# 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

$H = \sum_{\sigma}\epsilon_f f^{\dagger}_{\sigma}f_{\sigma} + \sum_{\sigma}t_{jj'} c^{\dagger}_{j\sigma}c_{j'\sigma} + \sum_{j,\sigma}(V_j f^{\dagger}_{\sigma}c_{j\sigma} + V_j^* c^{\dagger}_{j\sigma}f_{\sigma}) + Uf^{\dagger}_{\uparrow}f_{\uparrow}f^{\dagger}_{\downarrow}f_{\downarrow}$

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.

$H = \sum_{j\sigma}\epsilon_f f^{\dagger}_{j\sigma}f_{j\sigma} + \sum_{\sigma}t_{jj'}c^{\dagger}_{j\sigma}c_{j'\sigma} + \sum_{j,\sigma}(V_j f^{\dagger}_{\sigma}c_{j\sigma} + V_j^* c^{\dagger}_{\sigma}f_{j\sigma}) + U\sum_{j}f^{\dagger}_{j\uparrow}f_{j\uparrow}f^{\dagger}_{j\downarrow}f_{j\downarrow}$

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

$H = \sum_{i\sigma}\epsilon_f f^{\dagger}_{i\sigma}f_{i\sigma} + \sum_{\sigma}t_{ijj'} c^{\dagger}_{ij\sigma}c_{ij'\sigma} + \sum_{ij,\sigma}(V_j f^{\dagger}_{i\sigma}c_{ij\sigma} + V_j^* c^{\dagger}_{ij\sigma}f_{i\sigma}) + \sum_{i\sigma,i'\sigma '} \frac{U}{2}n_{i\sigma}n_{i'\sigma '}$

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