 Transverse spin order in a number eigenstate

For a two-species uniform boson number state, set $S_i^{+}=a_i^{†}{b}_i$. A single spin ladder operator changes the species occupation numbers, so its expectation value vanishes. At distinct sites, $⟨S_i^{+}{S}_j^{-}⟩=n_A{n}_B/L^2$, proving the displayed transverse two-point correlation function. For $n_A/L,n_B/L$ tending to nonzero constants, this correlation function has a nonzero large-distance limit despite the global U(1) symmetry. This diagnoses long-range order in the auxiliary state; its interpretation as a state of fixed physical spins still requires the Schwinger boson constraint.