The complete two-magnon spectrum of the ferromagnetic Heisenberg chain including NNN interactions

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Abstract
A study of two-magnon bound states in the next-nearest-neighbour (NNN) ferromagnetic Heisenberg chain begun by Majumdar in 1969 is extended by finding the bound states and continuum resonances for all pair wave-vectors for eta =J2/J1 in the ferromagnetic range of eta =-1/4 to infinity. An exact Green function formalism is combined with the exact analytic results recently obtained by the authors for the lattice Green functions of the system to afford a definitive analysis. The Heisenberg case exhibits a modified nearest-neighbour (NN) type bound state throughout the range of eta studied and for total pair wave-vector (K=k1+k2) throughout the zone, in contrast to Majumdar who concluded that this was only the case for positive eta . The continuum also possesses a rich structure which is displayed via a shaded (grey-level) intensity plot in the ( omega , K) plane. An unexpected result is the presence of a second (anti-)bound state just above the continuum for negative eta . The evolution of the bound state and resonance structure is also examined for the transition to dominant NNN interactions. A general formulation is given which includes Ising and uniaxial anisotropies as well as biquadratic exchange for arbitrary range, dimension and Bravais lattice.