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.
We derive the exact contribution of two spinons to the dynamical correlation function of the spin-\textonehalf{} Heisenberg model. For this, we use the isotropic limits of the exact form factors that have been recently computed through the quantum affine symmetry of the anisotropic Heisenberg model
XXZ.
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.
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.
The anisotropic Hamiltonian, $H=\ensuremath{-}\frac{1}{2}\ensuremath{\Sigma}({J}_{x}{{\ensuremath{\sigma}}_{l}}^{x}{{\ensuremath{\sigma}}_{l+1}}^{x}+{J}_{y}{{\ensuremath{\sigma}}_{l}}^{y}{{\ensuremath{\sigma}}_{l+1}}^{y}+{J}_{z}{{\ensuremath{\sigma}}_{l}}^{z}{{\ensuremath{\sigma}}_{l+1}}^{z})\ensuremath{-}m\mathcal{H}\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{l}}^{z},$ of the linear spin array in the Heisenberg model of magnetism is examined. The eigenstate and the partition function for the case
Jz=0 are obtained exactly for a finite system and for an infinite system with the aid of annihilation and creation operators, and the free energy
F of the latter is given by $\ensuremath{-}\frac{F}{\mathrm{NkT}}=(\frac{1}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\ensuremath{\pi}}\mathrm{ln}{2cosh{[{{K}_{x}}^{2}+{{K}_{y}}^{2}+2{K}_{x}{K}_{y}cos2\ensuremath{\omega}\ensuremath{-}2C({K}_{x}+{K}_{y})cos\ensuremath{\omega}+{C}^{2}]}^{\frac{1}{2}}}d\ensuremath{\omega},$ where
Kx=2kTJx,
Ky=2kTJy,
C=kTmH. The case
Jx=Jy=Jz=J is discussed with the aid of a high-temperature expansion and of analysis of small systems. Specific heats and susceptibilities in special cases: (i)
Jx=Jy=J,
Jz=0, (ii)
Jx=J,
Jy=Jz=0, (
iiif)
Jx=Jy=0,
Jz=J>0, (
iiia)
Jx=Jy=0,
Jz=J<0, (
ivf)
Jx=Jy=Jz=J>0, (
iva)
Jx=Jy=Jz=J<0 are compared and it is shown that (i), (
iiia), and (
iva) have the characteristic features of the observed parallel susceptibility of an antiferromagnetic substance, (ii) those of perpendicular susceptibility, and (
iiif) and (
ivf) those of paramagnetic susceptibility, even though they have no singularities. The distribution of the zeros of the partition function is also discussed.