Green's-Function Method for Antiferromagnetism

The statistical mechanics of simple antiferromagnets is studied over the entire temperature range by methods similar to those of Callen and Liu for ferromagnets. It is shown that the energy spectrum and the sublattice magnetization as functions of temperature are the solutions of a set of coupled equations. The equations are solved numerically for $S=\frac{1}{2}$, and the results compare very well with existing theories in both high- and low-temperature limits. The Néel temperature is also calculated for general spin value. Compared with other theoretical estimates, the Néel temperature obtained this way is appreciably higher for $S=\frac{1}{2}$ but approaches agreement in the large-spin limit. The transverse correlation functions of two spins and the correlation length for short-range order above the transition temperature are also calculated. The longitudinal correlation functions of two spins are calculated by extending Liu's method for ferromagnetism to antiferromagnetism. A calculation is carried out for $S=\frac{1}{2}$ using the Callen decoupling scheme for three-spin Green's functions. It is shown that the longitudinal correlation functions are related to the first-order response of the system to a space- and time-varying field. Thus, to calculate the longitudinal correlation functions, a perturbation calculation to first order is necessary and Callen's original decoupling scheme for the three-spin Green's functions has to be extended for Green's functions with perturbation. A satisfactory extension of the Callen decoupling scheme to the first-order equation of motion for Green's functions is found. The extended decoupling scheme leads to a number of desirable results: Rotational invariance of the correlation functions and susceptibilities at and above the Néel temperature and the validity of the sum rule for spin operators over the same temperature range. The parallel susceptibility is also calculated, as well as the internal energy and the specific heat above the Néel temperature. The groundstate energy turns out to be slightly lower than the values predicted by other theories. Numerical calculations of parallel and perpendicular susceptibilities for $S=\frac{1}{2}$ are carried out and the results are presented.