The thermodynamic behavior of the fcc Blume-Capel ferromagnet, $\mathcal{H}=\ensuremath{-}J\ensuremath{\Sigma}\stackrel{}{〈12〉}{S}^{z}(1){S}^{z}(2)+\ensuremath{\Delta}\ensuremath{\Sigma}\stackrel{}{1}{[{S}^{z}(1)]}^{2}\ensuremath{-}h\ensuremath{\Sigma}\stackrel{}{1}{S}^{z}(1), {S}^{z}=0, \ifmmode\pm\else\textpm\fi{}1,$ is studied by series-extrapolation techniques. By using both high- and low-temperature series, we are able to trace first- and second-order branches of the phase boundary and examine behavior at temperatures both above and below the phase transition. We find a tricritical point at $\frac{{k}_{B}{T}_{t}}{12J}=0.2615\ifmmode\pm\else\textpm\fi{}0.0070$, $\frac{{\ensuremath{\Delta}}_{t}}{12J}=0.4716\ifmmode\pm\else\textpm\fi{}0.0010$. Tricritical exponents are consistent with ${\ensuremath{\gamma}}_{t}={\ensuremath{\gamma}}_{t}^{\ensuremath{'}}=1$, ${\ensuremath{\beta}}_{t}=\frac{1}{4}$, ${\ensuremath{\phi}}_{t}={\ensuremath{\nu}}_{t}={\ensuremath{\alpha}}_{t}={\ensuremath{\alpha}}_{t}^{\ensuremath{'}}=\frac{1}{2}$, in good agreement with tricritical mean-field theory and the Gaussian tricritical fixed point of Riedel and Wegner.
Generalization of the random-phase-approximation (RPA) theory of ferromagnetism for spins S=1/2, based on a refinement of the decoupling for a relevant three-spin Green's function, is presented. With the reasonable choice of two characteristic functions of the present theory---the virtual spectrum \ensuremath{\Delta}(\ensuremath{\omega}) and the decoupling factor \ensuremath{\kappa}(\ensuremath{\sigma}), where \ensuremath{\omega} is an excitation energy and \ensuremath{\sigma} is the relative magnetization---the theory provides a fair description of magnetic phenomena not only at temperature T\ensuremath{\rightarrow}0, in the vicinity of the Curie temperature
TC, or at high T, but at all temperatures and external magnetic fields H. A relation between some parameters of the functions \ensuremath{\Delta}(\ensuremath{\omega}) and \ensuremath{\kappa}(\ensuremath{\sigma}), and the critical indices of the static scaling theory, has been established. The form of the universal scaling function W, following from our theory, has been found. For the power-law virtual spectrum \ensuremath{\Delta}\ensuremath{\propto}${\mathrm{\ensuremath{\omega}}}^{\mathit{s}}$ at \ensuremath{\omega}\ensuremath{\rightarrow}0, 0s1, we have found the nonanalytic increase of the magnetization versus the field at T
TC:\ensuremath{\sigma}(T,H)-\ensuremath{\sigma}(T,0)\ensuremath{\propto}
Hs. It reduces to the classic result of Holstein and Primakoff for the three-dimensional isotropic Heisenberg ferromagnet having s=1/2. Assuming this power-law spectrum at all energies up to some cutoff energy, we have obtained a rather good agreement of the present predictions for \ensuremath{\sigma}(T,H) with experimental data for Ni, for all temperatures and fields available. Some model virtual spectra are found to lead to interesting models of ferromagnetism, with the magnetization and the susceptibility functions given in terms of explicit analytic formulas.
A generalized random-phase approximation (RPA) for decoupling the double-time Green's functions (DTGF) equations of motion of spin systems is developed. The decoupling parameters are chosen by fixing the moments of both the commutator and anticommutator decoupled spectral functions, thus establishing a well-defined relationship between the approximation scheme and the fundamental properties of DTGF. It is shown that the standard RPA must be supplemented by a condition on correlations to provide a well-defined, complete approximation scheme. The procedure is demonstrated by application to the fully anisotropic S=1/2 Heisenberg model in a field.
Recent attempts to develop double-time Green's-function theories for the transverse Ising model (TIM) have been subject to certain difficulties; the appearance of zero-frequency poles in the commutator Green's function (CGF) led to the avoidance of CGF and an incomplete treatment based only upon anticommutator Green's functions (AGF), and ambiguities in the determination of some thermal averages. Upon decoupling the equations-of-motion hierarchy, we develop a consistent Green's-function theory of the TIM. The AGF and CGF are shown to provide consistent results, the problem of zero-frequency poles in CGF being treated appropriately. We resolve the ambiguity problem and obtain reasonable values for the Weiss critical field and the ground-state magnetization.
A recently developed moment theorem and an approximation scheme of Pryce and Stevens are used to compare 2nd-RPA and Tomita–Tanaka type decoupling procedures for Green function equations. The comparison is accomplished from the point of view of the moments of the paramagnetic resonance line shapes that the decouplings generate. An Ising system in an external magnetic field is taken as the model system. It is found that use of the more complicated 2nd RPA is justified only if accuracy to O(1/z2) is desired.