Necessary and Sufficient Conditions for the Validity of Luttinger's Theorem

We derive the minimum requirements for the application of Luttinger's Theorem to a general many-body fermionic system of arbitrary interaction strength. By redefining the Fermi surface to include zero solutions of the interacting Green's function (i.e.,"Luttinger surfaces"), we show that the only requirement that must be observed for the applicability of a"hard"Luttinger's theorem with trivial Luttinger count is the existence of a (D-1)-dimensional manifold of gapless chiral excitations. Examples of systems with Luttinger surfaces that obey Luttinger's theorem are given, and the explicit dependence of Luttinger's theorem on the momentum and frequency dependence of the self energy is discussed.

Introduction. Of fundamental importance to physics in both the IR and UV limits is the question of whether or not macroscopic phenomena can be described by the collective behavior of indivisible, well-defined particles that obey fundamental conservation laws. In the high-energy community, such an "independent-particle" approximation (IPA) [1] has lead to the successful prediction of new particles [2] and ultimately the creation of the presentday Standard Model [3][4][5]. The low-energy effective field theory of fermionic excitations also relies heavily upon an IPA, as the presence of a Fermi surface usually permits us to construct an isomorphism between the eigenstates of the non-interacting Fermi gas and the interacting Fermi system via either perturbative [6] or renormalization group [7][8][9][10][11] arguments. When a particle loses its mass, the IPA breaks down, resulting in the well-known scale invariant properties of photons and gauge bosons. On the contrary, the presence of free massive particles described by scale invariant physics is not predicted by the Standard Model. Such systems are described by an "un-particle" [12,13] approximation (UPA), with a continuous spectrum of mass replacing the discrete observables in the IPA [14]. This unparticle "stuff" has recently been embraced by condensed matter theorists as a possible description of the normal phase of the cuprates [15][16][17], leading to the possibility of an "un-Fermi liquid" state [18].
In the high-energy limit, unparticles may be found experimentally by detecting a loss of energy or momentum not accounted for by conservation laws [12,19]. Analogously, unparticles in the low-energy limit should correspond to "missing" degrees of freedom (DoF) once we turn on interactions. This latter scenario can be studied in a certain material by checking the applicability of Luttinger's theorem [20][21][22][23], which states that the direct relation between the D-dimensional volume contained within the Fermi surface and the total density of particles (1) * heathjo@bc.edu † kevin.bedell@bc.edu is invariant with respect to the particles' interaction. Because Luttinger's theorem is a non-perturbative theory, it is a statement that describes collective behavior beyond the vicinity of some cutoff near the Fermi surface, making it a more robust criterion of the IPA than Landau-Fermi liquid theory [24][25][26]. Unfortunately, the scope of when and where Luttinger's theorem is valid is somewhat unclear in the present literature, with some even claiming the very definition of the theorem is "clouded in folklore" [27]. This has led to a generalization of Luttinger's theorem into "hard" and "soft" variations, with the former being defined as in Eqn. (1) and the latter corresponding to those systems where the left-handside of Eqn. (1) is equal to some fraction of the total density, known as the "Luttinger count" [18,[28][29][30][31]. Because independent-particle behavior is only seen in systems that satisfy a hard Luttinger's theorem with trivial Luttinger count, it has since been accepted that the IPA breaks down whenever we lack a conventional Fermi surface or particle-hole symmetry [32]. This includes materials with a "Luttinger surface" [27], which corresponds to a (D −1)-dimensional manifold of zeros of the interacting Green's function G(k, ω) and which have been proposed to violate the fundamental assumptions of a hard Luttinger's theorem 1 [18,[28][29][30].
In this paper, we introduce the necessary and sufficient conditions in which we can safely consider the Luttinger count in any interacting fermionic system to be synonymous with the bare particle density. In other words, we outline the precise requirements where an independent particle description is valid in a many-body system of arbitrary interaction strength. We suggest that the existence of a generalized topologically-invariant boundary in momentum space always implies that the IPA is applicable, despite the presence of a Luttinger surface or the absence of particle-hole symmetry.
Generalization of the Fermi surface: Of central importance to Luttinger's theorem is the preservation of a Fermi surface [22]. By Fermi surface, we mean here some boundary in phase space i) that exactly overlaps with the Fermi surface of the non-interacting Fermi gas at {k F } in the isotropic case, ii) where G(k, ω) changes sign, and iii) which remains experimentally detectable for some finite interaction.
In a simple D-dimensional Landau-Fermi liquid, the presence of a discontinuity in the bare particle momentum distribution n(k) can be interpreted as a finite quasiparticle weight [33]: If 0 < Z k ≤ 1, the traditional proof of Luttinger's theorem follows 2 . However, a value of Z k ≥ 0 is not a strong indication for the applicability of Luttinger's theorem [34], nor is a vanishing Z k an indication of its failure [35]. A well-known example of the latter is the Tomonaga-Luttinger liquid [36][37][38][39], where perturbative methods [24] and the Lieb-Schultz-Mattis theorem [25] suggest that Luttinger's theorem is preserved in 1D metals despite the clear lack of a quasiparticle weight Z k . The Fermi surface at {k F } is then replaced by the set of Fermi points where the mth derivative of the bare distribution function becomes singular: This follows from the g-ology construction of the 1D fermionic distribution near the Fermi points [38]: where C 1 , C 2 , and α are positive constants [40]. Because the momentum distribution of the Landau-Fermi liquid also exhibits a singularity in the m = 1 derivative at k = k F , it appears that Eqn. (3) might be a nearly "universal" feature of systems that obey Luttinger's theorem. We propose that Eqn. (3) is indeed a universal feature of all systems that obey Luttinger's theorem. We begin by recalling the Kadanoff-Baym functional for some general interacting fermionic system [41][42][43][44][45]: where J is some external field and Φ is the Luttinger-Ward functional, defined as the sum of all skeleton diagrams 3 [21,44]: The Kadanoff-Baym functional is the effective action for the fermionic many-body state. The generating functional Z[J] = e iΓ [J] for the interacting Green's function is then [46,47] where Z classical [J] = exp (S[J]) is the classical generating functional. We know, by definition, that a Fermi surface exists when G −1 (k F , ω) = 0. Because the second term in the above can be simplified to ΣG = ((G 0 Σ) −1 − 1) −1 , which disappears at k = k F , the phase in Eqn. (7) is dominated by the divergent log(−G) term. This phase defines a covering map f (ℓ) = e iw(ℓ) , where f : S 1 → U (1) in the simplified D = 2 fermionic system and where ℓ is some contour in momentum space that winds around the singular manifold at {k F }. The mapping is characterized by the homotopy class When this winding number N = 0, then the system supports a Fermi surface. This definition is equivalent to the topological invariant introduced by Volovik [48,49] to provide a robust definition of the Fermi surface for Landau-Fermi liquids, Tomonaga-Luttinger liquids, and marginal Fermi liquids [15,50]. Because such a definition was inspired by the analogous topological singularities in superfluid 3 He-A (known as "boojums" [51,52]), we will refer to systems with N = 0 as "snarks" for conciseness 4 . The versatility of the snark description is that it is equivalent to an anomaly in quantum field theory. Such anomalies occur when the vacuum functional lacks some symmetry of the classical Lagrangian, which manifests itself as an anomalous phase in the Dirac operator [53,54]. The most well-known of these is the Adler-Bell-Jackiw (ABJ) or chiral anomaly [55,56], where the quantization of fermionic fields results in an apparent chiral symmetry breaking in the presence of a Dirac sea; i.e., an "upward" shift of energy levels for particles and a "downward" shift for anti-particles that remains uncompensated at the bottom of the sea in the continuum limit [57][58][59]. We see that the snark is a many-body "shortcut" to the chiral anomaly in the quantum fluid, due to its derivation from the fermionic Green's function which inherently includes the effects of Pauli correlation. The presence of chiral symmetry breaking is confirmed by invoking the Atiyah-Singer index theorem, which shows that the topological description (8) is equivalent to calculating the difference N = N + − N − in chiral zero modes (e.g., the index) of the Dirac operator [54,60,61]. A non-zero value of N in a D dimensional fermionic system is synonymous with the existence of a D − 1 dimensional manifold of gapless chiral modes at the Fermi surface. Chiral symmetry breaking is apparent in the conventional form of the latter due to the existence of a finite density of states at the Fermi level, where a non-zero condensate of particle-hole pairs with a linearized dispersion results in a violation of helicity and therefore chirality [62,63]. Note that this is fundamentally different from the anomalous current seen in Weyl semimetals, where a chiral symmetry is broken due to a negative longitudinal magnetoresistance in the crystal [64,65].
The mutual coexistence of gapless excitations with stable Fermi surfaces has been suggested via an Atiyah-Bott-Shapiro construction of the K-theory group K(R k ) = π k−1 (GL(N, C)) [66] as well as via conformal field theory arguments in the UV limit [67,68]. In a similar vein, the topological nature of the Luttinger count and Luttinger's theorem itself have previously been suggested in [69] and [26], respectively. However, unlike these studies, the snark description explicitly connects the microscopic details of the many-body propagator to the existence of a topologically robust manifold of chiral gapless excitations in momentum space.
Luttinger's Theorem and ω-dependence of Σ(k, ω): Because the snark solution is applicable to both the Fermi and Luttinger liquids, the existence of a manifold of zero modes at k F appears to be a promising "hard" requirement for Luttinger's theorem. However, Eqn. (8) tells us that a non-zero value of N may exist for zeros of G −1 (k, ω) or G(k, ω), the latter of which having been noted to contradict the fundamental postulates of Luttinger's theorem [18,[28][29][30]32]. For any fermionic system, the applicability of a hard Luttinger's theorem can be boiled down to two main principles: Given the requirements of Eqns. (9a) and (9b), we want to see if they are compatible with Eqn. (8). We start with the former condition. Because we can always write the fermionic Green's function in the Källen-Lehmann representation, we can easily see that G(k, |ω| → ∞) ∼ 1/ω regardless of whether or not the system is a Landau-Fermi liquid [70,71]. This makes sense, as the self energy cannot diverge at asymptotically large frequencies [44], and simplifies Eqn. (9a) to the condition that the lowfrequency phase of the retarded Green's function must disappear. This ultimately amounts to the imaginary part of the Green's function (and therefore the imaginary part of Σ(k, ω)) to converge faster than the real part as ω → 0. For some general system, we can relate the real and imaginary parts of the self energy to each other via a simple Kramers-Kronig relation [72], where we assume ω is small. If we consider some general case ℑΣ(k, ω) ∼ ω α , we find For α < 2 and α = 1, assuming some UV and IR cutoff ω c in the integration limits. Only the case 1 ≤ α satisfies Eqn. (9a) (with the case of α = 1 being the marginal Fermi liquid), and hence also satisfy Luttinger's theorem. A similar requirement for Luttinger's theorem is observed when we consider Eqn. (9b), where the integral will vanish only if we can write the self-energy as an exact differential of the Green's function; i.e., where we recognize Φ[G] as the Luttinger-Ward functional. For divergent frequency dependence in the selfenergy, we are unable to integrate the differential in the neighborhood of the Fermi surface and Luttinger's theorem is, once again, violated.
Whereas previous studies have connected the powerlaw coefficient in ℑΣ(k, ω) to some anomalous scaling of an unparticle propagator [17], the discussion above proves that the IPA is always preserved in the normal phase for optimal doping and above, independent of any other internal parameter. Because the cases where Luttinger's theorem fails correspond to the appearance of a (pseudo)gap, Eqn. (8) no longer yields a non-zero winding number as the Luttinger-Ward functional is illdefined and/or the chiral symmetry is at least partially restored at {k F }. In other words, the snark vanishes iff Luttinger's theorem fails. The power of this statement is that Eqn. (8) is the sole necessary and sufficient condition for the validity of Luttinger's theorem.
This result is reminiscent of Luttinger's original discussion, where empty bands fail to contribute to the Fermi surface volume and hence the Luttinger count [22]. Our conclusion is also supported by the more recent holographic description of strongly correlated matter, where Luttinger's theorem has been shown to be valid in all confined gauge theories with an "electron star" dual [83][84][85][86][87][88][89]. The formation of a gap has the same effect as a fractionalization of the fermionic DoF, as the former can be interpreted as some scalar hair on the black hole solution which violates Gauss' law in the bulk and subsequently violates Luttinger's theorem [90,91].
Moreover, the snark description resolves the issue of applying Luttinger's theorem at the Mott transition, where the onset of a correlation-induced insulating phase has led to the question of a Fermi gas-like state in these materials [92][93][94][95][96][97][98]. For Mott insulators with gapped excitations, it is well known that Luttinger's theorem is violated [97,99]. However, in models such as the large-U limit of the half-filled nearest-neighbor Hubbard model on the triangular lattice[100] and the weak-tunneling limit of intercoupled 1D Hubbard chains treated in the RPA [31,97], the gap either remains completely closed (as seen in the former) or negligible 5 See Appendix C for an explicit derivation.
compared to the bandwidth (as seen in the latter), supporting Kohn's original premise that the presence of an excitation gap is sufficient but not necessary for insulating behavior [96]. This is similarly supported by the proposal that the Mott transition in 1D and 2D Hubbard models in the U → ∞ limit is a Pokrovsky-Talapov (commensurate-incommensurate) transition, and are thus integrable [101].
Because Luttinger's theorem remains in the presence of a gapless Luttinger surface, we predict that the IPA remains applicable to this special class of insulators.
Luttinger's Theorem and k-dependence of Σ(k, ω): If a generic G(k, ω) = 0 solution beyond the marginal Fermi liquid is to obey Luttinger's theorem, the singular behavior of the self energy must lie in the momentum-dependence. When studying such behavior in strongly correlated matter, a local approximation These studies motivate us to consider some generalized momentum-dependence in the self energy that results in a Luttinger surface. To serve this purpose, we perform a Laurent expansion of some general self energy: By assuming the self energy is analytic about some annular region near k F , it should be clear that solutions of Eqn. (8) correspond to higher-order m-derivatives in Eqn. (3). This allows us to generalize the quasiparticle weight in Eqn. (2) to The snark can then be thought of as a "kink" in the bare particle distribution at k F . These kinks have previously been observed as "critical Fermi surfaces", and indicate non-Fermi liquid behavior in heavy fermion criticality, Mott criticality, and at optimal doping of the cuprates [77,112]. Much as in the case of a Tomonaga-Luttinger liquid, the existence of a critical Fermi sur- We now introduce the necessary nomenclature to categorize all possible snarks. We call the first mth order derivative of the bare particle distribution at {k F } that yields a non-zero Z (m) k the order of the snark. We include solutions of m = 0 in the above to account for the local Fermi liquid, which has no k-dependence [114][115][116]. Generic systems with Z (m) k ∈ R >0 for m > 0 are defined as quasi-local, and are said to exhibit snarks of the first kind (n = 1). Physically, quasi-local self energies correspond to some truncation in the Laurent expansion of a general self energy to order m for coefficients → ∞ for m > 0 are said to be snarks of the second kind (n = 2). Therefore, the snark of a local Fermi liquid would be defined as a 0th order Fermi surface of the 1st kind, while that of a Tomonaga-Luttinger liquid would be defined as a 1st order Luttinger surface of the 2nd kind (which follows directly from the form of the momentum distribution Eqn. (4)).
Our convention gives a systematic means of classifying all possible snarks from the momentum-dependence of Σ(k, ω). This exact dependence is derived in Appendix D, and is reproduced below: Note that, as ℜΣ(ω) ∼ ω for all cases that satisfy Luttinger's theorem, ∂ℜΣ(ω) ∂ω ∼ constant. Therefore, the set of all mth order snarks of the nth kind S (n) m defines all possible k-behavior in the self energy that satisfies Luttinger's theorem.
The divergent behavior discussed above has been hinted at in numerical studies of the Mott-Hubbard metal-insulator transition in the unfrustrated 2D Hubbard model [117] as well as in a functional renormalization group extension of DMFT applied to the 2D Hubbard model at half filling [118]. A more rigorous proposal of quasi-local behavior in 2D materials is seen in [119-121], where the applicability of the Bethe Ansatz in D > 1 allows us to describe excitations near the Fermi surface in terms of phase-shift variables. The presence of a unrenormalizable Fermi surface phase-shift results in the sudden collapse of the quasiparticle weight with the addition of even a single external particle; a phenomenon known as the "orthogonal catastrophe"[122-124]. A direct consequence of this is that the Landau parameter for this 2D system goes as f kk ′ ∼ 1/|k − k ′ |, and is therefore divergent for forward scattering. This interaction then leads to marginal Fermi liquid behavior in ℑΣ(k, ω) with the addition of a term ∼ log(q c v F ), where q c is an upper momentum cutoff [125,126]. Because we can always take a different branch cut in the low-ω integral of the logarithm, the Luttinger-Ward functional is still well-defined in any case of marginal Fermi liquid behavior of Σ(ω) (as expected [127]). Therefore, although the 2D Landau-Fermi liquid formalism might break down in the presence of forward-scattering near the Fermi surface, a 1st order Luttinger surface of the 2nd kind is present, and thus Luttinger's theorem and the IPA remains. This is in agreement with the work of Haldane, where the bosonized D ≥ 1 fermionic system is shown to obey Luttinger's theorem even when no Landau quasiparticle is present [35]. Our general result is similarly in agreement with experimental studies on dilute 2D materials (such as the low-disordered silicon metal-oxide semiconductor field-effect transistors), where evidence is found for a strongly-correlated metallic ground state despite the absence of a Landau-Fermi liquid-like quasiparticle[128-133].
Conclusion: Many condensed matter physicists study the properties of strongly interacting electron systems; how they interact with each other, themselves, and their environment. The presence of coherence might force the interacting regime to exhibit emergent phenomenon unlike anything seen in the non-interacting limit, but at the end of the day an independent-particle picture is always reduced to an extreme inconvenience rather than an absolute impossibility[134].
Luttinger's theorem is a powerful tool that tells us when an independent particle approximation is salvageable. Previous studies have suggested that such cases are rare, and instead an "un-particle" approximation must be used for the great majority of models where the IR limit loses any resemblance to the UV. Our work shows that Luttinger's theorem is synonymous with the existence of a (D − 1)-dimensional manifold of gapless chiral excitations at the Fermi momentum. The coexistence of Luttinger surfaces with a trivial Luttinger count is most likely in dimensions D ≤ 2, where quasilocal kdependence in the self energy is most probable. The existence of un-conventional, scale-invariant physics that breaks the IPA in the absence of a spectral gap would then be confined to noncompact dimensions much larger than our own, as already hinted in the work of Randall and Sundrum [135].
In light of recent experiments, numerics, and theoretical models regarding strongly correlated matter, we believe that our study brings Luttinger's theorem out of the "folklore" of recent years, and opens new avenues to solving the many-body problem with common-sense UV physics.
Acknowledgements: We thank Matthew Gochan and Tong Yang for useful discussions on the ideas presented in this paper. We also thank Krastan Blagoev for his guidance, encouragement, and input. One of the authors (J.

A Proof of Luttinger's Theorem in a Landau-Fermi Liquid
We now re-derive the well-known proof of Luttinger's theorem in a Landau-Fermi liquid. We hope that this will fill in certain gaps not appropriately addressed in the main body of the text.
We begin by recalling the form of the Green's function for a bare particle in the interacting system: where ξ k is measured with the respect to the chemical potential and an infinitesimal value of iδ is implied. The total density can be written as [71] N 2V To solve the above integral, we take the frequency derivative of the log of the Green's function, which yields This yields the following form of the Green's function: The particle density then becomes Which yields the integrals Eqns. (9a) and (9b), respectively. We'll start with the general solution of the first integral, which we'll solve by introducing the retarded Green's function G R (k, ω). Because there is only a pole in the upper half plane, any closed contour will then yield zero for G R , as we can shift the contour to the regime where ℑ(ω) is infinite. Therefore, the above integral becomes i 2π Note that, if ω > 0, then If we write G(k, ω) = e iφ(ω) |G(k, ω)|, we find that i 2π We therefore find that, as discussed in the main body of the text, that a key component of Luttinger's theorem is dependent upon the phase φ(ω) of the Green's function. Note that ℑ(G(k, ω)) > 0 when ω < 0, with ℑ(G(k, ω)) = 0 for ω = 0. ℑ(G(k, ω)) does not change sign, so we are confined in the upper half plane. Similarly, as ω → −∞, ℑ(G(k, ω)) falls off more rapidly then ℜ(G(k, ω)), because G(k, ω) ∼ 1/ω for ω → ±∞. This directly follows from the Källen-Lehmann representation, where the information from the self energy is contained in A s and B s . Therefore, the ratio of imaginary to real parts of the Green's function goes to 0. Now, from our definition of the phase above, we can easily see that ℜ(G(k, ω)) = cos(φ(ω))|G(k, ω)|, while ℑ(G(k, ω)) = sin(φ(ω))|G(k, ω)|. Hence, For this to go to zero as ω → −∞, φ(−∞) = π. This modifies Eqn. (9a) to the simpler form i 2π We are now left to solve for the phase of the Green's function at low frequency. Note that the above yields the well-known solution if we assume that the imaginary part of the Green's function "disappears" faster than the real component in the limit of ω → 0, which is equivalent to saying that the imaginary part of the self energy disappears faster than the real part in this limit. If this occurs, then tan(φ(0)) = 0, which occurs when φ(0) = 0 or φ(0) = π. The former case corresponds to G(k, 0) > 0, or when we are below the Fermi surface, while the latter case corresponds to G(k, 0) < 0 or when we are below the Fermi surface. Therefore, We then left with proving that the imaginary part of the self energy disappears faster than the real part at small frequency. This can be seen in the case of Landau-Fermi liquid by finding the explicit frequency-dependence of ℑΣ(k, ω), which can be done by first finding the lifetime of a quasiparticle near the Fermi surface. If we considered a free particle, the Green's function would be given by It is well known that the spectral function of the above is a perfect delta function. When considering a Landau quasiparticle, we include an additional component proportional to the quasiparticle lifetime τ : The spectral function of the above is a "widened" delta function with width 1/τ : Compare this with the form of the spectral function from the full Källen-Lehmann representation: Hence, we can easily see that ℑΣ(k, ω) ∼ 1/τ . Therefore, by calculating the lifetime, we can find the dependence of the self-energy on the frequency ω. This can be done by writing down Fermi's golden rule to find the decay rate towards n particle-hole pairs and subsequently replacing the scattering amplitude with a Fermi surface average 6 : where the primed terms denote the quasihole energies and we inserted the identity into the second line. Therefore, because the inverse of the lifetime 1/τ is given by the sum of all possibly allowed decay possibilities, ℑΣ(k, ω) ∼ ω 2 for the special case of a Landau-Fermi liquid (where this quasiparticle picture makes sense). From the Kramers-Kronig relation given in Eqn (11), it is then obvious that ℜΣ(k, ω) ∼ ω, and thus the imaginary part of the self energy disappears faster than the real component whenever a Landau quasiparticle picture is applicable, and thus Eqn. (27) remains valid. We now move onto the second integral, which is easier to solve: If we can write the self energy as an exact differential (as in Eqn. (12)), then the above integral disappears. To ensure this, we have to make sure that the self energy Σ(k, ω) doesn't exhibit any divergent frequency dependence. However, we have already proven this when solving for the previous integral! Therefore, we automatically have a well-defined Luttinger-Ward functional for the Landau-Fermi liquid, and we are left with Therefore, we come to the final form of Luttinger's theorem: The proof of Luttinger's theorem in the text is built form of the above derivation. From the calculation of Fermi's golden rule, we see that all non-Fermi liquid behavior is contained within the frequency dependence of the imaginary part of the self energy, where a deviation from the ω 2 behavior can be interpreted as a breakdown of the quasiparticle paradigm. Analogously, also note that only in the quasiparticle approximation of Fermi's golden rule did we assume any perturbative approximation in our derivation. In this way, we can apply the above to any system with a nonanalytic self energy as long as we take the diverging form after we perform the calculation and, as explained in the text, only if the non-analyticity is purely in the momentum-dependence of Σ(k, ω).

B Derivation of the Kadanoff-Baym functional and its connection to the snark
We will now illustrate how to obtain the form of the snark by solving for the general form of the Kadanoff-Baym functional. We will primarily follow the derivation given in [44]. For this reason, we will take the same notation and define arguments (n) = (x n , τ n ; σ n ), while an overbar means integrals over the space-time coordinates and spin sums We define the quantum action W [J] in terms of the partition function: where J is some source field. 7 Therefore, the Green function G(2, 1) is given by the functional derivative of F with respect to the external field: The effective action Γ[G] is then the Legendre transform of the above: The functional Γ[G] is the Kadanoff-Baym functional. The existence is dependent on the existence of a renormalization group picture and an appropriate cutoff Λ, such that Γ k→Λ ≈ S and Γ k→0 = Γ, where S is the un-quantized action. We can similarly find that To proceed, we need to simplify the r.h.s of the above. For this, we utilize the equation of motion for the Green's function: Now, let us define the inverse of the non-interacting Green function to be Plugging this into the equation of motion and simplifying, we have If we take the limit of J = 0, then we see that the above corresponds to the Dyson equation if we take Which can be re-written to give This simplifies the above equation for Γ[G] by solving the above to represent J(1, 2) in terms of Green's functions and the self energy: which yields zero in the limit of J = 0, as in that case Dyson's equation is exact.
We can now solve the above for the Baym-Kadanoff functional. It is easy to see that the above becomes where we have dropped the argument (2, 1) for conciseness. For the first term, we perform the simplification which can be seen with simple calculus. For the second term, we simplify it the following way: For the final term, recall that we can write the change of the Luttinger-Ward functional Φ as which is Tr [ΣδG]. Hence, the final term is just δΦ. This tells us that the differential of the Kadanoff-Baym functional is just The solution for the Kadanoff-Baym functional is now trivial: where S[J] is the classical action, and from which Eqn. (7) directly follows. Interestingly, because we have included the Luttinger-Ward functional and we assume that it is well-behaved, the above result for the effective action is exact as long as the Luttinger-Ward functional exists. In our case, we will also see the generating functional pick up some phase from a strictly quantum mechanical (i.e., unseen in the classical limit) behavior; namely, as discussed in the text, the presence of a Fermi surface defines an anomalous contribution to the generating functional. To see this, let's ask the question of what, exactly, happens when we are near the Fermi surface. We can simplify the second term easily: where, in all lines of the above, a trace over indices is implied. In the case of a traditional Fermi surface, at k = k F , G −1 0 → 0 while Σ remains finite. Therefore, this term asymptotically approaches negative one. Similar behavior is seen in the Luttinger-Ward functional Φ, which we will assume to be well-behaved and finite. This leaves log(−G). Ignoring the negative, this term becomes divergent at the Fermi surface, as G −1 = 0 at the boundary. Because the other contributions are well behaved, we can then simplify the above if we restrict the functional to k-points in the direct vicinity of the Fermi surface: This anomaly can by quantified by a winding number, as discussed in the text: where the contour C is taken about k F . We can easily simplify this contour to get the topological invariant of Volovik [48,49]: In Volovik's original argument, the existence of N is a direct result of the singularity in the interacting Green's function at k F . However, simple manipulation of Volovik's term given above yields a non-zero winding number for Luttinger surface solutions, where the Green's function itself has zeroes: As long as we assume the Green's function is holomorphic in the vicinity of the Fermi surface, the first integral disappears, and we are left with where we have changed the handedness of our contour from C to C ′ . Such a simplification of the original form of Volovik's topological invariant follows from basic calculus, and predicts a non-zero solution for the winding number N for all gapless systems with Luttinger surfaces. However, the existence of such solutions is not predicted by Volovik's original argument, which is directly dependent on the singularity of the Green's function at the Fermi surface. Only by interpreting the winding number as some index of the generating functional do Luttinger surface solutions beyond the marginal Fermi liquid case become apparent, as the topologically non-trivial behavior described above is now connected to singularities in log(G(k, ω)) as opposed to singularities in the Green's function itself.

C Derivation of self energy dependence on density of states
The main result of this paper concerns when the eigenstates of a generic many-body system can be approximated as the collective behavior of independent, interacting particles. This ultimately boils down to studying the regime of validity of a "hard" version of Luttinger's theorem, which we have shown is restricted to models with ℑΣ(k, ω) ∼ ω α where α ≥ 1. Experimentally, ARPES data tells us that this corresponds to either the overdoped phase (for α > 1) or the optimally-doped "strange-metal" phase (α = 1), both of which respects Luttinger's theorem, while the cases of 0 < α < 1 (the pseudogap phase) and α < 0 (the insulating phase) violate Luttinger's theorem. In this appendix we will briefly show that the condition α ≥ 1 is analogous to the existence of a well-defined, non-zero density of states at k = k F and ω = 0. Because this is also the frequency regime where the density of states is always well-defined, this further confirms that the snark definition given in Eqn. (8) is a necessary and sufficient condition for the validity of Luttinger's theorem by the Atiyah-Singer index theorem.
The more interesting case occurs when α < 1. This corresponds to ℑΣ(k, ω) ∼ ℜΣ(k, ω) ∼ ω α . The imaginary component of the Green's function then becomes We first deal with the regime of 0 < α < 1. Under such circumstances, we can ignore terms that go larger than O(ω α ). We can easily see that, for both ω > 0 and ω < 0, singularities and zeroes of the imaginary part of the Green's function occur when ω ∼ ±δ 1/α . Therefore, takign the limit ω → 0, δ → 0 no longer makes sense, and a density of states is not well-defined as it was for α > 1. Once again, this agrees with ARPES data, as 0 < α < 1 corresponds to the pseudogap state where a partial energy gap occurs.
For the case of α < 0, it is clear to see that ℑG(k F , ω) ∼ 1/|ω| α in the limit under consideration. Because α < 0, it vanishes as we approach ω = 0, and thus the density of states (and, hence, gapless chiral excitations) disappears for these self energies.

D Classification of self energy momentum dependencies that yield snark solutions
The goal of this appendix is to derive Eqn. (15). Before we can do this, let's recall the classification theme we have already introduced for snarks. The order m of the snark is the lowest k-derivative of the Landau quasiparticle weight Z (0) k that either yields a singularity or some real number. In principle, the order could be any natural number. The kind n of the snark tells us if the mth order derivative diverges or not. If it diverges, then it's a snark of the second kind. If the mth order derivative is a real number, then it's said to be of the first kind or quasi-local. Therefore, we have the constraint m ∈ [1, 2] by definition of the snark's kind. If the quasiparticle weight itself is non-zero, then it's said to be a Fermi surface. If the quasiparticle weight is vanishing, then it's said to be a Luttinger surface.
To more formally classify them, we introduce the shorthand F 2. Fermi surface of the 2nd kind: positive non-integer power law 3. Luttinger surface of the 1st kind: negative integer power law 4. Luttinger surface of the 2nd kind: negative non-integer power law A table illustrating the behavior of Z (0) k for snarks of different orders and kinds is given below.  We are now in a position to derive Eqn. (15). We start by looking at the 1st-order k-derivative of the quasiparticle weight of a 1st order snark of the 2nd kind: Remember that the first order snark has a singularity for Z k is always bounded by one, the divergent term must be the derivative of the self energy at the Fermi energy: We have assumed that the self-energy is analytic in frequency space, otherwise there will not exist a well-defined Luttinger-Ward functional. Therefore, ∂ℜΣ(k, ω)/∂ω is some finite value, and the divergent term must be the momentum derivative. For some general non-Fermi liquid system, however, Z k → 0, meaning ∂ℜΣ(k, ω)/∂ω → −∞ as the self energy approaches the Fermi energy. If the system is a non-Fermi liquid and has such divergent behavior in the frequency derivative of the self energy, then the condition for a first order Fermi boundary is that the term ∂ ∂k This condition will give us lim k→kF Z (1) k → ∞, or, in other words, lim k→kF (1/Z k ) → 0. For our perturbative Green's function approach to make sense, it's not the frequency derivative that diverges; rather, it's the momentum dependence and momentum derivative. In other words, if the frequency derivative diverges, then the above expansion of the self energy is invalid. Instead, we are saying that the momentum derivative of the self energy must diverge faster than the self energy itself at the Fermi energy; i.e., lim k→kF ∂ ∂k ∂ℜΣ(k, ω) ∂ω Note that we have to be careful how we take the limit in the above. The residue is only well-defined as k → k F . Because Z (m) k diverges for the mth derivative, the limit and the derivative might not commute. It is therefore implied that the above limit is taken after we take the derivative. If this is ensured, then the above defines the self-energy dependence for a 1st order Fermi surface of the 2nd kind. We can extend this idea to the 2nd order snark of the 2nd kind: One or both of these conditions is necessary for lim k→kF (1/Z k ) → 0. The former is a weaker condition than the case of the 1st order snark; namely, if the first order derivative diverges faster than the zeroth order derivative to the power of 2, then it will obviously diverge faster than the zeroth order derivative to the power 3/2. In other words, the first term tells us that a 1st order snark is automatically a second order snark. The first expression in the above is not the defining characteristic of the 2nd order snark. Instead, the unique condition for the 2nd order snark is given by We quote the next order derivative: If the system obeys the first condition, then it could also be a 1st or 2nd order snark, so the first condition is not unique for the 3rd order snark. Furthermore, if some 2nd order snark has the first order momentum derivative diverge faster than (1 − ∂Σ ∂ω ), then the second term is not unique for the 3rd order Fermi boundary. Therefore, the only unique condition for the 3rd order snark is that lim k→k F ω→0 where A m is some constant. The only unique constraint for some general mth order snark is the j = 1 term. Therefore, the general condition for some mth order snark of the second kind is for some integer m.
Of course, the above argument only makes sense for mth order Fermi and Luttinger surfaces of the second kind, as we have assumed that the mth order derivative diverges. From the form of Eqns. (60a) and (60c), we see that mth order snarks of the first kind are more complicated, as their mth order derivative is a constant. This can trivially be quantified for the 0th order Fermi surface of the first kind, and thus we begin with a 1st order snark of the first kind: All higher derivatives are clearly zero. However, we have to be careful here because we have singular behavior in Z k (or 1/Z k ). In the above calculation, the limit and derivative are interchangeable, as Z k is just some constant. This is easily seen from a back-of-the envelope calculation where we take the limit lim k→kF Z k first: Thus, the first derivative is a constant. However, for higher derivatives, we see that the above is zero. Because Z (1) k is only defined near k F , we take the above solution for higher derivatives, and hence Z (1) k = 0 for higher derivatives, rather than 1/Z (1) k = 0 as implied when we took the derivative first. Following Eqn. (60a) and (60c), we can now suggest a form of the self energy for 2nd order snarks of the 2nd kind: The quasilocal case of the snark condition is therefore given by lim k→k F ω→0 Note that the first derivative diverges, while all higher derivatives are zero. However, from the previous discussion, we know that all higher derivatives of Z (2) k are, in fact, zero, from the subtle issue of interchanging derivatives and limits. The first derivative of the above also goes to zero. In general, the condition for the mth order Fermi/Luttinger surface of the first kind becomes The form of Eqn. (15) follows by noting that terms in the above expression with parameters j < m are allowed for mth order snarks of the 1st kind. As such, we see that Eqn. (15) is the sole behavior the self energy Σ(k, ω) must observe if a snark is to be present and, hence, Luttinger's theorem preserved.