Total spin (${\mathbf{S}}_{\mathrm{tot}}={\sum }_n{\mathbf{S}}_n$)

The total spin is the vector sum of subsystem spin operators and generates their simultaneous rotation. For a rotationally invariant Heisenberg model, both its squared magnitude and any chosen component commute with the Hamiltonian. An anisotropic exchange or field can preserve only a component.