Kondo model
Encyclopedia
The Kondo model is a model for a quantum impurity coupled to a large reservoir of noninteracting electron
s. The quantum impurity is represented by a spin-1/2, which is coupled to a continuous band of noninteracting electrons by an antiferromagnetic exchange coupling, J. The Kondo model is used as a model for metals containing magnetic impurities, as well as quantum dot
systems.
where
is a spin-1/2 operator representing the impurity, and 
is the local spin-density of the noninteracting band at the impurity site (
are the Pauli matrices). J < 0, i.e. the exchange coupling is antiferromagnetic.
Jun Kondo applied third-order perturbation theory
to the Kondo model and showed that the resistivity
of the model diverges logarithmically as the temperature goes to zero. This explained why metal samples containing magnetic impurities have a resistance minimum (see Kondo effect
). The problem of finding a solution to the Kondo model which did not contain this unphysical divergence became known as the Kondo problem.
A number of methods were used to attempt to solve the Kondo problem. Phillip Anderson devised a perturbative renormalization group method, known as Poor Man's Scaling, which involves perturbatively eliminating excitations to the edges of the noninteracting band. This method indicated that, as temperature is decreased, the effective coupling between the spin and the band,
, increases without limit. As this method is perturbative in J, it becomes invalid when J becomes large, so this method did not truly solve the Kondo problem, although it did hint at the forward.
The Kondo problem was finally solve when Kenneth Wilson applied the Numerical renormalization group
to the Kondo model and showed that the resistivity goes to a constant as temperature goes to zero.
There are many variants of the Kondo model. For instance, the spin-1/2 can be replaced by a spin-1 or even a greater spin. The two-channel Kondo model is a variant of the Kondo model which has the spin-1/2 coupled to two independent noninteracting bands. One can also consider the ferromagnetic Kondo model (i.e. the standard Kondo model with J > 0).
The Kondo model is intimately related to the Anderson impurity model, as can be shown by Schrieffer-Wolff transformation.
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s. The quantum impurity is represented by a spin-1/2, which is coupled to a continuous band of noninteracting electrons by an antiferromagnetic exchange coupling, J. The Kondo model is used as a model for metals containing magnetic impurities, as well as quantum dot
Quantum dot
A quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei...
systems.
where
is a spin-1/2 operator representing the impurity, and is the local spin-density of the noninteracting band at the impurity site ( are the Pauli matrices). J < 0, i.e. the exchange coupling is antiferromagnetic.Jun Kondo applied third-order perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
to the Kondo model and showed that the resistivity
Resistivity
Electrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...
of the model diverges logarithmically as the temperature goes to zero. This explained why metal samples containing magnetic impurities have a resistance minimum (see Kondo effect
Kondo effect
In physics, the Kondo effect describes the scattering of conduction electrons in a metal due to magnetic impurities. It is a measure of how electrical resistivity changes with temperature....
). The problem of finding a solution to the Kondo model which did not contain this unphysical divergence became known as the Kondo problem.
A number of methods were used to attempt to solve the Kondo problem. Phillip Anderson devised a perturbative renormalization group method, known as Poor Man's Scaling, which involves perturbatively eliminating excitations to the edges of the noninteracting band. This method indicated that, as temperature is decreased, the effective coupling between the spin and the band,
, increases without limit. As this method is perturbative in J, it becomes invalid when J becomes large, so this method did not truly solve the Kondo problem, although it did hint at the forward.The Kondo problem was finally solve when Kenneth Wilson applied the Numerical renormalization group
Numerical renormalization group
The Numerical Renormalization Group is a technique devised by Kenneth Wilson to solve certain many-body problems where quantum impurity physics plays a key role. It is an inherently non-perturbative procedure, which was originally used to solve the Kondo model...
to the Kondo model and showed that the resistivity goes to a constant as temperature goes to zero.
There are many variants of the Kondo model. For instance, the spin-1/2 can be replaced by a spin-1 or even a greater spin. The two-channel Kondo model is a variant of the Kondo model which has the spin-1/2 coupled to two independent noninteracting bands. One can also consider the ferromagnetic Kondo model (i.e. the standard Kondo model with J > 0).
The Kondo model is intimately related to the Anderson impurity model, as can be shown by Schrieffer-Wolff transformation.