Low-energy physics in the critical phase of the bilinear-biquadratic spin-1 chain

We use an efficient density matrix renormalization group (DMRG) algorithm to compute precise dynamic structure factors for the bilinear-biquadratic spin-1 chain with Hamiltonian H = Σ_i [cosθ (S_i * S_i+1) + sinθ (S_i * S_i+1)²]. Here, we focus on explaining the physics in the extended critical phase (π/4 ≤ θ < π/2) of the model. The phase transition from the Haldane phase to the critical phase is marked by the SU(3)-symmetric ULS point (θ = π/4), where the elementary excitations are spinons that can be obtained from the Bethe ansatz solution. As we move deeper into the critical phase, the spinon continua contract, and new striking features appear at higher energies. In the vicinity of the transition point from the critical to the ferromagnetic phase, a dispersion with a surprisingly simple functional form emerges, suggesting integrability of the model in the limit θ → π/2^-.