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Physical review. B, Solid state

Analysis of hyperscaling in the Ising model by the high-temperature series method

George Allen Baker Jr.

Published Feb 1, 1977• 106 citations
Publisher

Abstract

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 $d=2$ 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.

DOI
10.1103/physrevb.15.1552

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Validity of hyperscaling for thed=3Ising model

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Critical universality and hyperscaling revisited for Ising models of general spin using extended high-temperature series

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The authors review exact studies on finite-sized 2 dimensional Ising models and show that the point for an infinite-sized model at the critical temperature is a point of nonuniform approach in the temperature-size plane. They also illuminate some strong effects of finite-size on quantities which do not diverge at the critical point. They then review Monte Carlo studies for 3 dimensional Ising models of various sizes (L = 2--100) at various temperatures. From these results they find that the data for the renormalized coupling constant collapses nicely when plotted against the correlation length, determined in a system of edge length L, divided by L. They also find that {zeta}{sub L}/L {ge} 0.26 is definitely too large for reliable studies of the critical value, g*, of the renormalized coupling constant. They have reasonable evidence that {zeta}{sub L}/L {approx} 0.1 is adequate for results that are within one percent of those for the infinite system size. On this basis, they have conducted a series of Monte Carlo calculations with this condition imposed. These calculations were made practical by the development of improved estimators for use in the Swendsen-Wang cluster method. The authors found from these results, coupled with a reversed limit computation (size increases with the temperature fixed at the critical temperature), that g* > 0, although there may well be a sharp downward drop in g as the critical temperature is approached in accord with the predictions of series analysis. The results support the validity of hyperscaling in the 3 dimensional Ising model.

•Sep 1, 1995•0 citations
High-temperature series expansions are derived for (\ensuremath∂2\ensuremathχ\ensuremath∂H2\frac{{\ensuremath{\partial}}^{2}\ensuremath{\chi}}{\ensuremath{\partial}{H}^{2}}\ensuremath∂H2\ensuremath∂2\ensuremathχ​) 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\ensuremathΔ\ensuremath−d\ensuremathν\ensuremath−\ensuremathγ=\ensuremath−0.028\ifmmode±\else\textpm\fi0.003(d=3)2\ensuremath{\Delta}\ensuremath{-}d\ensuremath{\nu}\ensuremath{-}\ensuremath{\gamma}=\ensuremath{-}0.028\ifmmode\pm\else\textpm\fi{}0.003 (d=3)2\ensuremathΔ\ensuremath−d\ensuremathν\ensuremath−\ensuremathγ=\ensuremath−0.028\ifmmode±\else\textpm\fi0.003(d=3), \ensuremath−0.302\ifmmode±\else\textpm\fi0.038(d=4)\ensuremath{-}0.302\ifmmode\pm\else\textpm\fi{}0.038 (d=4)\ensuremath−0.302\ifmmode±\else\textpm\fi0.038(d=4). The triviality of hyper-strong-coupling, Euclidean, boson, :\ensuremathφ4:3,4{:{\ensuremath{\varphi}}^{4}:}_{3,4}:\ensuremathφ4:3,4​ field theory follows. The expected location of the zero of the \ensuremathβ\ensuremath{\beta}\ensuremathβ function in the Callan-Symanzik equation is computed in two and three dimensions.
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.
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 coefficients in the Callan-Symanzik equations for a three-dimensional, continuous spin Ising model with an $\mathrm{exp}(\ensuremath{-}A{s}^{4}+B{s}^{2})$ spin-weight factor are expanded in the dimensionless, renormalized coupling constant. These series are summed by the Pad\'e-Borel method to yield the critical indices $\ensuremath{\gamma}=1.241\ifmmode\pm\else\textpm\fi{}0.002$, $\ensuremath{\eta}=0.02\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\nu}=0.63\ifmmode\pm\else\textpm\fi{}0.01$, and ${\ensuremath{\Delta}}_{1}=0.49\ifmmode\pm\else\textpm\fi{}0.01$.
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.
We have extended through ${\ensuremath{\beta}}^{23}$ the high-temperature expansion of the second field derivative of the susceptibility for Ising models of general spin, with nearest-neighbor interactions, on the simple cubic and the body-centered cubic lattices. Moreover, the expansions for the nearest-neighbor correlation function, the susceptibility, and the second correlation moment have been extended up to ${\ensuremath{\beta}}^{25}.$ Taking advantage of these new data, we can improve the accuracy of direct estimates of critical exponents and of hyperuniversal combinations of critical amplitudes such as the renormalized four-point coupling gr{g}_{r}gr​ or the quantity usually denoted by ${R}_{\ensuremath{\xi}}^{+}.$ In particular, we obtain $\ensuremath{\gamma}=1.2371(1),$ $\ensuremath{\nu}=0.6299(2),$ ${\ensuremath{\gamma}}_{4}=4.3647(20),$ gr=1.404(3),{g}_{r}=1.404(3),gr​=1.404(3), and ${R}_{\ensuremath{\xi}}^{+}=0.2668(5).$ We have used a variety of series extrapolation procedures and, in some of the analyses, we have assumed that the leading correction-to-scaling exponent $\ensuremath{\theta}$ is universal and roughly known. We have also verified, to high precision, the validity of the hyperscaling relation and of the universality property both with regard to the lattice structure and to the value of the spin.
The authors review exact studies on finite-sized 2 dimensional Ising models and show that the point for an infinite-sized model at the critical temperature is a point of nonuniform approach in the temperature-size plane. They also illuminate some strong effects of finite-size on quantities which do not diverge at the critical point. They then review Monte Carlo studies for 3 dimensional Ising models of various sizes (L = 2--100) at various temperatures. From these results they find that the data for the renormalized coupling constant collapses nicely when plotted against the correlation length, determined in a system of edge length L, divided by L. They also find that {zeta}{sub L}/L {ge} 0.26 is definitely too large for reliable studies of the critical value, g*, of the renormalized coupling constant. They have reasonable evidence that {zeta}{sub L}/L {approx} 0.1 is adequate for results that are within one percent of those for the infinite system size. On this basis, they have conducted a series of Monte Carlo calculations with this condition imposed. These calculations were made practical by the development of improved estimators for use in the Swendsen-Wang cluster method. The authors found from these results, coupled with a reversed limit computation (size increases with the temperature fixed at the critical temperature), that g* > 0, although there may well be a sharp downward drop in g as the critical temperature is approached in accord with the predictions of series analysis. The results support the validity of hyperscaling in the 3 dimensional Ising model.