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

TL;DR

This paper examines the conditions under which Luttinger's theorem holds in scale-invariant fermionic models, finding it generally does not apply unless specific symmetries are present, with implications for understanding fermionic systems.

Contribution

The study identifies key properties that determine the validity of Luttinger's theorem in scale-invariant fermionic models, highlighting when the theorem applies or fails.

Findings

Luttinger's theorem generally does not hold for fermions with power-law Green functions.

Particle-hole symmetry and zero imaginary part of Green function at zero frequency are sufficient for the theorem's validity.

Luttinger's theorem is valid in Luttinger liquids due to specific symmetry properties.

Abstract

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. However, Ref. \cite{Blagoev1997,Yamanaka1997} showed that Luttinger liquids, another scale-invariant fermionic model, 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 $\mathrm{Im} G(\omega=0,-\infty)=0$. We conjecture that these two properties represent sufficient, but not necessary, conditions for the validity of the Luttinger sum rule in condensed matter systems.