Including the second term yields a realistic model that also predicts a transition from conductor to insulator as the ratio of interaction to hopping, U / t , is varied. This ratio can be modified by, for example, increasing the inter-atomic spacing, which would decrease the magnitude of t without affecting U . In the limit where U / t ≫ ≫ 1 , the chain simply resolves into a set of isolated magnetic moments. If U / t is not too large, the overlap integral provides for superexchange interactions between neighboring magnetic moments, which may lead to a variety of interesting magnetic correlations, such as ferromagnetic, antiferromagnetic, etc. depending on the model parameters. The one-dimensional Hubbard model was solved by Lieb and Wu using the Bethe ansatz. Essential progress was achieved in the 1990s: a hidden symmetry was discovered, and the scattering matrix, correlation functions, thermodynamic and quantum entanglement were evaluated.