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Journal of Physics A Mathematical and General

On hyperscaling in the Ising model in three dimensions

Bernie G. Nickel, Bruce Sharpe

Published Oct 1, 1979• 38 citations
Publisher

Abstract

The spin- 1 / 2 Ising model high-temperature series are re-examined by an analysis that parallels the method used to analyse the continuum phi 4 spin model. Results of the Ising model in three dimensions do not agree with results in the literature based on more conventional methods of analysis. The authors cannot decide using only the series terms presently available, which of the methods is to be preferred. They conclude that previous error assessments are unduly optimistic and that the three-dimensional, spin- 1 / 2 , Ising model may satisfy hyperscaling in agreement with the continuum results. An attempt is made to estimate the number of additional high-temperature series terms necessary to resolve the hyperscaling question.

DOI
10.1088/0305-4470/12/10/027

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Journal de physique•Jan 1, 1979•86 citations
The spin- 1 / 2 Ising model high-temperature series are re-examined by an analysis that parallels the method used to analyse the continuum phi 4 spin model. Results of the Ising model in three dimensions do not agree with results in the literature based on more conventional methods of analysis. The authors cannot decide using only the series terms presently available, which of the methods is to be preferred. They conclude that previous error assessments are unduly optimistic and that the three-dimensional, spin- 1 / 2 , Ising model may satisfy hyperscaling in agreement with the continuum results. An attempt is made to estimate the number of additional high-temperature series terms necessary to resolve the hyperscaling question.
High-temperature series expansions are derived for ($\frac{{\ensuremath{\partial}}^{2}\ensuremath{\chi}}{\ensuremath{\partial}{H}^{2}}$) for arbitrary dimension. These series and others already available are analyzed to determine if hyperscaling holds. Hyperscaling is found to hold for d=2d=2d=2 but fails in three and four dimensions as $2\ensuremath{\Delta}\ensuremath{-}d\ensuremath{\nu}\ensuremath{-}\ensuremath{\gamma}=\ensuremath{-}0.028\ifmmode\pm\else\textpm\fi{}0.003 (d=3)$, $\ensuremath{-}0.302\ifmmode\pm\else\textpm\fi{}0.038 (d=4)$. The triviality of hyper-strong-coupling, Euclidean, boson, ${:{\ensuremath{\varphi}}^{4}:}_{3,4}$ field theory follows. The expected location of the zero of the $\ensuremath{\beta}$ function in the Callan-Symanzik equation is computed in two and three dimensions.
The validity of the hyperscaling relation 2 Delta -dv- gamma =0 is studied for the four-dimensional spin- 1 / 2 Ising model. High-temperature series expansions are derived for the fourth-field derivative chi 0 (2) of the free energy on the four-dimensional hyper-face-centred cubic (HFCC) and hyper-body-centred cubic (HBCC) lattices to order 9 and 11 respectively. These are analysed, together with other series already available for the susceptibility chi 0 and correlation length xi for the HFCC, HBCC and the hyper-simple cubic (HSC) lattices. All these series are found to behave consistently with the asymptotic form t -q mod lnt mod p , where t is a reduced temperature variable and q is the appropriate mean-field exponent (so that hyperscaling is satisfied automatically). The best estimates for p are as follows: p=0.30+or-0.05 (HFCC), 0.32+or-0.05 (HBCC) for xi 0 (with q=1= gamma ) and p=0.33+or-0.05 for xi 2 (with q=1=2v). These estimates are in good agreement with the renormalisation group (RG) prediction of p= 1 / 3 . Results for chi 0 (2) are more slowly convergent, but are still consistent with p= 1 / 3 for q=4=2 Delta + gamma .
The availability of extended series expansions for ${\ensuremath{\chi}}_{0}^{(2)}$=${\ensuremath{\partial}}^{2}$\ensuremath{\chi}/\ensuremath{\partial}H2{H}^{2}H2, accurate Monte Carlo estimates of the critical temperature for the sc spin-(1/2) Ising model, and refined methods of series analysis allow the hyperscaling relation d\ensuremath{\nu}=2\ensuremath{\Delta}-\ensuremath{\gamma} to be severely tested. From analysis of ${\ensuremath{\chi}}_{0}^{(2)}$ and ${\ensuremath{\chi}}_{0}^{(2)}$/${\ensuremath{\chi}}_{0}^{2}$ we estimate \ensuremath{\gamma}=1.240\ifmmode\pm\else\textpm\fi{}0.006 and \ensuremath{\Delta}=1.565\ifmmode\pm\else\textpm\fi{}0.004. The estimate of \ensuremath{\Delta} is in complete agreement with the prediction of hyperscaling using both the best extant series and renormalization-group estimates of the exponents \ensuremath{\gamma} and \ensuremath{\nu}. With the assumption of hyperscaling, our analysis gives \ensuremath{\nu}=0.630\ifmmode\pm\else\textpm\fi{}0.002.
High-temperature series for the second spherical moment ${M}_{2}=\ensuremath{\Sigma}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{}{x}^{2}〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}〉$ are presented to 15 terms for the spin-12\frac{1}{2}21​ Ising model, on both the simple cubic and body-centered-cubic lattices. These results are combined with previously known series to study hyperscaling to 15 terms on the simple cubic lattice. Some of the evidence suggests a violation of hyperscaling.
High temperature series expansion for the critical exponents of the Ising model are reanalysed using a modified ratio method. The analysis shows that a minor modification of the ratio method yields for all lattices a value, for the exponent γ in three dimensions, close to 1.245, therefore lower than the quoted value 1.250, and much closer to the renormalization group (R.G.) value 1.241. The exponent is analysed in two ways : in one method γ is estimated directly while in the other one Tc, is calculated first. With these new values of Tc, the exponent α is recalculated and found to be very close to the R.G. value 0.110. The value of ν is not modified (0.638) and is therefore still a problem for hyperscaling, and is in disagreement with the R.G. value (0.630).