Two-magnon excitations in the Heisenberg ferromagnet with uniaxial anisotropy

Two-magnon excitations are examined in detail for the nearest-neighbor simple-cubic Heisenberg ferromagnet with various values of uniaxial anisotropy and all total wave vectors ($\overrightarrow{\mathrm{K}}$) in the [111] direction. Attention is focused on the changes brought about by the addition of a single-ion-pair mode to the basic exchange-pair modes of the isotropic case. The propagator for exchange pairs separates into a contribution from the $d$-wave mode and a combined effect from $s$-wave exchange and single-ion pairings. The latter has the same denominator as the propagator for single-ion pairs but exhibits very little of the strong single-ion resonance due to the vanishing of the numerator at the position of the single-ion Ising level. Lattice Green's functions approaching analytic quality have been used to evaluate various spectral components in order to establish how the single-ion mode behaves in the two-magnon continuum. For values of uniaxial anisotropy such that the single-ion Ising level crosses the $s$-wave bound-state mode of the isotropic case, we find that the single-ion resonance correlates well with the corresponding Ising level in the continuum but evolves continuously to the $s$-wave exchange bound state at the zone boundary.