Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.
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
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.
High-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heinsenberg model are calculated for various lattices and anisotropies through order ${T}^{\ensuremath{-}8}$ (close-packed lattices) and ${T}^{\ensuremath{-}9}$ (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices $\ensuremath{\gamma}$ (susceptibility), $\ensuremath{\nu}$ (correlation range), and $\ensuremath{\alpha}$ (specific heat) as functions of anisotropy. Our results are consistent with the hypothesis that the critical indices change only when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended.
The authors present the results of a study of the free energy and specific heat of the classical vector model of ferromagnetism. High-temperature series for the free energy are presented as far as the 12th term in general spin dimensionality (D) and the 13th term for the case D=3. The techniques used to derive these series are discussed in some detail. The general series is shown to reduce to that for the Ising Model for D=1, and to agree with the expansion of the exact result for the spherical model for the case D= infinity . The cases D=2 and D=3 correspond to the classical planar and classical Heisenberg models, respectively. It is demonstrated that the general series reproduces previous results for both of these models, with the exception of a small disagreement with the calculated 11th terms of Ferer et al. (1971, 1973). In addition, the general series extends the series for each of these models by one term, and independent work for D=3 provides one further term.
Exact expressions for the partition function, spin pair correlation function, and susceptibility of the onedimensional isotropic classical Heisenberg model are obtained in zero external field with cyclic boundary conditions. It is shown that the methods used to derive these results enable the partition functions and susceptibilities of finite clusters of interacting classical spins to be evaluated in terms of the $3n\ensuremath{-}j$ symbols of Wigner. Exact results in one dimension are also obtained for the partition function and susceptibility of a "planar" classical Heisenberg model. In this model the spin vectors interact via a Heisenberg coupling but each spin vector is restricted to lie in a plane.The anisotropic classical Heisenberg model described by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{(\mathrm{ij})}2({{J}_{\mathrm{ij}}}^{x}{{s}_{i}}^{x}{{s}_{j}}^{x}+{{J}_{\mathrm{ij}}}^{y}{{s}_{i}}^{y}{{s}_{j}}^{y}+{{J}_{\mathrm{ij}}}^{z}{{s}_{i}}^{z}{{s}_{j}}^{z})\ensuremath{-}mH\ensuremath{\Sigma}\stackrel{N}{j=1}{{s}_{i}}^{z},$ where
six,
siy, and
siz are components of the unit vector
si, is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions
Jijx=Jijy=J and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.
Exact expressions are given for the free energy of the classical Heisenberg model in one lattice dimension and two or three spin dimensions with general anisotropy; the two-dimensional result, already known, is in terms of Mathieu functions and the three-dimensional result in terms of Lame wavefunctions. The method presented solves the eigenvalue problem for an appropriate transfer matrix by relating it to solutions of the Helmholtz equation in suitable curvilinear coordinate systems. High and low temperature limits are discussed for the three-dimensional spin case and the connection given with already known results for partial anisotropy.