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)\left(\tau _{i}^{\alpha }-{\frac {1}{2}}\right)\left(\tau _{j}^{\alpha }-{\frac {1}{2}}\right)+\left(\tau _{i}^{\alpha }+{\frac {1}{2}}\right)\left(\tau _{j}^{\alpha }+{\frac {1}{2}}\right)-1\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5368d23978a0e2b5527775470bbe69b070132628)
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