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.
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.
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.
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.
Based solely on the analytical properties of the single-particle Green's function of fermions at finite temperatures, we show that the generalized Luttinger theorem inherently possesses topological aspects. The topological interpretation of the generalized Luttinger theorem can be introduced because (i) the Luttinger volume is represented as the winding number of the single-particle Green's function and, thus, (ii) the deviation of the theorem, expressed with a ratio between the interacting and noninteracting single-particle Green's functions, is also represented as the winding number of this ratio. The formulation based on the winding number naturally leads to two types of the generalized Luttinger theorem. Exploring two examples of single-band translationally invariant interacting electrons, i.e., simple metal and Mott insulator, we show that the first type falls into the original statement for Fermi liquids given by Luttinger, where poles of the single-particle Green's function appear at the chemical potential, while the second type corresponds to the extended one for nonmetallic cases with no Fermi surface such as insulators and superconductors generalized by Dzyaloshinskii, where zeros of the single-particle Green's function appear at the chemical potential. This formulation also allows us to derive a sufficient condition for the validity of the Luttinger theorem of the first type by applying the Rouche's theorem in complex analysis as an inequality. Moreover, we can rigorously prove in a nonperturbative manner, without assuming any detail of a microscopic Hamiltonian, that the generalized Luttinger theorem of both types is valid for generic interacting fermions as long as the particle-hole symmetry is preserved. Finally, we show that the winding number of the single-particle Green's function can also be associated with the distribution function of quasiparticles, and therefore the number of quasiparticles is equal to the Luttinger volume. This implies that the fundamental hypothesis of the Landau's Fermi-liquid theory, the number of fermions being equal to that of quasiparticles, is guaranteed if the Luttinger theorem is valid since the theorem states that the number of fermions is equal to the Luttinger volume. All these general statements are made possible because of the finding that the Luttinger volume is expressed as the winding number of the single-particle Green's function at finite temperatures, for which the complex analysis can be readily exploited.
Breakdown of Landau's hypothesis of adiabatic continuation from noninteracting to fully interacting electrons is commonly believed to bring about a violation of Luttinger's theorem. Here, we elucidate what may go wrong in the proof of Luttinger's theorem. The analysis provides a simple way to correct Luttinger's expression of the electron number in single-band models where perturbation theory breaks down through the birth of a Luttinger surface without symmetry breaking. In those cases, we find that the Fermi volume only accounts for the doping away from half-filling. In the hypothetical circumstance of a non-symmetry-breaking Mott insulator with a Luttinger surface, our analysis predicts the noteworthy existence of quasiparticles whose Fermi surface is just the Luttinger one. Therefore, those quasiparticles can be legitimately regarded as spinons, and the Mott insulator with a Luttinger surface as realization of a spin-liquid insulator.
Luttinger's theorem for Fermi liquids equates the volume enclosed by the Fermi surface in momentum space to the electron filling, independent of the strength and nature of interactions. Motivated by recent momentum balance arguments that establish this result in a nonperturbative fashion [M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000)], we present extensions of this momentum balance argument to exotic systems which exhibit quantum number fractionalization focusing on Z2 fractionalized insulators, superfluids and Fermi liquids. These lead to nontrivial relations between the particle filling and some intrinsic property of these quantum phases, and hence may be regarded as natural extensions of Luttinger's theorem. We find that there is an important distinction between fractionalized states arising naturally from half filling versus those arising from integer filling. We also note how these results can be useful for identifying fractionalized states in numerical experiments.