Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

ClassicalO(N) Heisenberg model: Extended high-temperature series for two, three, and four dimensions

P. Butera, M. Comi, G. Marchesini

Published Jun 1, 1990• 31 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.41.11494

Related papers

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

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.

arXiv:hep-lat/9710092•Physical review. B, Condensed matter•Dec 1, 1996•50 citations

The high-temperature susceptibility of the classical Heisenberg model in four dimensions

Sati McKenzie, Cyril Domb, D L Hunter

The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K 9 for the hyper face centred (HFC) lattice, to order K 11 for the hyper body centred cubic (HBCC) lattice and to order K 12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t -1 mod In t mod p , predicted by the renormalisation group theory (t=1-K c /K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.

Journal of Physics A Mathematical and General•Dec 1, 1982•4 citations

Extended high-temperature series for the classical Heisenberg model in three dimensions

Sati McKenzie, Cyril Domb, D L Hunter

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.

Journal of Physics A Mathematical and General•Dec 1, 1982•30 citations

The high temperature susceptibility of the classical Heisenberg model in four dimensions

Sati McKenzie, Cyril Domb, D L Hunter

The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K9 for the hyper face centred (HFC) lattice, to order K11 for the hyper body centred cubic (HBCC) lattice and to order K12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t-1 mod In t mod p, predicted by the renormalisation group theory (t=1-Kc/K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.

Journal of Physics A Mathematical and General•Dec 1, 1983•2 citations

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

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.

arXiv:hep-lat/9703018•Physical review. B, Condensed matter•Oct 1, 1997•139 citations

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

P. Butera, M. Comi

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.

arXiv:hep-lat/9805025•Physical review. B, Condensed matter•Nov 1, 1998•59 citations
We present simple tables of integers from which it is possible to reconstruct the high-temperature series coefficients through \ensuremathβ14{\mathrm{\ensuremath{\beta}}}^{14}\ensuremathβ14 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.
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.
The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K 9 for the hyper face centred (HFC) lattice, to order K 11 for the hyper body centred cubic (HBCC) lattice and to order K 12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t -1 mod In t mod p , predicted by the renormalisation group theory (t=1-K c /K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.
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.
The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K9 for the hyper face centred (HFC) lattice, to order K11 for the hyper body centred cubic (HBCC) lattice and to order K12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t-1 mod In t mod p, predicted by the renormalisation group theory (t=1-Kc/K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.
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.