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. It is often applied to the description of Kondo effect-type 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 The Anderson impurity model, named after Philip Warren Anderson, is a Hamiltonian that is used to describe magnetic impurities embedded in metals. It is often applied to the description of Kondo effect-type 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 ↓ {displaystyle H=sum _{sigma }epsilon _{f}f_{sigma }^{dagger }f_{sigma }+sum _{langle j,j^{prime } angle sigma }t_{jj'}c_{jsigma }^{dagger }c_{j'sigma }+sum _{j,sigma }(V_{j}f_{sigma }^{dagger }c_{jsigma }+V_{j}^{*}c_{jsigma }^{dagger }f_{sigma })+Uf_{uparrow }^{dagger }f_{uparrow }f_{downarrow }^{dagger }f_{downarrow }} , where the f {displaystyle f} operator corresponds to the annihilation operator of an impurity, and c {displaystyle c} corresponds to a conduction electron annihilation operator, and σ {displaystyle sigma } labels the spin. The on–site Coulomb repulsion is U {displaystyle U} , which is usually the dominant energy scale, and t j j ′ {displaystyle t_{jj'}} is the hopping strength from site j {displaystyle j} to site j ′ {displaystyle j'} . A significant feature of this model is the hybridization term V {displaystyle V} , which allows the f {displaystyle f} electrons in heavy fermion systems to become mobile, although they are separated by a distance greater than the Hill limit.