Kugel–Khomskii coupling

Kugel–Khomskii coupling describes a coupling between the spin and orbital degrees of freedom in a solid; it is named after the Russian physicists Kliment I. Kugel (Климент Ильич Кугель) and Daniel I. Khomskii (Daniil I. Khomskii, Даниил Ильич Хомский). The Hamiltonian used is:

${\displaystyle H={\frac {t^{2}}{U}}\sum _{\langle i,j\rangle }\left[4\left({\overrightarrow {S_{i}}}\cdot {\overrightarrow {S_{j}}}\right)(\tau _{i}^{\alpha }-{\frac {1}{2}})(\tau _{j}^{\alpha }-{\frac {1}{2}})+(\tau _{i}^{\alpha }+{\frac {1}{2}})(\tau _{j}^{\alpha }+{\frac {1}{2}})-1\right]}$