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Physical Review

Further Applications of the Padé Approximant Method to the Ising and Heisenberg Models

George Allen Baker Jr.

Published Jan 1, 1963• 79 citations
Publisher

Abstract

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 $\frac{1}{(T\ensuremath{-}{T}_{c})}$.

DOI
10.1103/physrev.129.99

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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 ) .

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The eigenvalue problem of three-dimensional Ising model is solved in successive screw approximation on a computer. Suitable extrapolation of the results suggests that the maximum value of the specific heat of the simple cubic lattice tends to infinity, in proportion to the logarithm of the number of spins making up each layer. It does not seem unreasonable to consider that the temperature at which the specific heat has a maximum gradually approaches to the critical temperature predicted from the Padé approximant method as the number of spins in each layer increases.

Progress of Theoretical Physics•Nov 1, 1974•1 citation
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−Tc∣log|T\ensuremath{-}{T}_{c}|log∣T\ensuremath−Tc​∣ 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 1(T\ensuremath−Tc)\frac{1}{(T\ensuremath{-}{T}_{c})}(T\ensuremath−Tc​)1​.
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 ) .
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(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 ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{fcc}}\ensuremath{\approx}{[12.5(0.664658\ensuremath{-}{z}^{2})]}^{0.3}$, ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{bcc}}\ensuremath{\approx}{[10.4(0.5326607\ensuremath{-}{z}^{2})]}^{0.3}$, ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{sc}}\ensuremath{\approx}{[10.9(0.411940\ensuremath{-}{z}^{2})]}^{0.3}$, where $z=\mathrm{exp}(\ensuremath{-}\frac{2J}{\mathrm{kT}})$ and again the last place quoted is somewhat uncertain. The numbers 54\frac{5}{4}45​ and 74\frac{7}{4}47​ 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.
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.
Abstract The method of successive Padé approximants is applied to the high-temperature series for the susceptibility of Heisenberg systems with nearest-neighbour interactions only. It is concluded that the susceptibility of a ferromagnet diverges near the critical temperature with a law (T —Tc)-r, where r is either exactly 4/3 or indistinguishable from 4/3 by the method used with the power series at present available. The susceptibility of an antiferromagnet is also discussed. The specific heat near the critical point is considered but the results are inconclusive.
High-temperature series expansions of the partition functions for the Ising and Heisenberg models are analyzed for various values of the spin sss. The fcc lattice is used for which successive coefficients are sufficiently regular for estimates of critical behavior to be made with confidence. It is suggested that the magnetic susceptibility above the Curie temperature is of the form $A{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{\ensuremath{-}\frac{4}{3}}$ for the Heisenberg model for all sss, instead of ${A}^{\ensuremath{'}}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{\ensuremath{-}\frac{5}{4}}$ for the Ising model. Critical estimates of energy and entropy show that the magnitude of the "tail" of the specific heat anomaly is insensitive to the value of sss, and is about 2.5 times larger for the Heisenberg than for the Ising model. The sharpness of the anomaly at the Curie point increases as sss increases, and on passing from the Heisenberg to the Ising model. A brief reference is made to experimental results.
The eigenvalue problem of three-dimensional Ising model is solved in successive screw approximation on a computer. Suitable extrapolation of the results suggests that the maximum value of the specific heat of the simple cubic lattice tends to infinity, in proportion to the logarithm of the number of spins making up each layer. It does not seem unreasonable to consider that the temperature at which the specific heat has a maximum gradually approaches to the critical temperature predicted from the Padé approximant method as the number of spins in each layer increases.