Incommensurability in the magnetic excitations of the bilinear-biquadratic spin-1 chain

We study the magnetic excitation spectrum of the spin-1 chain with Hamiltonian H = Σi cos θSi · Si+1 + sin θ(Si · Si+1)2. We focus on the range 0 ≤ θ ≤ + π/4 where the spin chain is in the gapped Haldane phase. The excitation spectrum and static structure factor is studied using direct Lanczos diagonalization of small systems and density-matrix renormalization group techniques combined with the single-mode approximation. The magnon dispersion has a minimum at k = π until a critical value θc = 0.38 is reached at which the curvature (velocity) vanishes. Beyond this point, which is distinct from the VBS point and the Lifshitz point, the minimum lies at an incommensurate value that goes smoothly to k = 2π/3 when θ approaches π/4, the Lai-Sutherland point. The mode remains isolated from the other states: there is no evidence of spinon deconfinement before the point θ = +π/4. These findings explain recent observation of the behavior of the magnetization curve M ≈ (H − Hc)1/4 for θ = θc.