A Luttinger's theorem revisited

For uniform systems of spinless fermions in d spatial dimensions with d > 1 , interacting through the isotropic two-body potential nu(r-r'), a celebrat ed theorem due to Luttinger (1961) states that under the assumption of vali dity of the many-body perturbation theory the self-energy Sigma(k ; epsilon ), with 0 less than or equal to k less than or similar to 3k(F) (where k(F) stands for the Fermi wavenumber), satisfies the following universal asympt otic relation as e approaches the Fermi energy epsilon(F) : Im [Sigma(k ; e psilon)] similar to -/+alpha(k)(epsilon - epsilon(F))(2), epsilon greater t han or less than epsilon(F) with alpha(k) greater than or equal to 0. As th is is, by definition, specific to self-energies of the 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 necessari ly break down. We demonstrate that Im [Sigma(k ; epsilon)] similar to -/+al pha(k)(epsilon - epsilon(F))(2 epsilon) greater than or less than epsilon(F ), is implicit in Luttinger's proof and that, for d > 1, in principle nothi ng prohibits a non-Fermi-liquid-type land, in particular a Luttinger-liquid -type) Sigma(k ; epsilon) from being obtained within the framework of the m any-body perturbation theory. We in addition indicate how seemingly innocuo us 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 oth erwise. Proofs that a certain metallic system is in a Fermi-liquid state, b ased on such expansions, are therefore tautologies.