Anderson impurity model – Wikipedia

Hamiltonian used in quantum physics

The Anderson impurity model, named after Philip Warren Anderson, is a Hamiltonian that is used to describe magnetic impurities embedded in metals.^[1]
It is often applied to the description of Kondo effect-type problems,^[2] such as heavy fermion systems^[3] and Kondo insulators^{[citation needed]}. In its simplest form, the model contains a term describing the kinetic energy of the conduction electrons, a two-level term with an on-site Coulomb repulsion that models the impurity energy levels, and a hybridization term that couples conduction and impurity orbitals. For a single impurity, the Hamiltonian takes the form^[1]

${displaystyle H=sum _{k,sigma }epsilon _{k}c_{ksigma }^{dagger }c_{ksigma }+sum _{sigma }epsilon _{sigma }d_{sigma }^{dagger }d_{sigma }+Ud_{uparrow }^{dagger }d_{uparrow }d_{downarrow }^{dagger }d_{downarrow }+sum _{k,sigma }V_{k}(d_{sigma }^{dagger }c_{ksigma }+c_{ksigma }^{dagger }d_{sigma })}$

,

where the

${displaystyle c}$

operator is the annihilation operator of a conduction electron, and

${displaystyle d}$

is the annihilation operator for the impurity,

${displaystyle k}$

is the conduction electron wavevector, and

${displaystyle sigma }$

labels the spin. The on–site Coulomb repulsion is

${displaystyle U}$

, and

${displaystyle V}$

gives the hybridization.

Regimes[edit]

The model yields several regimes that depend on the relationship of the impurity energy levels to the Fermi level

${displaystyle E_{rm {F}}}$

:

In the local moment regime, the magnetic moment is present at the impurity site. However, for low enough temperature, the moment is Kondo screened to give non-magnetic many-body singlet state.^[2][3]

Heavy-fermion systems[edit]

For heavy-fermion systems, a lattice of impurities is described by the periodic Anderson model.^[3] The one-dimensional model is

${displaystyle H=sum _{k,sigma }epsilon _{k}c_{ksigma }^{dagger }c_{ksigma }+sum _{j,sigma }epsilon _{f}f_{jsigma }^{dagger }f_{jsigma }+Usum _{j}f_{juparrow }^{dagger }f_{juparrow }f_{jdownarrow }^{dagger }f_{jdownarrow }+sum _{j,k,sigma }V_{jk}(e^{ikx_{j}}f_{jsigma }^{dagger }c_{ksigma }+e^{-ikx_{j}}c_{ksigma }^{dagger }f_{jsigma })}$

,

where

${displaystyle x_{j}}$

is the position of impurity site

${displaystyle j}$

, and

${displaystyle f}$

is the impurity creation operator (used instead of

${displaystyle d}$

by convention for heavy-fermion systems). The hybridization term allows f-orbital electrons in heavy fermion systems to interact, although they are separated by a distance greater than the Hill limit.

Other variants[edit]

There are other variants of the Anderson model, such as the SU(4) Anderson model^{[citation needed]}, which is used to describe impurities which have an orbital, as well as a spin, degree of freedom. This is relevant in carbon nanotube quantum dot systems. The SU(4) Anderson model Hamiltonian is

${displaystyle H=sum _{k,sigma }epsilon _{k}c_{ksigma }^{dagger }c_{ksigma }+sum _{i,sigma }epsilon _{d}d_{isigma }^{dagger }d_{isigma }+sum _{i,sigma ,i’sigma ‘}{frac {U}{2}}n_{isigma }n_{i’sigma ‘}+sum _{i,k,sigma }V_{k}(d_{isigma }^{dagger }c_{ksigma }+c_{ksigma }^{dagger }d_{isigma })}$

,

where

${displaystyle i}$

and

${displaystyle i’}$

label the orbital degree of freedom (which can take one of two values), and

${displaystyle n}$

represents the number operator for the impurity.

See also[edit]

References[edit]