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arXiv:hep-lat/9805025 · Physical review. B, Condensed matter

Renormalized couplings and scaling correction amplitudes in theN-vector spin models on the sc and the bcc lattices

P. Butera, M. Comi

Published Nov 1, 1998• 59 citations
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Abstract

For the classical N-vector model, with arbitrary $N,$ we have computed through order ${\ensuremath{\beta}}^{17}$ the high-temperature expansions of the second field derivative of the susceptibility ${\ensuremath{\chi}}_{4}(N,\ensuremath{\beta})$ on the simple cubic and on the body centered cubic lattices. [The N-vector model is also known as the $O(N)$ symmetric classical spin Heisenberg model or, in quantum field theory, as the lattice $O(N)$ nonlinear $\ensuremath{\sigma}$ model.] By analyzing the expansion of ${\ensuremath{\chi}}_{4}(N,\ensuremath{\beta})$ on the two lattices, and by carefully allowing for the corrections to scaling, we obtain updated estimates of the critical parameters and more accurate tests of the hyperscaling relation $d\ensuremath{\nu}(N)+\ensuremath{\gamma}(N)\ensuremath{-}2{\ensuremath{\Delta}}_{4}(N)=0$ for a range of values of the spin dimensionality $N,$ including $N=0$ (the self-avoiding walk model), $N=1$ (the Ising spin 1/2 model), $N=2$ (the $\mathrm{XY}$ model), $N=3$ (the classical Heisenberg model). Using the recently extended series for the susceptibility and for the second correlation moment, we also compute the dimensionless renormalized four point coupling constants and some universal ratios of scaling correction amplitudes in fair agreement with recent renormalization group estimates.

DOI
10.1103/physrevb.58.11552

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For the classical N-vector model, with arbitrary N,N,N, we have computed through order \ensuremathβ17{\ensuremath{\beta}}^{17}\ensuremathβ17 the high-temperature expansions of the second field derivative of the susceptibility \ensuremathχ4(N,\ensuremathβ){\ensuremath{\chi}}_{4}(N,\ensuremath{\beta})\ensuremathχ4​(N,\ensuremathβ) on the simple cubic and on the body centered cubic lattices. [The N-vector model is also known as the O(N)O(N)O(N) symmetric classical spin Heisenberg model or, in quantum field theory, as the lattice O(N)O(N)O(N) nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model.] By analyzing the expansion of \ensuremathχ4(N,\ensuremathβ){\ensuremath{\chi}}_{4}(N,\ensuremath{\beta})\ensuremathχ4​(N,\ensuremathβ) on the two lattices, and by carefully allowing for the corrections to scaling, we obtain updated estimates of the critical parameters and more accurate tests of the hyperscaling relation d\ensuremathν(N)+\ensuremathγ(N)\ensuremath−2\ensuremathΔ4(N)=0d\ensuremath{\nu}(N)+\ensuremath{\gamma}(N)\ensuremath{-}2{\ensuremath{\Delta}}_{4}(N)=0d\ensuremathν(N)+\ensuremathγ(N)\ensuremath−2\ensuremathΔ4​(N)=0 for a range of values of the spin dimensionality N,N,N, 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), N=3N=3N=3 (the classical Heisenberg model). Using the recently extended series for the susceptibility and for the second correlation moment, we also compute the dimensionless renormalized four point coupling constants and some universal ratios of scaling correction amplitudes in fair agreement with recent renormalization group estimates.
We have computed through order ${\ensuremath{\beta}}^{21}$ the high-temperature expansions for the nearest neighbor spin correlation function $G(N,\ensuremath{\beta})$ of the classical N-vector model, with general N, on the simple cubic and on the body centered cubic lattices. For this model, also known in quantum field theory as the lattice O(N)O(N)O(N) nonlinear $\ensuremath{\sigma}$ model, we have presented in previous papers extended expansions of the susceptibility, of its second field derivative, and of the second moment of the correlation function. Here we study the internal specific energy and the specific heat $C(N,\ensuremath{\beta}),$ obtaining updated estimates of the critical parameters and therefore a more accurate direct test of the hyperscaling relation $d\ensuremath{\nu}(N)=2\ensuremath{-}\ensuremath{\alpha}(N)$ on a range of values of the spin dimensionality N, 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), N=3N=3N=3 (the classical Heisenberg model). By the newly extended series we also compute the universal combination of critical amplitudes usually denoted by ${R}_{\ensuremath{\xi}}^{+}(N),$ in fair agreement with renormalization group estimates.
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.
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 ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$
High-temperature expansions for the susceptibility and the second correlation moment of the classical N-vector model [also known as the O(N) symmetric Heisenberg classical spin model] on the square lattice are extended from order ${\mathrm{\ensuremath{\beta}}}^{14}$ to ${\mathrm{\ensuremath{\beta}}}^{21}$ for arbitrary N. For the second field derivative of the susceptibility the series expansion is extended from order ${\mathrm{\ensuremath{\beta}}}^{14}$ to ${\mathrm{\ensuremath{\beta}}}^{17}$. For -2\ensuremath{\le}N2, a numerical analysis of the series is performed in order to compare the critical exponents \ensuremath{\gamma}(N), \ensuremath{\nu}(N), and \ensuremath{\Delta}(N) to exact (though nonrigorous) formulas and to compute the ``dimensionless four point coupling constant'' g${\mathrm{^}}_{\mathit{r}}$(N). For N>2, we present a study of the analyticity properties of \ensuremath{\chi}, \ensuremath{\xi}, etc. in the complex \ensuremath{\beta} plane and describe a method to estimate the parameters which characterize their low-temperature behaviors. We compare our series estimates to the predictions of the perturbative renormalization group theory, to exact (but nonrigorous or conjectured) formulas and to the results of the 1/N expansion, always finding a good agreement. \textcopyright{} 1996 The American Physical Society.
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.
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.