Electronic Correlation - Crystalline Systems

Crystalline Systems

In condensed matter physics, electrons are typically described with reference to a periodic lattice of atomic nuclei. Non-interacting electrons are therefore typically described by Bloch waves, which correspond to the delocalized, symmetry adapted molecular orbitals used in molecules (while Wannier functions correspond to localized MOs). A number of important theoretical approximations have been proposed to explain electron correlations in these crystalline systems.

The Fermi liquid model of correlated electrons in metals is able to explain the temperature dependence of resistivity by electron-electron interactions. It also forms the basis for the BCS theory of superconductivity, which is the result of phonon-mediated electron-electron interactions.

Systems that escape a Fermi liquid description are said to be strongly-correlated. In them, interactions plays such an important role that qualitatively new phenomena emerge. This is the case, for example, when the electrons are close to a metal-insulator transition. The Hubbard model is based on the tight-binding approximation, and can explain conductor-insulator transitions in Mott insulators such as transition metal oxides by the presence of repulsive Coulombic interactions between electrons. Its one-dimensional version is considered an archetype of the strong-correlations problem and displays many dramatic manifestations such as quasi-particle fractionalization. However there is no exact solution of the Hubbard model in more than one dimension.

The RKKY Interaction can explain electron spin correlations between unpaired inner shell electrons in different atoms in a conducting crystal by a second-order interaction that is mediated by conduction electrons.

The Tomonaga Luttinger liquid model approximates second order electron-electron interactions as bosonic interactions.