Notes on Fermion Parity

QuantumHamiltonianParticle uses Wigner-Jordan (WJ) transformatoin to represent fermion creation and annihilation operators math``` c{i}^{\dagger} = \sigma^{-} \prod{j < i} \sigma^{z}_{i}

with the WJ string between site i and site 1.

The basis states are also defined using the σ⁻.
For example, the basis state |00110⟩ in a 5-site fermion system, with sites numbered 1 to 4 from the right, is defined as
math```
\vert 00110 \rangle \equiv \sigma_{2}^{-} \sigma_{3}^{-} \vert 00000 \rangle

Since the σ's on different sites commute with each other, the order of the σ⁻'s in the product does not matter. This defines a natural convention for the states in terms of the fermion creation operators. Since the WJ string runs toward site 1, applying creation operators to the vacuum state |00...0⟩ starting with larger site indices gives basis states consistent with the basis states defined in terms of the σ⁻'s. The basis state in the above examples can be written as math\vert 00110 \rangle \equiv c_{2}^{\dagger} c_{3}^{\dagger} \vert 00000 \rangle.