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

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 K<J, (J>0). 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.