The author establishes a relation between discrete excitation models at low temperature and the anisotropic rotator, XY and Heisenberg models at high temperature for dimensions D>or=2. The correlation functions of these models are shown to decay exponentially for short-range interactions. A qualitative discussion of the phase transitions is given.
The author establishes a relation between discrete excitation models at low temperature and the anisotropic rotator, XY and Heisenberg models at high temperature for dimensions D>or=2. The correlation functions of these models are shown to decay exponentially for short-range interactions. A qualitative discussion of the phase transitions is given.
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.
We consider the zero-temperature properties of four different spin 1/2 models on two-dimensional lattices: the XY ferromagnet, the XY antiferromagnet, the Heisenberg antiferromagnet, and the Dzyaloshinsky–Moriya models. Most of this article is a review of previously published work, but a few previously unpublished results are included. The relation between three of the models on bipartite lattices is described. The properties of the XY ferromagnet in two dimensions, especially those derived from extrapolation of finite lattice results, are reviewed. A numerical factor by which spin-wave and finite-lattice estimates of the long-range order parameter differ is discussed. For frustrated models on the triangular lattice the possibility of a chirality phase transition instead of, or in addition to, a magnetic phase transition is considered.
A theoretical approach to the one-dimensional isotropic Heisenberg model at low temperatures is presented which uses rotationally invariant combinations of spin operators only and is thus adapted to treat correctly effects resulting from the absence of long-range order in these systems. Analytic expressions for space- and time-dependent spin correlation functions are given for small values of JS2T and S1. The results include an expression for the spin correlation function in the antiferromagnetic ground state and a discussion of changes in line shape, due to quantum effects, for short-wavelength magnons as observed in neutron scattering experiments.
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.
We consider a “generalized Heisenberg model”, with a Hamiltonian given by ℋ︁(D) = –JΣij S(D)j, where the quantities Si are isotropically-interacting D-dimensional classical spins and the summation is restricted to nearest-neighbor pairs of sites <i,j>. Thus when D = 1, 2, and 3, this model reduces to the S = 1/2 Ising, plane rotator, and classical Heisenberg models which are useful for describing phase transitions in, respectively, liquid-gas systems, Bose fluids, and magnetic materials. Unfortunately this Hamiltonian is exactly soluble only in the limit D → ∞, in which case the solution corresponds to the spherical model of Berlin and Kac. For finite D, we must consider approximation procedures in order to obtain any useful information. In this paper we develop a diagrammatic representation, valid for arbitrary D, for successive terms in the high-temperature expansions of the free energy and the spin-spin correlation function—from which we can obtain series expansions for all the thermodynamic properties of the system.
The structure of the space-time-dependent spin-correlation functions for an anisotropic Heisenberg paramagnet with cylindrical symmetry at elevated temperature is analyzed in terms of a two-parameter Gaussian representation representation of the generalized diffusivity. The parameters are chosen so as to exactly preserve the frequency moments, i.e., ${{〈{\ensuremath{\omega}}^{2n}〉}_{\mathrm{K}}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ ($\ensuremath{\alpha}=z, x, or y$; n=0,1,2) of the longitudinal and the transverse spectral functions. These moments are calculated for arbitrary spin S, arbitrary range of the exchange interactions, and arbitrary dimensionality. The results for the correlation functions are worked out for one, two, and three dimensions. In the extreme transverse limit, the results are compared with the exactly soluble XY model for a one-dimensional spin-\textonehalf{} system with only nearest-neighbor (transverse) exchange. The agreement of the results is found to be satisfactory.