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In a Fermi liquid the momentum distribution shows a jump at the Fermi surface, i.e. \begin{equation}\langle n_{k_F-\delta k} - n_{k_F+\delta k}\rangle = Z_{k_F}\end{equation} with $Z_k$ the strength of the pole in the Green's function $G\left(k,\omega\right) = \frac{Z_{k}}{\omega - \epsilon_k + i \delta \mathrm{sgn}\left(k-k_F\right)} + g\left(k,\omega\right)$ and where $g$ is supposed to be regular near $k_F$. By relating $\langle n_k \rangle$ to the Green's function and closing the contour in the complex plane this is straightforward to show - the integral over $g$ cancels and the only remaining contribution is $Z_k$ for $\langle n_{k_F-\delta k}\rangle$.

However I don't see how this is reflected in writing the spectral representation (I'm using Coleman's book) \begin{equation} G\left(k,\omega\right) = \sum_\lambda \frac{|M_{\lambda} \left(k\right)|^2}{w- \epsilon_{\lambda} + i \delta \mathrm{sgn} \left(\epsilon_\lambda\right)} \end{equation} which would contribute $\sum_{\lambda'} |M_{\lambda'} \left(k\right)|^2$ to $\langle n_k \rangle$, where $\lambda'$ is such that $\epsilon_{\lambda'} < 0$. This suggests

\begin{equation} \langle n_{k_F-\delta k} - n_{k_F+ \delta k} \rangle = \sum_{\lambda'} |M_{\lambda'}\left(k_F-\delta k\right)|^2 - |M_{\lambda'} \left(k_F+ \delta k\right)|^2. \end{equation}

I wonder if we can actually use the spectral representation, because it seems to suggest a series of sharp poles, but I would expect them to broaden into a continuum especially away from the Fermi surface.

So: can we actually use the spectral representation written this way? If so, how does the last equation simplify to $Z_{k_F}$?

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The spectral representation is very general, while the Fermi liquid at T=0 is just one specific ground state. If we want to learn anything from the spectral representation about the Fermi liquid we would have to make a sufficient number of physical assumptions (e.g., the ground state is not degenerate, has zero magnetization, is spatially homogeneous, is ungapped, etc.) and somehow combine them to learn about the properties of the matrix elements of the spectral representation.

In this case, the matrix elements will somehow form the discontinuity at the Fermi level as a pole and the smooth incoherent part as a continuum of poles in the thermodynamic limit. However, without the proper assumptions, we can not learn how that happens. (I am curious which assumptions are sufficient to obtain the Fermi liquid form of the Green's function from the spectral representation but I do not know it.)

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