  We study spontaneous dimerization and emergent criticality in a spin-3/2
chain with antiferromagnetic nearest-neighbor $J_1$, next-nearest-neighbor
$J_2$ and three-site $J_3$ interactions. In the absence of three-site
interaction $J_3$, we provide evidence that the model undergoes a remarkable
sequence of three phase transitions as a function of $J_2/J_1$, going
successively through a critical commesurate phase, a partially dimerized gapped
phase, a critical floating phase with quasi-long-range incommensurate order, to
end up in a fully dimerized phase at very large $J_2/J_1$. In the field theory
language, this implies that the coupling constant of the marginal operator
responsible for dimerization changes sign three times. For large enough $J_3$,
the fully dimerized phase is stabilized for all $J_2$, and the phase
transitions between the critical phases and this phase are both
Wess-Zumino-Witten (WZW) SU(2)$_3$ along part of the boundary and turn first
order at some point due to the presence of a marginal operator in the WZW
SU(2)$_3$ model. By contrast, the transition between the two dimerized phase is
always first order, and the phase transitions between the partially dimerized
phase and the critical phases are Kosterlitz-Thouless. Finally, we discuss the
intriguing spin-1/2 edge states that emerge in the partially dimerized phase
for even chains. Unlike their counterparts in the spin-1 chain, they are not
confined and disappear upon increasing $J_2$ in favour of a reorganization of
the dimerization pattern.
