Does a pole in the self-energy imply a violation of the Luttinger theorem?

Luttinger theorem relates Fermi surface volume to electron density per spin $n$. It holds in an arbitrary Fermi liquid but is widely believed to break when a fermionic self-energy has a pole. We show that this is not necessarily the case. As an example, we consider a fermionic system consisting of two gapped bands, originating from a self-energy $\mathrm{\Sigma }(\omega ,\mathbf{k})$ with a pole, and a narrow residual quasiparticle band crossing $\omega =0$, with a small residue $Z\ll 1$ (a residual Fermi liquid). We consider Model I, in which $Z$ also renormalizes quasiparticle bandwidth $W$ to $ZW$ and Model II, in which $W$ remains unrenormalized. The Green’s function for such a system $G(\omega ,\mathbf{k})$ contains two zeros and three poles — one from the quasiparticle band and two from the gapped bands. Yet, we show that in Model I, the Luttinger-Ward analysis is fully applicable for any $Z>0$, and the results are exactly the same as if the gapped bands did not exist. For Model II, the Luttinger-Ward analysis recovers once $Z$ exceeds some critical value, which depends on fermionic density.