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 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

$H=\sum _{\sigma }\epsilon _{f}f_{\sigma }^{\dagger }f_{\sigma }+\sum _{\langle j,j^{\prime }\rangle \sigma }t_{jj'}c_{j\sigma }^{\dagger }c_{j'\sigma }+\sum _{j,\sigma }(V_{j}f_{\sigma }^{\dagger }c_{j\sigma }+V_{j}^{*}c_{j\sigma }^{\dagger }f_{\sigma })+Uf_{\uparrow }^{\dagger }f_{\uparrow }f_{\downarrow }^{\dagger }f_{\downarrow }$ ,

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 $t_{jj'}$ 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.

For heavy-fermion systems, a lattice of impurities is described by the periodic Anderson model:

$H=\sum _{j\sigma }\epsilon _{f}f_{j\sigma }^{\dagger }f_{j\sigma }+\sum _{\langle j,j^{\prime }\rangle \sigma }t_{jj'}c_{j\sigma }^{\dagger }c_{j'\sigma }+\sum _{j,\sigma }(V_{j}f_{j\sigma }^{\dagger }c_{j\sigma }+V_{j}^{*}c_{j\sigma }^{\dagger }f_{j\sigma })+U\sum _{j}f_{j\uparrow }^{\dagger }f_{j\uparrow }f_{j\downarrow }^{\dagger }f_{j\downarrow }$

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

$H = \sum_{i\sigma}\epsilon_f f^{\dagger}_{i\sigma}f_{i\sigma} + \sum_{<j, j'>\sigma}t_{ijj'} c^{\dagger}_{ij\sigma}c_{ij'\sigma} + \sum_{ij,\sigma}(V_j f^{\dagger}_{i\sigma}c_{ij\sigma} + V_j^* c^{\dagger}_{ij\sigma}f_{i\sigma}) + \sum_{i\sigma,i'\sigma '} \frac{U}{2}n_{i\sigma}n_{i'\sigma '}$

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

Footnotes

References

Anderson, P. W. (1961). "Localized Magnetic States in Metals". Phys. Rev. 124 (1): 41–53. Bibcode:1961PhRv..124...41A. doi:10.1103/PhysRev.124.41.
Hewson, A. C. (1993). The Kondo Problem to Heavy Fermions. New York: Cambridge University Press.

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Anderson impurity model

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References