Critical properties of a lattice model of isotropically interacting 'continuous spins' are studied using high-temperature series expansions. Both the spin-spin and spin-field interactions are different from the planar classical Heisenberg model, yet the ground-state symmetry is the same. The model is found to behave in most respects like the planar classical Heisenberg model, and in three dimensions the following values of critical indices are found: gamma =1.32+or-0.04 and alpha =-0.04+or-0.06. In two dimensions there is flimsy evidence for a divergent susceptibility with an exponent gamma approximately=3'. The results support the belief that for finite-range interaction and fixed dimensionality, unrenormalized critical indices are determined by the underlying symmetry.
Critical properties of a lattice model of isotropically interacting 'continuous spins' are studied using high-temperature series expansions. Both the spin-spin and spin-field interactions are different from the planar classical Heisenberg model, yet the ground-state symmetry is the same. The model is found to behave in most respects like the planar classical Heisenberg model, and in three dimensions the following values of critical indices are found: gamma =1.32+or-0.04 and alpha =-0.04+or-0.06. In two dimensions there is flimsy evidence for a divergent susceptibility with an exponent gamma approximately=3'. The results support the belief that for finite-range interaction and fixed dimensionality, unrenormalized critical indices are determined by the underlying symmetry.
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.
A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].
For pt.I see J. Phys. C: Solid St. Phys., vol.6, p.2691 (1973). The critical behaviour of the step model of phase transitions on the square, simple cubic and body-centred cubic lattices is investigated by the method of exact series expansions. The model is found to be in agreement with two aspects of universality, namely the dependence of critical exponents on spin-space dimensionality and the lattice independence of critical exponents for fixed lattice and spin-space dimensionality. Series expansions for the three-dimensional susceptibility and specific heats are generated and analysed, and the critical exponents are found to be gamma =1.335+or-0.01 and alpha =-0.04+or-0.10 respectively. In two dimensions the existence and nature of a phase transition remains an open question. No evidence of either a conventional algebraic singularity or a vortex induced essential singularity is found.
The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.
The high-temperature susceptibility and internal energy series have been reexamined for four two-dimensional isotropic spin models on a triangular lattice. They are the planar classical Heisenberg (PCH), infinite spin X-Y, classical Heisenberg and step models. For all four models evidence of a phase transition is found. This evidence is good for the first two models, and weak for the last two. A method of series analysis is developed which permits us to rule out, with some degree of confidence, an algebraic singularity for either the susceptibility or the specific heat for all four models. For the PCH model the susceptibility fits the form suggested by Kosterlitz (1974) while for the X-Y model susceptibility a similar conclusion may be drawn, with a lesser degree of confidence.
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.