diff --git a/docs/source/mf_details.md b/docs/source/mf_details.md deleted file mode 100644 index 7c5579b37687d642212a9f6970d8c0a6e6c8d025..0000000000000000000000000000000000000000 --- a/docs/source/mf_details.md +++ /dev/null @@ -1,69 +0,0 @@ -# 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: - -$$ -c_i^\dagger c_j^\dagger c_l c_k \approx \\ -\langle c_i^\dagger c_k \rangle c_j^\dagger c_l - \langle c_i^\dagger c_l \rangle c_j^\dagger c_k - \langle c_j^\dagger c_k \rangle c_i^\dagger c_l + \langle c_j^\dagger c_l \rangle c_i^\dagger c_k + \langle c_i^\dagger c_j^\dagger \rangle c_l c_k + \langle c_l c_k \rangle c_i^\dagger c_j^\dagger -\\ -- \langle c_i^\dagger c_l \rangle \langle c_j^\dagger c_k \rangle + \langle c_i^\dagger c_k \rangle \langle c_j^\dagger c_l \rangle + \langle c_i^\dagger c_j^\dagger \rangle \langle c_l c_k \rangle. -$$ diff --git a/docs/source/mf_notes.md b/docs/source/mf_notes.md index a8f55bffbcdd33f814cef0d3822c3e53430def8a..39409caab64e7f97c9b4ab31a1f1b093e9434a94 100644 --- a/docs/source/mf_notes.md +++ b/docs/source/mf_notes.md @@ -36,12 +36,6 @@ The ground-state density matrix reads: ::: where $\beta = 1/ (k_B T)$ is the inverse temperature, $\mu$ is the chemical potential, and $\hat{N} = \sum_i c_i^\dagger c_i$ is the number operator. -:::{admonition} Derivation of the mean-field Hamiltonian with Wicks theorem -:class: dropdown info -```{include} mf_details.md -``` -::: - ### Finite tight-binding grid To simplify the mean-field Hamiltonian, we assume a finite, normalised orthogonal tight-binding grid defined by the single-particle basis states: