Hamiltonian used in quantum physics
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
formula_1,
where the formula_2 operator is the annihilation operator of a conduction electron, and formula_3 is the annihilation operator for the impurity, formula_4 is the conduction electron wavevector, and formula_5 labels the spin. The on–site Coulomb repulsion is formula_6, and formula_7 gives the hybridization.
Regimes.
The model yields several regimes that depend on the relationship of the impurity energy levels to the Fermi level formula_8:
In the local moment regime, the magnetic moment is present at the impurity site. However, for low enough temperature, the moment is Kondo screened to give non-magnetic many-body singlet state.
Heavy-fermion systems.
For heavy-fermion systems, a lattice of impurities is described by the periodic Anderson model. The one-dimensional model is
formula_14,
where formula_15 is the position of impurity site formula_16, and formula_17 is the impurity creation operator (used instead of formula_3 by convention for heavy-fermion systems). The hybridization term allows "f"-orbital electrons in heavy fermion systems to interact, although they are separated by a distance greater than the Hill limit.
Other variants.
There are other variants of the Anderson model, such as 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
formula_19,
where formula_20 and "formula_21" label the orbital degree of freedom (which can take one of two values), and "formula_22" represents the number operator for the impurity.
References.
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