Fermion Spin: FermionSpin
FermionSpin is a convenience sector for $SU(2)$ spin together with the matching fermion parity. It is implemented as SU2Irrep ⊠ FermionParity, with half-integer spin assigned odd parity.
Sector type
TensorKitSectors.FermionSpin — Type
const FermionSpin = SU2Irrep ⊠ FermionParity
FermionSpin(j::Real)Represents the fermion spin as the direct product of a $SU₂$ irrep j and a fermion parity, with the restriction that the fermion parity is odd if 2 * j is odd.
See also: SU2Irrep, FermionParity
Construct FermionSpin(j) from an integer or half-integer spin. The parity is odd exactly when 2j is odd:
\[j \mapsto (SU(2)\text{ spin }j,\; 2j \bmod 2).\]
The unit is FermionSpin(0), and all sectors are self-dual because the $SU(2)$ irreps and parity sectors are self-dual.
Fusion Rules
The $SU(2)$ component fuses by angular-momentum addition:
\[j_1 \otimes j_2 = \bigoplus_{j = |j_1-j_2|}^{j_1+j_2} j.\]
The parity component is fixed by the output spin. Quantum dimensions are inherited from SU2Irrep:
\[d_j = 2j + 1.\]
The parity component supplies fermionic signs, so exchanging two half-integer-spin sectors has the odd-parity sign in addition to the SU2Irrep representation-category data.
Because this is a product sector, most fusion and topological data is inherited componentwise from SU2Irrep and FermionParity.