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An antisymmetric exchange interaction between spins, proportional to ${\mathbf{D}}_{ij}\cdot ({\mathbf{S}}_i\times {\mathbf{S}}_j)$ with a bond-dependent vector ${\mathbf{D}}_{ij}$. The overall sign is conventional and must be specified. It favors a handed twist rather than simple parallel or antiparallel alignment. For a uniform chain term $-D({\mathbf{S}}_j\times {\mathbf{S}}_{j+1}{)}_z$, common spin rotations about $z$ and time reversal remain symmetries, although reversing the bond orientation changes the sign of $D$.

The antisymmetric exchange statistics of identical fermions. Interchanging two one-particle labels reverses the total state, and fermionic creation operators anticommute. Occupation of a one-particle mode is therefore zero or one, giving the Pauli exclusion principle; thermal equilibrium leads to the Fermi-Dirac distribution. Color, flavor, spin and orbital factors must together have the required antisymmetry, even if one factor is symmetric.

A fermionic atom is an atom whose total quantum state has half-integer angular momentum and obeys Fermi-Dirac statistics as a composite particle. Counting the parity of its fermionic constituents distinguishes fermionic and bosonic isotopes when the atom remains intact.

For example, neutral lithium-6 is fermionic, while lithium-7 is bosonic. Identical fermionic atoms in the same internal state have antisymmetric exchange symmetry, constraining their allowed low-energy collision states. The label concerns the complete atom's exchange behavior, not whether each of its constituent particles is a fermion.