  We provide an exact construction of interaction Hamiltonians on a
one-dimensional lattice which grow as a polynomial multiplied by an exponential
with the lattice site separation as a matrix product operator (MPO), a type of
one-dimensional tensor network. We show that the bond dimension is $(k+3)$ for
a polynomial of order $k$, independent of the system size and the number of
particles. Our construction is manifestly translationally invariant, and so may
be used in finite- or infinite-size variational matrix product state
algorithms. Our results provide new insight into the correlation structure of
many-body quantum operators, and may also be practical in simulations of
many-body systems whose interactions are exponentially screened at large
distances, but may have complex short-distance structure.
