We show that the Luttinger-Ward functional can be formulated as an operator insertion in the path integral and hence can be thought of as a generalized symmetry. The key result is that the associated charge, always quantized, defines the homotopy, not the physical charge. The disconnect between the two arises from divergences in the functional or, equivalently, zeros of the single-particle Green's function. Such divergences produce an anomaly of the triangle-diagram type. As a result of this anomaly, we are able to account for the various deviations of the Luttinger count from the particle density. As a consequence, non-Fermi liquids can be classified generally by the well-known anomaly structures in particle physics. Charges descending from generalized symmetries, as in the divergence of the Luttinger-Ward functional, are inherently nonlocal, their key experimental signature.
We show that the Luttinger-Ward functional can be formulated as an operator insertion in the path integral and hence can be thought of as a generalized symmetry. The key result is that the associated charge, always quantized, defines the homotopy, not the physical charge. The disconnect between the two arises from divergences in the functional or, equivalently, zeros of the single-particle Green's function. Such divergences produce an anomaly of the triangle-diagram type. As a result of this anomaly, we are able to account for the various deviations of the Luttinger count from the particle density. As a consequence, non-Fermi liquids can be classified generally by the well-known anomaly structures in particle physics. Charges descending from generalized symmetries, as in the divergence of the Luttinger-Ward functional, are inherently nonlocal, their key experimental signature.
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.
One-dimensional metals have a particular symmetry associated with the discrete structure of the Fermi surface: separate charge conservation in low-energy-scattering processes for particles near the left and right Fermi points, respectively. The field-theoretical renormalization group allows for an efficient exploitation of the Ward identities following from this symmetry. As a first application we prove that the \ensuremath{\beta} function of the Luttinger model vanishes identically. The same symmetry ensures the finiteness of the compressibility, thus making possible the existence of stable metallic phases with anomalous dimensions in d=1.
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.
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.
Using the density matrix renormalization group method, we systematically investigate the evolution of the Luttinger integral in the one-dimensional generalized t-V model as a function of filling and interaction strength, and identify three representative phases. In the weak-coupling regime, the zero-frequency Green's function exhibits a branch-cut structure at the Fermi momentum, and the Luttinger integral accurately reflects the particle density, indicating that the Luttinger theorem holds. As the interaction increases, the spectral weight near the Fermi momentum is gradually suppressed. Interestingly, in the strong coupling regime near half-filling, this singularity is progressively destroyed, accompanied by the emergence of momentum-space zeros in the real part of the Green's function, leading to a novel non-Fermi liquid metallic phase beyond the classic Luttinger liquid paradigm, where the Luttinger surface is no longer defined by a single singularity. While finite spectral weight remains at the original Fermi momentum, the singularity gradually diminishes. Meanwhile, zeros with negligible spectral weight appear away from this momentum, significantly affecting the integral. At exact half-filling, a single-particle gap opens, and the Green's function becomes nearly vanishing across the entire momentum space, indicating the complete suppression of low-energy electronic states consistent with the nature of an insulating charge-density-wave phase. These results suggest that the breakdown of the Luttinger theorem is not triggered by a single mechanism, but rather results from the interplay between interaction-driven evolution of excitation modes and the breaking of particle-hole symmetry, ultimately leading to a continuous reconstruction of the generalized Fermi surface from topologically protected to correlation-driven.
For a system of correlated electrons, the Luttinger-Ward functional provides a link between static thermodynamic quantities on the one hand and single-particle excitations on the other.The functional is useful in deriving several general properties of the system as well as in formulating the thermodynamically consistent approximations.Its original construction, however, is perturbative as it is based on the weak-coupling skeleton-diagram expansion.Here, it is shown that the Luttinger-Ward functional can be derived within a general functional-integral approach.This alternative and non-perturbative approach stresses the fact that the Luttinger-Ward functional is universal for a large class of models.