XXZ model

Example

Let $L \ge 2$ and let $\mathcal{H} = (\C^2)^{\otimes L}$, with spin operators $S^a_j = \tfrac12 \sigma^a_j$ for $a = x, y, z$ and periodic boundary conditions $S^a_{L+1} = S^a_1$. For $J > 0$ and an anisotropy parameter $\Delta \in \R$, the XXZ chain has Hamiltonian $$H = J \sum_{j=1}^{L} \big( S^x_j S^x_{j+1} + S^y_j S^y_{j+1} + \Delta\, S^z_j S^z_{j+1} \big).$$ It is the anisotropic quantum Heisenberg model with couplings $J_x = J_y = J$ and $J_z = J\Delta$, and $\Delta = 1$ gives the isotropic antiferromagnet. Check: $H$ commutes with the total magnetization $S^z = \sum_j S^z_j$. Indeed each term $S^x_j S^x_{j+1} + S^y_j S^y_{j+1} = \tfrac12(S^+_j S^-_{j+1} + S^-_j S^+_{j+1})$ moves one unit of $S^z$ between neighbours. For instance the fully polarized state $\lvert\uparrow\cdots\uparrow\rangle$ is an eigenstate with energy $\tfrac14 J\Delta L$.