  We study Hilbert space fragmentation in the extended Fermi-Hubbard model with
nearest and next-nearest-neighbor interactions. Using a generalized spin/mover
picture and saddle point methods, we derive lower bounds for the scaling of the
number of frozen states and for the size of the largest block preserved under
the dynamics. We find fragmentation for strong nearest- and
next-nearest-neighbor repulsions as well as for the combined case. Our results
suggest that the involvement of next-nearest-neighbor repulsions leads to an
increased tendency for localization. We then model the dynamics for larger
systems using Markov simulations to test these findings and unveil in which
interaction regimes the dynamics becomes spatially localized. In particular, we
show that for strong nearest- and next-nearest-neighbor interactions random
initial states will localize provided that the density of initial movers is
sufficiently low.
