Spin-spin interaction ($H_{ij}\propto {\mathbf{S}}_i\cdot {\mathbf{S}}_j$)

A spin-spin interaction couples two spin angular momenta. In an isotropic model it is proportional to their dot product, whose expectation in a total-pair-spin state follows from $2{\mathbf{S}}_i\cdot {\mathbf{S}}_j=S_{ij}^2-S_i^2-S_j^2$. Magnetic interactions generate examples in atoms and effective colour-magnetic interactions in hadron models.