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We investigate the properties of the Heisenberg S=1 chain with bilinear andbiquadratic interactions in a magnetic field using the Density MatrixRenormalization Group, Bethe ansatz and field theoretical considerations. In alarge region of the parameter space, we identify a magnetized ferroquadrupolarLuttinger liquid consisting of a quasi-condensate of bound magnon pairs. Thisliquid undergoes a continuous pair unbinding transition to a more conventionalLuttinger liquid region obtained by polarizing the system above the Haldane gapregion. This pair unbinding transition is shown to be in the Ising universalityclass on top of a Luttinger liquid, leading to an effective central charge 3/2.We also revisit the nature of the partially polarized Luttinger liquid aroundand above the Uimin-Lai-Sutherland point. Our results confirm that this is atwo-component liquid and rule out the formation of a single-component vectorchiral phase.

Phase diagram and continuous pair-unbinding transition of the bilinear-biquadratic S=1 Heisenberg chain in a magnetic field