Forward-scattering for a single impurity in an infinite system
I'm slightly confused with the following situation:
Suppose you have an electron in a tight-binding model, and let's say we are in one dimension with $N$ lattice sites.
Add to this a single impurity at a specific site $i$.
The pure system is translationally invariant and has as eigenfunctions the plain waves. As a consequence, the probability to find the (free) electron on the impurity site should scale as $1/N$ and I'd imagine that the full forward-scattering Green function for the problem has some form
$$G(k,\omega) = G_0(k,\omega) + \frac{1}{N} T(k,\omega) G_0(k,\omega)^2$$
where $T(k,\omega)$ would be some function independent of $N$.
As a consequence, in the limit of an infinite system $N \rightarrow \infty$ I would expect to not see the impurity effect at all as a consequence of that $1/N$ factor.
However, if I imagine that the impurity is a temperature effect (an example would be a flipped local spin in a ferromagnetic ground state, or an excited phonon) then the impurity could occur on any of the $N$ sites of the system. If I confine myself to very low temperatures where I assume that the entire system contains at most one impurity, the temperature averaged propagator would then look like
$$G_T(k,\omega) = 1/Z \times \left[ \left\langle \phi_0 | \hat G(k,\omega) |\phi_0 \right\rangle + \sum_i e^{-\beta E_I} \left\langle i | \hat G(k,\omega + E_I) | i \right\rangle \right]$$
where $|\phi_0\rangle$ is the free system's ground state and $| i \rangle$ is the state with an impurity at site $i$ and $E_I$ is the energy cost of the impurity.
$Z$ is the partition function and in our simple example just gives $1 + Ne^{-\beta E_I}$.
If that's all correct, then the finite temperature Green's function should just be $$G(T,k,\omega) = \frac{G_0(k,\omega) + Ne^{-\beta E_I} \left( G_0 + G_0^2 1/N T_{kk}(\omega)\right)}{1 + Ne^{-\beta E_I}}$$ which can be simplified to $$G_0(k,\omega) + \frac{e^{-\beta E_I} G_0^2 T_{kk}}{1 + Ne^{-\beta E_I}}$$
What confuses me here is that if we let $N$ go to infinity, the impurity term will again vanish at small temperature and thus seems to play no role at all in determining the Green function or the spectral function, but clearly that cannot be correct, since we know that impurities can, for example, lead to bound states.
Am I missing something really obvious here?
EDIT: With your answers so far, let me just add what I think I could do right now.
At low temperature, I'd assume a low density of thermally excited impurities, so I will do an expansion to first order in that density. The partition function is $$Z = (1 + e^{-\beta E_I})^N = \sum_{n}^N \begin{pmatrix}N\n\end{pmatrix} e^{-\beta n E_I}.$$
The numerator for $\langle G \rangle$ becomes approximately, only accounting for interactions with one impurity at a time, justified by the low density: $$\sum_n e^{-\beta n E_I} \begin{pmatrix}N \ n \end{pmatrix} \left[ G_0(\omega) + n T(\omega) G_0^2(\omega)\right]$$ Now the $G_0$ don't depend on anything we're summing over, so we can pull them out of the sum and obtain
$$\langle G \rangle = G_0 + \langle n \rangle T G_0^2$$ where I omit the momentum and frequency arguments because they're the same everywhere anyway. 680