Statistical Mechanics of Ferromagnetism
Authors: Callen Herbert B. 
Published: April 2005

The fact that reliable statistical mechanical theories of ferromagnetism existed only in the very low temperature region (spin wave theory) and in the very high temperature region (Kramers and Opechowski) has seriously hampered the development of magnetic theory. The Weiss molecular field theory neglects all spin-spin correlations and therefore gives too high a Curie temperature, misrepresents the details of the phase transition, and is unreliable as a basis for analyzing any effect depending on spin interactions. Cluster methods attempt to introduce correlation, but they are both inadequate and sometimes run into severe difficulties, such as the appearance of an anti-Curie temperature (in the Bethe-Peierls-Weiss theory). The source of such difficulties is demonstrated by the formalism of P. W. Kasteleijn and J. Van Kranendonck [Physica 22, 317 (1956)] who showed that the cluster theories correspond to the choice of a self-inconsistent two-particle density matrix. Furthermore the spin wave theory indicates that the interactions which are dominant in establishing spin-spin correlations are iterated interactions through enormously many long and complicated paths connecting the spins. Cluster methods are unable to include these long paths, and stress instead the direct short-path interactions. The need for a many-body theory is therefore indicated.
One such many-body theory is the Green function treatment of Bogoliubov and Tyablikov. That treatment is equivalent to the random phase approximation. The simplest formulation starts with the spin wave theory of Dyson. The destruction operator for a Dyson spin wave is ak≡12NS12  ∑ j S(Rj) exp(ik·Rj). Similarly Dyson introduces the Fourier components of the z components of spin Skz≡ ∑ j Sz(Rj) exp(ik·Rj). Unfortunately the spin waves defined in terms of these operators are neither orthogonal nor are they eigenstates of the Hamiltonian. In fact the commutation relations are [Skz,aλ]=ak+λ,  [ak,aλ+]=(1/NS)Sk−λzand [H,ak]=−μHak−(2SN)−1 Σλ (ελ·k−ελ)(Sλzak−λ−aλSk−λz),where εk is the energy of a simple spin wave, expressed as a function of k; for small k it is proportional to k2. The random phase approximation ``decouples'' the awkward higher-order terms in the right-hand members of these commutation relations. In lowest order this decoupling consists of replacing Sλz by N〈Sz〉δλ0, where 〈Sz〉 is the average value of the z component of spin, to be evaluated self-consistently. Then the commutation relations become [ak,aλ+]=(1/S)〈Sz〉δλkand [H,ak]=−[μH+εk〈Sz〉/S]ak. The first of these states that the quasi-particles (magnons) produced by the creation operators ak+ are bosons. The second commutation-relation states that the energy of such a quasi-particle is μH+εk〈Sz〉/S. Thus the average number of such magnons, of wave vector k, is 〈nk〉=〈Sz〉/S[expβ(μH+εk〈Sz〉S−1)−1]−1,which is the result given by Tyablikov1 and by Englert. It differs from the simple spin wave theory by reducing the energy of each spin wave proportionally to the average magnetization.
The Green function method (or RPA) gives a result which forms a useful interpolation throughout the entire temperature range. It agrees to second order in 1/T with the Kramers-Opechowski series in the high temperature (paramagnetic) region. Near the Curie temperature the theory reduces to the Weiss theory, and in the very low temperature region it agrees with the simple spin wave theory. The corrections to the simple spin wave theory at slightly higher temperatures are incorrectly given, however, as the RPA approximation predicts a correction to the magnetization varying as T3, whereas Dyson has shown that the lowest order correction varies as T4. This error in the theory arises from the decoupling procedure, which ignores spin-wave correlation effects in replacing Sλz by zero (if λ≠0).
A direct approach to a statistical theory consists in a series expansion of the partition sum, a diagrammatic representation of the terms, and a partial summation of those terms corresponding to the long, circuitous paths connecting and correlating two spins. Such a theory has been carried out by Brout and co-workers and by Horwitz and Callen. The results have been published, or are in publication in detailed form elsewhere for the Ising model, and the extension to the Heisenberg model will be presented in a separate more extensive publication. Again the theory provides a useful interpolation between the low and high temperature region. Evaluation of the theory at low temperatures, by R. Stinchcomb, indicates that it properly gives the Dyson dynamical (T4) correction.