Exact solution of the two-magnon problem in the $k=-\pi/2$ sector of a finite-size anisotropic spin-1/2 frustrated ferromagnetic chain

Abstract

The two-magnon problem in the $k=-\pi/2$ sector of a \emph{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 $\Delta_1$ and $\Delta_2$ 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 $J_1/(|J_2|\Delta_2)$-$\Delta_1$ 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 $\Delta_1$ and $\Delta_2$ are small (large) enough. Our results for the isotropic case $\Delta_1=\Delta_2=1$ are consistent with an early work [Ono I, Mikado S and Oguchi T 1971 \emph{J. Phys. Soc. Japan} \textbf{30} 358].