We investigate the validity of Luttinger’s theorem (or Luttinger sum rule) in two scale-invariant fermionic models. We find that, in general, Luttinger’s theorem does not hold in a system of fermions with power-law Green functions which do not necessarily preserve particle-hole symmetry. However, Ref. [1,2] showed that Luttinger liquids, another scale-invariant fermionic model, respect Luttinger’s theorem. To understand the difference, we examine the spinless Luttinger liquid model. We find two properties which make the Luttinger sum rule valid in this model: particle-hole symmetry and lmG(\omega=0,-\infty)=0 l m G ( ω = 0 , − ∞ ) = 0 . We conjecture that these two properties represent sufficient, but not necessary, conditions for the validity of the Luttinger sum rule in condensed matter systems.
We investigate the validity of Luttinger’s theorem (or Luttinger sum rule) in two scale-invariant fermionic models. We find that, in general, Luttinger’s theorem does not hold in a system of fermions with power-law Green functions which do not necessarily preserve particle-hole symmetry. However, Ref. [1,2] showed that Luttinger liquids, another scale-invariant fermionic model, respect Luttinger’s theorem. To understand the difference, we examine the spinless Luttinger liquid model. We find two properties which make the Luttinger sum rule valid in this model: particle-hole symmetry and lmG(\omega=0,-\infty)=0 l m G ( ω = 0 , − ∞ ) = 0 . We conjecture that these two properties represent sufficient, but not necessary, conditions for the validity of the Luttinger sum rule in condensed matter systems.
Abstract Luttinger’s theorem is a major result in many-body physics that states the volume of the Fermi surface is directly proportional to the particle density. In its ‘hard’ form, Luttinger’s theorem implies that the Fermi volume is invariant with respect to interactions (as opposed to a ‘soft’ Luttinger’s theorem, where this invariance is lost). Despite it is simplicity, the conditions on the fermionic self energy under which Luttinger’s theorem is valid remains a matter of debate, with possible requirements for its validity ranging from particle-hole symmetry to analyticity about the Fermi surface. In this paper, we propose the minimal requirements for the application of a hard Luttinger’s theorem to a generic fermionic system of arbitrary interaction strength by invoking the Atiyah–Singer index theorem to quantify the topologically-robust behavior of a generalized Fermi surface. We show that the applicability of a hard Luttinger’s theorem in a D-dimensional system is directly dependent on the existence of a (D − 1)-dimensional manifold of gapless chiral excitations at the Fermi level, regardless of whether the system exhibits Luttinger or Fermi surfaces (i.e., manifolds of zeroes of the Green’s function and inverse Green’s function, respectively). The exact form of the self-energy which guarantees validity of a hard Luttinger’s theorem is derived, and agreement with current experiments, numerics, and theories are discussed.
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.
The validity of the Luttinger sum rule is investigated within the prototype tight-binding model of interacting fermions in one dimension, i.e., the $t\text{\ensuremath{-}}V$ model including the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, in order to break the particle-hole symmetry. Scaling analysis of finite-system results at half filling reveals evident breakdown of the sum rule in the large gap regime at V⪢t while the sum rule appears to recover together with vanishing of the Mott-Hubbard gap.
We show exactly with an SU(N) interacting model that even if the ambiguity associated with the placement of the chemical potential, μ, for a T=0 gapped system is removed by using the unique value μ(T→0), Luttinger's sum rule is violated even if the ground-state degeneracy is lifted by an infinitesimal hopping. The failure stems from the nonexistence of the Luttinger-Ward functional for a system in which the self-energy diverges. Since it is the existence of the Luttinger-Ward functional that is the basis for Luttinger's theorem which relates the charge density to sign changes of the single-particle Green function, no such theorem exists. Experimental data on the cuprates are presented which show a systematic deviation from the Luttinger count, implying a breakdown of the electron quasiparticle picture in strongly correlated electron matter.
In this Comment, we first present general arguments showing that the absence of the Luttinger theorem (LT) for the SU(N) model of Dave, Phillips and Kane (DPK) is rooted in the non-uniqueness of the ground state (GS) of this model for 0 < n < N, where n denotes the number of particles in the GS; the validity of the Luttinger theorem for n = N/2, when N even, is accidental, a consequence of particle-hole symmetry. Consequently, by supplementing the Hamiltonian of the SU(N) model with a perturbation Hamiltonian that removes the GS degeneracy, the LT is to apply also for the SU(N) model in the limit of the coupling constant, λ, of this perturbation approaching zero, where the limit λ--> 0 is clearly to be taken subsequent to taking the zero-temperature limit of the thermal single-particle Green function in the expression for the Luttinger number N_L. We explicitly establish the validity of this statement for the case of N=4. The details of the relevant calculations being distinctly transparent, one can readily convince oneself that our observation is valid for arbitrary N. It follows that the issues raised by DPK, such as non-existence of the Luttinger-Ward functional and "breakdown of the elemental particle picture in strongly correlated electron matter", are all inessential to the observed failure of the LT. As regards the singularity of the self-energy Σ(ω) on the real ω-axis, observed by DPK, we demonstrate that this also is a direct consequence of the non-uniqueness of the GS of the SU(N) model for 0 < n < N. In the light of the above observations, we are in a position to state that to this date no case has come to light indicative of the failure of the LT under the conditions for which it has been deduced. [Abridged Abstract]
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.