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Physical Review Letters

Partial Differential Approximants for Multicritical Singularities

Michael Ellis Fisher, Robert M. Kerr

Published Sep 5, 1977• 42 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.39.667

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Physical Review•Jan 10, 1969•48 citations
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-XY\mathrm{XY}XY crossover in a three-dimensional classical ferromagnet. Encouraging results are obtained for the unbiased estimation of the bicritical point, of the multicritical exponents \ensuremathγ\ensuremath{\gamma}\ensuremathγ and \ensuremathφ\ensuremath{\varphi}\ensuremathφ, and of the slopes of the scaling axes.
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 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 xxx, fff 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 $\frac{{a}_{\ensuremath{\xi}}}{{a}_{\ensuremath{\chi}}}=0.83(5)$.
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.
Scaling and renormalization-group theories of crossover in criticality induced by random interactions are searchingly tested for Ising (n=1) systems. Extrapolation of 15th-order series in two variables for the susceptibility of the \ifmmode\pm\else\textpm\fi{}J simple-cubic Ising model yields a crossover exponent \ensuremath{\varphi}=0.125\ifmmode\pm\else\textpm\fi{}0.025. This agrees well with the specific-heat exponent, \ensuremath{\alpha}\ensuremath{\simeq}0.11. Consistency between free-energy and susceptibility scaling with \ensuremath{\varphi}=\ensuremath{\alpha} is cross-checked and leading universal scaling-function coefficients and nonuniversal scaling field amplitudes are determined.
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.