Magnon Bound States in Anisotropic Linear Chains

Abstract

The recent observation of two-, three-, four-, and five-magnon bound states in the linear chains of Co${\mathrm{Cl}}_2$·2${\mathrm{H}}_2$O has prompted a theoretical examination of such states in an anisotropic linear ferromagnetic chain with $S=\frac{1}{2}$. A new method called the Ising-basis-function (IBF) method is developed. This method treats the conventional, localized Ising wave functions as Wannier functions, from which a complete, orthonormal set of Bloch functions (IBF's) is formed. Using these IBF's as basis functions, we obtain the expression for the energy of the two-magnon bound state originally found by Orbach for general longitudinal exchange anisotropy. Furthermore, we can calculate the energy of the ($n>\mathrm{}2\mathrm{}$)-magnon bound states for the case of strong longitudinal anisotropy. The method is also applied to describe the effect of transverse exchange anisotropy. It is shown that this anisotropy causes an interaction between bound states, particularly important near zero field, and gives rise to a finite probability of exciting the bound states by photon absorption. The generalization of this method to treat bound states in two and three dimensions and for $S>\mathrm{}\frac{1}{2}$ is also discussed. The method is simple and has a direct physical interpretation. As an example, a physical description of the two-magnon bound state in a general system is given. Since the IBF method automatically contains some of the magnon-magnon interactions in zero order, it should be useful in other problems where these interactions are important.