Natural super-orbitals representation of many-body operators

We introduce the concept of natural super-orbitals for many-body operators, defined as the eigenvectors of the one-body super-density matrix associated with a vectorized operator. We first analyze the general analytical properties of these objects in various contexts, including the time-evolution operator of non-interacting Hamiltonians and Haar-random unitaries. We then perform a numerical investigation of the natural super-orbitals corresponding to both the time-evolution operator and a time-evolved local operator, focusing on two many-body systems: the fermionic $t\text{-}V$ chain and an impurity model, using tensor network simulations. Our results reveal that the $t\text{-}V$ model lacks a preferred super-orbital basis, while in the impurity model, the occupations of the natural orbitals for both operators decay exponentially at all times. This indicates that only a small number of orbitals contribute significantly to quantum correlations, enabling a compact matrix-product-operator representation in the natural orbital basis. Finally, we examine the spatial spread of the natural orbitals for time-evolved local operators in the impurity model and show that the complexity of this operator in the natural orbital basis saturates over time. This new framework opens the door to future research that leverages the compressed structure of operators in their natural super-orbital basis, enabling for instance the computation of out-of-time-order correlators in large interacting systems over extended time scales.