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

Abstract

The excitation spectra of the T=0 dynamic structure factors for the spin, dimer, and trimer fluctuation operators as well as for the newly defined center fluctuation operator in the one-dimensional S=1 Heisenberg model wi th isotropic bilinear $(J\cosθ)$ and biquadratic $(J\sinθ)$ exchange are investigated via the recursion method for systems with up to N=18 site s over the predicted range, $-π/4<θ\lesssimπ/4$, of the topologically ordered Haldane phase. The four static and dynamic structure factors probe t he ordering tendencies in the various coupling regimes and the elementary and composite excitations which dominate the T=0 dynamics. At $θ= \arctan{1/3}$ (VBS 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 $q=π$ ar e found to consist of single modes with $N$-independent excitation energies $ω_λ^D/|e_0|=5$ and $ω_λ^T/|e_0|=6$, 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 ($θ=-π/4$), the two-magnon excitations turn into two-spinon excitations.