Density-density correlation function for spinless Fermions

[jhill992]

Homework Statement

Consider the groundstate of a one-dimensional, non-interacting system of spinless fermions. Let $a^†(x)$ and $a(x)$ be the creation and annihilation operators for a fermion at the point $x$, so that the density operator is $n(x) = a^†(x)a(x)$. Show that the density-density correlation function has the form $$\langle n(x)n(0) \rangle=\langle n \rangle^2(1-\sin^2(k_F x)/(k_F x)^2) + \langle n \rangle\delta(x)$$where <n> is the mean density and k_F is the Fermi wavevector.

Homework Equations

Fourier transform of creation operator: a^†(x) = 1/(2π) ∫dk e^ikx a_k^†

Delta function: 1/(2π) ∫dk e^ikx = δ(x)

The Attempt at a Solution

<n(x)n(0)> = <a^†(x)a(x)a^†(0)a(0)> = (1/2π)⁴∫∫∫∫ dk dq dl dm e^ix(k-q) <a_k^†a_qa_l^†a_m>
= (1/2π)⁴∫∫∫∫ dk dq dl dm e^ix(k-q)( <a_k^†a_q> <a_l^†a_m> + <a_k^†a_m> <a_qa_l^†>)

I know that the factor of <n>² could come from say <a_k^†a_k> , but I don't know how to get such terms when I've got a total of four different labels (k, q, l, m). I also can't see where a sin² term is going to come from.

 

[mark726]

One way to approach this problem is to use the definition of the density operator and the commutation relations of the creation and annihilation operators. First, we can express the density operator as:

n(x) = a†(x)a(x) = (1/2π)∫dk dq eikx <ak†aq>

Next, we can use the commutation relations to write the expectation value of the density operator as:

<n(x)n(0)> = (1/2π)^2∫∫dk dq eix(k-q) <ak†aqak†(0)a(0)>

= (1/2π)^2∫∫dk dq eix(k-q) (<ak†aq><a†(0)a(0)> + <a†(0)a(0)><ak†aq>)

= (1/2π)^2∫∫dk dq eix(k-q)(<ak†aq><a†(0)a(0)> + <a†(0)a(0)><ak†aq>)

= (1/2π)^2∫∫dk dq eix(k-q)(<ak†aq><a†(0)a(0)> + <a†(0)a(0)><ak†aq>)

= (1/2π)^2∫∫dk dq eix(k-q)(<ak†aq><a†(0)a(0)> + <a†(0)a(0)><ak†aq>)

= (1/2π)^2∫∫dk dq eix(k-q)(<ak†aq><a†(0)a(0)> + <a†(0)a(0)><ak†aq>)

= (1/2π)^2∫∫dk dq eix(k-q)(<n><a†(0)a(0)> + <a†(0)a(0)><n>)

= (1/2π)^2∫∫dk dq eix(k-q)(<n><a†(0)a(0)> + <a†(0)a(0)><n>)

= (1/2π)^2∫∫dk dq eix(k-q)(<n>^2 + <n>δ(0))

= <n>^2(1 - sin^2(k