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In a previous paper [Phys. Rev. B 85,195144 (2012)], variational Monte Carlomethod (based on Gutzwiller projected states) was generalized to $S=1$ systems.This method provided very good trial ground states for the gapped phases of$S=1$ bilinear-biquadratic (BLBQ) Heisenberg chain. In the present paper, weextend the approach to study the low-lying elementary excitations in $S=1$chains. We calculate the one-magnon and two-magnon excitation spectra of theBLBQ Heisenberg chain and the results agree very well with recent data inliterature. In our approach, the difference of the excitation spectrum betweenthe Haldane phase and the dimer phase (such as the even/odd size effect) can beunderstood from their different topology of corresponding mean field theory. Weespecially study the Takhtajan-Babujian critical point. Despite the fact thatthe `elementary excitations' are spin-1 magnons which are different from thespin-1/2 spinons in Bethe solution, we show that the excitation spectrum,critical exponent ($\eta=0.74$) and central charge ($c=1.45$) calculated fromour theory agree well with Bethe ansatz solution and conformal field theorypredictions.

Gutzwiller Approach for Elementary Excitations in $S=1$ Antiferromagnetic Chains