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Journal of Physics C Solid State Physics

Derivation of series expansions for the spin-spin correlation function of the Ising model

Douglas S. Ritchie, J W Essam

Published Oct 14, 1977• 0 citations
Publisher

Abstract

A method is described for obtaining perturbation expansions of the pair correlation function about the ferromagnetically ordered state of the S=1/2 Ising model. The expansions take the form of a double power series in temperature and field variables and may be used in any region of the T-H plane of the ferromagnetic and in the disordered region of the antiferromagnet. Coefficients are calculated for the face-centred cubic lattice which allow the low-temperature expansions in zero field to be obtained through u31 and the high-field expansions along the critical isotherm to be obtained through mu 6. Agreement is obtained with existing high-temperature expansions.

DOI
10.1088/0022-3719/10/19/017

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A method is described for obtaining perturbation expansions of the pair correlation function about the ferromagnetically ordered state of the S=1/2 Ising model. The expansions take the form of a double power series in temperature and field variables and may be used in any region of the T-H plane of the ferromagnetic and in the disordered region of the antiferromagnet. Coefficients are calculated for the face-centred cubic lattice which allow the low-temperature expansions in zero field to be obtained through u31 and the high-field expansions along the critical isotherm to be obtained through mu 6. Agreement is obtained with existing high-temperature expansions.
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 .
High-temperature series for the inverse spin-spin correlation function of the s=12s=\frac{1}{2}s=21​ Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at Tc{T}_{c}Tc​ and in the Ornstein---Zernike region are determined. The Fisher---Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.
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.
High-temperature series expansions for the Ising model's spin-spin correlation function are found to order (JkT)12{(\frac{J}{\mathrm{kT}})}^{12}(kTJ​)12 on the simple cubic, bcc, and fcc latices. Analysis of moment series indicates that the scaling form of the correlation function is correct, at least in the limit $\ensuremath{\kappa}r$ fixed, $\ensuremath{\kappa}\ensuremath{\rightarrow}0$. Values are obtained for the correlation indices $\ensuremath{\nu}$ and $\ensuremath{\eta}$-the latter by two methods, thereby providing a direct check on the scaling law.
The high-temperature expansion coefficients of the ordinary and the higher susceptibilities of the spin-1/2 nearest-neighbor Ising model are calculated exactly up to the 20th order for the general d-dimensional (hyper)simple-cubical lattices. These series are analyzed to study the dependence of critical parameters on the lattice dimensionality. Using the general d expression of the ordinary susceptibility, we have more than doubled the length of the existing series expansion of the critical temperature in powers of 1/d.