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.
Motivated by the lack of direct evidence with inelastic neutron scattering of the well documented bound state of Heisenberg ferromagnets, we use the time-dependent thermal density matrix renormalization group algorithm to study the temperature dependence of the dynamical spin structure factor of Heisenberg ferromagnetic spin chains. For spin 1/2, we show that the bound state appears as a well defined excitation with significant spectral weight in the temperature range $J/12\ensuremath{\lesssim}T\ensuremath{\lesssim}J/3$, pointing to the possibility of detecting it with inelastic neutron scattering near $k=\ensuremath{\pi}$ provided the temperature is neither too low nor too high---at low temperature, the spectral weight only grows as
T3/2 and, at high temperature, the bound state peak merges with the two-magnon continuum. For spin 1, the situation is more subtle because the bound state with two neighboring spin flips competes with an antibound state with two spin flips on the same site. As a consequence, the relative spectral weight of the bound state is smaller than for spin 1/2 and a weak resonance due to the antibound state appears in the continuum. A clearer signature of the bound state (antibound state) can be obtained if a negative (positive) biquadratic interaction is present.
Results are presented and interpreted for the zero- and finite-temperature excitation spectrum of ferromagnetic Ising-Heisenberg chains, as probed by dynamic form factors. Finite-chain calculations demonstrate the occurrence of a thermally induced resonance centered around zero frequency, and a broadening of the magnon and two-magnon bound-state resonances at low
T. The occurrence of bound states in the associated transitions demonstrates the failure of finite-order magnon perturbation theory.
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.
Understanding the low-temperature properties of the Quantum Heisenberg Ferromagnet is central to a mathematical understanding of magnetism. The Quantum Heisenberg Ferromagnet can naturally be reformulated in terms of interacting bosons (called spin-waves or magnons) as an expansion in the inverse spin size. We construct a trial state which provides the first order upper bound to the free energy. We also discuss the temperature dependence and conclude that optimal temperature dependence requires cancellations in higher orders of the expansion.