  We investigate the topological gauge-theory terms and quantum criticality in
spin-1 Kitaev chain with generic single-ion anisotropies (SIAs). The
ground-state phase diagram, including Kitaev spin liquid (KSL) phase and
gapless dimer phase, is determined by the infinite time evolving block
decimation (iTEBD) method. By varying the strength of uniaxial SIA, there is
quantum phase transition between the KSL phase and the dimer phase, which is
equivalent to the confinement-deconfinement transition in the lattice Schwinger
model with a topological $\theta$-angle of $\pi$. Here, we demonstrate an added
rhombic SIA to the $\mathbb{Z}_2$ symmetric model shifts topological angle
$\theta$ away from $\pi$, leading to the emergence of $y$-ferroquadrupole and
$x$-ferroquadrupole phases for negative and positive values of rhombic SIAs,
respectively. By adjusting these rhombic SIAs, the system from the
$y$-ferroquadrupole to the $x$-ferroquadrupole phases undergoes either a
crossover through the KSL phase or a genuine phase transition through the
deconfined line, thus providing an example for unnecessary phase transitions.
We find the spin-1 Hamiltonian can be exactly mapped to the spin-1/2 effective
extended PXP Hamiltonian coupled to all positive $\mathbb{Z}_2$ gauge charges,
where the uniaxial SIA is equivalent to the uniform detuning while the rhombic
SIA plays a role of staggered detuning. We then examine the hierarchical
fragmentation of the Hilbert space and its associated dynamics in specific
limiting cases. The quantum many-body scar (QMBS) arises from the Fibonacci
constraint under a weak field of SIAs. When the staggered detuning is
sufficiently large, the resultant slow dynamics can be nominally understood
using a second-order effective Hamiltonian derived by the Schrieffer-Wolff
transformation.
