Strongly Correlated Electronic Materials
The Simplest model of a periodic solid is a periodic array of valence orbitals embedded in a matrix of atomic cores
If correlations between different orbitals may be neglected, solving the problem of one of the orbitals is often equivalent to solving the whole system
These orbitals hybridize to form a valence band
If we may ignore nonlocal Coulombic correlations, these states are filled by electrons starting from the bottom of the band. This simple one-electron picture ignores both intersite and even some intrasite correlations between the electrons. Nevertheless, this approach has been very successful at describing many of the properties of periodic solids. However, electronic correlations are responsible for some of the most fascinating properties of materials.
Superconductivity: For example, in conventional superconductors, such as lead, the phonons mediate an attraction between the electrons. As illustrated below, first one electron moves quickly through the lattice, causing the positively charged lattice ions to distort towards its path. Due the the different time scales of the lattice and the electrons, once the lattice is fully distorted, the first electron is far away (more than 1000 Angstroms). Thus a second electron can be attracted to the region of positive charge along the path of the first electron, without feeling its Coulomb repulsion. At sufficiently low temperatures this attractive interaction causes the electrons to simultaneously pair into bosons and condense into a superconducting state.
Magnetism: Electronic correlations are also responsible for both the formation of magnetic moments and their ordering in magnetic materials. To understand the former, note that higher angular momentum states are precluded from sampling the region with the highest ionic potential by the angular momentum barrier. As a result, higher angular momentum states are more strongly screened, and are therefore promoted in energy as shown in the potential energy sketch below. In elemental Ce, for example, the 4f, 5d and 6s states all participate in the valence shell. However, since the principle quantum number n determines the size of the orbital, the 4f orbitals are far smaller, and hence more correlated than the 5d or 6s orbitals. These correlations move the doubly occupied (singlet) 4f state to higher energy. Thus, 4f electrons tend to not be paired into singlets and so form a magnetic moment.
Adjacent moments can become antiferromagnetically correlated through superexchange. If two orbitals have a hybridization overlap t, then moments on adjacent sites tend to have opposite spin so that they can gain hybridization energy.
High Temperature Superconductivity: The cuprate high temperature superconductors are basically doped antiferromagnets, or nearly antiferromagnetic metals. We can gain some idea of how holes in an antiferromagnet pair by considering a classical spin system pictured below. Holes on adjacent sites cost less exchange energy than do separated holes, since they break fewer antiferromagnetic bonds. This mediates a pairing potential between the holes.
Heavy Fermion Behavior: Electronic correlations are also responsible for Heavy Fermion behavior in which certain materials act as conventional metals with a strongly enhanced electronic mass (which can exceed 1000 electron masses). This behavior is observed in materials with a periodic array of moments embedded in a metallic matrix, and results from the competition between the antiferromagnetic exchange energy which tends to localize the conduction electrons into singlets and the kinetic energy of these states at the Fermi surface