In microscopic theories of phase transitions occurring in itinerant-electron systems, physical phenomena are generally considered in the random-phase (RPA) or mean-particle-field approximation. We describe here a many-body theoretical method of calculating the appropriate order-parameter susceptibility function $\chi (\mathrm{q},{\omega }^{+})$ which goes beyond the RPA. A diagrammatic analysis of the equation of motion for a quantity related to $\chi (\mathrm{q},{\omega }^{+})$ is made, and it is shown how one can systematically and self-consistently include the effect of order-parameter fluctuations on $\chi (\mathrm{q},{\omega }^{+})$. The method is applied here to the paramagnetic phase of an itinerant-electron ferromagnet. A mean-fluctuation-field approximation (MFFA) which includes the contribution of one internal spin fluctuation to $\chi (\mathrm{q},{\omega }^{+})$ is discussed in detail. Its temperature-dependent contribution to ${\chi }^{-1\mathrm{}}$ goes roughly as ${\left(k_{B}T{\rho }_{{\epsilon }_F}\right)}^{\frac{4}{3}}$. A self-consistent solution of the MFFA equation for $\chi \mathrm{}(0,0)\mathrm{}$ leads to a Curie-Weiss-like behavior for it. We make an explicit comparison of our results with experimental values for Ni, and find good agreement in the range $0.1\le \frac{(T-T_c)}{T_c}=\epsilon \le 0.6$. In the Stoner or RPA theory the Curie-Weiss law is ascribed to the $T^2$ part of the particle-field term $-U\int f^{-\prime }\mathrm{}\left(E\right)\mathrm{}\rho \mathrm{}\left(E\right)\mathrm{}dE$. This is smaller than the MFF term by a factor ${\left(k_{B}T{\rho }_{{\epsilon }_F}\right)}^{-\frac{2}{3}}$, and for Ni, is only 5% of the latter. The Curie-Weiss-like law observed in metallic paramagnets is therefore due to the mean spin-fluctuation field, as also realized by Murata and Doniach, and by Moriya and Kawabata. Going beyond the MFFA, we calculate the contribution of the simplest spin-fluctuation correlation diagram. The contribution of this diverges logarithmically as $\epsilon \to 0$. When this term becomes comparable to the MFFA, we are well in the critical regime which cannot be conveniently discussed by this method. This criterion is used to provide a first-principles estimate of the static and dynamic critical regimes in Ni. The former obtains for $\epsilon \lesssim 0.05$ and the latter for $\epsilon \lesssim 0.06\left(\frac{q}{k_F}\right)$. We show how spin fluctuations suppress ferromagnetism in a two-dimensional system and plot ${\chi }_{2\mathrm{d}}(0,0)$ vs $T$ for a Ni-like film in the MFFA. The method developed here can be applied to discuss fluctuation effects in the ferromagnetic phase, in superconductivity, and in itinerant-electron antiferromagnetism.

• Received 14 February 1974

DOI:https://doi.org/10.1103/PhysRevB.10.4014