Interpolation Approach to the Green Function Theory of Ferromagnetism

The difficult problem of decoupling the Green functions of a Heisenberg ferromagnet is approached from an ad hoc point of view. As indicated by Wortis, the Green functions pertaining to spin problems do not obey a simple Dyson equation; the higher order Green function is proportional to the lower order Green function (the proportionality factor being the "mass operator") plus an anomalous additive term. The forms of both the mass operator and the anomalous term are here evaluated by requiring agreement of the theory with rigorous results known from low-temperature and high-temperature expansions. The mass operator is found to be almost precisely that proposed by Callen on heuristic grounds. At low temperatures, the anomalous term makes a particularly significant correction to the results for spin ½, but becomes relatively unimportant for higher spins. Near the Curie temperature, however, the contribution of the anomalous term is important even for larger spin values. The resultant theory agrees with the Curie temperature estimates of Rushbrooke and Wood or Domb and Sykes to within about 1%. In addition to the imposed agreement with the same authors to order ${\left(\frac{T_c}{T}\right)}^4$ at high temperatures and with Dyson to order ${\left(\frac{T}{T_c}\right)}^4$ at low temperatures, the estimates of the critical value of the magnetic energy are also in close agreement with those of Domb and Sykes. The critical behavior of the susceptibility, as $T$ approaches $T_c$ from above, and of the magnetization, as $T$ approaches $T_c$ from below, is investigated. It is found that within the random phase and the Callen approximations, the susceptibility obeys a relation of the form $\chi =\mathrm{const}{(1-\frac{T_c}{T})}^{-2\mathrm{}}$, whereas the magnetization approaches zero as ${(1-\frac{T}{T_c})}^{\frac{1}{2}}$. However, when the anomalous term is taken into account consistently, the theory predicts that if the susceptibility is set to agree with Domb and Sykes' result, $\chi =\mathrm{const}{(1-\frac{T_c}{T})}^{-\frac{4}{3}}$, the magnetization below the Curie point would approach zero as ${(1-\frac{T}{T_c})}^{\frac{1}{3}}$.