t–J model

Example

Let $\Lambda$ be a finite graph with edge set $E$, with fermion operators $c_{j\sigma}$ and $n_j = n_{j\uparrow} + n_{j\downarrow}$. Let $P$ be the projection onto states with no doubly occupied site. Let $\mathbf{S}_j = \tfrac12 \sum_{\alpha, \beta} c_{j\alpha}^\dagger \boldsymbol{\sigma}_{\alpha\beta} c_{j\beta}$ be the spin at site $j$, where $\boldsymbol{\sigma}$ is the vector of Pauli matrices. For $t > 0$ and $J > 0$, the t–J model acts on the range of $P$ with Hamiltonian $$H = -t \sum_{\set{i, j} \in E} \sum_{\sigma} P\big(c_{i\sigma}^\dagger c_{j\sigma} + c_{j\sigma}^\dagger c_{i\sigma}\big)P + J \sum_{\set{i, j} \in E} \Big(\mathbf{S}_i \cdot \mathbf{S}_j - \tfrac14 n_i n_j\Big).$$ Check on two sites with two electrons: the hopping term has no allowed move, so $H = J(\mathbf{S}_1 \cdot \mathbf{S}_2 - \tfrac14)$. This gives $-J$ for the singlet and $0$ for the triplet. This agrees with the Hubbard model on two sites for large $U$, where the singlet lies at about $-4t^2/U$, when $J = 4t^2/U$.