Crystalline solids have a long-range order (where symmetry is broken) but liquids have only a short-range order (where no symmetry is broken). Ferromagnets have a long-range magnetic order while a paramagnet lacks it. The converse also seems to be true, for example, in the Kosterlitz-Thouless transition there is no symmetry breaking and there is no long-range order (but quasi long-range). By long-range order, I understand that below some critical temperature $T_c$ the two-point correlation function of the order parameter (density) becomes a constant (independent of position).
Is this a generic feature? In other words, does long-range order necessarily imply the symmetry-breaking? And does the symmetry-breaking necessarily imply the long-range order?
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