  Affleck, Kennedy, Lieb and Taski (AKLT) constructed an exemplary spin-3/2
valence-bond solid (VBS) state on the hexagonal lattice, which is the ground
state of an isotropic quantum antiferromagnet and possesses no spontaneous
magnetization but finite correlation length. This is distinct from the N\'eel
ordered state of the spin-3/2 Heisenberg model on the same lattice. Niggemann,
Kl\"umper and Zittartz then generalized the AKLT Hamiltonian to one family
invariant under spin rotation about the z-axis. The ground states of this
family can be parameterized by a single parameter that deforms the AKLT state,
and this system exhibits a quantum phase transition between the VBS and N\'eel
phases, as the parameter increases from the AKLT point to large anisotropy. We
investigate the opposite regime when the parameter decreases from the AKLT
point and find that there appears to be a Berezinskii-Kosterlitz-Thouless-like
transition from the VBS phase to an XY phase. Such a transition also occurs in
the deformation of other types of AKLT states with triplet-bond constructions
on the same lattice. However, we do not find such an XY-like phase in the
deformed AKLT models on other trivalent lattices, such as square-octagon, cross
and star lattices. On the star lattice, the deformed family of AKLT states
remain in the same phase as the isotropic AKLT state throughout the whole
region of the parameter. However, for two triplet-bond generalizations, the
triplet VBS phase is sandwiched between two ferromagnetic phases (for large and
small deformation parameters, respectively), which are characterized by
spontaneous magnetizations along different axes. Along the way, we also discuss
how various deformed AKLT states can be used for the purpose of universal
quantum computation.
