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Quantum impurity problems
Quantum impurity problems describe interaction between impurities with
internal degrees of freedom (spin, orbital quantum number, positional
pseudo-spin, etc.) and a continuum of states. The continuum can, for
example, be fermionic conduction band formed by valence electrons in a
metal, or bosonic bands of collective excitations (spin waves, phonons). The
example is Kondo's exchange (s-d) model of a magnetic impurity in a
metal. This model was used to explain the anomalous thermodynamic and
transport properties of metallic samples (such as copper) with diluted
magnetic impurities (such as cobalt), known as the Kondo effect. The
anomalous behavior appears due to increased spin-flip scattering of
conduction band electrons at low temperatures. In this process, the local
moment of the magnetic impurity is
screened below a characteristic Kondo temperature,
TK, and the impurity becomes effectively non-magnetic.
The research in quantum impurity problems has gained momentum when the Kondo
effect has been observed in nanostructures, such as quantum
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)
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
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).
Dynamical mean-field theory (DMFT) and lattice problems
The dynamical mean-field theory (DMFT) consists of neglecting spatial
correlations in a lattice model, while retaining the quantum (time)
correlations on a lattice site. In this manner, a lattice problem (such as
the Hubbard model) can be mapped to a quantum impurity model (for the
Hubbard model, this turns out to be the Anderson model).
It was shown that this method becomes exact in the limit of infinite
connectivity (each site couples to all other sites); it is expected that
results obtained in this manner are qualitatively correct for the original
model with a finite number of neighboring sites.
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
We briefly present several theoretical approaches to the quantum impurity
problems which led to the most significant conceptual break-throughs.
Good understanding of the conventional Kondo problem was established in the
period from 1970s to 1990s using a variety of non-perturbative techniques.
P. W. Anderson has led the way by establishing the renormalization group
approach to this problem in early 1970s. He had demonstrated the scaling
properties of the Kondo system, which paved the way to Wilson's numerical renormalization group (NRG) technique which provided
accurate numerical results for the thermodynamic properties (magnetic
susceptibility, heat capacity). NRG was later extended to calculations of
dynamic properties (spectral functions, resistivity, thermopower, thermal
conductivity, magnetoresistance, etc.) by T. A. Costi, A. C. Hewson and V.
Zlatic in 1980s. The greatest achievement of the NRG method is to show how
the impurity, that is weakly interacting at high temperatures, has a
profound effect on the conduction band as the temperature is lowered.
Surprisingly, the low temperature strong coupling fixed point is simple: it
corresponds to a simple Fermi liquid with modified boundary conditions (a
phase shift), as predicted by P. Nozieres in 1960s. The NRG technique is the
only approach which can demonstrate how the system crosses over from the
high temperature weakly perturbed Fermi liquid to a low temperature Fermi
liquid. It is, in fact, the intermediate temperature cross-over region which
is the most intriguing and non-trivial.
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
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.
Bosonization and refermionization
One-dimensional quantum field theories (and the Kondo model can be cast in
this form) have very special properties due to reduced dimensionality. In
particular, massless fermionic fields can be bosonized. By this we mean that
fermionic fields can be expressed as exponential functions of bosonic
fields. In this picture, charge and spin degrees of freedom of a physical
electron can be described by different bosonic fields. It was shown that a
Kondo-type magnetic impurity couples exclusively with the spin degrees of
freedom, while the charge degrees of freedom are decoupled. For a particular
choice of parameters, the model can then be transformed back into a
fermionic model (resonant level interacting with a continuum) which can be
exactly solved. Apart from providing interesting analytical results,
bosonization and spin-charge separation have provided an important
conceptual tool to think about fermionic field theories.
Boundary conformal field theory
Conformal field theory (CFT) also takes advantage of special mathematical
properties of one-dimensional field theories. It was shown that such fields
are, in some cases, conformally invariant in the two-dimensional plane of
space and time. Conformal transformation conserve angles: they include
scaling, rotations and translations. Such a high symmetry greatly simplifies
problems of this class, which can often be reduced to an exercise in group
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,
Localized magnetic states in metals
, Phys. Rev., 124 41 (1961).
P. W. Anderson,
A poor man's derivation of scaling laws for the Kondo problem
, J. Phys. C: Solid St. Phys., 3 2436 (1970).
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