The research in quantum impurity problems has gained momentum when the Kondo
effect has been observed in **nanostructures**, such as **quantum
dots**, **carbon
nanotubes**, and **metal-organic complexes**. In addition, the advances in the
low-temperature scanning tunneling microscopy (STM) and spectroscopy have
enabled direct observation of the signatures of the Kondo effect in
**individual magnetic atoms** adsorbed on surfaces of metals. Using
the STM, atoms can be controllably moved around on surfaces and integrated
in atomic-scale nanostructures. It is, for example, possible to build
**magnetic clusters** made out of few atoms, make equally spaced (Kondo)
**chains**
or two-dimensional **lattices**, etc. A considerable amount of effort
is invested in building and studying such structures, and improving
the theoretical understanding of this class of systems.

It is important to note that the Kondo effect is a very general
phenomenon. By no means is it limited to magnetic problems. The impurity
degree of freedom can be position (as occurs in **two-level systems** and
for ions which have several degenerate equilibrium positions), orbital
quantum number (in the **orbital Kondo effect**), charge (**charge-Kondo
effect** in
quantum dots), etc. Related problems are also the X-ray absorption edge
problem and the diffusion of heavy particles (hydrogen, muons) in metals.

Kondo physics is strongly related to the **orthogonality catastrophe**:
as the spin flips, the new ground state of the continuum is orthogonal to
the previous one (in the continuum limit).

DMFT is an **iterative method**. The spectral function of the quantum
impurity model needs to be obtained, for example using quantum Monte Carlo
or numerical renormalization group. It is then used as an input in a
**self-consistency equation** that yields as output the effective density of
states, which represents the **combined effect of all other lattice sites** on a
given site. This density of states is then used in the new iteration to
obtain the corresponding spectral functions. This is repeated until
convergence.

The NRG method has since been generalized to study **two-channel
Kondo problem** (with its non-trivial **non-Fermi liquid ground state**),
**two-impurity Kondo problem** (which features a **quantum phase
transition**
of second order between an inter-impurity singlet phase and a Kondo phase,
separated by a critical point with non-Fermi liquid properties),
**multiple-orbital Anderson problem**, **Anderson-Holstein problem** (impurity interacting with localized
**phonon** modes) and other more complex systems.

Another important aspect of the NRG is that the continuum is put on the
front stage, not the impurity. The Kondo effect is something that occurs *in
the continuum* due to the presence of impurity.

The Kondo problem is a conformal field theory with a **boundary** (the
boundary corresponding to the impurity). Building on John Cardy's work, Ian
Affleck and Andreas Ludwig have applied the CFT techniques of to the
standard Kondo problem, two-channel Kondo problem and two-impurity Kondo
problem to study their ground states. The technique allows to determine the
**fixed point spectra**, as well as the low-temperature thermodynamic and
dynamic properties (**Green's functions**). The CFT predictions are in full
agreement with NRG calculations.

An important result of the CFT work is that all quantum impurity problems
appear to behave in a similar way. As the internal degrees of freedom at the
impurity site are screened, the boundary condition changes. This modifies
the properties of the field: in the simple Kondo case, the periodic boundary
condition in spatial direction is replaced by an anti-periodic boundary
condition, which corresponds to a change of scattering phase shift from 0 to
π/2. For generalized Kondo problems the boundary condition can, however,
become highly nontrivial. Apart from modifying the boundary condition, the
impurity degrees of freedom drop out of the problem at low temperatures. In
a matter of speaking, the **impurity is dissolved in the conduction band**.

P. W. Anderson, A poor man's derivation of scaling laws for the Kondo problem , J. Phys. C: Solid St. Phys., 3 2436 (1970).

I. Affleck, A current algebra approach to the Kondo effect , Nucl. Phys. B, 336 517 (1990).

I. Affleck and A. W. W. Ludwig, Exact critical theory of the two-impurity Kondo model , Phys. Rev. Lett. 68 1046 (1992).

J. von Delft, G. Zarand and M. Fabrizio, Finite-size bosonization of 2-channel Kondo model: A bridge between numerical renormalization group and conformal field theory , Phys. Rev. Lett. 81 196 (1998).

A. Georges, G. Kotliar, W. Krauth and M. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions , Rev. Mod. Phys., 68 13 (1996).

Last modified: 22.1.2007