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 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.
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.
Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.
For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.
We analyze the original proof by Luttinger and Ward of the Luttinger theorem, according to which for uniform ground states of systems of (interacting) fermions, which may be metallic or insulating, the number of k points corresponding to non-negative values of G_s(k;mu) is equal to the total number of particles with spin index s in these ground states. Here G_s(k;mu) is the single-particle Green function of particles with spin index s at the chemical potential mu. For the cases where the two-body interaction potential is short-range, and in particular for lattice models, we explicitly demonstrate that this theorem is unconditionally valid, irrespective of the strength of the bare interaction potential. We arrive at this conclusion by amongst other things demonstrating that the perturbation series expansion for self-energy in terms of skeleton diagrams, as encountered in the proof of the Luttinger-Ward identity, is uniformly convergent for almost all momenta and energies. We further investigate the mechanisms underlying some reported instances of failure of the Luttinger theorem. With one exception, for all the cases considered in this paper, we show that the apparent failures of the Luttinger theorem can be attributed either to shortcomings of the employed single-particle Green functions or to misapplication of this theorem. The one exceptional case brings to light the possibility of a genuine failure of the Luttinger theorem for insulating ground states, which we show to be brought about by a false limit that in principle can be reached on taking the zero-temperature limit without the value of mu coinciding with the zero-temperature limit of the chemical potential satisfying the equation of state at finite temperatures; no such ambiguity can arise for metallic states.
Luttinger theorem relates Fermi surface volume to electron density per spin n. It holds in an arbitrary Fermi liquid but is widely believed to break when a fermionic self-energy has a pole. We show that this is not necessarily the case. As an example, we consider a fermionic system consisting of two gapped bands, originating from a self-energy Σ(ω,k) with a pole, and a narrow residual quasiparticle band crossing ω=0, with a small residue Z≪1 (a residual Fermi liquid). We consider Model I, in which Z also renormalizes quasiparticle bandwidth W to ZW and Model II, in which W remains unrenormalized. The Green's function for such a system G(ω,k) contains two zeros and three poles --- one from the quasiparticle band and two from the gapped bands. Yet, we show that in Model I, the Luttinger-Ward analysis is fully applicable for any Z>0, and the results are exactly the same as if the gapped bands did not exist. For Model II, the Luttinger-Ward analysis recovers once Z exceeds some critical value, which depends on fermionic density.