Low-Temperature Behavior of the Planar Heisenberg Ferromagnet

Abstract

We have examined the low-temperature properties of the cubic-planar Heisenberg ferromagnet with nearest-neighbor exchange which is defined by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}{i,j}^{}{J}_{\mathrm{ij}}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}+\ensuremath{\Sigma}{i,j}^{}({J}_{\mathrm{ij}}\ensuremath{-}{K}_{\mathrm{ij}}){S}_{i}^{x}{S}_{j}^{x}$, where $\ensuremath{-}J\ensuremath{\le}K\ensuremath{\le}J$ ($J$ positive). We find that as the exchange-anisotropy parameter $\ensuremath{\theta}=\frac{(J\ensuremath{-}K)}{J}$ ranges over the planar ferromagnetic stability limits $0\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}2$, the behavior of the system changes from that of the isotropic ferromagnet at $\ensuremath{\theta}=0$ into that of the isotropic antiferromagnet at $\ensuremath{\theta}=2$. 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 $\ensuremath{\theta}$ 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 $\ensuremath{\theta}=2$ the planar ferromagnet be unitarily equivalent to the isotropic antiferromagnet.