Locality optimization for parent Hamiltonians of tensor networks

Tensor Network states form a powerful framework for both the analytical and numerical study of strongly correlated phases. Vital to their utility is that they appear as exact ground associated parent Hamiltonians, where canonical proof techniques guarantee controlled space structure. Yet, while those Hamiltonians are local by construction, known often yield complex which act on rather large number spins. In this paper, we present an algorithm systematically simplify breaking them down into any given basis elementary interaction terms. The underlying optimization problem semidefinite program, thus optimal solution can be found efficiently. Our method exploits degree freedom in construction -- excitation spectrum terms over it optimizes such obtain best possible approximation. We benchmark our AKLT model Toric Code model, show (acting 3 or 4 12 sites, respectively) broken 2-body 4-body then apply paradigmatic Resonating Valence Bond (RVB) kagome lattice. Here, simplest previously Hamiltonian acts all spins one star. With algorithm, vastly simpler Hamiltonian: find RVB state whose products at most four Heisenberg interactions, range further constrained, providing major improvement 12-body Hamiltonian.