Herbrand structure

The Anderson Impurity Model is a Hamiltonian model that is often used to describe heavy fermion systems and Kondo insulators. The model contains a narrow resonance between a magnetic impurity state and a conduction electron state. The model also contains an on-site repulsion term as found in the Hubbard model between localized electrons. For a single impurity, the Hamiltonian takes the form

$H=\sum _{\sigma }{ϵ}_f{f}_{\sigma }^{†}{f}_{\sigma }+\sum _{<j,j^{\prime }>\sigma }{t}_{j{j}^{\prime }}{c}_{j\sigma }^{†}{c}_{j^{\prime }\sigma }+\sum _{j,\sigma }(V_j{f}_{\sigma }^{†}{c}_{j\sigma }+V_j^{*}{c}_{j\sigma }^{†}{f}_{\sigma })+U{f}_{\uparrow }^{†}{f}_{\uparrow }{f}_{\downarrow }^{†}{f}_{\downarrow }$

where the $f$ operator corresponds to the annihilation operator of an impurity, and $c$ corresponds to a conduction electron annihilation operator, and $\sigma$ labels the spin. The onsite Coulomb repulsion is $U$, which is usually the dominant energy scale, and $t_{j{j}^{\prime }}$ is the hopping strength from site $j$ to site $j^{\prime }$. A significant feature of this model is the hybridization term $V$, which allows the $f$ 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.

$H=\sum _{j\sigma }{ϵ}_f{f}_{j\sigma }^{†}{f}_{j\sigma }+\sum _{<j,j^{\prime }>\sigma }{t}_{j{j}^{\prime }}{c}_{j\sigma }^{†}{c}_{j^{\prime }\sigma }+\sum _{j,\sigma }(V_j{f}_{\sigma }^{†}{c}_{j\sigma }+V_j^{*}{c}_{\sigma }^{†}{f}_{j\sigma })+U\sum _j{f}_{j\uparrow }^{†}{f}_{j\uparrow }{f}_{j\downarrow }^{†}{f}_{j\downarrow }$

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

$H=\sum _{i\sigma }{ϵ}_f{f}_{i\sigma }^{†}{f}_{i\sigma }+\sum _{<j,j^{\prime }>\sigma }{t}_{ij{j}^{\prime }}{c}_{ij\sigma }^{†}{c}_{i{j}^{\prime }\sigma }+\sum _{ij,\sigma }(V_j{f}_{i\sigma }^{†}{c}_{ij\sigma }+V_j^{*}{c}_{ij\sigma }^{†}{f}_{i\sigma })+\sum _{i\sigma ,i^{\prime }{\sigma }^{\prime }}\frac{U}{2}{n}_{i\sigma }{n}_{i^{\prime }{\sigma }^{\prime }}$

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

See also

• Kondo effect
• Kondo model

Bibliography

• P.W. Anderson, Phys. Rev. 124 (1961), p. 41 http://dx.doi.org/10.1103/PhysRev.124.41
• A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, New York, N.Y., 1993.