where $|\textrm{vac}\rangle$ is the vacuum state, $b_i^\dagger$ is the creation operator of eigenstate $i$ with energy $E_i$ of $\hat{H}_\text{mf}$ and $\mu$ is the Fermi level.
We relate the $b_i^\dagger$ particles to our original basis via a unitary transformation:
$$
c_i^\dagger = \sum_{k} U_{ik} b_k^\dagger.
$$
## Normal ordering and contractions
Before proceeding, we define the *normal ordering* operation, $:ABC...:$, as a sorting of operators such that all the creation operators $b_i^\dagger$ below the Fermi level are to the left of the annihilation $b_i$ operators below the Fermi levels, and vice versa for the operators above the Fermi level.
Whenever the normal ordered operators acts on the ground state, it gives zero:
$$
:ABC...: | 0 \rangle = 0.
$$
Lastly, we define the *contraction* of two operators $A$ and $B$ as:
$$
\overline{AB} = \hat{A}\hat{B} - :AB:.
$$
## Expansion of the interaction term
We utilize Wick's theorem to expand the interaction term:
where the first term in the second equality is zero due to the normal ordering operation and the second term is zero since we assume deviations from the mean-field ground state are small.
Next, we apply the mean-field approximation to the second term in the expansion: