= Transverse spin order in a number eigenstate
{title2=$\langle S_i^x\rangle=0,\quad\langle S_i^xS_j^x\rangle=n_An_B/(2L^2)\quad(i\ne j)$}
For a <two-species uniform boson number state>, set $S_i^+=a_i^\dagger b_i$. A single <spin ladder operator> changes the species <occupation numbers>, so its <expectation value> vanishes. At distinct sites, $\langle S_i^+S_j^-\rangle=n_An_B/L^2$, 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>.
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