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arXiv:hep-lat/9903010 · Physical review. B, Condensed matter

Critical specific heats of theN-vector spin models on the simple cubic and bcc lattices

P. Butera, M. Comi

Published Sep 1, 1999• 39 citations
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Abstract

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 $O(N)$ 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 $N=0$ (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). 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.

DOI
10.1103/physrevb.60.6749

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We have computed through order \ensuremathβ21{\ensuremath{\beta}}^{21}\ensuremathβ21 the high-temperature expansions for the nearest neighbor spin correlation function G(N,\ensuremathβ)G(N,\ensuremath{\beta})G(N,\ensuremathβ) 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 \ensuremathσ\ensuremath{\sigma}\ensuremathσ 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,\ensuremathβ),C(N,\ensuremath{\beta}),C(N,\ensuremathβ), obtaining updated estimates of the critical parameters and therefore a more accurate direct test of the hyperscaling relation d\ensuremathν(N)=2\ensuremath−\ensuremathα(N)d\ensuremath{\nu}(N)=2\ensuremath{-}\ensuremath{\alpha}(N)d\ensuremathν(N)=2\ensuremath−\ensuremathα(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\ensuremathξ+(N),{R}_{\ensuremath{\xi}}^{+}(N),R\ensuremathξ+​(N), in fair agreement with renormalization group estimates.
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 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.
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.
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 ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$
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We have extended, from order 12 through order 25, the high-temperature series expansions (in zero magnetic field) for the spin-spin correlations of the spin-SSS Ising models on the square, simple-cubic and body-centered-cubic lattices. On the basis of this large set of data, we confirm accurately the validity of the scaling and universality hypotheses by resuming several tests which involve the correlation function, its moments and the exponential or the second-moment correlation lengths.