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Physical review. B, Condensed matter

Confluent-singularity analysis of high-temperature series for thermodynamic functions of the S=12 XY model, the S=∞ XY model, and the planar-rotator model on the fcc lattice

Jos Rogiers, Martin V. Ferer, E. Rodney Scaggs

Published Feb 1, 1979• 23 citations
Publisher

Abstract

Leading singularity analysis for the susceptibility index $\ensuremath{\gamma}$ of these planar models had indicated a nonuniversal variation in the index between the classical spin models and the spin-\textonehalf{} model in three dimensions. Using existing methods of analysis which allow for confluent corrections to the leading singularity, we have reanalyzed the existing susceptibility series for $\ensuremath{\gamma}$ as well as the second-moment series for the correlation length index $\ensuremath{\nu}$ for the classical models. Our results are consistent with the following universal values of the indices $\ensuremath{\gamma}=1.333\ifmmode\pm\else\textpm\fi{}0.010$ and $\ensuremath{\nu}=0.678\ifmmode\pm\else\textpm\fi{}0.005$. In addition, our analysis suggests that the nonanalytic correction to scaling is absent in the spin-\textonehalf{} model and that the correction to scaling exponent ${\ensuremath{\Delta}}_{1}$ may be different for the two classical models investigated, being 0.78\ifmmode\pm\else\textpm\fi{}0.08 for the planar rotator model and 0.60\ifmmode\pm\else\textpm\fi{}0.08 for the $s=\ensuremath{\infty}$ $\mathrm{XY}$ model. Analysis of existing series for the second field derivative of the susceptibility to determine the gap index $\ensuremath{\Delta}$ has indicated a slight difference between the $s=\frac{1}{2}$ and the classical spin values, 1.70\ifmmode\pm\else\textpm\fi{}0.02 and 1.67\ifmmode\pm\else\textpm\fi{}0.01, respectively. This small difference probably reflects a defect in the analysis, since our analysis clearly indicates that assuming confluent corrections does not satisfactorily account for the scaling corrections in these second-field derivative series.

DOI
10.1103/physrevb.19.1644

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Physical review. B, Solid state•Dec 1, 1973•129 citations
Leading singularity analysis for the susceptibility index \ensuremathγ\ensuremath{\gamma}\ensuremathγ of these planar models had indicated a nonuniversal variation in the index between the classical spin models and the spin-\textonehalf{} model in three dimensions. Using existing methods of analysis which allow for confluent corrections to the leading singularity, we have reanalyzed the existing susceptibility series for \ensuremathγ\ensuremath{\gamma}\ensuremathγ as well as the second-moment series for the correlation length index \ensuremathν\ensuremath{\nu}\ensuremathν for the classical models. Our results are consistent with the following universal values of the indices \ensuremathγ=1.333\ifmmode±\else\textpm\fi0.010\ensuremath{\gamma}=1.333\ifmmode\pm\else\textpm\fi{}0.010\ensuremathγ=1.333\ifmmode±\else\textpm\fi0.010 and \ensuremathν=0.678\ifmmode±\else\textpm\fi0.005\ensuremath{\nu}=0.678\ifmmode\pm\else\textpm\fi{}0.005\ensuremathν=0.678\ifmmode±\else\textpm\fi0.005. In addition, our analysis suggests that the nonanalytic correction to scaling is absent in the spin-\textonehalf{} model and that the correction to scaling exponent \ensuremathΔ1{\ensuremath{\Delta}}_{1}\ensuremathΔ1​ may be different for the two classical models investigated, being 0.78\ifmmode\pm\else\textpm\fi{}0.08 for the planar rotator model and 0.60\ifmmode\pm\else\textpm\fi{}0.08 for the s=\ensuremath∞s=\ensuremath{\infty}s=\ensuremath∞ XY\mathrm{XY}XY model. Analysis of existing series for the second field derivative of the susceptibility to determine the gap index \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ has indicated a slight difference between the s=12s=\frac{1}{2}s=21​ and the classical spin values, 1.70\ifmmode\pm\else\textpm\fi{}0.02 and 1.67\ifmmode\pm\else\textpm\fi{}0.01, respectively. This small difference probably reflects a defect in the analysis, since our analysis clearly indicates that assuming confluent corrections does not satisfactorily account for the scaling corrections in these second-field derivative series.
We present new twelfth-order high-temperature series for the susceptibility, correlation length, and free energy of the nearest-neighbor classical Heisenberg ferromagnet in zero field on a fcc lattice. Checks corroborating the correctness of the series are discussed. Standard analysis of these series produces results in essential agreement with the renormalization-group calculations and several other series analyses. However, a more reliable confluent singularity analysis suggests slightly larger values, \ensuremath{\gamma}${=1.40}_{\mathrm{\ensuremath{-}}0.01}^{+0.03}$ and \ensuremath{\nu}=0.72\ifmmode\pm\else\textpm\fi{}0.01, as well as a confluent correction of just the size to cause the observed standard analysis underestimate.
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 ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of sss. 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=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 ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of sss, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
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.
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