Heisenberg model with higher-order exchange: Ground-state properties and excitations

Properties of Heisenberg models with higher-order exchange, $H=\frac{1}{2} \Sigma {l=1\mathrm{}}^{l_0}\Sigma {n,m}^{}{J}_{\mathrm{nm}}^{\mathrm{}\left(l\right)\mathrm{}}{\left({\overrightarrow{\mathrm{S}}}_n·{\overrightarrow{\mathrm{S}}}_m\right)}^l$, are discussed. It is found that there is a one-to-one relation between the set of all coupling constants and the set of the energies of all $\nu$-magnon states with $\nu \le 2s$, where $s$ is the spin at each site. Necessary and sufficient conditions for a ferromagnetic ground state are derived for systems with only nearest-neighbor interactions as well as for systems which also include next-nearest-neighbor interactions. These conditions are qualitatively different in the quantum and in the classical case. Discussing the elementary excitations, a classification of the $\nu$-magnon bound states is found: an extreme case is the pure exchange bound state, where the $\nu$ spin deviations extend around the nearest $\nu$ sites; the opposite extreme case is the pure single-ion bound state where the $\nu$ spin deviations are most likely at the same site. Between these two cases there exist all combinations of both, if higherorder exchange with $l_0\ge \nu$ is present. The case $\nu =2\mathrm{}$ is discussed in more detail.