# 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.

• Side note: these are not three Hubbard models, but three approximations of the Hubbard model – Adam Apr 13 '16 at 19:41
• You may wish to look at Figure 6.1 of "Condensed matter field theory" by Altland and Simons. It talks about the three decoupling channels by Hubbard-Stratanovich transformation. – leongz Apr 14 '16 at 1:48
• thank you @leongz I will take at look at it, though from first glance the treatment there goes beyond my current knowledge of field theory (haven't got into Feynman path integrals yet). Do you think I may find the answer to my question there? Also, is there any way to answer my question without involving such advanced treatment? – Sr Incerteza Apr 14 '16 at 18:39

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 $n_{\uparrow} n_{\downarrow}\to n_{\uparrow} \langle n_{\downarrow}\rangle+n_{\downarrow} \langle n_{\uparrow}\rangle-\langle n_{\uparrow}\rangle \langle n_{\downarrow}\rangle$.
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).
• I'm concentrating only on the parts that involve operators, that's why I left out the constant term. Still, I don't see how the third version describes a moment-moment interaction. As I said, I know it yields the same result as in the first, hence my question of why they both describe different things. Lastly, by modifying the electron operators by hole operators I don't get the same expression as in the second case ( a term $1/2 n_{i\sigma}$ is missing). – Sr Incerteza Apr 13 '16 at 18:55
• Version (iii), without mean field is simply $U (n^2 - m^2)$, where the $m$-part would describe an interaction of a moment with itself, which, also to me, is a bit of an odd interpretation. Usually, in the Hubbard model, you get an interaction of the moment on site $i$ with the moment on a different site $j$ via the hopping term, $t_{ij}$. You then do perturbation theory in the hopping $t$, to end up with a moment-moment interaction with a coupling constant $J=4 t^2/U$. – Stesh Apr 14 '16 at 6:54