  Stark many-body localization (SMBL) is a phenomenon observed in interacting
systems with a nearly uniform spatial gradient applied field. Contrasting to
the traditional many-body localization phenomenon, SMBL does not require
disorder. Here we investigate SMBL in a spin-$1/2$ described by a Heisenberg
model including a next-nearest-neighbor exchange coupling. By employing an
exact diagonalization approach and time evolution calculation we analyze both
level spacing ratio (LSR) statistics of the Hamiltonian model as well as the
dynamics of the system from a given initial state. Our results reveals that for
zero field in our finite system, LSR statistics suggest localization while the
dynamics shows thermalization, which has been attributed to a finite-size
effect. Slightly nonuniform field gradient, LSR statistic predictions agree
very well with the dynamics of the physical quantities indicating
delocalization and localization for small and large field gradient,
respectively. More interestingly, we find that localization is robust in the
presence of next-nearest-neighbor coupling in the Hamiltonian. Moreover, this
coupling can be tuned to enhance SMBL in the system, meaning that localized
regimes can be obtained for smaller field gradient as compared to the
traditional nearest-neighbor isotropic Heisenberg model.
