Anderson model

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The Anderson Model is a Hamiltonian model that is often used to describe Heavy Fermion systems. 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

$H = \sum_{\sigma}\epsilon_f f^{\dagger}_{\sigma}f_{\sigma} + \sum_{<j, j'>\sigma}t_{jj'} c^{\dagger}_{j\sigma}c_{j'\sigma} + \sum_{j,\sigma}(V_j f^{\dagger}_{\sigma}c_{j\sigma} + V_j^* c^{\dagger}_{j\sigma}f_{\sigma}) + Uf^{\dagger}_{\uparrow}f_{\uparrow}f^{\dagger}_{\downarrow}f_{\downarrow}$

where the $f$ operator corresponds to the annihilation operator of an impurity, and $c$ corresponds to a conduction electron annihilation operator, and $σ$ labels the spin. The onsite Coulomb repulsion is $U$ , which is usually the dominant energy scale, and $t j j'$ 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, 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.

$H = \sum_{j\sigma}\epsilon_f f^{\dagger}_{j\sigma}f_{j\sigma} + \sum_{<j, j'>\sigma}t_{jj'}c^{\dagger}_{j\sigma}c_{j'\sigma} + \sum_{j,\sigma}(V_j f^{\dagger}_{\sigma}c_{j\sigma} + V_j^* c^{\dagger}_{\sigma}f_{j\sigma}) + U\sum_{j}f^{\dagger}_{j\uparrow}f_{j\uparrow}f^{\dagger}_{j\downarrow}f_{j\downarrow}$