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.
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