Second-Order Perturbation in electron gas

Question

I was trying to figure out the solution to exercise 1.4 on Quantum theory of many particles systems by Fetter & Walecka and I read through this question and its answer. But a point made in both the answer and the question confused me a lot.

We have this $$⟨m|H_1|Φ_0⟩ = ⟨m| a^{\dagger}_{\vec{k'}+\vec{q'},λ'_1} a^{\dagger}_{\vec{p'}-\vec{q'},λ'_2} a_{\vec{p'},λ'_2} a_{\vec{k'},λ'_1} |Φ_0⟩$$ expression and it is said that $|\vec{k'}+\vec{q'}|$ and $|\vec{p'}-\vec{q'}|$ should be above the Fermi level otherwise $⟨m|H_1|Φ_0⟩$ will be zero. My confusion is why we need $|\vec{k'}+\vec{q'}|$ AND $|\vec{p'}-\vec{q'}|$ to be above fermi level. I think, if we have either one of them above the Fermi level and the other below, we would still get an excited state with 1 electron excited above fermi level so I feel like it should be $|\vec{k'}+\vec{q'}|$ OR $|\vec{p'}-\vec{q'}|$ above the Fermi level.

I am sorry if this is not correct question formatting but this thing is really stuck in my mind and I really need a conclusion.

Answer 1

The ground state $|\Phi_0\rangle$ is defined as a state filled up to the Fermi level, that is $$ a_\mathbf{k}|\Phi_0\rangle=0 \text{ if } \epsilon_\mathbf{k} > \epsilon_F,\\ a_\mathbf{k}^\dagger|\Phi_0\rangle=0 \text{ if } \epsilon_\mathbf{k} < \epsilon_F. $$

Thus, the matrix element $\langle m|a^\dagger_{\mathbf{k}+\mathbf{q}}a^\dagger_{\mathbf{p}-\mathbf{q}}a_\mathbf{p}a_\mathbf{k}|\Phi_0\rangle$ is non-zero, only if

• the creation operators create electrons above the Fermi surface
  OR
• when either (or both) of them fills the hole created by one of the annihilation operators on the right

The process where the created holes are filled will be the essential one when you get to Wick's theorem, and calculating matrix elements $\langle \Phi_0|a^\dagger_{\mathbf{k}+\mathbf{q}}a^\dagger_{\mathbf{p}-\mathbf{q}}a_\mathbf{p}a_\mathbf{k}|\Phi_0\rangle$ (the same ground state in the bra and in the ket).

However, in the calculation you are dealing with now, there is a phase space factor: there is huge amount of states above the Fermi level, and only few possibilities to fill the two holes. Thus, these processes could be safely neglected... in fact, they disappear exactly in thermodynamic limit, where one takes $V\rightarrow +\infty$ (check in the book, that there is a volume factor in front): integration over the states above the Fermi level cancels the volume, but the contribution with filling holes vanishes as $\sim1/V$.