Few-magnon excitations in a frustrated spin-S ferromagnetic chain with single-ion anisotropy

We study few-magnon excitations in a finite-size spin-S chain with ferromagnetic nearest-neighbor (NN) interaction J>0 and antiferromagnetic next-nearest-neighbor (NNN) interaction J′<0, in the presence of the single-ion (SI) anisotropy D. We first reveal the condition for the emergence of zero-excitation-energy states. In the isotropic case with Δ=Δ′=1 (Δ and Δ′ are the corresponding anisotropy parameters), a threshold of J/|J′|, above which the ground state is ferromagnetic, is determined by exact diagonalization for short chains up to 12 sites. Using a set of exact two-magnon Bloch states, we then map the two-magnon problem to a single-particle one on an effective open chain with both NN and NNN hoppings. The whole two-magnon excitation spectrum is calculated for large systems, and the commensurate-incommensurate transition in the lowest-lying mode is found to exhibit different behaviors between S=1/2 and higher spins due to the interplay of the SI anisotropy and the NNN interaction. For the commensurate momentum k=-π, the effective lattice is decoupled into two NN open chains that can be exactly solved via a plane-wave ansatz. Based on this, we analytically identify in the Δ′-D/|J′| plane the regions supporting the SI or NNN exchange two-magnon bound states near the edge of the band. In particular, we prove that there always exists a lower-lying NN exchange two-magnon bound state near the band edge for arbitrary S≥1/2. Finally, we numerically calculate the n-magnon spectra for S=1/2 with n≤5 by using a spin-operator matrix element method. The corresponding n-magnon commensurate instability regions are determined for finite chains, and consistent results with prior literature are observed.