 Off-diagonal long-range order

For a homogeneous bosonic condensate, the one-body density matrix retains a nonzero long-distance component. More generally, macroscopic occupation is identified by an eigenvalue of that density matrix proportional to particle number. This gives a number-conserving description even when the one-point annihilation-field expectation vanishes. In a paired fermionic system the corresponding macroscopic order belongs to the two-particle density matrix, rather than an electron single-particle condensate.