We investigate spin chains with bilinear-biquadratic spin interactions as a function of an applied magnetic field $h$. At the Uimin-Lai-Sutherland (ULS) critical point we find a remarkable hierarchy of fractionalized excitations revealed by the dynamical structure factor $S(q,omega)$ as a function of magnetic field yielding a transition from a gapless phase to another gapless phase before reaching the fully polarized state. At $h=0$, the envelope of the lowest energy excitations goes soft at two points $q_1=2pi/3$ and $q_2=4pi/3$, dubbed the A-phase. With increasing field, the spectral peaks at each of the gapless points bifurcate and combine to form a new set of fractionalized excitations that soften at a single point $q=pi$ at $h_{c1}approx 0.94$. Beyond $h_{c1}$ the system remains in this phase dubbed the B-phase until the transition at $h_{c2}=4$ to the fully polarized phase. We discuss the central charge of these two gapless phases and contrast the behavior with that of the gapped Haldane phase in a field.
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