Chirality operators for Heisenberg spin systems.

The ground state of closed Heisenberg spin chains with an odd number of sites has a chiral degeneracy, in addition to a twofold Kramers degeneracy. A nonzero chirality implies that the spins are not coplanar, and is a measure of handedness. The chirality operator, which can be treated as a spin-1/2 operator, is explicitly constructed in terms of the spin operators, and is given as commutator of permutation operators.