Long-range order

Long-range order means that an appropriate order-sensitive correlation function remains nonzero as the separation becomes arbitrarily large in the thermodynamic limit. In a ferromagnet, a spin-spin correlation can approach a positive constant, indicating a persistent relation between distant spins. An antiferromagnet requires an alternating sign convention, captured by staggered magnetization, to reveal its corresponding order.

The choice of average matters. A finite zero-field equilibrium sample can preserve symmetry and have zero mean magnetization, while distant spin correlations still diagnose the approach to an ordered thermodynamic phase. Within a selected symmetry-broken state, the disconnected contribution can carry the order and the connected correlation can instead decay. One must therefore state the observable, state and order of limits before applying a nonzero-correlation criterion.

This also explains a locality constraint: a circuit with bounded depth and bounded-range gates cannot generate arbitrary-distance connected correlations from a product state. Its finite propagation range limits which separated regions can share correlations.

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.