Discontinuity of Fermi liquid occupancy

Question

In Fermi liquid theory, the electron spectral function is often represented by

$$A(k,\omega) = Z\delta(\omega-\epsilon_k)\ + \text{incoherent background}$$ where $Z$ is the weight in the quasiparticle peak. Consequently, the zero-temperature occupancy $$n(k)=\int_{-\infty}^0 d\omega A(k,\omega)$$ has a discontinuity of $Z$ at the Fermi level.

However, since the spectral function is actually more accurately described by a Lorentzian with a non-zero width (except right at the Fermi level), is the occupancy, in fact, continuous, and the purported discontinuity only approximate?

This post imported from StackExchange Physics at 2017-02-18 11:09 (UTC), posted by SE-user leongz

Comments

The discontinuity is exact in the thermodynamic limit, which is usually presupposed.

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Seconding Arnold, it means it depends on the number of electrons in question.