PHYSICAL REVIEW B VOLUME 35, NUMBER 1 1 JANUARY 1987 Bethe ansatz for two-magnon bound states in anisotropic magnetic chains of arbitrary spin N. Papanicolaou Department of Physics, University of Crete and Research Center of Crete, P. O. Box 1527, Heraklion, Crete GR 711 -10, Greece G. C. Psaltakis Research Center of Crete, Institute of Electronic Structure and Laser, P. O. Box 1527, Heraklion, Crete GR 711 -10, Greece (Received 14 April 1986; revised manuscript received 11 August 1986) The two-magnon spectra in anisotropic magnetic chains of arbitrary spin are derived by an ele- mentary method which is an extension of Bethe's original ansatz for spin- —, systems. Applications are presented for quantum spin chains with uniaxial or Ising Anisotropy as well as chains with quartic exchange interactions. The nature of certain collective modes emerging in a 1/n expansion is also clarified. I. INTRODUCTION The study of bound states of magnons has been a sub- ject of considerable interest since the early work of Bethe. ' The Bethe ansatz for multimagnon states provided essen- tially complete information concerning the bound-state spectrum of spin- —, ' magnetic chains. Moreover a variety of one-dimensional problems have been solved exactly by suitable generalizations of the original ansatz. The applicability of the Bethe ansatz is, however, limit- ed to dynamical systems that are completely integrable. This limitation becomes evident already in the context of anisotropic magnetic chains of general spin S& —, , for which integrability is doubtful except in very special cases. Therefore more conventional methods had to be employed. For the two-magnon spectra, a Green's- function approach initiated by Wortis has practically dominated all studies. This approach works in any di- mension, provided the ground state is ferromagnetic, and also yields expressions for measurable quantities, namely the dynamic structure factors. Extensive calculations of the latter in one dimension have been carried out by Schneider and co-workers. Nevertheless we note that although lack of complete integrability would be crucial for three-magnon and multimagnon states, it puts little restriction on the dynamics of two-magnon states. Indeed, a Bethe-ansatz type of technique for evaluating the two-magnon bound- state frequencies of the completely isotropic ferromagnetic chain of arbitrary spin has already been used. In our present work we show that in fact the original Bethe an- satz may be extended to the calculation of the two- magnon states also in anisotroptc magnetic chains of arbi- trary spin. Hence we are able to provide an alternative to Wortis's method for the special case of one-dimensional spin systems. The present calculation furnishes a simple expression for the two-magnon wave function together with the ener- gy spectrum. Details are given in Sec. II for a chain with an easy axis of magnetization due to a uniaxial anisotro- py. The appearance of a single-ion bound state, in addi- tion to the usual exchange bound state, is confirmed and clarified. Section III extends the calculation to a variety of spin chains. Thus we study the effect of Ising anisotro- py, planar chains in a spin-flop phase caused by a magnet- ic field and Heisenberg models incorporating quartic ex- change interactions. A by-product of this analysis is the illumination of the nature of certain collective modes emerging in a 1/n ex- pansion, which are identified with the single-ion bound states. This and related issues are discussed in Sec. IV. Our conclusions are summarized in Sec. V. II. THE TWO-MAGNON STATE HC O=EOC 0 Eo: — N(J +D)S (2.2) Let now N„denote the state obtained from No by de- creasing the azimuthal spin by one unit at the site n. The one-margon eigenstate is constructed as N 4, = g e'""0&„, H+) E)4)— — n=1 Ei ED+co, co=2SJ— — (l — cosk)+ (2S — 1 )D . (2.3) Here cu is the one-magnon excitation energy developing a mass gap for S & —, due to the anisotropy. In order to gain some insight about the nature of two- magnon states we consider first the extreme limit of large D, or J=O. The moments in (2. 1) uncouple in this limit. Two-magnon states can be constructed either by reducing the azimuthal spin by two units at a single site or by reducing its value by one unit at two different sites. In The method will be illustrated in detail for a magnetic chain described by the Hamiltonian N H =— g [J(S„. S„+))+D (S„') ], n=1 (2. l) S„=S(S+1), J, D) 0, assuming periodic boundary conditions. Since both J and D are positive, the ground state No is fully ordered in the z direction and its energy is given by 35 342 1987 The American Physical Society