Hubbard Model

Matrix for two sites and one electron:
$$ \begin{matrix}
0 &-\gamma
\\ -\gamma &0
\\ \end{matrix} $$

Matrix for two sites and two electrons:
$$ \begin{matrix}
U, &\gamma &-\gamma &0 &0 &0
\\ \gamma &0 &0 &\gamma &0 &0
\\ -\gamma &0 &0 &-\gamma &0 &0
\\0 &\gamma &-\gamma &U &0 &0
\\0 &0 &0 &0 &0 &0
\\0 &0 &0 &0 &0 &0 
\\ \end{matrix} $$

Things to think about:
What is the basis in each case?
What is the lowest eigenvalue in each case and what is its eigenstate?
What is the expectation value of its kinetic energy? (The lowest eigenstate)
What are the other eigenvectors and eigenvalues?
3 electrons and 2 sites: what would be an appropriate matrix (and basis) for that?

what about something like? (what should the signs of the \(\gamma\)s be? What is the basis?

$$ \begin{matrix}
U, &\gamma &0 &0
\\ \gamma &U &0 &0
\\0 &0 &U &\gamma
\\0 &0 &\gamma &U
\\ \end{matrix} $$