Kondo model

Example

Let $c_{k\sigma}$ be fermion operators for the conduction electrons of a metal, with energies $\varepsilon_k$. Let $\psi_\sigma = N^{-1/2}\sum_k c_{k\sigma}$ annihilate an electron at the impurity site, where $N$ is the number of momenta. Let $\mathbf{S}$ be the spin-$\tfrac12$ operator of a localized impurity. Let $\mathbf{s} = \tfrac12 \sum_{\alpha, \beta} \psi_\alpha^\dagger \boldsymbol{\sigma}_{\alpha\beta} \psi_\beta$ be the conduction-electron spin at the impurity site, with $\boldsymbol{\sigma}$ the Pauli matrices. For a coupling $J \in \R$, the Kondo model has Hamiltonian $$H = \sum_{k, \sigma} \varepsilon_k c_{k\sigma}^\dagger c_{k\sigma} + J\, \mathbf{S} \cdot \mathbf{s}.$$ The coupling is antiferromagnetic for $J > 0$. Check on the interaction alone, with a single conduction electron at the impurity site: $\mathbf{S} \cdot \mathbf{s} = \tfrac12\big((\mathbf{S} + \mathbf{s})^2 - \tfrac32\big)$. So the singlet has energy $-\tfrac34 J$ and the triplet $\tfrac14 J$. For $J > 0$ the conduction electron tends to screen the impurity spin.