A Pairwise Approximation in the Hubbard Model

The Hubbard Hamiltonian is projected onto a representation consisting of electron pairs characterised by the momentum of their centre of mass. Within this approximation the electron gas can be viewed as a collection of subsets, each of which contains a constant number of electron pairs, all having the same centre of mass momentum. As these subsets are decoupled, the Hubbard Hamiltonian is diagonalised to give two types of many-body eigenstates: correlated and uncorrelated. The uncorrelated pairs build up an ideal Fermi gas. Excellent agreement is found for the uncorrelated energy calculated at zero temperature in one dimension between this model and the Bethe ansatz, for arbitrary electron concentration and magnitude of the electron interaction. The correlated states turn out to be of the BCS type.