Variational Treatment of the Heisenberg Antiferromagnet

Abstract

A form of the Peierls free-energy variational theorem is applied to the Heisenberg Hamiltonian for a three-dimensional system with nearest-neighbor antiferromagnetic interaction. For a large magnetic field ($h\equiv \frac{g\mu H}{4SJz}\approx 1$) we find a phase boundary separating a region of antiferromagnetic order from one of ferromagnetic order. At low temperatures ($\theta \equiv \frac{\mathrm{kT}}{2SJz}\ll 1$) the phase boundary has the leading behavior: $h=1-a{\theta }^{\frac{3}{2}}$ with $a=\frac{2\zeta \left(\frac{3}{2}\right) {\left(\frac{3}{2\pi }\right)}^{\frac{3}{2}}}{S}$ for a simple cubic antiferromagnetic lattice (e.g., RbMn${\mathrm{F}}_3$). At the phase boundary the magnetization is continuous; whereas a discontinuity in the susceptibility is suggested but not firmly established by this treatment. Low-temperature expressions are given for the magnetization, susceptibility, and specific heat above the boundary. Numerical calculations show that, for the approximation used, the phase boundary extends to a maximum $\theta$ at which the magnetization is nonzero. For the limiting case of $h=0\mathrm{}$ we obtain Keffer and Loudon's renormalized spectrum and magnetization for a ferromagnet and for an antiferromagnet from a single variational calculation. Attention is also given to a reduced Hamiltonian which, when treated by the variational method, exhibits the properties of an antiferromagnetic molecular field model previously proposed by Garrett for $S=\frac{1}{2}$.