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I'm looking at the tight binding Hamiltonian which describes non interacting fermions on a bravis lattice, and has the form (in real space):

$H=-t\sum_{<ij>,\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+c^{\dagger}_{j\sigma}c_{i\sigma})$

where c and $c^{\dagger}$ are the fermionic annihilation and creation operators, t is the hopping parameter, the sum is over the nearest neighbors $<ij>$ and spin $\sigma=\downarrow ,\uparrow$.

Now, if I look for example at the first term, it tells me that I destroy an electron on the site j with spin $\sigma$, and I create it at site i again with spin $\sigma$.

My question is: would it be possible in principle to consider other terms in the hamiltonian with different spin indices, e.g: $-t\sum_{<ij>,\sigma, \sigma'}c^{\dagger}_{i\sigma}c_{j\sigma'}$? So in other words can I consider in the model the possibility that the electron changes spin while hopping from one site to another? Or is this forbidden for some reason?

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Sure, it is possible. It is certainly mathematically allowed. For the model to be physical, we'd want there to be an underlying process by which an electron traveling through the system changes spin. This can occur due to spin-orbit coupling, which is indeed often implemented in tight-binding models as such spin-non-conserving hopping. For condensed matter applications, you'll typically want to ensure the Hamiltonian is Hermitian, but recent interest in non-Hermitian Hamiltonians highlights that even this condition isn't always necessary.

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