  We investigate the Zeeman-field-driven quantum phase transitions between
singlet spin liquids and algebraically ordered O(2) nematic spin liquids of
spin-one bosons in one-dimensional optical lattices. We find that the critical
behavior is characterized by condensation of hardcore bosons instead of ideal
magnons in high dimensional lattices. Critical exponents are strongly
renormalized by hardcore interactions and critical states are equivalent to the
free Fermion model up to the Friedal oscillations. We also find that the
algebraically ordered nematic spin liquids close to critical points are fully
characterized by the Luttinger liquid dynamics with Luttinger liquid parameters
magnetically tunable. The Bethe Ansatz solution has been applied to determine
the critical magnetization and nematic correlations.
