(S. Östlund, A. Stolin, B. Mehlig)
An important problem in condensed matter theory is to understand why certain material are conductors (metals) and others are insulators. The basic idea, put forth by Sir Neville Mott in the 1940's, shows there is a competition for electrons to delocalize and minimize the quantum mechanical kinetic energy and thereby make a metal, or to be localized to minimize the local interactions within the atom. In the latter case, the tendency is to form an insulator. The ``Mott transition'' occurs in a material when these tendencies are equally important.
A consequence of the widely successful band theory model is that in principle, an insulator can only exist if the electrons occupy precisely an integer number of bands. Since electrons exist in two spin types, it follows that an insulator must have an even number of electrons per unit cell. With this as a fundamental principle, one has relied on symmetry breaking mechanisms to describe metals where insulators occur that would otherwise not obey this simple counting rule. By breaking translational symmetries, one has a larger unit cell, thereby being able to create a band gap for systems that would otherwise have odd valence. Only in one dimension has one been able to get around this restriction and in this case, exact calculations show that a so-called Luttinger liquid can result. Similar calculations in two and three dimensions have not been possible, and many researchers believe a fundamental explanation is still lacking for metallic and/or insulating properties occur a number of such ``Mott insulators''.
We have been able to show that a special type of ``canonical transformation'' can mathematically map a Mott insulator, which is a localized and dense electronic system that is difficult to analyze, to a new quantum mechanical vacuum to which it is mathematically easy to add electrons. This approach entails exchanging bosonic and fermionic degrees of freedom, an approach that appears to be relatively untested.
We have recently exploited this method to analyze a Mott transition for a spin 3/2 system; this analysis strongly suggests there is a relation between Mott transitions and superconductivity. We have also looked at the Kondo lattice model where a similar fermion-fermion transformation can be applied. This a new approach to deal with strong electron correlations that have been difficult to model with conventional approaches. A clear pattern emerges, where the approach to Mott insulators with odd valence should be with boson-fermion transformations whereas even valence Mott insulators can be analyzed with fermion-fermion transformations.