Static and Dynamic Structure Factors in the Haldane Phase of the Bilinear-Biquadratic Spin-1 Chain

The excitation spectra of the $T=0$ dynamic structure factors for the spin, dimer, and trimer fluctuation operators as well as for the center fluctuation operator in the one-dimensional $S=1$ Heisenberg model with isotropic bilinear $(J\mathrm{cos}\ensuremath{\theta})$ and biquadratic $(J\mathrm{sin}\ensuremath{\theta})$ exchange are investigated via the recursion method for systems with up to $N=18$ sites over the predicted range, $\ensuremath{-}\ensuremath{\pi}/4l\ensuremath{\theta}\ensuremath{\lesssim}\ensuremath{\pi}/4,$ 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 $T=0$ dynamics. At $\ensuremath{\theta}=\mathrm{arctan}\frac{1}{3}$ (valence-bond solid point), the dynamically relevant spectra in the invariant subspaces with total spin ${S}_{T}=0,1,2$ 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 ${(S}_{T}=0,2).$ The dimer and trimer spectra at $q=\ensuremath{\pi}$ are found to consist of single modes with $N$-independent excitation energies ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{D}/|{e}_{0}|=5$ and ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{T}/|{e}_{0}|=6,$ 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 $(\ensuremath{\theta}=\ensuremath{-}\ensuremath{\pi}/4),$ the two-magnon excitations turn into two-spinon excitations.