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arXiv:hep-lat/9710092 · Physical review. B, Condensed matter

Perturbative renormalization group, exact results, and high-temperature series to order 21 for theN-vector spin models on the square lattice

P. Butera, M. Comi

Published Dec 1, 1996• 50 citations
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Abstract

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.

DOI
10.1103/physrevb.54.15828

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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 \ensuremathβ14{\mathrm{\ensuremath{\beta}}}^{14}\ensuremathβ14 to \ensuremathβ21{\mathrm{\ensuremath{\beta}}}^{21}\ensuremathβ21 for arbitrary N. For the second field derivative of the susceptibility the series expansion is extended from order \ensuremathβ14{\mathrm{\ensuremath{\beta}}}^{14}\ensuremathβ14 to \ensuremathβ17{\mathrm{\ensuremath{\beta}}}^{17}\ensuremathβ17. 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.
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.
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.
We present simple tables of integers from which it is possible to reconstruct the high-temperature series coefficients through ${\mathrm{\ensuremath{\beta}}}^{14}$ for the susceptibility, for the second correlation moment, and for the second field derivative of the susceptibility of the O(N) classical Heisenberg model on a simple (hyper)cubic lattice in dimension d=2, 3, and 4 and for any N. To construct the tables we have used the recent extension of the high-temperature series by M. Luscher and P. Weisz and some analytic properties in N that we have derived from the Schwinger-Dyson equations of the O(N) model. We also present a numerical study of these series in the d=2 case. The main results are: (a) the extended series give further support to the Cardy-Hamber-Nienhuis exact formulas for the critical exponents when -2N2; (b) for N\ensuremath{\ge}3 there are no indications of any critical point at finite \ensuremath{\beta}; (c) the series are consistent with the low-temperature asymptotic forms predicted by the perturbative renormalization group.
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.
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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