Non-perturbative approach to Luttinger's theorem in one dimension

Jutta Ortloff, M. Balzer, Michael Potthoff

Behnam Farid

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 Sigma(k;epsilon), with 0 ,< epsilon_F, with alpha_k >,= 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 Im[Sigma(k;epsilon)] ~ -,+ alpha_k (epsilon-epsilon_F)^2, epsilon >,< epsilon_F, 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) 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.

Meng Gao, Yin Zhong

N. Kawakami, Sung-Kil Yang

Antoine Praz, Joel Feldman, Horst Knörrer +1 more

A rigorous and simple perturbative proof of Luttinger's theorem is sketched for Fermi liquids in two and three dimensions. It is proved that in the finite volume, the quasi-particle density is independent of the interaction strength. The thermodynamic limit is then controlled to all orders in perturbation theory.

Behnam Farid

For uniform systems of spinless fermions in d spatial dimensions with d > 1, interacting through the isotropic two-body potential ν(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 ∑(k;ϵ), with 0 ⩽ k ≲ 3k F (where k F stands for the Fermi wavenumber), satisfies the following universal asymptotic relation as ε approaches the Fermi energy εF: Im [∑(k; ϵ)]∼ ∓ α k (ϵ - ϵF) 2, ϵ ≶ ϵF, with α k ⩾ 0. As this 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 necessarily break down. We demonstrate that Im [∑(k; ϵ)]∼ ∓ α k (ϵ - ϵF) 2, 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 a Luttinger-liquid-type) ∑(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 Fermiliquid state, based on such expansions, are therefore tautologies.