  We introduce a one-dimensional correlated-hopping model of spinless fermions
in which a particle can hop between two neighboring sites only if the sites to
the left and right of those two sites have different particle numbers. Using a
bond-to-site mapping, this model involving four-site terms can be mapped to an
assisted pair-flipping model involving only three-site terms. This model shows
strong Hilbert space fragmentation (HSF). We define irreducible strings (IS) to
label the different fragments, determine the number of fragments, and the sizes
of fragments corresponding to some special IS. In some classes of fragments,
the Hamiltonian can be diagonalized completely, and in others it can be seen to
have a structure characteristic of models which are not fully integrable. In
the largest fragment in our model, the number of states grows exponentially
with the system size, but the ratio of this number to the total Hilbert space
size tends to zero exponentially in the thermodynamic limit. Within this
fragment, we provide numerical evidence that only a weak version of the
eigenstate thermalization hypothesis (ETH) remains valid; we call this
subspace-restricted ETH. To understand the out-of-equilibrium dynamics of the
model, we study the infinite-temperature time-dependent autocorrelation
functions starting from a random initial state; we find that these exhibit a
different behavior near the boundary compared to the bulk. Finally we propose
an experimental setup to realize our correlated-hopping model.
