Field-induced phase transitions of symmetry protected topological states in integer spin chains

We investigate spin chains with bilinear-biquadratic spin interactions in the gapped symmetry protected topological (SPT) state and at the critical Uimin-Lai-Sutherland (ULS) point as a function of an applied magnetic field. From calculations of the dynamical structure factor S(q,ω) using time-dependent density matrix renormalization group (DMRG), we observe a phase transition from the gapped to a gapless phase driven by a field in the Haldane phase. In the ULS critical phase, we see a phase transition at h_c1 from a gapless phase with two degenerate soft modes at 2π/3 and 4π/3 to another gapless phase with a soft mode at π. From S(q,ω) we observe the fractionalization of modes below h_c1 and their recombining to form new modes above h_c1. We show that some aspects of the dispersion of these modes can be qualitatively understood within a single-mode approximation.