Density-density correlation function for spinless Fermions

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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 kF is the Fermi wavevector.

Homework Equations


Fourier transform of creation operator: a(x) = 1/(2π) ∫dk eikx ak

Delta function: 1/(2π) ∫dk eikx = 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 eix(k-q) <akaqalam>
= (1/2π)4∫∫∫∫ dk dq dl dm eix(k-q)( <akaq> <alam> + <akam> <aqal>)

I know that the factor of <n>2 could come from say <akak> , 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.
 

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