# Fermionic static structure factor

1. Oct 1, 2015

### Catria

1. The problem statement, all variables and given/known data

Calculate the static structure factor for noninteracting fermions

$S(\vec{q})=\frac{1}{N}\langle \phi_0|\hat{n}_{\vec{q}}\hat{n}_{-\vec{q}} | \phi_0\rangle$

where $\hat{n}_\vec{q}=\sum_{\vec{k},\sigma} a^\dagger_{\vec{k}\sigma}a_{\vec{k}+\vec{q}\sigma}$ is the particle density operator in the momentum representation and $|\phi_0\rangle$ is the ground state. Take the continuum limit $\sum_{\vec{k},\sigma} \to 2V\int \frac{d^3 k}{(2\pi)^3}$ and calculate $S(\vec{q})$ explicitly.

Hint: Consider the $\vec{q}=0$ and $\vec{q}\neq 0$ separately.

2. Relevant equations

The spin-independent one-particle operator: $\langle\vec{q}|O|\vec{k}\rangle =\frac{1}{V}\int d\vec{r} e^{-i(\vec{q}-\vec{k})\cdot\vec{r}} O(\vec{r})$

3. The attempt at a solution

The case $\vec{q}=0$:

In that case, the operator $\hat{n}_\vec{q}\hat{n}_{-\vec{q}}=\sum_{\vec{k},\sigma} a^\dagger_{\vec{k}\sigma}a_{\vec{k}+\vec{q}\sigma}a^\dagger_{\vec{k}\sigma}a_{\vec{k}-{\vec{q}}\sigma} =\sum_{\vec{k},\sigma} a^\dagger_{\vec{k}\sigma}a_{\vec{k}\sigma}a^\dagger_{\vec{k}\sigma}a_{\vec{k}\sigma}$

$S(0)=\frac{2V}{N}\int \frac{d^3 k}{(2\pi)^3} n^2_{\vec{k}\sigma} = 1$.

The physical process, in the $\vec{q}\neq 0$ case, is as follows: there is a particle at location $\vec{k}$ in momentum space within the Fermi sphere of radius $k_F$ that gets annihilated, and is created back with momentum $\vec{k}-\vec{q}$, which lies outside of the Fermi sphere, due to Pauli's exclusion principle. Said particle is annihilated and created back at the hole it first left behind.

Due to this constraint, if there was some spherical-coordinate integral to evaluate in the $\vec{q}\neq 0$ case, the radial integration limits would probably be $k_F-|\vec{q}|,k_F$. But this is as far as I have taken the integral:

$S(\vec{q})=2\int \frac{d^3 k}{(2\pi)^3} e^{-i\vec{k}\cdot\vec{q}} a^\dagger_{\vec{k}\sigma}a_{\vec{k}+\vec{q}\sigma}a^\dagger_{\vec{k}\sigma}a_{\vec{k}-{\vec{q}}\sigma}$

I'd like to think I am getting stuck at a rather advanced stage...

2. Oct 6, 2015