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Physical review. B, Solid state

High-temperature series for the susceptibility of the spin- s Ising model: Analysis of confluent singularities

William J. Camp, John P. VAN DYKE

Published Apr 1, 1975• 148 citations
Publisher

Abstract

We have extended the series for the zero-field susceptibility of the spin-$s$ Ising model through tenth order in the reduced inverse temperature $K$ on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients ${h}_{n}(s)$ are expressed as simple polynomials in $X=s(s+1)$. Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of $s$. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For $s=\frac{1}{2}$ only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of $s$, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.

DOI
10.1103/physrevb.11.2579

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High-temperature series for the susceptibility of the spin- S Ising model

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Extension to order β23 of the high-temperature expansions for the spin- 12 Ising model on simple cubic and body-centered cubic lattices

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Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$

arXiv:hep-lat/0006009•Physical review. B, Condensed matter•Dec 1, 2000•40 citations
We have extended the series for the zero-field susceptibility of the spin-sss Ising model through tenth order in the reduced inverse temperature KKK on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients hn(s){h}_{n}(s)hn​(s) are expressed as simple polynomials in X=s(s+1)X=s(s+1)X=s(s+1). Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent \ensuremathγ1{\ensuremath{\gamma}}_{1}\ensuremathγ1​ equals 1.25, independent of sss. The exponent of the leading correction term is estimated to be \ensuremathγ2\ensuremath≃0.75\ifmmode±\else\textpm\fi0.08{\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08\ensuremathγ2​\ensuremath≃0.75\ifmmode±\else\textpm\fi0.08 in good agreement with renormalization-group theory. For s=12s=\frac{1}{2}s=21​ only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that \ensuremathγ1=74{\ensuremath{\gamma}}_{1}=\frac{7}{4}\ensuremathγ1​=47​, independent of sss, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
We have extended the series for the zero-field susceptibility of the spin-SSS Ising model to eighth order in the reduced temperature KKK, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series hn(S){h}_{n}(S)hn​(S) are expressed as simple polynomials in X=S(S+1)X=S(S+1)X=S(S+1). For the face-centered-cubic lattice, an accurate polynomial fit to the critical point Kc(S){K}_{c}(S)Kc​(S) is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all SSS. However, the large-SSS series seems to exhibit more rapid apparent convergence.
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.
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.
The high-temperature susceptibility of the spin-SSS Ising model on the triangular net is analyzed as a function of SSS using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.
Extended series expansions are derived for the high-temperature susceptibility of the classical Heisenberg model, on three-dimensional lattices. Series coefficients are presented to twelfth order for the simple cubic (SC) and face centred cubic (FCC) lattices and to eleventh order for the body centred cubic (BCC) lattice. The results are in agreement with earlier calculations apart from a small discrepancy at the tenth order on the FCC lattice. Extrapolation studies on the extended series are used to obtain revised estimates for the critical points (K c ) and the susceptibility exponent ( gamma ). On the FCC lattice, the authors also investigate the possibility of a confluent non-analytic correction to the dominant singularity. While the coefficients are consistent with the presence of such a correction term with an exponent ( Delta 1 ) of 0.55, as predicted by renormalisation group (RG) calculations, the amplitude of the correction term appears to be very small compared with that of the first analytic correction term.
Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$