The energy density is classically positive, a property that fails to be true in quantum theory. However, if it were indefinitely
negative, it might produce macroscopic violations of the second law of Thermodynamics or allow the existence of exotic spacetimes configurations (such as wormholes, time machines and warp drives). Fortunately, lower bounds exist and have been proved for linear fields (free quantum field theory), with few results for interacting models of QFT. In a class of quantum integrable models, on the level of one-particle states, we prove the existence of lower bounds for local averages of the energy density, establishing a first result for QFTs with interaction.