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The basic tight-binding hamiltonian consists of terms of the form $$\hat{c}^{\dagger}_{i\uparrow} \hat{c}^\phantom{\dagger}_{i+1\uparrow}+\text{h.c.}$$ (where $\text{h.c.}$ denotes the adjoint of the preceding term). A crucial feature is that it involves only interactions of electrons of the same spin.

Therefore the question: does it make any sense to study a generalized hamiltonian with terms of the form $$\hat{c}^{\dagger}_{i\uparrow} \hat{c}^\phantom{\dagger}_{i+1\downarrow}+\text{h.c.}$$ or similar? The Hubbard hamiltonian used to model highly correlated systems already includes terms of the form $\hat{n}_{i\uparrow}\hat{n}_{j\downarrow},$ so those other terms don't look like too crazy an idea, right? Is there any conceptual reason not to consider them ever?

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    $\begingroup$ The kind of spin-fliping hopping is a type of spin-orbit coupling term. The effect of spin-orbit coupling terms has been extensively studied in condensed matter theory. $\endgroup$ – Everett You Mar 8 '17 at 23:48
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Terms like $\hat{c}^{\dagger}_{i\uparrow} \hat{c}_{i+1\downarrow}+c.c$ are perfectly possible in a Hamiltonian. They respresent the probability of an electron hopping to the next site while flipping it's spin. Just as a term $\hat{c}^{\dagger}_{i\uparrow} \hat{c}_{i\downarrow}+c.c$ is possible and would represent on site spin flipping.

These transitions do not preserve magnetic moment so you would need some external source like a varying magnetic field for these transitions to occur.

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