Anderson impurity model

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]

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

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.