Hubbard model within mean-field: three different approaches While reading doi:10.1016/j.carbon.2012.03.009 , the authors mention three types of Hubbard models within mean-field approximation. The first one describes the electron-electron interaction, and to my understanding is the standard way of writting up the model and reads (only interaction term):
$H_{int}=U\sum_{i,\sigma} n_{i\sigma}\langle{n_{i-\sigma}}\rangle$
The second version seems to describe electron-hole interaction and reads:
$H_{int}=U\sum_{i,\sigma}(\langle{n_{i-\sigma}}\rangle-\frac{1}{2})n_{i\sigma}$
And a third one seems to describe moment-moment interaction and reads: 
$H_{int}=\frac{U}{2}\sum_{i} n_{i}\langle{n_i}\rangle-U\sum_i 2m_i\langle{m_i}\rangle$
where $n_i=n_{i\uparrow}+n_{i\downarrow}$ and $m_i=\frac{1}{2}(n_{i\uparrow}-n_{i\downarrow})$. 
My question is how can I see that the two last 'versions', describe what they are supposed to. To my understanding, all of them are the same version except for a shift on the Fermi level for case 2 which shifts the half-filling below $E=0$, as opposed to $E=U/2$. Case 1 and 3 are the same. 
 A: To me, the Hubbard interaction per site is defined as $H_{int}=U n_{\uparrow} n_{\downarrow}$. (I suppressed the $i$, and also there is a factor of two because of your spin sum.)
Then, the mean-field approximation is defined as $n_{\uparrow} n_{\downarrow}\to n_{\uparrow} \langle n_{\downarrow}\rangle+n_{\downarrow} \langle n_{\uparrow}\rangle-\langle n_{\uparrow}\rangle \langle n_{\downarrow}\rangle$.
So your version (i) lacks the constant term.
Per site, you have two operators, $n_{\uparrow}$ and $n_{\downarrow}$, i.e. the occupancies for each species of electrons, which, of course, you can trade against total charge, $n=n_{\uparrow}+n_{\downarrow}$ and moment $m=n_{\uparrow}-n_{\downarrow}$ (beware, again a factor of two with your definition). This makes you end up with version (iii).
For version (ii), you need a so-called electron-hole transformation, i.e. for one spin species, say $\downarrow$, you replace the electron destruction operator by a hole creation operator $c_{\downarrow}\to a^{+}_{\downarrow}$, and vice versa. (The $a$-operators satisfy the same fermionic algebra as the original $c$-operators.) So $n_{\downarrow}=c^{+}_{\downarrow}c_{\downarrow}\to a_{\downarrow}a^{+}_{\downarrow} =1-a^{+}_{\downarrow}a_{\downarrow}$. This last guy, $a^{+}_{\downarrow}a_{\downarrow}$, you call it $n_{\downarrow}$ again, but remember, it counts holes now.  And you end up with the electron-hole interaction in (ii).
