Hubbard model

Example

Let $\Lambda$ be a finite graph with edge set $E$, and let $c_{j\sigma}, c_{j\sigma}^\dagger$ be fermion operators for site $j \in \Lambda$ and spin $\sigma \in \set{\uparrow, \downarrow}$, with $n_{j\sigma} = c_{j\sigma}^\dagger c_{j\sigma}$. For a hopping amplitude $t > 0$ and an on-site repulsion $U \ge 0$, the Hubbard model is the lattice model with Hamiltonian $$H = -t \sum_{\set{i, j} \in E} \sum_{\sigma} \big(c_{i\sigma}^\dagger c_{j\sigma} + c_{j\sigma}^\dagger c_{i\sigma}\big) + U \sum_{j \in \Lambda} n_{j\uparrow} n_{j\downarrow}.$$ The first term is the tight-binding model kinetic energy, and the second penalizes two electrons on the same site. Check on two sites with two electrons and total spin $0$. In the basis of the covalent singlet and the symmetric ionic state, $H$ is the matrix $\begin{pmatrix} 0 & -2t \\ -2t & U \end{pmatrix}$. So the ground-state energy is $$E_0 = \frac{U}{2} - \sqrt{\frac{U^2}{4} + 4t^2}.$$ The three triplet states cannot hop, since hopping would put two equal spins on one site, so they have energy $0$.