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Physical review. B, Solid state

Critical-Algebra Coefficients for the Three-Dimensional Ising Model

Reginald Rutherford, Michael Wortis

Published Aug 1, 1973• 0 citations
Publisher

Abstract

High-temperature series expansions are used to test the reduction hypothesis of Kadanoff's operator algebra for the three-dimensional fcc Ising model with spin $s=\frac{3}{2},2,\ensuremath{\infty}$. Agreement with our data is convincing. Numerical values are given for the coupling coefficients between the order parameter $\ensuremath{\mu}$ and ${\ensuremath{\mu}}^{3}$ and between between ${\ensuremath{\mu}}^{2}$ and the local energy density.

DOI
10.1103/physrevb.8.1167

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Journal of physics. A, Proceedings of the Physical Society. General•May 1, 1972•139 citations
High-temperature series expansions are used to test the reduction hypothesis of Kadanoff's operator algebra for the three-dimensional fcc Ising model with spin s=32,2,\ensuremath∞s=\frac{3}{2},2,\ensuremath{\infty}s=23​,2,\ensuremath∞. Agreement with our data is convincing. Numerical values are given for the coupling coefficients between the order parameter \ensuremathμ\ensuremath{\mu}\ensuremathμ and \ensuremathμ3{\ensuremath{\mu}}^{3}\ensuremathμ3 and between between \ensuremathμ2{\ensuremath{\mu}}^{2}\ensuremathμ2 and the local energy density.
The energy-density-energy-density correlation function is studied in the three-dimensional Ising model in zero external field, for various spins on the fcc lattice. An extension of the linked-cluster technique is used to obtain high-temperature series expansions for the correlation function and its associated functions, to at least eighth order in the high-temperature interaction variable. Analysis of the series provides values for their critical indices and amplitudes. These values are used to verify the predictions of several theories of critical behavior. The critical index of the coherence length of the energy density correlation function is found to be equal to that of the spin-spin correlation function for all spins within the accuracy obtained here. For spins higher than ~ this critical index appears to obey strong scaling. An analysis of universality indicates that only one scale factor is needed to transform the energy density correlation function from one spin to another. This prediction is checked here and found to be good within our present accuracy. The current-algebra hypothesis predicts definite relations between the leading singularities of various correlation functions at the critical point. This prediction is checked directly and consistent values of the leading coefficients are obtained. The shape of the energy density correlation function at large distances from the origin, both at and above the critical point, is determined. The results are sufficiently accurate to rule out the Ornstein-Zernike form in zero field for T > T , but not sufficient to eliminate the possibility c of anisotropy at T = T .
The susceptibility of the spin-sss nearest-neighbor fcc Ising model is studied at zero magnetic field by twelve-term high-temperature series. We find that the data near criticality are consistent with $\ensuremath{\chi}=A(s){t}^{\frac{\ensuremath{-}5}{4}}[1+B(s){t}^{{\ensuremath{\Delta}}_{1}}]$, $t=1\ensuremath{-}\frac{{T}_{c}(s)}{T}$. The correction-to-scaling exponent has the universal value ${\ensuremath{\Delta}}_{1}=0.50\ifmmode\pm\else\textpm\fi{}0.05$ in good agreement with $\ensuremath{\epsilon}$-expansion estimates. The confluent amplitude B(s)B(s)B(s) vanishes at s=12s=\frac{1}{2}s=21​ to within uncertainties. Values of the critical parameters are tabulated for a variety of spins.
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.
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.
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.