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Journal of Physics C Solid State Physics

Critical behaviour of two Ising models with four-spin interactions

J. Oitmaa, R. W. Gibberd

Published Jul 9, 1973• 39 citations
Publisher

Abstract

Two two-dimensional Ising models with two- and four-spin interactions are studied. Evidence from high temperature series expansions suggests that the susceptibility exponent gamma remains constant at 1.75, independent of J 4 , the strength of the four- spin interaction. This is in agreement with the general 'universality' hypothesis. Mean field theory predicts that for J 4 large enough the transition becomes first-order, but we find no evidence for this from the series expansions.

DOI
10.1088/0022-3719/6/13/008

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The Monte Carlo technique is applied to a study of the phase transitions and the critical behavior of the spin-\textonehalf{} Ising model on an fcc lattice with mixtures of two- (${J}_{2}$) and four - (${J}_{4}$) spin interactions. In the limit ${J}_{2}=0$ the model exhibits a first-order transition. The transition remains of first order for $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\gtrsim}\frac{1}{2}$, but a crossover to continuous transitions is found around $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\approx}\frac{1}{4}\ensuremath{-}\frac{1}{2}$ indicating that the model exhibits tricritical behavior. A modified mean-field theory is presented leading to an approximate description of the tricritical behavior in agreement with the Monte Carlo calculations. In the region of continuous transitions. $0<~\frac{{J}_{4}}{{J}_{2}}\ensuremath{\lesssim}\frac{1}{4}$, the critical exponent $\ensuremath{\beta}$ pertaining to the order parameter derived from the Monte Carlo data retains the Ising value, in accordance with the universality hypothesis. Our findings show that the four-spin interactions do not lead to nonuniversal critical behavior, contrary to the conclusions made by Griffiths and Wood from a series analysis.

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Low-temperature expansions for Ising models with four-spin interactions

J. Oitmaa

Low-temperature series for the spontaneous magnetization are derived for two Ising models with four-spin interactions-(i) a model studied previously by the author using high temperature series, (ii) a model equivalent to Baxter's (1973) eight-vertex model. In the former case the exponent beta appears to remain constant at 1/8, independent of J 4 , the strength of the four-spin interaction; for the eight-vertex model beta varies continuously with J 4 as expected.

Journal of Physics C Solid State Physics•Jan 21, 1974•14 citations
Two two-dimensional Ising models with two- and four-spin interactions are studied. Evidence from high temperature series expansions suggests that the susceptibility exponent gamma remains constant at 1.75, independent of J 4 , the strength of the four- spin interaction. This is in agreement with the general 'universality' hypothesis. Mean field theory predicts that for J 4 large enough the transition becomes first-order, but we find no evidence for this from the series expansions.
We have calculated the high-temperature series expansion of the zero-field susceptibility of the square-lattice Ising model with first and second neighbour interactions to the 20th order by computer. Our results extend the previous calculation by Hsiao and Lin to two more orders. We use the Padé approximants to estimate the critical exponent γ and the critical temperature. Our result 1.747 < γ < 1.753 supports the universality conjecture that all two-dimensional Ising models have the same critical exponent γ = 1.75.
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.
For pt.II see ibid., vol.9, p.479 (1976). An earlier study of the conventional nearest-neighbour Ising model on the tetrahedron lattice is extended to include four-spin interactions about each tetrahedron. The problem of deriving high-temperature susceptibility series can be converted into a form for which previous graphical techniques can be readily applied. Analysis of the series by Pade approximants and the ratio method suggests that the exponent gamma has the constant value of 1.25 independent of the strength of the four-spin coupling, in agreement with universality.
The Monte Carlo technique is applied to a study of the phase transitions and the critical behavior of the spin-\textonehalf{} Ising model on an fcc lattice with mixtures of two- (J2{J}_{2}J2​) and four - (J4{J}_{4}J4​) spin interactions. In the limit J2=0{J}_{2}=0J2​=0 the model exhibits a first-order transition. The transition remains of first order for $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\gtrsim}\frac{1}{2}$, but a crossover to continuous transitions is found around $\frac{{J}_{4}}{{J}_{2}}\ensuremath{\approx}\frac{1}{4}\ensuremath{-}\frac{1}{2}$ indicating that the model exhibits tricritical behavior. A modified mean-field theory is presented leading to an approximate description of the tricritical behavior in agreement with the Monte Carlo calculations. In the region of continuous transitions. $0<~\frac{{J}_{4}}{{J}_{2}}\ensuremath{\lesssim}\frac{1}{4}$, the critical exponent $\ensuremath{\beta}$ pertaining to the order parameter derived from the Monte Carlo data retains the Ising value, in accordance with the universality hypothesis. Our findings show that the four-spin interactions do not lead to nonuniversal critical behavior, contrary to the conclusions made by Griffiths and Wood from a series analysis.