  The principles of ergodicity and thermalization constitute the foundation of
statistical mechanics, positing that a many-body system progressively loses its
local information as it evolves. Nevertheless, these principles can be
disrupted when thermalization dynamics lead to the conservation of local
information, as observed in the phenomenon known as many-body localization.
Quantum spin chains provide a fundamental platform for exploring the dynamics
of closed interacting quantum many-body systems. This study explores the
dynamics of a spin chain with $S\geq 1/2$ within the Majumdar-Ghosh model,
incorporating a non-uniform magnetic field and single-ion anisotropy. Through
the use of exact numerical diagonalization, we unveil that a nearly
constant-gradient magnetic field suppress thermalization, a phenomenon termed
Stark many-body localization (SMBL), previously observed in $S=1/2$ chains.
Furthermore, our findings reveal that the sole presence of single-ion
anisotropy is sufficient to prevent thermalization in the system.
Interestingly, when the magnitudes of the magnetic field and anisotropy are
comparable, they compete, favoring delocalization. Despite the potential
hindrance of SMBL by single-ion anisotropy in this scenario, it introduces an
alternative mechanism for localization. Our interpretation, considering local
energetic constraints and resonances between degenerate eigenstates, not only
provides insights into SMBL but also opens avenues for future experimental
investigations into the enriched phenomenology of disordered free localized
$S\geq 1/2$ systems.
