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Proceedings of the Physical Society

The critical isotherm and critical exponents of the three-dimensional ising ferromagnet

David S. Gaunt

Published Sep 1, 1967• 64 citations
Publisher

Abstract

With the aid of exact seriesexpansions the shape of the critical isotherm of an Ising ferromagnet (or fluid) is re-analysed and it is conjectured that δ = 5 for all three-dimensional lattices. (The analysis strengthens previous conjectures that δ = 15 for all two-dimensional lattices.) Estimates are given for the amplitude of the singularity for the various lattices. Evidence is presented to support the conjecture that α' = α = 1/8, β = 5/16 and γ' = γ = 1¼ for all three-dimensional lattices.

DOI
10.1088/0370-1328/92/1/321

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Critical exponents in a new integral representation for the Ising ferromagnet

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Low-Temperature Critical Exponents from High-Temperature Series: The Ising Model

David S. Gaunt, George Allen Baker Jr.

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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.

Physical review. B, Solid state•Feb 1, 1970•113 citations
With the aid of exact seriesexpansions the shape of the critical isotherm of an Ising ferromagnet (or fluid) is re-analysed and it is conjectured that δ = 5 for all three-dimensional lattices. (The analysis strengthens previous conjectures that δ = 15 for all two-dimensional lattices.) Estimates are given for the amplitude of the singularity for the various lattices. Evidence is presented to support the conjecture that α' = α = 1/8, β = 5/16 and γ' = γ = 1¼ for all three-dimensional lattices.
Using extended high-field series expansions and more precise estimates of the critical temperature the critical isotherm of the spin 1 / 2 Ising ferromagnet is re-analysed. The conjectures delta =15 (d=2) and delta =5 (d=3) are confirmed to within 1 / 2 % and 1%, respectively. Good agreement is found, particularly in two dimensions, between series estimates of the critical amplitudes and the predictions of the lattice-lattice scaling hypothesis. Some tentative conclusions regarding the coincident and nonphysical singularities are drawn.
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.
The rational values beta = 5 / 16 , delta =5, alpha = 1 / 8 , nu = 5 / 8 , and eta =0 of the critical exponents for the three-dimensional spin 1 / 2 Ising ferromagnet are shown to be consistent with a recently-derived linearised equation for the Ising two-spin correlation function. This equation arises from an exact integral representation which previously, on application of a straightforward approximation for higher-order cumulant averages, was shown to yield transition temperatures T c to within 2% of their correct values for the cubic lattices. In the search for beta , the requirement is imposed that the correction terms for the magnetisation and for M 2 , a quantity closely related to the system internal energy, be analytic in some power of T c -T (which turns out to be 1 / 8 ) when the external magnetic field h is zero.
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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.