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arXiv:hep-lat/9703018 · Physical review. B, Condensed matter

N -vector spin models on the simple-cubic and the body-centered-cubic lattices: A study of the critical behavior of the susceptibility and of the correlation length by high-temperature series extended to order β21

P. Butera, M. Comi

Published Oct 1, 1997• 139 citations
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Abstract

High-temperature expansions for the free energy, the susceptibility, and the second correlation moment of the classical $N$-vector model [also known as the $O(N)$ symmetric classical spin-Heisenberg model or as the lattice $O(N)$ nonlinear $\ensuremath{\sigma}$ model] on the simple-cubic and the body-centered-cubic lattices are extended to order ${\ensuremath{\beta}}^{21}$ for arbitrary $N$. 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 $N$, including $N=0$ (the self-avoiding walk model), $N=1$ (the Ising spin-1/2 model), $N=2$ (the $\mathrm{XY}$ model), and $N=3$ (the classical Heisenberg model). For all values of $N$ 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.

DOI
10.1103/physrevb.56.8212

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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 \ensuremathσ\ensuremath{\sigma}\ensuremathσ model] on the simple-cubic and the body-centered-cubic lattices are extended to order \ensuremathβ21{\ensuremath{\beta}}^{21}\ensuremathβ21 for arbitrary NNN. The series for the second field derivative of the susceptibility is extended to order \ensuremathβ17{\ensuremath{\beta}}^{17}\ensuremathβ17. 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.
For the classical N-vector model, with arbitrary N,N,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)O(N)O(N) symmetric classical spin Heisenberg model or, in quantum field theory, as the lattice O(N)O(N)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,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.
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 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.
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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.