Anderson impurity model

The model yields several regimes that depend on the relationship of the impurity energy levels to the Fermi level ${\displaystyle E_{\rm {F}}}$:

• The empty orbital regime for ${\displaystyle \epsilon _{d}\gg E_{\rm {F}}}$ or ${\displaystyle \epsilon _{d}+U\gg E_{\rm {F}}}$, which has no local moment.
• The intermediate regime for ${\displaystyle \epsilon _{d}\approx E_{\rm {F}}}$ or ${\displaystyle \epsilon _{d}+U\approx E_{\rm {F}}}$.
• The local moment regime for ${\displaystyle \epsilon _{d}\ll E_{\rm {F}}\ll \epsilon _{d}+U}$, which yields a magnetic moment at the impurity.

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]

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

${\displaystyle H=\sum _{k,\sigma }\epsilon _{k}c_{k\sigma }^{\dagger }c_{k\sigma }+\sum _{j,\sigma }\epsilon _{f}f_{j\sigma }^{\dagger }f_{j\sigma }+U\sum _{j}f_{j\uparrow }^{\dagger }f_{j\uparrow }f_{j\downarrow }^{\dagger }f_{j\downarrow }+\sum _{j,k,\sigma }V_{jk}(e^{ikx_{j}}f_{j\sigma }^{\dagger }c_{k\sigma }+e^{-ikx_{j}}c_{k\sigma }^{\dagger }f_{j\sigma })}$,

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.

There are other variants of the Anderson model, such as 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

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

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.

• Hubbard model
• Kondo effect
• Kondo model
• Anderson localization