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 = ∑ σ ϵ f f σ † f σ + ∑ ⟨ j , j ′ ⟩ σ t j j ′ c j σ † c j ′ σ + ∑ j , σ ( V j f σ † c j σ + V j ∗ c j σ † f σ ) + U f ↑ † f ↑ f ↓ † f ↓ ,

where the f operator corresponds to the annihilation operator of an impurity, and c corresponds to a conduction electron annihilation operator, and σ labels the spin. The onsite Coulomb repulsion is U , which is usually the dominant energy scale, and t j j ′ 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, although 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 = ∑ j σ ϵ f f j σ † f j σ + ∑ ⟨ j , j ′ ⟩ σ t j j ′ c j σ † c j ′ σ + ∑ j , σ ( V j f j σ † c j σ + V j ∗ c j σ † f j σ ) + U ∑ j f j ↑ † f j ↑ f j ↓ † f j ↓

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 = ∑ i σ ϵ f f i σ † f i σ + ∑ < j , j ′ > σ t i j j ′ c i j σ † c i j ′ σ + ∑ i j , σ ( V j f i σ † c i j σ + V j ∗ c i j σ † f i σ ) + ∑ i σ , i ′ σ ′ U 2 n i σ n i ′ σ ′

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