Europhysics Prize 2006: Dynamical Mean-Field Theory

Heavy fermion compounds, high temperature superconductors and many other materials with unusual properties like colossal magnetoresistance in manganites, revived the studies of strongly correlated electron systems. This field was already very active in the 60's, in particular for the study of the Mott metal to insulator transition, experimentally observed in materials like Vanadium oxide. The full explanation of this phenomenon is one of the main achievements obtained with the method introduced by the winners of the 2006 Agilent prize.

The main theoretical paradigms previously available to describe metallic phases, like band theory and Fermi liquid theory, were inadequate to deal with strongly correlated systems. Even if the insulating phase could be described in terms of electrons localized at atoms, strongly correlated systems are generically in the intermediate regime where the localizing electron-electron interaction is comparable and competes with the delocalizing kinetic energy. These two terms are usually schematized via the Hubbard Hamiltonian with an on-site repulsion and with a hopping term between neighbour sites. The competing effect leads to a variety of physical properties and to rich phase diagrams. The difficulty in dealing with these systems, even when they are schematized in terms of the simplest model Hamiltonian, is due to the intrinsic non-perturbative nature of the problem in the absence of the simplifying aspects of universality available for instance in classical critical phenomena. Solvable limits with a well-defined controlling parameter are therefore of invaluable help in understanding these systems.

Walter Metzner and Dieter Vollhardt introduced the method of dealing with correlated fermions on a lattice by a suitable rescaling of the hopping in the large dimensionality limit or better in the large lattice coordination number, whose inverse is the controlling parameter. In this way they succeeded to maintain the dynamical competition between the kinetic energy and the Coulomb interaction along with the discovery of the main simplification of the method, namely the locality of perturbation theory. This dynamical mean field theory (DMFT) is a well defined starting point to deal with finite dimensional correlated systems in the same spirit as the cavity mean field is for classical statistical systems.

Antoine Georges and Gabriel Kotliar introduced a considerable technical and conceptual improvement of the DMFT that produced many applications to physical systems. By relating DMFT to the single impurity Anderson model, the full quantum many body problem of correlated materials on a lattice or on the continuum was reduced to an impurity self-consistently coupled to a bath of electrons.

The single site problem retains the full dynamics of the original problem. In analogy with the classical mean field theory where a single degree of freedom (e.g. a spin on a site) is immersed in the self-consistent effective field (the Weiss field) of the remaining degrees of freedom, here a local set of quantum mechanical degrees of freedom on a single site are linked to the reservoir of the electrons via a frequency dependent function which plays the role of the self-consistent mean field and allows the electrons to be emitted and absorbed in the atom. A local description of correlated systems is achieved, which is amenable to calculations while the main features of competition between itinerancy and locality are still present.

Various extensions of the method are now considered e.g.: Realistic one-particle and Coulomb interaction aspects are included by combining local density approximation method and DMFT; Short range space-correlations are introduced by switching from a single atom to a cluster.

The very successful applications of the method have covered numerous phenomena at the heart of the present research activity. To quote just few of them we can mention the metal-insulator transition, the doped Mott insulator, the competition of spin, charge and orbital order, the interplay between correlation and electron-phonon interaction, the phonon spectrum of delta Plutonium and some general features related to quantum criticality.

In conclusion the Dynamical Mean-Field Theory represents one of the most powerful approaches to strongly correlated electron systems. In addition to the number of successes of DMFT in model systems and realistic calculations, the applications of the method are still increasing and many extensions and developments are nowadays the object of the research of several groups.