Relationship between Imaginary Time Green's function and Average Occupancy

[a2009]

Hello everyone,

In Fermi Liquid Theory, I'm trying to understand what the relationship is between the Green's function and the average occupancy of levels. In my lecture they gave the relation

$$\left\langle n_k \right\rangle = G(k,\tau\rightarrow 0^+)$$

Anyone know where this comes from?

Thanks!

 

[Bill_K]

This is almost by definition. The Green's function is G(x, x') = -i <T(ψ(x)ψ⁺(x'))>. The particle density is n(x) = -i <ψ(x)ψ⁺(x)>. If you let x→x' and t→t' in G, you get n!

 

[a2009]

Thanks for the reply. Actually I was referring to the average occupation. In the non-interacting case

$$n=(e^{\beta (\epsilon -\mu)}+1)^{-1}$$

In the interacting case it is not so simple. But this relationship was stated. I think it has to do with the spectral function.

Thanks again for any help.

 

[Bill_K]

Just rewrite what I said in momentum space.