An Approximate Quantum Theory of the Antiferromagnetic Ground State

A careful treatment of the zero-point energy of the spin-waves in the Kramers-Heller semiclassical theory of ferromagnetics leads to surprisingly exact results for the properties of the ground state, as shown by Klein and Smith. An analogous treatment of the antiferromagnetic ground state, whose properties were unknown, is here carried out and justified. The results are expected to be valid to order $\frac{1}{S}$ or better, where $S$ is the spin quantum number of the separate atoms.The energy of the ground state is computed and found to lie within limits found elsewhere on rigorous grounds. For the linear chain, there is no long-range order in the ground state; for the simple cubic and plane square lattices, a finite long-range order in the ground state is found. The fact that this order can be observed experimentally, somewhat puzzling since one knows the ground state to be a singlet, is explained.