Kugel-Khomskii coupling

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Kugel-Khomskii coupling describes a coupling between the spin and orbital degrees of freedom in a solid. The Hamiltonian used is:

H=\frac{t^2}{U}\sum_{<i,j>}\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]

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