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Journal of Physics A Mathematical Nuclear and General

Critical behaviour of the Ising model along the coexistence curve and the critical isotherm

Donald D. Betts, C. F. S. Chan

Published Mar 21, 1974• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/7/5/014

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Journal of Physics C Solid State Physics•Mar 14, 1973•62 citations
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.
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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
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.
We perform a Monte Carlo analysis of the Ising model on many three-dimensional lattices. By means of finite-size scaling we obtain the critical points and determine the scaling dimensions. As expected, the critical exponents agree with the three-dimensional Ising universality class for all models. The irrelevant field, as revealed by the correction-to-scaling amplitudes, appears to be relatively large. Combining the Monte Carlo results for the hydrogen peroxide lattice with those for five other three-dimensional lattices, we obtain a set of data covering a wide range of the irrelevant temperature field. This is helpful in the determination of the parameters describing the corrections to scaling. As a consequence, new results are obtained for the universal parameters describing Ising criticality in three dimensions, with reduced error margins in comparison with earlier Monte Carlo analyses. The critical exponents describing the thermodynamic singularities are determined by the temperature renormalization exponent yt = 1.58693(9) and the magnetic renormalization exponent yh = 2.48178(5). The corrections to scaling are governed by the irrelevant exponent y1 = −0.821(5).
The high-temperature series expansions for the locus of the maximum susceptibility are calculated from the high-temperature series for the magnetic field for various two- and three-dimensional lattices of the spin- 1 / 2 Ising model. The authors examine their critical behaviour and determine the critical temperature nu e and the exponent Delta (= beta delta ) using the ratio method and the Pade approximant method. The values obtained for nu c are in good agreement with exact values and values obtained from standard series method and the Delta also agrees with beta delta obtained from other estimates. These results are another new supporting evidence for the universality hypothesis.