Anderson impurity model

The Anderson impurity model is a Hamiltonian that is used to describe magnetic impurities embedded in metallic hosts. It is often applied to the description of Kondo-type of problems, such as heavy fermion systems and Kondo insulators. 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

$H=\sum _{\sigma }\epsilon _{f}f_{\sigma }^{\dagger }f_{\sigma }+\sum _{<j,j'>\sigma }t_{jj'}c_{j\sigma }^{\dagger }c_{j'\sigma }+\sum _{j,\sigma }(V_{j}f_{\sigma }^{\dagger }c_{j\sigma }+V_{j}^{*}c_{j\sigma }^{\dagger }f_{\sigma })+Uf_{\uparrow }^{\dagger }f_{\uparrow }f_{\downarrow }^{\dagger }f_{\downarrow }$

where the $f$ operator corresponds to the annihilation operator of an impurity, and $c$ corresponds to a conduction electron annihilation operator, and $\sigma$ labels the spin. The onsite Coulomb repulsion is $U$, which is usually the dominant energy scale, and $t_{jj'}$ is the hopping strength from site $j$ to site $j'$. A significant feature of this model is the hybridization term $V$, which allows the $f$ electrons in heavy fermion systems to become mobile, despite the fact they are separated by a distance greater than the Hill limit.

In heavy-fermion systems, we find we have a lattice of impurities. The relevant model is then the periodic Anderson model.

$H=\sum _{j\sigma }\epsilon _{f}f_{j\sigma }^{\dagger }f_{j\sigma }+\sum _{<j,j'>\sigma }t_{jj'}c_{j\sigma }^{\dagger }c_{j'\sigma }+\sum _{j,\sigma }(V_{j}f_{\sigma }^{\dagger }c_{j\sigma }+V_{j}^{*}c_{\sigma }^{\dagger }f_{j\sigma })+U\sum _{j}f_{j\uparrow }^{\dagger }f_{j\uparrow }f_{j\downarrow }^{\dagger }f_{j\downarrow }$

There are other variants of the Anderson model, for instance the SU(4) Anderson model, 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

$H=\sum _{i\sigma }\epsilon _{f}f_{i\sigma }^{\dagger }f_{i\sigma }+\sum _{<j,j'>\sigma }t_{ijj'}c_{ij\sigma }^{\dagger }c_{ij'\sigma }+\sum _{ij,\sigma }(V_{j}f_{i\sigma }^{\dagger }c_{ij\sigma }+V_{j}^{*}c_{ij\sigma }^{\dagger }f_{i\sigma })+\sum _{i\sigma ,i'\sigma '}{\frac {U}{2}}n_{i\sigma }n_{i'\sigma '}$

where i and i' label the orbital degree of freedom (which can take one of two values), and n represents a number operator.

See also

• Kondo effect
• Kondo model

References

• Anderson, P. W. (1961). "Localized Magnetic States in Metals". Phys. Rev. 124: 41. doi:10.1103/PhysRev.124.41.
• Hewson, A. C. (1993). The Kondo Problem to Heavy Fermions. New York: Cambridge University Press.