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Physical Review

High-Temperature Specific Heat of the Ising Model

Richard A. Farrell, Paul H. E. Meijer

Published Apr 10, 1969• 5 citations
Publisher

Abstract

Pad\'e-approximant techniques are applied to the high-temperature series for the temperature derivative of the specific heat of the Ising model. This method indicates a critical exponent $\ensuremath{\alpha}$ in the neighborhood of $\frac{1}{8}$ for those Pad\'e approximants which yield an estimate of the critical temperature in substantial agreement with that determined from the susceptibility series.

DOI
10.1103/physrev.180.579

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For pt. III, see abstr. A21890 of 1970. A new low-temperature expansion variable for the Ising model is given. Expansions in the new variable, x=1-tanh(J/kT), then converge right up to the critical point. Power series in the new variable are derived, and then analysed by the Pade method and the method of N-point fits. For the spontaneous magnetization of the Ising model in three dimensions. beta =0.312+or-0.002 for the simple cubic lattice, beta =0.3125+or-0.0015 for the body-centred cubic lattice, and beta =0.312+or-0.004 for the face-centred cubic lattice. For the low-temperature susceptibility gamma =1.30+or-0.02, gamma =1.28+or-0.04, and gamma '=1.29+or-0.04 for the simple cubic, body-centred cubic and face-centred cubic lattices respectively. For the low-temperature specific heat the authors estimate 1 / 16 <or= alpha =0.2 for all three lattices.

Journal of Physics C Solid State Physics•Aug 1, 1970•14 citations
Pad'e-approximant techniques are applied to the high-temperature series for the temperature derivative of the specific heat of the Ising model. This method indicates a critical exponent \ensuremathα\ensuremath{\alpha}\ensuremathα in the neighborhood of 18\frac{1}{8}81​ for those Pad'e approximants which yield an estimate of the critical temperature in substantial agreement with that determined from the susceptibility series.
Pad\'e approximants have been used to estimate both the critical temperature and the critical-point exponents of the Ising model. In the standard Pad\'e analysis the approximants near the diagonal are given more credence. In the present study of two-dimensional models more credence is given to those entries for which the estimated critical temperature is close to the exact value. For three-dimensional models an analogous method is employed using the "exact" critical temperature estimated from the high-temperature compressibility series. The rationale for selecting these particular approximants is based on the fact that one is estimating critical-point behavior by extrapolating from a series representation. Thus, if the estimated critical point is shifted from its true value, one should expect a corresponding shift in the estimated exponent. Indeed, this is what is found.
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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
A high-temperature series expansion in the cumulant form of Brout (1959, 1960) is applied to mixed-spin Ising models on simple quadratic, simple cubic and body-centred cubic lattices. Motivation is provided primarily by an investigation of critical phenomena in Ising models with less than the usual translational symmetry and of relevance to the study of ferrimagnetism. Secondary motivation, appropriate to the simple quadratic lattice, is derived from a desire to obtain results that can be compared with those of renormalisation-group calculations performed elsewhere by the authors. Series expansions of the susceptibility and specific heat are obtained to seventh and tenth order, respectively. The derived expansion coefficients are analysed through both ratio and Pade approximant methods and exponent values are found to be in good agreement with those suggested by the universality hypothesis.
For pt. III, see abstr. A21890 of 1970. A new low-temperature expansion variable for the Ising model is given. Expansions in the new variable, x=1-tanh(J/kT), then converge right up to the critical point. Power series in the new variable are derived, and then analysed by the Pade method and the method of N-point fits. For the spontaneous magnetization of the Ising model in three dimensions. beta =0.312+or-0.002 for the simple cubic lattice, beta =0.3125+or-0.0015 for the body-centred cubic lattice, and beta =0.312+or-0.004 for the face-centred cubic lattice. For the low-temperature susceptibility gamma =1.30+or-0.02, gamma =1.28+or-0.04, and gamma '=1.29+or-0.04 for the simple cubic, body-centred cubic and face-centred cubic lattices respectively. For the low-temperature specific heat the authors estimate 1 / 16 <or= alpha =0.2 for all three lattices.