  We investigate critical phenomena of a spin-1 chain in the vicinity of the
SU(3) symmetric critical point, which we already specified in the previous
study. We numerically diagonalize a Hamiltonian combining the
bilinear-biquadratic (BLBQ) Hamiltonian with the trimer Hamiltonian. We then
discuss the numerical results based on the conformal field theory (CFT) and the
renormalization group. As a result, we firstly verify that the critical point
found in our previous study is the tri-critical point among the Haldane phase,
trimer phase, and the trimer liquid (TL) phase. Secondly, with regard to the
TL--trimer transition and TL--Haldane transition, we find that the critical
phenomena around this tri-critical point belong to the
Berezinskii--Kosterlitz--Thouless (BKT)-like universality class. We thirdly
find the boundary between the Haldane phase and the trimer phase, which is
illustrated by the massive self-dual sine-Gordon (SDSG) model.
