by Dr. Gene Bickers
One of the most active fields of research in theoretical condensed matter (CM) physics concerns the behavior of correlated electrons in solids. Whereas atomic and molecular (AM) physics treats the detailed quantum mechanics of systems of a few (less than ten to perhaps a hundred) electrons, CM physics treats the properties of electrons in macroscopic numbers (on the order of Avogradro's number in a bulk sample). Despite this difference a common feature in both AM and CM theory is the need to approximate accurately the effects of electron-electron correlations.
If each electron in a molecule or a solid could be treated independently, the quantum mechanics of these systems would be quite simple. One could write down a separate Schrodinger equation describing the interaction of each electron with an array of positively charged nuclei, then solve for the behavior of the full system by summing over one-electron results. Unfortunately the physics of many-electron systems is not nearly so simple. The reasons are as follows: (1) Electrons obey Fermi-Dirac statistics, implying many-electron wave functions must change sign when any two electron labels are interchanged. (2) Electrons strongly repel each other through a long-range Coulomb interaction.
These two features of electron systems may be crudely taken into account without giving up the simplifying framework of separate Schrodinger equations for each electron. One approximates each electron's dynamical interaction with its companions by a ``mean-field'' interaction with a time-averaged negative charge distribution. If this distribution is determined self-consistently and one-electron energy levels are filled according to the Pauli Principle, the resulting description is the so-called Hartree-Fock approximation. This is the simplest approach which incorporates the effects of (1) and (2) above. All corrections to an AM or CM theory beyond the Hartree-Fock approximation are conventionally categorized as ``electron correlations,'' since these corrections describe electrons coordinating, or correlating, their motions to lower the system energy.
The Hartree-Fock approximation is the basis for the description of electrons in atoms using one-electron orbitals (1s, 2p, 4f, and so forth). In solids the Hartree-Fock approximation also provides a useful starting point, and in most cases its predictions are at least qualitatively correct.
In some solids, however, Hartree-Fock predictions are completely wrong. For example, Hartree-Fock may predict that a solid is a metallic paramagnet, while experiments show it to be an antiferromagnetic wide-gap insulator. The neglect of electron correlations is particularly suspect in solids with small one-electron orbitals (3d, 4d, 4f, and 5f). In such systems, which include transition metal, rare earth, and actinide compounds, the electronic energy is significantly lowered by keeping electrons apart at short distances. The Hartree-Fock approximation, which concerns itself only with average charge distributions, ignores these important short-distance correlations.
Since only a handful of models for correlated electrons in solids can be solved exactly (and then only in one spatial dimension), this problem has been studied using a great number of approximate techniques. These techniques include (1) exact diagonalization of the quantum Hamiltonian matrix for small clusters; (2) Monte Carlo simulation of the quantum statistical partition function; (3) perturbation theory about the weak- or strong-correlation limits; (4) variational estimation of the ground-state energy; and (5) reformulation and study of the problem using various field-theoretical methods.
A strong motivation for perfecting these techniques has come from experiment. In particular, the discovery of high-temperature superconductivity in a large number of copper oxide compounds during the late 1980's and early 1990's has increased and sustained interest in this area.
Our own group at USC has studied a variety of models using self-consistent extensions of the Hartree-Fock approximation which incorporate the effect of electronic charge and spin fluctuations (category 3 above). Our studies have been among the earliest to suggest the presence of an exotic ``d-wave'' superconducting transition in the most widely investigated model for the high-temperature superconductors. Experiments have now unambiguously demonstrated that superconductivity in the copper oxides is of the d-wave variety, though the electronic pairing mechanism remains controversial. Our most recent work (a synthesis of approaches from categories 2 and 3) has suggested a new and conceptually simple way to describe the crossover of some systems from antiferromagnetic wide-gap insulators to unconventional metals with electron doping.
The discovery of materials with novel electronic properties will continue to motivate problems in correlated electron physics in the future. It remains a theoretical goal to predict electronic properties and trends in new materials before experiments are performed, but this quantitative ability will not be achieved for some time to come.
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