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.
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.
We investigate the relation between magnon, magnon bound states, and the classical soliton solutions in the isotropic, the anisotropic-exchange, and the easy-axis ferromagnetic Heisenberg chains. The Dyson-Maleev boson representation is used to study the interaction between magnons, and bound states are investigated in terms of two-spin Green's functions. For easy-axis and anisotropic-exchange magnets, a mapping to the Bose gas with attractive $\ensuremath{\delta}$-function interactions is established, and it yields the eigenvalues of the magnon and the
mth bound state in the weak-coupling and continuum limits. The classical limit of the two-spin Green's function reveals that, to leading order in temperature, the bound-state resonance and the associated effects on the two-magnon continuum survives. An important result is a
T2 dependence of the "binding" energy of the bound-state resonance. As a consequence, in a classical description the bound states enter only in order
T2. Finally, we quantize the soliton solutions according to the Bohr-Sommerfeld and de Broglie rules. This approach is found to be exact for the
s=21 isotropic Heisenberg chain and for sufficiently small wave numbers for all
s values. In the anisotropic-exchange and easy-axis models, it agrees with the results obtained from the mapping on the Bose gas, for large quantum numbers and in the easy-axis case for $s\ensuremath{\gg}\frac{1}{2}$ in addition. On this basis, we conclude that the envelope solitons considered here lead to bound-state resonances and associated effects in a classical treatment. Moreover, their semiclassical quantization gives remarkably accurate energy levels.
Abstract The two-magnon problem in the k = − π /2 sector of a finite-size spin-1/2 chain with ferromagnetic nearest-neighbor (NN) interaction ( J 1 > 0) and antiferromagnetic next-nearest-neighbor (NNN) interaction ( J 2 < 0) and anisotropy parameters Δ 1 and Δ 2 is solved exactly by combining a set of exact two-magnon Bloch states and a plane-wave ansatz. Two types of two-magnon bound states (BSs), i.e., the NN and NNN exchange BSs, are revealed. We establish a phase diagram in the J 1 /(∣ J 2 ∣Δ 2 )-Δ 1 plane where regions supporting different types of BSs are analytically identified. It is found that no BSs exist (the two types of BSs coexist) when both Δ 1 and Δ 2 are small (large) enough. Our results for the isotropic case Δ 1 = Δ 2 = 1 are consistent with an early work [Ono I, Mikado S and Oguchi T 1971 J. Phys. Soc. Japan 30 358].