  The phase diagram of the spin-1 chain with bilinear-biquadratic and
next-nearest neighbor inter- actions, recently investigated by Pixley, Shashi
and Nevidomskyy [Phys. Rev. B 90, 214426 (2014)], has been revisited in the
light of results we have recently obtained on a similar model. Combining
extensive Density Matrix Renormalization Group (DMRG) simulations with
conformal-field theory arguments, we confirm the presence of the three phases
identified by Pixley et al, a Haldane phase, a next-nearest neighbor (NNN)
Haldane phase, and a dimerized phase, but we come to significantly different
conclusions regarding the nature of the phase transitions to the dimerized
phase: i) We provide numerical evidence of a continuous Ising transition
between the NNN-Haldane phase and the dimerized phase; ii) We show that the
tri-critical end point, where the continuous transition between the Haldane
phase and the dimerized phase turns into a first order transition, is distinct
from the triple point where the three phases meet; iii) Finally, we demonstrate
that the tri-critical end point is in the same Wess-Zumino-Witten (WZW) SU(2)
level 2 universality class as the continuous transition line that ends at this
point
