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Journal of Physics A Mathematical and General

The Ising model on the tetrahedron lattice. I. An extended susceptibility series

J Ho-Ting-Hun, J. Oitmaa

Published Dec 1, 1975• 12 citations
Publisher

Abstract

The authors have calculated the first 16 terms of the series expansion for the high-temperature susceptibility of the Ising model for a lattice made up of the B sites of the crystobalite or spinel structures (the 'tetrahedron' lattice). By means of a transformation technique which allows the enumeration of graphs to be carried out on the simpler diamond lattice six new terms have been added to previously published results. They estimate that the critical temperature is given by nu c =0.23300+or-0.00001 and that the exponent gamma is gamma =1.250+or-0.001, a result which gives further confirmation of the universality hypothesis.

DOI
10.1088/0305-4470/8/12/009

Related papers

The Ising model on the tetrahedron lattice. II. Amplitudes and confluent singularities

J. Oitmaa, J Ho-Ting-Hun

For pt.I see ibid., vol.8, p.1920 of 1975. The authors have carried out a further analysis of the recently derived 16 term high temperature susceptibility series for the spin 1/2 Ising model on the tetrahedron lattice (crystobalite or B-site spinel lattice). They have found convincing evidence for the asymptotic form chi =C 0 (1-K/K c -5 4/+C 1 (1-K/K c -1 4/+X 0 +C 2 (1-K/K c ) 3 4/+... and have obtained estimates for the amplitudes C 0 , C 1 , C 2 and X 0 . They have also used recent lattice-lattice scaling hypotheses to calculate the specific heat and magnetization amplitudes.

Journal of Physics A Mathematical and General•Mar 1, 1976•5 citations

The Ising model on the tetrahedron lattice. III. Four-spin interactions

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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.

Journal of Physics A Mathematical and General•Dec 1, 1976•5 citations

The critical behaviour of the maximum susceptibility locus

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The critical exponent gamma for the three-dimensional Ising model

J. Oitmaa, J Ho-Ting-Hun

The authors have re-examined the high-temperature susceptibility series of the spin- 1 / 2 Ising model on the three-dimensional tetrahedron lattice. They conclude that the series data provide strong evidence for the result gamma =1.250 and are inconsistent with the recent renormalisation group prediction gamma =1.2402+or-0.0009.

Journal of Physics A Mathematical and General•Oct 1, 1979•8 citations

High-temperature specific heat series of the Ising model on the crystobalite lattice

R. W. Gibberd

Betts and Ditzian have recently published the first 11 coefficients of the exact high-temperature series for the specific heat of the spin 1/2 Ising model on a crystobalite lattice. In this paper the exact coefficients for the next 8 terms are derived by making use of an approximate transformation between the Ising partition function of the crystobalite and diamond lattices. The series is analyzed by using the ratio and Padé approximant methods, but a reliable estimate for α has not been obtained.

Canadian Journal of Physics•Feb 1, 1970•5 citations

High-temperature series for the B-site spinel and diamond lattices and the question of universality

D.N. Lambeth, M. Howard Lee, Eugene Stanley

The high-temperature series for the two-spin correlation functions on both the B-site spinel and diamond lattices have been calculated through tenth order for the Hamiltonian, H=−JΣ〈 ij 〉 Σα=1n(n≤ D) SiαSjα. Here Siα is the αth component of the D-dimensional classical spin and we define n as the spin space symmetry parameter. By <ij> we limit the first summation to nearest-neighbor pairs of sites i and j on the lattice. The spin space symmetry parameter n is ≤D. We consider all six models resulting from D=1–3. Our analysis of the high-temperature series indicates that while the exponent estimates for the more loosely packed diamond lattice (q=4) indicate agreement with the universality hypothesis, the B-site spinel lattice (q=6) series are not yet displaying their critical behavior. Some conjectures concerning this peculiarity of the series for the two lattices are made.

The Journal of Chemical Physics•Feb 1, 1974•3 citations
The authors have calculated the first 16 terms of the series expansion for the high-temperature susceptibility of the Ising model for a lattice made up of the B sites of the crystobalite or spinel structures (the 'tetrahedron' lattice). By means of a transformation technique which allows the enumeration of graphs to be carried out on the simpler diamond lattice six new terms have been added to previously published results. They estimate that the critical temperature is given by nu c =0.23300+or-0.00001 and that the exponent gamma is gamma =1.250+or-0.001, a result which gives further confirmation of the universality hypothesis.
For pt.I see ibid., vol.8, p.1920 of 1975. The authors have carried out a further analysis of the recently derived 16 term high temperature susceptibility series for the spin 1/2 Ising model on the tetrahedron lattice (crystobalite or B-site spinel lattice). They have found convincing evidence for the asymptotic form chi =C 0 (1-K/K c -5 4/+C 1 (1-K/K c -1 4/+X 0 +C 2 (1-K/K c ) 3 4/+... and have obtained estimates for the amplitudes C 0 , C 1 , C 2 and X 0 . They have also used recent lattice-lattice scaling hypotheses to calculate the specific heat and magnetization amplitudes.
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 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.
Betts and Ditzian have recently published the first 11 coefficients of the exact high-temperature series for the specific heat of the spin 1/2 Ising model on a crystobalite lattice. In this paper the exact coefficients for the next 8 terms are derived by making use of an approximate transformation between the Ising partition function of the crystobalite and diamond lattices. The series is analyzed by using the ratio and Padé approximant methods, but a reliable estimate for α has not been obtained.
The high-temperature series for the two-spin correlation functions on both the B-site spinel and diamond lattices have been calculated through tenth order for the Hamiltonian, H=−JΣ〈 ij 〉 Σα=1n(n≤ D) SiαSjα. Here Siα is the αth component of the D-dimensional classical spin and we define n as the spin space symmetry parameter. By <ij> we limit the first summation to nearest-neighbor pairs of sites i and j on the lattice. The spin space symmetry parameter n is ≤D. We consider all six models resulting from D=1–3. Our analysis of the high-temperature series indicates that while the exponent estimates for the more loosely packed diamond lattice (q=4) indicate agreement with the universality hypothesis, the B-site spinel lattice (q=6) series are not yet displaying their critical behavior. Some conjectures concerning this peculiarity of the series for the two lattices are made.