A Luttinger's theorem revisited

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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 theₐssumption of validity of the many-body perturbation theory the self-energy Sigma(k;epsilon), with 0 <,= k <, 3 kF (where kF stands for the Fermi wavenumber), satisfies the following universal asymptotic relation as epsilon approaches the Fermi energy epsilonF: Im[Sigma(k;epsilon)] -,+ alphaₖ (epsilon-epsilonF)², epsilon >,< epsilonF, with alphaₖ >,= 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 shouldₙecessarily break down. We demonstrate that Im[Sigma(k;epsilon)] -,+ alphaₖ (epsilon-epsilonF)², epsilon >,< epsilonF, isᵢmplicit in Luttinger's proof and that, for d > 1, in principle nothing prohibits a non-Fermi-liquid-type (and, in particular Luttinger-liquid-type) Sigma(k;epsilon) 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, epsilon 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.