Absence of Luttinger's Theorem

We show exactly with an $SU(N)$ interacting model that even if the ambiguity associated with the placement of the chemical potential, $\mu$, for a $T=0$ gapped system is removed by using the unique value $\mu(T\rightarrow 0)$, Luttinger's sum rule is violated. The failure stems from the non-existence of the Luttinger-Ward functional for a system in which the self-energy diverges. Since it is the existence of the Luttinger-Ward functional that is the basis for Luttinger's theorem which relates the charge density to sign changes of the single-particle Green function, no such theorem exists. Experimental data on the cuprates are presented which show a systematic deviation from the Luttinger count, implying a breakdown of the electron quasiparticle picture in strongly correlated electron matter.