Is first-order perturbation always equal to Hartree-Fock approximation?

Question

Let us have a system of homogenous electronic system. The Coulomb interaction is given as $$ H_{int} = \frac{1}{2\mathcal{V}} \sum_{k_1 k_2 q}\sum_{\sigma_1 \sigma_2} V(q) c_{k_1+q,\sigma_1}^\dagger c_{k_2-q,\sigma_2}^\dagger c_{k_2,\sigma_2} c_{k_1,\sigma_1} $$

If we assume spin symmetry i.e. $\langle c_{k\uparrow}^\dagger c_{k\uparrow}\rangle = \langle c_{k\downarrow}^\dagger c_{k\downarrow}\rangle$ then we can prove that Hartree-Fock is equal to first order perturbation theory.

My question is if Hartree-Fock approximation of any system always equal to first order perturbation approximation?

Answer 1

My question is if Hartree-Fock approximation of any system always equal to first order perturbation approximation?

Generally speaking, no, since the perturbation theory is applicable to many situations other than Coulomb gas. Hartree-Fock is also a rather general term - e.g., one can "dress" the electron lines by summing the infinite series of diagrams and it will still be Hartree-Fock. Finally, it also depends in the ground state wave function - e.g., it will not be manifestly Hartree-Fock in BCS regime.

For the detailed perturbative treatment of the electron gas, and comparison with the diagrammatic derivation I recommend Fetter&Walecka's book.