Non-Fermi liquids as ersatz Fermi liquids: general constraints on compressible metals

A system with charge conservation and lattice translation symmetry has a well-defined filling $\nu$, which is a real number representing the average charge per unit cell. We show that if $\nu$ is fractional (i.e. not an integer), this imposes very strong constraints on the low-energy theory of the system and give a framework to understand such constraints in great generality, vastly generalizing the Luttinger and Lieb-Schultz-Mattis theorems. The most powerful constraint comes about if $\nu$ is continuously tunable (i.e. the system is charge-compressible), in which case we show that the low-energy theory must have a very large emergent symmetry group -- larger than any compact Lie group. An example is the Fermi surface of a Fermi liquid, where the charge at every point on the Fermi surface is conserved. We expect that in many, if not all, cases, even exotic non-Fermi liquids will have the same emergent symmetry group as a Fermi liquid, even though they could have very different dynamics. We call a system with this property an"ersatz Fermi liquid". We show that ersatz Fermi liquids share a number of properties in common with Fermi liquids, including Luttinger's theorem (which is thus extended to a large class of non-Fermi liquids) and periodic"quantum oscillations"in the response to an applied magnetic field. We also establish versions of Luttinger's theorem for the composite Fermi liquid in quantum Hall systems and for spinon Fermi surfaces in Mott insulators. Our work makes connection between filling constraints and the theory of symmetry-protected topological (SPT) phases, in particular through the concept of"'t Hooft anomalies".

In condensed matter physics, systems with prescribed microscopic degrees of freedom (usually electrons) can exhibit varied and exotic emergent behavior at low energies. In general, it is extremely difficult, either analytically or numerically, to predict the nature of the emergent low-energy behavior [as described by an "IR (infra-red) theory"] from the properties of the microscopic degrees of freedom. For this reason, it is invaluable to have general results that constrain the nature of the IR theory, given microscopic properties of the system.
One such result is the Lieb-Schultz-Mattis theorem [1], which states (in the formulation of interest to us here [2]) that in a system in one spatial dimension with a conserved U(1) charge and discrete translation symmetry, if the average charge per unit cell (which we call the "filling") is not an integer, then either the IR theory is gapless or else it spontaneously breaks one of the symmetries. This result was later generalized to higher dimensions by Oshikawa [3] and Hastings [4]; in higher dimensions there is also the possibility that a system at fractional filling can be gapped but with non-trivial topological order [5].
A related result is Luttinger's theorem [6], which states that if the IR theory is a Fermi liquid, then the volume enclosed by the Fermi surface (modulo the volume of the Brillouin Zone) is determined entirely by the fractional part of the filling, and in particular is independent of the interaction strength. Originally proven perturbatively by Luttinger, the result was later established through a nonperturbative argument (but still assuming that the IR theory is describable by Fermi liquid theory) by Oshikawa [7]. A generalized version of Luttinger's theorem even holds in a class of phases -known as Fractionalized Fermi Liquids [8,9] -which are distinct from conventional Fermi liquids. In such phases a gapless Fermi liquid co-exists with non-trivial topological order. The generalization to Luttinger's theorem can then exactly be determined from the interplay of the symmetry with the topological order [9][10][11].
These results and others [12][13][14] raise the question: what is the most general statement that one can make about the relation between the microscopic filling and properties of the IR theory? In the present work, we will answer this question, showing that the microscopic filling is always completely determined by a few properties of the IR theory, namely: (a) its emergent symmetries; (b) the relation between the microscopic translation and U(1) symmetries and the emergent symmetries of the IR theory; and (c) a property of the IR theory called its " 't Hooft anomaly". Some connections between filling and 't Hooft anomalies have previously been explored in Refs. [15][16][17][18].
Our work has important implications for the study of "non-Fermi liquids", which are systems that are metallic down to zero temperature but for which the IR physics cannot be described by Fermi liquid theory. Motivated by the results on filling constraints just mentioned, we will introduce the concept of an ersatz Fermi liquid (EFL) as a general framework to understand non-Fermi liquids. An EFL is a system which has the same kinematic properties in the deep infrared as a Fermi liquid, though it might have very different dynamical properties. By "kinematic" properties we mean properties that relate to the structure of the Hilbert space that describes the ground state and the low-energy excitations, as opposed to "dynamical" properties which relate to the Hamiltonian that acts in this Hilbert space. More precisely, the kinematic properties comprise properties (a), (b) and (c) described in the previous paragraph and in more detail in the next section. Further, it will will turn out that the kinematic properties we will discuss have a strong topological flavor, so in a suitably vague sense we can say that an EFL is a system that is "topologically equivalent" to a Fermi liquid. (However, since the dynamical properties of an EFL can be sharply different to that of a Fermi liquid, it will generally not be the case that an EFL can be continuously deformed into a Fermi liquid. ) We will show that many (though by no means all) well-known aspects of Fermi liquid phenomenology are, in fact, purely kinematic in nature, and therefore apply equally well to any EFL. In particular, we will show that any EFL has a Fermi surface that hosts long-lived excitations (though these excitations may not be Landau quasiparticles). In fact, our approach leads to a very general perspective on what it means for a system to have a Fermi surface. Furthermore, we will find that this Fermi surface must obey Luttinger's theorem (or a generalization thereof, analogously to the "fractionalized Fermi liquids" mentioned above); and that, if the Fermi surface geometry is such that a Fermi liquid with that geometry would exhibit quantum oscillations in the dependence of physical properties on magnetic field, then any EFL with the same Fermi surface geometry is also expected to display quantum oscillations with the same periodicity.
Since none of the kinematic properties that define an EFL require that the system be weakly coupled or have a description in terms of quasiparticles, we expect a wide variety of exotic non-Fermi liquid phenomena to be realizable within the class of EFLs. In fact, we will argue based on the general theory of filling constraints that any IR theory which describes a compressible metal, i.e. the filling can be continuously tuned, must have a very non-trivial emergent symmetry group, larger than any compact Lie group. Such a property is indeed satisfied by EFLs (due to the infinitely many conserved quantities associated with the Fermi surface); whether it could be satisfied in a different way that leads to fundamentally different kinematic properties is an important open problem.
At the very least, however, it is clear that a number of non-Fermi liquid metals can be fruitfully discussed from the perspective developed in this paper. The simplest are non-Fermi liquid metals that arise when a Fermi surface is coupled to a critical boson that represents a fluctuating order parameter. Within the standard framework for such quantum critical points, they will be seen to be EFLs. A closely related system is a Fermi surface coupled to a gapless U(1) gauge field, which arises in the theory of composite Fermi liquid metals in the quantum Hall regime, and in some insulating quantum spin liquids with Fermi surfaces of emergent electrically neutral quasiparticles. In these states we will find that, unlike a strict EFL, the kinematic properties of the IR theories can differ from those of a Fermi liquid (indeed, unlike a Fermi liquid, these are not states in which the microscopic density can be continuously tuned) though still closely related. The main point will still stand, though, that the kinematic properties of these IR theories provide a powerful framework to thinking about their universal behavior. In particular we will show that in such systems a version of Luttinger's theorem is still satisfied.
As a final application of our results, we will examine the possibility for systems to exhibit disconnected Fermi arcs instead of a closed Fermi surface. We will find that, assuming the translational symmetry is unbroken, such a scenario is inconsistent with the IR theory being an EFL, except when the system exists on the boundary of a gapless bulk (such as a Weyl semimetal). This provides strong evidence for the impossibility of Fermi arcs. In fact, we obtain a stronger constraint: in an EFL, the Fermi surface must enclose a volume in the Brillouin zone; this volume is the generalization to EFLs of the "electron sea" in a weakly interacting system. Before proceeding, let us make a final technical remark. We note that previous work [19][20][21] generalized the original perturbative proof of Luttinger's theorem to situations where there are a number of fermion and boson fields with U (1) symmetries (either global or gauge). This proof relies on the existence of a Luttinger-Ward functional of the exact Green's functions of the fermions/bosons from which the self-energies can be extracted by functional derivatives. These works lead to the expectation that Luttinger's theorem will be satisfied by some classes of non-Fermi liquids. However, as the quasiparticle is destroyed in the IR in such problems, perturbation theory (even to all orders) should be used with caution. In a non-perturbative context, despite the formal existence [22] of a Luttinger-Ward functional, the conventional proof of Luttinger's theorem fails. This is dramatically illustrated by the fractionalized Fermi liquid phases. It is thus desirable to have a more general nonperturbative argument for Luttinger's theorem in this context, which we provide in this paper.

II. KINEMATIC PROPERTIES OF THE IR THEORY
Among all the features of of the IR theory describing the low-energy properties of a system, there are certain ones that we will refer to as "kinematic", and these are the subject of this section.
Before describing the first kinematic property, let us note that the microscopic system will have a group of symmetries G UV . We will specifically be interested in systems where this includes a global U(1) corresponding to charge conservation, and translation symmetries (possibly on a lattice).
The first kinematic property of the IR theory is the set of emergent symmetries. We thus introduce the group G IR of emergent symmetries 1 of the IR theory. G IR is, in general, not the same as the microscopic symmetry group G UV . Nevertheless, each element g ∈ G UV of the microscopic symmetry group gets mapped into an element ϕ(g) ∈ G IR of the emergent symmetry group, such that ϕ(g 1 )ϕ(g 2 ) = ϕ(g 1 g 2 ) for all g 1 , g 2 ∈ G. There could, of course, be elements of the emergent symmetry group G IR that do not correspond to any microscopic symmetry. Also, a microscopic symmetry g ∈ G UV could act trivially in the IR theory, in which case ϕ(g) = 1 (the identity element in G IR ); an example of the latter case would be for systems with a charge gap, in which case the microscopic U(1) symmetry acts trivially in the IR. All these statements can be expressed in a compact mathematical way by saying that ϕ defines a group homomorphism from G UV to G IR that need not be injective or surjective.
Note that for many of the arguments in this paper, we will specifically want G IR to represent the internal symmetries of the system (that is, the symmetries which do not move space-time points around). In general, the IR 1 For most of the paper we will focus on the most familiar kind of emergent symmetry; what has retroactively been renamed "0form symmetry", i.e. symmetries that act everywhere in spacetime at once. Recently, the importance of so-called "higher-form symmetries" has started to be understood [23]. Loosely speaking the restriction to the ordinary 0-form symmetries means that there are no emergent "fractionalized excitations". We say a point-like excitation is "fractionalized" if it cannot be created by a point-like creation operator, but rather is created at the endpoints of an open string by an operator supported on the string. There are similar definitions for higher-dimensional excitations. Fractionalized excitations emerge, for instance, in phases where there is topological order and/or deconfined emergent gauge fields. For such systems, there will be emergent higherform symmetries that will play an important role, but in the interest of simplifying the exposition we will defer discussion of this point to later, and for the moment we assume that all the emergent symmetries are 0-form symmetries. We will eventually return to this point in Section X. theory will also have "trivial" emergent space-time symmetries such as continuous translation symmetry, which we do not include in G IR . It is important to note that the microscopic translation symmetry in general will not map into these trivial translation symmetries, but rather into the internal symmetry G IR . One way to think about this is that since there is a spatial rescaling transformation associated with passing to the IR theory, a microscopic translation symmetry in fact has trivial translation action in the IR limit. An alternative perspective is that we imagine that a microscopic translation symmetry acts like a product of an internal symmetry and a "trivial" translation symmetry, and we only worry about the internal part in defining the map ϕ.
The other property of the IR theory that will be pertinent is the extent to which the full emergent symmetry group G IR can be 'naturally' realized in some realization (not necessarily the original microscopic lattice model) of the IR theory. This property is formalized by the concept of the 't Hooft anomaly. Such an anomaly in d + 1 space-time dimensions is an obstruction to UVregularizing the theory on a lattice in d spatial dimensions with the full emergent symmetry group G IR realized as an "on-site" microscopic symmetry [24,25]. A powerful alternative but formal characterization of an 't Hooft anomaly [26,27] is that the conservation law corresponding to the G IR symmetry is broken upon coupling to a background gauge field for the symmetry G IR . We will consider to the 't Hooft anomaly of the emergent symmetry G IR to also be a 'kinematic' property of the IR theory.
't Hooft anomalies are also closely related to the theory of symmetry-protected topological (SPT) phases [28][29][30]. These are gapped phases of matter in a system with an unbroken global symmetry G with the following property: the ground state cannot be continuously deformed into a trivial product-state ground state while preserving the symmetry without closing the gap if the symmetry G is preserved, but can be if the symmetry G is lifted. Examples of SPT phases include the celebrated topological insulators [31] and the Haldane phase of the spin-1 antiferromagnetic chain in d = 1 [32][33][34].
The connection between 't Hooft anomalies and SPTs is that a G SPT phase in d + 1 spatial dimensions must have a non-trivial boundary theory, and the boundary theory carries a 't Hooft anomaly. We can say that the 't Hooft anomaly of the boundary theory is "canceled" by inflow from the bulk, in the sense that the conservation laws of the bulk+boundary system are preserved in the presence of background gauge fields. Thus, the classification of 't Hooft anomalies in d spatial dimensions is precisely equivalent to the classification of SPT phases in d + 1 spatial dimensions 2 . Such classification has been explored at great depth, from a variety of perspectives, ranging from physical considerations to very formal ones [24,25,29, The aspects of SPT phases that we will need to use in this paper, however, will be simple enough, at least if we want to understand filling in systems of low spatial dimension d ≤ 2, that a reader unfamiliar with this literature should still be able to follow our paper.
A. Example: Luttinger liquid in 1 spatial dimension Let us illustrate the above general considerations in the case of a system of spinless electrons in a lattice in one spatial dimension. Thus, the microscopic symmetry group G UV is comprised of a U(1) symmetry generated by the total electron numberQ, and a Z symmetry generated by the lattice translation operator T.
Let us now assume that the IR theory of the electrons is a Luttinger liquid. Thus, the low-energy physics takes place at the two Fermi points at momenta k L and k R . Excitations with momentum close to k L are left-movers, and those with momentum close to k R are right-movers. At low energies, the numbersN L andN R of left-and right-movers are separately conserved. Therefore, they generate the emergent symmetry group Next, we need to specify how the microscopic symmetry acts on the IR theory, which we can do by expressing the generators of the microscopic symmetry in terms of the generators of the emergent symmetry. Indeed, we haveQ where the tilde "∼" refers to an equivalent action on the IR theory. The emergent symmetry group G IR in a Luttinger liquid has the well known axial anomaly. This is an example of the 't Hooft anomaly mentioned in the previous subsection. The signature is that, if we turn on an electric field E for the U(1) symmetry generated byN L +N R [which is equivalent to a microscopic electric field, by Eq. (1)], then left-and right-moving charges are no longer separately conserved; instead, if we let j µ L , µ = 0, 1 be the current density for the left-moving charge, and similarly for j µ R , we have One can easily understand this equation in the case of a non-interacting Fermi gas. In that case, in the absence of electric field the electrons occupy single-particle states labeled by momentum k. But applying an electric field pointing to the right causes an overall flow of electrons in momentum space according tok = E. This causes a charge excess to accumulate at k R and a corresponding charge deficit at k L . As we mentioned above, there is always a bulkboundary correspondence that relates a 't Hooft anomaly for a symmetry G in d spatial dimensions to an SPT in d + 1 spatial dimensions whose boundary theory carries the 't Hooft anomaly. In this case, we have d = 1 and G = U(1)×U (1). The corresponding SPT phase in d = 2 is realized by a "quantum spin Hall" state corresponding to putting spin-up electrons in a quantum Hall state with quantized Hall conductance σ xy = 1 and spin-down electrons in a quantum Hall state with σ xy = −1. Here the U(1) × U(1) symmetries correspond to the separate conservation of spin-up and spin-down electrons. If we now consider a system with boundary, and apply an electric field parallel to the boundary, due to the Hall conductance this will generate a current of spin-up electrons incident onto the boundary, and a current of spin-down electrons with opposite sign. This precisely accounts for for the charge non-conservation in the boundary theory due to the 't Hooft anomaly.
Finally, let us remark that a useful way to think about SPT phases (and hence 't Hooft anomalies) is in terms of topological terms describing the response of the SPT phase to background gauge fields. For example, consider the quantum spin Hall state described above. We can theoretically couple to background gauge fields of the two U(1) symmetries, which in the one-dimensional Luttinger liquid were interpreted as the conservation of left-and right-movers; hence, we denote the gauge fields by A L and A R . The response of the quantum spin Hall state is then described by a Chern-Simons action on (2 + 1)-D spacetime:

III. FILLING CONSTRAINTS
Now we will turn to the question of how to understand constraints on the IR theory resulting from the microscopic filling, i.e. the average charge per unit cell, which is a real number ν. Only the fractional part of ν (i.e. ν mod 1), should be expected to be detectible in the IR theory, because an atomic insulator (whose IR theory is completely trivial) can have any integer filling. The fundamental observation we will make is that the microscopic filling is completely fixed by the kinematic properties of the IR theory, i.e. the emergent symmetry group and the 't Hooft anomaly, along with the mapping from the microscopic symmetry group into the emergent symmetry group. This is a rare example of a precise relation between a microscopic quantity (the filling) and properties of the IR theory. Such relations are extremely useful, given that it is usually very difficult to determine the IR theory from the microscopic Hamiltonian, either analytically or numerically.
The reason why such a UV-IR correspondence is possible in this case is because the UV and IR are linked through the homomorphism ϕ that implements the microscopic symmetry inside the emergent IR symmetry. Naively, we can imagine arguing as follows. The lattice filling is defined in systems that have at least a microscopic U(1) symmetry (charge conservation), and a lattice translation symmetry Z d (where d is the space dimension). So we set the microscopic symmetry G UV = U(1) × Z d . This G UV is embedded into the symmetries of the IR theory, and so we can talk about the 't Hooft anomaly of the IR theory thought of as a G UV -symmetric theory (which can be computed from the G IR 't Hooft anomaly in light of the homomorphism ϕ : G UV → G IR ). Then we invoke "UV-IR anomaly matching" to relate the filling ν to this G UV 't Hooft anomaly.
Unfortunately, this argument is not quite right. In fact, fractional filling cannot correspond to a nontrivial 't Hooft anomaly for G UV in the usual sense, because there are no candidate SPTs with G UV = U(1) × Z d symmetry which could constitute the "bulk", for which a system with fractional filling could be the boundary, as we explain in Appendix A. Indeed, a system with fractional filling seems perfectly well-defined on its own without any need for a higher-dimensional bulk.
Instead, we will give the correct version of the argument below, in various spatial dimensions. The careful reader might notice that they still appear reminiscent of "anomaly matching", albeit for a 't Hooft anomaly that is trivial with respect to the microscopic symmetries. How exactly one could make such a notion precise, we leave as an open question. Nevertheless, the arguments are self-contained and hold without any need for such an interpretation.

A. One spatial dimension
In one spatial dimension, we imagine putting the system on a circle and then very slowly threading 2π flux of the microscopic U(1) symmetry through the circle [3,7,60], By a standard argument, this transforms the ground state into a low-lying excited state with a different momentum; if we label states by their eigenvalues of the translation operator T, i.e. T|ψ = e −ip |ψ , then the momentum gets shifted by e −ip → e −i(p+2πν) .
Meanwhile, since the microscopic U(1) symmetry, generated byQ, corresponds to a U(1) symmetry of the IR theory, whose generator we callQ IR , we can also imagine performing the 2π flux insertion in the IR theory. Now consider the IR symmetry τ = ϕ(T) ∈ G IR corresponding to microscopic translation T. In the IR theory, the ground state can get transformed into a low-lying excited state with a different eigenvalue of τ ; that is, the eigenvalue of τ gets shifted according to exp(−iθ) → exp (−i[θ + α]) for some α [defined mod 2π]. One can argue that α depends only on the 't Hooft anomaly of G IR and on the choice of τ andQ IR ; see for instance the example below. We write α = α(Q IR |τ ) (the dependence on the 't Hooft anomaly is kept implicit). Now, the key point is that the processes described in the two paragraphs above are in fact the same process, just described in two different ways. Therefore, we must equate

Example: Luttinger liquid in one spatial dimension
We can apply the general framework described above to the particular example of a Luttinger liquid in one spatial dimension, as discussed in Section II A. In this caseQ IR and τ are defined by the right-hand sides of Eqs. (1) and (2).
Threading the flux ofQ IR generates an electric field ofQ IR by Faraday's law, and then, from the anomaly equation Eq. (3) we see that the 2π flux threading creates −1 charge of U(1) L and +1 charge of U(1) R . Therefore, in light of Eq. (2), the momentum transforms according to Then, from Eq. (5) we find that which is nothing other than Luttinger's theorem for a Luttinger liquid in one spatial dimension.

B. Two spatial dimensions
Here the idea is to consider a "2π flux" of the microscopic U(1) symmetry; that is, a very weak background magnetic field spread out over some very large region, such that the total flux is 2π. We can then consider how such a 2π flux transforms under translation symmetry. We claim that, in the presence of fractional filling ν, such 2π fluxes exhibit translational symmetry fractionalization; that is, acting on a 2π flux, the lattice x and y translations T x and T y obey in the limit as the 2π flux becomes infinitely spread out spatially) the magnetic algebra Heuristically, this is clear because a 2π flux sees a background charge density as an effective magnetic field. We give some more careful arguments in Appendix B. This result is also closely connected to the translational symmetry fractionalization of a monopole in an insulator in 3 spatial dimensions in the presence of polarization, as discussed in Ref. [61]. Meanwhile, in the IR theory we can consider 2π flux configurations of the IR symmetry generated byQ IR that corresponds to the microscopic U(1) symmetry. The homomorphism ϕ maps T x and T y into some elements τ x , τ y ∈ G IR . In the presence of a 't Hooft anomaly, such 2π fluxes indeed can carry a projective representation, i.e.
where V (g) denotes the action of a group element g ∈ G IR on the 2π flux, and α = α(Q IR |τ x , τ y ) [defined mod 2π] depends onQ IR , τ x and τ y , and on the 't Hooft anomaly of G IR . Therefore, we must identify The natural example to consider to illustrate this constraint will be a Fermi liquid. However, by contrast to the one-dimensional case, the 't Hooft anomaly of a Fermi liquid in two spatial dimensions has not previously been discussed. This is the subject of Section V.

C. General space dimension
The formulation of filling constraints discussed above can be generalized to arbitrary space dimension d. The idea is to generalize the functions α(Q IR |τ ) (in d = 1) and α(Q IR |τ x , τ y ) (in d = 2) to a function α(Q IR |τ 1 , · · · , τ d ) that determines the filling in general spatial dimension d. A convenient way to express this function is in terms of the "topological action" that describes the SPT phase in d + 1 spatial dimensions which cancels the G IR 't Hooft anomaly in d spatial dimensions by inflow on the boundary. We give the details in Appendix C.

IV. CONSEQUENCES OF THE FILLING CONSTRAINTS FOR COMPRESSIBLE STATES
The systematic theory of filling constraints described in the previous section has a very important corollary. We want to consider IR theories which can exist at generic filling; that is, they are not pinned to a particular filling but instead the filling can be continuously tuned. In other words, the IR theory represents a "compressible" state. What we will show is that in this case, for spatial dimensions ≥ 2, the emergent symmetry G IR cannot be a compact Lie group. [In one dimension, recall that the Luttinger liquid example discussed in Section II A achieves generic filling with only a compact Lie group emergent symmetry G IR = U(1) × U(1).] Note that compact Lie groups include finite groups as special cases, since we do not require the Lie group to be connected; also, in this paper when we refer to Lie groups, we will always assume they are finite-dimensional. Therefore, what we find is that the emergent symmetry for IR theories that exist at generic filling must be a very large one, larger than any compact Lie group 3 . Indeed, this is the case for Fermi liquids, as we describe in the next section.
Our main result is the following: Theorem 1. Suppose the emergent symmetry group of the IR theory is some compact Lie group G IR . Then for any spatial dimension d ≥ 2, the filling ν is constrained to be an integer multiple of 1/N GIR , for some finite integer N GIR that depends only on the group G IR and the dimension.
The proof is straightforward based on the framework of filling constraints discussed in the previous section. We give the details, and reveal what determines N GIR , in Appendix F, and also give a more formal point of view in Appendix G.
Let us mention a simple way to understand this result for d = 2. As mentioned in the introduction, the 't Hooft anomaly of a theory in d spatial dimensions implies that the theory can be realized as the boundary of an SPT phase protected by the same symmetry G IR in d+1-space dimensions. Now suppose that the SPT is such that n copies of it is trivial, with n a finite positive integer. For SPTs protected by a compact Lie group G IR in space dimension d + 1 = 3, this is known to be always true. For the d-dimensional boundary theory of interest, this means that for n copies, there is no 't Hooft anomaly. If now we consider n copies of the microscopic lattice system, we see that it has a total filling nν. Since at this filling the IR theory has no anomaly, it follows that νn = p with p an integer, which is essentially the claim of Theorem 1. In d = 3, this simple argument does not work because there are SPTs in d + 1 = 4 space dimensions such that there is no finite n for which n copies become trivial. (This is also why the argument does not work for d = 1). However, Theorem 1 still holds for any d ≥ 2 as we show in Appendices F and G.

V. THE KINEMATIC PROPERTIES OF FERMI LIQUIDS
In this section, we will return to a familiar IR theory: a Fermi liquid in two spatial dimensions (we will briefly discuss higher dimensions as well) and analyze its kinematic properties, in the language introduced previously. We will see how Fermi liquids, by virtue of having an emergent symmetry group that is "larger" than a compact Lie group, are able to evade the theorem of the previous section and exist at generic filling.

A. Emergent symmetry group
The first step is to identify the emergent symmetry group G IR . We invoke the following well-known property of Fermi liquids: non-forward scattering terms are irrelevant in the RG sense at low energies, so the quasiparticle number at each point on the Fermi surface is separately conserved at low energies. Thus, Fermi liquids have a very large emergent symmetry group [62]. Roughly, we can say that G IR = "U(1) ∞ ". However, let us be a bit more precise about how one approaches the "∞".
We parameterize the Fermi surface by a continuous parameter θ (we do not require that θ literally represents a geometrical angle), which is periodic, i.e. it lives on a circle. Imagine that we place an IR cutoff on the system (that is, place it in finite volume), which since the Fermi surface exists in momentum space, corresponds to a short-distance cutoff on θ, i.e. θ now takes discrete values. To each such θ value, we associate a U(1) emergent symmetry generated by an integer-valued operatorN θ . Hence, a general symmetry operator will take the form where we identify f θ ∼ f θ + 2π. Now we send the IR cutoff (the spatial volume) to infinity, which corresponds to sending the spacing between discrete θ values to zero. What we want is to consider symmetry operators that do not depend too sensitively on the precise way in which the short-distance cutoff in θ gets sent to zero. In order to achieve this, we require that, in this continuum limit, the f θ parameters become smooth functions f (θ). Therefore, in the limit, the emergent symmetries are in one-to-one correspondence with smooth functions from the circle into U(1). The group of all such functions is called the loop group [63] of U(1) and we denote it by LU(1). Hence we conclude that G IR = LU(1).
We emphasize that the group structure of LU(1) differs somewhat from naive conceptions of what "U(1) ∞ " would mean; in particular, LU(1) has only one U(1) subgroup, whose elements correspond to taking f (θ) to be a constant function. Physically, this is because only the total charge on the Fermi surface is quantized to be an integer; there is no well-defined concept of the (quantized) charge at a single point on the Fermi surface, only of the linear charge density with respect to θ. Accordingly, we can represent the loop group LU(1) formally by introducing a density operatorn(θ) such that the number of quasiparticles between θ and θ + dθ is measured byn(θ)dθ. Technically,n(θ) is not really an operator in itself, but an operator-valued distribution which should be integrated against a test function. The elements of the emergent symmetry group can be expressed as where f (θ) is any smooth function of θ. Note that, because we identify f (θ) ∼ f (θ)+2π, we are allowed to consider functions f with non-trivial winding number around the circle, such that 1 2π ∂ θ f (θ)dθ is any integer.

B. The homomorphism GUV → GIR
Now it should be clear how to embed the microscopic symmetries into the emergent symmetry group G IR = LU (1). Indeed, ifQ is the generator of the microscopic U(1) symmetry, we havê where the integer q is the charge of a Landau quasiparticle. Of course, for a Fermi liquid of electrons, q = 1, but in principle one can imagine Fermi liquid-like states where the quasiparticles carry a different charge. (For example, the quasiparticles could be bound states of an odd number of electrons). Meanwhile, if k x (θ) and k y (θ) represent the components of the lattice momentum of the point on the Fermi surface parameterized by θ, then we have where α = x, y, and T x , T y are the lattice translation operators.

C. The 't Hooft anomaly
Now we are in a position to discuss the 't Hooft anomaly for the emergent LU(1) symmetry. As usual, the 't Hooft anomaly can be understood by coupling to a background gauge field for the symmetry. But first, we must ask, what is a gauge field for an LU(1) symmetry? Since, roughly speaking, an LU(1) symmetry means there is a U(1) symmetry for each point on the circle, we can naively say that a gauge field for an LU(1) symmetry should be a space-time vector field A µ (θ) for each point θ on the circle, with gauge transformations parameterized by a scalar field λ(θ), and gauge transformation (where the derivative is respect to spacetime, not θ). Moreover, we require that A µ (θ) and λ(θ) be smooth functions of θ. However, there is in fact an additional ingredient that is required, that is unique to loop groups. To see this, we can appeal to the quasiparticle picture of the Fermi liquid. The spatial components A i (θ) describe the Aharanov-Bohm phase (which can be interpreted as a Berry phase) picked up as a spatially localized quasiparticle, localized near position θ on the Fermi surface, is transported in space. But we can also keep a quasiparticle fixed in real space, and transport it along the Fermi surface in momentum space. Therefore, the gauge field needs an additional component A θ to describe the Berry phase associated with such a process. We emphasize, though, that even going beyond Fermi liquids, A θ represents an intrinsic part of what it means to couple to a gauge field for an LU(1) symmetry, regardless of any quasiparticle picture. We will return to this point in Section VI.
The A θ component of the gauge field transforms under gauge transformations as A θ → A θ + ∂ θ λ. Therefore, if we now combine the vector field A µ with A θ , we obtain a vector field A on the D +1-dimensional manifold M ×S 1 , where M is the space-time manifold, D = dimM is the space-time dimension, and S 1 is the circle on which the θ variable lives. Moreover, A transforms under gauge transformations precisely as would a U(1) gauge field on M × S 1 , Hence we arrive at our conclusion: An LU(1) gauge field on M is equivalent to a U(1) gauge field on M × S 1 . From this point on, we will denote this combined vector field as A µ , taking the convention that indices such as µ vary both over space-time directions and the θ direction. Now we can discuss 't Hooft anomalies. One way to characterize an 't Hooft anomaly in two spatial dimensions is in terms of the topological term describing the response of the corresponding SPT in three spatial dimensions to background gauge fields. [As usual, the relation is that on a (3+1)-D spacetime M + whose boundary is a (2+1)-D manifold M , the SPT on M + gives rise to the 't Hooft anomaly on the space-time M by anomaly inflow.] But from the above discussion, a topological term for LU(1) gauge fields on a (3+1)-D spacetime M + is equivalent to a topological term for U(1) gauge fields on the (3+1+1)-D manifold M + × S 1 . Hence, we conclude that the appropriate topological term is the 5D Chern-Simons action where the level m is quantized to be an integer 4 . Below, we will show that such a topological term, with m = ±1, indeed reproduces many known properties of spinless Fermi liquids. (Note that the sign of m is only defined relative to a choice of orientation for the Fermi surface, because redefining θ → −θ sends m → −m). 4 To see the quantization, observe that the total charge carried by the SPT, under the U(1) subgroup of LU(1) corresponding to setting the function f (θ) in Eq. (12) to be independent of θ, is given by the integral (over a fixed time-slice) where C[A] is the second Chern number and is quantized to be an integer. Hence, m must be an integer in order for the total charge of the SPT to always be an integer.
From the topological term Eq. (16) we can determine the anomaly equation on the boundary by computing the current j = δS δA and then considering the current incident onto the boundary. We find that, on the boundary, the continuity equation is violated according to Here the current j depends both on space-time coordinates and on θ. Its space-time components describe the space-time current of the charge at position θ on the Fermi surface, i.e. of the symmetry generated byn(θ). However, recalling that indices are supposed to vary over the θ direction as well as space-time directions, we have to introduce the component j θ = δS δA θ , which describes flow of charge along the Fermi surface. An example of a case where j θ = 0 is a Fermi liquid in a magnetic field, described in the next subsection.
Finally, let us mention that there is an alternative picture to understand inflow of the 't Hooft anomaly that is sometimes helpful. Instead of considering a system with LU(1) symmetry in a (3+1)-D space-time M + with boundary, we can consider a system with L T 2 U(1) symmetry in a (2+1)-D space-time M without boundary. Here L T 2 U(1) is the group of smooth maps from the Brillouin zone (thought of as a torus T 2 ) into U(1). In other words, we imagine that the charge is conserved not just at each point on the Fermi surface, but also at every kpoint in the whole Brillouin zone. This is the case, for example, in a non-interacting Fermi gas. In an interacting Fermi liquid, the physical Hamiltonian presumably has nonzero scattering rate for quasiparticles in the interior of the Fermi surface, but we can, theoretically, imagine an extension of the quasiparticle Hamiltonian from the vicinity of the Fermi surface to its interior in a manner that preserves conservation of quasiparticle number at every k-point. This is a familiar construction in Fermi liquid theory. (The point is that the physics on the Fermi surface will ultimately not depend on the precise form of the Hamiltonian in the interior.) Then by similar arguments to before, a gauge field for the L T 2 U(1) symmetry is equivalent to a U(1) gauge field on M × T 2 . Since this is also a 5-dimensional manifold, we can write a similar Chern-Simons term to Eq. (16). Specifically, we write where D ⊆ T 2 is the volume "occupied by electrons", that is, the volume enclosed by the Fermi surface 5 . This gives rise to the same anomaly equation Eq. (17) on the boundary ∂(M × D) = M × ∂D, where ∂D is the Fermi surface. Note that in this interpretation, the components A 0 , A x , A y of A µ are the usual ones while the components A kx , A ky are k-space gauge fields. Then the 5 dimensional manifold has the interpretation of being "phase space" (that is, the space where points are labeled by position and momentum) plus time, and A µ can be considered a gauge field in phase space.

D. Filling constraint and Luttinger's theorem
We consider a theory in two spatial dimensions whose 't Hooft anomaly is canceled by inflow from Eq. (16). According to the discussion of Section III B, we first need to determine how the emergent symmetry LU(1) gets represented projectively in such a theory in the presence of a 2π flux (of the microscopic charge U(1) symmetry). This is something that can be derived from the 't Hooft anomaly that was characterized in the previous subsection. We show in Appendix D that this leads to a projective representation on a 2π flux described by the commutation relations wheren(θ) are the operators introduced in Section V A, and δ is the derivative of the Dirac delta function. Note that Eq. (19) is precisely the so-called Kac-Moody algebra satisfied by the local density operator of a chiral fermion in one spatial dimension [such as appears, for example, at the boundary of a integer quantum Hall state in two spatial dimensions]. Indeed, this relationship is not a coincidence; one can see roughly how it comes about in the case of a Fermi liquid at the level of the semiclassical theory of electron transport. Suppose, for simplicity, that we switch off the interactions so that the Fermi liquid becomes a non-interacting Fermi gas. In the presence of a spatially-dependent magnetic field B(x), the semiclassical equations of motion take the form [64] where Ω(k) is the Berry curvature of the Bloch states, v(k) = ∂E(k) ∂k , E(k) is the dispersion relation, and we have omitted terms beyond linear order in the magnetic field strength. In particular, Eq. (20) implies that electrons develop a circulation in momentum space along contours of constant energy, in particular along the Fermi surface. In two spatial dimensions the Fermi surface is onedimensional and the circulation along the Fermi surface is unidirectional (set by the sign of the magnetic field), which indeed resembles a chiral fermion.
This intuitive argument, however, does not fix the coefficient of the right-hand side of Eq. (19). In Appendix E we give a more careful derivation of Eq. (19) from the semiclassical theory of electron transport, confirming that m = ±1 for a spinless Fermi liquid.
From Eq. (19) we can compute the projective representation of the translation symmetry in light of the embedding Eq. (14). We find where V F is the volume in momentum space enclosed by the Fermi surface. (Here we have chosen a particular convention to define the orientation of the Fermi surface). Hence, from Eq. (10), we conclude that which (if we set m = q = 1) is precisely Luttinger's theorem for a spinless Fermi liquid in two spatial dimensions. We wish to emphasize, however, that, in general, Luttinger's theorem Eq. (24) follows directly from Eq. (19), which in turn follows directly from the 't Hooft anomaly. It was not necessary to assume anything about the dynamical properties of the Fermi liquid, e.g. the existence of quasiparticles. Thus, Luttinger's theorem also holds (with a possible integer multiplicative factor mq) for any IR theory that has the same emergent symmetry as a Fermi liquid.
It is interesting to reconsider Eq. (19) and its relation with the 5D Chern-Simons term from the viewpoint that the 5-dimensional manifold can be thought of as 4dimensional phase space together with the time direction (see the last two paragraphs of Section V C). In the presence of a static 2π-strength magnetic flux in the x, y components of A µ (with the corresponding components A x and A y independent of k x ,k y ,t), the 5D Chern-Simons term Eq. (18) reduces to a 3D Chern-Simons term for the remaining components A I = (A 0 , A kx , A ky ), assuming that they are independent of x and y: where k x and k y are integrated over the volume D, i.e. over the interior of the Fermi surface. Now the claim is that on the Fermi surface, there is a chiral mode carrying the Kac-Moody algebra In particular since a 2π flux of the microscopic U(1) corresponds to a 2πq flux of A by Eq. (13), we recover Eq. (19).
In the phase space interpretation, Eq. (25) describes an integer quantum Hall effect in momentum space in the interior of the Fermi surface. In other words we think of the rigid interior of the Fermi surface as hosting an integer quantum Hall state in momentum space when we apply a 2π flux in real space. The Fermi surface is the boundary in momentum space of the interior, and hence hosts a chiral edge state.

E. Extension to higher dimensions
The description of the anomaly extends straightforwardly to higher dimensional Fermi liquids. Indeed, if M is the (d + 1)-D space-time manifold, and F is a (d − 1)dimensional manifold parameterizing the Fermi surface, which is a codimension-1 surface in the d-dimensional Brillouin zone, then the emergent symmetry group is L F U(1), i.e. the group of smooth maps from F to U(1). Including the components of the Berry connection on the Fermi surface promotes the L F U(1) gauge field to a U(1) gauge field on the (2d)-dimensional manifold M × F .
Then we can write down a topological term on a (d+2)-D spacetime M + describing the SPT whose inflow generates the anomaly of the Fermi liquid, given by the (2d + 1)-dimensional Chern-Simons action:

VI. ERSATZ FERMI LIQUIDS AND THEIR PHENOMENOLOGY
As we showed previously (Section IV), any IR theory that can exist at generic filling must have a very large symmetry group, "larger" than any compact Lie group. In the simplest cases, this emergent symmetry group will be the same as that of a Fermi liquid, i.e. G IR = LU(1). We refer to an IR theory with this emergent symmetry as an ersatz Fermi liquid (EFL). We note that EFLs represent a class of theories; there could be many different EFLs with different dynamical properties. Here we will examine the properties that all EFLs have in common. In particular, as a Fermi liquid is an example of an EFL, this section will provide a fresh perspective on many aspects of Fermi liquid phenomenology, showing that they arise directly from the emergent symmetry and its 't Hooft anomaly without any need to invoke the detailed dynamical properties of a Fermi liquid.
Going beyond Fermi liquids, an important example of states which we expect will be EFLs are associated with quantum critical points in metals which are not tied to a particular electron density. As a concrete example consider a putative quantum critical point associated with the onset of Ising nematic order 6 in a metal in d = 2. There is no particular electron density at which this transition will happen in any given system. Indeed, as microscopic parameters are changed continuously, we expect that the electron density at the transition will also change continuously without a change of universality class. The universal critical properties of this transition are described by a theory of electrons near the Fermi surface coupled to the fluctuating order parameter modes. (A model with similar structure also describes insulating quantum spin liquid phases with a spinon Fermi surface coupled to a dynamical U (1) gauge field). In these models the resulting IR fixed point is not a Fermi liquid [65][66][67][68][69][70][71][72][73][74][75][76]. Nevertheless, from our point of view, these metallic quantum critical points are expected to be EFLs. (We will discuss neutral Fermi surfaces in insulators separately in Sec. VIII C.) We may understand why such critical points should be EFLs as follows. In previous papers [68,[71][72][73] the ultimate IR fixed point was accessed through a 'patch construction' which begins by by dividing the Fermi surface into small patches. It was then argued that the important coupling of the fermions within a single patch is to boson fluctuations with momentum tangential to the local Fermi surface. This enables treating the full system by focusing on a pair of antipodal patches and ignoring their coupling to other such pairs of patches. The patch width is taken to zero at the end. In this scheme, the number of fermions within each patch is conserved. The assumption that this patch description captures the IR fixed point then implies that the linear charge densityn(θ) at each point of the Fermi surface is conserved at the fixed point. (There is, potentially, a dangerous inter-patch BCS coupling in the pairing channel that could destroy these conservation laws. For the Fermi surface coupled to a gauge field, a weak pairing interaction is irrelevant at the IR fixed point [77] while in the Ising nematic quantum critical point it is relevant. In the latter case the non-Fermi liquid metallic fixed point is preempted by the superconducting instability. Our discussion then applies at a scale above this instability. ) We also expect that more complex quantum critical points associated with the death of a Fermi surface can be subsumed under the umbrella of EFLs. Such critical points have been argued [78] to possess a critical Fermi surface even in the absence of Landau quasiparticles. As we will see below, the EFL description, if it indeed applies to such quantum critical points, enables inferring many of their general properties.

A. General properties of EFLs
Let us return to a general EFL. Then, translation symmetry must embed into G IR somehow: that is, we have for some U(1)-valued functions k α (θ). The values of k α (θ) can be interpreted as momenta in the Brillouin zone, so this defines a codimension-1 surface in the Brillouin zone. We can take this to be the definition of a Fermi surface in a general EFL. Of course, this "Fermi surface" may or may not have any signature in, say, angle-resolved photoemission spectroscopy (ARPES) measurements, and if it does the signature might be different in character from that of a Fermi liquid. Nevertheless, the fact thatn(θ) is a conserved operator for every θ tells us that, at any point θ on the Fermi surface, there are infinitely long-lived excitations which are forbidden from scattering away from that point. To see this, first observe that sincen(θ) is conserved, excitations can be labeled by their corresponding eigenvalue n(θ). An excitation at point θ on the Fermi surface is characterized by for some N . One can show that N is quantized to an integer, because by the definition of LU(1), the operator N = n(θ)dθ (30) must have integer eigenvalues, since exp(−2πiN ) is the identity operator. Henceforth, we will refer to such excitations as Fermi surface quanta. From Eq. (28) and Eq. (29), we see that a single Fermi surface quantum carries momentum k(θ). Fermi surface quanta are the generalization of Landau quasiparticles to a general ersatz Fermi liquid. Their dynamics, however, could be very different from Landau quasiparticles. Moreover, apart from the fact that their number is quantized, the Fermi surface quanta need not have particularly "quasiparticlelike" properties. Nevertheless, the Berry's phase of a spatially localized Fermi surface quantum as it is moved in space or over the Fermi surface is still a well-defined quantity, which supplies the general interpretation of the gauge field A µ (including the A θ component) discussed in Section V C. Next, we can consider how the microscopic U(1) symmetry embeds into LU(1). The quantization of charge, i.e. the requirement that e i(2π)Q = 1 (whereQ is the microscopic charge operator), constrains the embedding to be of the formQ for some integer q. We can interpret q as the charge of a single Fermi surface quantum. Finally, we can consider the 't Hooft anomaly of the LU(1) symmetry. There is not much freedom, since as we saw in Section V the 't Hooft anomalies are just classified by the level m of the 5D Chern Simons term Eq. (16) which is quantized to be an integer. In summary, once we have fixed the shape of the Fermi surface, the kinematic properties of an ersatz Fermi liquid (in the sense of Section II) are captured by the integers q and m. (Note that there is a freedom that sends q → −q and m → −m simultaneously by redefining the generators of LU(1); moreover, as we mentioned earlier, choosing the reverse orientation of the Fermi surface sends m → −m while leaving q fixed).
Next we observe the arguments of Section V D can be applied in any EFL. Therefore, we immediately conclude that the Fermi surface in any EFL satisfies Luttinger's theorem in the form Eq. (24).
In the remainder of this section we will consider various aspects of Fermi liquid phenomenology and argue that they hold equally well in any EFL.

B. Response to electric fields
A property of Fermi liquids is that if a uniform electric field is applied, the system responds in essentially the same way as it would in a non-interacting Fermi gas, which is to say that the momenta of quasiparticles get shifted according to k → k + Et. This causes the density operatorsn(θ) to be no longer conserved in the presence of the electric field. Specializing for simplicity to a Fermi liquid in two spatial dimensions, and assuming that electric field E is applied along the x direction, the total charge on a segment [θ, θ + dθ] of the Fermi surface gets shifted according tô where dk y (θ) = k y (θ + dθ) − k y (θ), and L x and L y are the linear dimensions of the system (normalized by a x and a y respectively, the unit cell dimensions) in the x and y direction. We have to divide by the denominator in the second term in Eq. (32) to take into account the density of single-particle states in momentum space in a finite-size system. We will now show that Eq. (32) indeed follows from the 't Hooft anomaly. In order to do that, we will want to reformulate Eq. (32) in a way that removes the explicit dependence on L y . We write Eq. (32) as Next we identify the difference with the density shift rate associated with applying a uniform electric field in the x direction in the presence of a "flux of y translation" symmetry around the y direction.
To compute this shift from the 't Hooft anomaly, we start with the anomaly equation Eq. (17) and consider a configuration where The motivation for our choice of A y is that it ensures that the LU(1) flux around the y direction is given by which coincides with the expression Eq. (14) for the translation symmetry operator T y . Substituting into Eq. (17) and integrating over x and y gives where J θ = j θ dxdy, and we can identify j t dxdy with the expectation value n(θ) = n(θ) . Hence, provided that ∂ θ J θ = 0, we recover Eq. (35) if we set m = 1.
Our assumption that ∂ θ J θ = 0, even in the presence of a background electric field, requires a bit more explanation. J θ represents a charge circulation along the Fermi surface. As we mentioned in Section V D, in a Fermi liquid a magnetic field induces a chiral flow of quasiparticles along the Fermi surface, in which case J θ ∝ Bn(θ) and generally ∂ θ J θ = 0. So we need to address why the same thing could not happen with an electric field. Of course, in a Fermi liquid, one can easily convince oneself that it does not, but we want an argument that holds more generally in any EFL.
In a quantized formulation of the IR theory, the current operatorĴ θ is defined bŷ whereĤ is the Hamiltonian of the IR theory. Then we know that, in the absence of an electromagnetic field,Ĵ θ must satisfy (as an operator identity) the conservation law On the other hand, in the absence of an external electromagnetic field, by assumptionn(θ) should be conserved for each θ, since it is the generator of the emergent symmetry. Therefore, in the absence of electromagnetic field, the current operator identically satisfies Now observe that because the electric field E enters into the Hamiltonian "temporally", i.e. through the timederivative of A, it is not possible forĴ θ defined by Eq. (42) to depend on E because the Hamiltonian is defined on a single time slice of space-time. Therefore,Ĵ θ continues to satisfy Eq. (44) even with an applied electric field. On the other hand, it is possible forĴ θ to depend on the applied magnetic field, as the Fermi liquid example demonstrates.

C. Quantum oscillations
Recall that Fermi liquids display "quantum oscillations" when a weak magnetic field B is applied; that is, various physical properties are periodic in 1/B [79]. In the case that the physical property under consideration is resistivity, for example, this is known as the Shubnikovde Haas effect. (What we mean by "weak magnetic field" is that the magnetic flux per unit cell should be much less than 1. To observe the oscillations at finite temperature T , it is also necessary that the cyclotron energy E c should satisfy E c T , which for T > 0 also places a lower bound on the magnetic fields for which the oscillations are observable).
It is sometimes stated that observing quantum oscillations is evidence for a quasiparticle description, i.e. of a Fermi liquid. Here, however, we will show that there are very general reasons to expect any EFL to display the same periodicity of the quantum oscillations. We will not, however, make any statement about the amplitude of the quantum oscillations, and it remains possible that this amplitude and its dependence on parameters such as temperature will still allow for Fermi liquids and non-Fermi liquids to be distinguished.
Our task is complicated by the fact that the quantum oscillations are non-perturbative in the magnetic field, and therefore, it is not clear that one expects them to be directly describable in terms of the deep IR theory. Instead, we will find it necessary to make appeal to a UV completion.

Two spatial dimensions
Let us first consider the case of a system in two spatial dimensions which microscopically has continuous translation symmetry, with a microscopic charge density ρ. In such a case, in the presence of a magnetic field the x and y translation generators P x and P y fail to commute, but we can define a "magnetic unit cell" of volume b x × b y , such that the flux per magnetic unit cell is 2π and the discrete translations T x = exp(−ib x P x ) and T y = exp(−ib y P y ) do commute. Then we can define the magnetic filling ν M = (b x b y )ρ = 2πρ/B, which is the charge per magnetic unit cell. Then we can apply all the usual results on filling with respect to this discrete translation symmetry; in particular, if ν M is not an integer then the Lieb-Schultz-Mattis-Oshikawa-Hastings theorem [1][2][3][4] forbids the system from having a trivial (i.e. not topologically ordered) gapped ground state, whereas such a state is permitted for integer ν M . Moreover, if we assume that the ground state at a given ν M is itself an EFL with a single Fermi surface, then Luttinger's theorem for EFLs implies that at integer ν M the Fermi volume must fill all of the Brillouin zone corresponding to the magnetic unit cell, i.e. the Fermi surface becomes degenerate, presumably leading to an instability. Most generally, the point is that since ν M mod 1 always reflects properties of the IR theory according to our general framework of filling constraints, the nature of the IR theory must vary with ν M mod 1.
These considerations motivate our assumption that, in general, observable properties of the ground state vary periodically with ν M , with period 1 (The periodicity refers to the behavior for ν M 1, and where ν M varies over an interval ∆ν M ν M . On longer scales there will be some envelope function governing the amplitude of the oscillations.) This behavior is well known in the Fermi liquid case, where integer ν M (non-integer ν M ) corresponds to fully filled (partially filled) Landau levels respectively. Our periodicity assumption immediately implies that, if we keep ρ fixed and vary B, then the periodicity with respect to 1/B is Next we can consider a system in two spatial dimensions that microscopically only has discrete translation symmetry. Here we recall that, even for Fermi liquids, there is a condition on the Fermi surface in order to observe quantum oscillations. For a Fermi surface as depicted in Figure 1, no quantum oscillations will be observed; physically one can think of this as coming from the fact that the semiclassical orbits of this Fermi surface in a magnetic field are not closed in position space. The general condition for a Fermi liquid to display quantum oscillations is that each connected component of the Fermi surface should have trivial winding number on the Brillouin torus.
Let us therefore examine what we can say about a general EFL that satisfies this condition. To avoid some subtleties in the argument, we will assume that q = 1. The key point is that when the Fermi surface does not have nontrivial winding number, then there is a consistent way to define the Fermi momentum k(θ) without any mod 2π ambiguity. Then we can define the filling that is relevant for quantum oscillations as where V F is the volume enclosed by the Fermi surface, and this formula holds without any mod 1 equivalence. Note that ν o is not necessarily the same as the microscopic filling ν, although by Luttinger's theorem it must differ from it by an integer. Moreover, we can define emergent symmetry generatorŝ with a x , a y the dimensions of the unit cell, such that exp(−ia α P α ) ∼ T α . This leads us to believe (although we will not give a rigorous proof) that, even though the actual microscopic theory may have had only a discrete translation symmetry, there exists a UV completion of the IR theory with continuous translation symmetry whose action on the IR theory is generated byP α . Assuming that this is the case, invoking the continuous translations version of Luttinger's theorem shows that the density of U(1) charge in this UV completion must be ρ = ν o /(a x a y ). Then with ρ so defined we again find that the periodicity of quantum oscillations is given by Eq. (45).

Two spatial dimensions: multiple Fermi surfaces
As a warm-up to going to three spatial dimensions, we will extend the results of the previous section to the case where we have several disconnected components of the Fermi surface, instead of just one. For simplicity we will consider the case of two components, although the arguments can easily be generalized. In that case, the emergent symmetry is L F U(1), i.e. the group of smooth maps from F into U(1), where F = S 1 S 1 , i.e. the disjoint union of two circles. We can represent this by writing the generators of the emergent symmetry asn (λ) (θ), where θ ∈ S 1 and λ ∈ {1, 2} labels the two components of the Fermi surface. Similarly we write the momentum of the Fermi surface as k (λ) (θ). We assume that each component satisfies the condition of no winding described earlier, which allows us to define the momentum operator The crucial point is that the fact of two connected components of the Fermi surface allows us to define two different U(1) emergent symmetries, generated byQ (1) and Q (2) , withQ In the same spirit as the argument of the previous section, therefore, we will postulate that there is a UV completion with both continuous translation symmetry andQ (1) and Q (2) realized microscopically. Defining ρ (1) and ρ (2) to be the corresponding charge densities, by similar arguments as earlier we have a generalized Luttinger's theorem . This implies that if we vary B while keeping ρ (λ) fixed, and assuming that the ratio ρ (1) /ρ (2) is an irrational number, then O varies quasiperiodically in 1/B with base frequencies 2πρ (1) and 2πρ (2) .

Three spatial dimensions
Now we can consider a 3D system. By dimensional reduction, we can consider a 3D system as a 2D system; but the Fermi surface of the 2D system will consist of infinitely many components corresponding to taking slices through the 3D Fermi surface with fixed k z . By generalizing the previous discussion to N components, and then taking the limit as N → ∞, we find that any observable property should be expressible as where F is some functional of ν ⊥ , and ν ⊥ is a function into R/Z (i.e. the real line with ν ∼ ν + 1 identified), defined by where V ⊥ (k z ) is the two-dimensional area enclosed by the intersection of the Fermi surface with a plane of fixed k z . We show in Appendix H that this implies, so long as the functional F is sufficiently regular, that the dependence of O on B at small B is dominated by the extremal Fermi surface cross-sections, and we obtain that O varies periodically or quasiperiodically with 1/B, with base frequencies 2πρ ⊥ (k * z ), k * z ∈ Σ, where Σ is the set of solutions to d dkz V ⊥ (k z ) = 0. This is the same result as for a Fermi liquid.

D. Anomalous Hall effect
The anomalous Hall effect refers to a Hall conductance that exists in a time reversal broken system in zero external magnetic field. In general there can be many different effects that contribute to this effect. For a clean free electron system with a Fermi surface, there is an interesting contribution that arises due to the net Berry curvature Φ B of the filled Fermi sea. In d = 2, we have This contribution can alternately be re-expressed as a Fermi surface property in terms of the Berry gauge connection: Note that the integrand (as opposed to the integral) on the right hand side -while only involving the Fermi surface -is not by itself invariant under k-space gauge transformations. Going beyond non-interacting systems, for an interacting Fermi liquid it has been argued [80] that there will be additional Fermi surface contributions to the anomalous Hall conductance that, by contrast to the Berry phase contribution above, involve an integral over gauge invariant quantities defined in local patches of the Fermi surface. Next we show that the 't Hooft anomaly of an EFL directly implies the Berry phase contribution to the anomalous Hall effect. To that end we start from Eq. (18) and consider a configuration where the gauge field A has a flux Φ B through the interior of the Fermi surface in the (k x , k y ) plane, and take the corresponding gauge field components (A kx , A ky ) to be independent of (t, x, y). Similar to the discussion surrounding Eq. (25), we now find that the 5D Chern-Simons term reduces (assuming that A 0 , A x , A y are independent of k x and k y ) to: for the I, J, K = (0, x, y). This then directly corresponds to a contribution to the Hall response σ xy = Φ B 2π . We can write this directly in terms of the boundary theory at the Fermi surface as Φ B = dθA θ . In the free fermion case this is exactly the Berry phase contribution to the anomalous Hall effect discussed above. In a general EFL, we should regard this as an unavoidable contribution due to the 't Hooft anomaly of the IR theory. Like in the interacting Fermi liquid, the full measured anomalous Hall effect may include other contributions that are 'local' on the Fermi surface. Note that Eq. (57) has the structure of a 3D Chern-Simons but with a coefficient that is not quantized to be an integer multiples of 1 4π . This is allowed here because the boundary gauge fields are coupled to the gapless modes associated with the Fermi surface. The full boundary action that includes both the Fermi surface modes and the unquantized Chern-Simons action of Eq. (57) will be properly gauge invariant (including under large gauge transformations).

E. Chiral magnetic effect
The chiral magnetic effect refers to a phenomenon in a three-dimensional Fermi liquid where if the Chern number of the Berry curvature on the two-dimensional Fermi surface is nonzero (which, in a non-interacting Fermi gas, would occur when the Fermi surface encloses a Weyl point), then the total charge becomes nonconserved in the presence of both an electric field and magnetic field, with E · B = 0 [81]. Recall that that the Berry gauge field on the Fermi surface is still defined in a general EFL. Here we will show that when this Berry gauge field has non-trivial Chern number in a 3D EFL, then the system exhibits the chiral magnetic effect.
We start from the anomaly equation that is the analog for an EFL in three spatial dimensions of Eq. (17) (which was stated for an EFL in two spatial dimensions); for convenience we write it in an index-free form as: where d is the exterior derivative on forms, " * " is the Hodge star operator, and J is the current expressed as a 1-form, and F is the 2-form gauge curvature (which we can write locally as F = dA, although A may not be globally defined). Note that all these forms live on a 6dimensional space, M × F, where M is the 4-dimensional space-time manifold and F is the two-dimensional manifold that parameterizes the Fermi surface. Now suppose that we define the total charge current j EM , which is a 1-form on M , by integrating j over the Fermi surface F. Suppose furthermore that we write F = π * F (F EM ) + π * M (F Berry ), where F EM are 2-forms on M and F respectively, and π * F and π * M are the pull-back operators associated with the projections π F : F × M → F and π M : F × M respectively. In index notation, this would just be saying that F is the sum of two terms, each of which only depends on, and only has components in, M and F respectively. Then by integrating Eq. (58) we find that where is the Chern number of the Berry curvature. We can also write Eq. (59) as indicating that the charge is not conserved when the right-hand side is nonzero.

VII. EXTENSION TO SPINFUL SYSTEMS
Throughout this paper, we have assumed that the only microscopic internal symmetry is U(1), the charge conservation symmetry. Physically, one often wants to consider systems which also have an SU(2) spin rotation symmetry. In that case, the full internal symmetry group is U(2), which acts by matrix multiplication on the vector (ψ † ↑ , ψ † ↓ ) of microscopic spin-up and spin-down electron creation operators. Here we will briefly discuss how the general considerations above get extended in that case. (One is still allowed to use the results of previous sections by simply ignoring the spin rotation symmetry, but taking into account the additional symmetry will lead to stronger constraints.) For simplicity we will focus on the case of d = 2 spatial dimensions.
Firstly we observe that U(2) has a subgroup U(1) ↑ × U(1) ↓ ≤ U(2) corresponding to the diagonal unitary matrices. We write the generators of U(1) ↑ and U(1) ↓ aŝ Q ↑ andQ ↓ ; they measure the total number of up-spin and down-spin electrons respectively. The total charge is the sumQ =Q ↓ +Q ↑ . They both have corresponding fillings ν ↑ and ν ↓ in the ground state; however, U(2) invariance of the ground state immediately implies that ν ↑ = ν ↓ := ν. The total charge density is then ρ = 2ν. Now, if we consider a spinful version of a Fermi liquid, the quasiparticle charge at each Fermi surface point will be conserved, as will the total quasiparticle spin. Note in particular that the quasiparticle spin at each Fermi surface point is not separately conserved due to the presence of Landau interactions in the spin channel. Thus, for a spinful Fermi liquid, the emergent symmetry group G IR is a quotient of LU(1) × U(2), where the quotient corresponds to identifying two U(1) subgroups: the U(1) subgroup of U(2) corresponding to unitary matrices e iθ I 2 , where I 2 is the (2 × 2) identify matrix; and the U(1) subgroup of LU(1) generated byN = n(θ)dθ, wherê n(θ) are the generators of LU(1) as in previous sections. Therefore, we define a spinful EFL to be a system which has the same emergent symmetry group G IR . We assume that the translation symmetry continues to embed into LU(1) according to Eq. (14), which defines the Fermi surface of the EFL. Moreover, the mapping of the microscopic U(2) into G IR is induced from a homomorphism ϕ : U(2) → U(2) IR , where the target U(2) IR is the corresponding subgroup of G IR . One can show (for example, by considering the induced homomorphism on the Lie algebras), that such maps are characterized by an odd 7 integer q, which again we interpret as the charge of a 7 The restriction to odd integers did not come up in Section VI, but we would have found the same restriction if we assumed that microscopically the fermion parity is (−1)Q whereQ is the total charge (as is the case in an electronic system), and moreover required (as is the case is all known examples) that an elementary Fermi surface quantum be fermionic. In the spinful case, the restriction to odd q comes directly from assumed form of G IR , Fermi surface quantum (which in a Fermi liquid would be a Landau quasiparticle). Then, we have, in particular that the microscopic chargeQ corresponds in the IR theory toQ whereN ↑ andN ↓ are the generators of the U(1) ↑,IR × U(2) ↓,IR ≤ U(2) IR symmetry corresponding to the diagonal matrices in U(2) IR . Now we can discuss the form that Luttinger's theorem must take in a spinful EFL. We imagine inserting a 2π flux of U(1) ↑,IR or U(1) ↓,IR ; this leads to a projective representation of LU(1) described by Eq. (26) with some integer anomaly coefficient m ↑ or m ↓ respectively. U(2) invariance again implies that m ↑ = m ↓ := m. Therefore, repeating the argument for Luttinger's theorem, and taking into account Eq. (62), we find where V F is the volume enclosed by the Fermi surface. With respect to the total charge density ρ = 2ν, on the other hand, we have The extra factor of two takes into account the two possible spin values, and agrees (setting m = 1) with the usual result for spinful Fermi liquids. (Thus, if we had defined m with respect to the total charge as we did in Section VI, we would have found m = 2 for a spinful Fermi liquid/EFL). Our discussion goes through with little modification for a Kondo lattice in which itinerant electrons couple to a local spin-1/2 moment in each unit cell. In this case, we can define a U(1) ↑ symmetry generated byQ tot ↑ := S z + N/2 +Q ↑ , whereŜ z is the total spin component of the local moments,Q ↑ is defined as before with respect to the itinerant electrons, and N is the total number of unit cells. Assuming no spin ordering of the local moments, we have that S z = 0 and hence the total filling ofQ tot ↑ is ν tot ↑ = 1/2 + ν ↑ , where ν ↑ is the contribution from the itinerant electrons. If we now apply similar arguments to before, we find that Luttinger's theorem for a putative IR Fermi liquid requires a "large" Fermi surface that counts the local moments as part of the Fermi sea, as previously discussed with the flux threading argument in Ref. [7].
For an alternative point of view, we can think of the Kondo lattice model as being obtained from an Anderson lattice model with a correlated half-filled band of electrons coupled to a separate weakly correlated partially filled band. Then, by definition, the charge ρ per although in principle one could get even q if we replaced U(2) with SO(3) × U(1) in the definition of G IR . unit cell includes all the microscopic electrons, in particular the contribution from the correlated band. If the IR theory is a spinful Fermi liquid (or spinful EFL), then Luttinger's theorem in the form Eq. (64) shows that there Fermi surface must be "large" as in the previous paragraph. Since our discussion (as well as that of Ref. 7) of Luttinger's theorem does not rely on perturbation theory in the interaction strength, it will hold even in the Kondo limit of the Anderson model.
Note that the arguments of this section assume that all the relevant symmetries -charge conservation, spin rotation and translation symmetry -are preserved and not spontaneously broken, and that the IR theory is a spinful Fermi liquid or at least a spinful EFL with the same emergent symmetries. If any of these conditions are violated then the appropriate statement of Luttinger's theorem will be modified and in such cases, a "small" Fermi surface could be permitted in the Kondo lattice model. The case of 'Fractionalized Fermi Liquids' which have a small Fermi surface while preserving all the symmetries requires separate discussion but can be easily understood within the framework of this paper. We illustrate this for simple class of such phases in Section X.

VIII. FERMI SURFACES COUPLED TO DYNAMICAL GAUGE FIELDS
In this section, we will consider cases where the IR theory comprises a Fermi surface coupled to a dynamical gauge field. As we will see, these do not constitute EFLs in the strict sense of Section VI, but they are still closely related and it still turns out to be helpful to think of the behavior of these systems in terms of their "kinematic" properties.
We will consider three main examples. The first two examples relate to a system of interacting electrons in two spatial dimensions with continuous translational symmetry in a magnetic field B, such that the magnetic filling ν = 2πρ/B (where ρ is the electron density) is equal to ν = 1/2. Such a system of electrons is believed to form a "composite Fermi liquid" with a Fermi surface of "composite fermions" which are distinct from the microscopic electrons. There are two competing proposals for the IR theory of the system, which we consider separately: the one proposed by Halperin, Lee and Read (HLR) [67]; and the one proposed by Son [82]. In the HLR and Son theories, the system is not quite a conventional Fermi liquid, even when expressed in terms of the composite fermions, because in the IR theory the composite fermions still couple to a dynamical gauge field.
The final example relates to a Mott insulator in two spatial dimensions with discrete translation symmetry and no magnetic field. In certain circumstances, it is believed that the IR theory of such a system can have a "spinon Fermi surface" consisting of emergent fermions coupled to a dynamical gauge field.

A. HLR theory of composite Fermi liquid
We begin by discussing the microscopic symmetries of electrons in 2d in a uniform magnetic field. It is well known that in the presence of such a magnetic field, the translation operatorsP x andP y do not commute. Rather they satisfy [P x ,Q] = 0, [P y ,Q] = 0, whereQ is the operator for the total electric charge (generator of the global U (1) symmetry). This symmetry algebra will have to be matched by any putative IR theory. In an infinite system, we think of these operatorsQ,P x ,P y as the generators of the symmetry action on local observables; hence, there is an ambiguity in their definition in that one is free to shift any of them by a constant without changing the action on local observables. Thus, one is free to fix them to have zero eigenvalue in the ground state.
Let us now consider the HLR theory of the composite Fermi liquid ground state at ν = 1 2 . Our starting point is the Lagrangian for the HLR theory, which can be written as [67,83] where a and b are dynamical U(1) gauge fields, and A is the background gauge field for the microscopic U(1) symmetry. Here χ is a fermion field carrying unit gauge charge of a, and L[χ, a] is the Lagrangian describing a Fermi surface of these fermions coupled to the gauge field a. This is the corrected version (see e.g., Ref. [83]) of the Lagrangian initially written by HLR [67], which suffered from an improperly quantized Chern-Simons term.
Varying with respect to the temporal components a 0 and b 0 gives the constraintŝ wheren χ is the density of the χ fermions, andB a and B b are the magnetic fields of the a and b gauge fields respectively. Meanwhile, varying with respect to A 0 allows us to identify the microscopic charge densityρ aŝ We define the excess charge according tô where n χ is the charge density in the ground state. We also define the effective magnetic flux that the χ fermions experience (they only couple directly to a) according tô Using 2n χ = B and the constraints above, it follows that Now we need to consider what form the IR symmetry takes. Let us first consider what happens if we treat the gauge fields a and b at the mean-field level. In that case, the χ fermions do not experience a magnetic field in the ground state and therefore form a Fermi liquid for which the charge at each point of the Fermi surface is conserved. Thus, the emergent symmetry group is still something resembling LU(1) with generatorsn(θ). It is not exactly LU(1), however, because if we consider low-energy excitations that carry a net chargeN = n(θ)dθ = 0, wherê n(θ) are the generators of the emergent symmetry group, then from Eq. (74) we see that the χ fermions feel a net magnetic flux. From the discussion of Section V D we expect the magnetic flux to induce a projective representation of LU(1) 8 . Hence, the emergent symmetry group is LU(1), where LU(1) is obtained from LU(1) by replacing the commutation relation [n(θ),n(θ )] = 0 obeyed by the generatorsn(θ) of LU(1) with: From Eqs. (69) and (71), it is clear that the microscopic chargeQ embeds into the IR symmetry group according toQ ∼N . Meanwhile, the continuous translation symmetry, with generatorsP x andP y , should embed according toP which defines the Fermi surface k(θ). We will not attempt to extend the general discussion of topological terms as in Section V C to this more nontrivial symmetry group LU(1). However, we can at least argue for a version of Luttinger's theorem for the composite Fermi liquid. Unlike in the case of no magnetic field, it will turn out not to be necessary to invoke the 't Hooft anomaly of the IR theory; the result actually follows directly from the structure of the emergent symmetry group.
From the action of translations on the low-energy theory, we find that 8 One might ask why the magnetic flux does not also induce a chiral flow of quasiparticles along the Fermi surface, violating conservation ofn(θ). The point is that in that for a low-energy excitation, the variation of the local charge density, and therefore of the flux of a, should be spread over an arbitarily large volume, and hence the effective local magnetic field, which determines the velocity of the chiral flow, actually goes to zero. where is the volume enclosed by the Fermi surface. We emphasize that hereN is not just a number but an operator which acts non-trivially on the states of the IR theory.
On the other hand, microscopically we know that we must have the magnetic translation algebra of Eqn. 65 with which we must impose compatibility of Eq. (77). This gives which is Luttinger's theorem for the composite Fermi liquid. Now we should discuss to what extent we expect these considerations to survive once we include gauge fluctuations. The only property of the IR theory that we needed to derive Luttinger's theorem was that the emergent symmetry group is LU(1). The question is whether this emergent symmetry survives the inclusion of gauge fluctuations. Let us argue that, at any rate, the emergent symmetry group cannot be a compact Lie group, which suggests that the full LU(1) symmetry is preserved. We cannot apply the results of Section IV to show this, because if we define the magnetic unit cell such that the discrete translations with respect to this unit cell commute, then the filling with respect to this unit cell is supposed to be ν = 1/2 for the composite Fermi liquid, a rational number.
However, we can instead argue as follows. Suppose that the emergent symmetry group G IR is a compact Lie group. Then letp x ,p y ,q be the elements of the Lie algebra of G IR corresponding to the microscopic translation symmetriesP x andP y and the microscopic charge conservation symmetryQ. Then since the microscopic symmetries obey the algebra of Eqs. (65)(66)(67). it follows that the same algebra must be satisfied byp x ,p y ,q. Eqs. (65)(66)(67) constitute the Heisenberg algebra and cannot be implemented inside of a compact Lie group unlessq is identically zero. To see this, suppose that G IR is a compact Lie group. Then it admits a finite-dimensional faithful unitary representation. Sincep x ,p y andq generate a subgroup of G IR , this induces a finite-dimensional representation of the Heisenberg algebra Eqs. (65)(66)(67). Now we invoke the famous fact that the Heisenberg algebra does not admit any finite-dimensional representations in whichq acts non-trivially 9 .
Therefore, we conclude that, if G IR is a compact Lie group then the charge conservation symmetry acts trivially on the IR theory; in other words, the system has a charge gap and is an electrical insulator. This would be inconsistent with the physics expected of a composite Fermi liquid. Thus, we expect that the emergent symmetry group remains LU(1) in the presence of gauge fluctuations. Of course, in principle it could go to some totally different group that is larger than any compact Lie group, but this does not seem very plausible, and moreover, as we we discussed in Section VI, the charge is still conserved at every point in the Fermi surface in the conventional description of the fixed point for a Fermi surface coupled to dynamical gauge field. Thus, assuming that the emergent symmetry group remains LU(1), we find that the composite Fermi liquid indeed obeys Luttinger's theorem, contrary to a recent suggestion [84].

B. Son theory of composite Fermi liquid
We start from the Lagrangian [82,83] where a and b are dynamical U(1) gauge fields, and the first term represents the Lagrangian for a massless relativistic Dirac fermion field χ coupled to the gauge field a 10 . The "· · · " represents higher-order terms. This is the corrected version [83] of the Lagrangian originally written by Son [82].
Proceeding similarly to Ref. [83], from this Lagrangian one obtains the constraintŝ whereN is the charge of the χ fermions,Φ a andΦ b are the total magnetic fluxes of the a and b fields respectively, Q is the total microscopic charge, and all ofN ,Q,Φ a ,Φ b represent the excess values compared to the ground state. From these equations we find thatN = 0 andΦ a = 2Φ b . At the mean-field level the χ fermions form a Fermi liquid, so the charge at each point on the Fermi surface is conserved. However, we have to impose the constraint thatN = n(θ)dθ = 0. Assuming for the moment that the fermions feel no magnetic flux, this would imply that the emergent symmetry group is ΩU(1) := LU(1)/U(1). 10 We follow the regularization convention adopted in Ref. [83], where the action for the massless Dirac fermion implicitly includes the so-called "η invariant". Physically this amounts to assuming that the system also contains an additional massive Dirac fermion in addition to the massless one explicitly represented in the Lagrangian. Now if we take into account that the fermions feel an effective magnetic flux given byΦ a = 2Φ b , we must have the commutation relation Since then(θ)'s are conserved, it must also be the case that the magnetic fluxΦ b is conserved and (since it is quantized in units of 2π) generates a U(1) flux symmetry. Hence, the full emergent symmetry group G IR is a central extension of ΩU(1) by U(1) flux . We note that this is a different group compared to the HLR case, which reflects a physical difference between the Son and HLR theories. Eq. (83) tells us how the microscopic U(1) symmetry embeds into G IR , and as before the continuous translation should embed according to Eq. (76), which defines the Fermi surface. By similar arguments to the HLR case one again finds Luttinger's theorem Eq. (79) for the composite Fermi liquid. Since, again, the only property of the IR theory that we used was the nature of the emergent symmetry group, we expect that this result continues to hold in the presence of gauge fluctuations.
Note that our argument for Luttinger's theorem in the Son theory is very different from the one given by Son [82], who noted that the density of composite fermions is given by ρ CF = B/(4π), and then invoked Luttinger's theorem for a Fermi liquid to relate this to the volume of the Fermi surface. Of course, the composite Fermi liquid is not a Fermi liquid beyond mean-field theory, but the arguments of this paper have shown that Luttinger's theorem is in fact far more general. However, from this point of view it remains unclear whether one ought to have expected a Luttinger's theorem to hold with respect to ρ CF , since the composite fermions do not carry charge under any global U(1) symmetry. The argument we gave above does not suffer from these difficulties.

C. Spinon Fermi surface
Finally, let us consider a system in 2 spatial dimensions where the IR theory consists of a spinful Fermi surface coupled to a dynamical U(1) gauge field. This should apply, for example, to Mott insulators exhibiting a "spinon Fermi surface".
If we treat the dynamical gauge field at the mean-field level, then the fermions form a spinful Fermi liquid described by the considerations of Section VII. Thus, we have conserved Fermi surface densityn(θ). In particular, if we use the observation of that section that a spinful Fermi surface has anomaly coefficient m = 2 with respect to the total charge U(1), we conclude that then(θ)'s must satisfy whereΦ is the operator that measures the magnetic flux of the dynamical gauge field. Moreover, we have to im-pose that the total gauge charge is zero, so n(θ)dθ = 0. We see that the emergent symmetry group G IR takes the form G IR = G IR,charge × SO (3), where the charge part G IR,charge is identical to the emergent symmetry group of the Son theory as described above [although for an electrical insulator the microscopic charge U(1) will act trivially on the IR theory instead of embedding into the emergent U(1) flux ], while the SO(3) part accounts for to the spin-rotation symmetry of the spinons 11 . Unlike in the quantum Hall systems discussed above, for a Mott insulator one microscopically has only a discrete (commuting) translation symmetry. The discrete translations must embed into G IR in the usual way: which defines the Fermi surface. From Eq. (85), the requirement that T x and T y commute give a non-trivial constraint: where V F is the volume enclosed by the Fermi surface.
We can obtain more information by considering the 't Hooft anomaly of the IR theory and invoking the filling constraints of Section III B. We assume that the system microscopically has spin-rotation symmetry, so that the full microscopic internal symmetry group is U(2), and in particular there is a U(1) ↑ × U(1) ↓ subgroup as discussed in Section VII, and we can define the corresponding fillings ν ↑ and ν ↓ which satisfy ν ↑ = ν ↓ := ν and ρ = 2ν, where ρ is the filling of the total charge. For an insulating state without symmetry fractionalization (see Section X), the microscopic charge U(1) acts trivially on the IR theory and hence from the general theory of filling constraints we find that ρ must be an integer. On the other hand, for a band insulator, ρ must be an even integer (taking into account the two spin components). Since a Mott insulator by definition is an insulator that exists at fillings where band insulators are impossible, we conclude that ρ for a Mott insulator must be an odd integer and hence ν = 1/2 (mod 1).
From the general discussion of Section III B, we now see that if we apply a flux Φ ↑ of U(1) ↑ and flux Φ ↓ of U(1) ↓ , it must be the case that translations act projectively on it, Now we need to see how this comes about in the IR theory. We return to Eq. (85) but take into account that, at the mean-field level where we can treat the dynamical gauge field as a background, the fermions now feels an effective U(1) ↑ flux ofΦ a + Φ ↑ and an effective U(1) ↓ flux ofΦ a + Φ ↓ . Hence, Eq. (85) gets generalized to 12 which indeed gives Eq. (88) provided that which is Luttinger's theorem for a spinon Fermi surface As usual, we must consider to what extent we expect these results to hold beyond mean-field theory. The result Eq. (87) only depends on the structure of G IR , so we must ask whether G IR remains the same upon including gauge fluctuations. Since we are assuming the system is a Mott insulator, with integer filling of the microscopic U(1) charge, we cannot use the results of Section IV. Still, as we mentioned in the introduction to Section VI, the charge on each patch of the Fermi surface remains conserved in the usual approach to Fermi surfaces coupled to a dynamical gauge field, suggesting that G IR is indeed robust to gauge fluctuations and that Eq. (87) is satisfied generally. To get the stronger result Eq. (90), we also have to assume that the 't Hooft anomaly of the IR theory, captured by Eq. (89), remains unchanged by gauge fluctuations. However, the classification of 't Hooft anomalies is discrete -more concretely, there is no way to continuously deform Eq. (89) without leading to inconsistencies -so it seems unlikely that the 't Hooft anomaly would be affected by gauge fluctuations. Moreover from Eq. (87) the only possibility other than Eq. (90) would be V F /(2π) 2 = 0 (mod 1), in which case the Fermi surface is singular and presumably unstable.

A. Fermi arcs
In the celebrated pseudogap normal phase of the cuprate high temperature superconducting materials, 12 We remark that if we set Φ ↑ = Φ ↓ = 2π, then the effect on Eq. (89) can be absorbed by redefiningΦa →Φa − 2π. SinceΦa always has such an ambiguity in terms of its action on a background flux (such ambiguities are the reason why fluxes are able to carry projective representations in the first place), we conclude that Φ ↑ = Φ ↓ = 2π does not carry a non-trivial projective representation. This corresponds to the statement that the projective representations of G IR,charge are Z 2 classified, whereas the projective representations of LU(1) (see Section V D) were Z classified. These results are consistent with the fact that the total microscopic charge U(1) acts trivially on the IR theory since it is an electrical insulator. ARPES measurements observe "Fermi arcs" rather than a closed Fermi surface (see, eg, Ref. [85] for a review). One possible explanation is that these materials still have a closed Fermi surface, but for some reason parts of the Fermi surface are not easily visible to the ARPES probe. On the other hand, 3-dimensional Weyl semimetals exhibit Fermi arcs at their 2-dimensional boundaries [86], but these can only exist at the boundary of a gapless 3dimensional bulk. It is an important fundamental question to ask whether a system with true Fermi arcs (not related to the above mechanisms) can ever exist.
Specifically, in this section we will ask whether it is possible to extend the framework of EFLs to substitute a closed Fermi surface with a Fermi arc. We will find that there is an obstruction to doing so. Therefore, in light of the results of Section IV, the only remaining route to Fermi arcs, assuming that they can exist in systems at generic filling, would be to find some other way to have an emergent symmetry group larger than any compact Lie group, different from what occurs in EFLs.
Let us consider a theory in d spatial dimensions described by a suitable generalization of the EFL class discussed in Section VI. Specifically, we imagine that the Fermi surface is a parameterized by a (d−1)-dimensional manifold F ; but, instead of requiring that F be a closed manifold as before, we allow it to be a manifold with boundary. Then we assume that the emergent symmetry group is L F U(1), the space of smooth maps from F to U(1). We can discuss 't Hooft anomalies as before, and again we find that the 't Hooft anomaly should be described by inflow from a Chern-Simons term on a higher-dimensional manifold, for example for d = 2: where M + is an extension of the d + 1-dimensional spacetime manifold to a d + 2-dimensional manifold. Now, however, we immediately encounter a problem if F has a boundary ∂F , because the Chern-Simons term is not gauge-invariant -that is, the system has a 't Hooft anomaly -in the presence of boundaries (unless m = 0). At the boundary of M + , the gauge variation is canceled by the 't Hooft anomaly of the EFL, but since the interior of M + is supposed to correspond to a gapped SPT bulk, there is no way to cancel the anomaly at M + × ∂F . Should we be worried about this anomaly? In general, an uncanceled 't Hooft anomaly signals a violation of charge conservation. We can see this concretely in spatial dimension d = 2 if we assume a particular model of a Fermi arc which has Fermi liquid-like quasiparticles. Then in the presence of a magnetic field, the quasiparticles have a chiral flow along the Fermi surface as discussed in Section V D; but when they reach the end of the Fermi arc they have no choice but to simply disappear. This violates conservation of microscopic charge (if the microscopic charge q of a quasiparticle is nonzero) and conservation of momentum (even when q = 0). Beyond this simple model, the 't Hooft anomaly indicates the same issue with lack of charge/momentum conservation, and therefore we must impose that, if F has a boundary, then m = 0. However, by the discussion of Section V D this would imply the filling ν = 0 mod 1, so it cannot correspond to a state that exists at generic filling.
Of course, this is only a proof of the impossibility of Fermi arcs if we assume that the system is described by the EFL framework. We cannot make more general statements, except to recall from Section IV that any system that can exist at generic filling yet is not an EFL needs to find some other way of having an emergent symmetry group that is larger than any compact Lie group.

B. More general constraints
Returning to EFLs, we can can also obtain some more general constraints than just the impossibility of Fermi arcs. Indeed, the most general statement will be that the Fermi surface must "enclose" a volume in a suitable generalized sense. This obviously fails for Fermi arcs, but it also rules out Fermi surfaces which are closed but nevertheless fail to enclose a volume in the Brillouin zone, e.g. see Figure 2(c).
Let us imagine generally that we construct the Fermi surface out of a collection of "patches" which might be glued together along their boundaries and might not all have the same values of the anomaly coefficient m. Then the requirement that the Chern-Simons term be gaugeinvariant places a constraint on how patches are allowed to be glued together. For example, in two spatial dimensions the rule is that the sum of the anomaly coefficients for all the segments of Fermi surface (taking into account the orientation of the segment) intersecting at a given point must be zero. This is the most general sense in which the Fermi surface must be "closed". Thus, the configuration shown in Figure 2(f) is allowed, but the one shown in Figure 2(b) is not.
In fact, we can make an even stronger statement: configurations such as the one shown in Figure 2(c), even though the Fermi surface is closed, are also disallowed (except possibly in the case where the Fermi surface quanta are uncharged, i.e. q = 0). In this case, the Chern-Simons term is gauge-invariant, but there is still an issue with the implementation of translation symmetry. Recall that for d = 2 the translation symmetry T x and microscopic charge operatorQ can be expressed in terms of the LU(1) generatorsn(θ) as In light of the commutation relations Eq. (19), when acting on a 2π flux we have FIG. 2. Allowed () and disallowed () Fermi surface configurations in a two-dimensional Brillouin zone. Each Fermi surface segment has an associated anomaly coefficient m, shown by the number next to the segment. An arbitary choice of orientation of the segment (denoted with arrows) determines the convention for the sign of m. For allowed configurations, it is always possible to define an integer function n(k) -depicted here by the different shading colors of different regions -such that the boundary between two regions carries the anomaly coefficient determined by the difference of n(k) between the two regions. In a Fermi gas, n(k) represents the number of occupied bands at the point k. where is a winding number of the Fermi surface on the Brillouin torus. However, microscopically the translation operator should commute with the charge, so we must impose that mqW x = 0. Another way to see this (which generalizes more easily to higher dimensions) would be impose conservation of microscopic charge on the electric field response discussed in Section VI B. From such considerations, we can obtain the most general constraint on the Fermi surface. We find that it must be the case that each point k in the Brillouin zone can be assigned an integer n(k), such that the boundary between two regions with different n(k) carries a Fermi surface with anomaly coefficient m given by the difference ∆n. In a Fermi gas, n(k) has a natural interpretation: it describes the number of bands that are occupied at the point k. What we have found is that such an n(k) can always be defined in any consistent EFL. Note that in general the Fermi surface only determines n(k) up to addition of a k-independent integer. Repeating the analysis of Section V D in this more general setting, we can also now formulate the most general version of Luttinger's theorem: where the integral is over the whole Brillouin zone. Finally, let us note that a compact mathematical way to state the above results is if we think of the Fermi surface, together with associated anomaly coefficients m of the different patches, as defining a chain (in the homology theory sense) ω ∈ C d−1 (T d , Z), where T d represents the Brillouin zone. Then the requirement that the Chern-Simons term be gauge invariant implies that this chain is closed, i.e. it has trivial boundary, ∂ω = 0; whereas, from the winding number arguments, we find that this chain must be exact, ω = ∂κ for some κ.

X. THE ROLE OF FRACTIONALIZATION
The analysis of Sections III and IV is not sufficient to describe systems which admit "fractionalized" excitations, i.e. localized excitations which cannot be created locally. To see this, consider a specifc and familiar example, namely a system of bosons on a lattice in two spatial dimensions with microscopic filling ν = 1/2. A symmetry preserving ground state that is allowed is one where the IR theory is equivalent to a Z 2 gauge theory [87][88][89]. Let us first recall the physical properties of this state and how it is allowed at filling ν = 1/2 filling. The key property is that the ground state is topologically ordered and admits fractionalized excitations, which we can label as 1, e, m, f , where e is the Z 2 gauge charge, m is the Z 2 gauge flux, and f is the composite of e and m. Moreover, these excitations can exhibit "symmetry fractionalization" of the microscopic symmetries, which roughly means that they carry fractional quantum numbers. Specifically, in the ν = 1/2 Mott insulator, one of the excitation types (say e) carries half-quantized charge under the microscopic U(1) symmetry. This is allowed because these excitations are fractionalized and can only be created in pairs. Moreover, the m particle also experiences translational symmetry fractionalization [89][90][91], which is to say that T x T y T −1 x T −1 y = −1 when acting on a single m excitation. Again, this does not contradict the fact that translations commute microscopically, because the global number of m particles is always even.
Physically, one can think of the translational symmetry fractionalization as saying that the ground state has an e particle in each unit cell; the −1 phase factors then comes from the −1 braiding phase for m going around e. But then since the e particle has 1/2 charge under the microscopic U(1), these e particles contribute a filling ν = 1/2. It is also instructive to ask how this state is compatible with the momentum balance constraints from flux threading introduced by Oshikawa [7]. The point is that adiabatic threading of a 2π flux through one cycle of a torus shifts the ground state from one topological sector to another which is at a different total crystal momentum [5]. This change of the ground state momentum is then able to match what is expected from flux threading.
How should we think about the emergent IR symmetry of this state? Deep in the IR, at scales below the gap to all quasiparticle excitations, we can describe it as a purely topological theory. Naively the only emergent symmetry of the resulting theory might seem to be G IR = Z 2 , generated by the "electromagnetic duality" which exchanges a Z 2 gauge charge with a gauge flux; however, for the theory under consideration, neither translations nor the microscopic charge U(1) map into this duality symmetry. Therefore, following the analysis of Section III we would immediately conclude that ν = 0 mod 1. How then, are we to make sense, in the context of our general framework, of the fact that such a IR theory actually can exist at ν = 1/2?
We have to think about what went wrong in the argument of Section III B. The assumption we made there is that a 2π flux of the microscopic symmetry generated by the microscopic chargeQ will correspond in the IR theory to a 2π flux ofQ IR , the generator of the corresponding IR symmetry. In any system theory with a charge gap, we will have thatQ IR is identically zero, so we would conclude that such an object would be completely trivial, leading to a trivial filling. However, in the Z 2 gauge theory example, a 2π flux of the microscopic symmetry is evidently a non-trivial object since acting on it we have T x T y T −1 x T −1 y = −1. In fact, as a defect of the IR theory it is equivalent to an m particle. In general, an interpretation of non-trivial symmetry fractionalization is precisely the statement that certain flux configurations of the microscopic symmetry look like topologically non-trivial excitations of the IR theory. Therefore, it will always necessary to take into account symmetry fractionalization when computing the filling.
Let us first recall a few cases from previous works where symmetry fractionalization of the microscopic symmetry affects the filling. The first case generalizes the deconfined Z 2 gauge theory for bosons at ν = 1/2 (as discussed above) to situations where the IR theory is an arbitrary gapped (2+1)-D gapped topological phase. The computation of ν has been discussed in Refs. [10,11]. The basic idea is that the translational symmetry fractionalization is described by a class in H 2 (Z × Z, A) ∼ = A, where A is the group of Abelian anyons. Roughly, we can imag-ine that there is a background a particle sitting in each unit cell. Meanwhile, the symmetry fractionalization of U(1) is described by an Abelian anyon b, which we can think of as the anyonic charge carried by a 2π flux. Because of the background of a particles, if b particles have non-trivial braiding phase S ab with a particles, then they carry a projective representation of translation symmetry. Thus, we find that the filling is given by A nice way to interpret this equation is that since a 2π flux has non-trivial braiding S ab with an a particle, then the latter must carry fractional charge q a = 1 2πi log S ab (mod 1) under the U(1) symmetry. Therefore, since we have an a particle in each unit cell, the average charge per unit cell is ν = q a . Building on the previous case, we can imagine a scenario where the IR theory is equivalent to stacking a gapless theory C gapless without any topologically non-trivial excitations and a gapped topological theory C gapped . In that case, the microscopic filling ν is simply the sum of the contributions from each component: where ν gapless is computed according to Section III and ν gapped is computed according to the previous paragraph.
An example of such a case is the FL* Fermi liquid discussed in [8,9]. Another case to consider is a superfluid state in d = 2 spatial dimensions (i.e. 3 space-time dimensions) where the microscopic U(1) symmetry is spontaneously broken. In the IR, there is a topologically non-trivial excitation, namely the vortex, and a 2π flux of the microscopic U(1) induces such a vortex on the IR theory. Since the vortex sees an effective "magnetic field" corresponding to the charge density, there is a projective action of translations on the vortex, which determines the microscopic filling ν [92,93].
If we want to construct a more general theory beyond these examples, we might start to become a bit more uncertain about what exactly we mean in general by a "topologically non-trivial excitation". To clarify our thinking on these issues, it is helpful to introduce a more abstract formalism. To motivate this formalism, note that in the Z 2 gauge theory example, G IR = Z 2 is not in fact the only emergent symmetry of the IR theory. Indeed, gapped topological phases in two spatial dimensions, such as Z 2 gauge theory, have emergent 1-form symmetries [23]. This is a general feature of topologically ordered phases; the topologically non-trivial excitations can be characterized by the fact that the string operators that create them are charged under the 1-form symmetry. (In general, operators that are charged are an n-form symmetry will be supported on n-dimensional regions, hence strings for a 1-form symmetry). Going beyond gapped topological phases, we will define topologically non-trivial excitations to be those whose n-dimensional creation operator is charged under an emergent n-form symmetry.
In fact, once one has introduced the emergent higherform symmetries, one can phrase symmetry fractionalization entirely in terms of these symmetries without needing to talk about excitations at all. In general, a d-dimensional system can have emergent 0-form, 1-form, ..., up to (d − 1)-form symmetries. These structures symmetries combine into a structure called a "d-group" [94][95][96][97], which we denote G IR . Meanwhile, the microscopic symmetries can also be thought of as a forming a d-group G UV , although normally we will consider the case where the microscopic symmetries are all 0-form and there are no non-trivial higher-form symmetries. Then the correct way to describe the action of the microscopic symmetries on the IR theory is through an n-group homomorphism One might think if the microscopic symmetries are all 0-form, then we do not need to worry about the higherform part of G IR , but it turns out that even in this case the homomorphism ρ contains not just the data of how the microscopic symmetries map into emergent 0-form symmetries, but also the data of the symmetry fractionalization of the microscopic symmetries, which can be interpreted as relating to the interplay between the microscopic symmetries and the emergent higher-form symmetries [97][98][99][100].
Armed with this point of view, we can now consider taking a Z n gauge theory such that the gauge charge carries fractional charge 1/n under the microscopic U(1) symmetry. Then we can imagine forming a Fermi liquidlike state where the Landau quasiparticles are composites of the gauge charge and a neutral microscopic fermion. This theory retains a Z n emergent 1-form symmetry generated by the string lines of the gauge charges (the dual emergent 1-form symmetry of the original gapped theory generated by the string operators of the gauge fluxes gets explicitly broken once we put in finite density of itinerant gauge charges). This makes clear that the only "topologically non-trivial excitations" we need to consider are the Z n gauge fluxes. The symmetry fractionalization of the microscopic U(1) symmetry is then classified by H 2 (U(1), Z n ) ∼ = Z n , which corresponds to the Z n gauge flux carried by a 2π flux of the microscopic U(1), so that if a 2π flux carries Z n gauge flux k, then by braiding we find the gauge charge carries fractional charge with fractional part k/n. Meanwhile, the symmetry fractionalization of the Z × Z symmetry would correspond to the gauge flux carried per unit cell, but we will assume that this is trivial. If we now consider the symmetry U(1) (n) generated by nQ, whereQ is the generator of the microscopic charge U(1), we conclude that the symmetry fractionalization of U(1) (n) must be trivial since it is n times that of U(1), and the symmetry fractionalization is Z n classified. Therefore, the overall symmetry fractionalization of the U(1) (n) × Z d symmetry is trivial, and we can compute the filling ν (n) with respect to U(1) (n) in the way described in the previous sections. In particular, for the Fermi liquid-like state considered here, the quasiparticles have charge 1 under U(1) (n) , and hence we have Luttinger's theorem On the other hand, we have ν (n) = nν, and so we conclude that which we can think of as Luttinger's theorem for a Fermi liquid of fractionally charged partons. Finally, it is necessary for us to address the question of whether symmetry fractionalization can allow for the spirit of Theorem 1 to bypassed; that is, is it possible for a system whose emergent symmetry is described by a compact Lie d-group to exist at generic filling? We conjecture that the answer is no, if we do not allow the microscopic U(1) symmetry or translation symmetry to be spontaneously broken. Firstly, we expect that in order to have generic filling, at the least the system needs to have an emergent continuous (d − 1)-form symmetry (e.g. a 1-form symmetry in d = 2 or a 2-form symmetry in d = 3). The usual way to get emergent continuous higher form symmetries would be to have an emergent deconfined gauge field. However, in d = 3, for example (we return to the more subtle d = 2 case below), we know that a deconfined U(1) gauge theory has two 1-form symmetries (electric and magnetic) but no 2-form symmetry. This corresponds to the fact that the "topologically non-trivial excitations" of such a theory are the electric charges and magnetic monopoles, both point particles. We can contrast this with gapped topological phases in d = 3, which typically have both a 1-form symmetry and 2-form symmetry, corresponding to line-like and pointlike topological excitations respectively.
On the other hand, a theory that does have a continuous (d − 1)-form symmetry in d spatial dimensions is a superfluid in which the microscopic U(1) symmetry is spontaneously broken. We saw above in the d = 2 case (without using the higher-form symmetry point of view) that such a theory indeed can have generic filling. This holds also in general dimension, and the (d−1)-form symmetry is generated by the winding number of the order parameter. Another way to achieve generic filling is by spontaneously breaking the translation symmetry with an incommensurate wavevector. The IR theory is then similar to the superfluid example above, with the gapless degrees of freedom now identified as phasons; in this case there is also an emergent continuous (d − 1)-form symmetry.
We also recall that a deconfined U(1) gauge theory in d = 2 spatial dimensions is dual to a superfluid, and that the only way for such a theory to be stable (other than by coupling to charged gapless fermions, which would explicitly break the 1-form symmetry) is if the monopole events (in the superfluid language, charge creation events) are forbidden by one of the microscopic symmetries, in which case this symmetry will be spontaneously broken and the theory is dual to one of those discussed in the previous paragraph.
Therefore it appears, at least in these well-known mechanisms for obtaining emergent higher-form symmetries, that there is no way to obtain generic filling without spontaneously breaking either the microscopic U(1) or translation symmetry. Whether there is some less familiar theory in d spatial dimensions with an emergent (d − 1)-form symmetry that can achieve this, we leave as an open question. In this work, we have argued that Luttinger's theorem, or a suitable generalization, in fact should hold in great generality, both for Fermi liquids and at least a certain class of non-Fermi liquids. Therefore, we must contend with past works that have reported a "violation of Luttinger's theorem" in various settings.
In certain cases [101][102][103], attempts to have to been made to formulate Luttinger's theorem in a way that could apply beyond perturbation theory in the interaction strength by expressing it in terms of the volume enclosed both by traditional Landau Fermi surfaces on which the electron Green's function G(k, ω) has a pole at ω = 0 and by "Luttinger surfaces" in which ReG(k, ω) changes sign via a zero of G(k, 0) instead of a pole. However, it has been shown that no such result holds in general [104,105]. In our work, we have defined a Fermi surface in a general system by its emergent symmetries, rather than by any particular properties of the electron Green's function. Our version of Luttinger's theorem is therefore very different from the one disproven by Refs. [104,105].
A potential violation of Luttinger's theorem has been discussed in the context of holographic models [106][107][108][109]. For a system that is holographically dual to a gravitational theory that includes a black hole, the charge hidden behind the event horizon does not contribute to the volume enclosed by the Fermi surface. It is not clear, however, whether such a scenario can ever actually occur at zero temperature, since an event horizon would be associated with a nonzero entropy density, seemingly in tension with the Third Law of Thermodynamics. If somehow the event horizon could be stabilized at zero temperature, however, one could envision it being associated with some exotic emergent symmetry group that is larger than any compact Lie group but different from that of a Fermi liquid, in which case a violation of Luttinger's theorem would be compatible with the considerations of this paper.
Next we discuss the numerical study of the t−J model in Ref. [110]. By an analytic continuation of a hightemperature series expansion, Ref. [110] computed the properties of the t−J model down to around temperature T ∼ 0.2J and found that, for several different definitions of the "Fermi surface" at finite temperature, the volume inside the Fermi surface appears to be be plateauing at a value less than the value predicted by Luttinger's theorem. However, our arguments in favor of Luttinger's theorem only applied at zero temperature. Since the ground state of the t−J model is unknown and could well exhibit exotic physics such as high-temperature superconductivity, the significance of these finite-temperature observations remains unclear, especially given the ambiguity of the definition of the Fermi surface at finite temperature and the fact that the results of Ref. [110] also contain hints of Fermi arcs (see Section IX A).
Finally, we discuss Ref. [111], which studied an SU(N ) generalization of the t − J model in a 1/N expansion where a Fermi liquid mean-field solution becomes exact in the limit as N → ∞. Ref. [111] found that the fluctuations appear to cause a violation of Luttinger's theorem to leading order in 1/N . According to our general arguments, a violation of Luttinger's theorem must correspond to a change in the emergent symmetry group (such that the system is no longer an EFL). However, the mean-field solution is an EFL, in particular a Fermi liquid, and it seems unlikely that fluctuations could give rise to a new emergent symmetry not already present while remaining perturbative. Therefore, assuming there is no error in the calculations of Ref. [111], it seems that we must interpret their results as signaling a breakdown of perturbation theory, such that the radius of convergence in 1/N is zero. In that case, there is no reason to believe that Luttinger's theorem is violated at all.

XII. CONCLUSION AND OUTLOOK
In this work, we have made significant progress towards an understanding of strongly correlated metals, going beyond both Fermi liquid theory and perturbation theory in the interaction strength. We have highlighted the essential role of the emergent symmetry of the IR theory and its 't Hooft anomaly.
Going forward, we believe that our work leads to a new perspective on studies of non-Fermi liquids: from our point of view, a crucial question one should ask about any potential non-Fermi liquid state is what its emergent symmetry group is, and if it is supposed to represent a compressible state, in what way the emergent symmetry group and its 't Hooft anomaly enable the system to satisfy the filling constraints developed in this paper for a continuously tunable filling ν. (In particular, in this case the emergent symmetry group must be larger than any compact Lie group).
What one might expect to find is that in fact, every non-Fermi liquid is simply an ersatz Fermi liquid (EFL) as defined in Section VI or a slight variant of it (for example including fractionalization as discussed in Section X). It is a very important open question to determine whether there might be a fundamentally different way to realize compressible metallic states, with a totally different emergent symmetry group (which must still be larger than any compact Lie group). If such a possibility were to be realized, we would still expect that consideration of the emergent symmetry group and its 't Hooft anomaly will still be a powerful way to determine properties of the system, as we found for EFLs.
In particular, it is quite energizing to imagine how our results apply to the strange metal phase in cuprates, assuming that it is possible for it to extend all the way down to a critical point at zero temperature, and assuming that it can exist in a translationally invariant system without disorder. The filling at which the critical point occurs does not appear to be near any particular rational filling. Therefore, it is likely that the filling is generic and can be continuously tuned depending on the microscopic parameters, which as we have noted, leads to very strong constraints on the IR theory. Although the strange metal has extremely exotic properties, there does not appear to be any obvious reason why it could not still be an EFL (In particular, we emphasize again that the "Fermi surface quantum" excitations of an EFL do not need to have the nature of quasiparticles, which are certainly expected to be absent in the strange metal.) Identification of the strange metal as being in the EFL class, or even a significant generalization of it, would have further profound implications which we intend to explore in a subsequent work [112]. Here we will explain why, contrary to what one might initially assume, filling in d spatial dimensions does not correspond to a boundary of a bulk SPT in d + 1 spatial dimensions, when viewed in terms of the microscopic symmetries. The relevant symmetry group is G IR = Z d × U(1). One can show [113][114][115] that the SPT classification for such a symmetry group in D spatial dimensions is given by where C m is the classification of U(1) SPTs in m spatial dimensions, C ×k denotes the product of k copies of C, and d r is the binomial coefficient. Physically, C D correspond to "strong" SPT phases, i.e. those protected by U(1) alone, while the remaining terms correspond to "weak" SPT phases that can be understood by stacking layers of lower-dimensional strong SPT phases. If filling in d spatial dimensions could somehow correspond to an SPT in D = d + 1 spatial dimensions, then it would have to be an SPT that inherently relies on all d translation symmetries. Therefore, the only term in Eq. (A1) that could be relevant is the r = d term, C 1 . The problem is that there are no 1-dimensional SPTs protected by U(1) alone, in either bosonic or fermionic systems. Therefore, C 1 = 0. So we conclude that there are no SPTs in d + 1 spatial dimensions that could correspond to filling in d dimensions. In this section, we will give a more detailed argument for why a 2π flux in a 2-D system with fractional filling should transform projectively under the translation symmetry, as discussed in Section III B.
First of all, let us recall that in presence of a magnetic field one should introduce gauge-invariant translation operators in the following way. For simplicity of notation, let us assume that the system is defined in continuous space (although with only a discrete translation symmetry); similar considerations can be made in a tight-binding model. We introduce the generator of gauge-invariant continuous translations according to where A(x) is the magnetic vector potential,ρ(x) is the local density operator, and P is the generator of translations when A = 0. These translation operators satisfy the commutation algebra where, where B(x) = ε αβ ∂ α A β (x) is the magnetic field. We then define the discrete translation operators T x and T y by where a (α) is the lattice translation vector in the α direction. Now consider any state |ψ that obeys the "cluster property" for some constants C and η > 0, and · = ψ| · |ψ denotes expectation values with respect to |ψ . Then it follows from Eq. (B2) that, if we fix that B(x)d 2 x = 2π but make the magnetic field more and more spread out, then the variance goes to zero in the limit where the magnetic field is infinitely spread out. Therefore, when acting on such a state |ψ , and in this limit, we can replace the commutator by its expectation value: where v is the volume of one unit cell and ρ(x) = ρ(x) . To get to the last identification, we used the fact that B(x) is very slowly varying, and that the integral of ρ(x) over one unit cell is ν. Finally, from Eq. (B6) and the definition Eq. (B3), we find that which is Eq. (8).
Appendix C: Filling constraints in general dimension from the topological action of SPTs Here we will discuss the sense in which the microscopic filling can be computed from the IR theory in general spatial dimension d, generalizing the discussion of Sections III A and III B. First of all, we know that there is a oneto-one correspondence between the 't Hooft anomaly of the IR theory and an SPT in d + 1 spatial dimensions. Since the SPT phase is topological, at long wave-lengths a G IR SPT phase in d + 1 spatial dimensions is described by a (Euclidean) topological action S[M, A] that depends on the d + 2-dimensional space-time manifold M and a gauge field A for the G IR symmetry. This action can be evaluated on any oriented space-time manifold M . (In some cases, there may be some additional structures required on M , for example fermionic systems require a spin structure or spin c structure.) This action is purely imaginary (i.e. has no real part), does not depend on a metric on M and, provided that G IR is a compact Lie group, depends only on the SPT phase and is independent of the microscopic details of the SPT ground state. In general, the action is only defined modulo 2πi since this does not affect the amplitude e −S .
Let τ 1 , · · · , τ d ∈ G IR be the elements of the IR symmetry group corresponding to microscopic translations.
We can define a generalization of the function α from the previous subsections in the following way. We set the space-time manifold M to be the d + 2-dimensional torus T d+2 . We require that over the first d of the d + 2 directions of the torus, the gauge field A is flat, with the fluxes through the associated 1-cycles given by τ 1 , · · · , τ d . Then over the final two directions on the torus, we require that there be a 2π flux of the U(1) subgroup of G IR (generated byQ IR ) corresponding to the microscopic U(1) symmetry. Finally, we define α(Q IR |τ 1 , · · · , τ d ) = −iS[M, A]. One can show that this agrees with the previous definitions of α in the cases d = 1, 2.
Thus, our hypothesis for the form of the filling constraint is We have not devised a physical derivation analogous to the d = 1 and d = 2 cases; however we expect that this should be possible, since α should relate to the anomalous action of translations on a 2π flux, which in d spatial dimensions is a (d − 2)-dimensional object.
Note that, as we mentioned, in certain cases the topological action could depend on other data (such as a spin structure for fermionic systems). However, we expect that if the action still depends non-trivially on this additional data for the particular space-time manifold M and gauge field A considered here, this corresponds to cases where the microscopic symmetry G UV = Z d × U(1) still has a non-trivial 't Hooft anomaly in d spatial dimensions. In this case, the system cannot be realized as lattice system in d spatial dimensions with translation symmetry and on-site U(1) symmetry, only as a boundary of an SPT in d + 1 spatial dimensions. In such systems, the filling of the d-dimensional system does not need to be well-defined. In this Appendix, we will show that Eq. (19) follows from the anomaly equation Eq. (17).
Let us work on a 4-dimensional space parameterized by coordinates (t, x, y, θ). We start by considering a configuration where the A x and A y components of the gauge field depend only on x and y, and the A t , A θ components depend only on z,t. Then, we integrate Eq. (17) over x and y. We obtain where the index µ now varies only over t and θ, and we have defined φ = 1 2π (∂ x A y − ∂ y A x )dxdy, which is the number of flux quanta passing through the system, and j µ = j µ dxdy. Effectively, what we have done is compactify our original system in 2 spatial dimensions to one in 0 spatial dimensions. This 0-D system has a 't Hooft anomaly that depends on φ, and is canceled by inflow from a 2-D SPT. Recall that such 't Hooft anomalies correspond to projective representations of the symmetry group. If we set φ = 1, this corresponds to the projective representation of the symmetry group acting on a 2π flux.
In our case, where the symmetry group is G = LU(1), we can express the projective representation through this central extension of the commutation algebra of the symmetry generatorsn(θ), i.e.
We should compare this with the anomaly equation Eq. (D1), in which we choose A θ to be independent of time, we have j θ = δH δA θ = 0, and we identify the timecomponent j t with the expectation value ofn(θ, t). We find that, to be consistent with Eq. (D3), it must be that Finally, to obtain the extra factor of q in the right-hand side of Eq. (19), we recall that Eq. (13) implies that φ, which we can interpret as the number of flux quanta of the IR U(1) symmetry generated by n(θ)dθ, is q times the number of flux quanta of the microscopic U(1) symmetry.
where we used the fact that the momenta k (1) obey the magnetic algebra 13 β } = −qB(x (1) )ε αβ (E10) (here we have chosen the convention that k carries units of momentum, not inverse length.) For general N , we have where in this equation both the left-and right-hand sides have an implicit dependence on the phase space variables (k (1) , · · · , k (N ) , x (1) , · · · , x (N ) ), on the right-hand side through n(x, k). Now let us write n(x, k) = n 0 (k) + δn(x, k), where n 0 (k) has no implicit dependence on the phase space variables and represents the equilibrium distribution of particles in the ground state (keeping in mind the Pauli exclusion principle). For low-energy configurations, the contribution of δn to Eq. (E11) will be negligible compared to n 0 . Hence, we can replace n with n 0 in Eq. (E11) and we find Now we observe that n 0 (k) = 1/(2π ) 2 in the occupied region of the Brillouin zone, and 0 outside. So by Stokes' theorem we obtain where m = ±1 (relative to an arbitrary choice of orientation of the Fermi surface, m = +1 or −1 corresponds to which side of the Fermi surface the occupied region of the Brillouin zone is on). This is the result we wanted to show.
For an alternative perspective that gives more precise bounds on N GIR , see the next appendix, Appendix G.
For even d, there are no generalized Chern-Simons invariants and we are done, so let us consider odd d instead. We will use a flat M so that there are no metric contributions to the Chern-Simons invariant. We will also use the fact that the remaining Chern-Simons invariants are defined for G IR bundles by taking trace in a unitary representation R : G IR → U (n) and is therefore equal to a Chern-Simons invariant U (n) bundle associated to it by the map R. Then, we use the fact that τ 1 , . . . , τ d and the U (1) generatorQ commute in G IR , hence their images in U (n) also commute and can be simultaneously diagonalized. This reduces the calculation to an Abelian Chern-Simons invariant for the subgroup U (1) n of diagonal matrices in U (n). Taking the trace reduces this to a calculation for a U (1) bundle.
We can extend our U (1) bundle from T d+2 = T d × T 2 to D 2 × T d−1 × T 2 , where D 2 is a disc whose boundary is the first coordinate cycle of T d . The extended gauge field A is defined so that it has curvature only along D 2 (integrating to Tr R (τ 1 ) as required by the extension) and along T 2 (where it has some Chern number). The Chern-Simons invariant is proportional to However, since the 2-form dA has only nonzero components in 4 directions, D 2 and T 2 , its power in the integrand above vanishes if d + 3 > 4, ie. if d > 1 16 . This proves the claim.
Proof of Theorem 3: We can get a handle on the maximum denominator N GIR , which we have proven divides |Ω d+2 (G IR ) tors |, using the Atiyah-Hirzebruch spectral sequence for computing the latter. This gives us an upper bound where G b IR = G IR /Z F 2 and Ω spin k is the usual spin bordism group, which in the low degrees of physical interest is This upper bound can be further improved by taking into account the known differentials of the dual spectral 16 We note that for d = 1, the Chern-Simons invariant is an integer multiple of Tr R (τ 1 ).
sequence for cobordism Ω d+2 (G IR ), which has isomorphic torsion, Ω d+2 (G IR ) tors = Ω d+2 (G IR ) tors . See [120] for a review. Another thing we can do is consider the special case G IR = H × U (1), where U (1) and H is compact. Because τ 1 , . . . , τ d andQ must commute, they are always contained in such a group. In this case, by taking considering d+2-manifolds of the form M ×T 2 with a arbitrary G IR bundle on M and a U (1) bundle with Chern number 1 on T 2 , our d + 2-dimensional cobordism invariant defines a d-dimensional cobordism invariant for M . By similar arguments as above, this gives us the further constraint that N GIR divides the element of highest order in Ω d (G IR ) tors = Ω d (G IR ) tors .
For d = 2, using the Atiyah-Hirzebruch spectral sequence for Ω 2 (G IR ), we find in the case G IR = H × U (1), One can show that since the fermion parity is a subgroup of the U (1) factor, that the first two factors do not contribute to the spin bordism because the spectral sequence has a nonzero differential there. The third part is |π 0 (G IR )| · |π 1 (G IR ) tors | torsion, so N GIR divides |π 0 (G IR )| · |π 1 (G IR ) tors |.
We note we can often get even better bounds if we better understand the cobordism invariants for G IR . For example, if G IR = Z r 2 × U (1), one can show the maximum denominator is N GIR = 2, even though this group has 2 r components.
To include a finite 1-form symmetry K, there is one more term which contributes to the spectral sequence, H 3 (B 2 K, Z) = K, which is |K| torsion.
Proof of Theorem 4: For d = 3, G IR = H × U (1), the contributions to the Atiyah-Hirzebruch spectral sequence for Ω 3 (G IR ) are As in d = 2, the first factor does not contribute. The second factor has one piece that can contribute, coming from the map H 2 (BU (1), Z) → H 2 (BG b IR , Z 2 ) but it turns out to give a spin Chern-Simons term for the U (1) factor and we can disregard it, ie. it does not contribute to the torsion in Ω 3 (G IR ). The third piece can be bounded using the Serre spectral sequence for the extension from which we find it is |π 0 (G IR )| · gcd(|π 1 (G IR ) tors |, |π 0 (G IR )|) torsion, so N GIR divides |π 0 (G IR )| · gcd(|π 1 (G IR ) tors |, |π 0 (G IR )|).
Note that when G IR is connected, we re-establish the result N GIR = 1: connected, compact G IR are incompatible with fractional filling in d = 3.
Finally we note that the bounds so derived hold for every subgroup G IR of the full emergent symmetry, so long as it contains the translation generators τ 1 , . . . , τ d and the U (1) particle-number symmetry. Thus, one does not need to know the entire emergent symmetry to obtain a bound. Further, by varying G IR , one can sometimes find a better bound for the problem at hand.

Appendix H: Asymptotic analysis for quantum oscillations
In this appendix, we will derive the asymptotic form for quantum oscillations stated in Section VI C 3. Abstracting out from the specific details, the situation is that we have some functional F[f ], where f is a function of a single variable into R/Z that can be expressed as and we we wish to extract the asymptotic dependence of F[f ] as t → ∞ while keeping the function g fixed.