  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 even if the ground-state degeneracy is lifted
by an infinitesimal hopping. 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.
