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arXiv:1208.6177

Free from spurious solutions integral equation for three-magnon bound states in 1D XXZ ferromagnet

P. N. Bibikov

Published Aug 30, 2012• 0 citations
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Abstract

A new integral equation for three-magnon bound states in XXZ spin chain is suggested. Unlike the one presented by C. K. Majumdar about 40 years ago this equation does not have spurious solutions. Such an advantage is a result of decomposition of the wave function in the Bloch basis rather than in the basis of flat waves.
DOI
10.48550/arxiv.1208.6177

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Few-magnon bound states in quantum spin chains have been long studied and attracted much recent attention. For a higher-spin ferromagnetic XXZ chain with single-ion anisotropy, several features regarding the evolution of the low-lying two-magnon bound states with varying wave number were observed in the literature. However, most of these observations are only qualitatively understood due to the lack of analytical tools. By combining a set of exact two-magnon Bloch states and a plane-wave ansatz, we achieve a complete solution of the two-magnon problem in such a system. We identify parameter regions that support different types of two-magnon bound states, with the boundaries defined by algebraic equations. We discover a narrow region in which two single-ion bound states coexist. We show that the phase diagrams for distinct wave numbers are similar to each other, which enables us to map the evolution of the bound states to the rectilinear movement of a representative point for given parameters in a rescaled phase diagram. This dynamic picture provides quantitative interpretations of the observed features.
The problem of three spin deviations from a fully aligned state is studied for the Heisenberg model with next-nearest-neighbour interactions for the case of spin 1. The method used is a straightforward generalization of the equation-of-motion method of Fukuda and Wortis, taking care of the unphysical states. The resulting integral equation is solved in one dimension and the dependence of the bound states on the next-nearest-neighbour interaction discussed. Numerical calculations have also been done for closed chains containing up to 40 spins. By using C N invariance of the Hamiltonian, the dimensionality of the space is substantially reduced. The results of the finite-chain calculation agree well with solutions obtained from the equation-of-motion method.
Recently several unphysical solutions of the three magnon bound state equation for the isotropic Heisenberg Hamiltonian have been found, and one unphysical solution for the Hamiltonian with longitudinal anisotropy has been computed. Here we complete the work for such unphysical solutions for all anisotropy from the Ising to the isotropic Heisenberg limit by directly solving the integral equation. Two types of wavefunctions are constructed. The eigenvalue of the first type satisfies a cubic equation in general, but gives only a real root in the Ising limit and a pair of complex conjugate roots in the isotropic limit. The other type has a single eigenvalue; this one, previously known numerically, is shown to have a simple analytic form.