Effect of the Hubbard U on Anderson localization

We discuss in this section the effect of short-range interaction on the Anderson-localized states of a Fermi glass described in Chapter 1, Section 7, and in particular the question of whether the states are singly or doubly occupied. Ball (1971) was the first to discuss this problem. In this section we consider an electron gas that is far on the metal side of the Wigner transition (Chapter 8) the opposite situation is described in Chapter 6, where correlation gives rise to a metal-insulator transition. We also suppose that Anderson localization is weak (cca 1) otherwise it is probable that all states are singly occupied. 

Our problem is to estimate A . It is the mean repulsive energy of a pair of charges at a distance or1 from each other. This will depend on the effective dielectric constant of the electron gas. This should be large for weak Anderson localization and will effectively screen out the repulsion, except when both electrons are in the same atom. We therefore write 

The term U will raise the doubly occupied states in a many-electron model, so that at the Fermi energy there should be a mixture of singly and doubly occupied states. Below A all are doubly occupied. 

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