Three-reversed-spin bound states in the Heisenberg model

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Abstract
In a one-, two-, or three-dimensional ferromagnetic system of localized spins described by the Heisenberg model, states occur in which two reversed spins in an otherwise aligned system are bound to one another; that is, localized relative to one another, but not with respect to any specific lattice site. The present work is concerned with the question of whether such bound states exist in the subspace of three reversed spins. A formulation of this problem is developed along the lines of recent investigations in the three-body bound-state problem which is valid for all dimensions. In order to display some complicating features which are peculiar to this problem, the two-reversed-spin subspace is discussed. The dependence of the pairwise interaction on the momentum of the pair is illustrated and the chosen method of dealing with the unphysical states which occur when a given spin is "raised" more than its spin magnitude allows is explained. The methods described in the two-reversed-spin case are applied to three reversed spins and the wave equation for the amplitudes in the reversed-spin basis is derived. Next, this equation is simplified using three-body methods in conjunction with symmetry properties of the amplitudes. In one dimension the final equation is a linear integral equation for a function of a single variable. This equation is solved numerically and Bethe's one-dimensional results—specialized to the three-reversed-spin case—are reproduced with the exception that the present work contains an additional set of shallow bound states.

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