Quantum Heisenberg model

Example

Let $\Lambda$ be a finite graph with edge set $E$, and give each vertex $j$ a spin $S$. Its spin operators $S^x_j, S^y_j, S^z_j$ act on the factor $\C^{2S+1}$ of the tensor product $\mathcal{H} = \bigotimes_{j \in \Lambda} \C^{2S+1}$. The quantum Heisenberg model is the lattice model with the self-adjoint Hamiltonian $$H = J \sum_{\set{i, j} \in E} \mathbf{S}_i \cdot \mathbf{S}_j, \qquad \mathbf{S}_i \cdot \mathbf{S}_j = S^x_i S^x_j + S^y_i S^y_j + S^z_i S^z_j,$$ with $\hbar = 1$. For $J > 0$ it is antiferromagnetic and for $J < 0$ ferromagnetic. Check on two spins $\tfrac12$: with $\mathbf{S} = \mathbf{S}_1 + \mathbf{S}_2$, one has $\mathbf{S}_1 \cdot \mathbf{S}_2 = \tfrac12\big(\mathbf{S}^2 - \tfrac32\big)$. The singlet ($\mathbf{S}^2 = 0$) has energy $-\tfrac34 J$ and the three triplet states ($\mathbf{S}^2 = 2$) have energy $\tfrac14 J$.