# Necessary and Sufficient Conditions for the Validity of Luttinger's Theorem

## TL;DR

This paper establishes the minimal topological and analytical conditions under which Luttinger's theorem holds in interacting fermionic systems, using the Atiyah-Singer index theorem to relate Fermi surface properties to robust gapless excitations.

## Contribution

It introduces a topological framework for the validity of Luttinger's theorem, linking it to the existence of gapless chiral excitations and deriving the specific self-energy conditions needed.

## Key findings

- Validity depends on a (D-1)-dimensional manifold of gapless chiral excitations.
- Derived the exact self-energy form ensuring Luttinger's theorem holds.
- Agreement with experiments, numerics, and existing theories discussed.

## Abstract

Luttinger's theorem is a major result in many-body physics that states the volume of the Fermi surface is directly proportional to the particle density. In its "hard" form, Luttinger's theorem implies that the Fermi volume is invariant with respect to interactions (as opposed to a "soft" Luttinger's theorem, where this invariance is lost). Despite it's simplicity, the conditions on the fermionic self energy under which Luttinger's theorem is valid remains a matter of debate, with possible requirements for its validity ranging from particle-hole symmetry to analyticity about the Fermi surface. In this paper, we propose the minimal requirements for the application of a hard Luttinger's Theorem to a generic fermionic system of arbitrary interaction strength by invoking the Atiyah-Singer index theorem to quantify the topologically-robust behavior of a generalized Fermi surface. We show that the applicability of a hard Luttinger's theorem in a D-dimensional system is directly dependent on the existence of a (D-1)-dimensional manifold of gapless chiral excitations at the Fermi level, regardless of whether the system exhibits Luttinger or Fermi surfaces (i.e., manifolds of zeroes of the Green's function and inverse Green's function, respectively). The exact form of the self-energy which guarantees validity of a hard Luttinger's theorem is derived, and agreement with current experiments, numerics, and theories are discussed.