Documentation

Construct the documentation for the release of the package.

docs/source/mf_details.md

+# Details
+
+## Ground state definition
+Before we proceed, we must find a way to simply represent the ground state of the system.
+Assume there exists a non-interacting system $\hat{H}_\text{mf}$ whose groundstate closely resembles the interacting groundstate $| 0 \rangle$:
+
+$$
+| 0 \rangle \approx \Pi_{E_i \leq \mu } b_i^\dagger |\textrm{vac}\rangle,
+$$
+
+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:
+
+$$
+c_i^\dagger c_j^\dagger c_l c_k = :c_i^\dagger c_j^\dagger c_l c_k: \\
++ \overline{c_i^\dagger c_k} :c_j^\dagger c_l: - \overline{c_i^\dagger c_l} :c_j^\dagger c_k: - \overline{c_j^\dagger c_k} :c_i^\dagger c_l: + \overline{c_j^\dagger c_l} :c_i^\dagger c_k: + \overline{c_i^\dagger c_j^\dagger} :c_l c_k: + \overline{c_l c_k} :c_i^\dagger c_j^\dagger: \\
+- \overline{c_i^\dagger c_l} \overline{c_j^\dagger c_k} +\overline{c_i^\dagger c_k} \overline{c_j^\dagger c_l}
++\overline{c_i^\dagger c_j^\dagger} \overline{c_l c_k}.
+$$
+
+## Mean-field approximation to Wick terms
+
+We are now able to apply the mean-field approximation to the Wick terms.
+Lets first apply this to the first term in the expansion:
+
+$$
+:c_i^\dagger c_j^\dagger c_l c_k: = \langle :c_i^\dagger c_j^\dagger c_l c_k: \rangle + \delta:c_i^\dagger c_j^\dagger c_l c_k: \approx 0,
+$$
+
+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:
+
+$$
+\overline{c_i^\dagger c_k} :c_j^\dagger c_l: \approx \\
+\langle \overline{c_i^\dagger c_k} \rangle :c_j^\dagger c_l: + \overline{c_i^\dagger c_k} \langle :c_j^\dagger c_l: \rangle - \langle  \overline{c_i^\dagger c_k} \rangle \langle :c_j^\dagger c_l: \rangle = \\
+\langle c_i^\dagger c_k \rangle :c_j^\dagger c_l:.
+$$
+
+Repeating this for all terms in the expansion and collecting we find:
+
+$$
+c_i^\dagger c_j^\dagger c_l c_k \approx \\
+\langle c_i^\dagger c_k \rangle c_j^\dagger c_l - \langle c_i^\dagger c_l \rangle c_j^\dagger c_k - \langle c_j^\dagger c_k \rangle c_i^\dagger c_l + \langle c_j^\dagger c_l \rangle c_i^\dagger c_k + \langle c_i^\dagger c_j^\dagger \rangle c_l c_k +  \langle c_l c_k \rangle c_i^\dagger c_j^\dagger
+\\
+- \langle c_i^\dagger c_l \rangle \langle c_j^\dagger c_k \rangle + \langle c_i^\dagger c_k \rangle  \langle c_j^\dagger c_l \rangle  + \langle c_i^\dagger c_j^\dagger \rangle \langle  c_l c_k \rangle.
+$$