Low-Temperature Behavior of the Planar Heisenberg Ferromagnet

We have examined the low-temperature properties of the cubic-planar Heisenberg ferromagnet with nearest-neighbor exchange which is defined by the Hamiltonian $\mathcal{H}=-\Sigma {i,j}^{}{J}_{\mathrm{ij}}{\overrightarrow{\mathrm{S}}}_i·{\overrightarrow{\mathrm{S}}}_j+\Sigma {i,j}^{}(J_{\mathrm{ij}}-K_{\mathrm{ij}})S_i^x{S}_j^x$, where $-J\le K\le J$ ($J$ positive). We find that as the exchange-anisotropy parameter $\theta =\frac{\mathrm{}(J-K)\mathrm{}}{J}$ ranges over the planar ferromagnetic stability limits $0\le \theta \le 2$, the behavior of the system changes from that of the isotropic ferromagnet at $\theta =0\mathrm{}$ into that of the isotropic antiferromagnet at $\theta =2\mathrm{}$. The system's noninteracting-spin-wave frequency, ground-state energy, zero-point spin deviation, and lowest-order renormalized frequency scale between isotropic ferromagnetic and antiferromagnetic values as $\theta$ goes from zero to two. Over most of the system's stability range, the planar ferromagnet exhibits a mixture of properties combining characteristics of its intrinsic ferromagnetism with those of the antiferromagnet. This behavior is discussed in terms of an isomorphic mapping symmetry for nearest-neighbor exchange in loose-packed lattices which requires that in the limit $\theta =2\mathrm{}$ the planar ferromagnet be unitarily equivalent to the isotropic antiferromagnet.