  We provide evidence for the mapping of critical spin-1 chains, in particular
the SU(3) symmetric bilinear-biquadratic model with additional interactions, to
free boson theories using exact diagonalization and the density matrix
renormalization group algorithm. Using the correspondence with a conformal
field theory with central charge c=2, we determine the analytic formulae for
the scaling dimensions in terms of four Tomonaga-Luttinger liquid parameters.
By matching the lowest scaling dimensions, we numerically calculate these
field-theoretic parameters and track their evolution as a function of the
parameters of the lattice model.
