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Journal of Physics C Solid State Physics

Critical behaviour of a generalized Ising model

J. Oitmaa

Published Feb 29, 1972• 46 citations
Publisher

Abstract

A high temperature expansion is obtained for the zero field susceptibility of a generalized Ising model, which can be used to describe the critical behaviour of various multicomponent systems. The first seven terms of the series are obtained for the face centred cubic lattice. Estimates of the critical temperature obtained from this series are considerably more precise than those obtained previously from the zero field free energy. Estimates are also obtained for the critical exponents of the susceptibility and the derivatives of the zero field free energy. The results are compared with the predictions of mean field theory.

DOI
10.1088/0022-3719/5/4/010

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Journal of Physics C Solid State Physics•Nov 1, 1969•32 citations
A high temperature expansion is obtained for the zero field susceptibility of a generalized Ising model, which can be used to describe the critical behaviour of various multicomponent systems. The first seven terms of the series are obtained for the face centred cubic lattice. Estimates of the critical temperature obtained from this series are considerably more precise than those obtained previously from the zero field free energy. Estimates are also obtained for the critical exponents of the susceptibility and the derivatives of the zero field free energy. The results are compared with the predictions of mean field theory.
For pt. I see ibid., vol. 5, no. 5, 624. Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K for the simple cubic lattice to v 17 , the body-centred cubic lattice to v 15 , and the face-centred cubic lattice to v 12 , inclusive. The coefficients are analysed by the ratio method and it is estimated that, subject to the implicit assumptions thereof, the critical temperatures correspond to v f =0.21813+or-0.00001 (SC), v f =0.15612+or-0.00003 (BCC) and v f =0.10174+or-0.00001 (FCC). The symptotic behaviour of the coefficients is studied in detail; for the loose-packed lattices the decay of the even-odd oscillation in the ratios is found to be consistent with the assumption that the critical index of the high temperature specific heat is very close to 1/8.
The first eleven terms of the high temperature susceptibility series for the anisotropic Ising model on a simple cubic lattice are calculated. The series is analysed to obtain estimates of the critical temperature and critical exponent for various values of the anisotropy parameter eta . Strong evidence is found for an apparent continuous variation of the exponent with eta which is in contradiction with the discontinuous change at eta =0 which has been generally believed to occur. The detailed variation of the critical temperature with eta is also in disagreement with predictions from generalized scaling laws.
Extended high-temperature series expansions are presented for the spin-s Ising model on the face-centred cubic lattice. Series coefficients for the zero-field internal energy (U 0 ) and the second moment ( mu 2 ) of the correlation function are derived, for 1 / 2 <or=s<or= 9 / 2 , to order 14 in the high-temperature expansion variable K(=J/kT). For s= 1 / 2 , the fourth field derivative of the free energy is calculated to order K 14 and the magnetisation series is given to order 13 and 27 in the temperature and magnetic field variables respectively. For s=1 to 9 / 2 , the zero-field susceptibility series are derived to order K 14 .
Both weak-coupling and strong-coupling series expansions are calculated for the Ising model in a transverse field at zero temperature in three dimensions. Series are obtained for the ground-state energy, the magnetization, the susceptibility, and the energy gaps, on the simple cubic, the body-centred cubic and the face-centred cubic lattices. The analysis of the critical behaviour is consistent with the behaviour predicted by renormalization group theory for the four-dimensional simple Ising model. There is a remarkable degree of universality between all three lattices.
Low temperature Ising model series in the usual variable z = exp(−4 J/kT ) are analysed. The positions of the singularities in the disk|z| <or=|z c | and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0·312 ± 0·002 for the simple cubic and face-centred cubic lattices, and β = 0·312 ± 0·004 for the body-centred cubic lattice. γprime = 1·31 ± 0·02 for the simple cubic lattice, γprime = 1·28 ± 0·04 for the body-centred cubic lattice, and γprime = 1·3 ± 0·1 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.