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Journal of Physics A Mathematical and General

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

Sati McKenzie, David S. Gaunt

Published Mar 1, 1980• 10 citations
Publisher

Abstract

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 .

DOI
10.1088/0305-4470/13/3/034

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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.

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arXiv:hep-lat/0112049•Physical review. B, Condensed matter•Apr 2, 2002•95 citations
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 authors investigate the spin- 1 / 2 Ising model with nearest-neighbour interactions on the four-dimensional simple hypercubic lattice. High-temperature series expansions are studied for the zero-field susceptibility chi 0 and the fourth-field derivative of the free energy Xi 0 (2) up to order nu 17 . The series are analysed for singularities of the form t -1 mod 1nt mod p where t is the reduced temperature. For chi 0 it is found that p=0.33+or-0.07 when q=1, in good agreement with the prediction p= 1 / 3 , q=1 of renormalisation group theory. The critical temperature is estimated to be nu c -1 =6.7315+or-0.0015. Results for chi 0 (2) are more slowly convergent but are not inconsistent with the renormalisation group prediction p= 1 / 3 , q=4.
The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K 9 for the hyper face centred (HFC) lattice, to order K 11 for the hyper body centred cubic (HBCC) lattice and to order K 12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t -1 mod In t mod p , predicted by the renormalisation group theory (t=1-K c /K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.
The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K9 for the hyper face centred (HFC) lattice, to order K11 for the hyper body centred cubic (HBCC) lattice and to order K12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t-1 mod In t mod p, predicted by the renormalisation group theory (t=1-Kc/K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.
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.
The high-temperature, zero-field susceptibility series of the spin- 1 / 2 Ising model on the face centred cubic lattice is analysed assuming the asymptotic form A 1 t - gamma +A 2 t - gamma +1 +Bt - gamma + Delta 1 where t is the reduced temperature. Good convergence is obtained for gamma =1.241 and Delta 1 =0.496, the values predicted by renormalisation group theory. Attempts to fit the series coefficients with gamma =1.25 and B=0, do not prove as successful. The amplitudes A 2 /A 1 and B/A 1 are estimated.
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.