Static and dynamic structure factors in the Haldane phase of the bilinear-biquadratic spin-1 chain

The excitation spectra of the $T=0\mathrm{}$ dynamic structure factors for the spin, dimer, and trimer fluctuation operators as well as for the center fluctuation operator in the one-dimensional $S=1\mathrm{}$ Heisenberg model with isotropic bilinear $\left(J\cos \theta \right)$ and biquadratic $\left(J\sin \theta \right)$ exchange are investigated via the recursion method for systems with up to $N=18\mathrm{}$ sites over the predicted range, $-\pi /4<\theta \lesssim \pi /4,$ of the topologically ordered Haldane phase. The four static and dynamic structure factors probe the ordering tendencies in the various coupling regimes and the elementary and composite excitations which dominate the $T=0\mathrm{}$ dynamics. At $\theta =\arctan \frac{1}{3}$ (valence-bond solid point), the dynamically relevant spectra in the invariant subspaces with total spin $S_T=0,1,2\mathrm{}$ are dominated by a branch of magnon states ${(S}_T=1),$ by continua of two-magnon scattering states ${(S}_T=0,1,2),$ and by discrete branches of two-magnon bound states with positive interaction energy ${(S}_T=0,2).\mathrm{}$ The dimer and trimer spectra at $q=\pi$ are found to consist of single modes with $N$-independent excitation energies ${\omega }_{\lambda }^D/|e_0|=5\mathrm{}$ and ${\omega }_{\lambda }^T/|e_0|=6,$ where $e_0{=E}_0/N$ is the ground-state energy per site. The basic structure of the dynamically relevant excitation spectrum remains the same over a substantial parameter range within the Haldane phase. At the transition to the dimerized phase $(\theta =-\pi /4),$ the two-magnon excitations turn into two-spinon excitations.