DMFT in a nutshell
Han-Yong Choi
Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea
Dynamical mean field theory (DMFT) is probably the most reliable method to describe an interacting many-particle system. The key idea of the DMFT is that, when physics of a problem was determined by the competition of two energy scales, the problem can be made tractable by simplifying other complicating aspects. For the Mott transition, it is the competition between the kinetic and Coulomb energies that determines physics. A prototype model that describes the situation is the Hubbard model. It can not be solved reliably. It is rendered tractable, however, in the seemingly unphysical limit of the infinite spatial dimension, where the competition is fully retained while the less essential momentum dependence is neglected. In that limit, an interacting problem on a lattice is mapped onto an interacting problem on a site. The methods to solve interacting problems on a site were well developed through the Kondo problem such as numerical renormalization group (NRG).
In this tutorial, we will first introduce the basic idea of the DMFT, and then will review how the long standing problem of the Mott transition was successfully understood by applying the DMFT to the Hubbard model. We will then consider the Hubbard-Holstein model, a simplest model that includes both the electron-electron and electron-phonon interactions. The results obtained in our group by applying NRG+DMFT to the Hubbard-Holstein model will be discussed.