A Luttinger’s theorem revisited

For uniform systems of spin-less fermions in d spatial dimensions with d > 1, interacting through the isotropic two-body potential v(r − r), a celebrated theorem due to Luttinger (1961) states that under the assumption of validity of the many-body perturbation theory the self-energy Σ(k; ε), with 0 ≤ k∼3kF (where kF stands for the Fermi wavenumber), satisfies the following universal asymptotic relation as ε approaches the Fermi energy εF : ImΣ(k; ε) ∼ ∓αk(ε− εF ) , ε<εF , with αk ≥ 0. As this is, by definition, specific to self-energies of Landau Fermi-liquid systems, treatment of non-Fermi-liquid systems are therefore thought to lie outside the domain of applicability of the many-body perturbation theory; that, for these systems, the many-body perturbation theory should necessarily break down. We demonstrate that ImΣ(k; ε) ∼ ∓αk(ε− εF ) , ε<εF , is implicit in Luttinger’s proof and that, for d > 1, in principle nothing prohibits a non-Fermi-liquid-type (and, in particular Luttinger-liquid-type) Σ(k; ε) from being obtained within the framework of the many-body perturbation theory. We in addition indicate how seemingly innocuous Taylor expansions of the self-energy with respect to k, ε or both amount to tacitly assuming that the metallic system under consideration is a Fermi liquid, whether the self-energy is calculated perturbatively or otherwise. Proofs that a certain metallic system is in a Fermi-liquid state, based on such expansions, are therefore tautologies.