Local spin constraint in the Schwinger boson representation ($(a_n^{†}{a}_n+b_n^{†}{b}_n)|\psi ⟩=2s|\psi ⟩$)

The Schwinger boson representation satisfies $S_n^2=(N_n/2)(N_n/2+1)$, where $N_n$ is the local total boson number operator. Thus a prescribed local spin $s$ requires the displayed operator constraint at every site. A condition on $⟨N_n⟩$ alone is insufficient: for $⟨N_n⟩=2s$, $⟨S_n^2⟩=s(s+1)+\mathrm{Var}(N_n)/4$. Projecting onto the required eigenspace at each site removes the unwanted local irreducible spin representations.