Statistical Mechanical Theory of Ferromagnetism. High Density Behavior

The partition function of the Ising model of ferromagnetism is examined in the limit of high density in the anticipation that in the limit of infinite density one recovers the Weiss molecular field. The formal parameter of expansion is $\frac{1}{z}$ where $z$ is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). In the absence of long-range order, only ring diagrams in the cluster expansion contribute. These give a divergence in the specific heat at $k{T}_c=\Sigma {j\ne i}^{}{v}_{\mathrm{ij}}$ where $v_{\mathrm{ij}}$ is the exchange potential. This is the molecular field value for the Curie point $T_c$. In the presence of a magnetic field the partition function is evaluated for fixed magnetic moment $M$ in the same approximation, $M$ being determined by minimization. This results in a susceptibility differing from the molecular field theory and hence an inconsistency in the theory.

The inconsistency is traced back to the observation that the acceptance of ring diagrams is equivalent to the gaussian model of Kac and Berlin which violates the sum rule ${{\Sigma }_{i=1\mathrm{}}}^N{{\mu }_i}^2=N$. Here ${\mu }_i$ is the "spin" per particle and $N$ is the total number of particles. This condition is reinstated by insuring the sum rule. The result leads to the spherical model. Thus, a consistent high density approximation to the Ising model is the spherical model. Below the Curie point or for fixed magnetic field, $M$ is again held fixed and only the Fourier components of the spin density with nonvanishing wave vector are "sphericalized." The result leads to a physically acceptable model which becomes the molecular field theory at low temperatures or high fields and deviates in $O\left(\frac{1}{z}\right)$ in general. Formally, the results are simply expressed in terms of a temperature dependent Weiss field. These results differ from the ordinary spherical model which is physically unacceptable below the Curie point. However, a molecular field modification of the spherical model due to M. Lax yields the same result when properly interpreted.

It is shown that the above results are also valid (to the same approximation) in the quantum mechanical Heisenberg model, for temperatures above the Curie point.