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Physical review. B, Condensed matter

Three-dimensional Heisenberg ferromagnet: A series investigation

Martin V. Ferer, A. Hamid Aidinejad

Published Nov 1, 1986• 28 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.34.6481

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arXiv:hep-lat/9703018•Physical review. B, Condensed matter•Oct 1, 1997•139 citations
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\ensuremath−0.01+0.03{=1.40}_{\mathrm{\ensuremath{-}}0.01}^{+0.03}=1.40\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.
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.
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 have derived a twelfth-order, high-temperature series for the axial next-nearest-neighbor Ising (ANNNI) models on the fcc and the sc lattices, and we present detailed results for one of the fcc models. We have determined the phase boundaries of these models and located the Lifshitz point from the intersection of the phase boundary with the ``disorder'' line, where the maximum in the susceptibility \ensuremath{\chi}(q) begins to occur at nonzero q signaling the onset of modulated correlations. As expected, the Lifshitz point appears to be an inflection point of the phase boundary. The effective susceptibility index shows the predicted universal values for definite regions of the parameter space, e.g., XY-like in the midrange modulated regime, with the expected smooth crossover near the multicritical points. Throughout the modulated regime from the Lifshitz point to the decoupling point, the variation in the critical wave vector shows only small deviations from mean-field predictions. Our best estimates for the Lifshitz-point indices, ${\ensuremath{\gamma}}_{\mathit{l}}$=1.62\ifmmode\pm\else\textpm\fi{}0.12, 4${\ensuremath{\nu}}_{\mathit{l}4}$=1.63\ifmmode\pm\else\textpm\fi{}0.10, and ${\mathrm{\ensuremath{\alpha}}}_{\mathit{l}}$=0.20\ifmmode\pm\else\textpm\fi{}0.15, differ from previous estimates. We are confident that our estimates are more reliable not only because we have longer series for a close-packed lattice but also primarily because we have analyzed our series along the appropriate scaling direction.
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 $s=\ensuremath{\infty}$ XY\mathrm{XY}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=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.
High-temperature expansions for the free energy, the susceptibility, and the second correlation moment of the classical NNN-vector model [also known as the O(N)O(N)O(N) symmetric classical spin-Heisenberg model or as the lattice O(N)O(N)O(N) nonlinear $\ensuremath{\sigma}$ model] on the simple-cubic and the body-centered-cubic lattices are extended to order ${\ensuremath{\beta}}^{21}$ for arbitrary NNN. The series for the second field derivative of the susceptibility is extended to order ${\ensuremath{\beta}}^{17}$. We report here on the analysis of the computed series for the susceptibility and the (second moment) correlation length which yields updated estimates of the critical parameters for various values of the spin dimensionality NNN, including N=0N=0N=0 (the self-avoiding walk model), N=1N=1N=1 (the Ising spin-1/2 model), N=2N=2N=2 (the XY\mathrm{XY}XY model), and N=3N=3N=3 (the classical Heisenberg model). For all values of NNN we confirm a good agreement with the present renormalization-group estimates. A study of the series for the other observables will appear in a forthcoming paper.