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Physical Review

Application of the Padé Approximant Method to the Investigation of Some Magnetic Properties of the Ising Model

George Allen Baker Jr.

Published Nov 1, 1961• 309 citations
Publisher

Abstract

On the basis of the Pad\'e approximant method we deduce from the exact series expansions for the Ising model that the reduced magnetic susceptibility behaves at the critical point as ${\ensuremath{\chi}}_{\mathrm{fcc}}\ensuremath{\approx}{[\frac{0.09923}{(0.101767\ensuremath{-}w)}]}^{\frac{5}{4}}$, ${\ensuremath{\chi}}_{\mathrm{bcc}}\ensuremath{\approx}{[\frac{0.152773}{(0.1561789\ensuremath{-}w)}]}^{\frac{5}{4}}$, ${\ensuremath{\chi}}_{\mathrm{sc}}\ensuremath{\approx}{[\frac{0.22138}{(0.218156\ensuremath{-}w}]}^{\frac{5}{4}}$, ${\ensuremath{\chi}}_{\mathrm{t}}\ensuremath{\approx}{[\frac{0.2432}{(2\ensuremath{-}\sqrt{3}\ensuremath{-}w)}]}^{\frac{7}{4}}$, ${\ensuremath{\chi}}_{\mathrm{sq}}\ensuremath{\approx}{[\frac{0.35724}{(\sqrt{2}\ensuremath{-}1\ensuremath{-}w)}]}^{\frac{7}{4}}$, and ${\ensuremath{\chi}}_{\mathrm{h}}\ensuremath{\approx}{[\frac{0.4506}{(\frac{1}{\sqrt{3}}\ensuremath{-}w)}]}^{\frac{7}{4}}$, where $w=tanh(\frac{J}{\mathrm{kT}})$ and the last figure quoted is somewhat uncertain. The spontaneous magnetization is found to behave as ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{fcc}}\ensuremath{\approx}{[12.5(0.664658\ensuremath{-}{z}^{2})]}^{0.3}$, ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{bcc}}\ensuremath{\approx}{[10.4(0.5326607\ensuremath{-}{z}^{2})]}^{0.3}$, ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{sc}}\ensuremath{\approx}{[10.9(0.411940\ensuremath{-}{z}^{2})]}^{0.3}$, where $z=\mathrm{exp}(\ensuremath{-}\frac{2J}{\mathrm{kT}})$ and again the last place quoted is somewhat uncertain. The numbers $\frac{5}{4}$ and $\frac{7}{4}$ have an error of at most ${10}^{\ensuremath{-}3}$, and 0.3 of at most ${10}^{\ensuremath{-}2}$. The lattices referred to are fcc, face-centered cubic; bcc, body-centered cubic; sc, simple cubic; t, triangular; sq, simple quadratic; and h, honeycomb.

DOI
10.1103/physrev.124.768

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On the basis of the Pad'e approximant method we deduce from the exact series expansions for the Ising model that the reduced magnetic susceptibility behaves at the critical point as \ensuremathχfcc\ensuremath≈[0.09923(0.101767\ensuremath−w)]54{\ensuremath{\chi}}_{\mathrm{fcc}}\ensuremath{\approx}{[\frac{0.09923}{(0.101767\ensuremath{-}w)}]}^{\frac{5}{4}}\ensuremathχfcc​\ensuremath≈[(0.101767\ensuremath−w)0.09923​]45​, \ensuremathχbcc\ensuremath≈[0.152773(0.1561789\ensuremath−w)]54{\ensuremath{\chi}}_{\mathrm{bcc}}\ensuremath{\approx}{[\frac{0.152773}{(0.1561789\ensuremath{-}w)}]}^{\frac{5}{4}}\ensuremathχbcc​\ensuremath≈[(0.1561789\ensuremath−w)0.152773​]45​, \ensuremathχsc\ensuremath≈[0.22138(0.218156\ensuremath−w]54{\ensuremath{\chi}}_{\mathrm{sc}}\ensuremath{\approx}{[\frac{0.22138}{(0.218156\ensuremath{-}w}]}^{\frac{5}{4}}\ensuremathχsc​\ensuremath≈[(0.218156\ensuremath−w0.22138​]45​, \ensuremathχt\ensuremath≈[0.2432(2\ensuremath−3\ensuremath−w)]74{\ensuremath{\chi}}_{\mathrm{t}}\ensuremath{\approx}{[\frac{0.2432}{(2\ensuremath{-}\sqrt{3}\ensuremath{-}w)}]}^{\frac{7}{4}}\ensuremathχt​\ensuremath≈[(2\ensuremath−3​\ensuremath−w)0.2432​]47​, \ensuremathχsq\ensuremath≈[0.35724(2\ensuremath−1\ensuremath−w)]74{\ensuremath{\chi}}_{\mathrm{sq}}\ensuremath{\approx}{[\frac{0.35724}{(\sqrt{2}\ensuremath{-}1\ensuremath{-}w)}]}^{\frac{7}{4}}\ensuremathχsq​\ensuremath≈[(2​\ensuremath−1\ensuremath−w)0.35724​]47​, and \ensuremathχh\ensuremath≈[0.4506(13\ensuremath−w)]74{\ensuremath{\chi}}_{\mathrm{h}}\ensuremath{\approx}{[\frac{0.4506}{(\frac{1}{\sqrt{3}}\ensuremath{-}w)}]}^{\frac{7}{4}}\ensuremathχh​\ensuremath≈[(3​1​\ensuremath−w)0.4506​]47​, where w=tanh(JkT)w=tanh(\frac{J}{\mathrm{kT}})w=tanh(kTJ​) and the last figure quoted is somewhat uncertain. The spontaneous magnetization is found to behave as (I0I\ensuremath∞)fcc\ensuremath≈[12.5(0.664658\ensuremath−z2)]0.3{(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{fcc}}\ensuremath{\approx}{[12.5(0.664658\ensuremath{-}{z}^{2})]}^{0.3}(I\ensuremath∞​I0​​)fcc​\ensuremath≈[12.5(0.664658\ensuremath−z2)]0.3, (I0I\ensuremath∞)bcc\ensuremath≈[10.4(0.5326607\ensuremath−z2)]0.3{(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{bcc}}\ensuremath{\approx}{[10.4(0.5326607\ensuremath{-}{z}^{2})]}^{0.3}(I\ensuremath∞​I0​​)bcc​\ensuremath≈[10.4(0.5326607\ensuremath−z2)]0.3, (I0I\ensuremath∞)sc\ensuremath≈[10.9(0.411940\ensuremath−z2)]0.3{(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{sc}}\ensuremath{\approx}{[10.9(0.411940\ensuremath{-}{z}^{2})]}^{0.3}(I\ensuremath∞​I0​​)sc​\ensuremath≈[10.9(0.411940\ensuremath−z2)]0.3, where z=exp(\ensuremath−2JkT)z=\mathrm{exp}(\ensuremath{-}\frac{2J}{\mathrm{kT}})z=exp(\ensuremath−kT2J​) and again the last place quoted is somewhat uncertain. The numbers 54\frac{5}{4}45​ and 74\frac{7}{4}47​ have an error of at most 10\ensuremath−3{10}^{\ensuremath{-}3}10\ensuremath−3, and 0.3 of at most 10\ensuremath−2{10}^{\ensuremath{-}2}10\ensuremath−2. The lattices referred to are fcc, face-centered cubic; bcc, body-centered cubic; sc, simple cubic; t, triangular; sq, simple quadratic; and h, honeycomb.
We use the Pad\'e approximant method to investigate the nature of the singularity in the specific heat for some three-dimensional Ising model lattices and to investigate the nature and location of the singularity in the magnetic susceptibility for some three-dimensional Heisenberg model lattices. We find that the three-dimensional Ising model specific heat becomes singular like $log|T\ensuremath{-}{T}_{c}|$ with a different coefficient above and below the singular point. For the Heisenberg model we find that the magnetic susceptibility does not behave like the Curie-Weiss law, but tends to infinity more rapidly than $\frac{1}{(T\ensuremath{-}{T}_{c})}$.
The Pade approximant method w a s applied to Clapp's spinspin correlation formulation for the magnetic susceptibility in an Ising square lattice, in order to investigate the nature of the critical point.For the exact cluster formulation it is found that as the cluster size increases, the critical point, namely the point at which the susceptibility has a singularity, approaches Onsager's exact value of 0.414, and that ~( t ) -?/( tt c ) .For the selfconsistent formulation, twenty Pade approximants were calculated.The value of the critical point converged very rapidly to 0.41862770 and it was found that X ( t ) l/(tt c ) .
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.
Series expansions for certain thermodynamic variables have been computed for various standard model lattices. The high-temperature series for the magnetic susceptibility of the Ising model of three-dimensional SC, BCC and FCC lattices allows estimates of the critical index gamma of each series to be made. Dramatic progress in this direction was made by the Pade Dlog method, and here a similar technique is introduced. Estimates of gamma from row sequences of generalised inverse vector-valued Pade approximants are made. The results point to the conclusion that gamma =1.2406+or-0.0017 is the critical index common to all three lattices.