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4C_{2}(G)\exp \left( \frac{3}{2}\beta \left( E_{in}-E_{out}\right)
\right) \right]$$
Fig. 1 (Colour online) A sketch of the process: a) the energy gain (in $%
\Delta E\ $) per one hopping is given in a spin representation; b) the energy cost in the form of an external field is shown.
It is worth noting that $C_{2}(G)\sim N^{2}/N$ and we see that the Nagaoka theorem is still valid for the Hubbard model with on-site correlated hopping.
Up to now we consider as the underlying group $G$ the group $G=SU(N)$ where $N$ is a positive integer. These results can be extended to other groups, in particular to the group $SU(2)$, and to the case of itinerant fermions in the spinor representation of the group $SU(2)$. In all these cases, for $%
t_{1}=t_{2}$ the condition $E_{in}=E_{out}$ can be reached only by the ground state of the system. Then the Nagaoka theorem is certainly valid.
Conclusions
===========
In conclusion we have developed a general method to overcome the Nagaoka theorem in the case of interacting electrons hopping in a crystal of correlated atoms. The “energy” gained by one hopping is represented in a spin representation: it is always possible to find a quantum state of the system where the gain in energy is exactly $\Delta E$, then the Nagaoka theorem is not valid. Then we have considered the generalization of the Nagaoka theorem to the case of non-orthogonal representations of the group. The theorem holds for the orthogonal and of course
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