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Journal of physics. A, Proceedings of the Physical Society. General

High temperature series for the susceptibility of the Ising model. I. Two dimensional lattices

M F Sykes, David S. Gaunt, P D Roberts, J A Wyles

Published May 1, 1972• 90 citations
Publisher

Abstract

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 triangular lattice to v 16 , for the square lattice to v 21 and for the honeycomb lattice to v 32 , inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.

DOI
10.1088/0305-4470/5/5/004

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High temperature series for the susceptibility of the Ising model. II. Three dimensional lattices

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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.

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Low-temperature series expansions for the (2+1)-dimensional Ising model

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We study complex-temperature singularities of the Ising model on the triangular and honeycomb lattices. We first discuss the complex-T phases and their boundaries. From exact results, we determine the complex-T singularities in the specific heat and magnetization. For the triangular lattice we discuss the implications of the divergence of the magnetization at the point (where ) and extend a previous study by Guttmann of the susceptibility at this point with the use of differential approximants. For the honeycomb lattice, from an analysis of low-temperature series expansions, we have found evidence that the uniform and staggered susceptibilities and both have divergent singularities at , and our numerical values for the exponents are consistent with the hypothesis that the exact values are . The critical amplitudes at this singularity were calculated. Using our exact results for and together with numerical values for from series analyses, we find that the exponent relation is violated at z = -1 on the honeycomb lattice; the right-hand side is consistent with being equal to 4 rather than 2. The connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed.

arXiv:hep-lat/9411023•Journal of Physics A Mathematical and General•Feb 21, 1996•41 citations
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 triangular lattice to v 16 , for the square lattice to v 21 and for the honeycomb lattice to v 32 , inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.
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.
By combining all available information it is possible to extend the numerical knowledge of the high- and low-temperature zero-field isothermal susceptibility of the Ising model. Estimates of the third most singular term of the ferromagnetic singularity are obtained, using both direct estimates and an extension of the Generalized Law of Corresponding States. A recent conjecture of Barouch, McCoy and Wu (1973) concerning the equality of the constant term in the asymptotic expansion of the high- and low-temperature ferromagnetic susceptibility is confirmed numerically. Finally, a non-physical critical exponent of the low-temperature susceptibility of the triangular lattice is identified.
We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.
The first two terms in the expansion near TC{\mathit{T}}_{\mathit{C}}TC​ of the zero-field susceptibility of the antiferromagnetic square Ising lattice are known exactly. We have isolated the next four terms in this expansion by carrying out a differential-approximant analysis of Nickel's 55-term high-temperature ferromagnetic series with the variable v=tanhJ/kB{\mathit{k}}_{\mathit{B}}kB​T replaced by -v. The four new terms are found to vary as t, t2{\mathit{t}}^{2}t2ln\ensuremath{\Vert}t\ensuremath{\Vert}, t2{\mathit{t}}^{2}t2, and t3{\mathit{t}}^{3}t3ln\ensuremath{\Vert}t\ensuremath{\Vert} where t=1-TC{\mathit{T}}_{\mathit{C}}TC​/T and values for their respective coefficients are obtained. It is established that to order t3{\mathit{t}}^{3}t3ln\ensuremath{\Vert}t\ensuremath{\Vert}, there are no confluent singularities associated with irrelevant variables for this system. A direct extrapolation of Nickel's series coefficients is also carried out and found to be consistent with the present results.
We study complex-temperature singularities of the Ising model on the triangular and honeycomb lattices. We first discuss the complex-T phases and their boundaries. From exact results, we determine the complex-T singularities in the specific heat and magnetization. For the triangular lattice we discuss the implications of the divergence of the magnetization at the point (where ) and extend a previous study by Guttmann of the susceptibility at this point with the use of differential approximants. For the honeycomb lattice, from an analysis of low-temperature series expansions, we have found evidence that the uniform and staggered susceptibilities and both have divergent singularities at , and our numerical values for the exponents are consistent with the hypothesis that the exact values are . The critical amplitudes at this singularity were calculated. Using our exact results for and together with numerical values for from series analyses, we find that the exponent relation is violated at z = -1 on the honeycomb lattice; the right-hand side is consistent with being equal to 4 rather than 2. The connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed.