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 prototypical 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, T_K, 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 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).

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 convergence.

Theoretical approaches

We briefly present several theoretical approaches to the quantum impurity problems which led to the most significant conceptual break-throughs.

Numerical renormalization group

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 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.

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 theory.

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.

Bibliography

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).
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