Fermion Number: FermionNumber
FermionNumber is a convenience sector for integer $U(1)$ charge together with the matching fermion parity. It is implemented as U1Irrep ⊠ FermionParity, with odd charge assigned odd parity.
Sector type
TensorKitSectors.FermionNumber — Type
const FermionNumber = U1Irrep ⊠ FermionParity
FermionNumber(a::Int)Represents the fermion number as the direct product of a $U₁$ irrep a and a fermion parity, with the restriction that the fermion parity is odd if and only if a is odd.
See also: U1Irrep, FermionParity
Construct FermionNumber(n) from an integer charge n. The underlying product label is
\[n \mapsto (U(1)\text{ charge } n,\; n \bmod 2).\]
The unit is FermionNumber(0), and duality negates the $U(1)$ charge while preserving the parity constraint.
Fusion Rules
Fusion adds charges:
\[n_1 \otimes n_2 = n_1 + n_2.\]
The fermion parity component is then fixed automatically by the resulting charge.
Topological data
All sectors have quantum dimension 1.
The $U(1)$ part has bosonic representation-category data, while the parity part supplies the fermionic exchange sign. Thus exchanging two odd-charge sectors contributes a minus sign.
Because this is a product sector, its fusion and topological data is inherited componentwise from U1Irrep and FermionParity.