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Journal of Physics A Mathematical and General

Applications of partial differential approximants: test functions and dimensional crossover in the Ising model

Jürgen F. Stilck, Silvio R. A. Salinas

Published Aug 1, 1981• 11 citations

Abstract

The method of partial differential approximants (PDA) has been introduced for approximating functions of two or more variables given a finite number of coefficients of their power series. It is supposed to be effective close to the multicritical points, where the thermodynamic functions are expected to behave according to the scaling hypothesis. In order to assess the performance of the method, the authors have undertaken the construction of PDAS for several test functions of two variables. For some simple functions, which are represented exactly by PDAS and which exhibit an analogue of a multicritical point, the numerical estimates yielded very good results even at lower orders. For other functions the estimates tended to improve as the order of the approximants is increased. For the best PDAS they constructed flow diagrams and estimated the scaling functions. Also, they have analysed the dimensional crossover (from d=3 to d=2) of the sign- 1 / 2 axial anisotropic Ising model on the FCC lattice. They considered series expansions for the direct susceptibility and also for the sum of the direct and the staggered susceptibilities.

DOI: 10.1088/0305-4470/14/8/026

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The method of partial differential approximants (PDA) has been introduced for approximating functions of two or more variables given a finite number of coefficients of their power series. It is supposed to be effective close to the multicritical points, where the thermodynamic functions are expected to behave according to the scaling hypothesis. In order to assess the performance of the method, the authors have undertaken the construction of PDAS for several test functions of two variables. For some simple functions, which are represented exactly by PDAS and which exhibit an analogue of a multicritical point, the numerical estimates yielded very good results even at lower orders. For other functions the estimates tended to improve as the order of the approximants is increased. For the best PDAS they constructed flow diagrams and estimated the scaling functions. Also, they have analysed the dimensional crossover (from d=3 to d=2) of the sign- 1 / 2 axial anisotropic Ising model on the FCC lattice. They considered series expansions for the direct susceptibility and also for the sum of the direct and the staggered susceptibilities.

A recently proposed approach---partial differential approximants---for analyzing power series in two (or more) variables for functions which exhibit singularities of multicritical, scaling character is tested by analyzing the susceptibility series for the bicritical point describing the Ising-Heisenberg-

\mathrm{XY}

crossover in a three-dimensional classical ferromagnet. Encouraging results are obtained for the unbiased estimation of the bicritical point, of the multicritical exponents $\ensuremath{\gamma}$ and $\ensuremath{\varphi}$, and of the slopes of the scaling axes.

High-temperature series for two bcc lattice models which interpolate between the Gaussian (or free-field) model and the

S=\frac{1}{2}

Ising model are analyzed by partial differential approximants. Series to order 21 in both $x\ensuremath{\propto}\frac{1}{T}$ and the interpolation parameter,

y

, 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.

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 $\frac{{a}_{\ensuremath{\xi}}}{{a}_{\ensuremath{\chi}}}=0.83(5)$.

Scaling with a parameter, in contrast to ordinary scaling theory, makes predictions concerning amplitude functions as well as critical-point exponents. We provide the first test of these predictions, by examining the dependence of the amplitude on the anisotropy parameter

R=\frac{{J}_{z}}{{J}_{xy}}

for the susceptibility and the second moment of the simple cubic and fcc Ising models with directional anisotropy. The "double-power-law" behavior found strongly supports the scaling predictions for thermodynamic functions and the two-spin-correlation function. Our analysis provides a measure of the ranges of

R

over which parameter scaling appears to hold for the simple cubic and fcc lattices. The relative domains of validity on the two lattices are interpretable in terms of the respective lattice structures.

The susceptibility of simple-cubic-lattice ferromagnetic Ising films of

n

two-dimensional lattice layers is studied by extrapolation of high-temperature series expansions (to eleventh and twelfth orders) for

n=3,4,\dots{},14

layers with periodic boundary conditions, and

n=3,4,\dots{},10

layers with free-surface conditions. The corresponding surface susceptibility series for sc, bcc, and fcc lattices are analyzed in the light of universality hypotheses. On the basis of finite-size scaling theory explicit scaling functions are constructed describing the crossover from two-dimensional to three-dimensional critical behavior in terms of the ratio $\frac{n}{\ensuremath{\xi}(T)}$, where $\ensuremath{\xi}\ensuremath{\sim}\ensuremath{\Delta}{T}^{\ensuremath{-}\ensuremath{\nu}}$ is the bulk correlation length.