Schwinger boson representation

Contracting two-mode oscillator bilinears with the Pauli matrices,gives $[T^a,T^b]=i{\epsilon }_{abc}{T}^c$. The subspace with total occupation $N$ is the spin-$N/2$ irreducible representation and has dimension $N+1$.

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.

For the two-dimensional isotropic harmonic oscillator, $H=\hslash \omega (N+1)$ and the Schwinger boson representation gives $J^2={\hslash }^{2}N(N+2)/4$. Substitution yields the displayed relation, including its quantum constant term. The fixed-total-occupation sector is a spin-$j=N/2$ representation, so energy is $\hslash \omega (2j+1)$ with degeneracy $2j+1$.

Effective exchange of internal states through virtual boson hopping into energetically costly occupation sectors. For two bosons initially one per site and charging energy $U{\Delta }^2/4$, the second-order operator is $-(t^2/U)(3+4{\mathbf{s}}_a\cdot {\mathbf{s}}_b)$ for dimensionless spins. The triplet is lowered and the singlet is unchanged in this example; signs for other energy functions require their actual denominators.