LROs

LROs, or long-range orders, refer to phases in condensed matter systems in which correlations between microscopic degrees of freedom persist at arbitrarily large separations, corresponding to a nonzero order parameter in the thermodynamic limit. Formally, a correlation function tends to a nonzero constant as the separation grows, indicating spontaneous symmetry breaking of a discrete or continuous symmetry.

Common examples include crystalline solids, where the particle density exhibits translational long-range order; magnets, where spin correlations align to give a nonzero magnetization; and superconductors or superfluids, where the pair or single-particle density matrix maintains a finite off-diagonal value (off-diagonal long-range order, ODLRO).

Two related concepts are short-range order, where correlations decay to zero at large distances, and quasi-long-range order, where correlations decay algebraically rather than tending to a constant. Quasi-long-range order is typical in two-dimensional systems with continuous symmetry, such as the 2D XY model below the Berezinskii–Kosterlitz–Thouless transition.

Dimensionality and symmetry play crucial roles. The Mermin-Wagner theorem forbids LRO at finite temperature for systems with continuous symmetry in one or two dimensions with short-range interactions, though LRO can occur in two dimensions for systems with discrete symmetry. LRO can also persist at zero temperature in lower-dimensional systems or in the presence of sufficiently long-range interactions.

Experimentally, LRO manifests as Bragg peaks in diffraction for crystals, a nonzero average magnetization for magnets, or a nonzero superconducting order parameter. The presence of LRO is associated with spontaneous symmetry breaking and, for continuous symmetries, with Goldstone modes.