for the diamond lattice: gamma =1.2506+or-0.0015 and gamma s =0.42+or-0.11. The other lattices give 0.10< gamma s <0.52. However, in all three-dimensional cases the amplitude of the subdominant singularity is less than a few per cent of the leading one. Therefore the uncertainties are large, but the subdominant singularity stays closer to the value Gamma -1=0.25 than to the field theoretical predicted one gamma s =0.75.
We analyze the 21st-order series of Nickel for the susceptibility and the correlation length of the spin-sss Ising model for the bcc lattice with the use of an unbiased method of confluent singularity analysis tailored to the loose-packed lattices. This modified five-fit method of analysis assumes a parametrization which includes one confluent correction, e.g., for the susceptibility $\ensuremath{\chi}={\mathrm{At}}^{\ensuremath{-}\ensuremath{\gamma}}(1+{\mathrm{Bt}}^{{\ensuremath{\Delta}}_{1}})$, where $t=1\ensuremath{-}\frac{{T}_{c}}{T}$. The method determines sequences for these five parameters, e.g., ${\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}{\ensuremath{\gamma}}_{n}=\ensuremath{\gamma}$, by equating five consecutive even or odd coefficients in the perturbation series with the five corresponding coefficients in the expansion of the assumed parametrization and by solving the five equations for the five parameters numerically. Our analysis of the spin-sss Ising-model series shows that the indices $\ensuremath{\gamma}$ and $2\ensuremath{\nu}$ are smaller than indicated by early analysis on shorter series and that the confluent correction is apparently not significant for spin-\textonehalf{}, validating the use of ratio methods for the spin-\textonehalf{} series. Considering the results of both the confluent singularity analysis and the standard analysis of the spin-\textonehalf{} series, we estimate $\ensuremath{\gamma}=1.242 (+0.003, \ensuremath{-}0.005)$ and $2\ensuremath{\nu}=1.268(+0.006, \ensuremath{-}0.008)$, in basic agreement with renormalization-group calculations, hyperscaling, and the biased analysis of Nickel's series. Test-function analysis demonstrates the reliability of the method. Estimates for the leading index represent a significant improvement over older methods, e.g., the ratio method. Estimates for the correction exponent and amplitude are consistently high, presumably because the method must try to have one correction mimic all the corrections to the leading singularity.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
A simple method is proposed for determining characteristic parameters of critical-point singularities from power-series expansions. It applies to a broad class of functions of physical interest and provides acceptable, simultaneous approximations to the critical point, exponent, and amplitude. Several examples of low- and high-temperature series expansions for thermodynamic properties of spin-(1/2) Ising models are studied, and quite accurate results are obtained for both close- and loose-packed arrays.
Power-series expansions of the spin-pair correlation functions of the square, sc, and bcc Ising lattices have been obtained using the semi-invariant approach for general field HHH and temperature TTT. In three dimensions P\'ade-approximant analyses indicate $2{v}^{\ensuremath{'}}=1.285\ifmmode\pm\else\textpm\fi{}0.020$, in agreement with scaling, but $2{v}^{c}\ensuremath{\simeq}0.835\ifmmode\pm\else\textpm\fi{}0.02$ which is some (1-3)% above the scaling prediction ${v}^{c}=\frac{v}{\ensuremath{\beta}\ensuremath{\delta}}\ensuremath{\simeq}0.823$. However, ratio techniques reveal that this discrepancy can be attributed to significant confluent critical singularities. Cubic and quintic parametric representations of the critical equation of state, and corresponding expressions for the correlation length, ${\ensuremath{\xi}}_{1}(H, T)$, are developed, which are considerable improvements over the linear model. The universality and spherical symmetry of the critical scattering intensity $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ is confirmed to within (1-2)% by estimating suitable invariant combinations of amplitudes. The deviations from Ornstein-Zernike theory for general HHH and TTT are found to be considerably greater than for H=0H=0H=0, T>TcT>{T}_{c}T>Tc​. Complete parametric scaling representations of $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ are developed for three dimensions; corresponding scaling approximants are constructed in two dimensions but only for T=TcT={T}_{c}T=Tc​ and for $T\ensuremath{\gtrless}{T}_{c}$ with H=0H=0H=0.