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We study in this paper a series of Gutzwiller Projected wavefunctions for S=1spin chains obtained from a fermionic mean-field theory for general S>1/2 spinsystems [Phys. Rev. B 81, 224417] applied to the bilinear-biquadratic (J-K)model. The free-fermion mean field states before the projection are 1D paringstates. By comparing the energies and correlation functions of the projectedpairing states with those obtained from known results, we show that theoptimized Gutzwiller projected wavefunctions are very good trial ground statewavefunctions for the antiferromagnetic bilinear-biquadratic model in theregime K<J, (J>0). We find that different topological phases of thefree-fermion paring states correspond to different spin phases: the weakpairing (topologically non-trivial) state gives rise to the Haldane phase,whereas the strong pairing (topologically trivial) state gives rise to thedimer phase. In particular the mapping between the Haldane phase and Gutwzillerwavefunction is exact at the AKLT point K=1/3. The transition point between thetwo phases determined by the optimized Gutzwiller Projected wavefunction is ingood agreement with the known result. The effect of Z2 gauge fluctuations abovethe mean field theory is analyzed.

Gutzwiller Projected wavefunctions in the fermonic theory of S=1 spin chains