Two-magnon spectra and Ising anisotropy: The relationship between resonances and bound states

Earlier studies of two-magnon bound states are complemented by calculations of the full density of states of two-magnon excitations. An exact formulation is developed at $T=0\mathrm{}°$ K for the $\overrightarrow{\mathrm{K}}=0\mathrm{}$ or Raman type of spectra corresponding to the excitation of two-spin deviations on the same or on nearest-neighbor sites for a ferromagnet with both exchange (Ising) and uniaxial (single-ion) anisotropies. Specific calculations are performed for the nearest-neighbor (nn) sc case in the presence of exchange anisotropy. The $K=0\mathrm{}$ spectra are followed as the anisotropy is increased from the isotropic limit until the bound states have been clearly separated from the two-magnon continuum. Spectra are exhibited for $S=\frac{1}{2},1,\mathrm{and}\frac{7}{2}$ for representative values of exchange anisotropy. These spectra satisfy derived sum rules very well in the absence of bound states and if resonances are not too sharp. In principle, these sum rules could be used to get the weights associated with the bound states. A relation noted by Wortis for the nn sc case is used to obtain $\overrightarrow{\mathrm{K}}$-dependent spectra along the (1,1,1) direction for the isotropic case $\overrightarrow{\mathrm{K}}$ is the total wave vector of the two magnons). This allows us to follow the resonant manifestation of the local (Ising) bound states, which exist at the zone corner (${\overrightarrow{\mathrm{K}}}_{max}$), through their transition to continuum resonances and deep inside the continuum as $K$ is decreased towards the Raman mode ($\overrightarrow{\mathrm{K}}=0\mathrm{}$). As a result, a comprehensive picture of the dynamics of two-magnon states emerges for both isotropic and anisotropic cases. A brief discussion of the effect of single-ion anisotropy, second-neighbor interactions, and other lattices is also given.