Anderson impurity model
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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
where the operator corresponds to the annihilation operator of an impurity, and corresponds to a conduction electron annihilation operator, and labels the spin. The on–site 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, although they are separated by a distance greater than the Hill limit.
For heavy-fermion systems, a lattice of impurities is described by 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 and label the orbital degree of freedom (which can take one of two values), and represents a number operator.