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Journal of Physics Condensed Matter

Three-magnon bound states in an S=1 linear chain with next-nearest-neighbour interaction

Charudatt Kadolkar, D K Ghosh, C. R. Sarma

Published Nov 30, 1992• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/4/48/019

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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.
The bound state problems associated with two magnons in a linear chain of spin 1/2 with nearest and next nearest neighbor exchange integrals, J and α J , respectively, are studied. The Hamiltonian is rewritten in terms of the Fermi operators in place of spin operators, and an integral equation to determine the bound state energies is obtained. In the cases when the total wave number K of two magnons is π or π/2, analytical solutions are easily obtained. A part of the present results disagrees with Majumdar's ones. Computer studies on the bound states are performed for a finite chain composed of spins up to twenty. They are consistent with the analytical results.
The spectra of both two- and three-magnon excitations in Heisenberg spin chains with next-nearest-neighbour (NNN) interactions are studied using scaling methods and the recursion method respectively. Both two-spin and three-spin couplings are considered for general spin S. In the three-magnon case, the asymptotic behaviour of the recursion coefficients can be used to directly identify the presence of bound states in the spectrum. Both integrable and non-integrable models can be studied, and the integrable models display special features in the bound-state spectra. Our results for the three-magnon bound states of S = 1 chains differ appreciably from those obtained in previous studies based upon an integral equation approach.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
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 =J 2 /J 1 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=k 1 +k 2 ) 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.
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.