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I was wondering what are the differences in application between the spinful and spinless Fermi-Hubbard model.

Hamiltonian for the spinless case:

$\hat H = -t \displaystyle\sum_{\langle i,j\rangle} (c^{\dagger}_ic_{j} + h.c.) +V\sum_{\langle i,j\rangle} n_in_j$

Hamiltonian for the spinful case:

$H=-t \displaystyle\sum_{\langle i,j\rangle \sigma}\left(c_{i \sigma}^{\dagger} c_{j \sigma}+\text { h.c. }\right)+\sum_{i} U n_{i \uparrow} n_{i \downarrow}, \quad n_i = c^{\dagger}_i c_i$

I understand that physically speaking the spinful case introduces more degrees of freedom into the system and therefore increases the Hilbert space where the quantum states are living.

But I am wondering how this applies in the real world and what are the different applications of the spinful case and the spinless case. Why are we interested in these different models?

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