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Physical review. B, Condensed matter

Validity of hyperscaling for thed=3Ising model

Anthony John Guttmann

Published Apr 1, 1986• 22 citations
Publisher

Abstract

The availability of extended series expansions for ${\ensuremath{\chi}}_{0}^{(2)}$=${\ensuremath{\partial}}^{2}$\ensuremath{\chi}/\ensuremath{\partial}${H}^{2}$, 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.

DOI
10.1103/physrevb.33.5089

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A test of hyperscaling for the spin-1/2Ising model in four dimensions

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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 .

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Series analysis of corrections to scaling for the spin-pair correlations of the spin- s Ising model: Confluent singularities, universality, and hyperscaling

William J. Camp, D. M. Saul, John P. VAN DYKE +1 more

We report a detailed study of twelve-term, high-temperature series for the second moment of spin-pair correlations ${\ensuremath{\mu}}_{2}(t)$ and the specific heat ${c}_{H}(t)$ of the nearest-neighbor spin-$s$ Ising model in zero magnetic field on the fcc lattice. Near criticality we find ${\ensuremath{\mu}}_{2}(t)={A}_{2}(s)t{(s)}^{\ensuremath{-}(\ensuremath{\gamma}+2\ensuremath{\nu})}[1 + {B}_{2}(s)t{(s)}^{{\ensuremath{\Delta}}_{1}}+ \ensuremath{\cdots}]$, {$t(s)=\frac{[T\ensuremath{-}{T}_{c}(s)]}{T}$}, showing a confluent correction to the dominant scaling singularity. To within uncertainties the exponents have the universal (i.e., spin-independent) values $\ensuremath{\nu}={0.638}_{\ensuremath{-}0.008}^{+0.002}$ (with $\ensuremath{\gamma}={1.250}_{\ensuremath{-}0.007}^{+0.003}$) and ${\ensuremath{\Delta}}_{1}=0.6\ifmmode\pm\else\textpm\fi{}0.1$. The confluent exponent ${\ensuremath{\Delta}}_{1}$ is in reasonable agreement with the correction-to-scaling index derived from earlier analysis of the susceptibility, as predicted by renormalization-group arguments. A similar analysis of the specific heat ${c}_{H}$ for the same model finds no detectable confluent singularities in rather noisy, high-temperature series and gives $\ensuremath{\alpha}=0.125\ifmmode\pm\else\textpm\fi{}0.020$ in general confirmation of earlier $s=\frac{1}{2}$ estimates. With $\ensuremath{\nu}$ as quoted above the hyperscaling relation $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$ at $d=3$ requires $\ensuremath{\alpha}={0.086}_{\ensuremath{-}0.006}^{+0.024}$ so the validity of hyperscaling remains problematical.

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

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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 ${g}_{r}$ or the quantity usually denoted by ${R}_{\ensuremath{\xi}}^{+}.$ In particular, we obtain $\ensuremath{\gamma}=1.2371(1),$ $\ensuremath{\nu}=0.6299(2),$ ${\ensuremath{\gamma}}_{4}=4.3647(20),$ ${g}_{r}=1.404(3),$ 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.

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Some Results on Hyperscaling in the 3d Ising Model

George Allen Baker Jr., Naoki Kawashima

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
The availability of extended series expansions for \ensuremathχ0(2){\ensuremath{\chi}}_{0}^{(2)}\ensuremathχ0(2)​=\ensuremath∂2{\ensuremath{\partial}}^{2}\ensuremath∂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 \ensuremathχ0(2){\ensuremath{\chi}}_{0}^{(2)}\ensuremathχ0(2)​ and \ensuremathχ0(2){\ensuremath{\chi}}_{0}^{(2)}\ensuremathχ0(2)​/\ensuremathχ02{\ensuremath{\chi}}_{0}^{2}\ensuremathχ02​ 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.
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 .
We report a detailed study of twelve-term, high-temperature series for the second moment of spin-pair correlations ${\ensuremath{\mu}}_{2}(t)$ and the specific heat cH(t){c}_{H}(t)cH​(t) of the nearest-neighbor spin-sss Ising model in zero magnetic field on the fcc lattice. Near criticality we find ${\ensuremath{\mu}}_{2}(t)={A}_{2}(s)t{(s)}^{\ensuremath{-}(\ensuremath{\gamma}+2\ensuremath{\nu})}[1 + {B}_{2}(s)t{(s)}^{{\ensuremath{\Delta}}_{1}}+ \ensuremath{\cdots}]$, {$t(s)=\frac{[T\ensuremath{-}{T}_{c}(s)]}{T}$}, showing a confluent correction to the dominant scaling singularity. To within uncertainties the exponents have the universal (i.e., spin-independent) values $\ensuremath{\nu}={0.638}_{\ensuremath{-}0.008}^{+0.002}$ (with $\ensuremath{\gamma}={1.250}_{\ensuremath{-}0.007}^{+0.003}$) and ${\ensuremath{\Delta}}_{1}=0.6\ifmmode\pm\else\textpm\fi{}0.1$. The confluent exponent ${\ensuremath{\Delta}}_{1}$ is in reasonable agreement with the correction-to-scaling index derived from earlier analysis of the susceptibility, as predicted by renormalization-group arguments. A similar analysis of the specific heat cH{c}_{H}cH​ for the same model finds no detectable confluent singularities in rather noisy, high-temperature series and gives $\ensuremath{\alpha}=0.125\ifmmode\pm\else\textpm\fi{}0.020$ in general confirmation of earlier s=12s=\frac{1}{2}s=21​ estimates. With $\ensuremath{\nu}$ as quoted above the hyperscaling relation $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$ at d=3d=3d=3 requires $\ensuremath{\alpha}={0.086}_{\ensuremath{-}0.006}^{+0.024}$ so the validity of hyperscaling remains problematical.
A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.
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.
Much attention has been paid to the d=3, s= 1 / 2 Ising model on the BCC lattice in recent years because the best high-temperature series expansions were available for this lattice. However, in order to compare series estimates of critical exponents and temperatures with independent evaluations via the Monte Carlo method, lattices of lower coordination number must be considered, since Monte Carlo studies are usually made on these. A study of extant susceptibility series on the FCC, SC, diamond and tetrahedron lattices is reported on and some comments are made on aspects of recent Monte Carlo analyses of the d=3, s= 1 / 2 Ising model on the SC and diamond lattices.
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.