Absence of Luttinger's theorem for fermions with power-law Green functions

We investigate the validity of Luttinger's theorem (or Luttinger sum rule) in two scale-invariant fermionic models. We find that, in general, Luttinger's theorem does not hold in a system of fermions with power-law Green functions which do not necessarily preserve particle-hole symmetry. This contrasts with the result by Ref. \cite{Blagoev1997,Yamanaka1997} that Luttinger liquids respect Luttinger's theorem. To understand the difference, we examine the spinless Luttinger liquid model. We find two properties which make the Luttinger sum rule valid in this model: particle-hole symmetry and $\mathrm{Im} G(\omega=0,-\infty)=0$. These two properties represent sufficient, but not necessary, conditions for the validity of the Luttinger sum rule in condensed matter systems.