Anderson impurity model

Example

Let $c_{k\sigma}$ be fermion operators for the conduction electrons of a metal, labelled by a momentum $k$ and a spin $\sigma \in \set{\uparrow, \downarrow}$, with energies $\varepsilon_k$. Let $d_\sigma$ be fermion operators for one localized orbital of an impurity atom, with $n_{d\sigma} = d_\sigma^\dagger d_\sigma$. For an orbital energy $\varepsilon_d \in \R$, an on-site repulsion $U \ge 0$ and hybridization amplitudes $V_k$, the Anderson impurity model (Anderson, 1961) has Hamiltonian $$H = \sum_{k, \sigma} \varepsilon_k c_{k\sigma}^\dagger c_{k\sigma} + \sum_\sigma \varepsilon_d n_{d\sigma} + U n_{d\uparrow} n_{d\downarrow} + \sum_{k, \sigma}\big(V_k d_\sigma^\dagger c_{k\sigma} + \overline{V_k}\, c_{k\sigma}^\dagger d_\sigma\big).$$ Check without hybridization, $V_k = 0$: the impurity has the four states empty, $\uparrow$, $\downarrow$ and doubly occupied, with energies $0$, $\varepsilon_d$, $\varepsilon_d$ and $2\varepsilon_d + U$. When $\varepsilon_d < 0 < \varepsilon_d + U$, the singly occupied states are lowest, so the impurity carries a local magnetic moment.