Our lecture notes described the action of the particle creation operator on a fermionic Fock state:

$$c_l^\dagger |n_1 n_2...\rangle = (-1)^{\sum_{j=1}^{l-1}n_j}|n_1 n_2 ... n_l+1 ...\rangle.$$

I am confused by why the $(-1)^{\sum_{j=1}^{l-1}n_j}$ factor is needed? Is it to maintain anti-symmetry of the wavefunction somehow?


1 Answer 1


Essentially yes. Recall the state for $ n_m=n_n=... =1$ with all intermediate occupation numbers vanishing, $$ |n_1 n_2...\rangle = c_m^\dagger c_n^\dagger ...|0\rangle.$$

If $\ell$ is one of the above occupied sites, $n_\ell=1$, nilpotence of the creation operators when $c_\ell^\dagger$ is applied on the state will net you 0, so you hardly care about about the phase multiplying it.

If not, then $n_\ell+1=1$, and, to get to that site, $c_\ell^\dagger$ has anticommuted with $c_ m^\dagger, c_n^\dagger, ...$ preceding it, to get to the $ \ell $ site, so it has picked up $\sum_{j=1}^{\ell-1}n_j$ minus signs, multiplying each other, $$c_\ell^\dagger |n_1 n_2...\rangle = (-1)^{\sum_{j=1}^{\ell-1}n_j}|n_1 n_2 ... n_\ell+1 ...\rangle.$$


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