Anderson impurity model

Heavy-fermion systems

For heavy-fermion systems, a lattice of impurities is described by the periodic Anderson model. The one-dimensional model is

${\displaystyle H=\sum _{k,\sigma }{ϵ}_k{c}_{k\sigma }^{†}{c}_{k\sigma }+\sum _{j,\sigma }{ϵ}_f{f}_{j\sigma }^{†}{f}_{j\sigma }+U\sum _j{f}_{j\uparrow }^{†}{f}_{j\uparrow }{f}_{j\downarrow }^{†}{f}_{j\downarrow }+\sum _{j,k,\sigma }{V}_{jk}(e^{ik{x}_j}{f}_{j\sigma }^{†}{c}_{k\sigma }+e^{-ik{x}_j}{c}_{k\sigma }^{†}{f}_{j\sigma })}$,

where ${\displaystyle x_j}$ is the position of impurity site ${\displaystyle j}$, and ${\displaystyle f}$ is the impurity creation operator (used instead of ${\displaystyle d}$ 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

${\displaystyle H=\sum _{k,\sigma }{ϵ}_k{c}_{k\sigma }^{†}{c}_{k\sigma }+\sum _{i,\sigma }{ϵ}_d{d}_{i\sigma }^{†}{d}_{i\sigma }+\sum _{i,\sigma ,i^{\prime }{\sigma }^{\prime }}\frac{U}{2}{n}_{i\sigma }{n}_{i^{\prime }{\sigma }^{\prime }}+\sum _{i,k,\sigma }{V}_k(d_{i\sigma }^{†}{c}_{k\sigma }+c_{k\sigma }^{†}{d}_{i\sigma })}$,

where ${\displaystyle i}$ and ${\displaystyle i^{\prime }}$ label the orbital degree of freedom (which can take one of two values), and ${\displaystyle n}$ represents the number operator for the impurity.