Confinement is a ubiquitous phenomenon in which constituent particles are bound together into a new quasiparticle. In condensed matter physics, magnon bound states can be regarded as a confinement of magnons, and an oscillating deconfinement crossover of two-magnon bound states scattered by a two-magnon continuum was revealed in the one-dimensional (1D) ferromagnetic chain by the spin entanglement entropy (SEE) analysis. Now, we extend the study to the two-dimensional (2D) and three-dimensional (3D) cases. Although the interactions between magnons in high dimensions are not as effective as in one dimension, two-magnon bound states can still survive around the corners of the Brillouin zone (BZ). In 2D or 3D square lattices, the bound states reduce their dimensionality to be 1D bound states at the BZ edges, which are bound only along the direction parallel to the BZ edge. This is verified by examining the relation between the magnon separation
d and the intercept
D of the SEE, which is found to satisfy the 1D relation
D=lnd+1, but in 2D and 3D lattices. Meanwhile, the 2D bound states satisfy
D=2lnd+1.3 in both 2D and 3D lattices. Unlike those in the 1D chain, some bound states will enter the continuum directly rather than cling to its bottom. We find that these immersed bound states are neither preserved nor deconfined completely when just meeting the continuum in finite-size systems, which is similar to what occurs in the 1D chain. Their SEE and magnon separation oscillate with respect to the system size. Therefore, these bound states undergo the oscillating deconfinement crossover in the continuum.
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.