A system with Hamiltonian H0 is assumed to undergo a phase transition at transition temperature T0. When a small perturbation term λH′ is added to H0, the transition temperature will depend on the coupling parameter λ and the critical indices will in general be changed. The change induced by the application of λH′ is classified into three types, and these types discussed for the cases of anisotropic Ising ferromagnets by means of thermodynamic perturbation theory and scaling law equation for spin correlation functions.
A system with Hamiltonian H0 is assumed to undergo a phase transition at transition temperature T0. When a small perturbation term λH′ is added to H0, the transition temperature will depend on the coupling parameter λ and the critical indices will in general be changed. The change induced by the application of λH′ is classified into three types, and these types discussed for the cases of anisotropic Ising ferromagnets by means of thermodynamic perturbation theory and scaling law equation for spin correlation functions.
Thermodynamic perturbation theory is applied to systems that undergo a phase transition. It is shown that certain conditions are necessary in order that the application of the perturbation does not change the qualitative features of the phase transition, e.g., the critical indices. The method is then applied to the two-dimensional-square Ising lattice with nearest-and next-nearest-neighbor interactions. The shift in the critical temperature is determined to two orders in the peturbation parameter, viz., the ratio of the strength of the next-nearest-neighbor force to the strength of the nearest-neighbor force. A similar calculation is also made for a two-dimensional-lattice potential of Kac and Helfand.
For pt.II see ibid., vol.14, p.1789 (1981). The phase transition of the anisotropic Ising lattices has been studied by the recently developed correlation reduction theory. The transition temperature and some other quantities at transition temperature for the anisotropic Ising lattices with spin 1 / 2 , 1, 3 / 2 are evaluated as functions of anisotropy parameter. The effect of anisotropy on these values is similar for the systems with three different spin values. Comparison with other calculations is made, and some physical significances are discussed.
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.
Expressions for general correlation functions in Ising ferromagnets are derived in terms of the Lee-Yang theorem. By the use of these expressions, the asymptotic forms of correlation functions in the presence of magnetic field are analyzed, assuming those in the absence of the external field. In particular, the critical indices of spin-spin, energy-density-energy-density, energy-density-spin correlation functions are related with the critical indices of thermal susceptiblity χT0, specific heat, susceptibility χH0, and magnetization. These results agree in form with those predicted by the scaling law approach.
The Hamiltonian of the form H=H0+λH1 is discussed, where λ is a parameter to change symmetry, dimensionality, or potential range. Scaling with the parameter λ is studied for thermodynamic quantities such as the free energy. By assuming the scaled form F(ε,λ)=ε2-αF(λ/εφ) for the singular part of the free energy near the critical point Tc(ε=(T-Tc)/Tc), the expression (or explicit value) of the critical exponent φ is obtained in each case of change of symmetry, dimensionality and potential range. In particular, the universal relation φ=γ is found for change of dimensionality, where γ is the critical exponent of the susceptibility in the unperturbed Hamiltonian H0. An extension to dynamical critical phenomena is also discussed briefly, particularly in connection with the critical slowing down. A possibility is suggested to derive this generalized scaling with the parameter λ by applying the usual scaling law to each term of the perturbational expansion with respect to the parameter λ.
The thermodynamic quantities of the Ising model are analyzed near Tc, using the asymptotic distribution function g(θ, t) of zeros for the partition function on a unit circle in the plane of the complex magnetic field. The distribution function is derived as g(θ, t) = tλf(θ/tλ + γ), assuming that the susceptibility behaves like χ0+ ∼ t-γ near Tc where t = T/Tc-1. It is shown that there are two types of the situation; type I, in which a critical angle θc(t) exists such that g(θ, t) = 0 for 0 ≦ |θ| ≦ θc, and type II, in which there is no critical angle. The critical indices are related to one another, using the above g(θ, t). The relation between the magnetic equation of state M = M(h, t) and the distribution function is given as follows: g(θ, t) = (1/2π)Re M(iθ, t), where h = 2mH/kT. g(θ, t) for an exactly soluble model near Tc is given by the equation θ2 ∼ t3 + a1t2g2 + a2tg4 + a3g6 (ai > 0). This is a good example for the general theory.