  Using a generalized Jordan-Wigner transformation combined with the defining
representation of the SO(N) spin, we map the SO(N) bilinear-biquadratic(BLBQ)
spin chain into the N-color bosonic particle model. We find that, when the
Jordan-Wigner transformation disentangles the symmetry-protected topological
entanglement, this bosonic model becomes negative-sign free in the context of
quantum Monte-Carlo simulation. For the SO(3) case, moreover, the
Kennedy-Tasaki transformation for the S=1 BLBQ chain, which is also a
topological disentangler, derives the same bosonic model through the dimer-R
bases. We present the temperature dependence of the energy, entropy and string
order parameter for the SO(N=3, 4, 5) BLBQ chains by a world-line Monte-Carlo
simulation for the N-color bosonic particle model.
