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Journal of Physics C Solid State Physics

Re-analysis of the critical isotherm of the Ising ferromagnet

David S. Gaunt, M F Sykes

Published Jun 26, 1972• 42 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/5/12/020

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A Test of Scaling Theory on the Critical Isotherm

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Canadian Journal of Physics•Dec 15, 1972•5 citations
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.
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.
The basic low temperature series expansion data for the spin one-half Ising model on the hydrogen peroxide lattice are used to obtain series in z=exp(-2J/k B T) along the coexistence curve for the specific heat C H , the magnetization M, and its first five derivatives delta 1 M/ delta mu 1 , where mu =exp(-2mH/k B T). The same data are used to derive series in mu along the critical isotherm for M and its first five derivatives delta l M/ delta z l . Ratio and Pade approximant analysis yield estimates of the critical exponents and critical amplitudes. On the whole the estimates of the critical exponents support scaling theory although a few of the exponent estimates are not in good agreement with scaling.
The critical isotherm of the Ising model for the four-dimensional simple hypercubic lattice is studied using a high-field expansion through mu 15 . The series for the magnetisation is analysed for singularities of the asymptotic form E(1- mu ) 1/3 mod ln(1- mu ) mod p , predicted by renormalisation group theory. Good convergence is obtained for values of p around 1 / 3 (the renormalisation group prediction) and we estimate p=0.30+or-0.05. Assuming p= 1 / 3 , the critical amplitude E is estimated to be 0.896+or-0.02.