Matrix product operator representation of polynomial interactions

Abstract

We provide an exact construction of particular Hamitonians on a one-dimensional lattice as matrix product operators, a type of tensor network. Namely, we consider Hamiltonians describing interactions between degrees of freedom at lattice sites whose strength grows with the lattice site separation as a polynomial multiplied by an exponential. 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.