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 $J^{\prime }<0$, 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 $\mathrm{\Delta }={\mathrm{\Delta }}^{\prime }=1$ ($\mathrm{\Delta }$ and ${\mathrm{\Delta }}^{\prime }$ are the corresponding anisotropy parameters), a threshold of $J/|J^{\prime }|$, 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 $k=-\pi$, 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 ${\mathrm{\Delta }}^{\prime }-D/\left|J^{\prime }\right|$ 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 $S\ge 1/2$. Finally, we numerically calculate the $n$-magnon spectra for $S=1/2$ with $n\le 5$ 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.

• Received 28 January 2024
• Revised 27 March 2024
• Accepted 17 April 2024

DOI:https://doi.org/10.1103/PhysRevB.109.174403