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 a 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
where the operator corresponds to the annihilation operator of an impurity, and corresponds to a conduction electron annihilation operator, and labels the spin. The onsite Coulomb repulsion is , which is usually the dominant energy scale, and is the hopping strength from site to site . A significant feature of this model is the hybridization term , which allows the 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.
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
where i and i' label the orbital degree of freedom (which can take one of two values), and n represents a number operator.