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.
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