chain of hydrogen atoms is thus predicted to be a conductor under conventional band theory. This 1D string is the only configuration simple enough to be solved directly. But in the case where the spacing between the hydrogen atoms is gradually increased, at some point the chain must become an insulator. Expressed using the Hubbard model, the Hamiltonian is made up of two terms. The first term describes the kinetic energy of the system, parameterized by the hopping integral, . The second term is the on-site interaction of strength that represents the electron repulsion. Written out in second quantization notation, the Hubbard Hamiltonian then takes the form where is the spin-density operator for spin on the -th site. The density operator is and occupation of -th site for the wavefunction is . Typically t is taken to be positive, and U may be either positive or negative, but is assumed to be positive when considering electronic systems. Without the contribution of the second term, the Hamiltonian resolves to the tight binding formula from regular band theory. Including the second term yields a realistic model that also predicts a transition from conductor to insulator as the ratio of interaction to hopping, , is varied. This ratio can be modified by, for example, increasing the inter-atomic spacing, which would decrease the magnitude of without affecting . In the limit where , the chain simply resolves into a set of isolated magnetic moments. If 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. More complex systems Although Hubbard is useful in describing systems such as a 1D chain of hydrogen atoms, it is important to note that more complex systems may experience other effects that the Hubbard model does not consider. In general, insulators can be divided into Mott–Hubbard insulators and charge-transfer insulators. A Mott–Hubbard insulator can be described as This can be seen as analogous to the Hubbard model for hydrogen chains, where conduction between unit cells can be described by a transfer integral. However, it is possible