We study in this paper a series of Gutzwiller Projected wavefunctions for S=1
spin chains obtained from a fermionic mean-field theory for general S>1/2 spin
systems [Phys. Rev. B 81, 224417] applied to the bilinear-biquadratic (J-K)
model. The free-fermion mean field states before the projection are 1D paring
states. By comparing the energies and correlation functions of the projected
pairing states with those obtained from known results, we show that the
optimized Gutzwiller projected wavefunctions are very good trial ground state
wavefunctions for the antiferromagnetic bilinear-biquadratic model in the
regime K0). We find that different topological phases of the
free-fermion paring states correspond to different spin phases: the weak
pairing (topologically non-trivial) state gives rise to the Haldane phase,
whereas the strong pairing (topologically trivial) state gives rise to the
dimer phase. In particular the mapping between the Haldane phase and Gutwziller
wavefunction is exact at the AKLT point K=1/3. The transition point between the
two phases determined by the optimized Gutzwiller Projected wavefunction is in
good agreement with the known result. The effect of Z2 gauge fluctuations above
the mean field theory is analyzed.Comment: 10 pages,7 figure